Terahertz System Analysis Using Zemax Prepared by Andrew Mueller for Ted Stinson and other members of the Basov Infrared Laboratory

I. Introduction The commercial optical design software Zemax was used to model and evaluate the performance of a THz Near Field microscope. In this system, geometric approximations were not valid due to the long of Thz radiation. So by using the Physical Optics Propagation (POP) tool build into the engineering version of Zemax, it was possible to model the beam using a paraxial approximation. The initial goals of this venture are listed below. For the most part, all were at least partially fulfilled, though some conclusions were realized late in the process of building the experiment, or did not easily suggest a revised experiment design.

Goals of the THz system analysis using Zemax:

. Determine the intensity and power of Thz radiation that could make it from the emitter and be focused onto the AFM tip.  Learn what simple modifications to the system’s geometry could yield even better focusing performance at the tip . Discover if the paraxial approximation of Thz radiation would show a situation where the light would not be focused as desired. . Develop a working knowledge of Zemax and its applicability to projects of this type, since it may be useful again in this lab.

Basic Layout

This image shows the basic geometry of the system. A broadband THz signal with ranging from 100um to 3000um is focused onto an AFM tip using parabolic mirrors. Some lengths, like length A and B as shown in the image, were not set to specific values in the Zemax file. The intention was to leave these lengths as variables so that changing them might improve the focus on the AFM tip. However, using the system with a broadband signal showed that it could not be optimized for specific paraxial beam waist locations. What worked best for one wavelength would not translate to others. After working with the system for some time, it appeared the best lengths for A and B were as short as possible, as permitted by hardware and spatial constraints. But the lengths still did not have a large impact on the system's performance. For example, increasing A from 30mm to 130mm caused the radius of the beam at the tip to increase by less than 2% both for 100um and 3000um light. Keeping A and B short has more of an effect on ratio of power that makes it to the tip, because of clipping at the various reflective surfaces. Increasing A from 30mm to 130mm decreased the power transmitted to the tip by less than 5% for wavelengths shorter than 1000um.

Objective of this Report

This report is focused on the modeling of an optical instrument and covers insights gained using Zemax with significance for the project. Though not a general guide on how to use Zemax, it does outline techniques that would be useful for modeling of similar systems. This report would be useful for any projects that involve off-axis layouts, parabolic mirrors, or Gaussian beam models of long wavelength light.

II. System Layout and Geometric Optics

A. System-Specific Layout

1. Sequential vs Non Sequential Zemax has two modes with different features and advantages for building and analyzing off axis systems: Sequential and Non- Sequential. The mode is chosen under the File menu. For sequential mode, the order that surfaces are listed in the Lens Data Editor matters. Rays are only traced from surface 1 to surface 2 and so on. This means rays will not interact with a surface twice and will pass through a surface if it is not the next in the list. Using sequential mode requires a good understanding of exactly how the rays are supposed to propagate. It was mainly used during this investigation because physical optics propagation is only supported for a set of surfaces in this mode.

For Non-Sequential mode, the order of surfaces in the Lens Data Editor does not matter, and rays will interact with any surface they encounter. Also in this mode, the incident rays can be defined in many more ways. As described in a later section, a point source of radiation can be modeled which was used for studying how light would propagate from the tip to the THz detector. Also, more arbitrary 3D surface shapes can be modeled or imported for use in Non-Sequential mode.

2. Aperture Type – Incidence Angle For the sake of viewing the experiment setup in the Zemax 3D layout viewer, correctly tracing some rays through the system can be helpful. However, there is not a strong correlation between how a beam is defined for raytracing and for Physical Optics Propagation. Defining the incident beam in the Zemax general settings only applies to the tracing of rays. Still, we modeled the incident beam geometrically using the Object Cone Angle Feature. This was useful because we did not know or care yet what the stop or smallest aperture was in our system. But we did know, from specifications of the Thz emitter, that most of the beam was inside a cone angle of 12.1 degrees.

3. Using Coordinate Breaks A three dimensional system of mirrors and lenses can be built and analyzed in Zemax while still using Sequential Mode. But because rays are propagated through surfaces in a specific order, the location of surfaces is defined in the context of this order. The location and orientation of a surface is not defined by a single universal coordinate system, but instead by the surface that preceded it. This way of building in 3D can be confusing for those with previous 3d modeling experience.

