Ray Optics Module Application Library Manual Ray Optics Application Library Manual

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Part number: CM024203 Created in COMSOL Multiphysics 5.3

Anti-Reflective Coating with Multiple Layers

This model is licensed under the COMSOL Software License Agreement 5.3. All trademarks are the property of their respective owners. See www.comsol.com/trademarks. Introduction

An anti-reflective coating is a set of thin, transparent films applied to the surface of an optical device such as a lens to reduce reflection. This reduction of reflected light leads to an increase in the efficiency of the optical system and minimizes stray light, which is important in many imaging applications. Anti-reflective coatings can also be applied to the surfaces of eyeglasses to reduce glare and make the eyes of the wearer more visible.

The simplest example of an anti-reflective coating is a quarter-wavelength layer, a single dielectric film with thickness equal to one quarter of the wavelength of the incident light. This layer can reduce the reflection coefficient to zero if the refractive index of the film is equal to the geometric mean of the refractive indices of the air (n0) and substrate (nS). For air (1.0) and common glass substrate (1.5) this optimal refractive index would be sqrt((1.0)(1.5)) or approximately 1.22.

Typically no material exists with a refractive index that yields a reflection coefficient of exactly zero. Another drawback of the quarter-wavelength layer is that while it can prevent reflection of light at one frequency, it reflects a substantial amount of radiation at any other frequency. The reflectance of the quarter-wavelength layer also depends heavily on the angle of incidence of the light.

An alternative is to use a coating that consists of multiple layers. Compared to single-layer coatings, a multi-layer coating is more likely to reduce the reflection coefficient across a band of wavelengths and can be produced using a wider variety of real materials.

Model Definition

Figure 1 shows the simple geometry used in this model. It consists of a box with an internal boundary separating the air and substrate domains. This boundary is also where the Thin Dielectric Film features are added.

The red arrow (activated in the Material Discontinuity feature) shows the sense in which the thin dielectric layers are stacked. The last Thin Dielectric Film in the Model Builder represents the topmost thin film in the model.

2 | ANTI-REFLECTIVE COATING WITH MULTIPLE LAYERS Figure 1: Geometry used to model multilayer anti-reflective coatings.

Simple anti-reflective coatings are designed to minimize the reflectance at a specified λ vacuum wavelength, 0. More sophisticated coatings can minimize reflectance across a relatively large band of wavelengths by employing multiple thin films with different material properties.

The simplest multilayer coating consists of two layers of different materials index on the surface of a glass substrate. Each layer has a thickness of 1/4 of the wavelength in the medium; therefore this design is called a quarter-quarter coating.

Theoretically, a quarter-quarter coating can reduce the reflectance to zero at the specified wavelength; however, because real materials with ideal refractive indices are seldom available, zero reflectance is normally not achieved.

In this example, the substrate consists of glass (nS = 1.5) and the first thin dielectric layer is made of magnesium fluoride, MgF2 (n1 = 1.38). The following expression can be used to determine the optimal refractive index for the second layer, n2, so that the reflectance can be reduced to zero:

2 n1ns n2 = ------na where the refractive index of the air, na, is taken to be 1.

3 | ANTI-REFLECTIVE COATING WITH MULTIPLE LAYERS Using this expression the optimal value for n2 is 1.69. From Table 1, cerium fluoride, CeF3, has a refractive index close to this value, 1.63. Using this material a minimum reflectance of less that 1% can be achieved.

TABLE 1: REFRACTIVE INDICES OF MATERIALS FREQUENTLY USED IN THIN FILMS

MATERIAL REFRACTIVE INDEX

Magnesium Flouride (MgF2)1.38

Silicon Dioxide (SiO2)1.46

Cerium Fluoride (CeF3)1.63

Zirconium Oxide (ZrO2)2.2 Silicon (Si) 3.5

The reflectance as a function of vacuum wavelength is shown in Figure 2. A noticeable drawback of the two-layer coating is that the reflectance is only significantly reduced in a narrow band around a single wavelength.

Figure 2: Reflectance of a quarter-quarter coating.

The reflectance can be reduced over a wider range of wavelengths by using a dielectric film with three or more layers. An example of a three-layer coating is the quarter-half-quarter coating, in which a thin layer of thickness λ/2 is placed between the two quarter- wavelength layers.

4 | ANTI-REFLECTIVE COATING WITH MULTIPLE LAYERS Results and Discussion

Figure 3 compares the reflectance of the quarter-quarter and quarter-half-quarter films. Because the refractive indices of real materials are used, the reflectance of the quarter- quarter coating does not decrease to zero. The quarter-half-quarter film exhibits slightly greater reflectance at the center of the band, but the reflectance is reduced over a much wider frequency range.

Figure 3: Reflectance response for a quarter-quarter and quarter-half-quarter coating configurations.

Application Library path: Ray_Optics_Module/Tutorials/ antireflective_coating_multilayer

Modeling Instructions

From the File menu, choose New.

NEW In the New window, click Model Wizard.

5 | ANTI-REFLECTIVE COATING WITH MULTIPLE LAYERS MODEL WIZARD 1 In the Model Wizard window, click 2D. 2 In the Select Physics tree, select Optics>Ray Optics>Geometrical Optics (gop). 3 Click Add. 4 Click Study. 5 In the Select Study tree, select Preset Studies>Ray Tracing. 6 Click Done.

GLOBAL DEFINITIONS

Parameters 1 On the Home toolbar, click Parameters. 2 In the Settings window for Parameters, locate the Parameters section. 3 In the table, enter the following settings:

Name Expression Value Description n_air 1 1 Refractive index of air n_glass 1.5 1.5 Refractive index of glass n_CeF3 1.63 1.63 Refractive index of CeF3 n_MgF2 1.38 1.38 Refractive index of MgF2 n_ZrO2 2.2 2.2 Refractive index of ZrO2 lam0 550[nm] 5.5E-7 m Vacuum wavelength

GEOMETRY 1

Square 1 (sq1) 1 On the Geometry toolbar, click Primitives and choose Square. 2 In the Settings window for Square, click to expand the Layers section. 3 In the table, enter the following settings:

Layer name Thickness (m) Layer 1 0.5

4 Click Build All Objects.

6 | ANTI-REFLECTIVE COATING WITH MULTIPLE LAYERS MATERIALS

Material 1 (mat1) 1 In the Model Builder window, under Component 1 (comp1) right-click Materials and choose Blank Material. 2 In the Settings window for Material, locate the Material Contents section. 3 In the table, enter the following settings:

Property Name Value Unit Property group Refractive index n n_air 1 Refractive index

Material 2 (mat2) 1 Right-click Materials and choose Blank Material. 2 Select Domain 1 only. 3 In the Settings window for Material, locate the Material Contents section. 4 In the table, enter the following settings:

Property Name Value Unit Property group Refractive index n n_glass 1 Refractive index

GEOMETRICAL OPTICS (GOP) 1 In the Model Builder window, under Component 1 (comp1) click Geometrical Optics (gop). 2 In the Settings window for Geometrical Optics, locate the Intensity Computation section. 3 From the Intensity computation list, choose Compute intensity and power. 4 Locate the Ray Release and Propagation section. Select the Allow frequency distributions at release features check box. 5 In the Maximum number of secondary rays text field, type 0.

Set up the quarter-quarter anti-reflective coating using two Thin Dielectric Film features.

It is useful to display the boundary normal in the Graphics window since this indicates the order in which the thin films are arranged. The arrow points in the direction away from the substrate, so the first layer that appears in the Model Builder is adjacent to the substrate and the last layer that appears is adjacent to the air domain.

7 | ANTI-REFLECTIVE COATING WITH MULTIPLE LAYERS 3 Select the Show boundary normal check box. 4 Locate the Coatings section. From the Thin dielectric films on boundary list, choose Add layers to surface. 5 Right-click Component 1 (comp1)>Geometrical Optics (gop)>Material Discontinuity 1 and choose Thin Dielectric Film.

The first layer (directly touching the glass substrate) is the CeF3 layer with a refractive index of 1.63. The thickness is set to be a quarter of the wavelength in this material.

Thin Dielectric Film 1 1 In the Settings window for Thin Dielectric Film, locate the Film Properties section. 2 In the n text field, type n_CeF3. 3 In the t text field, type lam0/(4*n_CeF3).

Material Discontinuity 1 Right-click Material Discontinuity 1 and choose Thin Dielectric Film.

The second layer (on top of layer 1) is the MgF2 layer with a refractive index of 1.38. The thickness of this layer is also set to a quarter of the wavelength.

Thin Dielectric Film 2 1 In the Settings window for Thin Dielectric Film, locate the Film Properties section. 2 In the n text field, type n_MgF2. 3 In the t text field, type lam0/(4*n_MgF2). Next, set up the Release From Grid feature to release a number of rays of different wavelengths from 400 nm to 800 nm.

3 In the qx, 0 text field, type 0.5.

4 In the qy, 0 text field, type 1.

5 Locate the Ray Direction Vector section. Specify the L0 vector as

0 x -1 y

8 | ANTI-REFLECTIVE COATING WITH MULTIPLE LAYERS 6 Locate the Initial Ray Frequency section. From the Distribution function list, choose List of values. 7 Click Range. 8 In the Range dialog box, choose Number of values from the Entry method list. 9 In the Start text field, type 3e8[m/s]/400[nm]. 10 In the Stop text field, type 3e8[m/s]/800[nm]. 11 In the Number of values text field, type 100. 12 Click Replace.

STUDY 1 Set up a Ray Tracing study step to compute the ray trajectories to a maximum optical path length of 1.1 m.

Step 1: Ray Tracing 1 In the Model Builder window, expand the Study 1 node, then click Step 1: Ray Tracing. 2 In the Settings window for Ray Tracing, locate the Study Settings section. 3 From the Time step specification list, choose Specify maximum path length. 4 In the Lengths text field, type range(0,0.01,1.1). 5 On the Home toolbar, click Compute.

RESULTS

1D Plot Group 2 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 In the Settings window for 1D Plot Group, type Reflectance in the Label text field. 3 Locate the Data section. From the Data set list, choose Ray 1. 4 From the Time selection list, choose Last. 5 Click to expand the Title section. From the Title type list, choose Manual. 6 In the Title text area, type Reflectance of Multilayer Films. 7 Locate the Plot Settings section. Select the x-axis label check box. 8 In the associated text field, type Vacuum wavelength (nm). 9 Select the y-axis label check box. 10 In the associated text field, type Reflectance (%).

9 | ANTI-REFLECTIVE COATING WITH MULTIPLE LAYERS Ray 1 1 On the Reflectance toolbar, click More Plots and choose Ray. Plot the percentage reflectance. 2 In the Settings window for Ray, locate the y-Axis Data section. 3 In the Expression text field, type 100*(gop.relg1.Q0-gop.Q)/gop.relg1.Q0. 4 Locate the x-Axis Data section. From the Parameter list, choose Expression. 5 In the Expression text field, type gop.lambda0. 6 From the Unit list, choose nm. 7 Click to expand the Legends section. Select the Show legends check box. 8 From the Legends list, choose Manual. 9 In the table, enter the following settings:

Legends Quarter-Quarter

10 On the Reflectance toolbar, click Plot. The plot should look like Figure 2.

Next, to simulate a quarter-half-quarter layer, add another Thin Dielectric Film feature. The material of the middle layer is chosen to be Zirconium Oxide, ZrO2, with refractive index 2.2. Set the thickness to half the specified wavelength.

GEOMETRICAL OPTICS (GOP)

Thin Dielectric Film 3 1 In the Model Builder window, under Component 1 (comp1)>Geometrical Optics (gop) right-click Material Discontinuity 1 and choose Thin Dielectric Film. This layer will sit between the other two thin layers so its node must be moved in the Model Builder. 2 In the Model Builder window, under Component 1 (comp1)>Geometrical Optics (gop)> Material Discontinuity 1 right-click Thin Dielectric Film 3 and choose Move Up. 3 In the Settings window for Thin Dielectric Film, locate the Film Properties section. 4 In the n text field, type n_ZrO2. 5 In the t text field, type lam0/(2*n_ZrO2).

Add another study so that the two films can be compared.

10 | ANTI-REFLECTIVE COATING WITH MULTIPLE LAYERS ADD STUDY 1 On the Home toolbar, click Add Study to open the Add Study window. 2 Go to the Add Study window. 3 Find the Studies subsection. In the Select Study tree, select Preset Studies>Ray Tracing. 4 Click Add Study in the window toolbar.

STUDY 2

Step 1: Ray Tracing 1 On the Home toolbar, click Add Study to close the Add Study window. 2 In the Model Builder window, under Study 2 click Step 1: Ray Tracing. 3 In the Settings window for Ray Tracing, locate the Study Settings section. 4 From the Time step specification list, choose Specify maximum path length. 5 In the Lengths text field, type range(0,0.01,1.1). 6 From the Stop condition list, choose No active rays remaining. 7 On the Home toolbar, click Compute.

RESULTS

Ray 2 1 In the Model Builder window, under Results>Reflectance right-click Ray 1 and choose Duplicate. 2 In the Settings window for Ray, locate the Data section. 3 From the Data set list, choose Ray 2. 4 From the Time selection list, choose Last. 5 Click to expand the Legends section. In the table, enter the following settings:

Legends Quarter-Half-Quarter

6 On the Reflectance toolbar, click Plot. Compare the result with Figure 3.

11 | ANTI-REFLECTIVE COATING WITH MULTIPLE LAYERS 12 | ANTI-REFLECTIVE COATING WITH MULTIPLE LAYERS Created in COMSOL Multiphysics 5.3

Corner Cube Retroreflector

This model is licensed under the COMSOL Software License Agreement 5.3. All trademarks are the property of their respective owners. See www.comsol.com/trademarks. Introduction

This tutorial model shows how to simulate the reflection of a bundle of rays at a corner cube retroreflector using the Geometrical Optics interface.

A corner cube retroreflector is used to reflect rays so that the reflected rays are antiparallel to the incident rays, regardless of the angle of incidence. A basic corner cube retroreflector consists of three orthogonal reflecting surfaces.

Model Definition

1000 rays are released into the corner cube retroreflector with a conical distribution. The geometry, shown in Figure 1, is an imported a built-in Part from the Part Library for the Ray Optics Module. The initial and final directions of the rays are used to confirm that the initial and final trajectories are parallel, regardless of the angle of incidence.

Figure 1: Geometry of a typical corner cube retroreflector.

Results and Discussion

Figure 2 shows the ray trajectories as they propagate through the geometry. The color expression corresponds to the ray index, which has a unique integer value for each ray. In Figure 3, the initial and final angles between each ray trajectory and the surface normal are plotted to confirm that the incident and reflected rays are parallel.

2 | CORNER CUBE RETROREFLECTOR Figure 2: Ray trajectories in the corner cube retroreflector.

Figure 3: Acute angle of incidence for the reflected rays as a function of the initial angle between the released rays and the surface normal.

3 | CORNER CUBE RETROREFLECTOR Application Library path: Ray_Optics_Module/Tutorials/ corner_cube_retroreflector

Modeling Instructions

From the File menu, choose New.

NEW In the New window, click Model Wizard.

MODEL WIZARD 1 In the Model Wizard window, click 3D. 2 In the Select Physics tree, select Optics>Ray Optics>Geometrical Optics (gop). 3 Click Add. 4 Click Study. 5 In the Select Study tree, select Preset Studies>Ray Tracing. 6 Click Done.

GEOMETRY 1 1 In the Model Builder window, under Component 1 (comp1) click Geometry 1. 2 In the Settings window for Geometry, locate the Units section. 3 From the Length unit list, choose mm.

PART LIBRARIES 1 On the Home toolbar, click Windows and choose Part Libraries. 2 In the Part Libraries window, select Ray Optics Module>3D>Retroreflectors> corner cube retroreflector 3d in the tree. 3 Click Add to Geometry.

GEOMETRY 1

Corner Cube Retroreflector 1 (pi1) 1 In the Model Builder window, under Component 1 (comp1)>Geometry 1 click Corner Cube Retroreflector 1 (pi1). 2 In the Settings window for Part Instance, locate the Input Parameters section.

4 | CORNER CUBE RETROREFLECTOR 3 In the table, enter the following settings:

Name Expression Value Description niy 1 1.0 Incident ray direction, y-component niz 1 1.0 Incident ray direction, z-component

GEOMETRICAL OPTICS (GOP) 1 In the Model Builder window, under Component 1 (comp1) click Geometrical Optics (gop). 2 In the Settings window for Geometrical Optics, locate the Ray Release and Propagation section. 3 In the Maximum number of secondary rays text field, type 0.

Medium Properties 1 1 In the Model Builder window, expand the Geometrical Optics (gop) node, then click Medium Properties 1. 2 In the Settings window for Medium Properties, locate the Medium Properties section. 3 From the n list, choose User defined. In the associated text field, type 1.5.

Wall 1 1 In the Model Builder window, right-click Geometrical Optics (gop) and choose Wall. 2 Select Boundaries 5–7 only. 3 In the Settings window for Wall, locate the Wall Condition section. 4 From the Wall condition list, choose Specular reflection.

Wall 2 1 Right-click Geometrical Optics (gop) and choose Wall. 2 Select Boundary 2 only.

Release from Grid 1 1 Right-click Geometrical Optics (gop) and choose Release from Grid. 2 In the Settings window for Release from Grid, locate the Initial Coordinates section.

3 In the qx, 0 text field, type -22/sqrt(3).

4 In the qy, 0 text field, type -22/sqrt(3)-5.

5 In the qz, 0 text field, type -22/sqrt(3)+5. 6 Locate the Ray Direction Vector section. From the Ray direction vector list, choose Conical.

7 In the Nw text field, type 1000.

5 | CORNER CUBE RETROREFLECTOR 8 Specify the r vector as

1 x 1.3 y 1 z

9 In the α text field, type pi/18.

STUDY 1

Step 1: Ray Tracing 1 In the Model Builder window, expand the Study 1 node, then click Step 1: Ray Tracing. 2 In the Settings window for Ray Tracing, locate the Study Settings section. 3 From the Time step specification list, choose Specify maximum path length. 4 From the Length unit list, choose mm. 5 Click Range. 6 In the Range dialog box, type 0.2 in the Step text field. 7 In the Stop text field, type 70. 8 Click Replace. 9 On the Home toolbar, click Compute.

RESULTS

Ray Trajectories (gop) In the Model Builder window, expand the Ray Trajectories (gop) node.

Color Expression 1 1 In the Model Builder window, expand the Results>Ray Trajectories (gop)> Ray Trajectories 1 node, then click Color Expression 1. 2 In the Settings window for Color Expression, locate the Expression section. 3 In the Expression text field, type gop.pidx. 4 Click the Zoom Extents button on the Graphics toolbar. 5 On the Ray Trajectories (gop) toolbar, click Plot. Compare the resulting plot to Figure 2.

Create a plot to display the angle of incidence of the reflected rays as a function of the angle of the incident rays with respect to the boundary normal. Use the at operator to get the angle of incidence at t = 0 s.

6 | CORNER CUBE RETROREFLECTOR 1D Plot Group 2 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 In the Settings window for 1D Plot Group, type Acute Angle of Incidence in the Label text field. 3 Locate the Data section. From the Data set list, choose Ray 1. 4 From the Time selection list, choose Last.

Ray 1 1 On the Acute Angle of Incidence toolbar, click More Plots and choose Ray. 2 In the Settings window for Ray, locate the y-Axis Data section. 3 In the Expression text field, type gop.phii. 4 Locate the x-Axis Data section. From the Parameter list, choose Expression. 5 In the Expression text field, type at(0,gop.phii).

Acute Angle of Incidence 1 In the Model Builder window, under Results click Acute Angle of Incidence. 2 In the Settings window for 1D Plot Group, click to expand the Title section. 3 From the Title type list, choose None. 4 Click the Zoom Extents button on the Graphics toolbar. 5 On the Acute Angle of Incidence toolbar, click Plot. Compare the resulting plot to Figure 3.

7 | CORNER CUBE RETROREFLECTOR 8 | CORNER CUBE RETROREFLECTOR Created in COMSOL Multiphysics 5.3

Czerny-Turner Monochromator

This model is licensed under the COMSOL Software License Agreement 5.3. All trademarks are the property of their respective owners. See www.comsol.com/trademarks. Introduction

This model simulates a grating spectrometer in a crossed Czerny-Turner configuration. The model uses the Geometrical Optics interface to compute the positions of incident rays on the detector plane, from which the instrument's spectral resolution can be derived.

A Czerny-Turner spectrometer spatially separates polychromatic light into a series of monochromatic rays. The configuration includes a slit source, a spherical collimating mirror, a planar diffraction grating, a spherical imaging mirror, and an array charge coupled device (CCD) detector, see Figure 1.

With a given optical component arrangement and with the knowledge of the detector properties it is possible to determine the wavelength calibration as well as the spectral resolution of the instrument.

Model Definition

A Czerny-Turner spectrometer is usually designed with F-number > 3 to avoid large aberrations of the image at the detector. Accordingly, this model treats the polychromatic light source as a cone-based release of rays with F-number =10. Note that the system’s F- number is related to its numerical aperture NA by:

1 F– number = ------2NA

Where the numerical aperture NA is related to the half-angle of the maximum cone of light that can enter the system Θ:

NA = nsin()Θ

Where n is the refractive index of the propagation medium (air, n =1).

The polychromatic source is simulated using twenty frequency values that are sampled from a uniform distribution. The corresponding free-space wavelength distribution has a minimum of λ=451 nm and a maximum of λ=894 nm.

The geometry of the spectrometer is shown in Figure 1. The rays are released from the front focal point of the collimating mirror. The collimating mirror is tilted with an angle θ c to direct the collimated light toward the diffraction grating. The rays of diffraction order m = 1 are then directed toward the focusing mirror; because the ray trajectories of nonzero diffraction order are frequency-dependent, the rays of different frequency arrive at the focusing mirror at different positions and with different angles of incidence.

2 | CZERNY-TURNER MONOCHROMATOR The focusing mirror reflects the rays onto the detector. The detector is composed of an = = μ array of N 3648 pixels of width wp 8 m. The geometrical parameters used in the model are listed in Table 1.

Figure 1: Typical crossed Czerny-Turner configuration. Numerical values are displayed in Table 1

TABLE 1: DEFINITION OF THE DESIGN PARAMETERS.

PARAMETERS VALUE DESCRIPTION θ g (deg) 28.76 Grating’s angle θ c (deg) 11.0 Collimating mirror’s angle θ i (deg) 77 Imaging mirror’s angle θ d (deg) 6.76 Detector’s angle

Qi (mm) (20.0,34.0) Coordinates, imaging mirror

Qc (mm) (40,16.161) Coordinates, collimating mirror

Qd (mm) (22.08,-24.12) Coordinates, detector

3 | CZERNY-TURNER MONOCHROMATOR TABLE 1: DEFINITION OF THE DESIGN PARAMETERS.

PARAMETERS VALUE DESCRIPTION

Ri (mm) 130 Radius of curvature, imaging mirror

Rc (mm) 100 Radius of curvature, collimating mirror The ray trajectories can be used to compute the spectrometer’s resolution. The pixel number pnum of a ray on the collector is

N q – Q p = ceil---- – ------x dx - num ()θ 2 wp cos d

Where N is the pixel number of the CCD, wp the pixel’s width, and qx the x-coordinate θ of the particle hitting the detector. Qdx and d are respectively the detector’s center x- coordinate and the angle of the CCD. The spectral (optical) resolution δλ of the element can then be estimated by

Δλw δλ ∝ ------i N wp

Δλ Where = 650 nm is the spectral range of the detector, and wi is the width of the slit’s image on the CCD.

The knowledge of the image width on the detector is given by the distance between the two rays delimiting a pencil of rays of uniform wavelength on the detector. Figure 2 shows how this distance is evaluated.

It is possible to evaluate the image width using the following expression:

gop.gopmaxop1(if(samefreq&&onccd, gop.gopmaxop1(if(samefreq&&onccd,distance,0)),0)) where gop.gopmaxop1 is a component coupling that returns the maximum value of an expression over all rays. The logical expression samefreq&&onccd is used to evaluate the maximum only over rays of a single frequency and to exclude rays other than those of diffraction order 0 that do not reach the CCD. These variables are defined as

samefreq=abs(gop.nu-dest(gop.nu))<1[Hz] onccd=qx>0[mm]&&qx<40[mm]&&qy>-32[mm]&&qy<-15[mm] Here the rectangle bounded by x = 0, x = 40 mm, y =−32 mm, and y =−15 mm is chosen because it encloses the CCD.

4 | CZERNY-TURNER MONOCHROMATOR λ j

ce Detector surfa

wi λ qj,min = (qx,qy) j,min λ qj,max = (qx,qy) j,max

Figure 2: Determination of the image width for a given wavelength.

Results and Discussion

Figure 3 shows the ray trajectories in the spectrometer for an F-number of 10. The free- space wavelength is indicated by the color expression. After the rays are reflected by the grating, rays of different frequency propagate in different directions and arrive at different locations on the CCD. The pixel numbers corresponding to incident rays of each wavelength are plotted in Figure 4. The spectral resolution of the image is shown in Figure 5. The spectral resolution of a system determines the maximum number of spectral peaks that the spectrometer can resolve.

5 | CZERNY-TURNER MONOCHROMATOR Figure 3: Wavelength separation on the CCD detector.

Figure 4: Wavelength calibration.

6 | CZERNY-TURNER MONOCHROMATOR Figure 5: Spectral resolution as a function of the wavelength.

Reference

1. K. Liu and F. Yu, “Accurate wavelength calibration method using system parameters for grating spectrometers,” Opt. Eng. vol. 52, no. 1, pp. 013603-1–013603-6, 2013.

Application Library path: Ray_Optics_Module/Polychromatic_Light/ czerny_turner_monochromator

Modeling Instructions

From the File menu, choose New.

NEW In the New window, click Model Wizard.

7 | CZERNY-TURNER MONOCHROMATOR MODEL WIZARD 1 In the Model Wizard window, click 2D. 2 In the Select Physics tree, select Optics>Ray Optics>Geometrical Optics (gop). 3 Click Add. 4 Click Study. 5 In the Select Study tree, select Preset Studies>Ray Tracing. 6 Click Done.

ROOT Insert the prepared geometry sequence from file. You can read the instructions for creating the geometry in the appendix.

GEOMETRY 1 1 On the Geometry toolbar, click Insert Sequence. 2 Browse to the model’s Application Libraries folder and double-click the file czerny_turner_monochromator_geom_sequence.mph. 3 Click the Zoom Extents button on the Graphics toolbar.

GLOBAL DEFINITIONS

Parameters 1 In the Model Builder window, under Global Definitions click Parameters. 2 In the Settings window for Parameters, locate the Parameters section. 3 In the table, enter the following settings:

Name Expression Value Description lam 600[nm] 6E-7 m Beam mean wavelength N 3648 3648 Number of pixels wp 8[um] 8E-6 m Pixel width Fnum 10 10 F-number NA 1/(2*Fnum) 0.05 Numerical aperture Srange 650[nm] 6.5E-7 m Spectral range

GEOMETRICAL OPTICS (GOP) 1 In the Model Builder window, under Component 1 (comp1) click Geometrical Optics (gop). 2 In the Settings window for Geometrical Optics, locate the Domain Selection section.

8 | CZERNY-TURNER MONOCHROMATOR 3 Click Clear Selection. Select the Allow frequency distributions at release features check box in order to release a distribution of rays of different free-space wavelengths. Also, because the Grating feature uses secondary rays for the higher diffraction orders, increase the Maximum number of secondary rays to 1200. 4 Locate the Ray Release and Propagation section. Select the Allow frequency distributions at release features check box. 5 In the Maximum number of secondary rays text field, type 1200. 6 In the Model Builder window, expand the Geometrical Optics (gop) node. 7 Right-click Component 1 (comp1)>Geometrical Optics (gop) and choose Release from Grid.

Release rays from a grid point located at the focus of the collimating mirror. Use a cone angle that corresponds to the numerical aperture of the system and select a uniform frequency distribution for the initial rays.

Release from Grid 1 1 In the Settings window for Release from Grid, locate the Initial Coordinates section.

2 In the qx, 0 text field, type -10.

3 In the qy, 0 text field, type 16.16104903340627. 4 Locate the Ray Direction Vector section. From the Ray direction vector list, choose Conical.

5 In the Nw text field, type 20. 6 In the α text field, type asin(NA). 7 Locate the Initial Ray Frequency section. From the Distribution function list, choose Uniform. 8 In the N text field, type 20. 9 In the μ text field, type c_const/lam. 10 In the σ text field, type 0.2*c_const/lam.

Wall 1 1 In the Model Builder window, right-click Geometrical Optics (gop) and choose Wall. 2 Select Boundary 8 only. 3 Right-click Geometrical Optics (gop) and choose Wall.

Select the Specular reflection wall condition for the curved mirrors.

9 | CZERNY-TURNER MONOCHROMATOR Wall 2 1 Select Boundaries 15 and 16 only. 2 In the Settings window for Wall, locate the Wall Condition section. 3 From the Wall condition list, choose Specular reflection. 4 Right-click Geometrical Optics (gop) and choose Grating.

Add a grating with a groove density of 600 lines/mm and include the first diffraction order. Note that in grating spectrometer systems, the first order spectra usually carry the primary diffraction intensity of the grating.

Grating 1 1 Select Boundary 3 only. 2 In the Settings window for Grating, locate the Device Properties section. 3 In the d text field, type 1[mm]/600. 4 From the Rays to release list, choose Reflected. 5 Right-click Component 1 (comp1)>Geometrical Optics (gop)>Grating 1 and choose Diffraction Order. Add a bounding box to stop the outgoing rays.

Ray Termination 1 1 Right-click Geometrical Optics (gop) and choose Ray Termination. 2 In the Settings window for Ray Termination, locate the Termination Criteria section. 3 From the Spatial extents of ray propagation list, choose Bounding box, from geometry.

MESH 1 1 In the Model Builder window, under Component 1 (comp1) click Mesh 1. 2 In the Settings window for Mesh, locate the Mesh Settings section. 3 From the Sequence type list, choose User-controlled mesh.

Refine the mesh around the curved surfaces.

Size 1 In the Model Builder window, under Component 1 (comp1)>Mesh 1 click Size. 2 In the Settings window for Size, locate the Element Size section. 3 Click the Custom button. 4 Locate the Element Size Parameters section. In the Minimum element size text field, type 0.002.

