Surface profiling of a transparent object by use of phase-shifting Talbot interferometry

Talbot interferometry is used to study the surface profile of a transparent object. Periodic patterns are produced by illuminating a grating with a collimated laser beam. The object is placed on the self-image plane of the grating. The deformed grating image, which interferes with another grating, results in the Talbot interferometric fringes. The fringe pattern is recorded on a CCD camera for subsequent analysis, and the phase variation is achieved by a linear translation stage. In this application two specimens are tested to demonstrate the validity of the method; one is a transparent object with a spherical shape with a height of less than 350 microm, and the other is a transparent object with an uneven surface of 50-microm average height. The experimental results are compared with the test results obtained with the mechanical stylus method.     

Measurement of focal length with phase-shifting Talbot interferometry

Phase-shifting interferometry with a Fourier fringe analysis technique is implemented to analyze Talbot interferometric fringes and to evaluate the focal length of a lens. A four-step algorithm is used to obtain the phase map of the lens. The slope of the phase map is related to the focal length, and it is from this relationship that we evaluate the focal length. Experimental results are presented. Our experimental study suggests that phase-shifting Talbot interferometry combined with a Fourier fringe analysis technique can be advantageously used to improve the accuracy of measurement.     

Measurement of transparent plates with wavelength-tuned phase-shifting interferometry

A wavelength-tuned Fizeau interferometer is applied to the problem of flatness testing of transparent plates. When the plate is positioned at a specific distance from the reference surface and an integer-math 13-frame phase-shifting algorithm is applied, the system directly filters out unwanted interference arising from backsurface reflections. The resulting front-surface profile exhibits less than 2 nm of residual error attributable to spurious reflections from within the plate.     

Measurement of changes in optical path length and reflectivity with phase-shifting laser feedback interferometry

The operating characteristics of a novel phase-shifting interferometer are presented. Interference arises by reflecting the light from a sample back into the cavity of a cw He-Ne laser. Changes in phase and fringe visibility are calculated from an overdetermined set of phase-shifted intensity measurements with the phase shifts being introduced with an electro-optic modulator. The interferometer is sensitive enough to measure displacements below 1 Hz with a rms error of approximately 1 nm from a sample that reflects only 3% of the 28 muW that is incident on its surface. The interferometer is applied to the determination of cantilever bending of a piezoelectric bimorph.     

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 liquid-film thickness by laser interferometry

Here the variation of a liquid-film thickness at small Reynolds numbers is discussed. The film thickness measurement by laser interferometry corresponds to the liquid flowing on the inner surface of a small-bore glass tube. An adequate theoretical background for the techniques used in this experiment is discussed to demonstrate the capability of the experimental technique. An advantage of this method is that it shows the shape of the thin film on the inner surface of the vertical tube at a point in a horizontal cross section. The results obtained from this experiment show that the flow of liquid films on vertical surfaces is inherently unstable and three dimensional even at a Reynolds number smaller than 1.     

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.     

Phase evaluation of speckle patterns during continuous deformation by use of phase-shifting speckle interferometry

A method for measurement of continuous displacements that uses phase-shifting speckle interferometry is presented. The initial random phase of the speckle pattern is evaluated by phase shifting before deformation. The changing phase thereafter is evaluated from only one image at a time by a least-squares algorithm. The technique can be used for measuring transients and other dynamic events, heat expansion as well as other phenomena, for which it is difficult to accomplish phase shifting during deformation. Theory along with computer simulations and experimental results are described.     

Digital image encryption and watermarking by phase-shifting interferometry

A method for both image encryption and watermarking by three-step phase-shifting interferometry is proposed. The image to be hidden is stored in three interferograms and then can be reconstructed by use of one random phase mask, several specific geometric parameters, and a certain algorithm. To further increase the security of the hidden image and confuse unauthorized receivers, images with the same or different content can be added to the interferograms, and these images will have no or only a small effect on the retrieval of the hidden image, owing to the specific property of this algorithm. All these features and the utility of this method for image retrieval from parts of interferograms are verified by computer simulations. This technique uses intensity maps as decrypted images for delivery, and both encryption and decryption can be conveniently achieved digitally. It is particularly suitable for the remote transmission of secret information via the Internet.     

Measurement of surface roughness profile by a triangulation method

Some aspects of measuring surface roughness parameters using a laser triangulation method are considered. A connection is established between surface roughness and scattering indicatrices with probing radiation polarization parameters of a semiconductor laser. A marked effect is noted of the form and position of the polarization plane on the measurement error for linear values. __________ Translated from Metrologiya, No. 6. pp. 21–27, June, 2008.     

An algorithm for profilometry by white-light phase-shifting interferometry

Abstract

In the case of white-light interferometry, the usual phase shifting methods cannot be used because the modulation in the interference intensity is due to both the phase variation and to the coherence envelope. We propose a practical approach in which the coherence envelope is considered as locally linear. The classical four-point calculation algorithm is adapted to a set of seven intensities which are weighted with adapted coefficients in order to compensate for the effect of the coherence envelope. The capabilities of this method are described theoretically.     