Coordinate breaks are ‘pseudo-surfaces’ that are added in the Lens Data editor. They are not true surfaces because they have no effect on rays propagated through the system. They only work to modify the coordinates system for the following surface in the Lend Data Editor list. To add a coordinate break, add another surface in the Lens Data editor with the Insert key. Double click the first column entry of this surface to bring up the surface properties. Inside the Surface Type dropdown menu the Coordinate Break option is found. Coordinate Breaks have six parameters as shown here in the Lens data Editor:

When the order parameter is set to zero, coordinate break applies the five transformations in this order: Decenters x and y, tilts about the local Z, tilts about the new Y, and then about the new x. When the order is set to any other number, then the transformations are applied in the opposite order. This can be useful when one needs one coordinate break to put a surface in an arbitrary configuration, and one coordinate break to bring the coordinate system back to what it was before that surface. Finally, the thickness of the coordinate break acts the same way as it does for any other surface. It is the length between the current surface and the next surface (or added after the transformations of a coordinate break). The image shows a simplified 2D representation of the order in which a coordinate break applies transformations.

4. Making custom shape fold mirror The aperture of a mirror or lens is the two dimensional shape that is projected over the surface of the element. The plane of the projection is normal to the (local) optical axis. Rays inside the shape are modified by the lens or mirror as normal; rays outside the shape are clipped or ignored. Essentially the lens or mirror is this shape when viewed along the optical axis. Custom shaped apertures for lenses and mirrors can be defined using a .UDA file which is selected for a particular surface on the Aperture tab of the surface properties menu.

Starting on page 78, the Zemax manual provides an excellent explanation on how to define a custom aperture using a list of User Defined Aperture Entities. Through this method, fold mirrors with one semicircular edge and three straight edges were added to the system.

5. Making an Off Axis Parabolic Mirror Zemax fully supports mirror surfaces, but the technique for making an Off Axis Parabolic Mirror (OAP) is rather involved. The procedure is to make the full parabolic mirror that the final OAP is part of. Then the parts of the mirror that are not needed are ‘cropped’ away.

First, add a Mirror type surface. Enter double the desired mirror focal length as the radius of curvature value. The focal length of a parabolic mirror is the closest distance from the mirror surface to the point of focus.

For a 90 degree off axis parabolic mirror, the distance from the focus to the center of the mirror is double the focal length of the ‘parent’ parabola. So if you know this distance, shown as the Reflected Focal Length in the image, then it can simply be entered in the radius of curvature value to build the parent parabola. In the Thz system, the first collimating OAP has a Parent Focal Length of 25.4mm and a Reflected Focal Length of 50.8mm. -50.8 was used as the Radius value for this surface. The radius is negative just so the mirror is oriented correctly in the system.

Second, enter -1 for the Conic value of the mirror surface in the Lens Data Editor. A parabola has a conic constant of -1. The Semi-Diameter of the mirror is its radius as measured from its optical axis. Make sure the Semi-Diameter is large enough so this ‘parent’ parabola includes the surface of the OAP. For example, if you wish to model a 90 degree OAP with a 100mm reflected focal length and a 30mm Diameter (see image above), then the parent parabola would need to have a Semi-Diameter of at least 115mm .

Third, under the Aperture tab of the surface properties, choose Circular Aperture. Enter the radius of the OAP as the Max Radius. For a 90 degree OAP, the reflected focal length can be entered for either the Aperture X-decenter or Aperture Y-

decenter, depending on where the OAP should be located. This is the distance from the center axis of the parent OAP to the center of the OAP aperture.

B. Angles of beams incident on AFM tip A collimated Thz beam is reflected off of the wide parabolic mirror surrounding the AFM tip. The resulting beam is a cone that focuses on the tip. Angles that describe the orientation of the central ray of this cone can be found. These angles were useful for constructing the entire Thz system layout in Zemax. In most of the Zemax project files used, a mirror was used where the AFM tip would be in the physical system. The mirror worked to reflect incident rays symmetrically so that the rays propagated after this mirror would represent the path of light radiated by the tip and collected by the Thz detector. The orientation of this mirror depended on a number or geometrical factors and was found using the method shown here.

