DOCSLIB.ORG

Jones Calculus in Phase-Only Liquid Crystal Spatial Light Modulators

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Jones Calculus in Phase-Only Liquid Crystal Spatial Light Modulators Appendix Jones Calculus in Phase-Only Liquid Crystal Spatial Light Modulators In Jones calculus, the polarization state of light is represented by a Jones vector, Ex EEE ⇒ (A.1) E y where Ex and Ey are normalized complex amplitude components along x- and y-axes, respectively. An optical element is represented by a Jones matrix, TT11 12 TTT ⇒ , (A.2) TT21 22 and light that impinges on it undergoes the transformation (out) (in) Ex 11 12 ETT x E(out) = TEE=TE (in) ⇒ = . (A.3) (out) TT (in) E y 21 22 E y A phase-only liquid crystal spatial light modulator (SLM) is a two-dimensional array of independently addressable uniaxial birefringent pixels. The pixels are characterized by a common ordinary refractive index, no, along one axis and individually voltage- controllable extraordinary refractive index, ne(V), along an orthogonal axis. To under- stand the various operation modes of an SLM, consider the optical setup shown in Fig. A.1. The SLM orientation is specified by the angle θ between the extraordinary axis and the y-axis. The Jones matrix for an SLM oriented with its extraordinary axis parallel to the y-axis (θ=0) is 2π expj dno 0 (0) λ TSLM = . (A.4) 2π 0 exp dne () x , y λ 306 Appendix: Jones Calculus in Phase-Only Liquid Crystal Spatial Light Modulators where d is the thickness of the SLM cell and the index modulation ne ( x, y) is controlled by the applied pixel voltages V= V( x, y) . To study the output polarization states, it is convenient to write the unrotated Jones matrix as 1 0 2π TTTSLM = expj dno , λ 0 expδ ()x , y 2π where δ ()x, y= d n() x, y− n describes a spatially varying phase retardation. λ e o Each SLM pixel acts as a programmable phase retarder, enabling various polarization states to be encoded on an incident light. Using this alternate expression with the rotation matrix, Rθ, we can apply a rotational transformation to get the new SLM Jones matrix for an arbitrary θ ()θ -1-1-1 (0) TRTRSLM= θ SLM θ 2 2 cosθ+ sin θ expj() δ sin θ cos θ− 1 + expj() δ . (A.5) = exp jφ ()o 2 2 sinθ cos θ− 1 + expj() δ sin θ+ cos θ expj() δ Note that the uniform phase offset, φ0 =(2 π λ )dno , which does not affect the polariza- tion, is commonly neglected in Jones calculus. polarization axis y x SLM axis θθθ Input light Polarizer SLM FigFigFig A.1 Optical setup for realizing different SLM operation modes. A linearly polarized (along the y-axis) light is incident on a spatial light modulator (SLM). The director axis of the liquid crystal molecules in the SLM is oriented at an angle, θ, with the incident polarization. A.1 Spatial Phase Modulation Consider a vertically polarized input light ()in 0 E = , (A.6) 1 A.2 Spatial Polarization Modulation 307 incident on an SLM oriented at θ=0. Using the SLM Jones matrix in Eq. (A.4), the output after SLM is a new vector EEE (out) given by 0 ()out (0)(0)(0) ()in ETE=SLM = exp jφ ()x , y . (A.7) 1 This is a vertically polarized output that contains a spatial phase modulation, 2π φ ()x, y= dne () x, y . (A.8) λ Thus, the SLM operates as a spatial phase-only modulator when its extraordinary axis is aligned with the incident polarization. A.2 Spatial Polarization Modulation o For an SLM oriented at θ = 45 , the rotationally transformed TSLM in Eq. (A.4) becomes 1+ expjδ 1 − exp j δ −1 1 ( ) ( ) RTR()()θSLM θ= exp()j φo . (A.9) 2 1− exp()()jδ 1 + exp j δ Using the same vertically polarized input EEE(in), the output field EEE(out) is − δ (out ) (0) 0 1 1 exp( j ) EET= TETSLM = exp()jφo 1 2 1+ exp()jδ (A.10) sinδ x , y /2 2φo + δ− π ()() = exp j . 