The Spatial Coherence Properties of Gaussian Schell-model Beams

The transverse and longitudinal spatial coherence properties of the light beams generated by planar gaussian Schell-model sources are discussed. It is found that for all gaussian Schell-model beams the ratio of the transverse coherence length to the beam width remains invariant upon propagation. An examination of the longitudinal coherence for both on-axis and off-axis pairs of points indicates that the longitudinal coherence will not, in general, die out as the separation between the points is increased. Rather, the degree of longitudinal coherence will approach a finite (non-zero) value as long as the source contains a finite coherence area, regardless of how small this area may be. Gaussian quasihomogeneous beams are studied as a limiting case. The relation of the present work to the analysis of speckle size is briefly discussed.     

Propagation of Gaussian Schell-model Beams in Dispersive and Absorbing Media

Abstract

The spectral properties and the coherence properties of Gaussian Schell-model beams, propagating in dispersive and absorbing media, are discussed. Unlike in free space, the ratio of the transverse spectral correlation length to the beam width is found to increase on propagation. Consequently upon propagation the beam becomes spatially more coherent at each frequency. Numerical results for the spectrum and for the degrees of spectral and of temporal coherence of the field are presented for some selected values of the beam parameters and for several values of the propagation distance. Propagation in a gain medium is also briefly discussed.     

Gaussian Schell-model beams propagating through polarization gratings

The effects of polarization gratings on partially coherent beams are investigated by studying a Gaussian Schell-model beam impinging on a linear polarizer whose transmission axis varies periodically along one transverse direction. Analytical expressions for the beam polarization-coherence matrix after the grating are obtained. In particular, the evolution of the degree of polarization upon propagation is analyzed. Different behaviors of the output beam, depending on the beam parameters and on the period of the grating, are exhibited. In particular, it is shown that, by suitably choosing the latter quantities, it is possible to obtain not only any desirable value of the degree of polarization of the output beam but also particular distributions of such parameters across the transverse sections of the beam.     

Fractional Fourier transform for partially coherent off-axis Gaussian Schell-model beam

The fractional Fourier transform (FRT) is applied to a partially coherent off-axis Gaussian Schell-model (GSM) beam, and an analytical formula is derived for the FRT of a partially coherent off-axis GSM beam. The corresponding tensor ABCD law for performing the FRT of a partially coherent off-axis GSM beam is also obtained. As an application example, the FRT of a partially coherent linear laser array that is expanded as a sum of off-axis GSM beams is studied. The derived formulas are used to provide numerical examples. The formulas provide a convenient way to analyze and calculate the FRT of a partially coherent off-axis GSM beam.     

Gaussian Process Modeling Using the Principle of Superposition

Partial differential equation (PDE) models of physical systems with initial and boundary conditions are often solved numerically via a computer code called the simulator. To study the dependence of the solution on a functional input, the input is expressed as a linear combination of a finite number of basis functions, and the coefficients of the bases are varied. In such studies, Gaussian process (GP) emulators can be constructed to reduce the amount of simulations required from time-consuming simulators. For linear initial-boundary value problems (IBVPs) with functional inputs as additive terms in the PDE, initial conditions, or boundary conditions, the IBVP solution is theoretically a linear function of the coefficients conditional on all other inputs, which is a result called the principle of superposition. Since numerical errors cause deviation from linearity and nonlinear IBVPs are widely solved in practice, we generalize the result to account for nonlinearity. Based on this generalized result, we propose mean and covariance functions for building GP emulators that capture the approximate conditional linear effect of the coefficients. Numerical simulations demonstrate the substantial improvements in prediction performance achieved with the proposed emulator. Matlab codes for reproducing the results in this article are available in the online supplement.     

Synthetic Incoherence via Scanned Gaussian Beams

Tomography, in most formulations, requires an incoherent signal. For a conventional transmission electron microscope, the coherence of the beam often results in diffraction effects that limit the ability to perform a 3D reconstruction from a tilt series with conventional tomographic reconstruction algorithms. In this paper, an analytic solution is given to a scanned Gaussian beam, which reduces the beam coherence to be effectively incoherent for medium-size (of order 100 voxels thick) tomographic applications. The scanned Gaussian beam leads to more incoherence than hollow-cone illumination.     