( ) Estimates:

The estimate values in blue are used in a recent Solidworks file of the Thz experiment. They were also used in the Zemax files.

( ) ( ) ( )

True focal length of this OAP:

( ) ( )

( ) ( ) ( ) ( )

( ) ( )( )

This angle is used by the coordinate breaks above and below the TipMirror surface in most project files. It is the angle between the normal of the mirror surface and the YZ plane of the 3D view.

III. Physical Optics Propagation

A. Gaussian beam equations and parameters

1. Rayleigh Range, The Rayleigh range is the distance from the waist to where the beam radius has increased by √ . For a circular beam, it is the distance from the waist to where the area has doubled.

(1.1) ( )

( ) (1.2) ( ) This image of a Gaussian beam visualizes most parameters in the equations.

2. Waist, w

(2.1) ( )

(2.2) ( ) √

3. Divergence Angle

(3.1) ( )

Valid for

B. Power transmitted to tip

1. Reevaluation of incident divergence angle The incident beam was initially modeled in Zemax using the values found in this image provided by the manufacturer of the Thz emitter:

At first, the divergence angle was set to the 12.1 angle from the image. But this led to clipping of a significant portion of the beam at the first OAP because the divergence angle as defined in Zemax is the the 1/e2 size of the beam. That is, the radius of a circle that traces 86% of the beam's total power. It was then decided that the 12.1 degree angle shown in the image is not the divergence angle, but instead an angle that traces the propagation of 99% of the beam's power. The A.E. Seigman book on lasers has a formula [2]:

( )

This implies that the angle tracing 99% of the beams power is related to the true divergence angle by:

( ) ( ) ( )

So the true divergence angle theta is:

(( ) ( ))

A divergence angle of 7.77 was used in the Physical Optics Propagating settings of the Thz system.

2. Separate X, Y and Polarization Turning on Separate X, Y in the POP settings makes Zemax define the beam using x and y coordinates rather than a radial coordinate. This setting allows the program to work better with non-radially symmetric beams. Since the entire Thz system incorporates coordinate breaks and other non-radially symmetric features, it is probably best to leave this setting on when simulating the most accurate propagation.

According to the Zemax manual, turning on the Use Polarization feature will "permit the modeling of effects of optical coatings on the phase and amplitude of the transmitted or reflected beam". Since all the reflective surfaces in the Thz system are gold which reflects very well in the Thz range, the effect of optical coatings on beam propagation was not thoroughly investigated. By keeping the Use Polarization feature turned off, all mirrors reflect perfectly and this assumption was adequate for this project. However, Zemax has the tools to model a system with more physically accurate mirrors if the added accuracy is ever needed. One would make a custom coating.dat file with the parameters specific to that mirror surface.

As shown by data later in this section, turning on the Use Polarization feature in the POP settings with no coating specified on the mirror surfaces will decrease transmitted intensity by a few percent per reflecting surface. This is because Zemax assumes mirrors to be bare aluminum with refractive index 0.7 + 7i – a non-ideal reflecting surface [3]. (A true aluminum surface would have a varying refractive index with wavelength, but in this case Zemax uses these fixed values which translate to about 94% reflectivity at normal incidence1). The default settings for coating in the properties menu of a mirror. The mirror will have refractive index 0.7 + 7.0i 3. Power transmitted to Tip – Initial Data Now with the system set up so Thz radiation is transmitted to the AFM tip through 2 OAP's and one fold mirror, Zemax can predict how much of the source's power will make it to the tip.

The following tables show data at the tip for several wavelengths emitted by the Thz source. The values of greatest interest are probably the Peak Irradiance and Size X,Y. The power initially emitted by the source in our simulation is exactly 1 Watt. This table included data updated since the discovery of the software glitch described in Section C of this chapter. The tilt angle used for the first OAP was 89.999 degrees.

The first column is data defining the Gaussian beam at the tip focus point with both settings described above turned off. Zemax is approximating the beam as radially symmetric throughout the system. Values for Waist X, Size X and Distance X are actually singular radial measurements for this column. The image shows what a non- radial Gaussian beam might look like and visualizes a few of the parameters in the data. However, the beam can also have different values for the Distance parameter in X and Y, which is not shown in the image.