2 jcos()δ ()x , y /2 The uniform phase offset, which is not relevant to the polarization state, can be neglected and we can describe the output polarization simply as sin(δ ( x , y) /2) EEE ()out = (A.11) jcos()δ ()x , y /2 In spatial amplitude modulation, an output polarizer isolates either the vertical or horizontal component of Eq. (A.11). Spatial polarization modulation utilizes the full output with both components. Spatially programmed polarization is achieved by controlling the spatial phase retardation of the SLM. Output polarization vectors for some values of δ are illustrated in Table A.1. Other δ -values yield elliptic polarization states with major axis aligned either along the horizontal or the vertical. 308 Appendix: Jones Calculus in Phase-Only Liquid Crystal Spatial Light Modulators Table A.A.A.1A.111 Polarization vector in Eq. (A.11) for various retardation, δ. δ ( x, y) E ()out 0 0 j ; 1 1 1 π ; /2 2 j 1 π ; 0 1 1 ; 3π/2 2 − j 0 − j ; 2π 1 Table A.2 A graphical representation of the output polarization ellipticity and orientation as a function of the phase shift implemented in corresponding regions of SLM-1 and SLM-2 A.3 Spatial Polarization Modulation with Arbitrary Axis 309 A.3 Spatial Polarization Modulation with Arbitrary Axis A polarizer and SLM can generate elliptically polarized light with major axis that can be vertical or horizontal. It is possible to generate an arbitrary state of polarization by taking this elliptically polarized light and rotating its major axis to a desired arbitrary orientation. This rotation can be realized using SLMs and two quarter wave plates, and the full polarization encoding system is schematically illustrated in Fig. A.2. Both SLMs are oriented at θ=45°. Elliptical generator (T1) Elliptical rotator (T2) 2D polarization Input light encoded light Polariser SLM-1 /4 SLM-2 /4 λ λ Fig. A.2Fig. A.2 An optical system for converting incident polarized light into an arbitrary state of elliptically polarized light with the major axis of the elliptically polarized light rotated an arbitrary angle. The lines denote the extraordinary axis of the SLMs, the quarter wave plates (λ/4) and the polarization direction of the linear polarizer. To verify the axis rotation, let us write the transformation matrix for the two quarter wave plates and SLM: 1 0 cos(δ2 (x ,/2 y) ) − j sin(δ2 ( x ,/2 y) ) 1 0 T = rotator 0 − j 0 j − jsin()δ2 () x , y /2 cos()δ2 ()x , y /2 (A.12) cos()δ2 ()x , y /2 sin()δ ()x , y /2 = . −sin()δ ()x , y /2 cos()δ2 ()x , y /2 This result shows that TTTrotator is, indeed, a rotation matrix with a rotation angle of φ2(x,y)/2. The net transformation matrix for the optical system in Fig. A.2 is (π /4) RTδ2 /2 SLM cos()δ2 ()x ,/2 y sin()δ2 ()x ,/2 y cos()δ1 ()x ,/2 y− j sin()δ1 () x ,/2 y = − δ δ − δ δ sin()2 ()x , y /2 cos()2 ()x , y /2 j sin()1 () x , y /2 cos()1 ()x , y /2 cos()δ1 ()x , y /2cos()δ2 ()x , y /2− j sin()δ1 () x , y /2sin()δ2 ()x , y /2 = . δ δ δ δ jsin()1 () x , y /2sin()() x , y /2 cos()1 ()x , y /2cos()2 ()x , y /2 (A.13) 