Coherence and polarization properties of far fields generated by quasi-homogeneous planar electromagnetic sources

In studies of radiation from partially coherent sources the so-called quasi-homogeneous (QH) model sources have been very useful, for instance in elucidating the behavior of fields produced by thermal sources. The analysis of the fields generated by such sources has, however, been largely carried out in the framework of scalar wave theory. In this paper we generalize the concept of the QH source to the domain of the electromagnetic theory, and we derive expressions for the elements of the cross-spectral density matrix, for the spectral density, the spectral degree of coherence, the degree of polarization, and the Stokes parameters of the far field generated by planar QH sources of uniform states of polarization. We then derive reciprocity relations analogous to those familiar in connection with the QH scalar sources. We illustrate the results by determining the properties of the far field produced by transmission of an electromagnetic beam through a system of spatial light modulators.     

Reciprocity Relations with Partially Coherent Sources

Two general reciprocity relations are formulated, relating to planar sources of any state of spatial coherence and the radiation fields they generate. One of them expresses a reciprocity between the effective coherence area of the source and the effective angular spread of the radiation. The other expresses a reciprocity between the effective intensity spread across the source and the effective angular domain subtended at the source by the coherence area of the far field. Two well-known reciprocity relations associated with coherent and incoherent sources are obtained as special cases. The general reciprocity relations take a particularly simple form when the source is quasi-homogeneous.     

Influence of atmospheric turbulence on the spatial correlation properties of partially coherent flat-topped beams

The analytical expression for the spectral degree of coherence of partially coherent flat-topped beams propagating through the turbulent atmosphere is derived, and the spatial correlation properties are studied in detail. In particular, we find that the oscillatory behavior and phase singularities of the spectral degree of coherence may appear when partially coherent flat-topped beams propagate through the turbulent atmosphere; this behavior is very different from the behavior of Gaussian Schell-model beams. But the oscillatory behavior becomes weaker with increasing turbulence and even disappears when the turbulence is strong enough. The width of the spectral degree of coherence is always smaller than that of the spectral density in the far field when the turbulence is strong enough, whereas the width of the spectral degree of coherence in free space can be either larger or smaller than that of the spectral density in the far field.     

Propagation of partially coherent flat-topped beams through a turbulent atmosphere

Based on the modified beam model for flat-topped beams and the Schell model for partially coherent light, an expression for partially coherent flat-topped (PCFT) beams has been proposed. The propagation characteristics of PCFT beams with circular symmetry through a turbulent atmosphere have been studied. By using the generalized Huygens-Fresnel integral and Fourier transform method, the expressions for the cross-spectral density function and the average intensity have been given and the analytical expression for the root-mean-square width has been derived. The effects of the beam order, the spatial coherence, and the turbulent parameter on the intensity distributions and beam spreading have been discussed in detail. Our results show that the on-axis intensity of the beams decreases with increasing turbulence and decreasing coherence of the source, whereas the on-axis intensity of the beams in the far field decreases slightly with increasing beam order. The relative spreading of PCFT beams is smaller for beams with a higher order, a lower degree of global coherence of the source, a larger inner scale, and a smaller outer scale of the turbulence.     

Quantitatively characterizing fluctuations of dielectric susceptibility of tissue with Fourier domain optical coherence tomography

A new model of Fourier domain optical coherence tomography (FDOCT) is proposed, valid within the first Born approximation, which takes the fluctuations of the dielectric susceptibility of tissue into account. It is shown that the spectral electrical power at the detector in the FDOCT system is proportional to the Fourier component of the spatial correlation function of the dielectric susceptibility of the tissue, proportional to the squares of the spectrum of the incident light field and the amplitude reflectance of the reference mirror. One possible application of the obtained result is to use the measured spectral data of the spatial correlation function of the dielectric susceptibility to quantitatively characterize properties of tissue.     

Radiance Functions of Partially Coherent Fields

Abstract

The behaviour of the spectral radiance in fields generated by planar, secondary, quasi-homogeneous sources is investigated. Examples are presented which show how both the spectral intensity and the degree of coherence of the source affect the spatial and the angular distribution of the spectral radiance. These examples clearly show the influence of source coherence on the radiometric behaviour of partially coherent optical fields.     

Variable-coherence tomography for inverse scattering problems

We propose a technique for determining the pair-correlation function of a quasi-homogeneous medium. The method uses the variation of the spatial-coherence properties of the incident beam to generate two separate volumes of coherence where the field is correlated. Using this specially prepared beam, we reconstruct experimentally the correlation function of a scattering potential by recording the scattered intensity in only one direction.     

