Effects of birefringence on Fizeau interferometr that uses polarization phase shifting technique Chunu Zhao, Dongel Kang and James H. Burge College of Optical Sciences, the Universit of Arizona 1630 E. Universit Blvd., ucson, AZ 8571 Abstract: Interferometers that use different states of polarization for the reference and test beams can modulate the relative phase shift using polarization optics in the imaging sstem. his allows the interferometer to capture simultaneous images that have a fied phase shift, which can be used for phase shifting interferometr. Since all measurements are made simultaneousl, the interferometer is not sensitive to vibration. Fizeau interferometers of this tpe have advantage over wman-green tpe sstems because the optics are in the common path of both the reference and test wavefronts, therefore errors in these optics affect both wavefronts equall and do not limit the sstem accurac. However, this is not strictl true for the polarization interferometer when both wavefronts are transmitted an optic that suffers from birefringence. If some of the components in the common path of the reference and test beams have residual birefringence, the two beams see different phases. herefore, the interferometer is not strictl common path. As a result, an error can be introduced in the measurement. In this paper, we stud the effect of birefringence on measurement accurac when different polarization techniques are used in Fizeau interferometers. We demonstrate that measurement error is reduced dramaticall for small amount of birefringence if the reference and test beams are circularl polarized rather than linearl polarized. Kewords: interferometr, polarization, birefringence 1. Introduction In a common path interferometer, e.g. a Fizeau interferometer, both the reference and test beam go through the same optics up to the reference surface, therefore, an defect in the common path affects the phases of reference and test beams equall and has no effect on the measurement accurac. 1 However, when the reference and test beams have different polarization states, if a component in the common path has residual birefringence, the reference and test beams see different phases and an error is introduced in the measurement. In this case, the interferometer is no longer strictl common path, though it still phsicall is. here now eist commercial phase-shifting Fizeau interferomters that use this polarization technique to simultaneousl take multiple frames to freeze vibration. 3,4 Birefringence is a concern because it is alwas present, especiall for big and thick optics. For eample, Schott specifies the residual birefringence of standard BK7 glass at <6nm/cm. If a test plate is 10cm thick, the birefringence is about 60nm which is significant if the required measurement accurac is high. But b using circular polarized light instead of linearl polarized light, the measurement error caused b the residual birefringence can be dramaticall reduced and can be eliminated. In Section, we give the maimum measurement error when the reference and test beams are linearl polarized. In Section 3, first we set up a model to stud the effect of birefringence when circularl polarized beams are used for the reference and test beams, and show analticall the combined beam before phase shifting is ellipticall polarized rather than linearl polarized. In Section 3., we Optical Manufacturing and esting VI, edited b H. Philip Stahl, Proc. of SPIE Vol. 5869 (SPIE, Bellingham, WA, 005) 077-786X/05/$15 doi: 10.1117/1.61551 Proc. of SPIE 58690X-1
stud the beam intensities after a general ellipticall polarized beam passes the linear polarizer. We show that the observed intensit is sinusoidal, but a phase retardation is introduced and fringe contrast is reduced. We use this result to analze the phase measurement error due to birefringence in Section 3.3. We present the simulation result in Section 3.4..Linear polarization case: o analze the effect of birefringence we assume a Fizeau interferometer as shown in Figure 1. As in all Fizeau interfermeters, the reference surface is the last surface in the sstem, so all of the transmissive optics are common path. If some optics have residual birefringence, we model the combined birefringence as a waveplate whose fast ais and retardation var from point to point in the pupil. he light reflected from the reference surface creates the reference wavefront and light transmitted through this surface, then reflected from the surface under test creates the test wavefront. he interference between the two is used to determine the error in the surface under test. Phase shift interferometr is conventionall performed moving the reference surface to cause a phase shift and capturing successive interferograms. Recentl, Fizeau interferometers that simultaneousl capture all of the different phase shifts