310 Appendix: Jones Calculus in Phase-Only Liquid Crystal Spatial Light Modulators Output polarization states from this arbitrary polarization encoding system are graphically illustrated in Table A.2. The phase retardations of SLM-1 and SLM-2 determine the type and direction of the output polarization. It is interesting to note that the direction of the elliptically polarized light changes from left handed to right handed when the phase modulation of SLM-1 is above π. This table shows that it is possible to generate an arbitrary state of elliptical polarization if both SLM-1 and SLM-2 can produce a phase modulation of at least 2π. Experimental demonstrations of arbitrary polarization encoding are available in Ref. [1]. Reference 1. R. L. Eriksen, P. C. Mogensen, and J. Glückstad, “Elliptical polarisation encoding in two dimensions using phase-only spatial light modulators,” Opt. Commun. 187187187,187 325-336 (2001). Index 3D optical trapping and manipulation, 176 Brownian motion, 200 4f system, 80, 106, 109, 183, 220, 225, 229, 233, chrome-on-glass mask, 256, 259 242, 244, 257, 264, 277, 285, 287, 288, circ-function, 181, 249, 286 289, 291, 293 colloidal constellations, 176 filtering, 157, 287, 288 colour-based sorting, 156 combined filter parameter, 2, 27, 30–32, 36–38, aberration, 1, 35, 176, 186, 233, 235, 255, 278, 40, 43–44, 46, 250–254, 257, 286, 288, 300 280 common-path interferometer (CPI), 1–4, 8, 14– Airy disc, 51 24, 27–32, 35–59, 61, 235, 248–255, 263, Airy function, 29 296, 300–304 annular intensity profile, 156, 159, 161 pair configuration, 48 aperture matching, 257 types, 2, 23, 300 Argand diagram, 250, 251–252, 255, 260–261, common-path interferometry, 1, 3, 8, 258 287–288 complex conjugate, 237, 274 array illumination, 220, 243 complex filter space, 37, 41, 58, 300 array illuminator, 221 plot, 37, 41, 58, 300 artefacts, 51 compression factor, 237, 243 Ashkin, Arthur, 151, 156, 167, 180, 203 compromise filter, 39 automation, 191, 196, 197, 212 computational axial and transverse stiffness, 209 cost, 58 axial control, 168, 171–172, 175, 211 load, 163 axial dependency of the trap stiffness, 194 overhead, 176, 238 axial dynamic range of manipulation, 209 computer-controlled polarization landscape, 174 axial force curve, 203, 207 computer-generated holography (CGH), 85–95, axial manipulation, 209–210 119, 123, 240 concurrent top- and side-view imaging, 178–179 bacteriorhodopsin, 218, 224–225 confinement stress, 165 binary coupling grating, 277, 280 constant intensity criterion, 250, 252 binary diffractive optical element, 234 contrast, see generalized phase contrast (GPC) binary phase mask, 288 phase contrast, reverse phase contrast (RPC) binary phase modulation, 247, 253, 263, 288 convolution, 51, 238 binary phase patterns counter-propagating beam trap, 168, 169 decrypting by amplitude, 296 C-parameter, 45 biologically
Load more

Recommended publications
  • Polarization Optics Polarized Light Propagation Partially Polarized Light
