Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes through complex ABCD optical systems

The change of coherence and polarization of an electromagnetic beam modulated by a random anisotropic phase screen passing through any optical system is found within the framework of complex ABCD-matrix theory This means that the formalism can treat imaging and Fourier transform and free-space optical systems, as well as fractional Fourier transform systems, with finite-size limiting apertures of Gaussian transmission shape. Thus, the current paper shall be considered as a continuation, extension, and generalization of a previous work by Shirai and Wolf [J. Opt. Soc. Am. A21, 1907 (2004)]. It will be shown that the inclusion of apertures in the optical system strongly influences not only the propagation of spatial coherence but also the degree of polarization of a propagating field. Analytical expressions of coherence and polarization propagation will be given in terms of the matrix elements for any complex optical system.     

Changes in the polarization and coherence of a random electromagnetic beam propagating through a misaligned optical system

On the basis of the generalized diffraction integral formula for misaligned optical systems in the spatial domain, an analytical propagation expression for the elements of the cross-spectral density matrix of a random electromagnetic beam passing through a misaligned optical system is derived. Some analyses are illustrated by numerical examples relating to changes in the spectral degree of polarization and in the spectral degree of coherence of an electromagnetic Gaussian-Schell-model beam propagating through such an optical system. We find that the degree of polarization in the neighboring areas of the focal plane is oscillating, and the effect of misalignment on coherence is not so evident as that on polarization.     

The effect of a moving diffuser on a random electromagnetic beam

The effects of a moving diffuser on the spectrum, on the spectral degree of polarization, and on the spectral degree of coherence of a random electromagnetic beam are investigated. It is found that while the spectrum and the spectral degree of coherence change on transmission, the degree of polarization does not.     

Stochastic electromagnetic beams focused by a bifocal lens

In this paper, we study the focusing of a stochastic electromagnetic beam by a bifocal lens. By taking the electromagnetic Gaussian Schell-model (EGSM) beam as an example, the changes in the spectral density, in the spectral degree of coherence, and in the spectral degree of polarization of the EGSM beam as the beam is focused by an unapertured bifocal lens are investigated. It is shown that the spectral density, the spectral degree of coherence, and the spectral degree of polarization of the focused electromagnetic EGSM beams depend upon the coherence lengths and focal lengths of the bifocal lens. The influence of the coherence lengths and the focal lengths on the focused spectral density, the spectral degree of coherence, and the spectral degree of polarization are investigated in great detail.     

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.     

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.     

Polarization changes in a partially coherent radially polarized doughnut beam through turbulent ocean

On the basis of the extended Huygens–Fresnel integral principle and unified theory of coherence and polarization of light, we studied the effects of oceanic turbulence on polarization properties of a partially coherent radially polarized doughnut (PCRPD) beam. The ocean-induced fluctuations in the refractive index are assumed be driven by temperature and salinity fluctuations. Numerical examples of changes in polarization properties, such as the degree of polarization, the degree of ellipticity, and the orientation angle in the oceanic turbulence for the PCRPD beam, are given. Our analysis demonstrates how polarization of the PCRPD beam is affected by statistical properties of the source and by several parameters of oceanic turbulence. We find that the propagation of the PCRPD beam is different from that of stochastic beams in oceanic turbulence. The degree of polarization for the PCRPD beam approaches a certain steady value, and the elliptical polarized state of the fully polarized portion of the beam will become fully linear in the far field.     

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.     

Phase-space representation and polarization domains of random electromagnetic fields

The phase-space representation of stationary random electromagnetic fields is developed by using electromagnetic spatial coherence wavelets. The propagation of the field's power and states of spatial coherence and polarization results from correlations between the components of the field vectors at pairs of points in space. Polarization domains are theoretically predicted as the structure of the field polarization at the observation plane. In addition, the phase-space representation provides a generalization of the Poynting theorem. Theoretical predictions are examined by numerically simulating the Young experiment with electromagnetic waves. The experimental implementation of these results is a current subject of research.     

Changes in the spectrum, in the spectral degree of polarization, and in the spectral degree of coherence of a partially coherent beam propagating through a gradient-index fiber

Expressions are derived for the cross-spectral density matrix of an electromagnetic Gaussian Schell-model beam propagating through a paraxial ABCD system. Using the recently developed unified theory of coherence and polarization of electromagnetic beams and the ABCD matrix for gradient-index fibers, we study the changes of the spectral density, of the spectral degree of polarization, and of the spectral degree of coherence of such a beam as it travels through the fiber. Effects of material dispersion are also considered.     

Reconstruction of a complex-valued object in double-passage coherent imaging through a random-phase screen

We consider the reconstruction of a complex-valued object that is coherently illuminated and viewed through the same random-phase screen. The reconstruction is based on two intensity measurements: the intensity of the Fourier transform of the image and the intensity of the Fourier transform of the image when modulated with an exponential filter. The illumination beam has a Gaussian intensity profile of arbitrary width, and the phase screen is assumed to be described by a Gaussian random process of large variance and arbitrary correlation length. Computer-simulated examples of the reconstruction of a two-dimensional complex object demonstrate that the reconstruction is robust.     

Changes in Polarization of a Light Beam with Arbitrary Autocoherence Propagating in a Birefringent Crystal

Based on the Mueller-Stokes matrix formalism and the Wolf autocorrelation functions, a general approach is developed for studying the polarization characteristics of a light beam with arbitrary autocoherence during its interaction with various media. This makes it possible to take into account the dispersion of the medium and the coherence properties of the light field. It is shown that the propagation of a polychromatic light beam with spectrally pure polarization in a certain medium can be described by an integrated Mueller matrix. For a linear phase plate, the matrix elements can be expressed by means of the modulus and the phase of the complex degree of autocoherence. The polarization of a light beam emitted by a black-body radiator and the influence of the dispersion in a KDP crystal in quasi-monochromatic light are discussed as applications.     

