Gaussian Schell-model Source Research Articles

The quasi-homogeneous approximation for a class of three-dimensional primary sources

We investigate the validity of the quasi-homogeneous approximation for three-dimensional primary Gaussian Schell-model sources. It is shown that the quasi-homogeneous approximation is not valid for such a source unless two conditions are satisfied, one of which depends only upon the spatial characteristics of the source, and the other depends upon the wavelength of the radiation. The second of these conditions appears not to have been appreciated in previous work relating to the quasi-homogeneous approximation, and its significance to the foundations of radiometry is discussed.

Effects of lens aperture on the average intensity in a fractional Fourier plane

Effects of the aperture of a lens which is used to conduct an optical fractional Fourier transform on the fractional average intensity are analysed numerically using a computer. The fractional average intensity means the average intensity of the optical field produced from a partially coherent source in a fractional Fourier plane. In the present study, the fractional average intensities of a monochromatic Gaussian Schell model (GSM) source are evaluated using the root-mean square (RMS) difference between those with and without the effects of the lens aperture. It is clearly shown from our analysis that the aperture size of the lens should be twice as large as the GSM source size or greater in order to satisfy the condition of the RMS difference .

Partially coherent sources with helicoidal modes

Abstract On the basis of the modal theory of coherence, we study partially coherent sources whose modes belong to the class of Laguerre-Gauss functions for which the Laguerre polynomial has zero order. These modes present a phase profile with a helicoidal structure, which is responsible for notable phenomena, such as the propagation of optical vortices, beam twisting, and the presence of dislocations in interference patterns. By suitably choosing the eigenvalues associated with such modes, different partially coherent sources are obtained: sources with a flattened Gaussian profile, twisted Gaussian Schell-model sources with a saturated twist, and a new class of sources having an annular profile. Owing to the shape-invariance property of the underlying modes, the fields radiated by these sources do not change their transverse profile through propagation, except for scale and phase factors. We also prove that, if any such source is covered by a circularly symmetric filter, the new modal structure can be found in a straightforward manner.

Partially coherent sources with helicoidal modes

On the basis of the modal theory of coherence, we study partially coherent sources whose modes belong to the class of Laguerre-Gauss functions for which the Laguerre polynomial has zero order. These modes present a phase profile with a helicoidal structure, which is responsible for notable phenomena, such as the propagation of optical vortices, beam twisting, and the presence of dislocations in interference patterns. By suitably choosing the eigenvalues associated with such modes, different partially coherent sources are obtained: sources with a flattened Gaussian profile, twisted Gaussian Schell-model sources with a saturated twist, and a new class of sources having an annular profile. Owing to the shape-invariance property of the underlying modes, the fields radiated by these sources do not change their transverse profile through propagation, except for scale and phase factors. We also prove that, if any such source is covered by a circularly symmetric filter, the new modal structure can be found in a straightforward manner.

Properties of the Gaussian Schell-model source field in a fractional Fourier plane

A fractional cross-spectral density is defined by use of an optical fractional Fourier transform based on the rotation of a Wigner distribution function within the accuracy of the paraxial approximation. The properties of the planar Gaussian Schell-model (GSM) source field in a fractional Fourier plane are analyzed theoretically. As a result, it becomes clear that the extent, the wave-front-curvature radius, and the spectral shift of the fractional Fourier field depend strongly on the fractional order of the Fourier transform, the Fresnel number associated with the GSM source size, and the scaled spatial-coherence length of the GSM source.

Propagation of partially coherent Gaussian Schell-model sources in nonlinear media

Propagation of mutual coherence functions of partially coherent Gaussian Schell-model beams in a medium with third-order nonlinearities is considered. It is shown that incoherence suppresses self-focusing. The spatial coherence of the axial part of the beam improves in the self-focusing regime, whereas the transverse correlation length of the beam decreases in the same region. At high powers, waveguide formation is also studied in media with saturating nonlinearity.