    The physics of polarization optics Polarized light propagation Partially polarized light Polarization Optics N. Fressengeas Laboratoire Mat´eriaux Optiques, Photonique et Syst`emes Unit´ede Recherche commune `al’Universit´ede Lorraine et `aSup´elec Download this document from http://arche.univ-lorraine.fr/ N. Fressengeas Polarization Optics, version 2.0, frame 1 The physics of polarization optics Polarized light propagation Partially polarized light Further reading [Hua94, GB94] A. Gerrard and J.M. Burch. Introduction to matrix methods in optics. Dover, 1994. S. Huard. Polarisation de la lumi`ere. Masson, 1994. N. Fressengeas Polarization Optics, version 2.0, frame 2 The physics of polarization optics Polarized light propagation Partially polarized light Course Outline 1 The physics of polarization optics Polarization states Jones Calculus Stokes parameters and the Poincare Sphere 2 Polarized light propagation Jones Matrices Examples Matrix, basis & eigen polarizations Jones Matrices Composition 3 Partially polarized light Formalisms used Propagation through optical devices N. Fressengeas Polarization Optics, version 2.0, frame 3 The physics of polarization optics Polarization states Polarized light propagation Jones Calculus Partially polarized light Stokes parameters and the Poincare Sphere The vector nature of light Optical wave can be polarized, sound waves cannot The scalar monochromatic plane wave The electric field reads: A cos (ωt kz ϕ) − − A vector monochromatic plane wave Electric field is orthogonal to wave and Poynting vectors
    [Show full text]
  • Outline • Types of Polarization • Jones' Matrices • Birefringence • Polarizing Optical Components • Polarization in Scattering
    Polarization Ivan Bazarov Cornell Physics Department / CLASSE Outline • Types of polarization • Jones’ matrices • Birefringence • Polarizing optical components • Polarization in scattering 1 Polarization P3330 Exp Optics FA’2016 Polarization ellipse For light traveling along z direction: k E = vB ⇥ with a complex amplitude: The electric field traces out an ellipse: 2 Polarization P3330 Exp Optics FA’2016 Polarization ellipse Polarization types: • linearly polarized light • circularly polarized light • unpolarized light (non-laser) 3 Polarization P3330 Exp Optics FA’2016 Linear & circular polarizations linearly polarized light circularly polarized light Right CCW Left CW 4 Polarization P3330 Exp Optics FA’2016 Unpolarized light? Unpolarized light means random (time-changing) polarization direction, e.g. excited atoms in a solid (a light bulb) emit randomly polarized light packets. 5 Polarization P3330 Exp Optics FA’2016 Jones vector We can represent any monochromatic wave polarization as a Jones’ vector: For normalized intensity : E.g. orthogonal polarizations whenever J0 J =0 1 · 2 6 Polarization P3330 Exp Optics FA’2016 Exercises Ex1: Check that HLP and VLP are orthogonal as well as RCP and LCP. Q: What does it mean? A: can use either as a basis to represent arbitrary polarization! Ex2: How to obtain the light intensity from its Jones vector (if in vacuum)? Ax 2 2 J0 J =[Ax⇤ Ay⇤] = Ax + Ay · Ay | | | | 7 Polarization P3330 Exp Optics FA’2016 Jones matrix (don’t work with unpolarized light!) Normal modes: TJ = µJ 8 Polarization P3330 Exp Optics FA’2016 Combining polarization devices & tilt Simply multiply the matrices in the reverse order: T1 T2 T3 T = T3T2T1 If polarization device is rotated, use: Proof: using , we get 9 Polarization P3330 Exp Optics FA’2016 Linear polarizer Linear polarizer in x-direction A polarizer rotated by angle q 10 Polarization P3330 Exp Optics FA’2016 Polarization retarders or wave plates These devices do not affect one polarization component (fast axis) but add a retarding phase to the other component (slow axis).