have been developed. hese sstems are configured so that the reference wavefront and the test wavefront have orthogonal polarization states. hrough clever use of geometr or coherence, the sstem can be configured so that onl the desired polarization states are measured. Waveplate: retardation φ Fast ais along Ref. Surf est Surf R Module of simultaneous phase shifter and detectors Figure 1. A simultaneous phase-shifting polarization Fizeau interferometer. If the reference and test beams are linearl polarized, e.g. the reference beam is -polarized and test beam is -polarized, and if the residual birefringence is φ (in radian) and its fast ais is along, then the test beam sees an additional phase φ due to this birefringence. It is mistakenl attributed to the surface figure error: Proc. of SPIE 58690X-
φ δ ma = λ. (1) π his would be the maimum measurement error caused b the residual birefringence in the common path of the reference and test beams. 3. Circular polarization case: When the reference and test beams are circularl polarized, a linear polarizer can be used as the phase shifter as shown in Figure. Rotation of the linear polarizer shifts the relative phase between the two polarizations. 5 4D echnologies uses the same principle but a piilated mask to achieve simultaneous phase shifting. 4 Retarder: Fast ais angle α, retardation φ Ref. Surf est Surf R Double pass OPD: γ Polarizer, ran. ais angle ω CCD Figure. A phase-shifting Fizeau interferometer with circularl polarized reference and test beams uses a linear polarizer as the phase-shifter. 3.1 he combined beam before the phase shifter If, for eample, the test beam is right hand circular (RHC) and the reference beam is left hand circular (LHC), there alwas eists a coordinate sstem where the reference and test beams are in phase. We define this coordinate sstem as the global coordinate sstem X G -Y G (see Figure 3). In this coordinate sstem, the Jones vectors 6 of the test beam (denoted as G ) and the reference beam (denoted as R G ) are 1 G = i and 1 R G =. () i Proc. of SPIE 58690X-3
As stated in Section, the residual birefringence is modeled as a waveplate. Assume its retardation is φ (angle), and its fast ais has an angle α with the global X G -ais. We choose its fast ais as the local -ais, denoted as X -ais. In the local coordinate X -Y (see Figure 3), the double pass Jones matri of the waveplate is 1 0 B =. (3) iφ 0 e Y Y G α X along the fast ais of the waveplate X G Reference beam est beam Figure 3. Illustration of the definitions of global coordinate sstem and the waveplate local coordinate sstem. X G -Y G is the global coordinate sstem where the incident reference and test beams are in phase. X -Y is the waveplate local coordinate sstem with the fast ais along the X direction. he test and reference beams are circularl polarized, and in phase in the global coordinate sstem. he have equal intensit (eaggerated in the figure). In the waveplate local coordinate sstem, the reference and test beam see a phase shift, e i iα = 1 and R e i iα = 1. (4) After the light is reflected from the reference and test surfaces, and passes the waveplate a second time, the reference and test beams become iγ 1 i(γ α ) ' = B e = e and iφ ie R 1 ie iα ' = B R = e, (5) iφ where γ is the single pass phase difference between the reference and test beams caused b the phsical separation between reference and test surfaces along a ra. he combined beam is Proc. of SPIE 58690X-4
i(γ α ) iα + e + e iγ cos( γ α) R = ' ' = e. (6) i(φ + α ) i(φ + γ α ) iφ ie ie sin( γ α) e So, when there is no birefringence, i.e. φ=0, the combined beam is linearl polarized with the electrical field vector oscillates along a direction which has an angle γ α with the waveplate s local X ais and an angle γ with the global X G ais. When the residual birefringence is nonzero, the combined beam is ellipticall polarized with the phase difference φ between the E- fields in Y and X directions (see Figure 4). Y Y G γ X α X G Figure 4. Illustration of the combined beam s polarization. he blue dashed line illustrates the ideal linearl polarized beam when no birefringence eists, while the green solid ellipsis illustrates the ellipticall polarized beam when birefringence eists. 3.. Intensit after an ellipticall polarized beam passes the phase shifter he reference and test beams both pass a linear polarizer before reaching the CCD. he polarizer combines the reference and test beams to obtain interference. It also serves as a phase shifter when it rotates b angle of ω the phase difference between the reference and test beam will increase b ω. 5 he interferometer can make simultaneous measurements with different phase shifts b creating multiple images of the pupil and viewing them through polarizers set at different angles. 4 When a general ellipticall polarized beam passes through a linear polarizer, the transmitted beam intensit is a function of incident beam parameters and the angle of linear polarizer s transmission ais. he Jones vector for general ellipticall polarized light is: Proc. of SPIE 58690X-5
