Department of Electrical and Computer Engineering University of Colorado at Boulder OPTICS LAB -ECEN 5606 Kelvin Wagner KW&K.Y. Wu 1994 KW&S.Kim 2007 Experiment No. 12 POLARIZATION and CRYSTAL OPTICS 1 Introduction Crystal optics studies the propagation of electromagnetic waves in anisotropic media and uses these affects as key components for manipulating, controlling, and analyzing the state of polarization. In this lab you will learn about the propagation of light through anisotropic dielectric crystals, and how to use the phenomena of birefringence and dichroism to manipulate the state of polarization of an optical wave. In addition you will learn how to determine the optical axes of an anisotropic crystal by using polarization holography to generate conoscopic interference figures. 2 2.1 Background Crystal Optics ~ is parallel to the electric field vector E, ~ In isotropic materials, the electric displacement vector D −12 ~ ~ ~ ~ related by D = ǫ0 ǫr E = ǫ0 E + P, where ǫ0 = 8.854 × 10 F/M in MKS and ǫr is the unit-less ~ is the material polarization vector. In an anisotropic crystal, relative dielectric constant, and P ~ and E ~ are no longer parallel and their relationship is determined by the tensor the vectors D ~ ~ relation, D = ǫ0 ǫr E, where ǫr is the relative dielectric susceptibility tensor, and the dependence of ~ on E ~ is P ~ = ǫ0 χE, ~ where χ is the unit-less susceptibility tensor. For non-absorbing polarization P crystals, there always exists a set of orthogonal coordinate axes (note that in low symmetry crystals these principal axis may be a function of temperature or wavelength), called principal dielectric axes, such that both the ǫ and the χ tensors assumes a diagonal form: ǫ11 0 0 χ11 0 0 ǫij = 0 ǫ22 0 χij = 0 χ22 0 0 0 ǫ33 0 0 χ33 The principal dielectric constants are ǫii = 1 + χii , and the principal indices of refraction experi√ enced by a wave purely polarized along the i principal axes are ni = ǫii (but note the refractive ~ and P ~ are not necessarily parallel to E, ~ but are parallel for principal index is not a tensor). D axes polarizations. This tensorial relation causes the index of refraction to vary with the direction of propagation and the state of polarization. When, χ11 = χ22 6= χ33 , the crystal is called uniaxial, and has a rotational symmetry of the 2nd rank properties around the 3 = z axis. Most of the crystals used in this lab and in most optical experiments are uniaxial. In a uniaxial crystal, the 1 z (0,0,n e) z k k ne θ De (0,no,0) x n e( θ) no x Do Figure 1: Optical indicatrix for a uniaxial crystal. For each direction of propagation, ~k the orthogonal ~e elliptical plane can be diagonalized to find the eigendirections for the electric displacement vectors, D ~ and Do , and the magnitudes of the index of refractions, no and ne (θ). unique element of the susceptibility tensor determines the optic axis, and by convention is always aligned in the 3 = z position. If the extraordinary index is less than (greater than) that along the two orthogonal axes, the crystal is termed negative(positive) uniaxial. The crystal structure determines the degree and nature of the anisotropy. As light refracts into an anisotropic dielectric medium it is resolved into two orthogonally polarized eigenmodes of polarization, which are characterized by two independent velocities of phase propagation. Arbitrary wavefronts can be Fourier resolved into a basis of plane wave components each propagating in a specific direction, and the double refraction of each plane wave component into the crystal can be used to decribe the properties of birefringent refraction. The simplest case to cosider is a uniaxial crystal that has a single optical axis of rotational symmetry along z. An ordinary eigenmode has its electric-field vector polarized normal to the optic axis, êo · ẑ = 0, and due to this symmetry has a phase velocity independent of direction, just as in an isotropic media. An extraordinary ray is an eigenmode whose electric-field vector is polarized in the same plane as ǫêe the optic axis, d̂e = |ǫê , d̂e · k̂ = 0 = d̂e · êo, and due to the anisotropy experiences a phase veloce| ity which depends on direction of propagation. Now suppose an unpolarized beam (or circularly polarized beam) is incident on an anisotropic crystal. The eigencomponents separately refract into and independently propagate through the crystal so that two beams polarized orthogonally to each other exit the crystal. This phenomenon is known as birefringence. There are numerous important consequences of birefringence for the manipulation and analysis of the polarization state of an optical beam. Important effects include birefringence, dichroism, anisotropic refraction, anisotropic reflection, walkoff, and optical activity. These effects can be used to make quarter wave plates, half wave plates, variable wave plates, dichroic polarizer, polarizing beamsplitting prisms, depolarizers, optoisolators, and other polarization manipulation devices. In addition, artificial birefringence can be induced by the application of an electric or magnetic field in appropriate non-centrosymmetric media or magnetic media respectively or by the application of a strain (or stress) or propagating acoustic wave in any media, which enable the production of useful electro-optic, magneto-optic, or acousto-optic light modulators. 