    [Show full text]
  • Instrumental Optics
    Instrumental optics lecture 3 13 March 2020 Lecture 2 – summary 1. Optical materials (UV, VIS, IR) 2. Anisotropic media (ctd) • Double refraction 2. Anisotropic media • Refractive index: no, ne() • Permittivity tensor • Birefringent materials – calcite, • Optical axis YVO4, quartz • Ordinary wave Polarization perpendicular to o.a.-푘 plane • Extraordinary wave Polarization in o.a.-푘 plane • Birefringent walk-off 3. Polarization of light • Wave vector ellipsoid • states of polarization Polarization of light linear circular elliptical 휋 휋 Δ휙 = 0, sin Θ = 퐸 /퐸 Δ휙 = , 퐸 = 퐸 Δ휙 ∈ (0, ) 푦 푥 2 푥 푦 2 Plane wave, 푘||푧, 푧 = const: • Fully polarized light 푖(휔푡+휙푥) 퐸푥(푡) 퐸0푥푒 퐸0푥 • Partially polarized light 푖(휔푡+휙 ) i(휔푡+휙푥) iΔ휙 퐸 푡 = 퐸푦(푡) = 퐸 푒 푦 =e 퐸 e • Fully depolarized light 0푦 0푦 0 0 0 Δ휙 = 휙푦 − 휙푥 Note: electric polarization 푃(퐸) vs. polarization of a wave! Linear polarizers Key parameter: Other parameters: 퐼 extinction coefficient, 푦 • Spectral range, 퐼푥 • damage threshold, (assumption: incident light is • angle of incidence, unpolarized!) • beam geometry For polarizing beamsplitters: 2 extinction coefficients, reflection typically worse • clear aperture Polarizer types: • Dichroic • Crystalline • Thin-film Dichroic polarizers Dichroic polarizer: • Made from material with anisotropic absorption (or reflection) coefficient long, „conductive” molecules/structures • Polaroid: polyvinyl alcohol + iodine – one polarization component is absorbed. • Cheap, large aperture, broadband, poor extinction, low damage threshold • Wire grid / nanowire:
    [Show full text]
  • Lab #6 Polarization
    EELE482 Fall 2014 Lab #6 Lab #6 Polarization Contents: Pre-laboratory exercise 2 Introduction 2 1. Polarization of the HeNe laser 3 2. Polarizer extinction ratio 4 3. Wave Plates 4 4. Pseudo Isolator 5 References 5 Polarization Page 1 EELE482 Fall 2014 Lab #6 Pre-Laboratory Exercise Bring a pair of polarized sunglasses or other polarized optics to measure in the lab. Introduction The purpose of this lab it to gain familiarity with the concept of polarization, and with various polarization components including glass-film polarizers, polarizing beam splitters, and quarter wave and half wave plates. We will also investigate how reflections can change the polarization state of light. Within the paraxial limit, light propagates as an electromagnetic wave with transverse electric and magnetic (TEM) field directions, where the electric field component is orthogonal to the magnetic field component, and to the direction of propagation. We can account for this vector (directional) nature of the light wave without abandoning our scalar wave treatment if we assume that the x^ -directed electric field component and y^ -directed electric field component are independent of each other. This assumption is valid within the paraxial approximation for isotropic linear media. The “polarization” state of the light wave describes the relationship between these x^ -directed and y^ -directed components of the wave. If the light wave is monochromatic, the x and y components must have a fixed phase relationship to each other. If the tip of the electric field vector E = Ex x^ + Ey y^ were observed over time at a particular z plane, one would see that it traces out an ellipse.
    [Show full text]
  • Modeling and Measuring the Polarization of Light: from Jones Matrices to Ellipsometry
    MODELING AND MEASURING THE POLARIZATION OF LIGHT: FROM JONES MATRICES TO ELLIPSOMETRY OVERALL GOALS The Polarization of Light lab strongly emphasizes connecting mathematical formalism with measurable results. It is not your job to understand every aspect of the theory, but rather to understand it well enough to make predictions in a variety of experimental situations. The model developed in this lab will have parameters that are easily experimentally adjustable. Additionally, you will refine your predictive models by accounting for systematic error sources that occur in the apparatus. The overarching goals for the lab are to: Model the vector nature of light. (Week 1) Model optical components that manipulate polarization (e.g., polarizing filters and quarter-wave plates). (Week 1) Measure a general polarization state of light. (Week 1) Model and measure the reflection and transmission of light at a dielectric interface. (Week 2) Perform an ellipsometry measurement on a Lucite surface. WEEK 1 PRELIMINARY OBSERVATIONS Question 1 Set up the optical arrangement shown in Figure 1. It consists of (1) a laser, (2) Polarizing filters, (3) A quarter-wave plate (4) A photodetector. Observe the variation in the photodetector voltage as you rotate the polarizing filters and/or quarter-wave plate. This is as complicated as the apparatus gets. The challenge of week 1 is to build accurate models of these components and model their combined effect in an optical system. An understanding of polarized light and polarizing optical elements forms the foundation of many optical applications including AMO experiments such as magneto-optical traps, LCD displays, and 3D projectors.
