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 200.0 mm convex and 100.0 mm concave test lenses are presented.

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 the surface form error of large aperture plane optical surfaces with a polarization phase-shifting liquid reference reflection Fizeau interferometer.

A polarization phase-shifting liquid reference reflection Fizeau interferometer has been proposed. A polarization cyclic path optical configuration along with a concave telescope mirror is used to produce a pair of expanded, collimated p and s polarized beams with a small angular separation between them. The collimated beams are deflected along a vertical direction toward a Fizeau interferometer cavity formed between a liquid surface that acts as a reference surface and a plane test surface. Either the p or s polarized beam is allowed to strike the liquid surface normally and the orientation of the test surface is adjusted to reflect the other beam, having orthogonal linear polarization, in the direction of the normally reflected reference beam from the liquid surface. A combination of a quarter-wave plate and linear polarizer is used to apply polarization phase shift between the test and reference beams, and quantitative surface form error is measured by applying phase-shifting interferometry. A method for elimination of the residual system aberration is discussed. Results obtained for an optically polished BK-7 disk of clear aperture diameter ≈160 mm 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.

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.

Measurement of the residual wedge angle of a nontransparent parallel plate with a modified cyclic path optical configuration.

A new technique for the measurement of the residual wedge angle (RWA) of a nontransparent nearly parallel plate (NPP), using a polarized He-Ne laser source, is proposed. A modified cyclic path optical configuration is used to produce counterpropagating collimated beams, which get reflected from the end surfaces of the NPP and are combined to produce two-beam Fizeau fringes, whose spacing is inversely proportional to the wedge angle of the NPP. The states of polarization of the beams reflected from the end surfaces of the NPP are transformed to opposite circular polarizations, and polarization phase shifts are applied between them. Phase shifting interferometry (PSI) has been used to determine the angular tilt between the interfering beams and, hence, the RWA of the NPP. Results obtained from a validation experiment are presented. The PSI-based technique does not suffer from any measurement ambiguities due to limitations in effective beam aperture.

White light differential interference contrast microscope with a Sagnac interferometer.

A new technique for producing a white light differential interference contrast (DIC) image using a lateral shearing, rotation phase shifting Sagnac interferometer (SI) is proposed. The SI, placed in the image space after the tube lens of a microscope system with spatially coherent white light Kohler illumination, splits the image forming beam into coherent components with small lateral shear. Phase shifts, between the interfering components, which can be considered as biased phase difference (BPD), are introduced by applying small angular rotation of the SI in its own plane. This variable BPD between the interfering white light components produces a uniform intensity colored background. The object related phase shift, due to the height difference between two close points on the object surface with separation on the order of least resolvable separation of the microscope objective, in addition to the BPD would produce a change in intensity/hue/color against a uniform background due to the BPD. Thus a DIC image is formed and the variable BPD provides an excellent means of improving the contrast of the image.

Determination of the surface form error of a spherical mirror with phase shifting Sagnac interferometer.

A polarization Sagnac interferometer (SI) is used to produce two laterally separated, identical, convergent emergent beams with linear orthogonal polarizations. The emergent p-polarized and s-polarized beams converge toward their respective focal points. The test and reference spherical mirrors are placed at confocal positions with respect to the s and p focal points so as to normally reflect back the test and reference beams through the SI that recombines the test and reference waves. Polarization phase shifting interferometry is applied to obtain the surface form error of the test surface with respect to the reference surface. A two-step measurement procedure eliminates the system aberrations. Results obtained for a concave spherical test surface with respect to a convex spherical reference surface are presented. The optical configuration is relatively less susceptible to external mechanical vibration.

Determination of the index inhomogeneity of transparent isotropic optical material with a dual Sagnac interferometer.

A technique for the measurement of inhomogeneity of optical glass, fused silica, etc., using a Sagnac interferometer (SI) has been presented. An SI produces a pair of laterally separated, mutually parallel, collimated beams with orthogonal planes of polarization, i.e., p and s polarizations from an expanded, linearly polarized (45°) collimated He-Ne (632.8 nm) input laser beam. The p and s beams pass through a liquid-filled cell with plane parallel glass windows. The test glass with plane parallel end surfaces is kept in the path of the p beam in the index matching liquid, while the s beam traverses a parallel path through the liquid. Another SI recombines the emergent p and s beams by removing the lateral shear. A quarter-wave plate transforms the state of polarization of the beams to opposite circular polarizations of which the components selected by a linear polarizer interfere to form Fizeau fringes. Polarization phase shifting interferometery has been applied to determine the optical path difference (OPD) variations. The OPD variation without the test glass is subtracted from that with test glass to eliminate the effect of system aberration. The results for a phosphate laser glass sample have been 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.

Simultaneous determination of refractive index and thickness of moderately thick plane-parallel transparent glass plates using cyclic path optical configuration setup and a lateral shearing interferometer.

We present a simple technique for simultaneous determination of refractive index and thickness of moderately thick plane-parallel transparent glass plates (GPs) using a cyclic path optical configuration (CPOC) setup and a wedge shear plate as lateral shearing interferometer. The CPOC setup is used to simultaneously focus the counterpropagating converging beams at a common point at its hypotenuse arm. The apparent thickness and real thickness of the test GP are determined by observing three retrocollimation positions of the GP surfaces with respect to the common focus point. The RI is obtained by dividing the real thickness with apparent thickness of the GP. Presented in this paper are the results obtained for a test GP with a thickness of 14.983 mm and a RI of 1.515.