2 s^ e θ ke ko Figure 2: Anisotropic k-space for a uniaxial crystal, showing the directions of the wavevectors, the corresponding polarizations, and the direction of the Poynting vector giving the power flow. The optical indicatrix construction (or index ellipsoid) shown in Fig. 1 is a useful graphical construction to determine the directions of the two polarized eigenmodes as well as their indices of refraction for a given direction of propagation. For a uniaxial crystal, the equation of the index ellipsoid is given by: x2 + y 2 z 2 + 2 =1 n2o ne where no is the ordinary index and ne is the extraordinary index. This is an ellipsoid of revolution with the circular symmetry axis parallel to z, as shown in Figure1(a). The direction of propagation k̂ is along the wavevector ~k at an angle θ to the optic (z) axis. Because of the circular symmetry about the z axis, we can choose the y axis to coincide with the projection of k̂ on the x − y plane without any loss of generality. The intersection ellipse of the plane normal to k̂ with the ellipsoid ~ are parallel to the major and minor is shaded in the figure. The two polarization eigen states of D ~ are axes of the ellipse, which correspond to the segments OR and OE. The eigenpolarizations of D perpendicular to k̂ as well as to each other. The wave which is polarized along OE is called the extraordinary wave. The index of refraction is given by the length of OE. It can be determined using Figure 1(b), which shows the intersection of the index ellipsoid with the y-z plane. Solving for the index of refraction as a function of direction of propagation yield two eigenvalues for each direction of propagation, each associated with a transverse polarization eigenvector. For a uniaxial crystal the ordinary index no is independent of direction of propagation, while the ordinary index only depends on the polar angle θ expressed in polar and cartesian components as: x2o + yo2 + zo2 =1 n2o no (θ, φ) = no cos2 θ sin2 θ + ne (θ, φ) = n2o n2e − 21 x2e + ye2 ze2 + 2 =1 n2e no These index of refractions determine the corresponding optical wavevectors ~ko = no ωc and ~ke = ne (θ) ωc which are often referred to as momentum vectors and are simply related to the actual optical momenta ~~ko and ~~ke . The concept of momentum conservation (phase matching) at the crystal boundary is very useful for finding the propagation direction(s) of the refracted wave in the 3 kz Uniaxial c λt αt kT λT αi αt αt αi αr λi Isotropic a) Real space b) k space Figure 3: Refraction from an isotropic medium into a uniaxial anisotropic medium with tilted optical axis with respect to the face normal. The periodicity of the wave crashing into the surface must be conserved and this is shown in a) real space and b) reciprocal Fourier space (also known as k-space or momentum space). anisotropic crystal. Boundary conditions require that the projection of the propagation vectors along the boundary plane must be equal for both the incident and refracted waves. This results in double refraction of a wave incident on the surface of an anisotropic crystal as shown in Figure3. We can therefore write the anisotropic generalization of Snell’s law: k0 sin θ0 = ko sin θo = ke sin θe The direction of the energy flow, however, is not always along the propagation direction of the phase of the plane wave, but it is normal to the wave-vector (or index) surface. Using this information, one can find the directions of the refracted beams in a given anisotropic crystal. 2.2 State of Polarization Two important mathematical representations of polarized light are the complex Jones vector and Jones calculus for coherent fields and the Stokes vector and Mueller calculus for general partially polarized fields, although the Stokes vectors and Poincaré sphere representation can also be utilized advantageously for coherent and purely polarized fields. 