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.     

Changes in the spectrum, polarization, and coherence of stochastic spatially and spectrally partially coherent electromagnetic J0-correlated Schell-model pulsed beams propagating in free space

Taking the stochastic electromagnetic J ₀-correlated Schell-model pulsed (JSMP) beam as a typical example of stochastic spatially and spectrally partially coherent electromagnetic pulsed beams, the analytical expressions for the cross-spectral density matrix, spectral density, spectral degree of polarization and spectral degree of coherence of stochastic electromagnetic JSMP beams propagating in free space are derived, and used to study the changes in the spectrum, polarization, and coherence of stochastic spatially and spectrally partially coherent electromagnetic JSMP beams. It is shown that the on-axis spectrum is blue-shifted in free-space propagation and is dependent on the pulse temporal coherence length and spatial correlation parameter. The distribution of the on-axis spectral degree of polarization depends on the frequency and spatial correlation parameter. The spectral degree of coherence increases with increasing pulse temporal coherence length. The results derived are interpreted physically.     

Changes in the state of polarization of apertured stochastic electromagnetic modified Bessel–Gauss beams in free-space propagation

By using the generalized Huygens–Fresnel diffraction integral, the analytical expressions for the cross-spectral density matrix, spectral degree of polarization, orientation angle and degree of ellipticity of polarization ellipse of apertured stochastic electromagnetic modified Bessel–Gauss beams (MBGBs) through a paraxial optical ABCD system are derived, and used to study the changes in the state of polarization of apertured stochastic electromagnetic MBGBs propagating in free space. The invariance of the on-axis state of polarization of unapertured stochastic electromagnetic MBGBs propagating through paraxial optical ABCD systems is illustrated analytically and numerically. For apertured stochastic electromagnetic MBGBs, the on-axis spectral degree of polarization, orientation angle and degree of ellipticity of polarization ellipse increase with increasing propagation distance, and approach asymptotic values when the propagation distance is large enough. There is a uniform distribution region of the state of polarization around the center of the beams whose range decreases with increasing truncation parameter. In addition, the state of polarization of apertured stochastic electromagnetic MBGBs upon propagation can be modulated by controlling the truncation parameter.     

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.     

Spectral properties of stochastic electromagnetic twist anisotropic Gaussian-Schell model beam truncated by a slit aperture propagating in turbulent atmosphere

With the help of the tensor method, the cross-spectral density matrix for the stochastic electromagnetic twist anisotropic Gaussian-Schell model (ETAGSM) beam truncated by a slit aperture propagating in turbulent atmosphere are derived. The spectral properties of this kind of beam are investigated in detail. It is shown by numerical results and analysis that the affection of the slit aperture on the spectral properties of the stochastic ETAGSM beam is obvious in the near field; while in the far field, the atmospheric turbulence plays an important role; the source beam's coherence can weaken the affection of the slit aperture and the atmospheric turbulence on the spectral properties of the stochastic ETAGSM beam truncated by a slit aperture propagating in turbulent atmosphere, while the twist properties of the source beam can strong the affection of the slit aperture on the spectral properties in the near field. Also, the spectral degree of polarization and normalized spectral density distributions and corresponding contour graphs of the stochastic ETAGSM beam truncated by a slit aperture propagating in turbulent atmosphere and free space at different propagation distances are investigated in detail.     

部分相干电磁光束的光谱交叉偏振度

基于光谱交叉偏振度新理论,研究了部分相干光束在传输过程中光谱交叉偏振度的变化情况.采用相干偏振统一理论和广义惠更斯-菲涅耳原理,推导部分相干电磁高斯-谢尔模型光束在自由空间传输时任意空间两点的交叉谱密度矩阵的解析式.研究表明,光谱交叉偏振度的值不再仅仅局限在0～1之间,而是可为任意的非负值.传输场中的光谱交叉偏振度与光源相关参量,初始偏振度和传输距离紧密相关.当光束经过足够长的传输距离后,轴上光谱交叉偏振度不再发生变化,而是趋向一个稳定值.保持光谱交叉偏振度不变的条件与一般偏振度相同.     