E A E = = iδ. (7) E Ae he electrical field vector at an point forms an ellipsis described b 7 E A E + A E A E A cos( δ ) = sin ( δ ). (8) Define an angle θ such that A tan(θ ) =. (9) A hen the ais of the ellipsis has an angle ψ with the -ais, as shown in Figure 5 where tan( ψ ) = tan(θ )cos( δ ). (10) Assume the ellipticall polarized beam passes through a linear polarizer whose transmission ais has an angle ω with -ais. he Jones matri for the polarizer is cos ω cosω sinω P =. cosω sinω sin ω ransmission ais of a linear polarizer A ω ψ θ A Figure 5. Illustration of definitions of the angles θ and ψ associated with an ellipticall polarized light. Also shown is a linear polarizer with transmission ais having an angle ω with -ais. Proc. of SPIE 58690X-6
After the linear polarizer, the transmitted E-field is A iδ cosω E trans = P iδ = ( A cosω + A sinω e ). (11) Ae sinω he intensit of the beam is 4 A + cos(δ ) + + A A A A A I = Etrans = + cos(ω ψ ), (1) where again tan( ψ ) = tan(θ )cos( δ ). Eq. (1) demonstrates that we can measure ψ, which is a characteristic of the incident ellipticall polarized beam, b phase shifting interferometr with a linear polarizer as phase shifter. When the incident beam is linearl polarized, i.e. δ = 0, then ψ = θ. When the beam is ellipticall polarized, compared to the linear polarization case, the beam intensit as a function of ω sees a phase shift = ( θ ψ ). (13) From Eq. (1), we also obtain the fringe contrast 4 C = A 4 + A A A cos(δ ) + A + A 4. (14) With the definition of θ, the fringe contrast can be rewritten: C = 1 sin (θ )sin δ. (15) Figure 6 plots the transmitted beam intensit as a function of ω for a linearl polarized beam and an ellipticall polarized beam. Proc. of SPIE 58690X-7
I θ ψ δ= 0 δ 0 I1( ω ) I( ω ) 0 150 ω 300 ω Figure 6. he transmitted light intensit as a function of the linear polarizer s rotation angle, after the waveplate the converts circular to linear polarization. he red dash line represents the case when there is no error and the light is linearl polarized, while the blue solid line represents the case when birefringence has caused the light to be ellipticall polarized. Note the intensit has a phase shift and a reduced contrast when the incident beam is ellipticall polarized compared to the linearl polarized. 3.3 Analsis In Section 3.1, we show that, when birefringence eists, the combined beam from reference and test surfaces is ellipticall polarized (see Eq. (6)). In Section 3., we show that we still measure a phase using phase shifting interferometr but with an error (see Eqs. (1) and (13)). he error depends on the characteristic of the ellipticall polarized beam. From Eqs. (6), (10) and (13), we get tan[( γ α)]sin φ tan( ) = tan(θ ψ ) =, (16) 1+ tan [( γ α)]cos φ Which indicates that the surface measurement error is a function of the retardation φ, the birefringence angle α and the actual phase γ. Since this is a function of the phase γ, the error will var when this phase is changed b making slight adjustments to the sstem. Assume π π < φ <, 4 4 then the maimum phase measurement error is Proc. of SPIE 58690X-8
sin φ tan( ma ) =. (17) cos φ For small φ approimation, φ ma. (18) he maimum surface figure measurement error is then φ =, (19) π 4π ma δma λ λ which indicates the maimum measurement error has quadratic dependence on the amount of birefringence. For comparison, we plot the maimum surface measurement errors as a function of birefringence when the reference and test beams are linearl and circularl polarized, respectivel, for small amount of residual birefringence. Ma. surf. measurement error (in nm) Linear polarization Circular polarization 0 ID 0 30 0 50 60 Birefringence (in nm) Figure 7. Plot of the maimum surface measurement error vs. birefringence for both linear and circular polarization cases. 3.4. Simulation of surface measurements for a sstem with birefringence he effect of birefringence is illustrated using a set of simulations. We perform the analsis for an interferometer that has 10 cm thick transmissive optic with birefringence of 6nm/cm. A map of the birefringence is shown in Figure 8(a). We evaluate the performance of this sstem as it measures a mirror that has 5 nm rms surface irregularit. We also assume the measurements to Proc. of SPIE 58690X-9
be made with 5 fringes of tilt due to alignment. he ideal and measured surface maps and measurement error maps are shown in Figure 8(b). he results verif that the measurement error is significantl smaller for small residual birefringence when circularl polarized beams, rather than linearl polarized beams, are used. I birefringence /1 / / /// ;J// // //// - _/ / /// // - - - 1- - - - - - (a) (b) Real Surface Interferogram Surface map Error in measurement (c) Simulated Measurements linear polarization case 5 nm rms circular polarization case 3 nm rms nm rms 6 nm rms 6 nm rms Proc. of SPIE 58690X-10
