An inverse problem in the theory of stochastic electromagnetic beams

The cross-spectral density matrix of an electromagnetic beam has been playing increasingly important role in studies of changes of spectra, of coherence and of polarization as the beam propagates. In this paper we derive solution to an inverse problem, which makes it possible to determine the cross-spectral density matrix of the beam in the source plane z=0, from the knowledge of the matrix in any cross-section z=z₀>0 in the half-space into which the beam propagates. We apply the result to the theory of so-called Stokes beams, which were introduced not long ago.     

Cross-spectral density matrix of the far field generated by a blackbody source

An expression is derived for the cross-spectral density matrix, valid in the paraxial region of the far field, generated by a planar blackbody source. With the help of it we determine the spectral degrees of polarization and of cross-polarization of the far field.     

Beam criteria for propagation of electromagnetic beams in the turbulent atmosphere

A recently introduced criterion [O. Korotkova, E. Wolf, Opt. Lett. 32 (2007) 2137] for testing whether a scalar beam preserves its beam-like shape after it propagates some distance through the turbulent atmosphere is generalized to the case of electromagnetic beam propagation. The generalized criterion applies to monochromatic as well as to stochastic electromagnetic beams. The analysis is illustrated by examples.     

Polarization properties of stochastic electromagnetic beams

The behavior of the degree of polarization of a Gaussian Schell-model beam propagating in free space is investigated. Contour diagrams for the degree of polarization, and for the spectral density (‘intensity’) of the polarized and the unpolarized portions of the beam are presented.     

Spectra, coherence and polarization in Young’s interference pattern formed by stochastic electromagnetic beams

New expressions are derived that predict the change of spectral density, the spectral degree of coherence and spectral degree of polarization in Young’s interference pattern when the pinholes are illuminated by an electromagnetic Gaussian Schell-mode beam. The spectral degree of polarization in the observation plane is found, in general, to take on values different from its values at each pinhole, and it changes with position at the same spatial frequency, as does the spectral density. When the light at the pinholes is completely coherent, there are singular points in the pattern, at which the spectral degree of polarization reaches minimum value.     

Stokes beams formed by superposition of a completely unpolarized and a completely polarized Gaussian Schell-model beam

Analytic expressions and computed examples are given to elucidate the coherence and polarization properties of Stokes beams, i.e. beams formed by superposition of a completely unpolarized and a completely polarized electromagnetic Gaussian Schell-model beam. We found that superposition of such two beams cannot form a Stokes beam with a constant state of polarization on propagation. An additional constraint on the source plane parameters of the two Gaussian Schell-model beams is proposed. The resultant Stokes beam with a constant state of polarization on propagation is found to be a Gaussian Schell-model beam with the same variances as the two constituent Gaussian Schell-model beams. However, the modulus of the Gaussian intensity distributions across the source planes of these beams may be different.     

Polarization vortices generation by diffraction from a four quadrant polarization mask

In this work we analyze the generation of an array of polarization singularities (polarization vortices) in the diffracted field produced by a simple four quadrant polarization mask, which transmits two orthogonal linear states in different quadrants. We use the Jones matrix method to analyze the polarization and diffraction characteristics of the diffracted field. We show that the same four quadrant mask can be used to generate either azimuth or ellipticity singularities simply by changing the input polarization. We experimentally verify the theory by generating the four quadrant polarization mask using a twisted nematic liquid crystal display. We show experimental results that confirm the theoretical predictions.     

Propagation properties of stochastic electromagnetic double-vortex beams in a turbulent atmosphere

Based on the extended Huygens-Fresnel principle,we study the propagation properties of stochastic electromagnetic double-vortex beams in a turbulent atmosphere.The result shows that the spreading of partially coherent double-vortex beams can be smaller than that of fully coherent ones.The degree of polarization of this kind of beam will experience change,which is dependent on the degree of polarization of the source plane,the atmospheric turbulence,topological charge,and the spatial coherence.The results may have applications in space optical communication.     

Spatial coherence properties of monochromatic electromagnetic beams and of laser modes

It is shown that, contrary to common belief, monochromatic light beams are, in general, not spatially completely coherent, i.e., they will, in general, not produce fringes of unit visibility in a Young's double pinhole interference experiment. We cite experiments with laser modes which confirm this result.     

Polarization ellipse and Stokes parameters of a stochastic electromagnetic Gaussian Schell-model beam propagating through a polarization grating

An analytical propagation formula for the cross-spectral density matrix of a stochastic electromagnetic Gaussian Schell-model (EGSM) beam after propagating through a polarization grating is derived with the help of a tensor method. The statistics properties, particularly the degree of polarization, polarization ellipse and Stokes parameters, of the EGSM beam on propagation after passing through a polarization grating are studied numerically. Our results clearly show that the statistics properties of the EGSM beam on propagation are closely determined by the initial parameters of the EGSM beam and the parameter of the polarization grating. The polarization grating provides one way for modulating the polarization properties of an EGSM beam.     