Spectral shift of an electromagnetic Gaussian Schell-model beam propagating through tissue

Based on the unified theory of coherence and polarization for stochastic electromagnetic beams, spectral changes of an electromagnetic Gaussian Schell-model (EGSM) beam propagating through tissue are investigated in detail. Numerical results show that the spectral shift of an electromagnetic GSM beam is closely determined by the parameters of the tissue (e.g. the fractal dimension of tissue and the ensemble-averaged variance of the refractive index fluctuations). The spectral shift changes from blue-shift to red-shift with the increase of the transverse coordinate.     

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.     

Evolution properties of a twisted Gaussian Schell-model beam in a uniaxial crystal

Paraxial propagation of a linearly polarised twisted Gaussian Schell-model (TGSM) beam in a uniaxial crystal is investigated. With the help of a tensor method, an analytical formula for the cross-spectral density (CSD) of a TGSM beam propagating in a uniaxial crystal orthogonal to the optical axis is derived. The evolution properties of the intensity distribution, the coherence widths and the twist factor of a TGSM beam in a uniaxial crystal are illustrated numerically. It is shown that the TGSM beam becomes an anisotropic TGSM beam in a uniaxial crystal and its evolution properties are closely related with its initial coherence, twist factor and the parameters of the crystal. The uniaxial crystal provides an effective way for modulating the properties of a TGSM beam, which will be useful in some applications, such as free-space optical communications and nonlinear optics, where a partially coherent beam with controlled beam profile and twist factor is required.     

Design of diffractive axicons producing uniform line images in Gaussian Schell-model illumination

We present a design method for diffractive axicons in spatially partially coherent Gaussian Schell-model illumination. The method of stationary phase applied to the Fresnel diffraction integral for on-axis intensity leads, on requiring a uniform axial image profile, to a second-order differential equation for the optimal axicon phase function. The first integral can be formally performed, and the phase function is subsequently obtained numerically. The correctness of the synthesized phase profiles is confirmed by numerical simulations using partially coherent Fresnel diffraction theory. The effects of input-beam spot size and coherence width are assessed, and influences of different forms of apodization, including asymmetric functions for narrow incident beams, in annular-aperture diffractive axicons are examined.     

The Intensity Distribution of a Gaussian Schell-model Beam Focused by an Aperture Lens

Abstract

Starting from the generalized Huygens-Fresnel diffraction integral, we study the intensity distribution of a Gaussian Schell-model (GSM) beam diffracted at an aperture lens. A great number of numerical calculations have been performed to illustrate the focused field characteristics. Isophote diagrams are given for systems of different Fresnel numbers, which focus GSM beams, and the related analysis is presented.     

Statistical properties of a radially polarized twisted Gaussian Schell-model beam in a uniaxial crystal

We derive the analytical formulas for the elements of the cross-spectral density matrix of a radially polarized partially coherent beam with a twist phase named radially polarized twisted Gaussian Schell-model (RPTGSM) beam propagating in a uniaxial crystal, and explore the statistical properties, such as the intensity distribution, the degree of polarization (DOP) and the state of polarization (SOP) of such beam in a uniaxial crystal with the help of the derived formulas. It is found that the statistical properties of a RPTGSM beam are closely related with the twist phase and the anisotropy of the uniaxial crystal, e.g. the twist phase leads to the rotation of the intensity, DOP and SOP distributions, and the anisotropy of the uniaxial crystal leads to the asymmetry distributions of these statistical properties. Our results will be useful for designing light field with prescribed intensity, DOP and SOP distributions, and may be useful in optical manipulations and free-space optical communications.     

Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams in dispersive and absorbing media

By adopting a new tensor method, we derived an analytical propagation formula for the cross-spectral density of partially coherent twisted anisotropic Gaussian Schell-model (GSM) beams through dispersive and absorbing media. Using the derived formula, we studied the evolution properties and spectrum properties of twisted anisotropic GSM beams in dispersive and absorbing media. The results show that the dispersive and absorbing media have strong influences on the propagation properties of twisted anisotropic GSM beams and their spectrum evolution. Our method provides a simple and convenient way to study the propagation of twisted anisotropic GSM beams in media with complex refractive index.     

