Moisés Cywiak

Noise in phase shifting interferometry

We present a theoretical analysis to estimate the amount of phase noise due to noisy interferograms in Phase Shifting Interferometry (PSI). We also analyze the fact that linear filtering transforms corrupting multiplicative noise in Electronic Speckle Pattern Interferometry (ESPI) into fringes corrupted by additive gaussian noise. This fact allow us to obtain a formula to estimate the standard deviation of the noisy demodulated phase as a function of the spectral response of the preprocessing spatial filtering combined with the PSI algorithm used. This phase noise power formula is the main result of this contribution.

Measurement of out-of-plane deformation by combination of speckle photography and speckle shearing interferometry

Summary We measure the derivative of out-of-plane deformation in presence of large in-plane rigid body displacement (RBD) by a combination of Electronic Speckle-Shearing Pattern Interferometry (ESSPI) and speckle photography. We use digital speckle photography in order to compensate for the in-plane RBD and eliminate any decorrelation effects. ESSPI techniques are then applied to the compensated speckle images to obtain the derivative of the out-of-plane deformation. The method is tested on an aluminum plate subjected to simultaneous out-of-plane deformation and large in-plane RBD. The results indicate that the proposed method preserves the fringe contrast observed in ESSPI.

Wave-front propagation by Gaussian superposition

Abstract A linear superposition of Gaussians is proposed for representing and propagating a plane or defocused scalar wave-front with an arbitrary amplitude distribution in a linear, homogeneous and isotropic medium. A comparison is made against a wave-front that is propagated using the discrete Fresnel diffraction integral. At distances of propagation where the quadratic phase in the discrete Fresnel integral oscillates slowly, both techniques give similar results. At distances where the quadratic phase undergoes a large number of oscillations, the number of samples when using the discrete Fresnel integral must be increased to fit within Nyquist criteria. In comparison, the wave-front expressed as a linear Gaussian superposition can be calculated precisely, without any approximation, making unnecessary to increase the number of samples. Then, shorter processing time can be attained without losing the resolution in computing the propagated field. Thus, the Gaussian superposition technique becomes especially efficient when complex multistage optical systems are analyzed.

Extreme shearing interferometry: theoretical limits with practical consequences.

In this work we analyze the frequency response, the spatial distribution and continuity of the recovered phase in Lateral Shearing Interferometry (LSI). This frequency content and topology of the recovered phase is analyzed for the forward LSI operator as well as its inverse LSI operator using one, two, or n two-dimensional sheared interferograms. The spatial frequency response of the shearing interferometer is well known and for the readers convenience, it is briefly revisited in a new perspective. It is however less well-known and more interesting to analyze the spatial distribution of the sheared data as well as the spatial topology of the recovered phase produced by some inverse LSI operators. Also we define a useful space of functions S with the property that any sheared data available, along any direction, may be used to recovered a smooth continuous phase with the bonus property of fully covering the pupil of the wavefront being tested. These combined aspects allow us to find the best possible wave-front reconstruction from the available sheared data using one, two or n sheared interferograms.

Far field optical imaging with subwavelength resolution

We describe a far field optical imaging technique for surface topography measurement with subwavelength lateral resolution. In this system, the surface under test is vibrated in a plane parallel to the plane of the surface and perpendicular to the incident probe beam. The phase of the reflected beam, modulated by surface irregularities, is detected by cutting half of the overall beam with a knife-edge and by using a lock-in amplifier whereby noise floor is lowered and transfer function profile is extended to higher spatial frequencies. We present experimental results showing subwavelength lateral resolution.

Fresnel-Gaussian shape invariant for optical ray tracing

We propose a technique for ray tracing, based in the propagation of a Gaussian shape invariant under the Fresnel diffraction integral. The technique uses two driving independent terms to direct the ray and is based on the fact that at any arbitrary distance, the center of the propagated Gaussian beam corresponds to the geometrical projection of the center of the incident beam. We present computer simulations as examples of the use of the technique consisting in the calculation of rays through lenses and optical media where the index of refraction varies as a function of position.

A technique for calculating the amplitude distribution of propagated fields by Gaussian sampling

We present a technique to solve numerically the Fresnel diffraction integral by representing a given complex function as a finite superposition of complex Gaussians. Once an accurate representation of these functions is attained, it is possible to find analytically its diffraction pattern. There are two useful consequences of this representation: first, the analytical results may be used for further theoretical studies and second, it may be used as a versatile and accurate numerical diffraction technique. The use of the technique is illustrated by calculating the intensity distribution in a vicinity of the focal region of an aberrated converging spherical wave emerging from a circular aperture.

Vibrating knife-edge technique for measuring the focal length of a microlens

A vibrating knife-edge technique is proposed for measuring the focal length of a microlens. The technique is based on the propagation properties of Gaussian beams. A laser beam with a Gaussian intensity profile is focused in front of the microlens under test. After being transmitted through the microlens, the beam propagates toward a detector, which consists of a photodiode that is half blocked by a knife-edge. The photodiode integrates approximately half the intensity of the transmitted beam. The knife-edge vibrates sinusoidally with small amplitude in a plane normal to the direction of propagation. Our analysis shows that the output signal at the photodiode consists of a dc component plus a temporal sinusoidal signal whose amplitude is proportional to the focal length of the microlens. After system calibration, the focal length is measured with an envelope detector or a lock-in amplifier.

New method for optical object derotation

A new optical method capable of measuring out-of-plane deformations of rotating objects based on the Fourier transform phase decoding technique is presented. Digital holography is used to test the method, which digitally derotates one of the holograms at its phase reconstruction stage to accurately remove object rotation fringes, uniquely rendering phase maps that quantitatively show the out-of-plane deformation. Commonly, object derotators are based on creating standing images of the rotating object under study. Typically, this is achieved by means of a rotating prism that has to be precisely synchronised with the object rotation. In contrast, this new method eliminates the need of using the expensive mechanical servomechanisms contained in the commercially available optomechanical derotators by using double pulsed digital holography in conjunction with Fourier optics.

Refractive index and geometrical thickness measurement of a transparent pellicle in air by Gaussian beam defocusing

We demonstrate that it is possible to measure the local geometrical thickness and the refractive index of a transparent pellicle in air by combining the diffractive properties of a Gaussian beam with the analytical equations of the light that propagates through a thin layer. We show that our measurement technique is immune to inherent piston-like vibrations present in the pellicle. As our measurements are based on characterizing properly the Gaussian beam in a plane of detection, a homodyne technique for this purpose is devised and described. The feasibility of our proposal is confirmed by measuring local geometrical thicknesses and the refractive index of a commercially available stretch film.