Polarization Phase-shifting Interferometry Research Articles

Two-dimensional refractive index distribution using polarization phase shifting interferometry and total internal reflection

An effective method to measure the 2-D refractive index distribution of homogeneous liquids: acetic acid and glycerol- water mixture at different concentration and a non- homogeneous mixture of liquids: water, acetic acid, acetone and glycerin but not limited to these liquids is presented. In the present setup a combination of polarization phase shifting interferometry (PPSI) and total internal reflection (TIR) is used. The s-polarized light strikes the boundary of a right angle prism and a tested liquid droplet. When TIR occurs on the interface, the incident light has a phase variation which depends on the refractive index of adhered liquid on prism surface. Two-dimensional index distributions can be easily calculated using the relation among reflection phase shift difference and the liquid index. The results of the tested mixed liquid by proposed technique show the refractive index and its distribution in range 1.433 ± 0.031 to 1.433 ± 0.124.

Fast and robust calibration method of liquid-crystal spatial light modulator based on polarization multiplexing.

We propose a fast and robust method for calibrating Spatial Light Modulators (SLMs) based on polarization phase-shifting interferometry. Our method effectively calibrates the SLM by addressing both the static aberration and nonlinear phase response, utilizing specially designed gray images loaded sequentially onto the SLM. Notably, we introduce a novel kinoform that effectively eliminates the influence of tilt phase shift between two shots of the polarization camera. This results in a highly accurate phase aberration map and phase modulation curve with exceptional stability, making it an ideal method to calibrate the SLM with exceptional efficiency and precision in real applications.

Dual-channel simultaneous spatial and temporal polarization phase-shifting interferometry.

Without the limitations of fringe number and fringe shape, a dual-channel simultaneous spatial and temporal polarization (DC-SSTP) phase-shifting interferometry system is proposed to achieve rapid and accurate phase retrieval through only one-time phase-shifting procedure with unknown phase shifts. First, an arbitrary phase shifts is simultaneously introduced into two channels of DC-SSTP system by a spatial light modulator (SLM). Second, by performing the subtraction operation between each pair of phase-shifting interferograms captured in the same channel, the background deduction of interferogram can be achieved easily, so the accurate phase can be retrieved rapidly. Especially, it is found even if the fringe number in interferogram is less than one, the proposed DC-SSTP method still reveals high accuracy of phase retrieval. Both the simulation and experimental results demonstrate the outstanding performance of proposed DC-SSTP method in phase measurement.

Interferometric focal length measurement of positive and negative lenses using a lateral-shearing cyclic path optical configuration setup and polarization phase-shifting interferometry.

We present a simple interferometric focal length measurement technique for measuring the focal length of positive and negative lenses using a lateral-shearing cyclic path optical configuration (CPOC) setup and polarization phase-shifting interferometry (PPSI). The technique requires an auxiliary lens, whose focal length need not be known accurately, for generating a focused point source at its focal plane. The focal point of the test lens is adjusted to be in the vicinity of the focus of the auxiliary lens. The radii of the resulting spherical wave fronts emerging from the test lens, due to the defocus, for two different longitudinal shifted positions of the test lens are measured using the lateral-shearing CPOC setup and PPSI. Focal length of the test lens is determined from the known longitudinal shift and from the measured radii of the spherical wave fronts. Results obtained for 200.0mm convex and 100.0mm concave test lenses are presented.

Measurement of surface form error of an optical surface with reduced interferometric spatial coherence artifacts using a ring source and polarization phase-shifting interferometry.

We present a technique for the measurement of surface form error of an optical surface using a ring source and polarization phase-shifting interferometry (PPSI). The ring source, generated using a refracting axicon, is projected onto a rotating diffuser to reduce the spatial coherence noise generated by the spatially coherent He-Ne laser and to enhance the interference fringe contrast. PPSI is applied to extract the phase [and hence the optical path difference (OPD)] from the interference fringes formed by a Fizeau cavity using the ring source. The OPD values are least-square fitted to a plane and the surface form errors are evaluated from the deviations of the OPD values from the fitted reference plane. A two-step subtraction method to reduce the system errors and the effect of the ring source diameter on the effective measurement area of the test surface are discussed. The main advantage of the technique is that the perturbations in the Fizeau cavity are negligible during the phase shifts as the phase shifts between the interfering beams are introduced outside the cavity.

