College of Optical Sciences The University of Arizona Study of birefringence effect on Fizeau interferometry: analytical and simulation results Chunyu Zhao Donyel Kang Jim Burge College of Optical Sciences, The University of Arizona

College of Optical Sciences The University of Arizona Outline The Problem we are facing Simulation results Analytical results Conclusion

College of Optical Sciences The University of Arizona 1. The problem we are facing When polarization technique is used in interferometry, and some optics exhibit birefringence, how would it affect the measurement accuracy?

College of Optical Sciences The University of Arizona Fizeau Interferometer Fizeau interferometers are common path interferometers, so only one high quality surface is needed.

College of Optical Sciences The University of Arizona Dynamic Polarization Fizeau Interferometer Polarization technique is used to differentiate reference and test beams. Linear polarizer is used as phase shifter

College of Optical Sciences The University of Arizona Birefringence effect When birefringence exists in any component in the common path, the polarization Fizeau interferometer is not common path any more.

College of Optical Sciences The University of Arizona What polarization to use? We can use linear or circular polarizations for the reference and test beams, but which one is advantageous when birefringence exists?

College of Optical Sciences The University of Arizona 2. Simulation Results Each pixel is modeled as a waveplate whose orientation and retardation vary from pixel to pixel Two cases are studied: –Reference and test beams are linearly polarized –Reference and test beams are circularly polarized

College of Optical Sciences The University of Arizona Linear polarization case In the extreme case, reference and test beams are aligned with the waveplate’s fast/slow axis, then the maximum phase measurement error caused by the birefringence equals the magnitude of the birefringence . Then surface figure error  is: R T CCD Phase Shifter Waveplate: Fast. axis angle , retardation  Ref Surf Test Surf

College of Optical Sciences The University of Arizona Circular polarization case Reference and test beams are circularly polarized – one is left hand circular, the other is right hand circular Both beams see the birefringence. Jim’s hunch is: the birefringence effect on measurement accuracy will be smaller than the linear polarization case.

College of Optical Sciences The University of Arizona Model The measurement error depends on many variables: ω, , , . R T CCDPolarizer, Tran. axis angle ω Waveplate: Fast. axis angle , retardation  OPD:  Ref Surf Test Surf

College of Optical Sciences The University of Arizona Simulation results  =30 deg  =55 deg Phase measurement error is indeed quadratic with birefringence. At certain combination of physical optical path difference and WP’s fast axis angle, there is no phase measurement error. The linear polarizer’s angle introduces no measurement error.

College of Optical Sciences The University of Arizona 3. Analytical approach Why take the analytical approach? –We want an independent result to compare with our simulation results –Because so many variables are involved, it’s hard for anybody to make sense of the simulation data

College of Optical Sciences The University of Arizona Choice of coordinate system There exist a global coordinate system in which incoming test and reference beams are in phase: α XGXG YGYG X T along the fast axis of the waveplate YTYT Test beam Referenc e beam Waveplate’s fast/slow axes are chosen as x-y axes of its local coordinate system, in which the waveplate’s Jones matrix is:

College of Optical Sciences The University of Arizona Return beams Deal the problem in the waveplate’s local coordinate where the incoming R/T beams are If the OPD between T and R is , the returned individual beams are The combined return beam:

College of Optical Sciences The University of Arizona Combined return beam The combined return beam: α γ/2 XGXG YGYG XTXT YTYT With no birefringence, i.e.  =0, the combined return beam is linearly polarized. With birefringence, it is elliptically polarized.

College of Optical Sciences The University of Arizona General elliptically polarized beam Described by an elliptical equation ω ψ  x yTransmission axis of a linear polarizer AxAx AyAy Define  and  :

College of Optical Sciences The University of Arizona Beam intensity after passing a linear polarizer The Jones matrix for the polarizer is The transmitted E-field is The observed intensity is

College of Optical Sciences The University of Arizona Phase shift and contrast reduction Phase shift: Contrast reduction: I  = 0  ≠ 0  Δ ψ ω

College of Optical Sciences The University of Arizona Maximum phase error Phase error Phase error is maximum when and the maximum error is Small angle approximation:, where

College of Optical Sciences The University of Arizona Comparison with simulation results Simulation data overlaps with strict analytical data small angle approximation Given the amount of birefringence, the maximum measurement error:

College of Optical Sciences The University of Arizona Comparison of two polarization cases Maximum error caused by using circular polarization is quadratic to the amount of birefringence. For small residual birefringence is advantageous than linear polarization. Birefringence (in nm) Linear polarization Circular polarization Max. surf. measurement error (in nm)

College of Optical Sciences The University of Arizona Simulation of a surface measurement: distribution of birefringence Fast axis angle varies linearly along y Retardation varies linearly along x x y

College of Optical Sciences The University of Arizona Simulation of a surface measurement: measurement error

College of Optical Sciences The University of Arizona 4. Conclusion Birefringence affects measurement accuracy of polarization Fizeau interferometry Equations are derived to calculate the measurement error caused by birefringence. The analytical results agree with the simulation results When the beams are circularly polarized, the maximum measurement error is quadratic to the retardation; when the beams are linearly polarized, the maximum measurement error equals the retardation With small amount of birefringence, measurement error is reduced dramatically when circular polarization rather than linear polarization is used When circular polarization is used, measurement error depends on the initial optical path difference btw reference and test surface. Therefore, in the presence of vibration, averaging multiple measurements help reduce measurement error caused by birefringence.