Theory of partial coherence for weakly periodic media

The second-order theory of partial coherence for scalar and TE or TM fields is developed for weakly periodic media, and the van Cittert-Zernike theorem of classical coherence theory is generalized for such media. The coherence properties of a wave field, generated by a quasi-homogeneous source distribution at the entrance plane of a finite weakly periodic medium, are calculated both inside such a structure and in the far field. The second-order theory of partial coherence for pulse propagation through weakly periodic media is also developed.     

Analysing the behaviour of partially coherent divergent Gaussian beams propagating through oceanic turbulence

Theoretical study of propagation behaviour of partially coherent divergent Gaussian beams through oceanic turbulence has been performed. Based on the previously developed knowledge of propagation of a partially coherent beam in atmosphere, the spatial power spectrum of the refractive index of ocean water, extended Huygens–Fresnel principle and the unified theory of coherence and polarization, analytical formulas for cross-spectral density matrix elements are derived. The analytical formulas for intensity distribution, beam width and spectral degree of coherence are determined by using cross-spectral density matrix elements. Then, the effects of some source factors and turbulent ocean parameters on statistical properties of divergent Gaussian beam propagating through turbulent water are analysed. It is found that beam’s statistical propagation behaviour is affected by both environmental and source parameters variations.     

A Study of the Propagation of Optical Spectra Using a Wavelength Independent Formulation of Diffraction Theory and Coherence Theory

Abstract

Basic relations in diffraction theory and coherence theory are rederived in spatial coordinates in units of a wavelength for each monochromatic component of the field. It is found that each component behaves identically in these coordinates independent of wavelength. This theory makes it clear that there is an intrinsic spectral shift that always occurs for any deterministic field as it propagates from a region near its sources into the far field. This theory also leads to a set of special conditions under which a partially coherent field has the same spectrum over a plane near its sources and also at every point in its far field. These conditions are believed to be satisfied by the fields radiated by natural thermal sources. Equations are given describing the relationships between the spectra of the source, the field near the source, and the radiation field under a wide range of conditions.     

Spatial correlation properties and correlation vortices of partially coherent vortex beams diffracted by an aperture

Taking the Gaussian–Schell model vortex beam as a typical example of partially coherent vortex beams, the spatial correlation properties and correlation vortices of partially coherent vortex beams diffracted by an aperture are studied. It is shown that the off-axis displacement and spatial coherence affects the spectral degree of coherence. The number and position of correlation vortices depend on the off-axis displacement, spatial coherence, aperture truncation and propagation distance, where the effect of aperture diffraction on the correlation vortices is stressed. The number of correlation vortices decrease as the truncation parameter increases. The correlation vortices in the diffracted field result from the vortex embedded in partially coherent beams at the source plane rather than from the aperture diffraction. The correlation vortices in the diffracted field appear even when the vortex core is stopped by the aperture.     

Radiation Efficiency of Partially Coherent Planar Sources with Various Intensity Distributions

Abstract

A general expression is derived for radiation efficiency, which is defined as a ratio of the radiated energy flux to the source-integrated intensity, of partially coherent planar sources with arbitrary intensity distributions. On the basis of the expression, radiation efficiencies of the sources with various intensity distributions are quantitatively investigated as a function of both the coherence condition and the intensity distribution across the source. By stemming from the present source model whose spatial coherence is characterized by Bessel functions, a physical origin of the variation in radiation efficiency is satisfactorily presented in connection with the spatial frequency spectrum of the source associated with evanescent waves. As a result, it is found that the broad spread in spatial frequency spectrum of the source results in the reduction of radiation efficiency for the sources with arbitrary intensity distributions including the quasi-homogeneous ones and that the behaviour of this kind is independent of the presence of evanescent waves, at least, for the quasi-homogeneous sources.     

Nonredundant array of apertures to measure the spatial coherence in two dimensions with only one interferogram

We propose to use a mask with a nonredundant array (NRA) of multiple apertures to measure spatial coherence in two dimensions. The spatial distribution of the apertures in the mask is made in such a way that we obtain a quasi-uniform sampling in the coherence domain. The spatial coherence is obtained by Fourier transform of the interferogram generated by the mask when it is illuminated by the light field under analysis.     

Coherence effects in the near-field

Abstract

Expressions are derived for the cross-spectral density and for the spectral degree of coherence of the near-field generated by a planar, secondary, homogeneous source of any spectral distribution and of any state of spatial coherence. The behaviour of the spectral degree of coherence and the spectral intensity in the immediate vicinity of the source are illustrated by computed examples.     