Two-wavelength phase-shifting interferometry insensitive to the intensity modulation of dual laser diodes

We have constructed two-wavelength phase-shifting interferometry that is insensitive to the intensity changes in interferograms associated with the current variations in two laser-diode (LD) sources by using a newly developed phase-extraction algorithm. The tested phase at a synthetic wavelength can be measured from six interferograms with different phase shifts. The algorithm becomes a simple form for seven interferograms and reduces to a minimum of five phase-shifted data in the proper conditions. We shifted the phases equally in opposite directions to one another by separately varying the stepwise currents in dual LD's on an unbalanced interferometer. The measurement accuracy has been improved compared with that of the two-wavelength four-step method. The phase error caused by the power changes in the dual LD's has been investigated theoretically and experimentally. The experimental results are shown to measure a step object with a 4.6-μm synthetic wavelength.     

Systematic errors of phase-shifting speckle interferometry

A theoretical and numerical investigation of the systematic phase errors in phase-shifting speckle interferometry is presented. The theoretical investigation analyzes the behavior of some systematic error induced by intensity variations in two cases of data-computing techniques. The first case deals with the technique in which the phase change is computed, unwrapped, and then linearly filtered; the second case deals with the technique in which the data are linearly filtered before the arctangent calculation and then unwrapped. With the first filtering technique it is shown that it is preferable when the phase change is of relatively low spatial frequency, leading to a particularly accurate method. With the second case it is demonstrated that an important parameter of speckle interferometry is the modulation factor of the interference frame that induces phase errors when the data are filtered before the arctangent calculation. We show that this technique is better than the first when the phase change is composed of high-spatial-frequency variations. The theoretical investigation of the two techniques is compared with numerical simulations, considering the frequency characteristics of the phase change, and this shows a good match between theory and simulations.     

Two-wavelength in-line phase-shifting interferometry based on polarizing separation for accurate surface profiling

We describe a configuration that can be used for two-wavelength phase-shifting in-line interferometry based on polarizing separation. The experiment is conducted on a sample with a step height of 1.34 μm nominally. In this paper, five- and seven-phase step algorithms have been compared for their effectiveness in reducing the noise in the phase maps. The noise is further reduced by the application of the flat fielding method. The recorded interferograms are processed using seven-phase step algorithm to obtain the phase map for each wavelength separately. The independent phase maps are subtracted and a phase map for the beat-wavelength is obtained and converted to height map. The results extracted from the seven-phase step algorithm have been compared with the results extracted from the single shot off-axis geometry and the results are in agreement.     

Numerical simulations of vibration in phase-shifting interferometry

Computer simulations predict the expected rms measurement error in a phase-shifting interferometer in the presence of mechanical vibrations. The simulations involve a numerical resolution of a nonlinear mathematical model and are performed over a range of vibrational frequencies and amplitudes for three different phase-shift algorithms. Experimental research with an interference microscope and comparison with analytical solutions verify the numerical model.     

Stabilized phase-shifting fringe analysis by use of current-induced frequency modulation of laser diodes

A closed-loop phase-shifting Fizeau-type interferometer was constructed that uses direct frequency modulation of a laser diode. The interferometer is servo controlled entirely in the phase domain, where optical phases are detected by two-frequency optical heterodyning. A detailed study of stabilization of the interferometer under feedback control was conducted both experimentally and theoretically. The interferometer showed good stability against vibration up to 200 Hz. The stabilization factors obtained experimentally are in good agreement with the theoretical calculations. The phase-shifting experiment was accomplished with high precision as well as with high stability against external disturbances. The profile measurement of a mirror surface was made with a phase-shifting analysis algorithm, and good measurement reproducibility of lambda/60 in the root-mean-square value was obtained for ten measurements within a period of 20 min.     

Multiperiod fringe projection interferometry using a backpropagation method for surface profile measurement

Interference fringes with different periods are projected on an object surface. There is a constant phase point where the phase of the fringe is kept at a constant value while the period is scanning. Multiple optical fields with different periods on the object surface are made from detected phases of the fringes. The multiple optical fields are backpropagated to the constant phase point of the phase where all of the phases of the multiple backpropagated fields become the same value and the amplitude of the sum of the multiple backpropagated fields becomes maximum. The distance of the backpropagation provides the position of the object surface. Some experiments show that this method can measure an object surface with discontinuities of several millimeters with high accuracy of several micrometers.     

Stabilization improvements of laser-diode closed-loop heterodyne phase-shifting interferometer for surface profile measurement

To stabilize the phase-shifting Fizeau-type interferometer against environmental disturbances (namely, vibration and temperature variations), the feedback scheme that uses the current-induced frequency modulation of a laser diode (lambda = 633 nm) and the two-frequency optical heterodyne method has been investigated, with particular attention to improvement of the achievable stabilization. It is demonstrated that introduction of the proportional-integral control into the feedback system improves stabilization against the proportional control case; e.g., stabilization is improved 5 times for 100-nm(p-p) vibration at the frequency range at 30 Hz. The surface profile measurement for a sample mirror was conducted with a reproducibility of 6.8 nm in the root mean square under the subwavelength-amplitude vibration at 100 Hz.