10 | CZERNY-TURNER MONOCHROMATOR 5 In the Curvature factor text field, type 0.002. 6 Click Build All.

DEFINITIONS Add the definition for the pixel number and for the image width of the entrance slit. The latter expression uses the gopmaxop component coupling, if conditions and the dest operator to determine the difference in position of extremum rays for a given wavelength.

1 In the Model Builder window, expand the Study 1 node.

Variables 1 1 Right-click Component 1 (comp1)>Definitions and choose Variables. 2 In the Settings window for Variables, locate the Variables section. 3 In the table, enter the following settings:

Name Expression Unit Description pnum ceil(N/2-(qx-Qdx)/(wp* Pixel number cos(theta_d))) wi gop.gopmaxop1(if(samefreq&&onccd, m Image width of gop.gopmaxop1(if(samefreq&&onccd, the entrance distance,0)),0)) slit samefreq abs(gop.nu-dest(gop.nu))<1[Hz] Are rays of same frequency? onccd qx>0[mm]&&qx<40[mm]&&qy>- Are rays close 32[mm]&&qy<-15[mm] to the detector? distance sqrt((qx-dest(qx))^2+(qy- m Distance dest(qy))^2) between the rays

STUDY 1

Step 1: Ray Tracing 1 In the Model Builder window, under Study 1 click Step 1: Ray Tracing. 2 In the Settings window for Ray Tracing, locate the Study Settings section. 3 In the Times text field, type range(0,0.01,0.8). 4 On the Home toolbar, click Compute.

11 | CZERNY-TURNER MONOCHROMATOR RESULTS

Ray 1 In the Model Builder window, expand the Results>Data Sets node.

Selection 1 Right-click Ray 1 and choose Duplicate. 2 On the Results toolbar, click Selection. 3 In the Settings window for Selection, locate the Geometric Entity Selection section. 4 From the Geometric entity level list, choose Boundary. 5 Select Boundary 8 only.

Ray Trajectories 1 1 In the Model Builder window, expand the Results>Ray Trajectories (gop) node, then click Ray Trajectories 1. 2 In the Settings window for Ray Trajectories, locate the Coloring and Style section. 3 Find the Line style subsection. From the Type list, choose Tube. 4 Select the Radius scale factor check box. 5 In the associated text field, type 0.025. 6 Find the Point style subsection. From the Type list, choose None.

Color Expression 1 1 In the Model Builder window, expand the Ray Trajectories 1 node, then click Color Expression 1. 2 In the Settings window for Color Expression, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1> Geometrical Optics>Ray properties>gop.lambda0 - Vacuum wavelength. 3 Locate the Coloring and Style section. From the Color table list, choose Spectrum. 4 Locate the Expression section. From the Unit list, choose nm. 5 On the Ray Trajectories (gop) toolbar, click Plot. Compare the resulting plot to Figure 3.

1D Plot Group 2 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 In the Settings window for 1D Plot Group, type Pixel Number in the Label text field. 3 Locate the Data section. From the Data set list, choose Ray 2. 4 From the Time selection list, choose Last.

12 | CZERNY-TURNER MONOCHROMATOR Ray 1 1 On the Pixel Number toolbar, click More Plots and choose Ray. 2 In the Settings window for Ray, locate the y-Axis Data section. 3 In the Expression text field, type pnum. 4 Locate the x-Axis Data section. From the Parameter list, choose Expression. 5 In the Expression text field, type gop.lambda0. 6 From the Unit list, choose nm. 7 Click to expand the Coloring and style section. Locate the Coloring and Style section. Find the Line markers subsection. From the Marker list, choose Circle. 8 From the Positioning list, choose In data points. 9 On the Pixel Number toolbar, click Plot. Compare the resulting plot to Figure 4.

1D Plot Group 3 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 In the Settings window for 1D Plot Group, type Device Resolution in the Label text field. 3 Locate the Data section. From the Data set list, choose Ray 2. 4 From the Time selection list, choose Last. 5 Click to expand the Title section. From the Title type list, choose None.

Ray 1 1 On the Device Resolution toolbar, click More Plots and choose Ray. 2 In the Settings window for Ray, locate the y-Axis Data section. 3 In the Expression text field, type Srange/N*wi/wp. 4 From the Unit list, choose nm. 5 Select the Description check box. 6 In the associated text field, type Spectral resolution. 7 Locate the x-Axis Data section. From the Parameter list, choose Expression. 8 In the Expression text field, type gop.lambda0. 9 From the Unit list, choose nm. 10 Click to expand the Coloring and style section. Locate the Coloring and Style section. Find the Line markers subsection. From the Marker list, choose Circle. 11 From the Positioning list, choose In data points. 12 On the Device Resolution toolbar, click Plot. Compare the resulting plot to Figure 5.

13 | CZERNY-TURNER MONOCHROMATOR Appendix — Geometry Instructions

On the Home toolbar, click Component and choose Add Component>2D.

GEOMETRY 1 1 In the Model Builder window, under Component 1 (comp1) click Geometry 1. 2 In the Settings window for Geometry, locate the Units section. 3 From the Length unit list, choose mm.

GLOBAL DEFINITIONS

Parameters 1 On the Home toolbar, click Parameters. 2 In the Settings window for Parameters, locate the Parameters section. 3 In the table, enter the following settings:

Name Expression Value Description theta_g 28.76[deg] 0.502 rad Grating’s angle theta_c 11[deg] 0.192 rad Collimating mirror’s angle theta_i 77[deg] 1.344 rad Imaging mirror’s angle theta_d 6.76[deg] 0.118 rad Detector’s angle Qix 20[mm] 0.02 m x coordinate Qi Qiy 34[mm] 0.034 m y coordinate Qi Qcx 40[mm] 0.04 m x coordinate Qc Qcy 40*tan(2*theta_c) 16.16 y coordinate Qc Qdx 22.08[mm] 0.02208 m x coordinate Qd Qdy -24.12[mm] -0.02412 m y coordinate Qd Ri 130[mm] 0.13 m Imaging mirror’s radius of curvature Rc 100[mm] 0.1 m Collimating mirror’s radius of curvature

GEOMETRY 1

Rectangle 1 (r1) 1 On the Geometry toolbar, click Primitives and choose Rectangle.

14 | CZERNY-TURNER MONOCHROMATOR 2 In the Settings window for Rectangle, locate the Size and Shape section. 3 In the Width text field, type 3. 4 In the Height text field, type 15. 5 Locate the Position section. From the Base list, choose Center. 6 In the x text field, type -1.5*cos(theta_g). 7 In the y text field, type -1.5*sin(theta_g). 8 Locate the Rotation Angle section. In the Rotation text field, type theta_g.

Rectangle 2 (r2) 1 On the Geometry toolbar, click Primitives and choose Rectangle. 2 In the Settings window for Rectangle, locate the Size and Shape section. 3 In the Width text field, type 3. 4 In the Height text field, type 15. 5 Locate the Position section. From the Base list, choose Center. 6 In the x text field, type Qcx. 7 In the y text field, type Qcy. 8 Locate the Rotation Angle section. In the Rotation text field, type theta_c.

Circle 1 (c1) 1 On the Geometry toolbar, click Primitives and choose Circle. 2 In the Settings window for Circle, locate the Size and Shape section. 3 In the Radius text field, type Rc. 4 Locate the Position section. In the x text field, type Qcx-Rc*cos(theta_c). 5 In the y text field, type Qcy-Rc*sin(theta_c).

Difference 1 (dif1) 1 On the Geometry toolbar, click Booleans and Partitions and choose Difference. 2 Click the Zoom Extents button on the Graphics toolbar. 3 Select the object r2 only. 4 In the Settings window for Difference, locate the Difference section. 5 Find the Objects to subtract subsection. Select the Active toggle button. 6 Select the object c1 only.

Rectangle 3 (r3) 1 On the Geometry toolbar, click Primitives and choose Rectangle.

15 | CZERNY-TURNER MONOCHROMATOR 2 In the Settings window for Rectangle, locate the Size and Shape section. 3 In the Width text field, type 3. 4 In the Height text field, type 30. 5 Locate the Position section. From the Base list, choose Center. 6 In the x text field, type Qix. 7 In the y text field, type Qiy. 8 Locate the Rotation Angle section. In the Rotation text field, type theta_i.

Circle 2 (c2) 1 On the Geometry toolbar, click Primitives and choose Circle. 2 In the Settings window for Circle, locate the Size and Shape section. 3 In the Radius text field, type Ri. 4 Locate the Position section. In the x text field, type Qix-Ri*cos(theta_i). 5 In the y text field, type Qiy-Ri*sin(theta_i).

Difference 2 (dif2) 1 On the Geometry toolbar, click Booleans and Partitions and choose Difference. 2 Select the object r3 only. 3 In the Settings window for Difference, locate the Difference section. 4 Find the Objects to subtract subsection. Select the Active toggle button. 5 Select the object c2 only.

Rectangle 4 (r4) 1 On the Geometry toolbar, click Primitives and choose Rectangle. 2 In the Settings window for Rectangle, locate the Size and Shape section. 3 In the Width text field, type 30. 4 In the Height text field, type 3. 5 Locate the Position section. From the Base list, choose Center. 6 In the x text field, type Qdx+1.5*sin(theta_d). 7 In the y text field, type Qdy-1.5*cos(theta_d). 8 Locate the Rotation Angle section. In the Rotation text field, type theta_d. 9 Click Build All Objects. 10 Click the Zoom Extents button on the Graphics toolbar.

16 | CZERNY-TURNER MONOCHROMATOR Created in COMSOL Multiphysics 5.3

Diffraction Grating

This model is licensed under the COMSOL Software License Agreement 5.3. All trademarks are the property of their respective owners. See www.comsol.com/trademarks. This tutorial uses the Wave Optics Module and the Ray Optics Module to simulate the propagation of rays through a diffraction grating at different angles of incidence. It uses the S-parameters computed by the Electromagnetic Waves, Frequency Domain interface on a unit cell of the grating to specify the reflectance and transmittance of each diffraction order in the Geometrical Optics interface, allowing ray propagation through the grating to be modeled over length scales much larger than the width of the unit cell.

Introduction

The Geometrical Optics interface includes a Grating feature that can be used to simulate propagation of electromagnetic waves on fully scaled optical devices without the need to spatially resolve the wavelength, which would be impractical in many cases due to the large number of mesh elements required.

Although the directions of propagation for the diffraction orders can be derived from the wavelength of radiation, the angle of incidence, and the width of a unit cell in the grating, reinitialization of the ray intensity requires prior calculation of the transmittance and reflectance for all diffraction orders as a function of angle of incidence. These quantities can be obtained by computing the S-parameters of each diffraction order for a single unit cell as a function of the angle of incidence using the Port and Diffraction Order features for the Electromagnetic Waves, Frequency Domain interface.

This 2D model is separated in two parts.

• First, the transmittance and reflectance of each diffraction order are computed using the Electromagnetic Waves, Frequency Domain interface on a single unit cell of the grating. For this part of the model it is necessary to fully resolve the wavelength. A Parametric Sweep is used to compute the transmittance and reflectance as functions of the angle of incidence. • The second part demonstrates how the transmittance and reflectance values can be used to generate a set of interpolation functions that can be used with the Grating feature of the Geometrical Optics interface.

Model Definition

λ = This model simulates the interaction of light of free-space wavelength 0 441 nm with a 5mm wide dielectric grating of grating constant (the distance between the grooves) d = 340 nm.

2 | DIFFRACTION GRATING NOTES ON DIFFRACTION ORDERS For a plane wave incident on a diffraction grating at angle of incidence α (SI unit: rad) as in Figure 1, the diffraction orders correspond to the angles at which the difference in optical path length for wavefronts from adjacent unit cells is an integer multiple of the β wavelength. A valid angle for a transmitted diffraction order m (SI unit: rad) must follow the relation

λ n sin()β – n sin()α = m-----0- β m α d where the diffraction order m (dimensionless) is an integer.

Air nα

β m

SiO2 d nβ

Figure 1: The geometric path lengths of two transmitted parallel rays.The shaded area represents a unit cell of the diffraction grating (SiO ). For this model the grating constant is 2 λ d = 340 nm and the monochromatic TE polarized light has a wavelength of 0 = 441 nm.

For m =0, the angle of refraction is described by Snell’s law,

n ()β α ()α sin 0 = ------sin nβ

For reflected rays, nα = nβ. For m =0, the equation for specular reflection is recovered,

()β ()α sin 0 = sin

Because the sine functions can only vary between -1 and 1, the existence of higher diffraction orders requires that

3 | DIFFRACTION GRATING mλ –()n + n <<------0 ()n + n α β d α β

In this example only the diffraction orders 0, 1, and -1 can be released, which means that

λ > ()()α 2 0 dnα sin + nβ

As mentioned in the introduction, the model consists of two parts: the S-parameter calculation using a single unit cell and the ray trajectory computation in an optically large modeling domain.

S-PARAMETER CALCULATION The transmittance and reflectance for the refraction, specular reflection, and first order diffraction of plane TE waves (electric field component in the z-direction, out of the xy- plane) are computed for a single unit cell.

The Electromagnetic Waves, Frequency Domain interface is used to model wave propagation in a single unit cell of the grating, as outlined in Figure 1. On either side of the unit cell, the Periodic Condition boundary condition with Floquet periodicity is used. This condition states that the solution on one side of the unit cell equals the solution on the other side multiplied by a complex-valued phase factor. The phase shift between the boundaries is evaluated from the perpendicular component of the wave vector. Note that due to the continuity of the field, the phase factor is the same for the refracted and reflected waves as for the incident wave.

Port boundary conditions are used to release the incident wave and to absorb the reflected and transmitted waves of order 0. To ensure that no non-physical reflections occur, Diffraction Order subnodes must be added to the Port nodes to absorb outgoing waves of each nonzero diffraction order.

The input to each periodic port is an electric field amplitude vector and an angle of incidence. In this example the angle of incidence is swept from 0° to 90° at 1° intervals.

RAY TRACING The Geometrical Optics interface computes the intensity of rays of each diffraction order using the transmittance and reflectance computed in the previous study. In order to see the effect of the angle of incidence on the ray trajectories and intensity, 901 rays are released from a point in a 90° cone with a source power density of 901 W/m, or 1W/m per ray. For each diffraction order, two rays may be released, one transmitted ray and one reflected ray. Because the transmitted ray of order 0 uses the same degrees of freedom as the incident ray, five extra degrees of freedom should be allocated per incident ray: one for

4 | DIFFRACTION GRATING the reflected ray of order 0 and two each for the reflected and transmitted rays of order m = 1 and m = -1. A total of 4505 secondary rays are allocated.

Results and Discussion

The electric field norm for a TE wave with an angle of incidence of 47.5° is shown in Figure 2. In order to get reliable results one has to use a very fine mesh to resolve the wavelength. To resolve a wave properly, it is necessary to use about 10 mesh elements per wavelength when using linear shape functions, or 5 elements per wavelength when using the default quadratic shape functions.

Figure 2: Norm of the electric field for a TE wave with an angle of incidence of 45 degrees.

The transmittance and reflectance of each diffraction order as functions of the angle of incidence are shown in Figure 3. Most of the radiation is transmitted at diffraction order 0, except at very large angles of incidence for which most of the radiation is reflected.

5 | DIFFRACTION GRATING Figure 3: Reflectance and transmittance of diffraction orders 0, 1, and -1 as functions of the angle of incidence.

The raw data from Figure 3 was used to define a series of six interpolation functions, each corresponding to the reflectance or transmittance of a diffraction order. These interpolation functions were used in the Geometrical Optics interface to define the reinitialized intensity of the transmitted and reflected rays.

In Figure 4 the total intensity of the reflected and transmitted rays, indicated by points at discrete angle intervals, is compared to the sum of the reflectance and transmittance functions defined with the solution data from full wave solution.

The curves for the Electromagnetic Waves, Frequency Domain interface and the Geometrical Optics interface agree closely, which is to be expected because the transmittance and reflectance of the grating in the latter are defined explicitly in terms of the solution to the former.

6 | DIFFRACTION GRATING Figure 4: The transmittance and reflectance computed by both the wave optics and ray optics models.

Application Library path: Ray_Optics_Module/Tutorials/diffraction_grating

Modeling Instructions

From the File menu, choose New.

NEW In the New window, click Model Wizard.

MODEL WIZARD 1 In the Model Wizard window, click 2D. 2 In the Select Physics tree, select Optics>Wave Optics>Electromagnetic Waves, Frequency Domain (ewfd). 3 Click Add.

7 | DIFFRACTION GRATING 4 Click Study. 5 In the Select Study tree, select Preset Studies>Frequency Domain. 6 Click Done.

GLOBAL DEFINITIONS

Parameters 1 On the Home toolbar, click Parameters. 2 In the Settings window for Parameters, locate the Parameters section. 3 In the table, enter the following settings:

Name Expression Value Description n_air 1 1 Refractive index air n_sio2 1.54874 1.549 Refractive index SiO2 d 340[nm] 3.4E-7 m Grating constant lam0 441[nm] 4.41E-7 m Vacuum wavelength of incident light f0 c_const/lam0 6.798E14 1/s Frequency of incident light alpha 0.0[deg] 0 rad Angle of incidence (input port)

Because this model uses two model Components with different geometries but the same material properties, it is convenient to define global materials before setting up the individual physics interfaces.

Material 1 (mat1) 1 In the Model Builder window, under Global Definitions right-click Materials and choose Blank Material. 2 In the Settings window for Material, type Air in the Label text field. 3 Click to expand the Material properties section. Locate the Material Properties section. In the Material properties tree, select Electromagnetic Models>Refractive Index> Refractive index, real part (n). 4 Click Add to Material.

8 | DIFFRACTION GRATING 5 Locate the Material Contents section. In the table, enter the following settings:

Property Name Value Unit Property group Refractive index, real n n_air 1 Refractive index part Refractive index, ki 0 1 Refractive index imaginary part

Material 2 (mat2) 1 Right-click Materials and choose Blank Material. 2 In the Settings window for Material, type SiO2 in the Label text field. 3 Click to expand the Material properties section. Locate the Material Properties section. In the Material properties tree, select Electromagnetic Models>Refractive Index> Refractive index, real part (n). 4 Click Add to Material. 5 Locate the Material Contents section. In the table, enter the following settings:

Property Name Value Unit Property group Refractive index, real n n_sio2 1 Refractive index part Refractive index, ki 0 1 Refractive index imaginary part

GEOMETRY 1 Create the geometry of a single unit cell in the grating.

Rectangle 1 (r1) 1 On the Geometry toolbar, click Primitives and choose Rectangle. 2 In the Settings window for Rectangle, locate the Size and Shape section. 3 In the Width text field, type d. 4 In the Height text field, type 6*d. 5 Locate the Position section. In the y text field, type -3*d.

9 | DIFFRACTION GRATING 5 Locate the Position section. In the y text field, type -3*d.

Rectangle 3 (r3) 1 On the Geometry toolbar, click Primitives and choose Rectangle. 2 In the Settings window for Rectangle, locate the Size and Shape section. 3 In the Width text field, type d/2. 4 In the Height text field, type d/4. 5 Locate the Position section. In the x text field, type d/4.

Union 1 (uni1) 1 On the Geometry toolbar, click Booleans and Partitions and choose Union. 2 Select the objects r2 and r3 only. 3 In the Settings window for Union, locate the Union section. 4 Clear the Keep interior boundaries check box. 5 Click Build All Objects. The geometry should look like the unit cell in Figure 1.

MATERIALS

Material Link 1 (matlnk1) 1 In the Model Builder window, under Component 1 (comp1) right-click Materials and choose Material Link. 2 Select Domain 2 only.

Material Link 2 (matlnk2) 1 Right-click Materials and choose Material Link. 2 Select Domain 1 only. 3 In the Settings window for Material Link, locate the Link Settings section. 4 From the Material list, choose SiO2 (mat2).

STUDY 1

Step 1: Frequency Domain It is convenient to specify the frequency in the sweep before setting up the physics, since it can then be used to automatically compute the diffraction orders for the Port boundary conditions.

1 In the Model Builder window, under Study 1 click Step 1: Frequency Domain. 2 In the Settings window for Frequency Domain, locate the Study Settings section.

10 | DIFFRACTION GRATING 3 In the Frequencies text field, type f0.

In this model the S-parameters of a TE wave are computed. Select Out-of-plane vector as the component of the electric field to be solved for. Let the mesh be generated automatically based on the frequency used in the study.

ELECTROMAGNETIC WAVES, FREQUENCY DOMAIN (EWFD) 1 In the Model Builder window, under Component 1 (comp1) click Electromagnetic Waves, Frequency Domain (ewfd). 2 In the Settings window for Electromagnetic Waves, Frequency Domain, locate the Components section. 3 From the Electric field components solved for list, choose Out-of-plane vector. 4 Locate the Physics-Controlled Mesh section. Select the Enable check box. Create a periodic input port. To model a TE wave, keep the Electric field as the Input quantity and enter the value 1 in the z-component field.

Port 1 1 Right-click Component 1 (comp1)>Electromagnetic Waves, Frequency Domain (ewfd) and choose Port. 2 Select Boundary 5 only. 3 In the Settings window for Port, locate the Port Properties section. 4 From the Type of port list, choose Periodic.

5 Locate the Port Mode Settings section. Specify the E0 vector as

0 x 0 y 1 z

6 In the α text field, type alpha. 7 Locate the Automatic Diffraction Order Calculation section. Clear the Include in automatic diffraction order calculation check box, as the Diffraction Order nodes need to be manually added for this port for normal incidence. 8 In the n text field, type n_air. Use Diffraction Order nodes to absorb the reflected waves of nonzero diffraction order.

Diffraction Order 1 1 Right-click Component 1 (comp1)>Electromagnetic Waves, Frequency Domain (ewfd)> Port 1 and choose Diffraction Order.

11 | DIFFRACTION GRATING 2 In the Settings window for Diffraction Order, locate the Port Mode Settings section. 3 From the Components list, choose Out-of-plane vector. 4 In the m text field, type -1.

Diffraction Order 2 1 Right-click Port 1 and choose Diffraction Order. 2 In the Settings window for Diffraction Order, locate the Port Mode Settings section. 3 From the Components list, choose Out-of-plane vector. 4 In the m text field, type 1. Add the output port. In this case the excitation is set to Off.

Port 2 1 In the Model Builder window, right-click Electromagnetic Waves, Frequency Domain (ewfd) and choose Port. 2 Select Boundary 2 only. 3 In the Settings window for Port, locate the Port Properties section. 4 From the Type of port list, choose Periodic.

5 Locate the Port Mode Settings section. Specify the E0 vector as

0 x 0 y 1 z

6 Locate the Automatic Diffraction Order Calculation section. In the n text field, type n_sio2.

Add the Diffraction Order nodes for the second periodic port by clicking the Compute Diffraction Orders button on the first periodic port.

Port 1 1 In the Model Builder window, under Component 1 (comp1)>Electromagnetic Waves, Frequency Domain (ewfd) click Port 1. 2 In the Settings window for Port, locate the Automatic Diffraction Order Calculation section. 3 Click Compute Diffraction Orders. Add the periodic boundary condition to the sides of the unit cell.

12 | DIFFRACTION GRATING Periodic Condition 1 1 In the Model Builder window, right-click Electromagnetic Waves, Frequency Domain (ewfd) and choose Periodic Condition. 2 Select Boundaries 1, 3, 10, and 11 only. 3 In the Settings window for Periodic Condition, locate the Periodicity Settings section. 4 From the Type of periodicity list, choose Floquet periodicity. 5 From the k-vector for Floquet periodicity list, choose From periodic port.

STUDY 1

Parametric Sweep 1 On the Study toolbar, click Parametric Sweep. 2 In the Settings window for Parametric Sweep, locate the Study Settings section. 3 Click Add. 4 In the table, enter the following settings:

Parameter name Parameter value list Parameter unit alpha range(0,1,90) deg

5 On the Study toolbar, click Compute.

RESULTS

Electric Field (ewfd) 1 In the Model Builder window, click Electric Field (ewfd). 2 In the Settings window for 2D Plot Group, locate the Data section. 3 From the Parameter value (alpha (deg)) list, choose 45. 4 On the Electric Field (ewfd) toolbar, click Plot. 5 Click the Zoom Extents button on the Graphics toolbar. Compare the resulting plot to Figure 2.

Global Evaluation 1 1 On the Results toolbar, click Global Evaluation. 2 In the Settings window for Global Evaluation, locate the Expressions section.

13 | DIFFRACTION GRATING 3 In the table, enter the following settings:

Expression Unit Description abs(ewfd.S11)^2 1 R0 abs(ewfd.S41)^2 1 T0 abs(ewfd.S21)^2 1 R-1 abs(ewfd.S51)^2 1 T-1 abs(ewfd.S31)^2 1 R1 abs(ewfd.S61)^2 1 T1

4 Click Evaluate. The resulting table shows the reflectance and transmittance values as functions of the angle of incidence.

TABLE 1 Go to the Table window. 2 Click Table Graph in the window toolbar.

RESULTS

Table Graph 1 1 In the Model Builder window, under Results>1D Plot Group 2 click Table Graph 1. 2 In the Settings window for Table Graph, click to expand the Legends section. 3 Select the Show legends check box.

1D Plot Group 2 1 In the Model Builder window, under Results click 1D Plot Group 2. 2 In the Settings window for 1D Plot Group, type Transmittance and Reflectance (ewfd) in the Label text field. 3 Click to expand the Title section. From the Title type list, choose None. 4 Locate the Plot Settings section. Select the x-axis label check box. 5 In the associated text field, type Angle of incidence (deg). 6 Select the y-axis label check box. 7 In the associated text field, type Transmittance and Reflectance. 8 Click to expand the Legend section. From the Position list, choose Middle left. 9 On the Transmittance and Reflectance (ewfd) toolbar, click Plot. Compare the resulting plot to Figure 3.

Now add a second model Component to compute the ray trajectories.

14 | DIFFRACTION GRATING ROOT On the Home toolbar, click Component and choose Add Component>2D.

GEOMETRY 2 In the Model Builder window, under Component 2 (comp2) click Geometry 2.

Rectangle 1 (r1) 1 On the Geometry toolbar, click Primitives and choose Rectangle. 2 In the Settings window for Rectangle, locate the Position section. 3 From the Base list, choose Center. 4 Locate the Size and Shape section. In the Width text field, type 5[mm]. 5 In the Height text field, type 1.35[mm].

Rectangle 2 (r2) 1 On the Geometry toolbar, click Primitives and choose Rectangle. 2 In the Settings window for Rectangle, locate the Size and Shape section. 3 In the Width text field, type 5[mm]. 4 Locate the Position section. From the Base list, choose Center. 5 Locate the Size and Shape section. In the Height text field, type 0.675[mm]. 6 Locate the Position section. In the y text field, type -0.675[mm]/2. 7 Click Build All Objects.

DEFINITIONS Use the reflectance and transmittance data from the previous study to define a series of interpolation functions for the large-scale geometrical optics analysis.

Interpolation 1 (int1) 1 On the Home toolbar, click Functions and choose Local>Interpolation. 2 In the Settings window for Interpolation, locate the Definition section. 3 From the Data source list, choose Result table. 4 Find the Functions subsection. In the table, enter the following settings:

Function name Position in file R0 1 T0 2 Rm1 3 Tm1 4

15 | DIFFRACTION GRATING Function name Position in file R1 5 T1 6 5 Locate the Units section. In the Arguments text field, type deg. 6 In the Function text field, type 1.

Now set up the Geometrical Optics interface.

ADD PHYSICS 1 On the Home toolbar, click Add Physics to open the Add Physics window. 2 Go to the Add Physics window. 3 In the tree, select Optics>Ray Optics>Geometrical Optics (gop). 4 Click Add to Component in the window toolbar. 5 On the Home toolbar, click Add Physics to close the Add Physics window.

MATERIALS

Material Link 3 (matlnk3) 1 In the Model Builder window, under Component 2 (comp2) right-click Materials and choose Material Link. 2 Select Domain 2 only.

Material Link 4 (matlnk4) 1 Right-click Materials and choose Material Link. 2 Select Domain 1 only. 3 In the Settings window for Material Link, locate the Link Settings section. 4 From the Material list, choose SiO2 (mat2).

GEOMETRICAL OPTICS (GOP) 1 In the Model Builder window, under Component 2 (comp2) click Geometrical Optics (gop). 2 In the Settings window for Geometrical Optics, locate the Intensity Computation section. 3 From the Intensity computation list, choose Compute intensity and power. 4 Locate the Ray Release and Propagation section. In the Maximum number of secondary rays text field, type 4505.

16 | DIFFRACTION GRATING Ray Properties 1 1 In the Model Builder window, under Component 2 (comp2)>Geometrical Optics (gop) click Ray Properties 1. 2 In the Settings window for Ray Properties, locate the Ray Properties section. 3 From the Ray property specification list, choose Specify frequency. 4 In the ν text field, type f0.

Define the angle of incidence as a function of the wave vector.

DEFINITIONS

Variables 1 1 In the Model Builder window, under Component 2 (comp2) right-click Definitions and choose Variables. 2 In the Settings window for Variables, locate the Variables section. 3 In the table, enter the following settings:

Name Expression Unit Description alpha_ro atan2(-ky,kx) rad

GEOMETRICAL OPTICS (GOP)

Grating 1 1 In the Model Builder window, under Component 2 (comp2) right-click Geometrical Optics (gop) and choose Grating. 2 Select Boundary 4 only. 3 In the Settings window for Grating, locate the Device Properties section. 4 In the d text field, type d. 5 In the R text field, type R0(alpha_ro). 6 In the T text field, type T0(alpha_ro). 7 Select the Store total transmitted power check box. 8 Select the Store total reflected power check box. These check boxes create variables that can be used to compute the total power of all transmitted rays and all reflected rays, respectively, for each angle of incidence. Now that the diffraction grating has been defined, use the Diffraction Order subnodes to release additional secondary rays of nonzero diffraction order.

17 | DIFFRACTION GRATING Diffraction Order 1 1 Right-click Component 2 (comp2)>Geometrical Optics (gop)>Grating 1 and choose Diffraction Order. 2 In the Settings window for Diffraction Order, locate the Device Properties section. 3 In the m text field, type -1. 4 In the R text field, type Rm1(alpha_ro). 5 In the T text field, type Tm1(alpha_ro).