Small roll angle measurement using lateral shearing cyclic path polarization interferometry.

We present a technique for the measurement of roll angular displacement of a rotary stage using a lateral shearing cyclic path optical configuration (CPOC) setup and polarization phase shifting interferometry (PPSI). The CPOC setup, aligned on the rotary stage, laterally shears the input plane polarized spherical beam into a pair of orthogonally polarized beams, which when brought to the same state of polarization by a polarizer produce interference fringes similar to Young's fringes. Rotation of the CPOC setup in its plane introduces a phase change between the orthogonally polarized lateral sheared beams due to the change in angle of incidence of the input beam. The change in the phase results in spatial displacement of the interference fringes. Using PPSI, the phase, or the optical path difference change between the laterally sheared beams that is related to the rotation angle of the CPOC setup, is measured.

Measurement of displacement and distance with a polarization phase shifting folded Twyman Green interferometer.

A Sagnac interferometer (SI), consisting of a polarization beam splitter (PBS), along with two equally spaced plane mirrors that are inclined at 45° to each other, is transformed into a folded Twyman Green interferometer (TGI) by placing a mirrored parallel plate (MPP) into the hypotenuse arm of the SI. The converging input beam produced by a telescope objective (TO) is split into reflected (s-polarized) and transmitted (p-polarized) components by the PBS. The p- and s-polarized focal spots are made to fall on the mirrored end surfaces of the parallel plate (PP). The retroreflected p- and s-polarized beams become collimated after passing through the TO. A linear shift of the PP in either (longitudinal) direction alters the positions of the p- and s-polarized focal spots and results in a set of converging and diverging spherical wavefronts that interfere to form concentric circular fringes. We applied polarization phase-shifting interferometry to obtain the optical path difference (OPD) variation of the interference field. The displacement is calculated from the OPD variation. A validation experiment has been carried out by introducing known shifts to the PP. The setup can be used for displacement as well as distance measurement.

Determination of the tangential surface profile of Toroidal beam line mirrors with polarization phase-shifting Twyman–Green interferometer

A new technique for the measurement of the tangential surface profile of Toroidal mirrors (TMs) used in synchrotron radiation beam lines is proposed. A cylindrical lens placed in the collimated input (He–Ne laser) beam of a polarization Twyman–Green interferometer (TGI) forms line foci in the reference and test arms. The effect of the sagittal curvature of the TM is eliminated by adjusting a line focus on the surface of the TM, along the central tangential section. Thus, a toric test wavefront (TTW) with sagittal curvature identical to that of the incident cylindrical wavefront and tangential curvature with slope twice that of the TM, is produced along the reflection direction. The TTW interferes with a reference cylindrical wavefront, generated by retroreflecting the incident cylindrical wavefront from a reference plane mirror in the reference arm of the TGI and having identical curvature in the sagittal direction. Polarization phase-shifting interferometry has been applied to obtain the tangential surface profile of the test TM. Results are compared with other methods. The advantages of the technique are that only a plane reference mirror is needed to generate the reference beam, and the technique is suitable for TMs with long tangential radii of curvature in the kilometer regime.

偏振正交性对偏振相移干涉检测精度的影响

The influence of polarization direction orthogonality on Polarization Phase Shifting Interferometry(PPSI)was analyzed to improve its measurement precision.Based on the analysis of PPSI principle,it points out that the measurement error is a function of Optical Path Difference(OPD)and the PV of measurement error is proportional to the non-orthogonality and the ratio of amplitudes of test beam to reference beam.The polarization non-orthogonality can be introduced by the reflection or refraction on a surface.Then several approaches were recommended to depress the non-orthogonality,such as proper coatings,incident angles and polarization directions.A configuration of Skip-Flat test with incident angle of 45°was set up.It shows that the distribution of measurement error is coincident with that of OPD.Moreover,the measurement error is 0.174 1λ(PV),which is in agreement with that of the theoretical analysis.Finally,the paper suggests that it is necessary to analyze individually each of the component under the test for making sure the measurement precision.