1 It may seem odd that the default no-coating setting would assume a non-ideal reflecting surface. But a Zemax staff member has explained on their forums that this was done to keep users from naively expecting 100% reflection. [4]

100um Off Separate XY Both Power (W) 9.98E-01 9.98E-01 7.90E-01 Peak Irradiance 2.67E+01 2.67E+01 2.06E+01 (W/mm2) Waist X (mm) 1.48E-01 1.49E-01 1.49E-01 Waist Y (mm) 1.48E-01 1.48E-01 Size X (mm) 1.49E-01 1.49E-01 1.49E-01 Size Y (mm) 1.48E-01 1.48E-01 Distance X (mm) -4.20E-02 -3.08E-02 -3.08E-02 Distance Y (mm) -5.31E-02 -5.31E-02

1000um Off Separate XY Both Power (W) 9.89E-01 9.89E-01 7.82E-01 Peak Irradiance 2.78E-01 2.78E-01 2.14E-01 (W/mm2) Waist X (mm) 1.26E+00 1.37E+00 1.37E+00 Waist Y (mm) 1.16E+00 1.16E+00 Size X (mm) 1.43E+00 1.49E+00 1.49E+00 Size Y (mm) 1.41E+00 1.41E+00 Distance X (mm) -2.72E+00 -2.57E+00 -2.57E+00 Distance Y (mm) -2.94E+00 -2.94E+00 Power (W) 9.89E-01 9.89E-01 7.82E-01

3000um Off Separate XY Both Power (W) 7.71E-01 7.64E-01 6.00E-01 Peak Irradiance 3.98E-02 4.00E-02 3.10E-02 (W/mm2) Waist X (mm) 2.55E+00 2.73E+00 2.73E+00 Waist Y (mm) 1.99E+00 1.99E+00 Size X (mm) 3.55E+00 4.50E+00 4.50E+00 Size Y (mm) 3.21E+00 3.21E+00 Distance X (mm) -6.59E+00 -1.02E+01 -1.02E+01 Distance Y (mm) -5.26E+00 -5.26E+00

C. Long wavelength problems

1. Introduction Once the Thz system was set up in Zemax, it was used to calculate the parameters of the Gaussian beam that was projected onto the AFM tip’s location. The system worked as expected for wavelengths below 2000um or so. The majority of the beam’s power would be collected by the first OAP, collimated, and then refocused on the tip. But for longer wavelengths, the power available near the tip dropped by multiple orders of magnitude. It was soon discovered the reason for this decrease was a very poorly collimated beam between the first OAP and fold mirror. The profile of the beam in this ‘collimated zone’ often looked similar to the following image. The beam was far from circular and had a very short Rayleigh range in at least one axis (when using the Separate X,Y feature). A well collimated beam does not change significantly along the direction of travel. For Gaussian beams, this is the case when the Rayleigh range is long compared with the propagation distance [7]. From early on, potential reasons for the poor collimation were categorized as follows:

a) A real-world phenomena Nothing was wrong with Zemax or its calculations and it was giving a valid prediction of the physical Thz system’s performance.

b) An incorrectly set up project file Perhaps the Physical Optics Propagation tool was not being used correctly, or Zemax was reporting relevant errors which were wrongly ignored.

c) An error in Zemax, or a limit of its programming

This seemed more likely if the Profile of the beam in the ‘collimated region’ with a 3000um signal. It problem appeared and disappeared often had this characteristic ‘butterfly’ shape. upon changing what should be inconsequential ways. A computational error also seemed more likely if we were using Zemax in especially unique ways that the original programmers would not have considered.

2. Clipping At first, it seemed one explanation for the collimation problems found for long wavelengths may have involved one of the mirrors clipping the beam. As long as the wavelength is not too large though, it was calculated before that the 1st OAP reflected 99% or more of the incident beam because the 99% angle of 12.1 degrees would project a circle of radius 10.9mm at the distance of 1 focal length (50.8mm). The 1st OAP has a radius of 12.7mm. However, for long wavelengths, the Rayleigh range is longer and the size of the beam does not seem to increase linearly with the divergence angle. The tangent of the 12.1 degrees times the distance from the waist is no longer a good approximation for the size of the beam.