Diffraction Order 2 1 Right-click Grating 1 and choose Diffraction Order. 2 In the Settings window for Diffraction Order, locate the Device Properties section. 3 In the R text field, type R1(alpha_ro). 4 In the T text field, type T1(alpha_ro).

3 In the qy, 0 text field, type 1e-6. The ray will be released an extremely short distance above the grating so that even rays at very large angles of incidence will reach the boundary fairly quickly. 4 Locate the Ray Direction Vector section. From the Ray direction vector list, choose Conical.

5 In the Nw text field, type 901. 6 In the α text field, type pi/4. 7 Specify the r vector as

1 x -1.01 y

Define a power density of 901 W/m so that each ray has a power density of 1 W/m.

8 Locate the Total Source Power section. In the Psrc text field, type 901[W/m]. 9 Locate the Initial Polarization section. From the Initial polarization type list, choose Fully polarized.

10 In the axy, 0 text field, type 0.

18 | DIFFRACTION GRATING 11 In the az, 0 text field, type 1. The released ray is S-polarized. This is consistent with the use of TE waves in the previous study.

ADD STUDY 1 On the Home toolbar, click Add Study to open the Add Study window. 2 Go to the Add Study window. 3 Find the Studies subsection. In the Select Study tree, select Custom Studies. 4 Find the Physicsinterfacesinstudy subsection. In the table, clear the Solve check box for the Electromagnetic Waves, Frequency Domain (ewfd) interface. 5 Find the Studies subsection. In the Select Study tree, select Preset Studies>Ray Tracing. 6 Click Add Study in the window toolbar. 7 On the Home toolbar, click Add Study to close the Add Study window.

STUDY 2

Step 1: Ray Tracing 1 In the Model Builder window, under Study 2 click Step 1: Ray Tracing. 2 In the Settings window for Ray Tracing, locate the Study Settings section. 3 From the Time unit list, choose ps. 4 In the Times text field, type range(0,0.1,1). 5 On the Home toolbar, click Compute.

RESULTS

Ray Trajectories (gop) The default plot shows the paths of the rays as they interact with the grating.

1D Plot Group 4 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 In the Settings window for 1D Plot Group, type Transmittance and Reflectance (ewfd and gop) in the Label text field. 3 Click to expand the Title section. From the Title type list, choose None. 4 Locate the Plot Settings section. Select the x-axis label check box. 5 In the associated text field, type Angle of incidence (deg). 6 Select the y-axis label check box. 7 In the associated text field, type Transmittance and Reflectance.

19 | DIFFRACTION GRATING 8 Click to expand the Legend section. From the Position list, choose Middle left.

Global 1 1 Right-click Transmittance and Reflectance (ewfd and gop) and choose Global. 2 In the Settings window for Global, locate the y-Axis Data section. 3 In the table, enter the following settings:

Expression Unit Description abs(ewfd.S11)^2+abs(ewfd.S21)^2+ 1 abs(ewfd.S31)^2 abs(ewfd.S41)^2+abs(ewfd.S51)^2+ 1 abs(ewfd.S61)^2

These expressions give the total reflectance and transmittance, respectively, for all diffraction orders. 4 Locate the x-Axis Data section. From the Parameter list, choose Expression. 5 In the Expression text field, type alpha. 6 From the Unit list, choose °. 7 Click to expand the Legends section. From the Legends list, choose Manual. 8 In the table, enter the following settings:

Legends Reflected WO Transmitted WO

Transmittance and Reflectance (ewfd and gop) In the Model Builder window, under Results click Transmittance and Reflectance (ewfd and gop).

Ray 1 1 On the Transmittance and Reflectance (ewfd and gop) toolbar, click More Plots and choose Ray. 2 In the Settings window for Ray, locate the Data section. 3 From the Data set list, choose Ray 1. 4 From the Time selection list, choose Last. 5 Locate the y-Axis Data section. In the Expression text field, type gop.Qgr. 6 Locate the x-Axis Data section. From the Parameter list, choose Expression. 7 In the Expression text field, type at(0,alpha_ro).

20 | DIFFRACTION GRATING 8 From the Unit list, choose °. 9 Click to expand the Coloring and style section. Locate the Coloring and Style section. Find the Line style subsection. From the Line list, choose None. 10 Find the Line markers subsection. From the Marker list, choose Point. 11 From the Positioning list, choose Interpolated. 12 In the Number text field, type 40. 13 Click to expand the Legends section. Select the Show legends check box. 14 From the Legends list, choose Manual. 15 In the table, enter the following settings:

Legends Reflected RO

Ray 2 1 Right-click Ray 1 and choose Duplicate. 2 In the Settings window for Ray, locate the y-Axis Data section. 3 In the Expression text field, type gop.Qgt. 4 Locate the Legends section. In the table, enter the following settings:

Legends Transmitted RO

5 On the Transmittance and Reflectance (ewfd and gop) toolbar, click Plot. 6 Click the Zoom Extents button on the Graphics toolbar. Compare the resulting plot to Figure 4.

21 | DIFFRACTION GRATING 22 | DIFFRACTION GRATING Created in COMSOL Multiphysics 5.3

Distributed Bragg Reflector

This model is licensed under the COMSOL Software License Agreement 5.3. All trademarks are the property of their respective owners. See www.comsol.com/trademarks. Introduction

A distributed Bragg reflector, or dielectric mirror, is a reflector used in waveguides and optical fibers. A distributed Bragg reflector has extremely low losses at optical and infrared frequencies compared to ordinary metallic mirrors. Its structure is formed from periodic thin layers of alternating materials with high and low refractive indices. Typically the stack would be made up of an odd number of layers where the first and last layers are chosen to have high refractive index.

Each layer boundary causes a partial reflection of an optical wave. When the wavelength is close to four times the optical thickness of the layers, the many reflected waves tend to interfere constructively, causing the layers to act as a high-quality reflector. The range of wavelengths in which most of the incident intensity is reflected is called the photonic stopband. In the limit in which the reflector contains a very large number of layers, radiation in this range of wavelengths cannot propagate into the structure.

Distributed Bragg reflectors are critical components in vertical cavity surface emitting lasers and other types of narrow-linewidth laser diodes such as distributed feedback lasers.

na ns

High refractive index (nH, tH)

Low refractive index (nL, tL)

Figure 1: Bragg reflector with 9 layers (N=4).

Model Definition

The model consists of a single domain containing a substrate with refractive index nS =1.5. The exterior of the modeling domain is air with refractive index na =1.0. At the default Material Discontinuity boundary condition on the surfaces of the substrate, a number of Thin Dielectric Film features are added to represent the alternating layers.

2 | DISTRIBUTED BRAGG REFLECTOR The layers of greater refractive index are made of ZnS with nH =2.32, and the layers of lower refractive index contain MgF2 with nL =1.38. The thicknesses of the layers are λ calculated such that nHtH = nLtL = 0/4. To show the response of the mirror across a span of wavelengths, a range of wavelengths (or frequencies) can be specified using the Release from Grid feature. This functionality is activated if the Frequency-dependent refractive indices is chosen in the interface properties.

Of particular interest in this devices, is the reflectance R of the device and what range of wavelengths it is effective over, Δλ. The reflectance for the distributed Bragg reflector is given by:

2 2 n 2N n 1 – ------H ------H - n n n R = ------L a b- (1) 2 n 2N n 1 + ------H ------H -  nL nanb where N is the number of pairs of dielectric layers; for example, N =5 implies that the reflector consists of 11 layers (five pairs plus an additional layer of high refractive index on top).

The bandwidth Δλ of the photonic stop-band is given by:

4λ n – n Δλ = ------0 asin------H L- (2) 0 π  nH + nL

λ where 0 is the central wavelength of the band.

Results and Discussion

Figure 2 shows the response of the dielectric mirror across a range of wavelengths from λ 400 nm to 800 nm. The vacuum wavelength 0 used in the specification of the layers is 550 nm. At this wavelength, a configuration with 2 unit cells (5 total layers) gives roughly 87 % reflectance whereas a configuration with 5 unit cells (11 total layers) gives roughly 99.5 % reflectance. The computed values of R agree with Equation 1.

The calculated stop-band is 180 nm using Equation 2. As shown in Figure 2 the reflectance approaches 100 % within the stop-band as the number of layers increases. For the maximum number of layers the reflectance is about 100 % for a range of free-space wavelengths from 475 nm to 655 nm for a stop-band of about 180 nm.

3 | DISTRIBUTED BRAGG REFLECTOR Figure 2: Response of the distributed Bragg grating for different numbers of layers, from a minimum of 5 total layers to a maximum of 41.

Application Library path: Ray_Optics_Module/Tutorials/ distributed_bragg_reflector

Modeling Instructions

From the File menu, choose New.

NEW In the New window, click Model Wizard.

MODEL WIZARD 1 In the Model Wizard window, click 3D. 2 In the Select Physics tree, select Optics>Ray Optics>Geometrical Optics (gop). 3 Click Add.

4 | DISTRIBUTED BRAGG REFLECTOR 4 Click Study. 5 In the Select Study tree, select Preset Studies>Ray Tracing. 6 Click Done.

GEOMETRY 1 Add some parameters for the refractive indices of the materials and the central wavelength.

GLOBAL DEFINITIONS

Parameters 1 On the Home toolbar, click Parameters. 2 In the Settings window for Parameters, locate the Parameters section. 3 In the table, enter the following settings:

Name Expression Value Description ns 1.5 1.5 Refractive index of substrate nh 2.32 2.32 Refractive index of ZnS nl 1.38 1.38 Refractive index of MgF2 lam0 550[nm] 5.5E-7 m Vacuum wavelength Nc 2 2 Number of unit cells

GEOMETRY 1 1 In the Model Builder window, under Component 1 (comp1) click Geometry 1. 2 In the Settings window for Geometry, locate the Units section. 3 From the Length unit list, choose mm.

Cylinder 1 (cyl1) 1 On the Geometry toolbar, click Cylinder. 2 In the Settings window for Cylinder, locate the Size and Shape section. 3 In the Radius text field, type 10. 4 In the Height text field, type 5. 5 Click Build All Objects.

In the Geometrical Optics interface, enable ray intensity calculation and allow distributions of ray frequency values to be released. Also set the number of secondary rays to zero as this saves memory when it is not necessary to compute the trajectories of the reflected rays.

5 | DISTRIBUTED BRAGG REFLECTOR GEOMETRICAL OPTICS (GOP) 1 In the Model Builder window, under Component 1 (comp1) click Geometrical Optics (gop). 2 In the Settings window for Geometrical Optics, locate the Intensity Computation section. 3 From the Intensity computation list, choose Compute intensity. 4 Locate the Ray Release and Propagation section. Select the Allow frequency distributions at release features check box. 5 In the Maximum number of secondary rays text field, type 0.

Specify the refractive index of the substrate.

MATERIALS

Property Name Value Unit Property group Refractive index n ns 1 Refractive index

Edit the Material Discontinuity settings to allow periodic thin dielectric films to be added to the surface.

GEOMETRICAL OPTICS (GOP)

Material Discontinuity 1 1 In the Model Builder window, under Component 1 (comp1)>Geometrical Optics (gop) click Material Discontinuity 1. 2 In the Settings window for Material Discontinuity, locate the Coatings section. 3 From the Thin dielectric films on boundary list, choose Add layers to surface, repeating. 4 In the N text field, type Nc. Add two Thin Dielectric Film features to the Material Discontinuity feature. To visualize the arrangement of the thin dielectric layers, show the boundary normal in the Graphics window. 5 Click the Wireframe Rendering button on the Graphics toolbar.

6 | DISTRIBUTED BRAGG REFLECTOR 6 Locate the Advanced Settings section. Select the Show boundary normal check box. The red arrow points in the direction of the stack of layers; the last Thin Dielectric Film feature in the Model Builder will be at the top of the stack. Add the layers and specify their refractive indices and thicknesses so that the optical thickness of each layer is equal to 1/4 of the vacuum wavelength.

Thin Dielectric Film 1 1 Right-click Component 1 (comp1)>Geometrical Optics (gop)>Material Discontinuity 1 and choose Thin Dielectric Film. 2 In the Settings window for Thin Dielectric Film, locate the Film Properties section. 3 In the n text field, type nh. 4 In the t text field, type lam0/(4*nh).

Thin Dielectric Film 2 1 Right-click Material Discontinuity 1 and choose Thin Dielectric Film. 2 In the Settings window for Thin Dielectric Film, locate the Film Properties section. 3 In the n text field, type nl. 4 In the t text field, type lam0/(4*nl).

The multilayer film will begin and end with the same layer. To allow any number of repeating unit cells to be applied, duplicate the first layer so that three Thin Dielectric Film nodes are present, two of which form the repeating unit cell.

Thin Dielectric Film 1 1 In the Model Builder window, under Component 1 (comp1)>Geometrical Optics (gop)> Material Discontinuity 1 right-click Thin Dielectric Film 1 and choose Duplicate. Exclude the first layer from the unit cell of the multilayer film. 2 In the Settings window for Thin Dielectric Film, click to expand the Repeating multilayer films section. 3 Locate the Repeating Multilayer Films section. Clear the Repeat layer in multilayer films check box. Add a Release from Grid feature and specify a range of wavelengths to be released. 4 In the Model Builder window, right-click Geometrical Optics (gop) and choose Release from Grid.

Release rays with a large number of frequency values within a given range.

7 | DISTRIBUTED BRAGG REFLECTOR Release from Grid 1 1 In the Settings window for Release from Grid, locate the Initial Coordinates section.

2 In the qz, 0 text field, type 10.

3 Locate the Ray Direction Vector section. Specify the L0 vector as

0 x 0 y -1 z

4 Locate the Initial Ray Frequency section. From the Distribution function list, choose List of values. 5 In the Values text field, type range(3e8[m/s]/400[nm],(3e8[m/s]/800[nm]- (3e8[m/s]/400[nm]))/999,3e8[m/s]/800[nm]).

STUDY 1

Parametric Sweep 1 On the Study toolbar, click Parametric Sweep. Run a Parametric Sweep over the number of unit cells in the dielectric mirror in order to observe the changes in the reflectance. 2 In the Settings window for Parametric Sweep, locate the Study Settings section. 3 Click Add. 4 Click to select row number 1 in the table. 5 In the table, enter the following settings:

Parameter name Parameter value list Parameter unit Nc 2 5 10 20

To show the reflectance response of the mirror as a function of wavelength, use a Ray plot.

8 | DISTRIBUTED BRAGG REFLECTOR RESULTS

1D Plot Group 2 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 In the Settings window for 1D Plot Group, type Reflectance in the Label text field. 3 Locate the Data section. From the Data set list, choose Ray 1. 4 From the Time selection list, choose Last. 5 Click to expand the Title section. From the Title type list, choose None. 6 Locate the Plot Settings section. Select the x-axis label check box. 7 In the associated text field, type Vacuum wavelength (nm). 8 Select the y-axis label check box. 9 In the associated text field, type Reflectance (%).

Ray 1 1 On the Reflectance toolbar, click More Plots and choose Ray. Compare the initial and current intensity to plot the reflectance at each wavelength. 2 In the Settings window for Ray, locate the y-Axis Data section. 3 In the Expression text field, type 100*(gop.relg1.I0-gop.I)/gop.relg1.I0. 4 Locate the x-Axis Data section. From the Parameter list, choose Expression. 5 In the Expression text field, type gop.lambda0. 6 From the Unit list, choose nm. 7 Click to expand the Legends section. Select the Show legends check box. 8 From the Legends list, choose Manual. 9 In the table, enter the following settings:

Legends Nc=2 (5 layers) Nc=5 (11 layers) Nc=10 (21 layers) Nc=20 (41 layers)

10 On the Reflectance toolbar, click Plot. 11 Click the Zoom Extents button on the Graphics toolbar. Compare the computed reflectance values with Figure 2.

9 | DISTRIBUTED BRAGG REFLECTOR 10 | DISTRIBUTED BRAGG REFLECTOR Created in COMSOL Multiphysics 5.3

Gravitational Lensing

This model is licensed under the COMSOL Software License Agreement 5.3. All trademarks are the property of their respective owners. See www.comsol.com/trademarks. Introduction

This model demonstrates how the sun causes 1.75 arcseconds of deflection for rays grazing the sun's surface as observed from the Earth. Einstein predicted this value after refining his theory of relativity during World War I (Ref. 1).

Model Definition

The gravitational lensing effect is modeled using a refractive index which varies continuously in space, also known as a graded medium. The refractive index, n, depends 3 2 on the gravitational constant G (SI unit: m /(kg s ), the solar mass m0 (SI unit: kg), the speed of light c (SI unit: m/s) and radial distance from the center of the sun r (SI unit: m):

2Gm n = 1 + ------0 c2r

The Gravitational constant is built-in, with name G_const and predefined value 6.67384e-11[m^3/(kg*s^2)]. For a list of all built-in constants, see Physical Constants in the COMSOL Multiphysics Reference Manual.

In this example, two rays are released which graze the surface of the sun, then continue until a distance of 150 million km is reached. At this point, the deflection angle of the rays from their initial direction is evaluated.

Results and Discussion

The angular change in the direction of the rays is plotted in Figure 1. After release, it takes the rays around 165 s to reach the sun. The rays then begin to deviate from their initial direction due to the gradient in the refractive index. The final value is about 1.75 arcseconds, consistent with Einstein’s prediction.

2 | GRAVITATIONAL LENSING Figure 1: Deflection angle in arcseconds caused by the sun’s gravitational field.

Reference

1. http://en.wikipedia.org/wiki/Gravitational_lens

Application Library path: Ray_Optics_Module/Graded_Media/ gravitational_lensing

Modeling Instructions

From the File menu, choose New.

NEW In the New window, click Model Wizard.

MODEL WIZARD 1 In the Model Wizard window, click 3D.

3 | GRAVITATIONAL LENSING 2 In the Select Physics tree, select Optics>Ray Optics>Geometrical Optics (gop). 3 Click Add. 4 Click Study. 5 In the Select Study tree, select Preset Studies>Ray Tracing. 6 Click Done.

GLOBAL DEFINITIONS

Parameters 1 On the Home toolbar, click Parameters. 2 In the Settings window for Parameters, locate the Parameters section. 3 In the table, enter the following settings:

Name Expression Value Description r0 7E5[km] 7E8 m Radius of the sun m0 2E30[kg] 2E30 kg Solar mass

GEOMETRY 1 1 In the Model Builder window, under Component 1 (comp1) click Geometry 1. 2 In the Settings window for Geometry, locate the Units section. 3 From the Length unit list, choose km.

Sphere 1 (sph1) 1 On the Geometry toolbar, click Sphere. 2 In the Settings window for Sphere, locate the Size section. 3 In the Radius text field, type r0.

Block 1 (blk1) 1 On the Geometry toolbar, click Block. 2 In the Settings window for Block, locate the Size and Shape section. 3 In the Width text field, type 2E8. 4 In the Depth text field, type 1E7. 5 In the Height text field, type 1E7. 6 Locate the Position section. In the x text field, type 0.5E8. 7 From the Base list, choose Center. 8 Click Build All Objects.

4 | GRAVITATIONAL LENSING 9 Click Go to Default View. 10 Click the Wireframe Rendering button on the Graphics toolbar.

DEFINITIONS

Variables 1 1 In the Model Builder window, under Component 1 (comp1) right-click Definitions and choose Variables. 2 In the Settings window for Variables, locate the Variables section. 3 In the table, enter the following settings:

Name Expression Unit Description r sqrt(x^2+y^2+z^2+eps) m Radial distance from center of the sun n 1+2*G_const*m0/(c_const^2*r) Refractive index

MATERIALS

Property Name Value Unit Property group Refractive index n n 1 Refractive index

Release from Grid 1 1 In the Model Builder window, right-click Geometrical Optics (gop) and choose Release from Grid. Release the rays so that they barely avoid contact with the sphere that represents the sun. 2 In the Settings window for Release from Grid, locate the Initial Coordinates section.

5 | GRAVITATIONAL LENSING 3 In the qx, 0 text field, type -0.5E8.

4 In the qy, 0 text field, type -7.01E5 7.01E5.

5 Locate the Ray Direction Vector section. Specify the L0 vector as

1 x 0 y 0 z

6 Clear the Suppress interaction with coinciding exterior boundaries check box.

MESH 1 Use a Finer mesh to improve the mesh resolution in the region surrounding the sun.

1 In the Model Builder window, under Component 1 (comp1) click Mesh 1. 2 In the Settings window for Mesh, locate the Mesh Settings section. 3 From the Element size list, choose Finer. 4 Click Build All.

STUDY 1

Step 1: Ray Tracing 1 In the Settings window for Ray Tracing, locate the Study Settings section. 2 From the Time step specification list, choose Specify maximum path length. 3 From the Length unit list, choose km. 4 Click Range. 5 In the Range dialog box, choose Number of values from the Entry method list. 6 In the Stop text field, type 2E8. 7 In the Number of values text field, type 100. 8 Click Replace.

Solution 1 (sol1) 1 On the Study toolbar, click Show Default Solver. 2 In the Model Builder window, expand the Solution 1 (sol1) node, then click Time- Dependent Solver 1. 3 In the Settings window for Time-Dependent Solver, click to expand the Time stepping section. 4 Locate the Time Stepping section. Select the Maximum step check box.

6 | GRAVITATIONAL LENSING 5 In the associated text field, type 1. 6 On the Study toolbar, click Compute.

RESULTS

Ray Trajectories (gop) In the Model Builder window, expand the Ray Trajectories (gop) node.

Color Expression 1 1 In the Model Builder window, expand the Results>Ray Trajectories (gop)> Ray Trajectories 1 node, then click Color Expression 1. 2 In the Settings window for Color Expression, locate the Expression section. 3 In the Expression text field, type 3600*asin(gop.niy). 4 From the Unit list, choose °. 5 On the Ray Trajectories (gop) toolbar, click Plot.

Global Evaluation 1 1 On the Results toolbar, click Global Evaluation. Compare the resulting value to Einstein’s prediction of 1.75 arcseconds. 2 In the Settings window for Global Evaluation, locate the Data section. 3 From the Time selection list, choose Last. 4 Locate the Expressions section. In the table, enter the following settings:

Expression Unit Description gop.gopaveop1(abs(3600* ° Average over rays (gop) asin(gop.niy)))

5 Click Evaluate.

1D Plot Group 2 1 On the Results toolbar, click 1D Plot Group. 2 In the Settings window for 1D Plot Group, type Deviation from Initial Direction in the Label text field. 3 Locate the Data section. From the Data set list, choose Ray 1. 4 Click to expand the Title section. From the Title type list, choose None. 5 Locate the Plot Settings section. Select the y-axis label check box. 6 In the associated text field, type Deflection angle in arcseconds.

7 | GRAVITATIONAL LENSING Ray 1 1 On the Deviation from Initial Direction toolbar, click More Plots and choose Ray. 2 In the Settings window for Ray, locate the y-Axis Data section. 3 In the Expression text field, type 3600*abs(asin(gop.niy)). 4 From the Unit list, choose °. 5 On the Deviation from Initial Direction toolbar, click Plot. 6 Locate the Data Series Operation section. From the Operation list, choose Average. 7 On the Deviation from Initial Direction toolbar, click Plot. Compare the resulting plot to Figure 1.

8 | GRAVITATIONAL LENSING Created in COMSOL Multiphysics 5.3

Transparent Light Pipe

This model is licensed under the COMSOL Software License Agreement 5.3. All trademarks are the property of their respective owners. See www.comsol.com/trademarks. Introduction

Light pipes are structures that can be used to transport light between different locations. Light pipes can be used to transport sunlight to locations that it would not otherwise reach, such as underground subway stations. In addition to facilitating the transport of light, it is possible to illuminate a large area by selectively leaking radiation out of the sides of the pipe. Because light may be reflected a large number of times before exiting a light pipe, it can also be used to homogenize a non-uniform light source.

In general, light pipes can be divided into two major groups: tubes lined with a reflective coating and transparent solids that contain light via total internal reflection. In this example, light is transported through a bent light pipe by total internal reflection. The effect of the pipe shape on the transmittance is investigated.

Model Definition

The model simulates the propagation of rays in a bent light pipe with a circular cross- section. The light pipe is composed of solid poly (methyl methacrylate). A parametric sweep over the radius of curvature of one of the pipe bends is used to measure the effect of pipe shape on transmittance. The rays are released from a point source at one end of the pipe with a cone angle of π/12.

The transmittance of the pipe is measured by using a Wall node with the Deposited Ray Power subnode to compute the total incident power that arrives at the opposite end of the pipe.

Results and Discussion

Figure 1 displays the ray trajectories in the pipe when the radius of curvature of the second bent section is 5mm. As the rays arrive at the second bend, a significant number of rays undergo reflection and refraction due to the lower angles of incidence. At the first bend, which has a much larger radius of curvature, most rays undergo total internal reflection. The transmittance of the light pipe as a function of the pipe bend radius of curvature is shown in Figure 2. The transmittance is approximately 100 % when the radius of curvature reaches the maximum value of 20 mm.

2 | TRANSPARENT LIGHT PIPE Figure 1: Ray trajectories in a bent light pipe. The color expression for the refracted rays indicates the transmittance as they exit the light pipe.

Figure 2: Transmittance as a function of the radius of curvature in the second pipe bend.

3 | TRANSPARENT LIGHT PIPE Application Library path: Ray_Optics_Module/Tutorials/light_pipe

Modeling Instructions

From the File menu, choose New.

NEW In the New window, click Model Wizard.

ROOT Insert the prepared geometry sequence from file. You can read the instructions for creating the geometry in the appendix.

GEOMETRY 1 1 On the Geometry toolbar, click Insert Sequence. 2 Browse to the model’s Application Libraries folder and double-click the file light_pipe_geom_sequence.mph.

ADD MATERIAL 1 On the Home toolbar, click Add Material to open the Add Material window. 2 Go to the Add Material window. 3 In the tree, select Optical>Organic Materials>poly (methyl methacrylate) (Sultanova). 4 Click Add to Component in the window toolbar.

4 | TRANSPARENT LIGHT PIPE MATERIALS poly (methyl methacrylate) (Sultanova) (mat1) On the Home toolbar, click Add Material to close the Add Material window.

GEOMETRICAL OPTICS (GOP) 1 In the Model Builder window, under Component 1 (comp1) click Geometrical Optics (gop). 2 In the Settings window for Geometrical Optics, locate the Intensity Computation section. 3 From the Intensity computation list, choose Compute intensity and power. 4 Locate the Ray Release and Propagation section. In the Maximum number of secondary rays text field, type 400. The secondary rays are needed to produce reflected rays whenever a ray escapes via refraction across the pipe wall.

Release from Grid 1 1 Right-click Component 1 (comp1)>Geometrical Optics (gop) and choose Release from Grid. 2 In the Settings window for Release from Grid, locate the Ray Direction Vector section. 3 From the Ray direction vector list, choose Conical.

4 In the Nw text field, type 2000. 5 Specify the r vector as

0 x 0 y 1 z

6 In the α text field, type pi/12. Use a Wall node with a Deposited Ray Power subnode to compute the transmittance of the light pipe.

Wall 1 1 In the Model Builder window, right-click Geometrical Optics (gop) and choose Wall. 2 Select Boundary 22 only.

Deposited Ray Power 1 Right-click Component 1 (comp1)>Geometrical Optics (gop)>Wall 1 and choose Deposited Ray Power.

5 | TRANSPARENT LIGHT PIPE The mesh should be fine on the curved surfaces of the light pipe, but can be coarse elsewhere. This can be controlled by specifying a low value for the Curvature factor.

MESH 1 1 In the Model Builder window, under Component 1 (comp1) click Mesh 1. 2 In the Settings window for Mesh, locate the Mesh Settings section. 3 From the Element size list, choose Finer. 4 From the Sequence type list, choose User-controlled mesh. Create another Size node to refine the mesh on the boundary where the deposited ray power will be computed.

Size 1 1 Right-click Component 1 (comp1)>Mesh 1 and choose Size. 2 In the Model Builder window, under Component 1 (comp1)>Mesh 1 right-click Size 1 and choose Move Up. 3 In the Settings window for Size, locate the Geometric Entity Selection section. 4 From the Geometric entity level list, choose Boundary. 5 Select Boundary 22 only. 6 Locate the Element Size section. Click the Custom button. 7 Locate the Element Size Parameters section. Select the Minimum element size check box. 8 In the associated text field, type 0.1. 9 Select the Curvature factor check box. 10 In the associated text field, type 0.1.

STUDY 1

Parameter name Parameter value list Parameter unit rb2

5 Click Range.

6 | TRANSPARENT LIGHT PIPE 6 In the Range dialog box, type 5 in the Start text field. 7 In the Step text field, type 2.5. 8 In the Stop text field, type 20. 9 Click Replace. 10 In the Settings window for Parametric Sweep, locate the Study Settings section. 11 Click to select row number 1 in the table. 12 In the table, enter the following settings:

Parameter name Parameter value list Parameter unit rb2 range(5,2.5,20) mm

Step 1: Ray Tracing 1 In the Model Builder window, under Study 1 click Step 1: Ray Tracing. 2 In the Settings window for Ray Tracing, locate the Study Settings section. 3 From the Time step specification list, choose Specify maximum path length. 4 From the Length unit list, choose mm. 5 Click Range. 6 In the Range dialog box, type 15 in the Step text field. 7 In the Stop text field, type 150. 8 Click Replace. 9 In the Settings window for Ray Tracing, locate the Study Settings section. 10 In the Characteristic group velocity text field, type c_const/1.5. 11 On the Study toolbar, click Compute.

RESULTS

Ray Trajectories (gop) The default plot of the ray trajectories shows that some of the light escapes from the pipe via refraction at the curved segments.

1 Click the Go to ZX View button on the Graphics toolbar.

Ray Trajectories 1 Render some additional time steps to resolve the ray-boundary interactions more clearly.

1 In the Model Builder window, expand the Ray Trajectories (gop) node, then click Ray Trajectories 1.

7 | TRANSPARENT LIGHT PIPE 2 In the Settings window for Ray Trajectories, locate the Extra Time Steps section. 3 From the Maximum number of extra time steps rendered list, choose Specified number of times. 4 In the Maximum number of extra time steps text field, type 200.

Color Expression 1 Plot the ratio of the final and initial intensity for each ray.

1 In the Model Builder window, expand the Ray Trajectories 1 node, then click Color Expression 1. 2 In the Settings window for Color Expression, locate the Expression section. 3 In the Expression text field, type gop.Q/1[W]*2000. 4 On the Ray Trajectories (gop) toolbar, click Plot.