Propagation properties of stochastic electromagnetic double-vortex beams in turbulent atmosphere

Based on the extended Huygens-Fresnel principle, we study the propagation properties of stochastic electromagnetic double-vortex beams in turbulent atmosphere. The result shows that the spreading of the partially coherent double-vortex beams can be smaller than that of the fully coherent ones. The degree of polarization of this kind of beam will experience change, which is dependent on the degree of polarization of the source plane, the atmospheric turbulence, topological charge, and the spatial coherence. The results may have applications in space optical communication.     

Polarization properties of stochastic spatially and spectrally partially coherent electromagnetic flat-topped pulses

The propagation behavior of the spectral degree of polarization of stochastic spatially and spectrally partially coherent electromagnetic flat-topped pulses in free space is studied. Numerical calculation results are given to illustrate the dependence of spectral degree of polarization on the beam order, pulse frequency, temporal coherence length, pulse duration and spatial correlation length. It is shown that with increasing beam order M the position z of the on-axis spectral degree of polarization P = 0 decreases, and the distribution of spectral degree of polarization is different for different values of M.     

Statistical properties of correlated radial stochastic electromagnetic array beams on propagation

A kind of array beam named the correlated radial stochastic electromagnetic array beam that is generated by an electromagnetic Gaussian Schell-model source is introduced by use of tensor method. The analytical expression for the cross-spectral density matrix of this array beam propagating through the turbulent atmosphere and in free space is obtained after performing vector integration. Some typical numerical calculations are illustrated for the changes in the spectral density, spectral degree of polarization, and spectral degree of coherence of the beam on propagation. We find that the atmospheric turbulence can destroy the correlated effect among the beamlets.     

Polarization and intensity correlations in stochastic electromagnetic beams upon interaction with devices of?polarization optics

Based on the recently formulated unified theory of coherence and polarization of light, we explore the behavior of the intensity–intensity correlations and the auxiliary quantity called the degree of cross-polarization in stochastic electromagnetic beams upon their passage through the devices of polarization optics. In particular, the effects of deterministic devices (such as polarizers, absorbers, compensators, and rotators) as well as of random devices (such as spatial light modulators) on passing beams are investigated. Our results may find applications in polarimetric communications, imaging and sensing.     

Thermal-lens effect induced changes of polarization, coherence and spectrum of a stochastic electromagnetic beam in a Gaussian cavity

Based on the unified theory of coherence and polarization, the interaction of a stochastic electromagnetic Gaussian Schell-model (EGSM) beam with a Gaussian cavity containing thermal-lens medium is analyzed. In particular, the effect of the thermal medium of the lens on the evolution properties of the degree of polarization, the degree of coherence and the spectral shift are investigated in detail. It is shown that the thermal-lens medium has a strong influence on the behavior of the EGSM beam especially in unstable resonators, thus it is necessary for us to take the thermal-lens effect into consideration in practice.     

Propagation characteristics of stochastic electromagnetic array beams

A stochastic electromagnetic array beam that is generated by an electromagnetic Gaussian Schell-model source is introduced by using tensor method. An analytical expression for the cross-spectral density matrix of the stochastic electromagnetic array beam propagating in a paraxial ABCD optical system is derived after performing vector integration. Some numerical calculations are illustrated for the propagation characteristics of such an array beam in free space and fractional Fourier transform (FRFT) system as application examples.     

Modulation in the statistical properties of stochastic electromagnetic beams through an electromagnetic induced transparency atomic vapor

We report analytical expressions for the elements of the 2 × 2 cross-spectral density matrix of a stochastic electromagnetic beam passing through an electromagnetic induced transparency (EIT) atomic vapor. By use of the derived formulas the changes in the spectral density, the spectral degree of coherence, and the spectral degree of polarization of such a beam on propagation can be studied in detail. Numerical examples show that the statistical properties of the stochastic electromagnetic beam can be modulated by the Rabi frequency of the control light when the beam propagates through the EIT atomic vapor.     

Interference law for coherent stochastic electromagnetic beams

Two-beam interference law for the superposition of stochastic, spatially coherent, electromagnetic beams is derived. Only a single phase is found to enter the interference law, in spite of the vector nature of the problem. The meaning of the phase is elucidated.     

Spectral and polarization properties of stochastic electromagnetic beams propagating in gain or absorbing media

Based on the generalized Huygens-Fresnel diffraction principle and on the unified theory of coherence and polarization of light, the analytical expressions for the cross-spectral density matrix of a class of stochastic electromagnetic beams in linear, deterministic media with gain or absorption, in which the wave number is generally complex, has been derived. In particular, through numerical examples based on our analytical formulas the behavior of the spectral density, and of the degree of polarization of the Electromagnetic Gaussian Schell-model is analyzed. It is demonstrated how these statistical properties of the beam propagating in such media depend on the properties of the source and on the wave number of the medium. Numerical examples relate to two model sources: with uncorrelated and partially correlated mutually orthogonal transverse components of the electric field. The results will impact potential applications involving natural absorbing media, such as oceanic waters, and man-made media such as active lasers, for instance.