应力双折射对偏振相移干涉检测的影响

The birefringence of transmission flat/sphere and other components introduces measurement error in Polarization Phase Shifting Interferometry(PPSI).Based on the principle of PPSI,the measurement error was analyzed by using the Jonse maxtrix formulations.It was found that the birefringence generated eight beams interference on the CCD,and the measurement error was a function of the ratio of amplitude of test beam and reference beam,the testing cavity phase,the amount and orientation of birefringence.The measurement error varied2periods in every fringe.The maximal PV of measurement error would be 4 times of birefringence and the measurement error would be compressed if the orientation of birefringence was consistent.When the birefringence was of the same orientation,the measurement error could be eliminated.A 12inches material was analyzed.The results show that the birefringence of the4inches area in the center was 18nm,which introduced the measurement error of 72nm(PV).The birefringence of the material at the edge was 22nm and the measurement error was24.9nm(PV)because of the consistence of birefringence orientation.

Simple technique for the accurate internal measurement of a right angle with a polarization phase-shifting, lateral-shearing, cyclic path optical configuration

A simple, new, self-referenced technique for the internal measurement of a right angle is presented. A polarization cyclic path optical configuration (CPOC) receives as the input beam a pair of laterally separated plane wave components, having angular separation of ψw = (π±4nδ), produced due to internal reflection of a linearly polarized plane wave on the surfaces enclosing the 90-deg (±δ) angle of a right-angled prism (where δ is the error in 90-deg angle and n is the index of refraction of the material of the prism). The CPOC is used to superpose the mutually inclined plane wave components by duplicating the input pair of plane wave fronts and introducing appropriate lateral shear between the identical pairs. The superposed inclined plane wave components interfere to produce two beam Fizeau fringes (FF), when brought to the same state of polarization. The error in the right angle can be calculated simply from the spacing of the resulting FFs or, more accurately, from the slope of the optical path difference variation obtained by applying polarization phase-shifting interferometry. Results obtained for a right-angled prism are presented.

Opaque optics thickness measurement using a cyclic path optical configuration setup and polarization phase shifting interferometry

Thickness measurement of an opaque optics using a cyclic path optical configuration (CPOC) setup and polarization phase shifting interferometry (PPSI) is presented. The CPOC setup is used to simultaneously focus two orthogonally polarized counterpropagating converging beams at its hypotenuse arm. The opaque optics is placed at the hypotenuse arm of the CPOC setup such that one of its surfaces reflects back one of the counterpropagating focusing beams. Because of the thickness of the opaque optics, the other focusing beam suffers a longitudinal shift in the beam focus. Applying PPSI, the longitudinal shift in the beam focus which is twice the thickness of the opaque optics is determined. The results obtained for a silicon plate of thickness 0.660 mm with a measurement uncertainty of 0.013 mm are presented.

Step height measurement using lateral shearing cyclic path optical configuration setup and polarization phase-shifting interferometry

We present a step height measurement technique using a lateral shearing cyclic path optical configuration (CPOC) setup and polarization phase shifting interferometry (PPSI). In the technique, the CPOC setup generates two orthogonally polarized, laterally sheared, converging beams that get focused at the back focal plane of a positive lens placed at the input beam of the CPOC setup. These focusing beams are reflected back into the CPOC setup by the step optics, positioned in such a way that one of the surfaces of the step optics coincides exactly with the focus point of one of the laterally sheared focusing beams. The CPOC recombines the beams reflected from the step surfaces and the beams re-pass through the lens, which produces collimated and spherical interfering wave fronts for the respective beams. Using PPSI, the radius of the spherical wavefront emerging from the lens, which is related to the focal shift introduced between the reflected beams due to the step height, is determined. The novelty of the technique is the introduction of the CPOC setup and the use of a single wavelength light source to determine the large step height. Results obtained for a step height of 1.340 mm are presented.