Ray Trajectories (gop) Compare the results for the maximum and minimum values of the pipe radius. Notice that the number of refracted rays increases as the radius of curvature is reduced.

1 In the Model Builder window, under Results click Ray Trajectories (gop). 2 In the Settings window for 3D Plot Group, locate the Data section. 3 From the Parameter value (rb2 (mm)) list, choose 5. 4 On the Ray Trajectories (gop) toolbar, click Plot. Compare the resulting plot to Figure 1.

Plot the transmittance of the light pipe for each value of the pipe bend radius of curvature.

1D Plot Group 2 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 In the Settings window for 1D Plot Group, type Transmittance in the Label text field. 3 Locate the Data section. From the Data set list, choose Study 1/ Parametric Solutions 1 (sol2). 4 From the Time selection list, choose Last. 5 Locate the Plot Settings section. Select the x-axis label check box. 6 In the associated text field, type Radius of curvature, bend 2.

Global 1 1 Right-click Transmittance and choose Global. 2 In the Settings window for Global, locate the y-Axis Data section.

8 | TRANSPARENT LIGHT PIPE 3 In the table, enter the following settings:

Expression Unit Description gop.wall1.bsrc1.Qp_int/(1[W]) 1 Transmittance

4 Locate the x-Axis Data section. From the Axis source data list, choose Outer solutions. 5 Click to expand the Legends section. Clear the Show legends check box. 6 On the Transmittance toolbar, click Plot. Compare the resulting plot to Figure 2.

Appendix A — Geometry Instructions

On the Home toolbar, click Component and choose Add Component>3D.

GEOMETRY 1 The geometry consists of three straight sections connected by two bent sections. To simplify the geometry setup, import a list of parameters from a text file.

Parameters On the Home toolbar, click Parameters.

GLOBAL DEFINITIONS

Parameters 1 In the Settings window for Parameters, locate the Parameters section. 2 Click Load from File. 3 Browse to the model’s Application Libraries folder and double-click the file light_pipe_parameters.txt.

GEOMETRY 1 1 In the Model Builder window, under Component 1 (comp1) click Geometry 1. 2 In the Settings window for Geometry, locate the Units section. 3 From the Length unit list, choose mm.

Cylinder 1 (cyl1) 1 On the Geometry toolbar, click Cylinder. 2 In the Settings window for Cylinder, locate the Size and Shape section. 3 In the Radius text field, type rpipe. 4 In the Height text field, type L1.

9 | TRANSPARENT LIGHT PIPE 5 Locate the Position section. In the x text field, type x0. 6 In the y text field, type y0. 7 In the z text field, type z0.

Torus 1 (tor1) 1 On the Geometry toolbar, click Torus. 2 In the Settings window for Torus, locate the Size and Shape section. 3 In the Major radius text field, type rb1. 4 In the Minor radius text field, type rpipe. 5 In the Revolution angle text field, type theta1. 6 Locate the Position section. In the x text field, type xc1. 7 In the y text field, type yc1. 8 In the z text field, type zc1. 9 Locate the Axis section. From the Axis type list, choose y-axis. 10 Locate the Rotation Angle section. In the Rotation text field, type 270-theta1.

Cylinder 2 (cyl2) 1 On the Geometry toolbar, click Cylinder. 2 In the Settings window for Cylinder, locate the Size and Shape section. 3 In the Radius text field, type rpipe. 4 In the Height text field, type L2. 5 Locate the Position section. In the x text field, type xL2. 6 In the y text field, type yL2. 7 In the z text field, type zL2. 8 Locate the Axis section. From the Axis type list, choose Cartesian. 9 In the x text field, type -sin(theta1). 10 In the z text field, type cos(theta1).

Torus 2 (tor2) 1 On the Geometry toolbar, click Torus. 2 In the Settings window for Torus, locate the Size and Shape section. 3 In the Major radius text field, type rb2. 4 In the Minor radius text field, type rpipe. 5 In the Revolution angle text field, type theta2.

10 | TRANSPARENT LIGHT PIPE 6 Locate the Position section. In the x text field, type xc2. 7 In the y text field, type yc2. 8 In the z text field, type zc2. 9 Locate the Axis section. From the Axis type list, choose y-axis. 10 Locate the Rotation Angle section. In the Rotation text field, type 90-theta1.

Cylinder 3 (cyl3) 1 On the Geometry toolbar, click Cylinder. 2 In the Settings window for Cylinder, locate the Size and Shape section. 3 In the Radius text field, type rpipe. 4 In the Height text field, type L3. 5 Locate the Position section. In the x text field, type xL3. 6 In the y text field, type yL3. 7 In the z text field, type zL3. 8 Locate the Axis section. From the Axis type list, choose Cartesian. 9 In the x text field, type sin(theta2-theta1). 10 In the z text field, type cos(theta2-theta1). 11 Click Build All Objects. 12 Click the Zoom Extents button on the Graphics toolbar.

Union 1 (uni1) 1 On the Geometry toolbar, click Booleans and Partitions and choose Union. 2 Click in the Graphics window and then press Ctrl+A to select all objects. 3 In the Settings window for Union, locate the Union section. 4 Clear the Keep interior boundaries check box. 5 On the Geometry toolbar, click Build All.

11 | TRANSPARENT LIGHT PIPE 12 | TRANSPARENT LIGHT PIPE Created in COMSOL Multiphysics 5.3

Linear Wave Retarder

This model is licensed under the COMSOL Software License Agreement 5.3. All trademarks are the property of their respective owners. See www.comsol.com/trademarks. Introduction

The intensity and polarization of light can be accurately controlled by transmitting it through various types of optical components. While it is possible to model complex changes in ray polarization by applying a customized Mueller matrix to rays at a single boundary, the same result can often be achieved by placing several simpler optical components in series. This tutorial model shows how the polarization of radiation can be manipulated using a combination of linear polarizers and linear wave retarders.

An ideal linear polarizer is a device that only transmits radiation for which the electric field lies within a plane. The intersection of this plane with the polarizer forms a line known as the transmission axis.

An ideal linear wave retarder is an optical device that applies a phase shift to radiation polarized in one direction with respect to radiation polarized in an orthogonal direction. The plane of linear polarization that corresponds to the direction of minimal phase retardation, when intersected with the surface of the wave retarder, yields a line known as the fast axis. The phase shift between radiation polarized parallel to and perpendicular to the fast axis is known as the retardance of the device.

By combining linear wave retarders and linear polarizers in series and varying their orientations and retardance values, it is possible to control the intensity of emitted radiation and to produce light in any state of linear, circular, or elliptical polarization.

Stokes Vectors and Optical Components

The intensity and polarization of a ray of light can be described by a 4-vector known as the Stokes vector, whose components are called the Stokes parameters. The first Stokes parameter is the ray intensity, the second and third parameters indicate linear polarization in various directions, and the fourth and final parameter corresponds to the degree of circular polarization. A more in-depth explanation of the physical meaning of the Stokes parameters can be found in the Ray Optics Module User’s Guide.

For example, a ray of natural (unpolarized) light with intensity I0 has Stokes vector

I0  0 s =  (1)  0 0

2 | LINEAR WAVE RETARDER The Stokes vector of linearly polarized light varies depending on the direction of polarization. If the light is polarized in the direction of the x-axis, the Stokes vector is

I0 I s = 0 (2)  0 0

For right-hand circularly polarized light the Stokes vector is

I0 0 s =  (3) 0  I0

Varying states of elliptical polarization can be defined by specifying nonzero values of all of the Stokes parameters. In general, the light ray is fully polarized if the L2 norm of the second, third, and fourth parameters equals the first parameter or ray intensity.

In the Geometrical Optics interface, optical components such as linear polarizers and wave retarders are implemented as boundary conditions that don’t affect the ray path but do change the Stokes parameters, which are stored as separate degrees of freedom for each ray. The effect of any optical component or system of optical components can be represented by a 44× Mueller matrix M. The Mueller matrix is multiplied by the Stokes vector of the incoming ray to produce a new Stokes vector for the transmitted ray,

–1 sR= MRsi where si is the Stokes vector of the incident ray. A rotation matrix R is included to account for the orientation of the optical component with respect to the coordinate system in which the ray’s Stokes parameters are defined.

LINEAR POLARIZER A linear polarizer transforms any incident light ray into a linearly polarized ray. Its Mueller matrix is

1100 1 M = --- 1100 (4) 2 0000 0000

3 | LINEAR WAVE RETARDER The orientation of the linear polarizer is specified by defining a direction called the transmission axis T. Light that is polarized in the direction parallel to the fast axis is completely transmitted, whereas light that is polarized in the orthogonal direction is completely blocked. The matrix R is then the rotation matrix from the coordinate system in which the Stokes parameters are defined to a coordinate system in which the transmission axis is parallel to the x-axis.

To see how the Mueller matrix in Equation 4 produces linearly polarized light, consider its effect on natural light (Equation 1):

0,5I 0,50,500I0 0   0 0,5I sMs==0,50,500 =0 (5) i   00000 0 00000 0

The transmitted light has half the intensity of the incident light and is linearly polarized. Linear polarization in any other direction can be achieved by including the rotation matrix R in Equation 5. This is equivalent to changing the orientation of the linear polarizer; that is, rotating the transmission axis.

LINEAR WAVE RETARDER A linear wave retarder applies a phase delay to linearly polarized light in one direction, relative to linearly polarized light in the orthogonal direction. Its Mueller matrix is

10 0 0 M = 01 0 0 (6) 00 cos()δ sin()δ 00–sin()δ cos()δ

The orientation of the linear wave retarder is specified by defining a direction called the fast axis F. Light that is polarized in the direction parallel to the fast axis is subjected to the minimum phase delay, whereas light that is polarized in the orthogonal direction is subjected to the maximum phase delay. The matrix R is then the rotation matrix from the coordinate system in which the Stokes parameters are defined to a coordinate system in which the fast axis is parallel to the x-axis. The relative phase delay δ is called the retardance.

A linear wave retarder has no discernible effect on natural light; this can be seen by combining Equation 1 and Equation 6 for any value of δ:

4 | LINEAR WAVE RETARDER 10 0 0 I0 I0   0 0 sMs==01 0 0  = i ()δ ()δ   00 cos sin 0 0 00–sin()δ cos()δ 0 0

A quarter-wave retarder (δπ= ⁄ 2 ) can convert linearly polarized light into circularly polarized light if the incident light is linearly polarized at a 45 degree angle with respect to the fast axis:

I I 10 0 00 0 0 0 sMs==01 0 0 = i I 0 00 0 10  00– 10 0 –I0

A half-wave retarder (δπ= ) can convert right-hand circularly polarized light into left- hand circularly polarized light and vice-versa:

I I 10 0 0 0 0 0 0 sMs==01 0 0  = i 0 0 00– 1 0   00 0– 1I0 –I0

COMBINING OPTICAL COMPONENTS A combination of several optical components in series can be modeled by defining a Mueller matrix for each layer. Alternatively, it is possible to specify a single Mueller matrix, which is the product of the Mueller matrices for all of the optical components; if there are

N optical components such that M1 is the Mueller matrix of the first component encountered and MN is the component of the last component encountered, the equivalent Mueller matrix Meq is … Meq = MNMN – 1MN – 2 M3M2M1

In this example, the individual polarizers and wave retarders are instead handled as separate entities, so that the effect of each optical component on the intensity and polarization of the transmitted light can be observed in greater detail.

5 | LINEAR WAVE RETARDER Model Definition

The model geometry consists of three parallel surfaces, all of which are parallel to the xy- plane. The first and last boundaries are assigned the Linear Polarizer boundary condition, one with a transmission axis parallel to the x-axis and the other with a transmission axis parallel to the y-axis. The middle boundary is assigned the Linear Wave Retarder boundary condition with a fast axis parallel to the line y=x; it makes a 45 degree angle with the transmission axes of both polarizers.

Without the linear wave retarder, no radiation would propagate through the assembly of polarizers. However, by varying the retardance, the linearly polarized ray transmitted by the first polarizer can be converted to a ray of circular polarization, elliptical polarization, or linear polarization in a different direction before reaching the second polarizer, thus allowing some of the light to be transmitted.

Results and Discussion

When the linear wave retarder is given zero retardance, the series of optical devices consists of two linear polarizers with orthogonal transmission axes. As shown in Figure 1, the first linear polarizer reduces the ray intensity by half, as indicated by the color expression. The transmitted ray is also linearly polarized, as indicated by the deformation of the ray trajectory. The second linear polarizer reduces the ray intensity to zero.

When the linearly polarized ray is transmitted through a quarter-wave retarder, the intensity of the ray transmitted by the second linear polarizer is nonzero, as shown in Figure 2. The quarter-wave retarder converts the incident linearly polarized ray to a circularly polarized ray without changing its intensity. Then half of the intensity of the circularly polarized ray is transmitted through the second polarizer, which returns it to a state of linear polarization.

When the quarter-wave retarder is replaced with a half-wave retarder, the incident linearly polarized ray is instead converted to linearly polarized radiation with an orthogonal polarization. As a result, the second linear polarizer does not cause any noticeable decrease in the ray intensity, as shown in Figure 3.

6 | LINEAR WAVE RETARDER Figure 1: A ray passing through two linear polarizers. The final intensity is zero.

Figure 2: Passage of a linearly polarized ray through a quarter-wave retarder between two linear polarizers. The final intensity is 1/4 of the initial intensity.

7 | LINEAR WAVE RETARDER Figure 3: Passage of a linearly polarized ray through a half-wave retarder between two linear polarizers. The final intensity is 1/2 of the initial intensity.

Application Library path: Ray_Optics_Module/Tutorials/linear_wave_retarder

Modeling Instructions

From the File menu, choose New.

NEW In the New window, click Model Wizard.

MODEL WIZARD 1 In the Model Wizard window, click 3D. 2 In the Select Physics tree, select Optics>Ray Optics>Geometrical Optics (gop). 3 Click Add. 4 Click Study.

8 | LINEAR WAVE RETARDER 5 In the Select Study tree, select Preset Studies>Ray Tracing. 6 Click Done.

GLOBAL DEFINITIONS

Parameters 1 On the Home toolbar, click Parameters. 2 In the Settings window for Parameters, locate the Parameters section. 3 In the table, enter the following settings:

Name Expression Value Description delta 0 0 Retardance

GEOMETRY 1 The geometry contains no domains and consists of three parallel surfaces, at which the boundary conditions will be applied.

Work Plane 1 (wp1) 1 On the Geometry toolbar, click Work Plane. 2 In the Settings window for Work Plane, locate the Plane Definition section. 3 In the z-coordinate text field, type 1.

Plane Geometry In the Model Builder window, under Component 1 (comp1)>Geometry 1> Work Plane 1 (wp1) click Plane Geometry.

Square 1 (sq1) 1 On the Work Plane toolbar, click Primitives and choose Square. 2 In the Settings window for Square, locate the Position section. 3 From the Base list, choose Center. 4 In the Model Builder window, click Geometry 1.

Array 1 (arr1) 1 On the Geometry toolbar, click Transforms and choose Array. 2 Select the object wp1 only. 3 In the Settings window for Array, locate the Size section. 4 From the Array type list, choose Linear. 5 In the Size text field, type 3.

9 | LINEAR WAVE RETARDER 6 Locate the Displacement section. In the z text field, type 1. 7 Click Build All Objects. 8 Click the Zoom Extents button on the Graphics toolbar.

GEOMETRICAL OPTICS (GOP) Since no reflection at material discontinuities occurs in this model, set the number of secondary rays to zero.

1 In the Model Builder window, under Component 1 (comp1) click Geometrical Optics (gop). 2 In the Settings window for Geometrical Optics, locate the Ray Release and Propagation section. 3 In the Maximum number of secondary rays text field, type 0. 4 Locate the Intensity Computation section. From the Intensity computation list, choose Compute intensity.

Release from Grid 1 1 In the Model Builder window, right-click Geometrical Optics (gop) and choose Release from Grid. 2 In the Settings window for Release from Grid, locate the Ray Direction Vector section.

3 Specify the L0 vector as

0 x 0 y 1 z

Create a pair of linear polarizers with orthogonal transmission axes.

Linear Polarizer 1 1 Right-click Geometrical Optics (gop) and choose Optical Devices>Linear Polarizer. 2 Select Boundary 1 only.

Linear Polarizer 2 1 Right-click Geometrical Optics (gop) and choose Optical Devices>Linear Polarizer. 2 Select Boundary 3 only. 3 In the Settings window for Linear Polarizer, locate the Device Properties section.

10 | LINEAR WAVE RETARDER 4 Specify the T vector as

0 x 1 y 0 z

Next, add a quarter-wave retarder to enable a fraction of the energy of the ray to propagate through the second polarizer.

Linear Wave Retarder 1 1 Right-click Geometrical Optics (gop) and choose Optical Devices>Linear Wave Retarder. 2 Select Boundary 2 only. 3 In the Settings window for Linear Wave Retarder, locate the Device Properties section. 4 Specify the F vector as

1 x 1 y 0 z

5 In the δ text field, type delta.

STUDY 1

Parametric Sweep 1 On the Study toolbar, click Parametric Sweep. The study uses three different retardance values. For δ = 0 the Linear wave retarder boundary condition has no effect. For δ =π /2 a quarter-wave retarder is used. For δ =π a half-wave retarder is used. 2 In the Settings window for Parametric Sweep, locate the Study Settings section. 3 Click Add. 4 In the table, enter the following settings:

Parameter name Parameter value list Parameter unit delta 0 pi/2 pi rad

Step 1: Ray Tracing 1 In the Model Builder window, under Study 1 click Step 1: Ray Tracing. 2 In the Settings window for Ray Tracing, locate the Study Settings section.

11 | LINEAR WAVE RETARDER 3 From the Time step specification list, choose Specify maximum path length. 4 Click Range. 5 In the Range dialog box, type 0.1 in the Step text field. 6 In the Stop text field, type 4. 7 Click Replace. 8 On the Study toolbar, click Compute.

RESULTS

Ray Trajectories (gop) 1 In the Model Builder window, click Ray Trajectories (gop). 2 In the Settings window for 3D Plot Group, locate the Data section. 3 From the Parameter value (delta (rad)) list, choose 0. 4 In the Label text field, type No Wave Retarder. 5 Locate the Plot Settings section. Clear the Plot data set edges check box.

Modify the default plot to indicate the intensity and polarization of the ray.

Color Expression 1 1 In the Model Builder window, expand the Ray Trajectories 1 node, then click Color Expression 1. 2 In the Settings window for Color Expression, locate the Coloring and Style section. 3 From the Color table list, choose Spectrum.

Ray Trajectories 1 1 In the Model Builder window, under Results>No Wave Retarder click Ray Trajectories 1. 2 In the Settings window for Ray Trajectories, locate the Coloring and Style section. 3 Find the Point style subsection. From the Type list, choose Ellipse. 4 In the Maximum number of ellipses text field, type 25. 5 Select the Ellipse scale factor check box. 6 In the associated text field, type 0.3.

12 | LINEAR WAVE RETARDER Surface 1 1 In the Model Builder window, under Results right-click No Wave Retarder and choose Surface. 2 In the Settings window for Surface, locate the Data section. 3 From the Data set list, choose Study 1/Solution 1 (sol1). 4 Locate the Coloring and Style section. From the Coloring list, choose Uniform. 5 From the Color list, choose Gray. 6 On the No Wave Retarder toolbar, click Plot. 7 Click Go to Default View. Compare the resulting image to Figure 1.

No Wave Retarder 1 1 Right-click No Wave Retarder and choose Duplicate. 2 In the Settings window for 3D Plot Group, type Quarter-Wave Retarder in the Label text field. 3 Locate the Data section. From the Parameter value (delta (rad)) list, choose 1.571. 4 On the Quarter-Wave Retarder toolbar, click Plot. Compare the resulting image to Figure 2. The quarter-wave retarder causes a phase shift between the electric field components parallel to and perpendicular to the fast axis. As a result, the linearly polarized ray becomes circularly polarized. Upon reaching the second linear polarizer, the ray intensity is reduced by half.

Quarter-Wave Retarder 1 1 In the Model Builder window, under Results right-click Quarter-Wave Retarder and choose Duplicate. 2 In the Settings window for 3D Plot Group, type Half-Wave Retarder in the Label text field. 3 Locate the Data section. From the Parameter value (delta (rad)) list, choose 3.142. 4 On the Half-Wave Retarder toolbar, click Plot. Compare the resulting image to Figure 3. While propagating through the half-wave retarder, the ray remains linearly polarized, but the direction of polarization is rotated. The ray then propagates through the second linear polarizer without any loss of intensity.

13 | LINEAR WAVE RETARDER 14 | LINEAR WAVE RETARDER Created in COMSOL Multiphysics 5.3

Luneburg Lens

This model is licensed under the COMSOL Software License Agreement 5.3. All trademarks are the property of their respective owners. See www.comsol.com/trademarks. Introduction

A Luneburg lens, in the most general sense, is a spherically symmetric thick lens with a variable-index refracting structure that forms perfect geometrical images of two concentric spheres onto each other. The Luneburg lens has a gradient of isotropic refractive index n radially out from its center. In the generalized Luneburg lens, there is a pair of conjugate foci outside the lens.

In the limiting case where one of the foci tends to infinity and the other one is located on the lens surface, the analytical solution for the index profile takes a very simple form. This is what is usually meant by “Luneburg lens” in the narrow sense. Such a lens focuses a parallel beam to a perfect point in the geometrical optics limit. The location of the focus is on the rim of the lens directly opposite to the incidence direction.

Unlike a conventional, constant-index lens, a Luneburg lens works perfectly for wide ray bundles, and not only for paraxial beams. Thus, the lens is said to have a 180 degree field of view and zero f-number (Ref. 1). These properties of a lens can only be achieved using gradient-index optics.

Luneburg lenses can be made from transparent dielectric media for essentially any wavelength of interest, and they can be extremely broadband (Ref. 1). At microwave frequencies, they are used as small form-factor focusing devices for high-fidelity satellite antennas. Unlike a parabolic reflector dish, a Luneburg lens can focus satellite signal arriving from any position in the sky, which makes it more suitable for satellite antennas mounted on moving objects, such as trains and ships.

Model Definition

The refractive index of a general Luneburg lens takes the form:

1 r 2 n = --- 1 + f2 – ---- f R where r is the radial coordinate from the center of the lens and R is the radius of the lens. The dimensionless parameter f determines whether rays are focused inside or outside the lens. For f = 1 the focal point lies on the surface of the lens.

Results and Discussion

The ray trajectories are plotted in Figure 1. The colors of the rays represent their optical path length. The color expression within the lens indicates the spherically symmetric

2 | LUNEBURG LENS refractive index distribution. The color in Figure 2 indicates the ray intensity, which reaches a maximum value close to the focal point. As expected, the rays travel in straight lines at the speed of light in vacuum and bend within the lens. In practice an image entering the lens would be flipped upside down, and magnified near the edges.

Figure 1: Ray trajectories in a Luneburg lens. The color indicates optical path length.

3 | LUNEBURG LENS Figure 2: Ray trajectories in a Luneburg lens. The color indicates the ray intensity.

References

1. N. Kundtz and D.R. Smith, Extreme-Angle Broadband Metamaterial Lens, Nature Materials Letters, 2009.

2. L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields, 4th ed., Butterworth- Heinemann, Oxford, 1975.

Application Library path: Ray_Optics_Module/Graded_Media/luneburg_lens_go

Modeling Instructions

From the File menu, choose New.

NEW In the New window, click Model Wizard.

4 | LUNEBURG LENS MODEL WIZARD 1 In the Model Wizard window, click 3D. 2 In the Select Physics tree, select Optics>Ray Optics>Geometrical Optics (gop). 3 Click Add. 4 Click Study. 5 In the Select Study tree, select Preset Studies>Ray Tracing. 6 Click Done.

GLOBAL DEFINITIONS Add some parameters for the geometry dimensions.

Parameters 1 On the Home toolbar, click Parameters. 2 In the Settings window for Parameters, locate the Parameters section. 3 In the table, enter the following settings:

Name Expression Value Description L 1[m] 1 m Box length R 0.4[m] 0.4 m Outer radius

GEOMETRY 1 The Luneburg lens is simply a sphere containing a graded-index medium.

Sphere 1 (sph1) 1 On the Geometry toolbar, click Sphere. 2 In the Settings window for Sphere, locate the Size section. 3 In the Radius text field, type R. 4 Click Build All Objects. 5 Click Go to Default View.

DEFINITIONS Add some expressions for the radius from the center of the lens. This will be used to define the refractive index later.

Variables 1 1 In the Model Builder window, under Component 1 (comp1) right-click Definitions and choose Variables.

5 | LUNEBURG LENS 2 In the Settings window for Variables, locate the Variables section. 3 In the table, enter the following settings:

Name Expression Unit Description r sqrt(x^2+y^2+z^2+eps) m Radial coordinate f 1.1 Focal shift parameter n sqrt(1+f^2-(r/R)^2)/f Refractive index

MATERIALS

Property Name Value Unit Property group Refractive index n n 1 Refractive index

GEOMETRICAL OPTICS (GOP) This model is only concerned with the transmitted (refracted) rays and not the reflected ones, so set the maximum number of secondary rays to 0. Since the optical path length traveled by each ray is of interest, activate the option to compute this. Also set the intensity computation to Compute intensity and power in graded media. This option will allow computing the ray intensity in the graded medium (lens).

1 In the Model Builder window, expand the Component 1 (comp1)>Geometrical Optics (gop) node, then click Geometrical Optics (gop). 2 In the Settings window for Geometrical Optics, locate the Ray Release and Propagation section. 3 In the Maximum number of secondary rays text field, type 0. 4 Locate the Intensity Computation section. From the Intensity computation list, choose Compute intensity in graded media. 5 Locate the Additional Variables section. Select the Compute optical path length check box. A grid based release mechanism is used. Rays are spread over the y-coordinate with a fixed initial x and z coordinate. The initial wave vector is only in the x-direction.

6 | LUNEBURG LENS Release from Grid 1 1 Right-click Geometrical Optics (gop) and choose Release from Grid. 2 In the Settings window for Release from Grid, locate the Initial Coordinates section.

3 In the qx, 0 text field, type -1.

4 In the qy, 0 text field, type range(-0.38,0.02,0.38).

5 Locate the Ray Direction Vector section. Specify the L0 vector as

1 x 0 y 0 z

Use a finer mesh on the curved surfaces of the sphere.

MESH 1 1 In the Model Builder window, expand the Study 1 node, then click Component 1 (comp1)>Mesh 1. 2 In the Settings window for Mesh, locate the Mesh Settings section. 3 From the Sequence type list, choose User-controlled mesh.

STUDY 1

Step 1: Ray Tracing 1 In the Settings window for Ray Tracing, locate the Study Settings section. 2 From the Time step specification list, choose Specify maximum path length. 3 Click Range. 4 In the Range dialog box, choose Number of values from the Entry method list. 5 In the Stop text field, type 3. 6 In the Number of values text field, type 201. 7 Click Replace.

7 | LUNEBURG LENS Solution 1 (sol1) 1 On the Study toolbar, click Show Default Solver. 2 In the Model Builder window, expand the Solution 1 (sol1) node, then click Time- Dependent Solver 1. 3 In the Settings window for Time-Dependent Solver, click to expand the Time stepping section. 4 Locate the Time Stepping section. Select the Maximum step check box. 5 In the associated text field, type 1e-12. 6 Click to expand the Output section. From the Times to store list, choose Specified values. 7 Click Compute.

RESULTS

Ray Trajectories (gop) 1 In the Model Builder window, under Results click Ray Trajectories (gop). 2 In the Settings window for 3D Plot Group, type Ray Trajectories, Optical Path Length in the Label text field.

Selection 1 On the Results toolbar, click More Data Sets and choose Solution. 2 On the Results toolbar, click Selection. 3 In the Settings window for Selection, locate the Geometric Entity Selection section. 4 From the Geometric entity level list, choose Boundary. 5 Click the Wireframe Rendering button on the Graphics toolbar. 6 Select Boundaries 1, 3, 5, and 7 only.

Ray Trajectories, Optical Path Length In the Model Builder window, expand the Results>Ray Trajectories, Optical Path Length node.

Color Expression 1 1 In the Model Builder window, expand the Results>Ray Trajectories, Optical Path Length> Ray Trajectories 1 node, then click Color Expression 1. 2 In the Settings window for Color Expression, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1> Geometrical Optics>Ray properties>gop.L - Optical path length. 3 Locate the Coloring and Style section. Clear the Color legend check box.

8 | LUNEBURG LENS Surface 1 1 In the Model Builder window, under Results right-click Ray Trajectories, Optical Path Length and choose Surface. 2 In the Settings window for Surface, locate the Data section. 3 From the Data set list, choose Study 1/Solution 1 (2) (sol1). 4 Locate the Coloring and Style section. From the Coloring list, choose Uniform. 5 From the Color list, choose Gray.

Slice 1 1 Right-click Ray Trajectories, Optical Path Length and choose Slice. 2 In the Settings window for Slice, locate the Expression section. 3 In the Expression text field, type n. 4 Locate the Plane Data section. From the Plane list, choose xy-planes. 5 In the Planes text field, type 1. 6 Locate the Coloring and Style section. From the Color table list, choose WaveLight. 7 Clear the Color legend check box. 8 Click the Go to XY View button on the Graphics toolbar. 9 On the Ray Trajectories, Optical Path Length toolbar, click Plot. 10 Click the Zoom Extents button on the Graphics toolbar. Compare the resulting plot to Figure 1.

Ray Trajectories, Optical Path Length 1 1 Right-click Ray Trajectories, Optical Path Length and choose Duplicate. 2 In the Settings window for 3D Plot Group, type Ray Trajectories, Intensity Logarithm in the Label text field.

Ray Trajectories, Intensity Logarithm In the Model Builder window, expand the Results>Ray Trajectories, Intensity Logarithm node.

Color Expression 1 1 In the Model Builder window, expand the Results>Ray Trajectories, Intensity Logarithm> Ray Trajectories 1 node, then click Color Expression 1. 2 In the Settings window for Color Expression, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1> Geometrical Optics>Intensity and polarization>gop.logI - Log of intensity.

9 | LUNEBURG LENS 3 On the Ray Trajectories, Intensity Logarithm toolbar, click Plot. 4 Click the Zoom Extents button on the Graphics toolbar. Compare the resulting plot to Figure 2.