2.2.1 Jones Calculus Jones vectors are a relatively compact 2-dimensional complex vector description of the state of polarization of a plane wave, but they describe the phase and amplitude of the electric field of a purely monochromatic wave, and thus can not capture the statistical properties of polychromatic or partially polarized light. The complex Jones vector is [Ex eiδx , Ey eiδy ]T , and when normalized to I = ~†·E ~ = 1 and referenced to the phase of the x component, simple forms include horizontal= 1 , E 0 1 1 , RHC= √12 1i , LHC= √12 −i , and general vertical= 01 45◦ linear= √12 11 , -45◦ linear= √12 −1 with orientation α and phase δ is given by eiδcossinαα . Jones matrices represent the transformation 4 z S3 S3 S3 P S out 2χ S1 2ψ S2 S in δ S δ =180 deg. S y S2 2θ S2 2θ S out F F S in x S1 S1 Figure 4: a) Geometry of the Poincaré sphere with the linear polarizations on the equator, circular polarizations at the poles, and elliptical polarizations in between. b) Operation of an arbitrary waveplate on an arbitrary polarization as a rigid body rotation of the Poincaré sphere. c) The operation of a half waveplate on a linear polarization as a 180 degree rotation of the SOP on the Poincaré sphere. of the state-of-polarization (SOP) as 2 by 2 linear operators: ′ Ex Ex jxx jxy = Ey jyx jyy Ey′ where the elements can be complex to represent a phase shift. To represent a rotated component in the laboratory frame, rotation operators are used J(θ) = R(−θ) J R(θ). An x-polarizer Jones 0 matrix is given by 10 00 while a half waveplate is 0i −i and a quarter waveplate with the fast iπ/4 1 0 axis vertical is e A train of components is represented by a series of Jones matrices 0 −i operating on the Jones vector in the sequence that the beam encounters the components, J = ~ A mirror flips the handedness of the coordinate system so must be represented by a J3 J2 J1 E. 0 special matrix M = 10 −1 and components after the mirror have the sign of their off diagonal jxx −jxy components reversed −j in order to account for the flipped right handed coordinates used yx jyy when propagating backwards. 2.2.2 Stokes Calculus and the Poincaré Sphere The Stokes vector can be written as [S0 , S1 , S2 , S3 ]T , where the real parameters are directly measurable and account for the statistical characterization of partially polarized light.. S0 is the intensity. S1 indicates a tendency for the polarization to resemble either a horizontal state (S1 = +1) or vertical state(S1 = −1), and is measured as the difference in transmission between horizontal and vertical polarizers. S2 indicates a tendency for the light to resemble a +45(S2 = +1) or -45(S2 = +1) polarization, and is measured as the difference in transmission between +45 and -45 polarizers. S3 describes a tendency of the beam toward right-handedness or left-handedness, reaching S3 = +1 for right hand circular (RHC) and S3 = −1 for left hand circular (LHC), and is measured as the difference in transmission between RHC and LHC polarizers. The geometrical representation of the Stokes parameters as the state of polarization is the Poincaré sphere of radius S0 , with the other three parameters forming three orthogonal axes intersecting at the center of the sphere as shown in Fig.4. For describing pure polarization transformations, often the Stokes vector components are normalized by S0 yielding new components [1, S1 /S0 , S2 /S0 , S3 /S0 ]T = [1, S1 , S2 , S3 ]T , 5 since the radius does not change as only the polarization is manipulated. The 4 x 4 Mueller matrices represent the transformation of Stokes parameters in an optical system, and obey a similar calculus to Jones, but involving 4 by 4 real matrices describing observable quantities instead of 2 by 2 complex matrices describing field amplitudes. They Mueller matrices correspond to rotations and other motions on the Poincaré sphere. The Stokes parameters are in terms of intensity and thus are capable of representing states of unpolarized and partially polarized light. 3 Preparation and Prelab Read the section on crystal optics in your favorite optics book. • Fowles, Modern Optics, Chap 6.7-8 • Born and Wolf, Principles of Optics, Chap 14 • Shubinikov, Principles of Optical Crystallography • Yariv and Yeh, Optical Waves in Crystals, chap 3,4,5 • E. Hecht, Optics, Chap 8 • H. J. Juretschke, Crystal Physics, Chap 9 Also review the appropriate summary of polarization calculus and the Poincaré sphere in classic references, • Polarized Light: Fundamentals and Applications (Optical Engineering, Vol 36) by Edward Collett • Polarization of Light by Serge Huard • Fundamentals of Polarized Light: A Statistical Optics Approach by Christian Brosseau • Field Guide to Polarization (SPIE Vol. FG05) by Edward Collett • Polarized Light in Optics and Spectroscopy by David S. Kliger and James W. Lewis • Introduction to Matrix Methods in Optics A. Gerrard and J. M. Burch. • Polarized light;: Production and use by William A Shurcliff where the last 2 are especially good introductory treatments. 6 3.1 Prelab 1. Wollaston prism optic axis Wollaston Prism n air ne no air Wollaston prism and the interpretation of its operation in k-space (a) Describe and analyze the operation of a Wollaston prism, which is made from two cemented prisms of uniaxial calcite, with a length to aperture L/A ratio of 1/3. (b) At a wavelength of 632.8 nm the ordinary index of calcite is no = 1.65566 and the extraordinary index is ne = 1.48518. What is the angular deviation between the two polarizations in this prism? Represent the wave vectors in momentum space. 