If the dashed lines represent the 12.1 degree 99% angle and the solid black lines the actual beam profile, then one can see how the beam will clip even if the circle traced by the 12.1 degree angle at 1 focal length is within the OAP aperture.

For lambda=100um, the 1st OAP reflects 99.8% of the incident beam. For lambda=3000um, it reflects 96.5% of the beam. The image shows the incident signal falling on the OAP at lambda=3000um. Notice the faint outline of the projection of the mirror.

So yes, less than 99% of the beam is reflected for long wavelengths, but 96.5% is still quite good. It is unlikely that from this small clipping is the cause of behavior seen in the collimated region for long wavelengths.

3. Zemax chooses the incorrect propagator Zemax employs two distinct mathematical methods or propagators for defining the phase of a beam’s electric field and commuting how it changes with distance. The Zemax manual has extensive information on how each propagator works, and when one should be used over the other. Before getting into how they are explicitly used by the program, the manual explains in what theoretical cases one propagator should work better than the other, and these cases depend on the dimensionless Fresnel number:

Where is the radius of of the beam at the first surface, is the distance between the surfaces, and is the wavelength.

The Angular Spectrum Propagator is valid when the Fresnel number between surfaces is large. This is the case when the surfaces are close together. But the propagator also works well over long distances and if the beam does not change size significantly. [1,603]

The Propagator is valid when the Fresnel number is small. It is most appropriate when the beam changes size significantly between one surface and the other. The Zemax manual provides some information on the mathematical methods that these propagators use.

In the case of the Thz system, all regions the beam passes through fulfill the requirements of the Angular Spectrum Propagator reasonably well. Zemax has an option on the Physical Optics tab of a surface’s properties to force use of the Angular Spectrum Propagator:

Checking this option for the first OAP in the Thz system does change results of the POP tool, suggesting Zemax was using the Fresnel Diffraction propagator in the collimated zone. But unfortunately the odd behavior persists while only using Angular Spectrum. For long wavelength light, the beam has a large divergence angle and strange asymmetrical cross section in the collimated zone. So the problem is not related to or not only caused by Zemax automatically choosing an inappropriate propagator.

4. Varying waist vs. varying divergence angle Zemax gives the option of specifying the waist or divergence angle when defining an incident Gaussian beam. If the waist is specified, then the divergence angle of the beam is proportional to its wavelength as shown by equation (3.1) If the divergence angle is specified, then the waist will vary with wavelength. After some deliberation, we finally came to the conclusion that fixing the divergence angle and letting the waist vary would best approximate the way in which the Thz source functioned.

5. Unexpected performance for zR f We tried to change parameters of our project file and discover in what specific situations the error would exist. We observed that if the beam was reflected by the 1st OAP and had a Rayleigh range very close to the focal length of the OAP, then collimation in the 'collimated zone' after the 1st OAP was quite poor. The behavior could be observed while propagating any wavelength of light with any Rayleigh range so long as the focal length of the mirror was adjusted accordingly. This graph shows how the Rayleigh range in the collimate zone after the first OAP changes with a varying mirror focal length. Data was collected in this way:

a) A wavelength and divergence angle for the incident Gaussian beam were chosen. The divergence angle for all data was constant: 7.77 degrees just like in the Thz system.

b) An 90 degree off axis parabolic mirror was added so that it reflected the beam. The reflected focal length of the mirror and distance between the mirror and the beam waist were varied together so that the reflected light would always be perfectly collimated if we were only tracing geometric rays. c) While the reflected focal length was varied over a range of values (from 80% to 120% of the beam’s incident Rayleigh range), the Rayleigh range after the mirror (in the collimated zone) was recorded. d) The process was repeated for several wavelengths of light. Rayleigh Range in Colliamted Zone

1.00E+02

1.00E+01

1.00E+00 100 um 80% 85% 90% 95% 100% 105% 110% 115% 120% 200 um 1.00E-01 500 um 1.00E-02 1000 um 3000 um

1.00E-03 Rayleigh Rayleigh Rangein Colliamted Zone 1.00E-04 Focal Length set to Percentage of Incident Rayleigh Range