10 | LUNEBURG LENS Created in COMSOL Multiphysics 5.3

Michelson Interferometer

This model is licensed under the COMSOL Software License Agreement 5.3. All trademarks are the property of their respective owners. See www.comsol.com/trademarks. Introduction

This model couples the Heat Transfer in Solids, Solid Mechanics, and Geometrical Optics interfaces to compute the effect of thermal expansion of optical components on the interference pattern displayed by a Michelson interferometer.

Note: This application also requires the Structural Mechanics Module

Michelson interferometers are used to precisely measure distances, wavelength, or index of refraction of optical components. In its simplest form, a Michelson interferometer is composed of five elements: two mirrors, one beam splitter, an imaging device (screen), and a coherent light source. Figure 1 shows and describes a basic Michelson interferometer arrangement.

As shown in Figure 1, the interference pattern is generated when the rays reflected by mirrors M1 and M2 arrive at the screen with different optical path lengths.

Changing, even slightly, the optical path length of either beam results in a change of the interference pattern at the screen. This change of optical path length can be accomplished by moving one of the mirrors. Unexpected changes in optical path length can also occur if any of the device’s optical components undergo deformation, such as thermal expansion. This model illustrates the effects of thermal expansion on the interference pattern obtained at the screen of the interferometer.

2 | MICHELSON INTERFEROMETER S

LS M2

Figure 1: An illuminated Michelson interferometer arrangement with the resulting interference pattern displayed on the screen (S). The light comes from the left (LS), hits the beam splitter (BS) before being equally diverted toward mirror 1 (M1) and mirror 2 (M2). Once reflected at the mirrors, the two composing beams return to the beam splitter (BS) where they recombine to travel toward the screen (S) or the light source (LS).

Model Definition

In this model the rays propagate through the beam splitter and the air enclosing the optical components. The mirrors are treated as Wall boundary conditions at which specular reflection occurs. Thermal expansion in one of the two mirrors is considered.

The mirror assemblies are composed of aluminum, Pyrex, and stainless steel screws. The thermal and mechanical properties of each material are listed in Table 1:

TABLE 1: THERMAL AND MECHANICAL PROPERTIES

MATERIALS ρα ν kCp E Aluminum 238 900 2700 23 70 0.33 Pyrex 1.11 738 2230 3.05 62.6 0.225 Stainless Steel 44.5 475 7850 12.3 205 0.28

Where the thermal conductivity k is in W/(m·K), the heat capacity at constant pressure Cp is in J/(kg·K), the density ρ is in kg/m3, the coefficient of thermal expansion α is in 1e-6 K-1, the Young’s modulus E is in 1e9 Pa, and the Poisson’s ratio ν is dimensionless.

3 | MICHELSON INTERFEROMETER The beam splitter (BS) is composed of two BK7 glass prisms separated by a thin dielectric film (BS interface). The beam splitter is surrounded by an anti-reflective (AR) coating. The AR coating and the BS interface are modeled by applying the Thin Dielectric Film subnode to the Material Discontinuity node.

The Settings window for the Material Discontinuity node includes an option to automatically set up a single-layer coating with the desired reflectance. In this case, a reflectance of 0.5 is desired for the interior boundary of the beam splitter so that an incoming ray is divided into two rays of equal intensity.

The light source used in the model is a He-Ne laser. For sake of simplicity only one ray is initially released from a point on a face of the enclosure. Note that a secondary ray is emitted for each reflection of the ray at the BS interface. In this model a total of three secondary ray are released from the BS interface. Due to the anti-reflective coatings on the exterior of the beam splitter, the intensity of the reflected rays at these boundaries is negligibly small, so the release of secondary rays is suppressed.

The initial ray is polarized such that it forms a s-polarized wave at the beam splitter surface. For the interference pattern to be accurate, it must correspond to a wavefront that subtends a very small solid angle. This can be accomplished by setting the initial radius of curvature of the wavefront to −1 m and setting the initial optical path length difference between the beam splitter and the two mirrors to a large multiple of the free-space wavelength.

The interference pattern obtained at the screen depends on the radii of curvature, phase, and angle of incidence of the two rays as they arrive at the screen.

For a spherical wavefront with principal radius of curvature r1,0 with normal incidence at a surface, the change in phase corresponding to a shift xp in position on the screen is

2 2 2 ΔΨ = kx+ r – r 1 p 10, 10, where k is the wave number. Similarly, the change in phase for a wavefront with principal radius of curvature r2,0 is

2 2 2 ΔΨ = kx+ r – r 2 p 20, 20,

For small values of xp, an approximate solution for the difference in phase between the two ΔΨ wavefronts can be obtained by taking a Taylor series expansion of the expressions for 1 ΔΨ and 2 and retaining terms of up to second order in xp,

4 | MICHELSON INTERFEROMETER 2 1 1 ΔΨ – ΔΨ ≅ kx ------– ------1 2 p 2r10, 2r20,

If the two rays interfere constructively at xp= 0, they also interfere constructively where the phase difference between the two wavefronts is an integer multiple of 2π. The first such point occurs where

1 1 –1 x = λ ------– ------(1) p 0 2r10, 2r20, which is the distance from the center of the interference pattern to a point of maximum intensity on the first fringe.

Figure 2: Interference of coherent light emitted from two point sources separated by a small distance.

The model is separated in these studies:

• Study 1: Computes the effect of moving mirror M2 on the interference pattern.

5 | MICHELSON INTERFEROMETER • Study 2: Computes the mechanical deformation of mirror M2 as a result of thermal expansion. A fixed temperature is applied at the back of the mirror and the assembly is cooled off by natural convection. • Study 3: Couples the deformation previously computed to the ray tracing model. The Moving Mesh interface is used to link the mirror deformation to the air surrounding the component.

Results and Discussion

Figure 3 shows the interference pattern for the undeformed configuration resulting from δ λ the combination of two rays having an optical path length difference of d=8000 0. As expected for spherical waves, the interference fringes are circular. On the figure it is possible to approximate the distance from the center of the screen to the radial position of greatest intensity on the first circular fringe.

δ λ Figure 3: Interference pattern obtained for an optical path length difference of d=8000 0.

Figure 4 shows the temperature distribution on mirror M2 when a fixed temperature of 294.15 K is maintained on the back of the aluminum mount. With only natural convection on the exterior faces of the assembly the temperature is nearly the same all over the mirror

6 | MICHELSON INTERFEROMETER ranging from 294.0 K on the tip of the Pyrex mirror to 294.15 K at the back of the aluminum mount.

Figure 4: Temperature distribution in mirror M2’s assembly when the back of the aluminum mount is heated with a fixed temperature of 294.15 K. The temperature is mostly uniform and range from 294.0 to 294.15 K.

Note that even for a small temperature increase of one degree Celsius, the different thermal expansions of the materials composing the mirror assembly creates deformations comparable to the wavelength of the light source (~600 nm). This is illustrated in Figure 5 where the total displacement and displacement field caused by thermal expansion is displayed.

7 | MICHELSON INTERFEROMETER Figure 5: Deformation caused by the thermal expansion on the mirror assembly.

Figure 6 shows the effect of the deformation on the interference pattern previously observed on Figure 3. Comparing Figure 3 and Figure 6 shows that for the same mirror δ λ positions ( d=8000 0) the circular fringes are shifted and widen. Figure 6 shows how small changes in optical path length difference can significantly affect the image formed by the Michelson interferometer.

8 | MICHELSON INTERFEROMETER Figure 6: Effect of the deformation on the interference pattern for mirror M2 located at 8000 λ 0 closer to the beam splitter than mirror M1.

Reference

1. M. Born and E. Wolf, Principle of Optics, 7th ed., Cambridge University Press, 2011.

Application Library path: Ray_Optics_Module/Industrial_Applications/ michelson_interferometer_thermal

Modeling Instructions

From the File menu, choose New.

NEW In the New window, click Model Wizard.

9 | MICHELSON INTERFEROMETER MODEL WIZARD 1 In the Model Wizard window, click 3D. 2 In the Select Physics tree, select Optics>Ray Optics>Geometrical Optics (gop). 3 Click Add. 4 Click Study. 5 In the Select Study tree, select Preset Studies>Ray Tracing. 6 Click Done.

GLOBAL DEFINITIONS

Parameters 1 On the Home toolbar, click Parameters. 2 In the Settings window for Parameters, locate the Parameters section.

10 | MICHELSON INTERFEROMETER 3 In the table, enter the following settings:

Name Expression Value Description lam 632.8[nm] 6.328E-7 m Wavelength of He-Ne laser delta_d 8000*lam 0.005062 m Optical path length difference th 0.12[in] 0.003048 m Thickness of mirrors dia 0.5[in] 0.0127 m Diameter of mirrors d1 2.35[in] 0.05969 m Distance between mirror M1 and the center of the beam splitter d2 d1-delta_d 0.05463 m Distance between mirror M2 and the center of the beam splitter dE 12[in] 0.3048 m Distance between the beam splitter and the screen T0 293.15[K]+1[K] 294.2 K Temperature applied to the back of miror M2 n_air 1 1 Refractive index of air n_bk7 1.52611 1.526 Refractive index of BK7 glass n_int 3.9641 3.964 Refractive index of the beam-splitter interface n_coat sqrt(n_bk7) 1.235 Refractive index of the anti-reflective coating

Create the mirror M2 assembly.

GEOMETRY 1

Block 1 (blk1) 1 On the Geometry toolbar, click Block. 2 In the Settings window for Block, locate the Size and Shape section. 3 In the Width text field, type 1[in]. 4 In the Depth text field, type 2*th. 5 In the Height text field, type 1[in]. 6 Locate the Position section. From the Base list, choose Center. 7 In the y text field, type -d2-th/2.

11 | MICHELSON INTERFEROMETER Block 2 (blk2) 1 On the Geometry toolbar, click Block. 2 In the Settings window for Block, locate the Size and Shape section. 3 In the Width text field, type 0.4[in]. 4 In the Depth text field, type 2*th. 5 In the Height text field, type 0.75[in]. 6 Locate the Position section. From the Base list, choose Center. 7 In the x text field, type 0.3[in]. 8 In the y text field, type -d2-th/2. 9 In the z text field, type 0.125[in].

Block 3 (blk3) 1 On the Geometry toolbar, click Block. 2 In the Settings window for Block, locate the Size and Shape section. 3 In the Width text field, type 1[in]. 4 In the Height text field, type 1[in]. 5 In the Depth text field, type 0.32[in]. 6 Locate the Position section. From the Base list, choose Center. 7 In the y text field, type -d2-th/2-0.45[in].

Cylinder 1 (cyl1) 1 On the Geometry toolbar, click Cylinder. 2 In the Settings window for Cylinder, locate the Size and Shape section. 3 In the Radius text field, type dia/2.5. 4 In the Height text field, type 3/2*th. 5 Locate the Position section. In the y text field, type -d2-th. 6 Locate the Axis section. From the Axis type list, choose y-axis.

Difference 1 (dif1) 1 On the Geometry toolbar, click Booleans and Partitions and choose Difference. 2 Select the object blk1 only. 3 In the Settings window for Difference, locate the Difference section. 4 Find the Objects to subtract subsection. Select the Active toggle button. 5 Select the objects blk2 and cyl1 only.

12 | MICHELSON INTERFEROMETER Cylinder 2 (cyl2) 1 On the Geometry toolbar, click Cylinder. 2 In the Settings window for Cylinder, locate the Size and Shape section. 3 In the Radius text field, type dia/2. 4 In the Height text field, type th. 5 Locate the Position section. In the y text field, type -d2-th. 6 Locate the Axis section. From the Axis type list, choose y-axis.

Difference 2 (dif2) 1 On the Geometry toolbar, click Booleans and Partitions and choose Difference. 2 Select the object dif1 only. 3 In the Settings window for Difference, locate the Difference section. 4 Find the Objects to subtract subsection. Select the Active toggle button. 5 Select the object cyl2 only.

Cylinder 4 (cyl4) 1 In the Model Builder window, under Component 1 (comp1)>Geometry 1 right-click Cylinder 2 (cyl2) and choose Duplicate. 2 On the Geometry toolbar, click Cylinder. 3 In the Settings window for Cylinder, locate the Size and Shape section. 4 In the Radius text field, type 0.078125[in]. 5 In the Height text field, type 0.52[in]. 6 Locate the Position section. In the x text field, type 0.375[in]. 7 In the y text field, type -d2-3/2*th-0.52[in]. 8 In the z text field, type -0.375[in]. 9 Locate the Axis section. From the Axis type list, choose y-axis.

Cylinder 5 (cyl5) 1 Right-click Cylinder 4 (cyl4) and choose Duplicate. 2 In the Settings window for Cylinder, locate the Position section. 3 In the x text field, type -0.375[in].

Cylinder 6 (cyl6) 1 Right-click Component 1 (comp1)>Geometry 1>Cylinder 5 (cyl5) and choose Duplicate. 2 In the Settings window for Cylinder, locate the Position section.

13 | MICHELSON INTERFEROMETER 3 In the z text field, type 0.375[in].

Work Plane 1 (wp1) 1 On the Geometry toolbar, click Work Plane. 2 In the Settings window for Work Plane, locate the Plane Definition section. 3 In the z-coordinate text field, type -0.5[in]. 4 Click Show Work Plane.

Circle 1 (c1) 1 On the Work Plane toolbar, click Primitives and choose Circle. 2 In the Settings window for Circle, locate the Size and Shape section. 3 In the Radius text field, type 0.1[in]. 4 Locate the Position section. In the yw text field, type -d2-th/2-0.45[in]. 5 In the Model Builder window, click Geometry 1.

Union 1 (uni1) 1 On the Geometry toolbar, click Booleans and Partitions and choose Union. 2 Select the objects dif2, cyl6, cyl5, cyl4, blk3, and cyl3 only.

Copy 1 (copy1) 1 On the Geometry toolbar, click Transforms and choose Copy. Create the assembly for mirror M1 by copying the geometry created above for mirror M2. 2 Select the object uni1 only. 3 In the Settings window for Copy, locate the Displacement section. 4 In the x text field, type d1. 5 In the y text field, type d2.

Rotate 1 (rot1) 1 On the Geometry toolbar, click Transforms and choose Rotate. 2 Select the object copy1 only. 3 In the Settings window for Rotate, locate the Rotation Angle section. 4 In the Rotation text field, type 90. 5 Locate the PointonAxisofRotation section. In the x text field, type d1. 6 Click the Zoom Extents button on the Graphics toolbar. Create the beam splitter.

14 | MICHELSON INTERFEROMETER PART LIBRARIES 1 On the Geometry toolbar, click Parts and choose Part Libraries. 2 In the Model Builder window, click Geometry 1. 3 In the Part Libraries window, select Ray Optics Module>3D>Beam Splitters> beam splitter cube in the tree. 4 Click Add to Geometry.

GEOMETRY 1

Beam Splitter Cube 1 (pi1) 1 In the Model Builder window, under Component 1 (comp1)>Geometry 1 click Beam Splitter Cube 1 (pi1). 2 In the Settings window for Part Instance, locate the Input Parameters section. 3 In the table, enter the following settings:

Name Expression Value Description d dia 0.0127 [m] Side length

Create the screen.

Block 4 (blk4) 1 On the Geometry toolbar, click Block. 2 In the Settings window for Block, locate the Size and Shape section. 3 In the Width text field, type 4*dia. 4 In the Depth text field, type th. 5 In the Height text field, type 4*dia. 6 Locate the Position section. From the Base list, choose Center. 7 In the y text field, type dE+th/2.

DEFINITIONS

Explicit 1 1 On the Definitions toolbar, click Explicit. 2 Click the Select Box button on the Graphics toolbar. 3 Select Domains 2–10 and 13–15 only. 4 In the Settings window for Explicit, type Deforming Mirror in the Label text field.

15 | MICHELSON INTERFEROMETER MATERIALS

Material 1 (mat1) 1 In the Model Builder window, under Component 1 (comp1) right-click Materials and choose Blank Material. 2 In the Settings window for Material, type BK7 in the Label text field. 3 Select Domains 11 and 12 only. 4 Locate the Material Contents section. In the table, enter the following settings:

Property Name Value Unit Property group Refractive index n n_bk7 1 Refractive index

Now define the physics for the ray tracing part of the model.

GEOMETRICAL OPTICS (GOP) 1 In the Model Builder window, under Component 1 (comp1) click Geometrical Optics (gop). 2 In the Settings window for Geometrical Optics, locate the Domain Selection section. 3 From the Selection list, choose Manual. 4 Select Domains 11 and 12 only. 5 Locate the Intensity Computation section. From the Intensity computation list, choose Compute intensity and power. 6 Locate the Additional Variables section. Select the Compute optical path length check box. 7 Locate the Intensity Computation section. Select the Compute phase check box. 8 Locate the Ray Release and Propagation section. In the Maximum number of secondary rays text field, type 3. Apply an anti-reflective coating to the outside of the beam splitter.

Material Discontinuity 1 1 In the Model Builder window, expand the Geometrical Optics (gop) node, then click Material Discontinuity 1. 2 In the Settings window for Material Discontinuity, locate the Coatings section. 3 From the Thin dielectric films on boundary list, choose Anti-reflective coating. λ 4 In the 0 text field, type lam. Use a second Material Discontinuity node for the interior boundary of the beam splitter.

16 | MICHELSON INTERFEROMETER Material Discontinuity 2 1 On the Physics toolbar, click Boundaries and choose Material Discontinuity. 2 Select Boundary 56 only. 3 In the Settings window for Material Discontinuity, locate the Coatings section. 4 From the Thin dielectric films on boundary list, choose Anti-reflective coating. 5 From the Thin dielectric films on boundary list, choose Single-layer coating, specified reflectance. 6 In the n text field, type n_int. 7 In the R text field, type 0.5. λ 8 In the 0 text field, type lam. θ 9 In the i text field, type 45[deg]. Ray Properties 1 1 In the Model Builder window, under Component 1 (comp1)>Geometrical Optics (gop) click Ray Properties 1. 2 In the Settings window for Ray Properties, locate the Ray Properties section. λ 3 In the 0 text field, type lam. Wall 1 1 On the Physics toolbar, click Boundaries and choose Wall. 2 Select Boundaries 63 and 108 only. 3 In the Settings window for Wall, locate the Wall Condition section. 4 From the Wall condition list, choose Specular reflection.

Wall 2 1 On the Physics toolbar, click Boundaries and choose Wall. 2 Select Boundary 2 only. Specify how the ray is going to be released. Release a single ray from the face of the enclosure facing mirror M1. Generate a fully polarized, -1 m radius of curvature spherical wave.

Release from Grid 1 1 On the Physics toolbar, click Global and choose Release from Grid. 2 In the Settings window for Release from Grid, locate the Initial Coordinates section.

3 In the qx, 0 text field, type -100[mm].

17 | MICHELSON INTERFEROMETER 4 Locate the Ray Direction Vector section. Specify the L0 vector as

1 x 0 y 0 z

5 Locate the Initial Radii of Curvature section. From the Wavefront shape list, choose Spherical wave.

6 In the r0 text field, type -1[m]. 7 Locate the Initial Polarization section. From the Initial polarization type list, choose Fully polarized. 8 From the Initial polarization list, choose User defined. 9 Specify the u vector as

0 x 0 y 1 z

Size Create a finer mesh size for the mirror assemblies.

1 In the Model Builder window, under Component 1 (comp1)>Mesh 1 click Size. 2 In the Settings window for Size, click to expand the Element size parameters section. 3 Locate the Element Size Parameters section. In the Curvature factor text field, type 0.01. 4 In the Minimum element size text field, type 0.0005.

Size 1 1 In the Model Builder window, under Component 1 (comp1)>Mesh 1 right-click Free Triangular 1 and choose Size. 2 Select Boundary 63 only. 3 In the Settings window for Size, locate the Element Size section. 4 Click the Custom button.

18 | MICHELSON INTERFEROMETER 5 Locate the Element Size Parameters section. Select the Maximum element size check box. 6 In the associated text field, type 0.0003. 7 Select the Minimum element size check box. 8 In the associated text field, type 0.0001. 9 Click Build All.

STUDY 1

Step 1: Ray Tracing 1 In the Settings window for Ray Tracing, locate the Study Settings section. 2 In the Times text field, type 0 2.2. 3 On the Home toolbar, click Compute.

RESULTS

Ray Trajectories (gop) Create a data set to display the interference pattern on the screen.

Cut Plane 1 1 On the Results toolbar, click Cut Plane. 2 In the Settings window for Cut Plane, locate the Data section. 3 From the Data set list, choose Ray 1. 4 Locate the Plane Data section. From the Plane list, choose xz-planes. 5 In the y-coordinate text field, type dE-eps.

2D Plot Group 2 1 On the Results toolbar, click 2D Plot Group. 2 In the Settings window for 2D Plot Group, type Interference Plot in the Label text field. 3 Locate the Plot Settings section. Clear the Plot data set edges check box. 4 Locate the Data section. From the Data set list, choose Cut Plane 1.

Interference Pattern 1 1 On the Interference Plot toolbar, click More Plots and choose Interference Pattern. 2 In the Settings window for Interference Pattern, locate the Coordinate Range section. 3 From the Origin location specification list, choose At ray of greatest intensity.

19 | MICHELSON INTERFEROMETER 4 On the Interference Plot toolbar, click Plot. Compare the resulting plot to Figure 3. Now add the physics to compute the thermal expansion of mirror M2’s assembly.

ADD PHYSICS 1 On the Home toolbar, click Add Physics to open the Add Physics window. 2 Go to the Add Physics window. 3 In the tree, select HeatTransfer>HeatTransferinSolids(ht). 4 Click Add to Component in the window toolbar.

ADD PHYSICS 1 Go to the Add Physics window. 2 In the tree, select Structural Mechanics>Solid Mechanics (solid). 3 Click Add to Component in the window toolbar. 4 On the Home toolbar, click Add Physics to close the Add Physics window. Add the multiphysics features.

MULTIPHYSICS

Thermal Expansion 1 (te1) 1 On the Physics toolbar, click Multiphysics and choose Domain>Thermal Expansion. 2 In the Settings window for Thermal Expansion, locate the Domain Selection section. 3 From the Selection list, choose All domains.

Temperature Coupling 1 (tc1) 1 On the Physics toolbar, click Multiphysics and choose Global>Temperature Coupling. Add the material properties for the thermal expansion model.

MATERIALS

Material 2 (mat2) 1 In the Model Builder window, under Component 1 (comp1) right-click Materials and choose Blank Material. 2 In the Settings window for Material, type Pyrex in the Label text field. 3 Select Domain 10 only.

20 | MICHELSON INTERFEROMETER 4 Locate the Material Contents section. In the table, enter the following settings:

Property Name Value Unit Property group Thermal k 1.11[W/(m*K)] W/(m·K) Basic conductivity Density rho 2230[kg/m^3] kg/m³ Basic Heat capacity at Cp 738[J/(kg*K)] J/(kg·K) Basic constant pressure Young’s modulus E 6.26e10[Pa] Pa Basic Poisson’s ratio nu 0.225 1Basic Coefficient of alpha 3.05e-6[1/K] 1/K Basic thermal expansion

Material 3 (mat3) 1 Right-click Materials and choose Blank Material. 2 In the Settings window for Material, type Aluminum in the Label text field. 3 Select Domains 2 and 3 only. 4 Locate the Material Contents section. In the table, enter the following settings:

Property Name Value Unit Property group Thermal k 238[W/(m*K)] W/(m·K) Basic conductivity Density rho 2700[kg/m^3] kg/m³ Basic Heat capacity at Cp 900[J/(kg*K)] J/(kg·K) Basic constant pressure Young’s modulus E 70e9[Pa] Pa Basic Poisson’s ratio nu 0.33 1Basic Coefficient of alpha 23e-6[1/K] 1/K Basic thermal expansion

Material 4 (mat4) 1 Right-click Materials and choose Blank Material. 2 In the Settings window for Material, type Stainless Steel in the Label text field. 3 Select Domains 4–9 and 13–15 only.

21 | MICHELSON INTERFEROMETER 4 Locate the Material Contents section. In the table, enter the following settings:

Property Name Value Unit Property group Thermal k 44.5[W/(m*K)] W/(m·K) Basic conductivity Density rho 7850[kg/m^3] kg/m³ Basic Heat capacity at Cp 475[J/(kg*K)] J/(kg·K) Basic constant pressure Young’s modulus E 205e9[Pa] Pa Basic Poisson’s ratio nu 0.28 1Basic Coefficient of alpha 12.3e-6[1/K] 1/K Basic thermal expansion

Set up the heat transfer part of the model.

HEAT TRANSFER IN SOLIDS (HT) 1 In the Model Builder window, under Component 1 (comp1) click Heat Transfer in Solids (ht). 2 In the Settings window for Heat Transfer in Solids, locate the Domain Selection section. 3 From the Selection list, choose Deforming Mirror.

Heat Flux 1 1 On the Physics toolbar, click Boundaries and choose Heat Flux. 2 In the Settings window for Heat Flux, locate the Boundary Selection section. 3 From the Selection list, choose All boundaries. 4 Locate the Heat Flux section. Click the Convective heat flux button. 5 In the h text field, type 10.

Temperature 1 1 On the Physics toolbar, click Boundaries and choose Temperature. 2 Select Boundary 7 only. 3 In the Settings window for Temperature, locate the Temperature section.

4 In the T0 text field, type T0. Set up the solid mechanics part of the model.

SOLID MECHANICS (SOLID) 1 In the Model Builder window, under Component 1 (comp1) click Solid Mechanics (solid).

22 | MICHELSON INTERFEROMETER 2 In the Settings window for Solid Mechanics, locate the Domain Selection section. 3 From the Selection list, choose Deforming Mirror.

Fixed Constraint 1 1 On the Physics toolbar, click Boundaries and choose Fixed Constraint. 2 Select Boundary 64 only.

MESH 1

Free Tetrahedral 2 1 In the Model Builder window, under Component 1 (comp1) right-click Mesh 1 and choose Free Tetrahedral. 2 In the Settings window for Free Tetrahedral, locate the Domain Selection section. 3 From the Geometric entity level list, choose Domain. 4 Select Domains 2–10, 13–15, and 17 only. 5 Click Build All.

ADD STUDY 1 On the Home toolbar, click Add Study to open the Add Study window. 2 Go to the Add Study window. 3 Find the Studies subsection. In the Select Study tree, select Preset Studies. 4 Find the Physicsinterfacesinstudy subsection. In the table, clear the Solve check box for the Geometrical Optics (gop) interface. 5 Find the Studies subsection. In the Select Study tree, select Preset Studies>Stationary. 6 Click Add Study in the window toolbar. 7 On the Home toolbar, click Add Study to close the Add Study window.

STUDY 2

Step 1: Stationary 1 On the Home toolbar, click Compute. Add a new data set to display the results obtained from the second study on the relevant domains.

RESULTS

Study 2/Solution 2 (sol2) In the Model Builder window, expand the Data Sets node.

23 | MICHELSON INTERFEROMETER Study 2/Solution 2 (3) (sol2) 1 Right-click Study 2/Solution 2 (sol2) and choose Duplicate. 2 In the Model Builder window, under Results>Data Sets right-click Study 2/ Solution 2 (3) (sol2) and choose Rename. 3 In the Rename Solution dialog box, type Study 2/Solution selection in the New label text field. 4 Click OK.

Selection 1 On the Results toolbar, click Selection. 2 In the Settings window for Selection, locate the Geometric Entity Selection section. 3 From the Geometric entity level list, choose Domain. 4 From the Selection list, choose Deforming Mirror.

Temperature (ht) 1 In the Model Builder window, under Results click Temperature (ht). 2 In the Settings window for 3D Plot Group, locate the Data section. 3 From the Data set list, choose Study 2/Solution selection (sol2). 4 Click the Go to ZX View button on the Graphics toolbar.

Max/Min Volume 1 1 On the Temperature (ht) toolbar, click More Plots and choose Max/Min Volume. 2 In the Settings window for Max/Min Volume, locate the Expression section. 3 In the Expression text field, type T. 4 On the Temperature (ht) toolbar, click Plot. 5 Click the Zoom Extents button on the Graphics toolbar. Compare the resulting plot to Figure 4.

Stress (solid) 1 In the Model Builder window, under Results click Stress (solid). 2 In the Settings window for 3D Plot Group, locate the Data section. 3 From the Data set list, choose Study 2/Solution selection (sol2).

Surface 1 1 Right-click Results>Stress (solid) and choose Arrow Volume. 2 In the Settings window for Surface, locate the Expression section.

24 | MICHELSON INTERFEROMETER 3 In the Expression text field, type solid.disp.

Arrow Volume 1 1 In the Model Builder window, under Results>Stress (solid) click Arrow Volume 1. 2 In the Settings window for Arrow Volume, locate the Expression section. 3 In the Xcomponent text field, type u. 4 In the Ycomponent text field, type v. 5 In the Zcomponent text field, type w. 6 Locate the Coloring and Style section. From the Color list, choose Black. 7 Click the Go to XY View button on the Graphics toolbar. 8 Click the Zoom Extents button on the Graphics toolbar. 9 On the Stress (solid) toolbar, click Plot. Compare the resulting plot to Figure 5. Now add a Moving Mesh interface to couple the deformation of mirror M2 to the Geometrical Optics interface.

ADD PHYSICS 1 On the Home toolbar, click Add Physics to open the Add Physics window. 2 Go to the Add Physics window. 3 In the tree, select Mathematics>Deformed Mesh>Moving Mesh (ale). 4 Click Add to Component in the window toolbar. 5 On the Home toolbar, click Add Physics to close the Add Physics window.

MOVING MESH (ALE) 1 In the Model Builder window, under Component 1 (comp1) click Moving Mesh (ale). 2 In the Settings window for Moving Mesh, locate the Domain Selection section. 3 In the list, select 3. 4 From the Selection list, choose Deforming Mirror.

Free Deformation 1 1 On the Physics toolbar, click Domains and choose Free Deformation. 2 In the Settings window for Free Deformation, locate the Domain Selection section. 3 From the Selection list, choose All domains.

25 | MICHELSON INTERFEROMETER Prescribed Mesh Displacement 1 1 In the Model Builder window, under Component 1 (comp1)>Moving Mesh (ale) click Prescribed Mesh Displacement 1. 2 In the Settings window for Prescribed Mesh Displacement, locate the Prescribed Mesh Displacement section.

