Arvind S. Marathay

Spatial correlation vortices in partially coherent light: theory

Spatial correlation vortex dipoles may form in the four-dimensional mutual coherence function when a partially coherent light source contains an optical vortex. Analytical and numerical investigations are made in near- and far-field regimes.

Matrix-Operator Description of the Propagation of Polarized Light through Cholesteric Liquid Crystals

The propagation of polarized light within the cholesteric structure along its helical axis is conveniently expressed in terms of four 2-by-2 matrix operators. This description is useful in studying the change in the state of polarization of a beam propagating along the helical axis. The well-known properties of the liquid crystals come out as natural consequences of the properties of these operators. In particular, for a certain region of wavelengths as the beam propagates along the helical axis, a vector representing the state of polarization in Stokes space precesses about the s3 axis. The s3 axis stands for circular polarization. Furthermore, owing to the properties of the matrix operators, if the propagation of polarized light within the liquid crystal is described in diagonal form for propagation in one direction along the helical axis, then it cannot have a diagonal form for propagation in the opposite direction.

Noncoherent-object hologram: its reconstruction and optical processing

Encoding a complex-amplitude diffraction pattern on a photographic film by addirtg a suitably chosen coherent reference beam forms a conventional hologram. A hologram of a spatially noncoherent object, referred to as a Γ hologram in this paper, is formed by encoding the complex-valued spatial-coherence function on a square-law detector such as photographic film. This record is made possible by means of a self-refetencing interferometer. Such a record behaves much as a hologram does; it permits reconstruction of the original object by illuminating it with a spatially noncoherent planar source of uniform (constant) intensity. If a conventional coherent-light setup is used with a Γ hologram, the intensity distribution of the reconstruction equals the square of the intensity of the original object. In the research reported in this paper, optical processing of spatially noncoherent objects is accomplished by using and modifying the spatial-coherence function. The Γ hologram is used to gain access to this function. This procedure opens new possibilities of noncoherent-object information processing. Examples of matched filtering, low-pass filtering, and high-pass filtering are discussed. The underlying theory has its roots in the fundamental Van Cittert–Zernike theorem of the theory of partial coherence.

Phase function of spatial coherence from second-, third-, and fourth- order intensity correlations

It is well known that the intensity interferometry of Hanbury Brown and Twiss measures the square of the absolute value of the normalized coherence function and that the phase of the function is lost. We show that the cosine and the sine of the phase can be determined using triple- and quadruple-intensity correlations. This information is used to derive the intensity distribution of the object. The procedure for the phase reconstruction described depends on the starting values of the phase function for the least separation of the mirrors or for the nearest neighbors in the two perpendicular directions of the array of mirrors. These starting values are determined by means of an amplitude correlation experiment, such as the Youngs two-slit type. The procedure of intensity correlations is then used for all other separations throughout the array. Computer simulation of the proposed procedure and a simple example of reconstruction of object intensity distribution are shown.

On the usual approximation used in the Rayleigh–Sommerfeld diffraction theory

The complete Rayleigh-Sommerfeld scalar diffraction formula contains (1 - ikR) in the integrand. Usually the wavelength is small compared with the distance of the observation point from the aperture and (1 - ikR) is approximated by -ikR alone. Other approximations usually made in the Rayleigh-Sommerfeld formula are addressed as well. Closed-form solutions, without approximations, are possible wherein interesting consequences of these approximations become apparent.

Image speckle patterns of weak diffusers

The use of weak phase-retarding diffusers in coherent imaging systems is analyzed. It is shown that ringing caused by diffraction from dust or blemishes can be significantly reduced by a diffuser that scatters light slightly beyond the first interference minimum of the diffraction pattern, provided that the specular transmission of the diffuser is small. Weak phase diffusers, whose phase standard deviations are a small part of a wavelength, accomplish this is an optimal manner. The characteristics of the speckle patterns that consititute their spatially filtered images are found to depend not only on the size of the aperture of the imaging system, but on the Wiener spectrum of the phase and the statistical law that the phase obeys. It is shown that random diffusers with normally distributed phase can easily be constructed that produce image speckle patterns whose Wiener spectra consist primarily of high spatial frequencies, if the size of the aperture of the imaging system is greater than a specified minimum. Speckle patterns are also found to exist near the image of a phase diffuser even when the aperture of the imaging system is large, and their Wiener spectra are functions of their distances from the image of the diffuser.

Young’s interference fringes with finite-sized sampling apertures

The theory of Young’s interference fringes is developed with particular attention to the finite size of the sampling apertures. A spatial Fourier transform of the product of the intensity distribution of the finite-sized source and the shifted intensity impulse response of the sampling aperture allows us to define a function Ĝ whose absolute value and phase dictate the visibility and the shift of the fringes, respectively. Alternatively, the function Ĝ may be expressed as a spatial Fourier transform of the spatial-coherence function across the sampling plane times the shifted transfer function of the sampling aperture. The effect of the finiteness of the sampling aperture becomes predominant in the neighborhood of the zeros of the coherence function or in regions where the coherence function is changing fast.

Diffraction of light from an aperture on a spherical surface

An appropriate Green’s function is constructed by use of the eigenfunctions of the Helmholtz equation for spherical geometry such that it vanishes on the sphere. The diffracted amplitude is expressed as an integral on the surface of the sphere that involves only the amplitude distribution in the aperture. In terms of the Laplace coefficients, the integral reduces to a product relation that allows us to define the transfer function of free space. The normal derivative of the Green’s function plays the role of the impulse response of the diffraction problem.

Instantaneous phase-shifted speckle interferometer for measurement of large optical structures

An instantaneous phase-shifting interferometer (PhaseCam, 4D Technology) was modified to a speckle phase-shifted interferometer. This interferometer was used to measure “diffused” objects such as a carbon fiber. Repeatability, accuracy, and dynamic range of the interferometer were measured. Different phase shifting algorithms were utilized to get rid of the high frequency speckle that modulates the low frequency fringes.

Rayleigh-sommerfeld diffraction theory and Lambert’s law

The Rayleigh-Sommerfeld diffraction theory is used to derive expressions for (1) the spectral irradiance on the surface of a hemisphere covering the aperture and (2) the spectral radiant intensity. For a uniform, noncoherent source-aperture, both calculations predict a cosθ angular variation, as is known to be the case of Lambertian sources. A cosine-fourth dependence of the spectral irradiance on a plane parallel to the aperture plane is also indicated.