2. Conoscopic pattern Screen Polarizer c^ Eout y^ x^ Ein z^ Crystal under test Strongly focused input beam Figure 5: Conoscopic Experimental setup, k-space interpretation, and resulting fringe pattern. Consider a crystallographically cut 1cm cube of negative uniaxial calcite illuminated with a tightly focused spherical wave of monochromatic vertically polarized light with λ=632.8nm, with the optical axis of the crystal parallel to the lens axis. (a) Sketch the state of polarization of the input as the optical ~k vector is rotated in a cone about the optical axis, and the decomposition into ordinary and extraordinary components, and sketch the state of polarization at the output of the crystal. (b) When the transmitted light is passed through a horizontal analyzer a set of circular fringes appears, modulated by a cross. Explain using your sketch of the state of polarization at the output of the crystal. (c) How can this effect be used to determine the optical axis of a uniaxial birefringent crystal. See Fig 5. 7 3. Producing an arbitrary state of polarization Design an experimental arrangement of waveplates that will produce an arbitrary state of polarization, characterized by an ellipticity b/a, where a is the major axis and b is the minor axis, and an orientation angle Φ of the major axis clockwise from the vertical. 4. Soleil-Babinet compensator [Not required] A Soleil-Babinet compensator is made from a movable uniaxial crystal wedge, and a fixed crystal wedge cemented to a compensating crystal plate with its optical axis orthogonal to that of the two wedges, and all optical axis orthogonal to the direction of propagation. The movable wedge is optically contacted to the fixed wedge with index matching oil so that as it is slid back and forth, a variable thickness anisotropic plate is synthesized. Show that when the two plates are of equal thickness a zero retardance waveplate is produced. When the wedge angle is α and the ordinary and extraordinary indices are no and ne at a wavelength λ, at what displacement from this zero retardance position of the movable wedge does the compensator become a half wave plate? What is the advantage of this compensator compared to the Babinet compensator, which does not have the compensating plate, and where the two wedges have their optic axes orthogonal to each other. 5. Opto-isolator PBS S3 slow axis y fast axis S out z F ( S’) S 2 S ( F’) 45 deg. x S in quarter−wave plate S1 reflective surface Figure 6: Optoisolator geometry and description on the Poincaré sphere. [Not required] Consider the system shown in Figure6 consisting of a perfect horizontal transmission polarizer, followed by a nominal quarter waveplate with its fast axis oriented at +45 degrees, followed by a retro-reflecting mirror. a) Assuming that waveplate is exactly quarter wave at the operating wavelength, what is the reflection backwards through the polarizer. Why? b) If the waveplate has a retardance that deviates away from δ = 90 degrees by a small deviation ∆, eg δ = 90 + ∆, solve approximately for the reflected power as a function of ∆ for small ∆. 8 3.2 • • • • • • • • • 4 Materials and Equipment 1 1 1 1 2 2 1 1 1 LiNbO3 crystal He-Ne laser Calcite (CaCO3 ) crystal Spatial filter Crystal mounts Irises l/4 wave plate (633 nm) Collimating lens l/2 wave plate (633 nm) • • • • • • • • 1 Microscope objective 3 Polarizers 1 Mirror 1 Polarizing Beamsplitter 1 Power meter 1 Oscilloscope 1 Optical detector Polarimeter (and eigenstate generator) Procedures 1. Polarizer Spatial filter a HeNe laser beam, collimate with a lens, and insert an iris. Then illuminate a pair of crossed polarizers with the HeNe laser beam. (a) What is the residual transmission in the crossed state, and what is the transmission when the two polarizers are aligned? (b) What is the extinction ratio of these polarizers at the HeNe wavelength. (c) Now insert a short focal length lens, and illuminate the crossed polarizers with a highly diverging beam. Can you observe any angular variation of the extinction ratio through the crossed polarizers? When done remove the lens. 2. Half wave plate (a) Obtain a half wave plate for the HeNe wavelength, and determine the optical axes. Describe your procedure for determining the axes. (b) Insert the half wave plate between the crossed polarizers using a plane wave illumination. Measure the maximum and minimum transmission through an analyzer as the waveplate is rotated by 10-15 degree increments with more samples concentrating around the null. Such selective sampling is an important time saving practice. Plot your results, and explain. 