Scale bar for all maps (in nm) Figure 8. Simulated Fizeau measurements for a sstem with birefringence in the common part of the sstem (a) Birefringence map. he fast ais angle has a linear distribution along -ais from 0 to 90 degrees. And the retardation has a linear distribution along -ais from 0 to 60nm. (b) Results of simulation for an ideal sstem with 5 nm rms surface irregularit (c) Results for simulated phase shift interferometr for the case with 60 nm birefringence with the spatial distribution shown in (a). he measurement error is calculated b subtracting the ideal surface error from the simulated measurement. Note the reduction in fringe contrast as well as the phase error for both cases. If we maintain the distribution of birefringence and var the magnitude of retardation, we epect to see the measurement error increases as a function of the maimum retardation. Figure 9 shows the RMS measurement error as a function of the maimum retardation for the linear and circular polarization cases. It is obvious that measurement error is linear to birefringence when linearl polarized beams are used. In contrast, the measurement error is quadratic to birefringence when circularl polarized beams are used instead. here is an important distinction between the form of the measurement errors for the two cases. When linear polarization is used, an error is created that will be proportional to the birefringence and constant for all interferograms. Superimposed is an error that depends on the alignment and shows up as ripples in the surface with two times the frequenc of the interferogram fringes. his component of the error will change as the alignment is adjusted, thus can be reduced b averaging, but the larger, fied component would remain as a real error in the test. For the case of the circular polarizarion, there is NO fied error, and onl the ripple tpe that depends on the interferogram alignment. herefore it is possible to reduce the effect of the birefringence for the case of circular polarization b averaging multiple maps with different alignment. RMS measurement error (in nm) ujiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiili:iiiiiiiiiiiiiiiii Linear polarization V d+*-,*4trffr7!t7f I I I 0 W 0 30 40 50 60 70 Maimum retardation (in nm) V Circular polarization Proc. of SPIE 58690X-11
Figure 9. Plot of RMS surface measurement error as a function of maimum retardation of the birefringence distribution shown in Figure 8(a), for both linear and circular polarization cases. he dots are theoretical calculation results using Eq. (16), which agree with the interferometric simulation results for the circular polarization case. 4. Summar Fizeau interferometers can use polarization techniques to create a phase shift between the reference and test beams. If some element in the common path ehibits residual birefringence, it can limit measurement accurac. We model the residual birefringence as a waveplate whose fast ais orientation and retardation var from point to point in the pupil. If the reference and test beams are linearl polarized and orthogonal, the measurement phase error can be as large as the amount of birefringence. We studied the case when the two beams are circularl polarized and orthogonal, and we derived a set of relations to calculate the measurement error and the fringe contrast when birefringence is present. For small amount of birefringence, we showed that the error is quadratic to the amount of birefringence. So, in the case of small birefringence, the measurement error is significantl smaller if circular polarization rather than linear polarization is used to differentiate reference and test beams. In addition, this error is a function of the phase difference between the reference and the test, so the error can be further reduced b averaging multiple measurements with slight phase shifts. References: 1. See Schott Glass Catalog.. M.V.R.K. Murt, Newton, Fizeau, and Haidinger Interferometers, in Optical Shop esting, D. Malacara ed., (Wile, New York, 1978), 1-45. 3. For eample, see www.4dtechnolog.com or www.odecorp.com. 4. J. Millerd, N. Brock, M. North-Morris, M. Novak, and J. Want, Pielated phase-mask dnamic interferometer, in Interferometr XII: echniques and Analsis, Proc. SPIE 5531, K. Creath and J. Schmit Eds. 304-314, (004). 5. G. Jin, N. Bao and P. S. Chung, Applications of a novel phase-shift method using a computer controlled polarization mechanism, Opt. Eng. Vol 33(8), 733-737 (1994). 6. E. Hecht, Optics, (Addison Wesle Longman, 1998), 368-371. 7. M. Born and E. Wolf, Principle of Optics (Pergamon, London, 1980), 5-3. Proc. of SPIE 58690X-1