Xinyue Du

Changes in the Statistical Properties of Stochastic Anisotropic Electromagnetic Beams on Propagation in the Turbulent Atmosphere

We report analytic formulas for the elements of the e 2 X2 cross-spectral density matrix of a stochastic electromagnetic anisotropic beam propagating through the turbulent atmosphere with the help of vector integration. From these formulas the changes in the spectral density (spectrum), in the spectral degree of polarization, and in the spectral degree of coherence of such a beam on propagation are determined. As an example, these quantities are calculated for a so-called anisotropic electromagnetic Gaussian Schell-model beam propagating in the isotropic and homogeneous atmosphere. In particular, it is shown numerically that for a beam of this class, unlike for an isotropic electromagnetic Gaussian Schell-model beam, its spectral degree of polarization does not return to its value in the source plane after propagating at sufficiently large distances in the atmosphere. It is also shown that the spectral degree of coherence of such a beam tends to zero with increasing distance of propagation through the turbulent atmosphere, in agreement with results previously reported for isotropic beams.

Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere

Analytical formulas for the elements of the 2x2 cross-spectral density matrix of a kind of stochastic electromagnetic array beam propagating through the turbulent atmosphere are derived with the help of vector integration. Two types of superposition (i.e. the correlated superposition and the uncorrelated superposition) are considered. The changes in the spectral density and in the spectral degree of polarization of such an array beam generated by isotropic or anisotropic electromagnetic Gaussian Schell-model sources on propagation are determined by the use of the analytical formulas. It is shown by numerical calculations that for the array beam composed by isotropic Gaussian-Schell model sources, the spectral degree of polarization in the sufficiently far field returns to the value of the array source; for the array beam composed by anisotropic sources, the spectral degree of polarization in the far field approaches a fixed value that is different from the source.

Polarization modulation of stochastic electromagnetic beams on propagation through the turbulent atmosphere

Analytic expression is derived for the cross-spectral density matrix of a stochastic electromagnetic beam truncated by a slit aperture and passing through the turbulent atmosphere. The new formula can be used in study of the modulation in the spectral degree of polarization of the electromagnetic Gaussian Schell-model beam on propagation. We find that the spectral degree of polarization in the output plane can be directly controlled by the width of the slit aperture. The effect of polarization shaping is also illustrated by numerical examples.

Fractional Fourier transform of truncated elliptical Gaussian beams.

Based on the fact that a hard-edged elliptical aperture can be expanded approximately as a finite sum of complex Gaussian functions in tensor form, an analytical expression for an elliptical Gaussian beam (EGB) truncated by an elliptical aperture and passing through a fractional Fourier transform system is derived by use of vector integration. The approximate analytical results provide more convenience for studying the propagation and transformation of truncated EGBs than the usual way by using the integral formula directly, and the efficiency of numerical calculation is significantly improved.

Propagation of decentered elliptical Gaussian beams in apertured and nonsymmetrical optical systems

A hard-edged elliptical aperture is described approximately by a tensor form, which can be expanded as a finite sum of complex Gaussian functions. An analytical propagation expression for a decentered elliptical Gaussian beam (DEGB) through an axially nonsymmetrical optical system with an elliptical aperture is derived by using vector integration. The approximate analytical results are compared with numerically integral ones, and it is shown that this method can significantly improve the efficiency of numerical calculation. Some numerical simulations are illustrated for the propagation properties of DEGBs through apertured and nonsymmetrical optical transforming systems.

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.

Propagation of elliptical Gaussian beams in apertured and misaligned optical systems

On the basis of the fact that a hard-edged elliptical aperture can be expanded approximately as a finite sum of complex Gaussian functions in tensor form, an analytical propagation expression for an elliptical Gaussian beam (EGB) through a misaligned optical system with an elliptical aperture is derived by use of vector integration. The approximate analytical results provide more convenience for studying the propagation and transformation of EGBs than the usual way by using a diffraction integral directly, and the efficiency of numerical calculation is improved. Some numerical simulations are illustrated for the propagation properties of EGBs through apertured optical transforming systems with misaligned thin lenses.

Criterion for keeping completely unpolarized or completely polarized stochastic electromagnetic Gaussian Schell-model beams on propagation.

The cross-spectral density matrixes of electromagnetic Gaussian Schell-model sources that are completely unpolarized or completely polarized are derived. We find that both the completely unpolarized stochastic electromagnetic Gaussian Schell-model beam and the completely polarized stochastic electromagnetic Gaussian Schell-model beam will keep their spectral degree of polarization or become partially polarized under different constraint conditions during their propagation in free space or through turbulent atmosphere. We give necessary theoretical explanation to the physical phenomena. They are considered as coherence-induced polarization changes and spectral density-induced polarization changes.

Propagation of elliptical Gaussian beams modulated by an elliptical annular aperture

An analytical propagation expression for a generalized astigmatic elliptical Gaussian beam (EGB) modulated by an elliptical annular aperture and passing through an axially nonsymmetrical optical system is obtained by the use of vector integration. The derived analytical results provide more convenience for studying the propagation and transformation of EGBs than the usual method of using the diffraction integral directly, and the efficiency of the numerical calculation is significantly improved. Some numerical simulations are illustrated for the propagation properties of EGBs passing through a free space with an elliptical annular aperture, an elliptical screen, or an elliptical aperture. Further extensions are also pointed out.

Propagation of broadband Gaussian Schell-model beams in the apertured fractional Fourier transformation systems

On the basis of the fact that a hard-edged aperture function can be expressed as finite matrices with different weighting coefficients, we obtain the analytical formula for the propagation of the broadband gaussian Schell-model (BGSM) beam through the apertured fractional Fourier transformation (AFrFT) system. It is shown by numerical examples that the intensity distribution in the plane of a small fractional order is obviously influenced by the bandwidth when the BGSM beams propagate through the AFrFT system. Further extensions are also pointed out.