3. Polarizing beamsplitter (a) Place a polarizing beamsplitter in the laser beam, and determine the polarization purity of the reflected and transmitted beams, eg for pure V input measure RV and TV as well 0 as RH and TH0 and for pure H input measure RH and TH as well as RV0 and TV0 , and determine the contrasts RH /RV and TV /TH . (b) Now follow the PBS with a quarter wave plate, and a mirror which retroreflects the light back through the quarter wave plate and PBS towards the laser. Measure the light bouncing off the beamsplitter as you rotate the waveplate. Plot your results. (c) Explain how to use this setup as an optoisolator that blocks reflections from going back into the laser cavity thereby disrupting the modal quality. What is the isolation level of your setup, and what limits this isolation. 9 4. Meadowlark polarimeter Calibrate the Meadowlark liquid crystal polarimeter. The calibration procedure requires 6 different states of polarization ( vertical, holizontal, +45, -45 linear polarizations, right and left circular polarization). Use a polarizer and quarter wave plate or the eigen-state generator. You can use the following procedure or devise your own, or you can follow the proceudere in the polarimeter manual if the eigenstate generator is available. (a) Check that the laser is pure vertical or horizontal with a Wollaston prism. Which is it? Insert a polarizer in the beam aligned to pass the laser polarization and clean up the state of polarization. (Typical lasers only have 100:1 polarization purity, while dichroic polarizers are 104 : 1 and crystal polarizers can be 105 : 1) (b) Follow with a 45◦ quarter waveplate to give circular polarization. How can you gaurantee it is circular? Reflective double passing through the waveplate and a PBS or Wollaston can be used to tweak the orientation/tilt to give pure circular, or you can use the fact that circular polarization will give no intensity variations when passed through a rotating linear polarizer, while any elliptical state will yield sinusoidal intensity variations. This will allow the insertion of various angles of linear polarizers without varying the intensity. (c) Use a rotatable polarizer or Glan or Wollaston prism to generate the 4 required linear polarizations as you run through the calibration procedure in the Polarimeter software. Check with a power meter that the power is constant for the different polarizations. When done, leave the polarizer at 45◦ . To make sure that your vertical and horizonal polarizations are aligned with the plane of the table, you can use an iris on a mag base to make sure that the deviated beam remains at a constant height as it propagates away from a Wollaston or PBS. Caution of possible eye hazards when rotating a beam deviating prism like a Wollaston, especially for the beam deviated upwards. (d) You can produce circular polarization by removing the polarizer in your setup, but that will give 3dB too much intensity. To compensate this, insert the 45◦ linear polarizer between the laser and cleanup polarizer. Check the power of the circular polarization. Can you determine which circular polarization you have, R or L? (e) Finally you can produce the other circular polarization by retroreflecting (or at a very small angle circular polarization off a mirror. Or you can flip the 45◦ oriented waveplate around its vertical axis by spinning the post in the post holder by 180◦ changing its orientation to -45◦. Double check that this is indeed circular with the right power and finish the software calibration procedure. 5. Tilted waveplate Variable birefringence retarders can be implemented by tilting a birefringent plate about an axis perpendicular to the optical beam and thereby varying the thickness of the path through the crystal. Align the input polarization at 45◦ to vertical, and orient a half wave plate with its principal axes horizontal and vertical. Measure and plot the ellipticity of the emerging wave that has passed through the half wave plate oriented with the optic axes at 45 degrees to the incoming polarization as you tilt the wave plate by 15 degree increments. The ellipticity can be measured by rotating a polarizer in the output and measuring the maximum and minimum transmission, the ratio giving the a2 /b2 ratio of major to minor 10 axes. Alternatively, the state of polarization can be easily determined using the calibrated polarimeter. 6. Producing arbitrary state of elliptical polarization Choose either approach a) two waveplate, or b) Soleil-Babinet to produce an arbitrary SOP. (a) Using your results from Prelab problem 3, set up a quarter wave plate and half wave plate in order to produce light of ellipticity of .25, and an orientation of 45 degrees away from the vertical. Use the polarimeter to verify that this is the state of polarization that you obtained. (b) The same state of polarization can be generated by transforming a polarized beam into the desired polarization using a Soleil-Babinet compensator. Vary the orientation and phase delay while observing the Poincaré sphere output from the polarimeter until you produce the desired state of polarization. Record the orientation and phase delay setting of the Soleil-Babinet compensator. (c) For both approaches there are two control parameters, orientation of the two waveplates, or orientation and phase delay of the Soleil Babinet compensator. Use the polarimeter and describe the trajectories on the Poincaré sphere for both of these control parameters as you spin them around, while holding the other fixed at a few separate settings.. Sketch in your lab book and interpret. 