    [Show full text]
  • OPTI 484/584: Polarized Light and Polarimetry Effective Spring 2015 Replaces: OPTI 623 Course Description
    OPTI 484/584: Polarized Light and Polarimetry Effective Spring 2015 Replaces: OPTI 623 Course Description: Polarized light and the Poincare sphere. Polarization in natural scenes and animal vision. Polarization elements: polarizers, retarders, and depolarizers. Jones and Mueller polarization calculus. Polarimetry: measuring the polarization properties of optical elements and materials. Polarization modulators and controllers. Polarization dependent loss and polarization mode dispersion in fiber optics. Advanced polarization issues in optical devices and systems. Pre-requisites: OPTI 501: Electromagnetic Waves/OPTI 330: Physical Optics II, or equivalent Number of Units/ component: 3 units/ lecture Locations and Times: T/R: 2:00 – 3:15 pm, Meinel Optical Sciences room 305 Instructor Information: Russell A. Chipman, Professor College of Optical Sciences, Room 737 The University of Arizona 1630 East University Boulevard Tucson, AZ 85721 Phone: 520-626-9435 [email protected] Expected Learning Outcomes: 1. Understand the relationship and differences between the Jones vector and Stokes parameter descriptions for polarized light. 2. Explain the operation of polarizers and retarders. 3. Be able to measure Stokes parameters and interpret Stokes images. 4. Understand the difference between linear retarders, elliptical retarders, and circular retarders and their Jones and Mueller matrices. 5. Describe five ways to electrically control polarization state. 6. Explain how liquid crystal displays can be fabricated from Twisted Nematic cells and VAN-mode cells. Required Texts: Course Notes: The instructor’s notes are used as the text and will be distributed periodically throughout the term. No other texts are required. Course notes are not available in electronic form. Notes are not to be copied or scanned into electronic form.
    [Show full text]
  • Constraints on Jones Transmission Matrices from Time-Reversal
    Constraints on Jones transmission matrices from time-reversal invariance and discrete spatial symmetries N.P. Armitage1 1The Institute for Quantum Matter, Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA (Dated: May 23, 2014) Optical spectroscopies are most often used to probe dynamical correlations in materials, but they are also a probe of symmetry. Polarization anisotropies are of course sensitive to structural anisotropies, but have been much less used as a probe of more exotic symmetry breakings in ordered states. In this paper a Jones transfer matrix formalism is discussed to infer the existence of exotic broken symmetry states of matter from their electrodynamic response for a full complement of possible broken symmetries including reflection, rotation, rotation-reflection, inversion, and time- reversal. A specific condition to distinguish the case of macroscopic time-reversal symmetry breaking is particularly important as in a dynamical experiment like optics, one must distinguish reciprocity from time-reversal symmetry as dissipation violates strict time-reversal symmetry of an experiment. Different forms of reciprocity can be distinguished, but only one is a sufficient (but not necessary) condition for macroscopic time-reversal symmetry breaking. I show the constraints that a Jones matrix develops under the presence or absence of such symmetries. These constraints typically appear in the form of an algebra relating matrix elements or overall constraints (transposition, unitarity, hermiticity, normality, etc.) on the form of the Jones matrix. I work out a number of examples including the trivial case of a ferromagnet, and the less trivial cases of magnetoelectrics and vector and scalar spin “chiral” states.
