Electromagnetic spatial coherence wavelets and the classical laws on polarization

Malus’ law, Fresnel-Arago interference laws and Stokes parameters are analyzed in the framework of the tensor theory of the electromagnetic spatial coherence wavelets. It is shown that these laws are suitable for each individual wavelet after regarding the correlation properties of the random electromagnetic field. Stokes parameters at the wavelet scale are introduced. They deal to generalized Stokes parameters which, in contrast to the classical ones, take into account the correlation properties of the random electromagnetic field on the aperture plane, that is the contributions of radiator pairs in surroundings of that plane.     

Generalized (two-point) stokes-parameters representation in terms of the radial and azimuthal field components

Within the paraxial framework, a generalization of the Stokes representation is introduced in terms of the radial and azimuthal field structure of the beam cross-section. These generalized parameters depend on two spatial variables and are closely related with the cross-spectral density (CSD) matrix of the field. The propagation law of such parameters through general first-order optical systems is also given. It allows to describe, in terms of the radial and azimuthal field components, the coherence and polarization behavior of partially-polarized, partially-coherent beams.     

大气湍流中部分相干光束上行和下行传输偏振特性的比较

以部分相干的电磁高斯-谢尔模型(electromagnetic Gaussian-Schell model, EGSM) 光束为研究对象, 根据相干和偏振的统一理论以及随机光束的Stokes参量, 推导出EGSM光束在大气湍流中斜程传输时的偏振度(degree of polarization, DoP)和偏振方向角的表达式, 研究了大气湍流中上行和下行传输时EGSM光束偏振特性的不同. 研究结果表明: 在相同条件下, EGSM 光束下行传输时整个光场DoP的分布比上行传输要集中; 下行传输时轴上点的DoP达到最大值所对应的传输距离长于上行传输. 可以看出, EGSM光束沿下行路径传输时, 探测器可以接收更远距离处的波束传输信息.     

Determination of the amplitude and the phase of the elements of electric cross-spectral density matrix by spectral measurements

The amplitudes and the phases of the elements of electric cross-spectral density matrix are determined experimentally for a pair of points in the cross-section of an expanded laser beam. A modified version of the Young’s interferometer is used as an experimental tool, which separates the beams emerging from the double-slit widely and provides ease in insertion of polarizers and half wave rotators in individual beams. To determine these complex elements of the cross-spectral density matrix, the experimentally obtained values of the spectral densities at an off-axis point are put in the mathematical expressions derived by us using the spectral interference law. The four complex generalized Stokes parameters are also determined using the linear combinations of the matrix elements. This unique but simple experimental approach for determining both the two-point parameters might provide a means to investigate the polarization and the coherence properties of the random electromagnetic beams on propagation.     

Generalized Stokes parameters of stochastic spatially and spectrally partially coherent electromagnetic pulsed beams

The analytical expressions for the generalized (two-point) Stokes parameters of stochastic electromagnetic Gaussian Schell-model pulsed (GSMP) beams propagating in free space are obtained. The changes in the spectral Stokes parameters in free space propagation are studied. The dependence of on-axis and transverse spectral Stokes parameters of stochastic electromagnetic GSMP beams on the pulse parameters including pulse duration and temporal coherence length is stressed and illustrated numerically. The results are interpreted physically.     

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.     

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.     

Contrasts of Stokes parameters in Young's interference experiment and electromagnetic degree of coherence

We analyze the modulation of the Stokes parameters in Young's two-pinhole interference experiment with a random electromagnetic beam. We demonstrate that the electromagnetic (spectral) degree of coherence put forward in Opt. Lett.29, 328 (2004) [or its space-time analog in Opt. Express 11, 1137 (2003)] is physically related to the contrasts of modulation in the four Stokes parameters. More explicitly, the electromagnetic degree of coherence is a measure of both the visibility of the intensity fringes and the modulation contrasts of the three polarization Stokes parameters. We also show that by using suitable wave plates the modulation in any Stokes parameter can be transformed into the form of intensity variation, and hence the electromagnetic degree of coherence can be obtained experimentally by four visibility measurements.     