The graph shows how the Rayleigh range in the collimated zone decreases by several orders of magnitude if the distance from the incident waist to the OAP is about the Rayleigh range of the incident beam (and the OAP is set up to collimated geometric rays correctly – waist to OAP is equal to reflected focal length). Given this information, it is highly probable the long wavelength issue has some relation to the Rayleigh range of the incident beam. Therefore, it helps to restate our hypotheses as such:

i) Rayleigh range has real-world significance Zemax is correctly modeling a real-world phenomenon related to the Rayleigh range. We could expect to see the results of the graph above in a physical experiment.

ii) Zemax computation methods use the concept of the Rayleigh range If Zemax algorithms calculate and use the Rayleigh range, then the error is due to a sort of discontinuity of processing methods. This is a computational error. a) Real-World significance of the Rayleigh Range The radius of curvature of a beam is minimized one Rayleigh range from its waist, and is equal to twice the Rayleigh range. At the waist, the wavefront is planar, while very far from the waist the wavefront has a radius equal to its distance from the waist (like that of a point source). This is one fact that supports the physical significance of the Rayleigh range. But it does not lend any useful insight into why reflecting the beam near this point would cause the effects seen in Zemax.

The math underlying propagation of a Gaussian beam was investigated. The Rayleigh range of a beam after it has passed through a thin on-axis lens can be found in terms of its original Rayleigh range[8].

( ) z is the distance between the incident waist and the lens; f is the focal length of the lens. We are investigating the case of z = f.

Using the definition of r and M from the image, the equation simplifies to:

Which would simply trace a parabola on the graph above. Clearly this on-axis theoretical model does not describe the unusual performance of Zemax with the OAP.

b) The Rayleigh range in Zemax computations As described above, Zemax will choose to the Angular Spectrum Propagator or the Fresnel Diffraction Propagator based on properties of the beam. The Zemax manual describes how the program chooses a propagator based on the properties of a pilot beam. The pilot beam is a simple Gaussian beam constructed by fitting Gaussian beam parameters to the initial (more complex) distribution at the first surface before a region. Zemax calculates the location of the Rayleigh range of this simple beam. It then uses this information to determine where the two propagators should be used for propagating the real beam and in what order [1,605]. The process is quite a bit more complicated and well described in the manual, but the only relevant knowledge gained is that certain Rayleigh ranges are found and used for intermediate calculations.

6. Fix Two ways to avoid the problem were found. First, under the Physical Optics tab of a surface’s properties, there is an option to Use Rays to Propagate to Next Surface. The problem is not apparent if Zemax traces rays from the mirror to the next surface, and then starts using Angular spectrum propagation after. It is not very clear if using this option will oversimplify the situation and not show us some physically significant result. But if the surface after the 1st OAP is very close to OAP, then the distance over which rays are used is short and possibly inconsequential.

When discussing the ray propagation feature, the Zemax manual states this:

….This is a very desirable property, because geometrical optics may be used to propagate through whole optical components that would be difficult to model with physical optics propagation. These include highly tilted surfaces and gradient index lenses, to name a few. [1,609]

Our 90 degree off axis parabolic mirror may qualify as a highly tilted surface.

Second, changing the tilt of the first coordinate break in the system from -90 to an acute but very close angle, like -89.999, will also cause the problem to disappear. This is persuasive evidence that the phenomena in the collimated zone with long wavelengths was a computational error all along. This is because we would not expect such a minute change of angle to cause such a significant change in the real reflected beam.

References [1] Zemax Development Corporation, "ZEMAX User's Guide" (2009)

[2] A. Siegman, Lasers (University Science Books, Mill Valley, New Edition, 1986), pp.669

[3] Nicholson, M. How is a MIRROR Without a Coating Handled?. Zemax (2007). at

[4] Zemax Staff,. MIRROR reflectivity in NS mode. Zemax Forums (2015). at

[5] Mathar, R. Solid Angle of a Rectangular Plate. (Max-Planck Institute of Astronomy, 2014). at

[6] Tocci, M. Demystifying the Off-Axis Parabola Mirror. Zemax (2006). at https://www.zemax.com/support/knowledgebase/demystifying-the-off- axis-parabola-mirror

[7] Paschotta, R. Collimated Beams. RP Photonics Encyclopedia. at http://www.rp-photonics.com/collimated_beams.html

[8]Saleh, B. Beam Optics. Fundamentals of Photonics (1991). at http://gautier.moreau.free.fr/cours_optique/chapter03.pdf