Optical Spatial Frequency Filtering of Radiographs

The frequency content of a photographic image can be considerably modified by using a suitable optical system. The Fraunhofer diffraction pattern (the Fourier transform) of a transparency is produced by using parallel light from a laser. This spatial frequency spectrum can be differentially filtered to produce a reconstructed image of arbitrary frequency content. The radiographic image is a convolution of the X-ray source with the object under examination. The filtering operation lessens the degrading effect of penumbra by means of a deconvolution with a filter that is representative of the Fourier transform of the X-ray source.

Manufacture of the spatial filter is described and results showing enhancement of detail in radiographs by reduction of penumbra are presented.     

Spectral changes of polychromatic stochastic electromagnetic vortex beams propagating through turbulent atmosphere

The unified theory of coherence and polarization and the propagation law of 2 × 2 cross-spectral density are employed to investigate spectral changes of the polychromatic stochastic electromagnetic vortex beam propagating in turbulent atmosphere. It is shown that the spectral changes of a polychromatic stochastic electromagnetic vortex beam in turbulent atmosphere differ from those of the beam without vortex. Specially, the on-axis relative spectral shifts exhibit not only blue-shift, but also red-shift. It is also shown that the topological charge, the correlation length and the refractive index structure constant influence the spectral changes of polychromatic stochastic electromagnetic vortex beams in a turbulent atmosphere.     

Dirac bra-ket in radiometry of quasi-homogeneous sources

The concept of "throughput" is used in traditional radiometry of Lambertian sources for computing and estimating the radiant flux passed through a pair of stops, in particular through the window and the pupil of an optical system. It is shown that in a more general case of quasi-homogeneous sources for energetic calculations of the perfect optical system, one must use instead of the throughput a functional that is similar to the famous "Dirac bra-ket." This functional takes into account the radiation pattern of the source. As the Dirac bra-ket satisfies the axioms of the inner product, powerful mathematical tools of functional analysis for the energy calculation of the optical systems are used. The main equations and principles of radiometry (the principle of reversibility and Maxwell's principle) are reformulated from the concept "throughput" into the concept "Dirac bra-ket." For generalization of Maxwell's principle to the class of quasi-homogeneous sources the concept of "effective stops" is introduced.     

Influence of spatial coherence on scattering by a particle

We analyze the effect of partial spatial coherence on the scattering of light by an arbitrary particle. We extend the definition of the extinction cross section to spatially partially coherent fields. We then discuss the effect of the partial coherence on the extinction scattering cross section by introducing the Wigner transform. It is shown that for rotationally invariant scatterers, the extinction cross section does not depend on the coherence of the incident field. The effect of partial coherence on the angular behavior of the scattered intensity is also discussed in the framework of the Wigner transform. The implications for practical applications are considered.     

FFT calculation of magnetic fields in air coils

The magnetostatic field produced by an air coil that possesses one-dimensional periodicity can be expressed as a one-dimensional discrete convolution of two functions. The first function expresses the field produced by a single turn of the coil. The second is a shape function; it expresses the spatial position and strength of current of each turn of the coil. The discrete convolution of these two functions gives the magnetostatic field produced by the coil. This paper presents one application of linear system theory to an air coil calculation, the use of the fast Fourier transform (FFT) in computing magnetostatic magnetic fields from air coils. A program is described which uses FFT convolution to perform this calculation.     

Two dimension-reduction probabilistic models for simulating nonstationary turbulent wind fields

The reasonable modeling of a nonstationary stochastic turbulent wind field is an important basis and premise for the analysis of the wind-induced response and reliability of engineering structures. In the present study, two dimension-reduction probabilistic models are established for simulating the multi-dimensional and multi-variable nonstationary turbulent wind fields based on the double proper orthogonal decomposition (DPOD) and the double spectral representation method (DSRM). Among them, the DPOD, originally used to simulate a stationary turbulent wind field, is extended to a nonstationary one, and the DSRM is a newly proposed method for a nonstationary turbulent wind field with a large number of simulation points. In essence, the DPOD is a discrete method with explicit physical significance and flexible spatial location of simulation points, while the DSRM is a continuous method, of which the simulation efficiency is theoretically independent of the number of simulation points. Furthermore, by introducing the dimension-reduction methods based on random function and POD-FFT (Fast Fourier transform) technique into the DPOD and the DSRM, the nonstationary stochastic turbulent wind field can be effectively described with merely three elementary random variables. Numerical examples of the nonstationary stochastic turbulent wind fields acting on a long-span bridge and a communication tower fully verify the effectiveness and superiority of the proposed methods.