3 In the dx text field, type u.

4 In the dy text field, type v.

5 In the dz text field, type w. Add a new study to compute the effect of the deformed geometry on the interference pattern.

Physics Geometrical Optics (gop) Heat Transfer in Solids (ht) Solid Mechanics (solid)

5 Find the Studies subsection. In the Select Study tree, select Preset Studies>Stationary. 6 Click Add Study in the window toolbar. 7 On the Home toolbar, click Add Study to close the Add Study window.

STUDY 1

Step 1: Ray Tracing 1 In the Model Builder window, expand the Study 1 node, then click Step 1: Ray Tracing. 2 In the Settings window for Ray Tracing, locate the Physics and Variables Selection section.

26 | MICHELSON INTERFEROMETER 3 In the table, clear the Solve for check box for the following interfaces:

Physics interface Heat Transfer in Solids (ht) Solid Mechanics (solid) Moving Mesh (ale)

STUDY 2

Step 1: Stationary 1 In the Model Builder window, expand the Study 2 node, then click Step 1: Stationary. 2 In the Settings window for Stationary, locate the Physics and Variables Selection section. 3 In the table, clear the Solve for check box for the Moving Mesh (ale) interface.

STUDY 3

Step 2: Ray Tracing 1 On the Study toolbar, click Study Steps and choose Time Dependent>Ray Tracing. 2 In the Settings window for Ray Tracing, locate the Study Settings section. 3 In the Times text field, type 0 2.2. 4 Locate the Physics and Variables Selection section. In the table, clear the Solve for check box for the following interfaces:

Physics interface Heat Transfer in Solids (ht) Solid Mechanics (solid) Moving Mesh (ale)

Update the physics solved in the previous studies for future use (for instance re- computing the solution).

Step 1: Stationary 1 In the Model Builder window, under Study 3 click Step 1: Stationary. 2 In the Settings window for Stationary, click to expand the Values of dependent variables section. 3 Locate the Values of Dependent Variables section. Find the Values of variables not solved for subsection. From the Settings list, choose User controlled.

27 | MICHELSON INTERFEROMETER 4 From the Method list, choose Solution. 5 From the Study list, choose Study 2, Stationary. Compute study 3. 6 On the Study toolbar, click Compute.

RESULTS

Ray Trajectories (gop) 1 1 In the Model Builder window, under Results click Ray Trajectories (gop) 1. 2 In the Settings window for 3D Plot Group, type Ray Trajectories, Deformed Geometry in the Label text field.

Cut Plane 2 1 On the Results toolbar, click Cut Plane. 2 In the Settings window for Cut Plane, locate the Data section. 3 From the Data set list, choose Ray 2. 4 Locate the Plane Data section. From the Plane list, choose xz-planes. 5 In the y-coordinate text field, type dE-eps.

2D Plot Group 7 1 On the Results toolbar, click 2D Plot Group. Create another interference pattern for the deformed geometry. Compare this pattern with the one obtained in Study 1. 2 In the Settings window for 2D Plot Group, type Interference Plot, Deformed Geometry in the Label text field. 3 Locate the Data section. From the Data set list, choose Cut Plane 2. 4 Locate the Plot Settings section. Clear the Plot data set edges check box.

Interference Pattern 1 1 On the Interference Plot, Deformed Geometry toolbar, click More Plots and choose Interference Pattern. 2 In the Settings window for Interference Pattern, locate the Coordinate Range section. 3 From the Origin location specification list, choose At ray of greatest intensity. 4 Click the Zoom Extents button on the Graphics toolbar. 5 On the Interference Plot, Deformed Geometry toolbar, click Plot. Compare the resulting plot to Figure 6.

28 | MICHELSON INTERFEROMETER 29 | MICHELSON INTERFEROMETER 30 | MICHELSON INTERFEROMETER Created in COMSOL Multiphysics 5.3

Newtonian Telescope

This model is licensed under the COMSOL Software License Agreement 5.3. All trademarks are the property of their respective owners. See www.comsol.com/trademarks. Introduction

This tutorial model shows how to trace rays of unpolarized light through a Newtonian telescope system. The incoming light is reflected off a parabolic mirror onto a flat secondary mirror which reflects the light into the focal plane. This type of telescope was first invented by Newton in 1668 and is still made today due to its low cost of assembly (Ref. 1).

Model Definition

The model simulates the propagation of rays coming from sources located at infinity (celestial objects) into a Newtonian telescope.

The telescope is composed of two mirrors and an ellipsoidal lens. The primary mirror (spherical) has a radius of curvature of 0.8 m. The secondary mirror (plane) forms a 45- degree angle with the rays reflected from the primary mirror. The secondary mirror directs the light towards the ellipsoidal lens before forming images of the objects upon the instrument output (focal surface), see Figure 1. Note that rays are only released from the telescope’s aperture such that they aren’t obstructed by the secondary mirror.

Results and Discussion

Figure 1 shows the ray trajectories in the telescope as well as the location of the sources’ images at the output of the instrument (focal surface).

A spot diagram is often used to characterize the performance of an optical system design. Given a distribution of rays launched from the object space (telescope’s aperture), the spot diagram shows where the rays hit the image plane. For a good (diffraction-limited) optical design, the spots should typically appear within a circle corresponding to the central diffraction spot from the system's aperture.

The spot diagram (Poincaré map) of rays in the image plane is shown in Figure 2. The color scale is used to show the ratio of ray intensity to initial intensity. Following magnification due to the curved mirror and the lens, the intensity of rays on the focal surface ranges from 65.3 to 88.8 times the intensity of the released rays.

2 | NEWTONIAN TELESCOPE Figure 1: Plot of the ray trajectories and intensity traveling through the telescope.

Figure 2: Spot diagram (Poincaré map) of the rays arriving at the focal surface. The color is the intensity divided by the initial intensity.

3 | NEWTONIAN TELESCOPE Reference

1. http://en.wikipedia.org/wiki/Newtonian_telescope.

Application Library path: Ray_Optics_Module/Tutorials/newtonian_telescope

Modeling Instructions

From the File menu, choose New.

NEW In the New window, click Model Wizard.

ROOT Insert the prepared geometry sequence from file. You can read the instructions for creating the geometry in the appendix.

GEOMETRY 1 1 On the Geometry toolbar, click Insert Sequence. 2 Browse to the model’s Application Libraries folder and double-click the file newtonian_telescope_geom_sequence.mph.

MATERIALS

Material 1 (mat1) 1 In the Model Builder window, under Component 1 (comp1) right-click Materials and choose Blank Material. 2 Select Domain 2 only.

4 | NEWTONIAN TELESCOPE 3 In the Settings window for Material, locate the Material Contents section. 4 In the table, enter the following settings:

Property Name Value Unit Property group Refractive index n 1.29 1 Refractive index

GEOMETRICAL OPTICS (GOP) Select only the domains containing materials other than air. By default, the Material Discontinuity boundary condition will be applied on boundaries adjacent to the selected domains.

1 In the Model Builder window, under Component 1 (comp1) click Geometrical Optics (gop). 2 In the Settings window for Geometrical Optics, locate the Domain Selection section. 3 In the list, select 2. 4 Click Clear Selection. 5 Select Domain 2 only. 6 Locate the Intensity Computation section. From the Intensity computation list, choose Compute intensity. 7 Locate the Additional Variables section. Select the Compute optical path length check box. 8 Locate the Ray Release and Propagation section. In the Maximum number of secondary rays text field, type 0.

Release from Grid 1 1 In the Model Builder window, right-click Geometrical Optics (gop) and choose Release from Grid. 2 In the Settings window for Release from Grid, locate the Initial Coordinates section. 3 Click y Range. 4 In the Range dialog box, choose Number of values from the Entry method list. 5 In the Start text field, type -0.025. 6 In the Stop text field, type 0.025. 7 In the Number of values text field, type 6. 8 Click Replace. 9 In the Settings window for Release from Grid, locate the Initial Coordinates section. 10 Click zRange.

5 | NEWTONIAN TELESCOPE 11 In the Range dialog box, choose Number of values from the Entry method list. 12 In the Start text field, type 0.04. 13 In the Stop text field, type 0.06. 14 In the Number of values text field, type 5. 15 Click Replace. 16 In the Settings window for Release from Grid, locate the Ray Direction Vector section.

17 Specify the L0 vector as

1 x 0 y 0 z

Release from Grid 2 1 Right-click Component 1 (comp1)>Geometrical Optics (gop)>Release from Grid 1 and choose Duplicate. 2 In the Settings window for Release from Grid, locate the Initial Coordinates section.

3 In the qz, 0 text field, type -range(0.04,0.02/4,0.06). Wall 1 1 In the Model Builder window, right-click Geometrical Optics (gop) and choose Wall. 2 Select Boundaries 13 and 27–30 only. 3 In the Settings window for Wall, locate the Wall Condition section. 4 From the Wall condition list, choose Specular reflection.

Wall 2 1 Right-click Geometrical Optics (gop) and choose Wall. 2 Select Boundary 8 only.

6 | NEWTONIAN TELESCOPE STUDY 1

Step 1: Ray Tracing 1 In the Settings window for Ray Tracing, locate the Study Settings section. 2 From the Time step specification list, choose Specify maximum path length. 3 Click Range. 4 In the Range dialog box, type 0.002 in the Step text field. 5 In the Stop text field, type 1.5. 6 Click Replace. 7 In the Settings window for Ray Tracing, locate the Study Settings section. 8 From the Stop condition list, choose No active rays remaining. 9 On the Home toolbar, click Compute.

RESULTS

Ray Trajectories (gop) In the Model Builder window, expand the Ray Trajectories (gop) node.

Color Expression 1 1 In the Model Builder window, expand the Results>Ray Trajectories (gop)> Ray Trajectories 1 node, then click Color Expression 1. 2 In the Settings window for Color Expression, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1> Geometrical Optics>Intensity and polarization>gop.logI - Log of intensity. 3 On the Ray Trajectories (gop) toolbar, click Plot. 4 Click the Zoom Extents button on the Graphics toolbar. 5 Click the Zoom In button on the Graphics toolbar. Compare the resulting plot to Figure 1.

Cut Plane 1 1 On the Results toolbar, click Cut Plane. 2 In the Settings window for Cut Plane, locate the Data section. 3 From the Data set list, choose Ray 1. 4 Locate the Plane Data section. From the Plane list, choose xy-planes. 5 In the z-coordinate text field, type -0.12.

7 | NEWTONIAN TELESCOPE 2D Plot Group 2 1 On the Results toolbar, click 2D Plot Group. 2 In the Settings window for 2D Plot Group, type Intensity Multiplication Factor in the Label text field.

Poincaré Map 1 1 On the Intensity Multiplication Factor toolbar, click More Plots and choose Poincaré Map. 2 In the Settings window for Poincaré Map, locate the Data section. 3 From the Cut plane list, choose Cut Plane 1. 4 On the Intensity Multiplication Factor toolbar, click Plot. 5 Locate the Coloring and Style section. Select the Radius scale factor check box. 6 In the associated text field, type 2E-4.

Color Expression 1 1 Right-click Poincaré Map 1 and choose Color Expression. 2 In the Settings window for Color Expression, locate the Expression section. 3 In the Expression text field, type gop.I/at(0,gop.I). 4 On the Intensity Multiplication Factor toolbar, click Plot. 5 Click the Zoom Extents button on the Graphics toolbar. Compare the resulting plot to Figure 2.

Appendix — Geometry Instructions

On the Home toolbar, click Add Component and choose 3D.

GEOMETRY 1

Cylinder 1 (cyl1) 1 On the Geometry toolbar, click Cylinder. 2 In the Settings window for Cylinder, locate the Size and Shape section. 3 In the Radius text field, type 0.075. 4 Locate the Axis section. From the Axis type list, choose x-axis.

Sphere 1 (sph1) 1 On the Geometry toolbar, click Sphere. 2 In the Settings window for Sphere, locate the Size section. 3 In the Radius text field, type 0.8[m].

8 | NEWTONIAN TELESCOPE 4 Locate the Position section. In the x text field, type -0.05. 5 Click Build All Objects.

Union 1 (uni1) 1 On the Geometry toolbar, click Booleans and Partitions and choose Union. 2 Click in the Graphics window and then press Ctrl+A to select both objects. 3 In the Settings window for Union, click Build All Objects.

Delete Entities 1 (del1) 1 In the Model Builder window, right-click Geometry 1 and choose Delete Entities. 2 In the Settings window for Delete Entities, locate the Entities or Objects to Delete section. 3 From the Geometric entity level list, choose Domain. 4 On the object uni1, select Domains 1 and 3 only. 5 Click Build All Objects.

Block 1 (blk1) 1 On the Geometry toolbar, click Block. 2 In the Settings window for Block, locate the Size and Shape section. 3 In the Width text field, type 0.01. 4 In the Depth text field, type 0.075. 5 In the Height text field, type 0.075. 6 Locate the Position section. From the Base list, choose Center. 7 In the x text field, type 0.39.

Rotate 1 (rot1) 1 On the Geometry toolbar, click Transforms and choose Rotate. 2 Select the object blk1 only. 3 In the Settings window for Rotate, locate the Rotation Angle section. 4 In the Rotation text field, type 45. 5 Locate the PointonAxisofRotation section. In the x text field, type 0.39. 6 Locate the Axis of Rotation section. From the Axis type list, choose y-axis. 7 Click Build All Objects.

Ellipsoid 1 (elp1) 1 On the Geometry toolbar, click More Primitives and choose Ellipsoid. 2 In the Settings window for Ellipsoid, locate the Size and Shape section.

9 | NEWTONIAN TELESCOPE 3 In the a-semiaxis text field, type 0.015. 4 In the b-semiaxis text field, type 0.015. 5 In the c-semiaxis text field, type 0.0075. 6 Locate the Position section. In the x text field, type 0.3975. 7 In the z text field, type -0.02-0.075. 8 Click Build All Objects. 9 Click the Wireframe Rendering button on the Graphics toolbar.

Cylinder 2 (cyl2) 1 On the Geometry toolbar, click Cylinder. 2 In the Settings window for Cylinder, locate the Size and Shape section. 3 In the Radius text field, type 0.075. 4 In the Height text field, type 0.12. 5 Locate the Position section. In the x text field, type 0.3975. 6 In the z text field, type -0.12. 7 Click Build All Objects.

Union 2 (uni2) 1 On the Geometry toolbar, click Booleans and Partitions and choose Union. 2 Select the objects del1 and cyl2 only. 3 In the Settings window for Union, locate the Union section. 4 Clear the Keep interior boundaries check box. 5 Click Build All Objects.

Difference 1 (dif1) 1 On the Geometry toolbar, click Booleans and Partitions and choose Difference. 2 Select the object uni2 only. 3 In the Settings window for Difference, locate the Difference section. 4 Find the Objects to subtract subsection. Select the Active toggle button. 5 Select the object rot1 only. 6 Click Build All Objects.

10 | NEWTONIAN TELESCOPE Created in COMSOL Multiphysics 5.3

Solar Dish Receiver

This model is licensed under the COMSOL Software License Agreement 5.3. All trademarks are the property of their respective owners. See www.comsol.com/trademarks. Introduction

A paraboloidal dish concentrator can focus incident solar radiation onto a target or cavity receiver, resulting in very high local heat fluxes. This can be used to generate steam, which can be used to power a generator, or hydrogen, which can be used directly as a fuel source. In some applications, such as hydrogen production via the solar thermal gasification of biomass in supercritical condition, the uniformity of the flux on the surface of the cavity receiver has a significant effect on the efficiency of hydrogen production (Ref. 1).

The basic concept behind the paraboloidal dish concentrator is shown in Figure 1. Solar radiation enters from the right and is reflected by the concentrator. The rays converge toward an extremely small area in the focal plane, where a cavity receiver can be positioned.

Figure 1: A simple solar concentrator system consisting of a parabolic dish and a small receiver. The color of the incident and focused rays corresponds to the ray intensity.

Of particular interest in evaluating the performance of solar collector-receiver systems is the concentration ratio, defined as the ratio of the incident flux to the ambient solar flux. A high concentration ratio usually means that the concentrator is capable of focusing solar radiation efficiently. When computing the concentration ratio, the incident flux can either be measured in the focal plane or on the surface of the cavity receiver.

A variety of computational methods to predict the concentration ratio are available. Shuai et al. (Ref. 1) compared the results of a Monte Carlo ray tracing code to compute the

2 | SOLAR DISH RECEIVER concentration ratios of several different cavity geometries. Jeter (Ref. 2) proposed a semi- analytical method in which the concentration ratio is computed via integration of the intensity distribution over the concentrator surface. A standard practice in many solar energy research institutions is to measure the concentrated solar flux using charge coupled device (CCD) imaging cameras (Ref. 3).

The ideal focusing system consists of a perfectly smooth paraboloidal dish that focuses collimated incident solar radiation onto a point in the focal plane that is infinitesimally small within the limit of the approximations of geometrical optics. However, several imperfections in this system cause the measured concentration ratio to deviate from the ideal case.

To accurately predict the concentration ratio, the finite size of the sun and the intensity distribution over the solar surface must be considered. The intensity profile on the solar disk is referred to as sunshape. Solar intensity is greatest at the center of the solar disk and decreases closer to the periphery of the disk, a phenomenon called solar limb darkening.

Integration Method for Ideally Smooth Solar Collectors

A method for computing the concentration ratio in the focal plane in the absence of surface roughness is described by Jeter (Ref. 2). Consider differential area elements on the surface of the concentrator at rc and on the focal plane of the receiver at r, as shown in Figure 2.

Figure 2: Diagram of the paraboloidal solar concentrator.

3 | SOLAR DISH RECEIVER where the surface normals at the concentrator and focal plane are rcnc and rn , respectively, and O is the focus. The following angles are defined:

δ ∠ θ ∠ θ ∠ 1 = Orcr c = rrcnc = rcrn

The concentration ratio at r is

1 fcos()θ cos() θ C()r = ------c dA (1) I  rr– 2 c 0Ω c

2 δψ≤ I ⁄ ()πψsin() m f()δ = 0 m (2)  δψ>  0 m where:

• f (SI unit: W/(m2 steradian)) is the radiant intensity, • Ω denotes surface integration over the collector surface, 2 • I0 (SI unit: W/m ) is the incident solar flux, ψ • m is the maximum solar disk angle, and 2 • dAc (SI unit: m ) is a differential area element on the surface of the collector. In Equation 2 it is assumed that the incident solar flux does not vary as a function of position on the solar disk; that is, no solar limb darkening is considered. However, it would be possible to extend Equation 2 to account for solar limb darkening by including a term dependent on the angle δ on the right-hand side.

In this model, Equation 1 is implemented using the dest() operator. The dest() operator evaluates a term in an integration component coupling on the destination side. For example, including the term u/((dest(x)-x)^2+(dest(y)-y)^2) in an integration component coupling gives the following function of x and y:

ux()', y' fxy(), = ------dyx'd ' ()xx– ' 2 + ()yy– ' 2

Model Definition

A parabolic solar dish concentrator with a focal length, f, of 3 m is constructed using a built-in Part from the Part Library for the Ray Optics Module. The geometry also includes a small cylinder, one surface of which lies in the focal plane. The incident flux on this surface will be computed, then used to compute the concentration ratio. By adjusting the

4 | SOLAR DISH RECEIVER shape of the cylinder it would be possible to compute the concentration ratios for various cavity geometries, as in Ref. 1, but for the present analysis, only the concentration ratio in the focal plane is computed.

If the dish was a perfect reflector (all of the incoming radiation reflected specularly), the dish was perfectly smooth, and the rays from the sun behaved as planar wavefronts from an infinitely distant point source, all of the incoming rays would be focused on a single point on the collector, at the focus of the paraboloid (within the limits of the geometrical optics approximation). However, in this model, several deviations from this idealized case are considered.

A dedicated boundary condition called Illuminated Surface is used to release rays directly from the surface of the dish, initializing their directions as if they were reflected from a distant plane wave source. The direction at which the rays are released from the surface of the dish depends on the incoming ray direction vector ni and the outward surface normal ns, according to the formula ()⋅ nr = ni – 2 ni ns ns (3)

The intensity of each individual ray can be computed along its trajectory; the evolution of ray intensity depends heavily on the curvature of the dish. More details on intensity computation can be found in the Ray Optics Module User’s Guide.

Each ray released is also assigned a fixed power, which is assigned an appropriate value based on the Source power setting for the Illuminated Surface feature. When the rays reach the surface of the solar collector, they are stopped by the Wall feature. The Deposited Ray Power subfeature computes the incident heat flux in the focal plane. By taking the ratio of the deposited flux to the incoming solar flux, the concentration ratio on the surface can be computed.

Some of the incoming radiation is absorbed by the dish itself. Even a newly installed dish absorbs a significant fraction of the incident radiation, and parts of the dish can degrade over the course of its lifetime, reducing its efficiency (Ref. 3). In this model, the absorption coefficient is set to 0.1, meaning that 90% of the incoming radiation is reflected.

An additional correction is included due to the finite size of the sun. Not all incident rays will be parallel; instead, the incident rays are sampled from a narrow cone with maximum ψ angle, m, of 4.65 mrad. In practice, some radiation is also emitted from the circumsolar region surrounding the solar disk, instead of the solar disk itself, but this radiation is neglected in the present model; that is, a circumsolar ratio (CSR) of zero is assumed. When rays are released from points other than the center of the solar dish, their initial intensity can be reduced to account for solar limb darkening effects.

5 | SOLAR DISH RECEIVER Since the surface of the dish is not perfectly smooth, the reflected rays are not all released at the exact direction given by Equation 3. Instead, the surface normal is perturbed by an additional angle that is sampled from a Rayleigh distribution:

φ φ2 P()φ = ------exp–------2 2 σφ 2σφ

1 where σφ (SI unit: rad) is the surface slope error .

The model includes two studies, each corresponding to a separate instance of the Illuminated Surface feature. For each study, rays are released from 100,000 distinct points. At each point, the incident ray direction is perturbed by a random angle; the probability ψ density of these perturbations is uniform within a cone of angle s. For the first study, no limb darkening model is used and the surface is assumed to be perfectly smooth and reflective. The resulting concentration ratio is compared to the semi- analytical method of Jeter (Ref. 2).

For the second study, a limb darkening model is used to reduce the intensity of solar radiation emitted from the edge of the solar dish. The built-in limb darkening model follows an exponential fit, with wavelength-dependent exponents as described in Ref. 5. The resulting concentration ratio in the focal plane is compared to results in Ref. 1.

For each study, the concentration ratio is computed on a small circular disc, centered at the origin, which lies in the focal plane.

Simply plotting the concentration ratio as a function of the radial distance and azimuthal angle in the focal plane is not sufficient to compare against the data of Refs. 1-2 because there is a significant amount of statistical noise in the model. This is due to the random nature of the incident ray direction vectors at each release point. To smooth some of this statistical noise, the average concentration ratio is taken over all azimuthal angles for each value of the radial coordinate in the focal plane:

π 1 2 ()ρ ------()θρθ, C = π C d (4) 2 0

Equation 4 is implemented using a General Projection component coupling. A General Projection component coupling evaluates a series of line or curve integrals on a source. In

1. The definition of the surface slope error used in the Illuminated Surface feature seems to differ from the definition used by Shuai et al. For the Illuminated Surface, the surface slope error is used to perturb the surface normal direction, not the incident ray direction. As a result, values of the surface slope error used in this model differ from the corresponding results in Ref.1 by a factor of 2.

6 | SOLAR DISH RECEIVER this example, the General Projection 1 node integrates the accumulated variable over concentric circles in the focal plane, centered at the origin.

Results and Discussion

The ray trajectories emanating from the solar dish can be seen in Figure 3. Almost every ray is stopped by the receiver, with only an extremely small number of propagating rays visible above the focal plane.

Figure 3: The ray trajectories emanate from the illuminated surface and hit the receiver.

The incident heat flux arriving on the surface of the collector is shown in Figure 4. The heat flux is extremely high, with an average value of about 23 W/mm2 near the center of the focal plane. The statistical noise is also apparent, since in some boundary mesh elements the incident heat flux exceeds 30 W/mm2. If smoothing is disabled in the plot, then in a very small number of boundary elements the incident flux is shown to be even greater, as high as 51 W/mm2. This demonstrates the need for averaging in the azimuthal direction to more consistently compare the concentration ratio to published values.

The azimuthally averaged concentration ratio is plotted in Figure 5 along with the semi- analytical solution of Jeter (Ref. 2). The data are shown to be in close agreement.

7 | SOLAR DISH RECEIVER Figure 4: Incident heat flux on the surface of the receiver, resulting from an ideally smooth, non-absorbing paraboloidal reflector. Solar limb darkening effects have also been neglected.

Figure 5: Comparison of the azimuthally averaged, computed concentration ratio in the focal plane to a semi-analytical solution. The two solutions are in close agreement.

8 | SOLAR DISH RECEIVER The ray trajectories resulting from the second study are shown in Figure 6. Compared to Figure 3, a substantial number of rays now miss the receiver and continue to propagate, reducing the efficiency of the cavity receiver.

Figure 6: Reflection of solar radiation by a paraboloidal dish. Surface roughness, absorption, and solar limb darkening have all been taken into account.

The flux distribution in the focal plane is shown in Figure 7. Compared to Figure 4, the distribution is much more widespread, lacking any well-defined plateau. The maximum flux has also been considerably reduced.

The comparison of the azimuthally averaged concentration ratio to Ref. 1 is shown in Figure 8. The two solutions are again shown to be in close agreement. Further statistical convergence could be achieved by increasing the number of rays and refining the mesh on the focal plane, at the cost of increased memory usage and solution time.

Finally, the incident heat flux distributions from the two studies are directly compared in Figure 9.

9 | SOLAR DISH RECEIVER Figure 7: Flux distribution in the focal plane when taking surface roughness, absorption, and solar limb darkening into account.

Figure 8: Comparison of the azimuthally averaged concentration ratio to published data.

10 | SOLAR DISH RECEIVER Figure 9: Direct comparison of the flux distributions when including roughness, absorption, and solar limb darkening (the “Real Collector”), and when neglecting these effects (the “Ideal Collector”).

References

1. Y. Shuai, X-L. Xia, and H-P. Tan, “Radiation performance of dish solar concentrator/ cavity receiver systems,” Solar Energy, vol. 82, pp. 13–21, 2008.

2. S. M. Jeter, “The distribution of concentrated solar radiation in paraboloidal collectors, ” Journal of Solar Energy Engineering, vol. 108, pp. 219-225, 1986.

3. G. Johnston, “Focal region measurements of the 20 m2 tiled dish at the Australian national university,” Solar Energy, Vol. 63, No. 2, pp. 117-124, 1998.

4. M. Schubnell, “Sunshape and its influence on the flux distribution in imaging solar concentrators,” Journal of Solar Energy Engineering, vol. 114, pp. 260-266, 1992.

5. D. Hestroffer and C. Magnan, “Wavelength dependency of the Solar limb darkening,” Astron. Astrophysl, vol. 333, pp. 338-342, 1998.

11 | SOLAR DISH RECEIVER Application Library path: Ray_Optics_Module/Industrial_Applications/ solar_dish_receiver

Modeling Instructions

From the File menu, choose New.

NEW In the New window, click Model Wizard.

GEOMETRY 1 Define some parameters for the geometry setup.

GLOBAL DEFINITIONS

Parameters 1 On the Home toolbar, click Parameters. 2 In the Settings window for Parameters, locate the Parameters section. 3 In the table, enter the following settings:

Name Expression Value Description f 3[m] 3 m Focal length phi 45[deg] 0.7854 rad Rim angle d 4*f*(csc(phi)-cot(phi)) 4.971 m Dish diameter A pi*d^2/4 19.4 m² Dish projected surface area

12 | SOLAR DISH RECEIVER Name Expression Value Description psim 4.65[mrad] 0.00465 rad Maximum solar disc angle sig 1.75[mrad] 0.00175 rad Surface slope error I0 1[kW/m^2] 1000 W/m² Solar irradiance

GEOMETRY 1

Cylinder 1 (cyl1) 1 On the Geometry toolbar, click Cylinder. 2 In the Settings window for Cylinder, locate the Size and Shape section. 3 In the Radius text field, type 30[mm]. 4 In the Height text field, type 100[mm]. 5 Click Build All Objects.

PART LIBRARIES 1 On the Geometry toolbar, click Parts and choose Part Libraries. 2 In the Model Builder window, click Geometry 1. 3 In the Part Libraries window, select Ray Optics Module>3D>Mirrors> paraboloidal reflector shell 3d in the tree. 4 Click Add to Geometry. 5 In the Select Part Variant dialog box, select Specify rim angle in the Select part variant list. 6 Click OK.

GEOMETRY 1

Paraboloidal Reflector Shell 3D 1 (pi1) 1 In the Model Builder window, under Component 1 (comp1)>Geometry 1 click Paraboloidal Reflector Shell 3D 1 (pi1). 2 In the Settings window for Part Instance, locate the Input Parameters section. 3 In the table, enter the following settings:

Name Expression Value Description phi phi 0.7854 [rad] Rim angle d2 0 0.0 [m] Center hole diameter F 3[m] 3.0 [m] Focal length

13 | SOLAR DISH RECEIVER Name Expression Value Description nix 0 0.0 Incident ray direction, x-component niz -1 -1.0 Incident ray direction, z-component 4 Locate the Position and Orientation of Output section. Find the Displacement subsection. In the zw text field, type -f. 5 Click to expand the Boundary selections section. Locate the Boundary Selections section. Click to select row number 1 in the table. 6 In the table, enter the following settings:

Name Contribute to Keep Physics All none √√

GLOBAL DEFINITIONS In the Model Builder window, collapse the Global Definitions node.

GEOMETRY 1 In the Model Builder window, collapse the Component 1 (comp1)>Geometry 1 node.

DEFINITIONS

Integration 1 (intop1) 1 On the Definitions toolbar, click Component Couplings and choose Integration. 2 In the Settings window for Integration, locate the Source Selection section. 3 From the Geometric entity level list, choose Boundary. 4 From the Selection list, choose All (Paraboloidal Reflector Shell 3D 1).

Interpolation 1 (int1) 1 On the Definitions toolbar, click Interpolation. Load the expected solution data from Ref. 1. 2 In the Settings window for Interpolation, locate the Definition section. 3 From the Data source list, choose File. 4 Click Browse. 5 Browse to the model’s Application Libraries folder and double-click the file solar_dish_receiver_reference.txt. 6 Click Import. 7 Locate the Units section. In the Arguments text field, type mm.