7. Polarization deviating Wollaston prism (a) Illuminate a Wollaston prism with the HeNe laser. Describe its operation. Measure the angles of refraction and the polarization of the refracted beams. Now reverse the prism and measure the angles and polarizations of the refracted beams. Are they the same or different? Why? (b) Now place a second Wollaston prism in one of the deflected beams and measure the isolation ratio of 2 crossed Wollaston prisms. (c) Rotate the second Wollaston by 45◦ around the optical axis and illuminate with one of the pure polarizations produced by the first Wollaston. How can it produce 2 new refracted beams with 2 orthogonal polarizations when we know that the incident state of polarization is purely linear? 8. z-axis conoscopic pattern Remove the wave plates and beamsplitter. Set up a conoscopic system to determine the z-axis of a crystalographically cut piece of LiNbO3 . Illuminate the crystal with a tightly focused beam. Use a short focal length objective lens. The shorter the focal length of the lens, the larger the range of angles, and the more fringes will be observed. Place a polarizer after the crystal and rotate the crystal about the vertical axis so that a set of concentric bright and dark circles appears, modulated by a cross(+). Rotate the crystal to center this pattern in the beam. Rotate the polarizer to give the pattern with the best contrast ratio. (a) Sketch this pattern in your lab book or capture a digital photograph. (b) Rotate the analyzer by 90 and sketch the new pattern. Does a contrast reversal occur as you rotate the output polarizer (analyzer)? 11 (c) Rotate the polarizer and analyzer to ±45◦ degrees to produce a rotated dark cross (×). Measure the spacing between the successive fringes at some plane a known distance away from the lens focus, and use this measurement along with the crystal thickness to determine the birefringence of the negative uniaxial crystal. 9. Thick waveplate conoscopic pattern Rotate the crystal about the vertical axis by 90 degrees so you are propagating in the orthgonal direction perpendicular to the optical axis (the same geometry as a thick waveplate) and look for a similar polarization interference pattern. Rotate the first polarizer such that the incident beam is 45 polarized, and rotate the analyzer. Sketch the interference pattern or take a digital photo? Describe and explain the shape of the fringes that you see. Choose one of the next two parts: EO modulator or Liquid Crystal modulator. 10. Liquid crystal variable retarder Hook up the liquid crystal variable retarder to the controller. Find the optical axis as before, between crossed polarizers. Now rotate the variable retarder by 45◦ . Vary the applied AC voltage and plot the transmittance versus AC voltage, and infer the phase retardance versus AC voltage. 11. Electro-optic modulator (EOM) (a) Identify if the EOM has any built in polarizers (amplitude modulator) or not (phase modulator). Send a slightly focusing beam through the electro-optic modulator (EOM). For an amplitude modulator, find the eigen axis (how?) and adjust the input polarization to be 45◦ from the eigenaxis, and analyze the output with a crossed analyzer, and adjust the input polarization to be 45◦ from the eigenaxis, and analyze the output with a crossed analyzer. (b) Do you have enough angular aperture to see the conoscopic pattern? Describe the motion of the fringes with variations of the applied field. (c) Measure the half wave voltage using the high voltage DC supply (careful!). Plot the transmission versus applied DC voltage. Do you see any fluctuations due to temperature changes? +45 input Electro optic transverse amplitude modulators −45 polarizer polarization constructed from KDP with light propagating y x along crystal Y and e-field along crystal Z exz lab frame perience net birefringence even in the absence Z X of a field. Since both ne and no are functions Z Y Y of temperature (and wavelength) then the net xtal 2 X xtal 1 polarization rotation and hence the transmitted amplitude can wander around as the ambient temperature changes. A 2 crystal transverse EO modulator shown below (where the two crystal are polished together and therefore exactly the same length) with the second crystal rotated by 90 degrees, and the field applied appropriately cancels the temperature dependence. 12

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