Coherence properties of a source array derived from a Gaussian Schell-model beam

The spatial coherence properties of a source array derived from a quasi-monochromatic Gaussian Schell-model (GSM) beam are analyzed. The method for realizing the source array by use of a GSM source and a Gaussian amplitude grating was originally proposed in our preceding paper [J. Opt. Soc. Am. A 11, 2112 (1994)], in which the intensity distribution was analyzed. One result that we obtain is that the degree of spatial coherence of each element in the source array is the same as that at the other elements. In particular, each element becomes equivalent to the GSM source and has a high degree of spatial coherence with the other elements if three conditions are satisfied simultaneously: (1) the spatial-coherence length of the GSM source is larger than the GSM source size, (2) the GSM source size normalized by a period of the grating is 0.3 to 0.5, and (3) each aperture size of the grating normalized by the period is 0.05 or less.

Reflection and transmission properties of holographic mirrors and holographic Fabry–Perot filters III Holographic Fabry–Perot filters

The reflection and transmission coefficients of holographic Fabry-Perot filters (HFPF's) are derived. Angular selectivities and wavelength sensitivities of the HFPF's on transmitted light are investigated. The effects of transmission by a HFPF on the spectrum and on the coherence properties of a partially coherent incident light are also analyzed. Numerical examples are presented for the case in which the incident light is generated by a Gaussian Schell-model source.

Realization of multiple images resulting from the partially coherent light from a Gaussian Schell-model source

Multiple images with spots similar to those produced by a Gaussian Schell-model (GSM) source are realized in the observation plane of the Fresnel diffraction region at a distance 4NZT (N, a positive integer; ZT, Talbot distance) from the GSM source. In this formation the grating with apertures of a Gaussian amplitude transmittance is placed midway between the GSM source and the observation plane. We base our analysis on coherence theory, using the propagation of the ambiguity function from which the average intensity can be derived. It is found that the intensity distribution of the spots in the multiple images becomes wider with decreases in the source size, the spatial-coherence length of the source, and each aperture size of the grating used.

Reflection and transmission properties of holographic mirrors and holographic Fabry–Perot filters II Holographic mirrors with partially coherent light

The reflection and transmission properties of holographic mirrors (HM's) under partially coherent illumination are investigated with emphasis on the properties of reflected light. The effects of a HM on the spectrum and on the coherence properties of partially coherent incident light are studied. We show that within angular and frequency-selectivity ranges of HM's (where their reflectivities are nearly uniform), the changes in the spectrum and in the coherence properties are negligible. Yet changes are expected when the spectrum of the incident light falls beyond frequency-selectivity ranges of the HM's or when the spectrum of the reflected light is observed beyond the angular-selectivity ranges of the mirrors. We illustrate the results by considering in detail the performance of HM's when the incident light is produced by planar secondary Gaussian Schell-model sources. Some computed curves are presented.

Equiambiguity-function ellipse of Gaussian Schell-model beams

The ambiguity function properties of a Gaussian Schell-model (GSM) beam are theoretically analyzed in the paraxial approximation. In analysis the behavior of an equi-ambiguity-function ellipse (trajectory of the ambiguity function at a constant value) is investigated in connection with the propagation properties of the GSM beam. As a result, its behavior is revealed in relation to the fundamental beam parameter, by which both the GSM beam radius and the wave-front–curvature radius of the beam are uniquely described as a function of the propagation distance. In addition, the effects of both the propagation distance and the spatial coherence of the GSM source on the behavior of the ellipse are investigated.

Spectral changes of light radiated by three-dimensional, anisotropic gaussian Schell-model sources

Expressions are derived for both the radiant intensity and the far-field spectrum produced by a three-dimensional, anistropic gaussian Schell-model source. On the basis of the spectrum expression, the shift of the spectral line on light propagation is investigated in some detail with reference to the correlation-induced spectral changes of the Wolf effect. As some numerical examples, the spectral shifts are illustrated in a polar diagram as a function of the observation angle. It is found from the figure that the angular distribution of the shift varies depending strongly on the three-dimensional anisotropy of the source.

Spectral changes in inhomogeneous media; a quasi-optical approach

In the past decade spectral changes depending on the spatial coherence of the source have been described by a cross-spectral density method. It is shown that for Gaussian-Schell model sources a quasi-optical approach is valid which uses Gaussian beams and ABCD matrices to describe the optical system.