Spectral switches of vector Gaussian Schell-model beams passing through a spherically aberrated lens

On the basis of the polarization matrix and propagation equation of the cross-spectral density matrix, the spectral changes for the on-axis spectrum of vector Gaussian Schell-model (GSM) beams focused by a spherically aberrated lens are studied, where the influence of the spherical aberration, correlation and polarizer on the behaviour of the spectral switch is stressed. It is shown that the spectrum of vector GSM beams passing through a spherically aberrated lens may be blueshifted and redshifted in comparison with the source spectrum, and a spectral switch may also take place. However, there are only blue shifts for two kinds of cases passing through an aberration-free lens and an astigmatic lens. The spectral switch arises from the oscillation of the spectral modifier with frequency.     

Experimental observation of truncated fractional Fourier transform for a partially coherent Gaussian Schell-model beam

The truncated fractional Fourier transform (FRT) is applied to a partially coherent Gaussian Schell-model (GSM) beam. The analytical propagation formula for a partially coherent GSM beam propagating through a truncated FRT optical system is derived by using a tensor method. Furthermore, we report the experimental observation of the truncated FRT for a partially coherent GSM beam. The experimental results are consistent with the theoretical results. Our results show that initial source coherence, fractional order, and aperture width (i.e., truncation parameter) have strong influences on the intensity and coherence properties of the partially coherent beam in the FRT plane. When the aperture width is large, both the intensity and the spectral degree of coherence in the FRT plane are of Gaussian distribution. As the aperture width decreases, the diffraction pattern gradually appears in the FRT plane, and the spectral degree of coherence becomes of non-Gaussian distribution. As the coherence of the initial GSM beam decreases, the diffraction pattern for the case of small aperture widths gradually disappears.     

Augmented ABCD Matrix for Non-specular Diffraction of Gaussian Beams

Abstract

Optical beams reflected or transmitted by a dielectric interface undergo several non-specular deformations: changes in the beam amplitude, lateral, focal and angular shifts of the beam axis and a magnification of the beam width. In this communication the non-specular interaction of the beam with, in general, multilayered, planar dielectric structure is described in terms of the ray transfer matrix formalism. A relationship between the beam deformations and a 3 × 3 ray transfer matrix of the structure is built. Transfer matrix elements are shown in a simple form dependent on the real deformations.     

Focal Shift Formula for Focused,Apertured Gaussian Beams

A formula is developed for fast computing focal shift in the diffraction field of a Gaussian beam that is focused by a thin lens filled in a circular aperture of any prescribed radius. This formula is a generalization of two previously known focal shift formulae, one for un-apertured Gaussian beam, another for plane-wave beam.     

Generalized huygens-fresnel diffraction integral for misaligned asymmetric first-order optical systems and decentered anisotropic Gaussian Schell-model beams

The generalized Huygens-Fresnel diffraction integral for misaligned asymmetric first-order optical systems is derived by using the canonical operator method, which enables us to study propagation properties of anisotropic Gaussian Schell-model (AGSM) beams through misaligned asymmetric first-order optical systems. It is shown that under the action of misaligned asymmetric first-order optical systems AGSM beams do not preserve the closed property. Therefore generalized partially coherent anisotropic Gaussian Schell-model beams called decentered anisotropic Gaussian Schell-model (DAGSM) beams are introduced, and AGSM beams can be regarded as a special case of DAGSM beams.     

Beam Quality Dependence on the Coherence Length of Gaussian Schell-model fields Propagating Through ABCD Optical Systems

Abstract

The quality behaviour of Gaussian Schell-model fields propagating through general (non-orthogonal) ABCD optical systems has been characterized by introducing a number of parameters. Such parameters have been given in terms of the Wigner distribution function of the field, closely connected with its second-order coherence features. Dependence of the focusing capabilities, transversal widths and beam asymmetry on the coherence length of the field has been investigated.     

Second-order Coherence Effects Due to the Superposition of Two Independent Thermal Light Beams

A photon-coincidence counting technique has been used to observe the second-order coherence in a plane receiving light from two small incoherently illuminated apertures. Measurements made for two wavelengths, 546·1 and 435·8 nm, gave results consistent with predictions from the van Cittert-Zernike theorem.