14 | SOLAR DISH RECEIVER 8 In the Function text field, type W/mm^2.

Add a selection for the surface of the paraboloidal dish, where the reflected rays are released.

Explicit 1 1 On the Definitions toolbar, click Explicit. 2 In the Settings window for Explicit, type Focal Plane in the Label text field. 3 Locate the Input Entities section. From the Geometric entity level list, choose Boundary. 4 Select Boundary 5 only.

Variables 1 1 In the Model Builder window, right-click Definitions and choose Variables. Define variables to compute the concentration ratio for an ideal parabolic solar concentrator, following Jeter (Ref. 2). To save time, these variables can be loaded from a file. 2 In the Settings window for Variables, locate the Variables section. 3 Click Load from File. 4 Browse to the model’s Application Libraries folder and double-click the file solar_dish_receiver_vars.txt.

General Projection 1 (genproj1) 1 On the Definitions toolbar, click Component Couplings and choose General Projection. 2 In the Settings window for General Projection, locate the Source Selection section. 3 From the Geometric entity level list, choose Boundary. 4 From the Selection list, choose Focal Plane. 5 Click Go to Default View. 6 Locate the Source Map section. In the x-expression text field, type r. 7 In the y-expression text field, type theta. 8 Locate the Destination Map section. In the x-expression text field, type r. 9 In the Model Builder window, collapse the Definitions node.

15 | SOLAR DISH RECEIVER 4 Locate the Intensity Computation section. From the Intensity computation list, choose Compute intensity and power. 5 Locate the Ray Release and Propagation section. In the Maximum number of secondary rays text field, type 0. Add two instances of the Illuminated Surface release feature. One of these features will include surface roughness and solar limb darkening. One release feature will be used in each of the two studies in this model.

Illuminated Surface 1 1 Right-click Component 1 (comp1)>Geometrical Optics (gop) and choose Illuminated Surface. 2 In the Settings window for Illuminated Surface, type Ideal Illuminated Surface in the Label text field. 3 Locate the Boundary Selection section. From the Selection list, choose All (Paraboloidal Reflector Shell 3D 1). 4 Locate the Initial Position section. From the Initial position list, choose Density. 5 In the N text field, type 100000.

6 Locate the Ray Direction Vector section. Specify the Li vector as

0 x 0 y -1 z

7 Locate the Angular Perturbations section. From the Corrections for finite source diameter list, choose Sample from conical distribution.

8 Locate the Total Source Power section. In the Psrc text field, type A*I0. ψ 9 Locate the Angular Perturbations section. In the m text field, type psim. Illuminated Surface 2 1 In the Model Builder window, right-click Geometrical Optics (gop) and choose Illuminated Surface. 2 In the Settings window for Illuminated Surface, type Real Illuminated Surface in the Label text field. 3 Locate the Boundary Selection section. From the Selection list, choose All (Paraboloidal Reflector Shell 3D 1). 4 Locate the Initial Position section. From the Initial position list, choose Density. 5 In the N text field, type 100000.

16 | SOLAR DISH RECEIVER 6 Locate the Ray Direction Vector section. Specify the Li vector as

0 x 0 y -1 z

7 In the α text field, type 0.1. 8 Locate the Angular Perturbations section. From the Corrections for finite source diameter list, choose Sample from conical distribution. ψ 9 In the m text field, type psim. 10 From the Limb darkening model list, choose Empirical power law. 11 Select the Include surface roughness check box.

12 In the σφ text field, type sig.

13 Locate the Total Source Power section. In the Psrc text field, type A*I0. 14 Locate the Initial Polarization section. From the Initial polarization type list, choose Unpolarized.

Wall 1 1 Right-click Geometrical Optics (gop) and choose Wall. 2 In the Settings window for Wall, type Focal Plane in the Label text field. 3 Locate the Boundary Selection section. From the Selection list, choose Focal Plane. 4 Click Go to Default View. Use the Deposited Ray Power node to compute the incident heat flux in the focal plane. 5 Right-click Component 1 (comp1)>Geometrical Optics (gop)>Focal Plane and choose Deposited Ray Power. 6 In the Model Builder window, collapse the Geometrical Optics (gop) node.

Size 1 1 In the Model Builder window, under Component 1 (comp1)>Mesh 1 right-click Free Triangular 1 and choose Size.

17 | SOLAR DISH RECEIVER 2 In the Settings window for Size, locate the Geometric Entity Selection section. 3 Click Clear Selection. 4 Select Boundary 5 only. 5 Locate the Element Size section. Click the Custom button. Use an extremely fine mesh on the focal plane to improve the resolution of the deposited ray power. 6 Locate the Element Size Parameters section. Select the Maximum element size check box. 7 In the associated text field, type 5E-4. 8 Select the Minimum element size check box. 9 In the associated text field, type 2E-4. 10 In the Model Builder window, collapse the Mesh 1 node. 11 Click Build All.

STUDY 1

Step 1: Ray Tracing 1 In the Settings window for Ray Tracing, locate the Study Settings section. 2 From the Time step specification list, choose Specify maximum path length. 3 In the Lengths text field, type 0 4. 4 Locate the Physics and Variables Selection section. Select the Modify physics tree and variables for study step check box. 5 In the Physics and variables selection tree, select Component 1 (comp1)> Geometrical Optics (gop)>Real Illuminated Surface. 6 Click Disable.

Solution 1 (sol1) 1 On the Study toolbar, click Show Default Solver. Specify a manual time step size to speed up the computation and reduce the file size. 2 In the Model Builder window, expand the Solution 1 (sol1) node, then click Time- Dependent Solver 1. 3 In the Settings window for Time-Dependent Solver, click to expand the Time stepping section. 4 Locate the Time Stepping section. From the Steps taken by solver list, choose Manual. 5 In the Time step text field, type 4[m]/c_const.

18 | SOLAR DISH RECEIVER 6 In the Model Builder window, collapse the Study 1 node. 7 On the Study toolbar, click Compute.

RESULTS

Ray Trajectories (gop) 1 In the Model Builder window, under Results click Ray Trajectories (gop). 2 In the Settings window for 3D Plot Group, type Ray Trajectories, Ideal Reflector in the Label text field. 3 Click to expand the Title section. From the Title type list, choose Manual. 4 In the Title text area, type Ray Trajectories, Ideal Reflector.

Ray Trajectories, Ideal Reflector In the Model Builder window, expand the Results>Ray Trajectories, Ideal Reflector node.

Color Expression 1 1 In the Model Builder window, expand the Results>Ray Trajectories, Ideal Reflector> Ray Trajectories 1 node, then click Color Expression 1. 2 In the Settings window for Color Expression, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1> Geometrical Optics>Intensity and polarization>gop.Q - Ray power. 3 On the Ray Trajectories, Ideal Reflector toolbar, click Plot. 4 Click the Zoom Extents button on the Graphics toolbar. Compare the resulting plot with Figure 3.

Ray Trajectories, Ideal Reflector In the Model Builder window, collapse the Results>Ray Trajectories, Ideal Reflector node.

Surface 1 1 On the Results toolbar, click More Data Sets and choose Surface. 2 In the Settings window for Surface, locate the Parameterization section. 3 From the x- and y-axes list, choose XY-plane. 4 Locate the Selection section. From the Selection list, choose Focal Plane. 5 Click Go to Default View.

2D Plot Group 2 1 On the Results toolbar, click 2D Plot Group.

19 | SOLAR DISH RECEIVER 2 In the Settings window for 2D Plot Group, type Deposited Power, Ideal Reflector in the Label text field. 3 Click to expand the Title section. From the Title type list, choose Manual. 4 In the Title text area, type Deposited Power, Ideal Reflector. 5 Locate the Data section. From the Data set list, choose Surface 1. 6 Locate the Color Legend section. Select the Show units check box.

Surface 1 1 Right-click Deposited Power, Ideal Reflector and choose Surface. 2 In the Settings window for Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1>Geometrical Optics> Accumulated variables>Boundary heat source comp1.gop.wall1.bsrc1.Qp> gop.wall1.bsrc1.Qp - Boundary heat source. 3 Locate the Expression section. In the Unit field, type W/mm^2. 4 Locate the Coloring and Style section. From the Color table list, choose ThermalLight. 5 Click to expand the Quality section. From the Resolution list, choose No refinement. 6 On the Deposited Power, Ideal Reflector toolbar, click Plot. 7 Click the Zoom Extents button on the Graphics toolbar. Compare the resulting plot with Figure 4.

Cut Line 3D 1 1 On the Results toolbar, click Cut Line 3D. 2 In the Settings window for Cut Line 3D, locate the Line Data section. 3 In row Point 1, set X to 1E-4. 4 In row Point 2, set X to 0.03-1E-4. 5 Select the Snap to closest boundary check box.

1D Plot Group 3 1 On the Results toolbar, click 1D Plot Group. 2 In the Settings window for 1D Plot Group, type Concentration Ratios, Ideal Reflector in the Label text field. 3 Click to expand the Title section. From the Title type list, choose Manual. 4 In the Title text area, type Concentration Ratios, Ideal Reflector. 5 Locate the Data section. From the Data set list, choose Cut Line 3D 1. 6 From the Time selection list, choose Last.

20 | SOLAR DISH RECEIVER Line Graph 1 1 Right-click Concentration Ratios, Ideal Reflector and choose Line Graph. Plot the azimuthally averaged, smoothed heat source on the cylinder using the projection component coupling previously defined. 2 In the Settings window for Line Graph, locate the y-Axis Data section. 3 In the Expression text field, type genproj1(gop.wall1.bsrc1.Qp)/genproj1(I0). 4 Click to expand the Quality section. From the Resolution list, choose No refinement. 5 Locate the x-Axis Data section. From the Parameter list, choose Expression. 6 In the Expression text field, type r. 7 Click to expand the Legends section. Select the Show legends check box. 8 From the Legends list, choose Manual. 9 In the table, enter the following settings:

Legends Ray Tracing

10 On the Concentration Ratios, Ideal Reflector toolbar, click Plot.

Line Graph 2 1 Right-click Results>Concentration Ratios, Ideal Reflector>Line Graph 1 and choose Duplicate. 2 In the Settings window for Line Graph, locate the y-Axis Data section. 3 In the Expression text field, type cr. 4 Click to expand the Title section. From the Title type list, choose None. 5 Locate the Legends section. In the table, enter the following settings:

Legends Jeter

6 On the Concentration Ratios, Ideal Reflector toolbar, click Plot. Compare the resulting plot with Figure 5. This figure shows the radial variation in the idealized concentration ratio. Now create a second study in which roughness, absorption, and limb darkening effects are considered.

ADD STUDY 1 On the Home toolbar, click Add Study to open the Add Study window. 2 Go to the Add Study window.

21 | SOLAR DISH RECEIVER 3 Find the Studies subsection. In the Select Study tree, select Preset Studies>Ray Tracing. 4 Click Add Study in the window toolbar.

STUDY 2

Step 1: Ray Tracing 1 On the Home toolbar, click Add Study to close the Add Study window. 2 In the Model Builder window, under Study 2 click Step 1: Ray Tracing. 3 In the Settings window for Ray Tracing, locate the Study Settings section. 4 From the Time step specification list, choose Specify maximum path length. 5 In the Lengths text field, type 0 4. 6 Locate the Physics and Variables Selection section. Select the Modify physics tree and variables for study step check box. 7 In the Physics and variables selection tree, select Component 1 (comp1)> Geometrical Optics (gop)>Ideal Illuminated Surface. 8 Click Disable.

Solution 2 (sol2) 1 On the Study toolbar, click Show Default SolverSpecify a manual time step size to speed up the computation and reduce the file size. 2 In the Model Builder window, expand the Solution 2 (sol2) node, then click Time- Dependent Solver 1. 3 In the Settings window for Time-Dependent Solver, click to expand the Time stepping section. 4 Locate the Time Stepping section. From the Steps taken by solver list, choose Manual. 5 In the Time step text field, type 4[m]/c_const. 6 In the Model Builder window, collapse the Study 2 node. 7 On the Study toolbar, click Compute.

RESULTS

Ray Trajectories (gop) 1 In the Model Builder window, under Results click Ray Trajectories (gop). 2 In the Settings window for 3D Plot Group, type Ray Trajectories, Real Reflector in the Label text field. 3 Click to expand the Title section. From the Title type list, choose Manual.

22 | SOLAR DISH RECEIVER 4 In the Title text area, type Ray Trajectories, Real Reflector.

Ray Trajectories, Real Reflector In the Model Builder window, expand the Results>Ray Trajectories, Real Reflector node.

Color Expression 1 1 In the Model Builder window, expand the Results>Ray Trajectories, Real Reflector> Ray Trajectories 1 node, then click Color Expression 1. 2 In the Settings window for Color Expression, locate the Expression section. 3 In the Expression text field, type gop.Q. 4 On the Ray Trajectories, Real Reflector toolbar, click Plot. 5 Click Go to Default View. Compare the resulting plot with Figure 6.

Ray Trajectories, Real Reflector In the Model Builder window, collapse the Results>Ray Trajectories, Real Reflector node.

Surface 1 Create duplicates of the Surface and Cut Line 3D data sets that point to Solution 2.

Surface 2 1 In the Model Builder window, under Results>Data Sets right-click Surface 1 and choose Duplicate. 2 In the Settings window for Surface, locate the Data section. 3 From the Data set list, choose Study 2/Solution 2 (sol2).

Cut Line 3D 2 1 In the Model Builder window, under Results>Data Sets right-click Cut Line 3D 1 and choose Duplicate. 2 In the Settings window for Cut Line 3D, locate the Data section. 3 From the Data set list, choose Study 2/Solution 2 (sol2).

Deposited Power, Ideal Reflector 1 1 In the Model Builder window, under Results right-click Deposited Power, Ideal Reflector and choose Duplicate. 2 In the Settings window for 2D Plot Group, type Deposited Power, Real Reflector in the Label text field. 3 Locate the Title section. In the Title text area, type Deposited Power, Real Reflector. 4 Locate the Data section. From the Data set list, choose Surface 2.

23 | SOLAR DISH RECEIVER 5 On the Deposited Power, Real Reflector toolbar, click Plot. 6 Click the Zoom Extents button on the Graphics toolbar. Compare the resulting plot with Figure 7.

Concentration Ratios, Ideal Reflector 1 1 In the Model Builder window, under Results right-click Concentration Ratios, Ideal Reflector and choose Duplicate. 2 In the Settings window for 1D Plot Group, type Concentration Ratios, Real Reflector in the Label text field. 3 Locate the Title section. In the Title text area, type Concentration Ratios, Real Reflector. 4 Locate the Data section. From the Data set list, choose Cut Line 3D 2.

Line Graph 1 1 In the Model Builder window, expand the Results>Concentration Ratios, Real Reflector node, then click Line Graph 1. 2 In the Settings window for Line Graph, locate the Legends section. 3 In the table, enter the following settings:

Legends Ray Tracing

Line Graph 2 1 In the Model Builder window, under Results>Concentration Ratios, Real Reflector click Line Graph 2. 2 In the Settings window for Line Graph, locate the y-Axis Data section. 3 In the Expression text field, type int1(r)/I0. 4 Locate the Legends section. In the table, enter the following settings:

Legends Shuai

5 On the Concentration Ratios, Real Reflector toolbar, click Plot. Compare the resulting plot with Figure 8.

Create another plot group to directly compare the flux distributions in the focal plane for the two solutions.

24 | SOLAR DISH RECEIVER Deposited Power, Ideal Reflector 1 1 In the Model Builder window, under Results right-click Deposited Power, Ideal Reflector and choose Duplicate. 2 In the Settings window for 2D Plot Group, type Deposited Power, Real and Ideal Reflectors in the Label text field. 3 Locate the Title section. In the Title text area, type Deposited Power, Real and Ideal Reflectors.

Surface 1 1 In the Model Builder window, expand the Results>Deposited Power, Real Reflector node. 2 Right-click Surface 1 and choose Copy.

Deposited Power, Real and Ideal Reflectors In the Model Builder window, expand the Results>Deposited Power, Real and Ideal Reflectors node.

Surface 2 1 Right-click Deposited Power, Real and Ideal Reflectors and choose Paste Surface. 2 In the Settings window for Surface, locate the Data section. 3 From the Data set list, choose Surface 2. 4 Click to expand the Inherit style section. Locate the Inherit Style section. From the Plot list, choose Surface 1. Use the Deformation feature to shift one of the plots so that they can be viewed side-by- side.

Deformation 1 1 Right-click Results>Deposited Power, Real and Ideal Reflectors>Surface 2 and choose Deformation. 2 In the Settings window for Deformation, locate the Expression section. 3 In the xcomponent text field, type 0.07. 4 In the ycomponent text field, type 0. 5 Locate the Scale section. Select the Scale factor check box. 6 In the associated text field, type 1.

Deposited Power, Real and Ideal Reflectors Create two Annotation features to identify the two plots in the Graphics window.

25 | SOLAR DISH RECEIVER Annotation 1 1 In the Model Builder window, under Results right-click Deposited Power, Real and Ideal Reflectors and choose Annotation. 2 In the Settings window for Annotation, locate the Annotation section. 3 In the Text text field, type Ideal Reflector. Enter the coordinates for the annotations. The exact coordinates may vary depending on the aspect ratio of the Graphics window. 4 Locate the Position section. In the x text field, type -0.015. 5 In the y text field, type 0.038. 6 Locate the Coloring and Style section. Clear the Show point check box. 7 Select the Show frame check box.

Annotation 2 1 Right-click Deposited Power, Real and Ideal Reflectors and choose Annotation. 2 In the Settings window for Annotation, locate the Annotation section. 3 In the Text text field, type Real Reflector. 4 Locate the Position section. In the x text field, type 0.055. 5 In the y text field, type 0.038. 6 Locate the Coloring and Style section. Clear the Show point check box. 7 Select the Show frame check box. 8 On the Deposited Power, Real and Ideal Reflectors toolbar, click Plot. 9 Click the Zoom Extents button on the Graphics toolbar. Compare the resulting plot with Figure 9.

26 | SOLAR DISH RECEIVER Created in COMSOL Multiphysics 5.3

Thermally Induced Focal Shift in High-Power Laser Focusing Systems

This model is licensed under the COMSOL Software License Agreement 5.3. All trademarks are the property of their respective owners. See www.comsol.com/trademarks. Introduction

Modern high-power industrial fiber laser systems can deliver up to 3kW of single-mode laser radiation onto surfaces to be cut, drilled, welded, or marked (Ref. 1). Even when the optical components used to focus the beam are almost completely transparent, the amount of heat generated in these optical components can degrade the ability of the system to correctly focus the beam.

The heat generated in a lens can change the paths of rays through several different mechanisms, including the following:

• Temperature dependence of the refractive index • Stress-optical effects resulting from thermal stress • Thermal expansion of the lenses

In this example, the temperature dependence of the refractive index and the thermal expansion of the lenses are considered, whereas the stress-induced changes in the refractive index are neglected. The Structural Mechanics Module and the Ray Optics Module are used to model thermally induced focal shift in a high-power laser focusing system.

Model Definition

A basic high-power laser focusing system consists of two identical silica glass plano-convex lenses. The first lens collimates the output of an optical fiber (numerical aperture of 0.1) and the second lens focuses the collimated beam at a target surface.

The model geometry consists of two 50 mm diameter fused silica glass lenses with an effective focal length of approximately 150 mm. The lenses are used to focus a laser beam λ = with free-space wavelength 0 1064 nm. The position of each lens is assumed to be fixed at three locations. The effects of changes in the lens temperature on the ray paths are modeled for two different values of the source power, 1W and 3kW. The thermal effects are negligible when the 1W source is used. When a 3kW beam is released, the change in temperature in the lenses causes a noticeable change in the position of the focal plane.

The model uses the Geometrical Optics interface to trace the paths of rays through the lens system. The Heat Transfer in Solids and Solid Mechanics interfaces are used to model the thermal expansion of the lenses.

ATTENUATION OF RAYS IN AN ABSORBING MEDIUM The intensity and power of a plane wave in an absorbing medium decay exponentially over time,

2 | THERMALLY INDUCED FOCAL SHIFT IN HIGH-POWER LASER FOCUSING SYSTEMS 2k κL II= exp–------0 0 n 2k κL PP= exp–------0 0 n where k0 is the free-space wave number,

2π ------k0 = λ 0

λ 0 is the free-space wavelength, L is the optical path length in the medium,

Lct= c is the speed of light in a vacuum, and t is the current time. The complex-valued refractive index is expressed as n −κi, where n and κ are dimensionless real numbers. Positive values of κ correspond to attenuating media whereas negative values indicate gain media.

In the Geometrical Optics interface it is possible to assign separate degrees of freedom for ray intensity and power. The ray power only changes due to absorption or gain by the surrounding media; the intensity is also affected by the convergence or divergence of each thin pencil of rays, when rays are reflected or refracted by curved surfaces for example. Whatever power is lost by the rays due to absorption becomes a heat source of equal magnitude on the underlying domain, through the Ray Heat Source multiphysics coupling feature.

COUPLING RAY OPTICS AND HEAT TRANSFER The ray trajectories and temperature distribution affect each other through a bidirectional, or two-way, coupling. In other words, the ray trajectories affect the temperature field, which in turn perturbs the ray trajectories, both directly and through the resulting structural deformation. To solve for the ray trajectories and temperature in a self-consistent manner, the dedicated Ray Heating interface and Bidirectionally Coupled Ray Tracing study step are used. The Bidirectionally Coupled Ray Tracing study step sets up a solver loop in which the ray trajectories and temperature are computed in alternating steps for a specified number of iterations, with the results of each iteration being used to assign the Values of variables not solved for in the iteration that immediately follows it. This iterative solver loop can also be set up manually by adding For and End For nodes to the solver sequence, but the Bidirectionally Coupled Ray Tracing study step adds these nodes to the solver sequence automatically.

For more details on the physics implementation and theory for the Geometrical Optics interface, see the Ray Optics Module User’s Guide.

3 | THERMALLY INDUCED FOCAL SHIFT IN HIGH-POWER LASER FOCUSING SYSTEMS Results and Discussion

The trajectories or rays in the 3kW beam are shown in Figure 1. The color expression indicates the logarithm of the ray intensity, which increases by several orders of magnitude as the beam is focused. The surface plot shows the temperature distribution, which is nearly identical in the two lenses. The maximum temperature in the lenses is approximately 510 K.

The von Mises stress and deformation resulting from absorption in the lenses are shown in Figure 2. A fixed color range has been used to more clearly show the stress distribution throughout the lenses. The maximum displacement obtained from the computation is approximately 3.3 μm.

The power deposited in the lenses and at the target surface is shown in Figure 3 and Figure 4, respectively. The heat source in the lenses reaches a maximum value of about 2.2 MW/m3 at the center. The boundary heat source at the target surface has a maximum value of about 36 MW/cm2. Note that the maximum deposited ray power on the boundary in Figure 4 appears to be significantly less than 36 MW/cm2; this is because the dependent variable for deposited ray power is discontinuous across boundaries between mesh elements, and the smoothing that is applied by default to surface plots can smear local maxima and minima across adjacent elements.

The change in the temperature of the lenses causes a change in their refractive indices, which is plotted in Figure 5 for the focusing lens. The figure displays the difference between the real part of the calculated refractive index (nr) and the refractive index at room temperature (n0). The change in the refractive index reaches a maximum at the center, where it is approximately 55 % greater than the change at the edges.

Figure 6 shows the percentage of the ray power absorbed by the system as the rays travel from the source to the target. Although the fused silica has a very low extinction coefficient (that is, the imaginary part of the refractive index is extremely small), each lens absorbs nearly 0.25 % of the beam power.

Figure 7 shows the average distance of the rays from the beam center in the region surrounding the focal plane. When the thermal effects are negligible (1W source case) the system’s focal plane is located at the target surface. Effects such as spherical aberration prevent the beam from being focused to a single point. When the beam power is increased to 3 kW, the focal plane is shifted away from the target surface.

4 | THERMALLY INDUCED FOCAL SHIFT IN HIGH-POWER LASER FOCUSING SYSTEMS Figure 1: Ray trajectories and surface temperature for the 3 kW source case.

Figure 2: Von Mises stress and deformation of the lenses when illuminated by the 3 kW source.

5 | THERMALLY INDUCED FOCAL SHIFT IN HIGH-POWER LASER FOCUSING SYSTEMS Figure 3: Volumetric heat source in the lenses due to attenuation of the 3 kW beam.

Figure 4: Boundary heat source generated in the focal plane by the 3 kW beam.

6 | THERMALLY INDUCED FOCAL SHIFT IN HIGH-POWER LASER FOCUSING SYSTEMS Figure 5: Change in the refractive index of the lens due to the 3 kW beam.

Figure 6: Percentage of the total source power absorbed by the lenses as the rays propagate from the optical fiber to the target.

7 | THERMALLY INDUCED FOCAL SHIFT IN HIGH-POWER LASER FOCUSING SYSTEMS Figure 7: Average radial displacement of the rays as a function of time. They reach the focal plane at about t = 1.386 ns.

Reference

1. O. Maerten, R. Kramer, H. Schwede, S. Wolf, and V. Brandl, “The Characterization of Focusing Systems for High-Power Lasers with High Beam Quality,” Laser+ Photonics (2009): pp. 60-64.

Application Library path: Ray_Optics_Module/Industrial_Applications/ thermally_induced_focal_shift

Modeling Instructions

From the File menu, choose New.

NEW In the New window, click Model Wizard.

8 | THERMALLY INDUCED FOCAL SHIFT IN HIGH-POWER LASER FOCUSING SYSTEMS MODEL WIZARD 1 In the Model Wizard window, click 3D. 2 In the Select Physics tree, select Optics>Ray Optics>Ray Heating. 3 Click Add. 4 In the Select Physics tree, select Structural Mechanics>Solid Mechanics (solid). 5 Click Add. 6 Click Study. 7 In the Select Study tree, select Preset Studies for Selected Physics Interfaces> Bidirectionally Coupled Ray Tracing. 8 Click Done.

Select a more appropriate length unit for the geometry.

GEOMETRY 1 1 In the Model Builder window, under Component 1 (comp1) click Geometry 1. 2 In the Settings window for Geometry, locate the Units section. 3 From the Length unit list, choose mm.

GLOBAL DEFINITIONS

Parameters 1 On the Home toolbar, click Parameters. Load the parameters for the geometry and physics setup from a file. 2 In the Settings window for Parameters, locate the Parameters section. 3 Click Load from File. 4 Browse to the model’s Application Libraries folder and double-click the file thermally_induced_focal_shift_parameters.txt.

Load a spherical plano-convex lens template from the built-in Part Library for the Ray Optics Module.

PART LIBRARIES 1 On the Home toolbar, click Windows and choose Part Libraries. 2 In the Model Builder window, under Component 1 (comp1) click Geometry 1. 3 In the Part Libraries window, select Ray Optics Module>3D>Spherical Lenses> spherical plano convex lens 3d in the tree.

9 | THERMALLY INDUCED FOCAL SHIFT IN HIGH-POWER LASER FOCUSING SYSTEMS 4 Click Add to Geometry. 5 In the Select Part Variant dialog box, select Specify radius of curvature and edge thickness in the Select part variant list. 6 Click OK.

GEOMETRY 1

Spherical Plano-Convex Lens 3D 1 (pi1) 1 In the Model Builder window, click Spherical Plano-Convex Lens 3D 1 (pi1). 2 In the Settings window for Part Instance, locate the Input Parameters section. 3 In the table, enter the following settings:

Name Expression Value Description d d 50.0 [mm] Diameter R R 68.8 [mm] Radius of curvature Te Te 3.0 [mm] Edge thickness nix 0 0.0 Incident ray direction, x component niy -1 -1.0 Incident ray direction, y component

4 Locate the Position and Orientation of Output section. Find the Displacement subsection. In the yw text field, type -dis/2.

Add cylinders to the geometry to create three circular boundaries along the perimeter of each lens. These surfaces will be used to apply fixed constraints when modeling the thermal expansion.

Cylinder 1 (cyl1) 1 On the Geometry toolbar, click Cylinder. 2 In the Settings window for Cylinder, locate the Size and Shape section. 3 In the Radius text field, type 0.75. 4 In the Height text field, type 20. 5 Locate the Position section. In the y text field, type -58. 6 In the z text field, type 10. 7 Right-click Cylinder 1 (cyl1) and choose Build Selected.

Rotate 1 (rot1) 1 On the Geometry toolbar, click Transforms and choose Rotate. 2 Select the object cyl1 only.

10 | THERMALLY INDUCED FOCAL SHIFT IN HIGH-POWER LASER FOCUSING SYSTEMS 3 In the Settings window for Rotate, locate the Rotation Angle section. 4 In the Rotation text field, type 120. 5 Locate the Axis of Rotation section. From the Axis type list, choose y-axis. 6 Locate the Input section. Select the Keep input objects check box. 7 Right-click Rotate 1 (rot1) and choose Build Selected.

Rotate 2 (rot2) 1 Right-click Rotate 1 (rot1) and choose Duplicate. 2 Select the object cyl1 only. 3 In the Settings window for Rotate, locate the Rotation Angle section. 4 In the Rotation text field, type -120. 5 Right-click Component 1 (comp1)>Geometry 1>Rotate 2 (rot2) and choose Build Selected.

Use the Partition Objects node to create surfaces where the cylinders intersect the lens.

Partition Objects 1 (par1) 1 On the Geometry toolbar, click Booleans and Partitions and choose Partition Objects. 2 Select the object pi1 only. 3 In the Settings window for Partition Objects, locate the Partition Objects section. 4 Find the Tool objects subsection. Select the Active toggle button. 5 Select the objects cyl1, rot1, and rot2 only.

Use the Union operation to remove some interior boundaries that are no longer needed.

Union 1 (uni1) 1 On the Geometry toolbar, click Booleans and Partitions and choose Union. 2 Select the object par1 only. 3 In the Settings window for Union, locate the Union section. 4 Clear the Keep interior boundaries check box. 5 Right-click Union 1 (uni1) and choose Build Selected.

Create the focusing lens, which is a mirror image of the collimating lens.

Mirror 1 (mir1) 1 On the Geometry toolbar, click Transforms and choose Mirror. 2 Select the object uni1 only. 3 In the Settings window for Mirror, locate the Input section.