    [Show full text]
  • Polarized Light
    Subscriptions Newsletter Site Demo Librarians Help/FAQs About Contributors Consulting Editors Contact Log In Advanced Search About Search Physics > Electromagnetic radiation > Physics > Optics > For further study: Print View | Email a Link | Bookmark Polarized light ENCYCLOPEDIA ARTICLES Dynamic nuclear polarization Contents [Hide] The creation of assemblies of nuclei - Types of polarized light - Analyzing devices whose spin axes are not oriented at - Law of Malus - Retardation theory - Linear polarizing devices - Bibliography random, and which are in a steady - Polarization by scattering state that is not a state of thermal...... - Production of polarized light Read Article View all 3 related articles... Light which has its electric vector oriented in a predictable fashion with respect to the propagation direction. In unpolarized light, the vector is oriented in a random, unpredictable fashion. Even in short time intervals, it appears to RESEARCH UPDATES be oriented in all directions with equal probability. Most light sources seem to be partially polarized so that some Frozen light fraction of the light is polarized and the remainder unpolarized. It is actually more difficult to produce a completely unpolarized beam of light than one which is completely polarized. Ultraslow light was first observed by L. V. Hau and her colleagues in 1998. The polarization of light differs from its other properties in that human sense organs are essentially unable to detect Light pulses were slowed in a the presence of polarization. The Polaroid Corporation with its polarizing sunglasses and camera filters has made Bose-Einstein condensate of sodium to millions of people conscious of phenomena associated with polarization. Light from a rainbow is completely linearly only.....
    [Show full text]
  • Generalized Jones Matrix Method for Homogeneous Biaxial Samples Noe Ortega-Quijano, Julien Fade, Mehdi Alouini
    Generalized Jones matrix method for homogeneous biaxial samples Noe Ortega-Quijano, Julien Fade, Mehdi Alouini To cite this version: Noe Ortega-Quijano, Julien Fade, Mehdi Alouini. Generalized Jones matrix method for homoge- neous biaxial samples. Optics Express, Optical Society of America - OSA Publishing, 2015, 23 (16), pp.20428-20438. 10.1364/OE.23.020428. hal-01181354 HAL Id: hal-01181354 https://hal.archives-ouvertes.fr/hal-01181354 Submitted on 8 Aug 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Generalized Jones matrix method for homogeneous biaxial samples Noe´ Ortega-Quijano,∗ Julien Fade, and Mehdi Alouini Institut de Physique de Rennes, CNRS, Universite´ de Rennes 1, Campus de Beaulieu, 35 042 Rennes, France ∗[email protected] Abstract: The generalized Jones matrix (GJM) is a recently introduced tool to describe linear transformations of three-dimensional light fields. Based on this framework, a specific method for obtaining the GJM of uniaxial anisotropic media was recently presented. However, the GJM of biaxial media had not been tackled so far, as the previous method made use of a simplified rotation matrix that lacks a degree of freedom in the three-dimensional rotation, thus being not suitable for calculating the GJM of biaxial media.
    [Show full text]
  • Polarization of Light Thursday, 11/09/2006 Physics 158 Peter Beyersdorf
    Polarization of Light Thursday, 11/09/2006 Physics 158 Peter Beyersdorf Document info 1 Class Outline Polarization of Light Polarization basis’ Jones Calculus 16. 2 Polarization The Electric field direction defines the polarization of light Since light is a transverse wave, the electric field can point in any direction transverse to the direction of propagation Any arbitrary polarization state can be considered as a superposition of two orthogonal polarization states (i.e. it can be described in different bases) 16. 3 Electric Field Direction Light is a transverse electromagnetic wave so the electric (and magnetic) field oscillates in a direction transverse to the direction of propagation Possible states of electric field polarization are Linear electric field Circular plane wave Elliptical Random 4 Examples of polarization states right hand circular horizontal (CW as seen from observer) left hand circular vertical (CCW as seen from observer) linear polarization at an elliptical arbitrary angle 16. 5 Linear Polarization Basis Any polarization state can be described as the sum of two orthogonal linear polarization states i(kz ωt+φx) Ex(z, t) = E0xˆie − i(kz ωt+φy ) Ey(z, t) = E0yˆje − ! iφxˆ iφy ˆ i(kz ωt) E(z, t) = Ex(z,!t) + Ey(z, t) = E0xe i + E0ye j e − " # ! !E0y=0 ! E0x=0 E0y=E0x E0y=-E0x y y y y x x x horizontal vertical diagonal diagonal φ φ φx=φy+π x= y 16. 6 Circular Polarization iφx iφy i(kz ωt) E(z, t) = E0xe ˆi + E0ye ˆj e − " # Fo!r the case |φx-φy|=π/2 the magnitude of the field doesn’t change, but the direction sweeps out a circle The polarization is said Left-Handed Right-Handed to be right-handed if y y it progresses clockwise as seen by an observer looking into the light.