Self-referenced technique for the determination of meridional figure errors of a toroidal mirror with a Sagnac interferometer.

A simple, self-referenced technique for the determination of meridional surface slope errors of toroidal mirrors (TMs) used in synchrotron radiation is presented. An expanded, collimated, linearly polarized laser beam is allowed to fall on a TM, and the reflected toric wavefront from the TM is allowed to enter a Sagnac interferometer based lateral shearing interferometer setup, which duplicates the beam into identical wavefront components having orthogonal linear polarizations, i.e., p and s polarizations, and introduces appropriate lateral shear between the identical pair of wavefronts along the meridional direction. The laterally sheared identical wavefronts interfere when brought to the same state of polarization. The optical path difference (OPD) variation along the direction of the lateral shear in the interference field is a map of the slope variation. Polarization phase shifting interferometery has been applied to find the OPD variation and thus the slope distribution and errors. Results obtained for a TM of moderately long average radius of curvature along the central meridional section and a relatively shorter radius of curvature along the sagittal section are presented and compared with that obtained with a long trace profiler.

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.

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 the surface profile of a curved optical surface with rotation phase-shifting lateral shear cyclic path optical configuration.

We present a new (to our knowledge) technique for introducing phase shifts between the laterally sheared emergent beam components of a cyclic path optical configuration (CPOC). The phase shifts are introduced by applying a small change in the angle of incidence of the incident beam due to the small angular rotation of the CPOC setup. Phase-shifting interferometry has been applied along with this phase-shifting technique for a CPOC with lateral shear to find the surface slope/profile of curved optical surfaces. Results for a spherical optical surface have been discussed. An optical setup for measurement of the surface profile of toroidal beam line mirrors of synchrotron radiation sources is proposed.

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.

Simultaneous measurement of refractive index and wedge angle of optical windows using Fizeau interferometry and a cyclic path optical configuration.

We present a new technique for the simultaneous measurement of refractive index and wedge angle of optical windows using Fizeau interferometry and a cyclic path optical configuration (CPOC). Two laterally separated beams are obtained from an expanded collimated beam using an aperture containing two rectangular openings. The test wedge plate is placed in one of the two separated beams. Using CPOC, these two beams are made to overlap and interfere, producing interference fringes in the overlapping region. The beams reflected from the front and back surfaces of the test wedge plate interfere and produce Fizeau fringes. The refractive index is related to the spacing of the above two beam fringes. The wedge angle is determined from the evaluated values of the refractive index and Fizeau fringe spacing. The results obtained for a BK-7 optical window are presented.

Technique for the focal-length measurement of positive lenses using Fizeau interferometry.

We present what we believe is a new technique for the focal-length measurement of positive lenses using Fizeau interferometery. The technique utilizes the Gaussian lens equation. The image distance is measured interferometrically in terms of the radius of curvature of the image-forming wavefront emerging from the lens. The radii of curvature of the image-forming wavefronts corresponding to two different axial object positions of known separation are measured. The focal length of the lens is determined by solving the equations obtained using the Gaussian lens equation for the two object positions. Results obtained for a corrected doublet lens of a nominal focal length of 200.0 mm with a measurement uncertainty of +/-2.5% is presented.

External measurement of dihedral right angles with cyclic optical configuration.

A new technique for external measurement of dihedral right angles is presented. An expanded, collimated, and linearly polarized He-Ne laser beam (632.8 nm) from a Fizeau interferometer is launched into a cyclic path optical configuration (CPOC) in which the counterpropagating p and s polarization components traverse the same optical path in opposite directions. A right-angled component (RAC), with its plane surfaces forming the right angle, is set to externally reflect the counterpropagating p and s components of the CPOC in nearly the same directions but with a lateral separation. In a plane normal to the right-angle edge of the RAC, the laterally separated collimated beams have angular separation, which is equal to twice the error in the dihedral right angle. Another CPOC setup is used to recombine the beams by reducing the lateral shear to zero. Error in right angle is calculated from the spacing of the resulting two-beam Fizeau fringes. Methods for overcoming the restriction of measurement accuracy due to beam aperture limitation and the effects of the positional tilt of the RAC have been discussed. Results of validation experiments are presented.

Measurement of two-dimensional small angle deviation with a prism interferometer.

A new technique for the measurement of two-dimensional small angular deviation is presented. A compound prism, which effectively produces a combination of two right-angled prisms in orthogonal directions, and plane reference surfaces have been utilized for the measurement of the orthogonal components of the angular tilt of an incident plane wavefront. Each orthogonal component of the angular tilt is separately measured from the angular rotation of the resultant wedge fringes between two plane wavefronts generated due to splitting of the incident plane wavefront by the corresponding set of right-angled prism and plane reference surface. The technique is shown to have high sensitivity for the measurement of small angle deviation. A monolithic prism interferometer, which is practically insensitive to vibration, is also proposed. Results obtained for the measurement of a known tilt angle are presented.