Changes in generalized Stokes parameters of stochastic electromagnetic beams on propagation through ABCD optical systems and in the turbulent atmosphere

Using the derived formulas for the transformation of 2 × 2 cross-spectral density matrix of the stochastic electromagnetic beams propagating through ABCD optical systems and in the turbulent atmosphere, the changes in the generalized Stokes parameters of the beams propagating under these conditions can be investigated directly. Some typical numerical calculations are illustrated relating to the electromagnetic Gaussian Schell-model beams passing through free space, focal system, dual-focus system, and the turbulent atmosphere with different structure parameters. Further extensions are also pointed out.     

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 of a stochastic electromagnetic Gaussian Schell-model beam through an optical system in turbulent atmosphere

A generalized diffraction integral formula for stochastic electromagnetic beams propagating through an optical system in turbulent atmosphere is derived with the help of tensor method. Some analyses are illustrated by a numerical example relating to changes in the average intensity and the degree of polarization of an electromagnetic Gaussian Schell-model beam propagating through a double-lenses system. It is shown that the optical system has strong influence on the propagation properties of the beam. The method used in this paper can be widely applied to the propagation of astigmatic beams through an optical system in turbulent atmosphere.     

Changes in the statistical properties of stochastic anisotropic electromagnetic beams propagating through the oceanic turbulence

On the basis of the extended Huygens?CFresnel principle, an analytical propagation expression for the elements of the cross-spectral density matrix of a stochastic anisotropic electromagnetic beam through oceanic turbulence is derived. From this formula the spectral density, spectral degree of coherence, spectral degree of polarization, orientation angle and the degree of ellipticity of such a beam on propagation are determined. Some numerical calculations are illustrated relating to the anisotropic electromagnetic Gaussian Schell-model beams propagating through oceanic turbulence. The results indicate that the spectral degree of coherence of stochastic anisotropic electromagnetic beams tends to zero with increasing propagation distance through oceanic turbulence, which is in agreement with results previously reported for turbulent atmosphere. It is also found that the changes in the statistical properties of the anisotropic source on propagation are qualitatively different from those of the isotropic source.     

Depolarization characteristics of incompletely polarized and partially coherent laser beams in slant atmospheric turbulence

Based on the cross-spectral density function of transmission theory and the unified theory of coherence and polarization, the depolarization characteristics of incompletely polarized and partially coherent laser propagation in slant atmospheric turbulence are investigated. According to extended Huygens–Fresnel principle, the analytical expressions for intensity and degree of polarization in the slant path are derived. The effects of the wavelength, the initial spot size and the transmission distance on the intensity and degree of polarization are described. The results show that a more stable distribution of the degree of polarization at the receiver is obtained with increasing wavelength for a certain receiver height. The conclusions play an important role in optical communications and target recognition.     

Diffraction properties of partially coherent electromagnetic four-petal Gaussian beams

The concept of existing scalar coherent four-petal Gaussian (FPG) beam is extended to the more general partially coherent electromagnetic four-petal Gaussian (PCFPG) beam, and the cross-spectral density matrix of the latter is derived in analytical expressions. Intensity distributions and degree of polarization in free space are investigated by numerical examples. Results show that propagation properties of the PCFPG beam are different from their scalar coherent and scalar partially coherent counterparts. Both intensity distributions and polarization strongly depend on transversal coherence lengths. When the propagation distance increases to a large value, intensity distributions of PCFPG beams convert into the Gauss profile. Polarization of PCFPG beams shows an oscillating characteristic upon propagation, and the oscillation would enhance with the increase of coherence lengths. These results may find potential applications in beam shaping and free space optical communications.     

Stokes parameters and degree of polarization of nonparaxial stochastic electromagnetic beams

On the basis of angular spectrum representation and the stationary-phase method, far-field expressions for generalized Stokes parameters of nonparaxial stochastic electromagnetic beams are derived, which permits us to study the changes in the ordinary Stokes parameters upon propagation, and the changes in the spectral degree of polarization of partially polarized nonparaxial stochastic electromagnetic beams. It is shown that the spectral degree of polarization changes across the section on beam propagation.     

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.     