Measurement of the surface profile of an axicon lens with a polarization phase-shifting shearing interferometer

We present a Twyman-Green interferometer (TGI)-based polarization phase-shifting shearing interferometric technique for testing the conical surface of an axicon (AX) lens. In this technique, the annular beam generated due to the passing of an expanded collimated laser beam traveling along the axis of revolution of the transparent glass AX element is split up into its reflected and transmitted components, having the plane of polarization in the orthogonal planes, by the polarization beam splitter (PBS) cube of the TGI-based optical setup. The split-up components are made to travel unequal paths along the two arms of the TGI and are recombined by the PBS. Because of the difference in path lengths traveled by the annular conical beams, a linear shear is introduced along the radial direction between the interfering components. Thus, the resulting interference pattern gives a map of the optical path difference (OPD) between two successive close points along a radial direction on the conical surface of the AX lens. The OPD map along radial directions, and hence the slopes/profiles of the conical surface, are obtained by applying polarization phase-shifting interferometry. Results obtained for an AX lens are presented.

Measurement of longitudinal displacement using lateral shearing cyclic path optical configuration setup and phase shifting interferometry

We present a technique for the measurement of longitudinal displacement using a lateral shearing cyclic path optical configuration (CPOC) setup and phase shifting interferometry. In the technique, a plane mirror mounted on a linear translation stage, placed slightly away from the focal plane of a lens, introduces a longitudinal focal shift to the incident focusing beam. The resulting spherical wavefront emerging from the lens is sheared into two orthogonally polarized beams using the CPOC setup. By applying polarization phase shifting interferometry (PPSI), the longitudinal focal shift of the beam focus is calculated by determining the slope of the optical path difference variation between the sheared beams. Similarly, the additional focal shift introduced due to longitudinal translation of the mirror, by an unknown amount, is determined using PPSI. Half of the difference between the two longitudinal focal shifts measured gives the longitudinal displacement of the mirror. The technique can be used for an extended range of distance measurement. The novelty of the technique is the introduction of CPOC for the distance measurement. The advantages of the technique compared to other related methods are discussed.

Thickness measurement of transparent glass plates using a lateral shearing cyclic path optical configuration setup and polarization phase shifting interferometry

We present a measurement technique to determine the thickness of a transparent glass plate (GP) by using a lateral shearing cyclic path optical configuration (CPOC) setup and polarization phase shifting interferometry (PPSI). In the technique, the GP introduces a longitudinal shift in the focus of the beam and, as a result, a spherical wavefront emerges from the lens, which is otherwise set for producing a collimated beam. Using CPOC, two laterally sheared orthogonally polarized beams are generated from the incident spherical wavefront. By applying PPSI, the slope of the optical path difference variation between the laterally sheared interfering beams is evaluated, and the radius of the spherical wavefront and the longitudinal shift of the beam focus are calculated. The thickness of the GP is determined from the standard relation between the longitudinal shift of the focus introduced by the GP and the thickness of the GP. Results obtained for a GP of 9.810mm thickness are presented.

Grating-based real-time polarization phase-shifting interferometry: error analysis.

A phase Ronchi grating-based real-time polarization phase-shifting method can be efficiently used for dynamic phase measurement in optical interferometry. A thorough error analysis is required for exhibiting how error sources influence phase-measurement results. We analyze the phase-measurement errors that are induced by the retardation error and azimuth angle error of the quarter-wave plate, the azimuth angle error of polarizers, the phase and intensity aberrations of diffractive wave fronts, and pixel mismatch of the interferometric patterns. The results will also be useful for evaluating the phase-measurement accuracy of other similar systems.

Generation of phase-modulated periodic optical signals by use of a polarization interferometer

Polarization phase-shifting interferometry is an established technique in optical metrology. In the present study it is shown that, by use of this technique, not only is it possible to realize any discrete magnitude of a predetermined phase difference (from 0 to 2pi) between two light beams but also phase-modulated periodic optical signals can be generated simply by rotation of a polarizer or a retarder or both placed at the input of a conventional two-beam interferometer. Some representative linear and nonlinear periodic polarization-induced phase-modulated optical signals are shown. A linear phase modulation of 0-2pi with constant output intensity is obtained in some cases. The Poincaré sphere representation is introduced as a convenient tool for visualizing the dynamics involved in the generation of polarization-phase-modulated waveforms and as a possible aid to intelligent modification of the generated waveform as required. This all-optical technique of continuous and periodic phase variation should prove useful for introducing phase modulation without the need for electro-optic devices.