    [Show full text]
  • PO18 Physical Optics 7 Polarization.Pdf
    Physical Optics Lecture 7: Polarization 2018-05-23 Herbert Gross www.iap.uni-jena.de 2 Physical Optics: Content No Date Subject Ref Detailed Content Complex fields, wave equation, k-vectors, interference, light propagation, 1 11.04. Wave optics G interferometry Slit, grating, diffraction integral, diffraction in optical systems, point spread 2 18.04. Diffraction G function, aberrations Plane wave expansion, resolution, image formation, transfer function, 3 25.04. Fourier optics G phase imaging Quality criteria and Rayleigh and Marechal criteria, Strehl ratio, coherence effects, two-point 4 02.05. G resolution resolution, criteria, contrast, axial resolution, CTF Energy, momentum, time-energy uncertainty, photon statistics, 5 09.05. Photon optics K fluorescence, Jablonski diagram, lifetime, quantum yield, FRET Temporal and spatial coherence, Young setup, propagation of coherence, 6 16.05. Coherence K speckle, OCT-principle Introduction, Jones formalism, Fresnel formulas, birefringence, 7 23.05. Polarization G components Atomic transitions, principle, resonators, modes, laser types, Q-switch, 8 30.05. Laser K pulses, power Basics of nonlinear optics, optical susceptibility, 2nd and 3rd order effects, 9 06.06. Nonlinear optics K CARS microscopy, 2 photon imaging Apodization, superresolution, extended depth of focus, particle trapping, 10 13.06. PSF engineering G confocal PSF Introduction, surface scattering in systems, volume scattering models, 11 20.06. Scattering L calculation schemes, tissue models, Mie Scattering 12 27.06. Gaussian beams G Basic description, propagation through optical systems, aberrations Laguerre-Gaussian beams, phase singularities, Bessel beams, Airy 13 04.07. Generalized beams G beams, applications in superresolution microscopy 14 11.07. Miscellaneous G Coatings, diffractive optics, fibers K = Kempe G = Gross L = Lu 3 Contents .
    [Show full text]
  • PDF (Chapter 1. Electromagnetic Theory)
    1 Electromagnetic Theory 1.0 INTRODUCTION In this chapter we derive some of the basic results concerning the propagation of plane, single-frequency, electromagnetic waves in homogeneous isotropic media, as well as in anisotropic crystal media. Starting with Maxwell's equa- tions we obtain expressions for the dissipation, storage, and transport of energy resulting from the propagation of waves in material media. We con­ sider in some detail the phenomenon of birefringence, in which the phase velocity of a plane wave in a crystal depends on its direction of polarization. The two allowed modes of propagation in uniaxial crystals—the "ordinary" and "extraordinary" rays—are discussed using the formalism of the index ellipsoid. 1.1 COMPLEX-FUNCTION FORMALISM In problems that involve sinusoidally varying time functions, we can save a great deal of manipulation and space by using the complex-function for­ malism. As an example consider the function (1.1-1) where ω is the circular (radian) frequency1 and Φa is the phase. Defining the complex amplitude of a(t) by (1.1-2) 1The radian frequency ω is to be distinguished from the real frequency v = ω/2π. 1 2 ELECTROMAGNETIC THEORY we can rewrite (1.1-1) as (1.1-3) We will often represent a(t) by (1.1-4) instead of by (1.1-1) or (1.1-3). This, of course, is not strictly correct so that when this happens it is always understood that what is meant by (1.1-4) is the real part of Aexp(iωt). In most situations the replacement of (1.1-3) by the complex form (1.1-4) poses no problems.
    [Show full text]