Effects of coherence on anisotropic electromagnetic Gaussian-Schell model beams on propagation

An analytical formula for the cross-spectral density matrix of the electric field of anisotropic electromagnetic Gaussian-Schell model beams propagating in free space is derived by using a tensor method. The effects of coherence on those beams are studied. It is shown that two anisotropic stochastic electromagnetic beams that propagate from the source plane z=0 into the half-space z>0 may have different beam shapes (i.e., spectral density) and states of polarization in the half-space, even though they have the same beam shape and states of polarization in the source plane. This fact is due to a difference in the coherence properties of the field in the source plane.     

随机电磁高阶Bessel-Gaussian光束在海洋湍流中的传输特性

利用广义惠更斯-菲涅耳衍射积分公式得到了随机电磁高阶Bessel-Gaussian光束在海洋湍流中传输的交叉谱密度矩阵的一般表达式,通过数值计算主要研究了随机电磁高阶Bessel-Gaussian光束在海洋湍流中传输时其在远场输出面的统计特性的变化,包括归一化光谱强度、光谱偏振度、两点的光谱相干度等.数值模拟结果显示海洋湍流能够对随机电磁高阶Bessel-Gaussian光束的归一化光谱强度分布产生影响,随着传输距离的增加,零阶Bessel-Gaussian光束中心出现凹陷,高阶Bessel-Gaussian光束中心会变平坦继而又凹陷下去,不管零阶还是高阶,当传输距离增加到足够远,光强分布都会演变成最终的类高斯分布.x轴上各点的偏振度改变与相干长度δ_(xx),δ_(yy)以及海洋湍流参数有关.x轴上任意一点和原点这两点的光谱相干度也随x的增加而呈振荡变化,并且海洋的均方温度耗散率χT对光谱相干度有影响.     

Propagation of random electromagnetic beams through axially nonsymmetrical optical systems

When random electromagnetic beams passing through axially nonsymmetrical ABCD optical systems, the analytical formula for the transformation of the elements of 2 × 2 cross-spectral density matrix is obtained with the help of vector integration. We derive analytical expressions of the spectral degree of polarization, the spectral degree of coherence, and the spectral density in any output plane z > 0. Some numerical calculations are illustrated relating to the electromagnetic Gaussian Schell-model beams propagating through such optical systems.     

Cross-spectral density matrix of a random electromagnetic beam propagating through an apertured axially nonsymmetrical optical system

Using the fact that a hard-edged rectangular aperture function can be approximated by a two-dimensional multi-Gaussian series, analytic formulas for the elements of the cross-spectral density matrix of a random electromagnetic beam truncated by a rectangular aperture and passing through an axially nonsymmetrical optical system is derived. The analysis is illustrated by numerical examples relating to changes in the spectral degree of polarization of electromagnetic Gaussian Schell-model beams propagating through free space, focal system and dual-focus system. In particular, the effect of the size of the hard aperture on the spectral degree of polarization of the beam along its optical axis is discussed.     

非傍轴双曲余弦-高斯脉冲电磁光束远场特性

利用矢量瑞利-索末菲衍射积分公式,推导出了非傍轴部分空间相干部分光谱相干双曲余弦-高斯(ChG)脉冲电磁光束在自由空间传输时交叉谱密度矩阵的远场解析公式,并用来表示脉冲电磁光束的光谱密度(光强)和偏振度。结果表明,对非傍轴远场部分空间相干部分光谱相干ChG脉冲电磁光束,其非傍轴性主要由参数f, f决定,而离心参数、脉冲宽度和时间相干长度影响其非傍轴行为。非傍轴部分空间相干部分光谱相干高斯-谢尔模型脉冲电磁光束的远场传输可作为特例处理。     

Partially coherent vortex beams propagation in a turbulent atmosphere

Based on the Rytov approximation and the cross-spectral density approximation for the mutual coherence function of the partially coherent field, the propagation properties of the partially coherent beams with optical vortices in turbulent atmosphere are discussed. The average intensity and the mutual coherence function of the partially coherent vortex beams propagation in weak turbulent atmosphere are obtained.It is shown that the vortex structure of the average cross-spectral density of partially coherent beams has the same helicoidally shape as that of the phase of the fully coherent Laguerre-Gauss beams in free space and the relative intensity of the beam is degraded by optical vortex.