Reduction of polarization-induced artifacts in grating-based spectrometers

An optical device that converts unpolarized light into a single polarization state is described. The device is based on a polarizing beam splitter that separates the two polarization directions. The beam splitter is combined with two pairs of equilateral prisms that are used to collimate the two beams in terms of both propagation and polarization directions. When it is used in combination with a blazed diffraction grating, this device is shown to effectively remove the polarization dependence of the first-order diffracted power. The device has an insertion loss of approximately 14% for purely s-polarized light. However, for unpolarized light incident upon the two gratings studied here, the increased throughput of the p-polarized component leads to an average relative gain in overall efficiency of 13%-19%, depending on the grating. In collimating the two polarization directions, the device may cause a reduction in spectral resolution for a rectangular entrance slit. As a result, the device is more likely to find use in spectrometers that have a circular aperture, such as that provided by an optical fiber.     

Liquid crystal over silicon device characteristics for holographic projection of high-definition television images

We discuss some fundamental characteristics of a phase-modulating device suitable to holographically project a monochrome video frame with 1280 x 720 resolution. The phase-modulating device is expected to be a liquid crystal over silicon chip with silicon area similar to that of commercial devices. Its basic characteristics, such as number of pixels, bits per pixel, and pixel dimensions, are optimized in terms of image quality and optical efficiency. Estimates of the image quality are made from the noise levels and contrast, while efficiency is calculated by considering the beam apodization, device dead space, diffraction losses, and the sinc envelope.     

Propagation properties of partially coherent radially polarized beam in a turbulent atmosphere

Based on the Huygens–Fresnel principle and the unified theory of coherence and polarization of partially coherent beams, we investigate the propagation characteristics of a partially coherent radially polarized doughnut (PCRPD) beam in a turbulent atmosphere. It is found that, after propagating through a turbulent atmosphere, the doughnut beam spot is changed into a circular Gaussian beam. Moreover, the degree of coherence, the degree of polarization and the degree of cross-polarization of the beam will change on propagation, and this change is dependent upon the degree of coherence of the source and atmospheric turbulence.     

Effect of spatial variability and propagation of seismic ground motions on the response of multiply supported structures

The response of lifelines, modelled as two- and three-span continuous symmetric beams of various lengths, subjected to partially and fully correlated seismic ground motions is examined. The partial correlation of the input motions consists of a term that characterizes the loss of coherence and a term that represents the apparent propagation (phase difference). The analysis suggests that the propagation effects may be neglected when the motions at the site exhibit loss of coherence; however, when the motions are coherent, phase differences may produce higher or lower response than the one induced by fully correlated motions, depending on whether the dominant modes are antisymmetric or symmetric at the location along the beam and for the response quantity (bending moment or shear force) under consideration.     

Electromagnetic spatial coherence wavelets

The recently introduced concept of spatial coherence wavelets is generalized to describe the propagation of electromagnetic fields in the free space. For this aim, the spatial coherence wavelet tensor is introduced as an elementary amount, in terms of which the formerly known quantities for this domain can be expressed. It allows for the analysis of the relationship between the spatial coherence properties and the polarization state of the electromagnetic wave. This approach is completely consistent with the recently introduced unified theory of coherence and polarization for random electromagnetic beams, but it provides further insight about the causal relationship between the polarization states at different planes along the propagation path.     

Spectral analysis of the device operators in polarization dynamics

Abstract

The spectral analysis of the device (instrument) operators, as an alternative approach to the dynamical polarization phenomena, is presented by means of an example of a classical time-varying optical device, the electrooptical modulator with longitudinal effect in crystals of KDP type. The polarization-spectral structure of the modulated light is analysed on this basis, both in terms of spectral Jones vectors and of spectral polarization matrices. Some experimental results are presented.     

Probing of a random phase screen by a focused spatially modulated laser beam. Diffraction at a large number of inhomogeneities

The contrast of the Young’s interference fringes formed in the diffraction field when a dynamic random phase screen is illuminated by a focused, spatially modulated laser beam is obtained analytically as a function of the statistical parameters of the screen. A threshold relative bleaching effect is established for a highly dispersive medium when a low-divergence illuminating beam is used. Pis’ma Zh. Tekh. Fiz. 23, 47–53 (October 12, 1997)     

Simple and accurate wavemeter implemented with a polarization interferometer

A simple and accurate wavemeter for measuring the wavelength of monochromatic light is described. The device uses the wavelength-dependent phase lag between principal polarization states of a length of birefringent material (retarder) as the basis for the measurement of the optical wavelength. The retarder is sandwiched between a polarizer and a polarizing beam splitter and is oriented such that its principal axes are 45 deg to the axis of the polarizer and the principal axes of the beam splitter. As a result of the disparity in propagation velocities between the principal polarization states of the retarder, the ratio of the optical power exiting the two ports of the polarizing beam splitter is wavelength dependent. If the input wavelength is known to be within a specified range, the measurement of the power ratio uniquely determines the input wavelength. The device offers the advantage of trading wavelength coverage for increased resolution simply through the choice of the retarder length. Implementations of the device employing both bulk-optic components and fiber-optic components are described, and the results of a laboratory test of a fiber-optic prototype are presented. The prototype had a wavelength accuracy of +/-0.03 nm.