A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams behaving like gaussian Schell-model beams

Based on the M 2-factor concept and the equivalence between a beam generated by a gaussian Schell-model (GSM) source and a mixed-mode laser (MML) beam, a simple method is presented to estimate the number of effectively oscillating modes and the corresponding weighting factors of MML beams simultaneously.

Gaussian Schell model sources in one-dimensional first-order systems with loss or gain: errata

A detailed examination of the propagation of Gaussian–Schell model sources in one-dimensional, possibly nonlossless, first-order systems is constructed. The laws of focusing are derived. The conditions for periodicity of the Gaussian–Schell model source are derived. This result generalizes the well-known result −2 ≤ A + D ≤ 2 for confinement of a perfectly coherent Gaussian beam to the partially coherent nonlossless case. When loss or gain is present several conditions must be satisfied simultaneously for periodicity. The self-consistent solutions are derived and the perturbation stability of the solutions is studied. A physical realization of an arbitrary nonlossless one-dimensional ABCD system is derived, which yields a convenient formula for deciding whether the ABCD system has loss or gain. Special attention is devoted to real and ripple systems.

Gaussian–Schell model sources in one-dimensional first-order systems with loss or gain

A detailed examination of the propagation of Gaussian-Schell model sources in one-dimensional, possibly nonlossless, first-order systems is constructed. The laws of focusing are derived. The conditions for periodicity of the Gaussian-Schell model source are derived. This result generalizes the well-known result -2 </= A + D </= 2 for confinement of a perfectly coherent Gaussian beam to the partially coherent nonlossless case. When loss or gain is present several conditions must be satisfied simultaneously for periodicity. The self-consistent solutions are derived and the perturbation stability of the solutions is studied. A physical realization of an arbitrary nonlossless one-dimensional ABCD system is derived, which yields a convenient formula for deciding whether the ABCD system has loss or gain. Special attention is devoted to real and ripple systems.

Changes in the coherence and spectral properties of partially coherent light reflected from a dielectric slab

The changes in the spectrum and in the degree of coherence of partially coherent light reflected from a stratified medium are studied. Reflection from a homogeneous dielectric slab is analyzed in detail for the case when the incident field is produced by a one-dimensional Gaussian Schell-model source. Results of numerical calculations are presented for several representative values of the parameters. Appreciable changes in the spectrum and in the degree of coherence are found to be produced in some cases.

First-order sources in first-order systems: second-order correlations: errata

The laws of propagation of the second-order correlation of a generalized Gaussian Schell-model source in first-order systems are deduced from the canonical operator formalism. A far-reaching generalization of Kogelnik's q parameter is derived, valid in n dimensions, for misaligned, lossy, and anisotropic sources and systems. Even in one dimension these results extend previous findings. The degree of global coherence is generalized and shown to be an invariant in n dimensions for anisotropic lossless systems. A new invariant, the degree of ripple, is found for a special class of lossy first-order system, which includes the Gaussian aperture.

First-order sources in first-order systems: second-order correlations

The laws of propagation of the second-order correlation of a generalized Gaussian Schell-model source in first-order systems are deduced from the canonical operator formalism. A far-reaching generalization of Kogelnik's q parameter is derived, valid in n dimensions, for misaligned, lossy, and anisotropic sources and systems. Even in one dimension these results extend previous findings. The degree of global coherence is generalized and shown to be an invariant in n dimensions for anisotropic lossless systems. A new invariant, the degree of ripple, is found for a special class of lossy first-order system, which includes the Gaussian aperture.

Wolf effect in homogeneous and inhomogeneous media

We derive expressions for the spectrum of the field produced by planar, secondary Gaussian Schell-model sources after propagation in free-space, homogeneous dispersive media, and graded-index fibers. Our results show, for the first time to our knowledge, the development of correlation-induced spectral changes (the Wolf effect) as a function of the propagation distance from the source plane. An important result of our study is the prediction of the enhancement of the Wolf effect for propagation in media of index of refraction larger than unity. In the case of graded-index fibers having a parabolic index profile, the source spectrum is shown to reproduce periodically at distances at which such fibers image the source.