11 | THERMALLY INDUCED FOCAL SHIFT IN HIGH-POWER LASER FOCUSING SYSTEMS 4 Select the Keep input objects check box. 5 Locate the Point on Plane of Reflection section. In the x text field, type 1. 6 Locate the Normal Vector to Plane of Reflection section. In the y text field, type -1. 7 In the z text field, type 0. 8 Click Build All Objects. 9 Click the Zoom Extents button on the Graphics toolbar.

Create a small square surface centered at the focal point. This surface will be finely meshed to resolve the deposited power in the focal plane.

Work Plane 1 (wp1) 1 On the Geometry toolbar, click Work Plane. 2 In the Settings window for Work Plane, locate the Plane Definition section. 3 From the Plane list, choose zx-plane. 4 In the y-coordinate text field, type dis/2+Tc+bfl. 5 Click Show Work Plane.

Square 1 (sq1) 1 On the Work Plane toolbar, click Primitives and choose Square. 2 In the Settings window for Square, locate the Size section. 3 In the Side length text field, type 2[mm]. 4 Locate the Position section. From the Base list, choose Center. 5 In the Model Builder window, click Geometry 1. 6 On the Home toolbar, click Build All.

DEFINITIONS

Variables 1 1 In the Model Builder window, under Component 1 (comp1) right-click Definitions and choose Variables. 2 In the Settings window for Variables, locate the Variables section.

12 | THERMALLY INDUCED FOCAL SHIFT IN HIGH-POWER LASER FOCUSING SYSTEMS 3 In the table, enter the following settings:

Name Expression Unit Description n_r n0+dndT*(T-T0) Real part of the refractive index n_i 3e-8 Imaginary part of the refractive index n n_r-i*n_i Refractive index of the plano-convex lens

The nonzero value of dndT makes the refractive index temperature dependent; this is by far the largest contributor to the focal shift in this model. By setting dndT to zero it is possible to isolate the effect of thermal deformation on the focal position.

COMPONENT 1 (COMP1)

Integration 1 (intop1) On the Definitions toolbar, click Component Couplings and choose Integration.

DEFINITIONS

Integration 1 (intop1) Click in the Graphics window and then press Ctrl+A to select both domains. This component coupling will be used later to compute the total deposited power in each lens.

Define some selections to make the boundary conditions easier to apply.

Explicit 1 1 On the Definitions toolbar, click Explicit. 2 In the Settings window for Explicit, locate the Input Entities section. 3 From the Geometric entity level list, choose Boundary. 4 Select Boundaries 1–10, 16, 17, and 19–24 only. 5 In the Label text field, type Exposed Lens Surfaces.

Explicit 2 1 On the Definitions toolbar, click Explicit. 2 In the Settings window for Explicit, locate the Input Entities section. 3 From the Geometric entity level list, choose Boundary. 4 Select Boundaries 11, 12, 14, 15, 18, and 25–27 only. 5 In the Label text field, type Fixed Lens Surfaces.

13 | THERMALLY INDUCED FOCAL SHIFT IN HIGH-POWER LASER FOCUSING SYSTEMS GEOMETRICAL OPTICS (GOP) 1 In the Model Builder window, under Component 1 (comp1) click Geometrical Optics (gop). 2 In the Settings window for Geometrical Optics, locate the Intensity Computation section. 3 From the Intensity computation list, choose Compute intensity and power. 4 Locate the Ray Release and Propagation section. In the Maximum number of secondary rays text field, type 0. Since an anti-reflective coating will be applied to the lens surfaces, it is not necessary to allocate secondary rays to model the reflection of stray light by the lens system.

Material Discontinuity 1 1 In the Model Builder window, expand the Geometrical Optics (gop) node, then click Material Discontinuity 1. 2 In the Settings window for Material Discontinuity, locate the Coatings section. 3 From the Thin dielectric films on boundary list, choose Anti-reflective coating. λ 4 In the 0 text field, type lam. The single-layer anti-reflective coating reduces the reflectance to zero for the specified free-space wavelength and angle of incidence.

Ray Properties 1 1 In the Model Builder window, under Component 1 (comp1)>Geometrical Optics (gop) click Ray Properties 1. 2 In the Settings window for Ray Properties, locate the Ray Properties section. λ 3 In the 0 text field, type lam. Release from Grid 1 1 In the Model Builder window, right-click Geometrical Optics (gop) and choose Release from Grid. 2 In the Settings window for Release from Grid, locate the Initial Coordinates section.

3 In the qy, 0 text field, type qy0. 4 Locate the Ray Direction Vector section. From the Ray direction vector list, choose Conical.

5 In the Nw text field, type 1000. 6 Specify the r vector as

0 x

14 | THERMALLY INDUCED FOCAL SHIFT IN HIGH-POWER LASER FOCUSING SYSTEMS 1 y 0 z 7 In the α text field, type theta.

8 Locate the Total Source Power section. In the Psrc text field, type Irms. Create a Wall boundary condition to stop rays as they reach the focal plane and compute the deposited ray power. 9 Right-click Geometrical Optics (gop) and choose Wall. 10 Select Boundary 13 only.

Deposited Ray Power 1 Right-click Geometrical Optics (gop) and choose Deposited Ray Power.

Next, set up boundary conditions for the temperature computation. Apply natural convection to the exposed surfaces of each lens.

HEAT TRANSFER IN SOLIDS (HT)

Heat Flux 1 1 In the Model Builder window, under Component 1 (comp1) right-click Heat Transfer in Solids (ht) and choose Heat Flux. 2 In the Settings window for Heat Flux, locate the Boundary Selection section. 3 From the Selection list, choose Exposed Lens Surfaces. 4 Locate the Heat Flux section. Click the Convective heat flux button. 5 In the h text field, type 10.

6 In the Text text field, type T0. Now set up the boundary conditions for the Solid Mechanics interface. Each lens is fixed in place at three locations and is subjected to thermal expansion.

SOLID MECHANICS (SOLID)

Fixed Constraint 1 1 In the Model Builder window, under Component 1 (comp1) right-click Solid Mechanics (solid) and choose Fixed Constraint. 2 In the Settings window for Fixed Constraint, locate the Boundary Selection section. 3 From the Selection list, choose Fixed Lens Surfaces.

15 | THERMALLY INDUCED FOCAL SHIFT IN HIGH-POWER LASER FOCUSING SYSTEMS MULTIPHYSICS

Thermal Expansion 1 (te1) 1 On the Physics toolbar, click Multiphysics and choose Domain>Thermal Expansion. 2 In the Settings window for Thermal Expansion, locate the Thermal Expansion Properties section.

3 In the Tref text field, type T0. 4 Click in the Graphics window and then press Ctrl+A to select both domains. Note that, even after adding the Thermal Expansion node, the ray trajectories are still computed in the undeformed geometry. To make the rays interact with the deformed surfaces of the lenses, it is important to select the Include geometric nonlinearity check box, described in the instructions for setting up the Study 1 node.

Temperature Coupling 1 (tc1) On the Physics toolbar, click Multiphysics and choose Global>Temperature Coupling.

MATERIALS

Material 1 (mat1) 1 In the Model Builder window, under Component 1 (comp1) right-click Materials and choose Blank Material. 2 In the Settings window for Material, type Silica glass in the Label text field. 3 Locate the Material Contents section. In the table, enter the following settings:

Property Name Value Unit Property group Refractive index n n 1 Refractive index Refractive index, ki 0 1 Refractive imaginary part index Thermal k 1.38[W/(m*K)] W/(m·K) Basic conductivity Density rho 2203[kg/m^3] kg/m³ Basic Heat capacity at Cp 703[J/(kg*K)] J/(kg·K) Basic constant pressure Young’s modulus E 73.1e9[Pa] Pa Basic

16 | THERMALLY INDUCED FOCAL SHIFT IN HIGH-POWER LASER FOCUSING SYSTEMS Property Name Value Unit Property group Poisson’s ratio nu 0.17 1Basic Coefficient of alpha 0.55e-6[1/K] 1/K Basic thermal expansion

MESH 1 1 In the Model Builder window, under Component 1 (comp1) click Mesh 1. 2 In the Settings window for Mesh, locate the Mesh Settings section. 3 From the Element size list, choose Extremely fine. 4 From the Sequence type list, choose User-controlled mesh. The default mesh sequence already includes a Size node. Add a second Size node to control the mesh resolution in the focal plane.

Size 1 1 Right-click Component 1 (comp1)>Mesh 1 and choose Size. 2 In the Model Builder window, under Component 1 (comp1)>Mesh 1 right-click Size 1 and choose Move Up. 3 In the Settings window for Size, locate the Geometric Entity Selection section. 4 From the Geometric entity level list, choose Boundary. 5 Select Boundary 13 only. 6 Locate the Element Size section. Click the Custom button. 7 Locate the Element Size Parameters section. Select the Maximum element size check box. 8 In the associated text field, type 0.05. 9 Select the Minimum element size check box. 10 In the associated text field, type 0.025.

STUDY 1

Step 1: Bidirectionally Coupled Ray Tracing 1 In the Model Builder window, under Study 1 click Step 1: Bidirectionally Coupled Ray Tracing. 2 In the Settings window for Bidirectionally Coupled Ray Tracing, locate the Study Settings section. 3 From the Time step specification list, choose Specify maximum path length. 4 From the Length unit list, choose mm.

17 | THERMALLY INDUCED FOCAL SHIFT IN HIGH-POWER LASER FOCUSING SYSTEMS 5 In the Lengths text field, type range(0,4,400),range(412,0.5,421). By using smaller optical path length intervals in the vicinity of the focal plane it will be easier to observe where the mean radial displacement of the rays reaches a minimum. 6 Select the Include geometric nonlinearity check box. When this check box is selected, rays are traced through the deformed geometry in which thermal expansion has been taken into account. If this check box is cleared, the temperature dependence of the refractive index still affects the ray trajectories, but the thermal expansion has no effect. 7 Locate the Iterations section. In the Number of iterations text field, type 3.

Parameter name Parameter value list Parameter unit Irms 1 3000

The first parameter value results in a very small change in temperature and a negligibly small focal shift; the larger value shows a substantial focal shift.

Set manual scaling for the displacement field components to improve convergence during the first iteration.

Solution 1 (sol1) 1 On the Study toolbar, click Show Default Solver. 2 In the Model Builder window, expand the Solution 1 (sol1) node. 3 In the Model Builder window, expand the Study 1>Solver Configurations> Solution 1 (sol1)>Dependent Variables 2 node, then click Displacement field (material and geometry frames) (comp1.u). 4 In the Settings window for Field, locate the Scaling section. 5 From the Method list, choose Manual. 6 In the Model Builder window, under Study 1>Solver Configurations>Solution 1 (sol1) click Time-Dependent Solver 1. 7 In the Settings window for Time-Dependent Solver, click to expand the Time stepping section. 8 Click to expand the Output section. From the Times to store list, choose Specified values.

18 | THERMALLY INDUCED FOCAL SHIFT IN HIGH-POWER LASER FOCUSING SYSTEMS 9 On the Study toolbar, click Compute.

RESULTS

Ray Trajectories (gop) In the Model Builder window, expand the Ray Trajectories (gop) node.

Ray Trajectories 1 1 In the Model Builder window, expand the Results>Ray Trajectories (gop)> Ray Trajectories 1 node, then click Ray Trajectories 1. 2 In the Settings window for Ray Trajectories, locate the Coloring and Style section. 3 Find the Point style subsection. From the Type list, choose None.

Color Expression 1 1 In the Model Builder window, under Results>Ray Trajectories (gop)>Ray Trajectories 1 click Color Expression 1. 2 In the Settings window for Color Expression, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1> Geometrical Optics>Intensity and polarization>gop.logI - Log of intensity. 3 Locate the Coloring and Style section. Clear the Color legend check box. 4 Click to expand the Range section. Select the Manual color range check box. 5 In the Minimum text field, type 6. 6 In the Maximum text field, type 10.

Use the Filter node to plot only a fraction of the rays, making them easier to see.

Filter 1 1 In the Model Builder window, under Results>Ray Trajectories (gop)>Ray Trajectories 1 click Filter 1. 2 In the Settings window for Filter, locate the Ray Selection section. 3 From the Rays to render list, choose Fraction. 4 In the Fraction of rays text field, type 0.1.

Add a Surface plot to view the temperature along with the ray trajectories.

Surface 1 1 In the Model Builder window, under Results right-click Ray Trajectories (gop) and choose Surface.

19 | THERMALLY INDUCED FOCAL SHIFT IN HIGH-POWER LASER FOCUSING SYSTEMS 2 In the Settings window for Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1>Heat Transfer in Solids> Temperature>T - Temperature. 3 Locate the Coloring and Style section. From the Color table list, choose ThermalLight. 4 On the Ray Trajectories (gop) toolbar, click Plot. 5 Click the Zoom Extents button on the Graphics toolbar. The plot should now look like Figure 1. Add a Selection to the solution data set to exclude the target surface, where the von Mises stress is not defined, from the following plot.

Study 1/Parametric Solutions 1 (sol2) In the Model Builder window, expand the Data Sets node, then click Study 1/ Parametric Solutions 1 (sol2).

Surface 1 1 In the Model Builder window, expand the Stress (solid) node, then click Surface 1. 2 In the Settings window for Surface, locate the Expression section. 3 From the Unit list, choose MPa. 4 Click to expand the Range section. Specify a manual color range to make the von Mises stress easier to see. 5 Select the Manual color range check box. 6 In the Minimum text field, type 0. 7 In the Maximum text field, type 10. 8 On the Stress (solid) toolbar, click Plot. 9 Click the Zoom Extents button on the Graphics toolbar. The plot should now look like Figure 2.

Create a plot of the deposited power in the lenses.

3D Plot Group 5 1 On the Home toolbar, click Add Plot Group and choose 3D Plot Group.

20 | THERMALLY INDUCED FOCAL SHIFT IN HIGH-POWER LASER FOCUSING SYSTEMS 2 In the Settings window for 3D Plot Group, type Deposited Ray Power (lenses) in the Label text field. 3 Locate the Data section. From the Data set list, choose Study 1/ Parametric Solutions 1 (sol2).

Volume 1 1 Right-click Deposited Ray Power (lenses) and choose Volume. 2 In the Settings window for Volume, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1>Heating and Losses> rhs1.Qsrc - Heat source. 3 Click to expand the Quality section. From the Resolution list, choose No refinement. 4 On the Deposited Ray Power (lenses) toolbar, click Plot. 5 Click the Zoom Extents button on the Graphics toolbar. The plot should now look like Figure 3.

Next create a separate data set to plot the deposited ray power in the focal plane.

Surface 1 1 On the Results toolbar, click More Data Sets and choose Surface. 2 In the Settings window for Surface, locate the Data section. 3 From the Data set list, choose Study 1/Parametric Solutions 1 (sol2). 4 Select Boundary 13 only.

2D Plot Group 6 1 On the Results toolbar, click 2D Plot Group. 2 In the Settings window for 2D Plot Group, type Deposited Ray Power (target) in the Label text field. 3 Locate the Data section. From the Data set list, choose Surface 1. 4 Locate the Plot Settings section. Clear the Plot data set edges check box.

Surface 1 1 Right-click Deposited Ray Power (target) and choose Surface. 2 In the Settings window for Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1>Geometrical Optics> Accumulated variables>Boundary heat source comp1.gop.wall1.bsrc1.Qp> gop.wall1.bsrc1.Qp - Boundary heat source. 3 Click to expand the Quality section. From the Resolution list, choose No refinement.

21 | THERMALLY INDUCED FOCAL SHIFT IN HIGH-POWER LASER FOCUSING SYSTEMS 4 On the Deposited Ray Power (target) toolbar, click Plot. The plot should now look like Figure 4.

Create another Surface data set to plot the change in the refractive index over one of the lens surfaces.

Surface 2 1 On the Results toolbar, click More Data Sets and choose Surface. 2 In the Settings window for Surface, locate the Data section. 3 From the Data set list, choose Study 1/Parametric Solutions 1 (sol2). 4 Select Boundary 10 only.

2D Plot Group 7 1 On the Results toolbar, click 2D Plot Group. 2 In the Settings window for 2D Plot Group, type Refractive Index in the Label text field. 3 Locate the Data section. From the Data set list, choose Surface 2.

Surface 1 1 Right-click Refractive Index and choose Surface. 2 In the Settings window for Surface, locate the Expression section. 3 In the Expression text field, type n_r-n0. 4 On the Refractive Index toolbar, click Plot. 5 Click the Zoom Extents button on the Graphics toolbar. The plot should now look like Figure 5.

Plot the amount of attenuated power in the lenses over time.

1D Plot Group 8 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 In the Settings window for 1D Plot Group, type Power Loss in the Label text field. 3 Locate the Data section. From the Data set list, choose Study 1/ Parametric Solutions 1 (sol2).

Global 1 1 Right-click Power Loss and choose Global. 2 In the Settings window for Global, locate the y-Axis Data section.

22 | THERMALLY INDUCED FOCAL SHIFT IN HIGH-POWER LASER FOCUSING SYSTEMS 3 In the table, enter the following settings:

Expression Unit Description intop1(rhs1.Qsrc)/gop.gopop1(gop.Q)*100

Power Loss 1 In the Model Builder window, under Results click Power Loss. 2 In the Settings window for 1D Plot Group, locate the Plot Settings section. 3 Select the x-axis label check box. 4 Select the y-axis label check box. 5 In the associated text field, type Percentage of absorbed power. 6 On the Power Loss toolbar, click Plot. 7 Click the Zoom Extents button on the Graphics toolbar. The plot should now look like Figure 6.

Finally, plot the average radial displacement of the rays as a function of time.

1D Plot Group 9 1 On the Home toolbar, click Add Plot Group and choose 1D Plot Group. 2 In the Settings window for 1D Plot Group, type Average Radial Displacement in the Label text field. 3 Locate the Data section. From the Data set list, choose Ray 1. 4 From the Time selection list, choose Manual. 5 Click Range. 6 In the Integer Range dialog box, type 103 in the Start text field. 7 In the Stop text field, type 118. 8 Click Replace.

Global 1 1 Right-click Average Radial Displacement and choose Global. 2 In the Settings window for Global, locate the y-Axis Data section. 3 In the table, enter the following settings:

Expression Unit Description gop.gopaveop1(sqrt(qx^2+qz^2)) mm

23 | THERMALLY INDUCED FOCAL SHIFT IN HIGH-POWER LASER FOCUSING SYSTEMS Average Radial Displacement 1 In the Model Builder window, under Results click Average Radial Displacement. 2 In the Settings window for 1D Plot Group, locate the Plot Settings section. 3 Select the x-axis label check box. 4 Select the y-axis label check box. 5 In the associated text field, type Average radial displacement. 6 On the Average Radial Displacement toolbar, click Plot. 7 Click the Zoom Extents button on the Graphics toolbar. The plot should now look like Figure 7.

24 | THERMALLY INDUCED FOCAL SHIFT IN HIGH-POWER LASER FOCUSING SYSTEMS Created in COMSOL Multiphysics 5.3

Vdara® Caustic Surface

This model is licensed under the COMSOL Software License Agreement 5.3. All trademarks are the property of their respective owners. See www.comsol.com/trademarks. Introduction

When the Vdara® hotel first opened in Las Vegas, visitors relaxing by the pool would experience intense periods of heat at certain times of the day and at certain times of the year. This intense heat was caused by the reflection of solar radiation from the curved, reflective surface on the South-facing side of the hotel. This model shows how a caustic surface is generated in the pool area around the time and date the problems were first reported.

Note: This application also requires the CAD Import Module.

Figure 1 below shows a small area of the CityCenter® complex which is the subject of this model. The concave surfaces of the Vdara® hotel are illuminated by sunlight, indicated by red arrows, at certain times of the day. The direction of the reflected rays depends on the direction of the incident solar radiation and the surface normal of the hotel.

Solar radiation

Pool area

Aria® Vdara® hotel

Parking lot

Figure 1: A solar flux incident on the concave surface of the Vdara® hotel is reflected down to the pool area beneath.

The Geometrical Optics interface can compute the intensity along individual ray paths by computing the principal radii of curvature of the associated wavefronts. When plane waves are reflected by the surface of the hotel, these principal radii of curvature are changed. When the rays are reflected by a concave surface, the radius of curvature decreases in

2 | VDARA® CAUSTIC SURFACE magnitude thereafter and eventually approaches zero. A continuous set of points at which either principal radius of curvature equals zero is called a caustic surface. In lens systems, the caustic surface often demarcates an envelope of rays. In the limit of geometrical optics, the ray intensity is infinite on a caustic surface. Practically, this corresponds to locations where the incident heat flux is extremely high, which can cause severe burns.

Model Definition

The model geometry includes the Vdara® hotel and several nearby buildings in the CityCenter® complex.

Figure 2: Imported CAD geometry of a section of the CityCenter® complex. The Vdara® hotel is shown at the top.

In order to avoid having to trace rays from the sun onto the surface of the hotel, a special boundary condition called the Illuminated Surface is employed. This boundary condition allows rays to be released from the surface of the hotel directly, significantly reducing the simulation time. The direction at which the rays are released from the surface of the hotel depends on the incoming ray direction vector n and the outward surface normal ns, according to the formula

()⋅ nr = ni – 2 ni ns ns

3 | VDARA® CAUSTIC SURFACE The principal radii of curvature of the released rays are also computed based on the radii of curvature of the incident wavefront and the curvature of the surface of the hotel. More details can be found in the Ray Optics Module User’s Guide.

When the rays arrive at the swimming pool area, the intensity value of each ray is projected onto the surface mesh. This allows for more convenient visualization of the intersection of the caustic surface with the boundary. The Accumulator feature is used to accomplish this by implementing the following equation:

Nt r R δ()rq b =  j – j j = 1 th where Rj is the value of an arbitrary source term for the j incident ray, qj is the position th of the j ray when it strikes the pool area, and rb is the value of the accumulated variable on a given boundary mesh element. Any expression for the source term Rj may be defined; = th for this example, Rj log(Ij) is used, where Ij is the intensity of the j ray.The sum is taken over all rays that reach a given boundary element. The logarithm is used to better visualize changes in the order of magnitude of the ray intensity.

The selections for the boundary conditions are shown below. The curved, reflective surfaces of the hotel that face the sun are shown in orange. The other surfaces of the hotel are shown in gold. The pool and the surrounding area are shown in blue.

Figure 3: Close-up view of the Vdara® hotel.

4 | VDARA® CAUSTIC SURFACE Results and Discussion

The trajectories of the rays can be seen in Figure 4. The rays begin to cross each other after they reflect off the surface of the hotel. The color represents the intensity, which becomes very high at specific locations, indicated by the green and red coloring.

Figure 4: Ray trajectories reflecting off the Vdara® hotel in September at 11:45 am. The arrows indicate the direction vector of the solar radiation.

The projection of the high-intensity regions onto the swimming pool area is plotted in Figure 5. As expected, for this specific time of month and day, there is a clearly visible caustic surface cutting directly across the pool yard.

5 | VDARA® CAUSTIC SURFACE Figure 5: Plot of the log of the intensity projected onto the swimming pool area. There is a region of very high intensity right across the swimming pool.

Reference

1. M. Vollmer and K-P. Möllmann, “Caustic effects due to sunlight reflections from skyscrapers: simulations and experiments,” Eur. J. Phys., vol. 33, pp. 1429–1455, 2012.

Application Library path: Ray_Optics_Module/Building_Science/ vdara_caustic_surface

Modeling Instructions

From the File menu, choose New.

NEW In the New window, click Model Wizard.

6 | VDARA® CAUSTIC SURFACE MODEL WIZARD 1 In the Model Wizard window, click 3D. 2 In the Select Physics tree, select Optics>Ray Optics>Geometrical Optics (gop). 3 Click Add. 4 Click Study. 5 In the Select Study tree, select Preset Studies>Ray Tracing. 6 Click Done.

GEOMETRY 1 1 In the Model Builder window, under Component 1 (comp1) click Geometry 1. 2 In the Settings window for Geometry, locate the Units section. 3 From the Length unit list, choose km. 4 This example uses an imported CAD geometry. Check that CAD kernel is selected from the Geometry representation list.

Import 1 (imp1) 1 On the Home toolbar, click Import. 2 In the Settings window for Import, locate the Import section. 3 Click Browse. 4 Browse to the model’s Application Libraries folder and double-click the file vdara_caustic_surface.x_b. 5 Click Import. Compare the imported geometry to Figure 2.

DEFINITIONS Create a Box selection that contains all of the surfaces of the hotel.

Box 1 1 On the Definitions toolbar, click Box. 2 In the Settings window for Box, type Hotel Surfaces in the Label text field. 3 Locate the Output Entities section. From the Include entity if list, choose Entity inside box. 4 Locate the Geometric Entity Level section. From the Level list, choose Boundary. 5 Locate the Box Limits section. In the x minimum text field, type 0.475. 6 In the x maximum text field, type 0.52.

7 | VDARA® CAUSTIC SURFACE 7 In the y minimum text field, type 0.38. 8 In the y maximum text field, type 0.5. 9 In the z minimum text field, type 0.01. 10 In the zmaximum text field, type 0.2. All of the surfaces of the hotel should be selected, including the orange and gold surfaces in Figure 3.

GEOMETRICAL OPTICS (GOP) 1 In the Model Builder window, under Component 1 (comp1) click Geometrical Optics (gop). 2 In the Settings window for Geometrical Optics, locate the Domain Selection section. 3 Click Clear Selection. 4 Locate the Ray Release and Propagation section. In the Maximum number of secondary rays text field, type 0. 5 Locate the Intensity Computation section. From the Intensity computation list, choose Compute intensity. Selecting the Store ray status data check box causes a variable for the final ray status available for postprocessing; this will be used to filter rays so that only the rays that reach the pool are viewed. 6 Locate the Additional Variables section. Select the Store ray status data check box.

Illuminated Surface 1 1 Right-click Component 1 (comp1)>Geometrical Optics (gop) and choose Illuminated Surface. 2 Select boundaries 351, 356, and 359, the curved surfaces of the hotel that face the sun. These surfaces are colored orange in Figure 3. 3 In the Settings window for Illuminated Surface, locate the Initial Position section. 4 From the Initial position list, choose Density. 5 In the N text field, type 50000. 6 Locate the Ray Direction Vector section. From the Incident ray direction vector list, choose Solar radiation. 7 From the Location defined by list, choose City. 8 In the table, enter the following settings:

Day Month Year 01 9 2014

8 | VDARA® CAUSTIC SURFACE 9 In the table, enter the following settings:

Hour Minute Second 11 45 0

Wall 1 1 In the Model Builder window, right-click Geometrical Optics (gop) and choose Wall. 2 Select boundaries 321, 331, and 345, the pool and the surrounding area. These surfaces are colored blue in Figure 3.

Accumulator 1 1 Right-click Component 1 (comp1)>Geometrical Optics (gop)>Wall 1 and choose Accumulator. 2 In the Settings window for Accumulator, locate the Accumulator Settings section. 3 From the Accumulate over list, choose Rays in boundary elements. 4 In the R text field, type gop.logI. The Source edit field will turn yellow and a tooltip warning will appear, indicating that the deduced unit does not match the expected unit. Fix this by specifying the dependent variable quantity. 5 Locate the Units section. Find the Dependent variable quantity subsection. From the list, choose None. 6 In the Unit text field, type m^-2.

Wall 2 1 In the Model Builder window, right-click Geometrical Optics (gop) and choose Wall. The second Wall condition allows rays to be reflected multiple times at different surfaces of the building. 2 In the Settings window for Wall, locate the Boundary Selection section. 3 From the Selection list, choose Hotel Surfaces. 4 Locate the Wall Condition section. From the Wall condition list, choose Specular reflection.

9 | VDARA® CAUSTIC SURFACE STUDY 1

RESULTS

Ray Trajectories (gop) In the Model Builder window, expand the Ray Trajectories (gop) node.

Filter 1 Plot only the rays with final status gop.fs==2. This is true for all rays that have hit a Wall with the Freeze condition; that is, all rays that have reached the pool. Filtering the rays makes the solution easier to visualize.

1 In the Model Builder window, under Results>Ray Trajectories (gop)>Ray Trajectories 1 click Filter 1. 2 In the Settings window for Filter, locate the Ray Selection section. 3 From the Rays to include list, choose Logical expression. 4 In the Logical expression for inclusion text field, type gop.fs==2.

Ray Trajectories 1 1 In the Model Builder window, under Results>Ray Trajectories (gop) click Ray Trajectories 1. 2 In the Settings window for Ray Trajectories, locate the Coloring and Style section.

10 | VDARA® CAUSTIC SURFACE 3 Find the Point style subsection. From the Type list, choose None. 4 On the Ray Trajectories (gop) toolbar, click Plot. 5 Click the Zoom to Selection button on the Graphics toolbar. Compare the resulting plot to Figure 4.

3D Plot Group 2 1 On the Home toolbar, click Add Plot Group and choose 3D Plot Group. 2 In the Settings window for 3D Plot Group, type Caustic Surface in Pool Area in the Label text field.

Surface 1 1 Right-click Caustic Surface in Pool Area and choose Surface. 2 In the Settings window for Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1>Geometrical Optics> Accumulated variables>Accumulated variable comp1.gop.wall1.bacc1.rpb> gop.wall1.bacc1.rpb - Accumulated variable rpb. 3 Click to expand the Quality section. From the Smoothing list, choose Everywhere. 4 From the Resolution list, choose No refinement. 5 Locate the Coloring and Style section. From the Color table list, choose ThermalEquidistant. Create another Surface plot to display the surfaces of the hotel. 6 On the Caustic Surface in Pool Area toolbar, click Surface.

Surface 2 1 In the Model Builder window, under Results>Caustic Surface in Pool Area click Surface 2. 2 In the Settings window for Surface, click to expand the Title section. 3 From the Title type list, choose None. 4 Locate the Coloring and Style section. From the Coloring list, choose Uniform. 5 From the Color list, choose Gray.

Selection 1 1 Right-click Results>Caustic Surface in Pool Area>Surface 2 and choose Selection. 2 In the Settings window for Selection, locate the Selection section. 3 From the Selection list, choose Hotel Surfaces. 4 On the Caustic Surface in Pool Area toolbar, click Plot.

11 | VDARA® CAUSTIC SURFACE 5 Click the Zoom to Selection button on the Graphics toolbar. Compare the resulting plot to Figure 5.

12 | VDARA® CAUSTIC SURFACE