Instantaneous coherent polarization and coherent polarimetric spectra

The statistics of polarization in electromagnetic waves or target vector has been studied deeply, generally using Stokes parameters. Unfortunately, all the data-processing includes the assumption that the sampling data are independent of time, without a consideration of the time coherence of the polarization data, such as instantaneous Stokes parameters. In this paper, the definitions of the instantaneous coherent polarization function and polarimetric spectrum are presented for the first time, which shows the coherences of the instantaneous Stokes parameters in both time and frequency domain, even in spatial coherence. The new formula of definition can be extended to spatial propagation coherence, both in free space or any linear medium.     

Polarization properties of polarized and partially coherent Electromagnetic Gaussian-Schell model pulse beams on slant path in turbulent atmosphere

This paper is based on the unified theory of coherence and polarization of stochastic electromagnetic beams and the extended Huygens–Fresnel principle, combined with the quadratic approximation of Rytov’s phase structure function and the generalized Stokes parameters. We have derived the novel expressions for the cross-spectral density matrix elements and the degree of cross-polarization of a class of elliptically polarized spatially and spectrally partially coherent Electromagnetic Gaussian-Schell model pulse (EGSMP) beams propagating through atmospheric turbulence along a slant path. Additionally, we calculate and analyze the effects of the turbulent intensity, the initial pulse duration, waist width of the beam, the spatial coherence length and temporal coherence length et al. on the polarization properties of fully polarized and partially coherent EGSMP beams. Finally, a comparison of the impact of those factors on the partially polarization beams is made. The results show that the influences of the turbulent intensity, the initial pulse duration, waist width of the beam, the spatial coherence length and temporal coherence length et al. on the polarization properties of fully polarized and partially coherent EGSMP beams are larger. While the effects of those parameters on the partially polarization and partially coherent EGSMP beams are smaller. It is noted that the results of this paper have established sound theoretical basis on the topic of improving performance of the laser system propagating through the atmospheric turbulence.     

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.     

Stokes parameters and polarization contrasts in Young's interference experiment

We derive a spectral interference law that governs the behavior of the four Stokes parameters in Young's two-pinhole experiment with a random electromagnetic beam. In addition to the visibility of intensity fringes, we introduce three new contrast parameters that describe the interference-induced changes in the field's state of partial polarization. The polarization modulation depends on the electric field correlations at the pinholes and is closely related to the two-point Stokes parameters. The results are expected to be particularly useful in polarization interferometry and electromagnetic coherence theory. The formalism is demonstrated with specific examples.     

部分相干洛伦兹-高斯光束在湍流大气中的峭度参数

基于广义惠更斯-菲涅耳衍射积分公式和洛伦兹函数的厄米-高斯展开,导出了部分相干洛伦兹-高斯光束在湍流大气中经傍轴ABCD光学系统的峭度参数的解析表达式,对该峭度参数进行了数值计算。结果表明:空间相干长度和结构常数对峭度参数的影响与部分相干洛伦兹-高斯光束本身的光束参数有关;当空间相干长度小于光束参数时,其影响显著;当洛伦兹部分的光束参数较大或高斯部分的光束参数较小时,结构常数的影响较明显且随传输距离的增大而增强。     

Spatiotemporal coherence properties of broadband Gaussian Schell-model beams propagating through atmospheric turbulence

The spatiotemporal coherence properties of broadband Gaussian Schell-model (GSM) beams with different spectral bandwidths propagating through atmospheric turbulence are numerically calculated and analyzed. The results show that although the spatial coherence properties of an intermediate-broadband GSM beam almost do not depend on the spectral bandwidth, those of its ultra-broadband counterpart do. The temporal coherence of an ultra-broadband GSM beam not only has radial dependence in the observation plane, but also varies with the increasing propagation distance; however, the same behavior does not hold for an intermediate-broadband GSM beam of which the temporal coherence remains nearly invariable as the radial distance of the observation point or propagation distance changes.     

Experimental verification of the electromagnetic spectral interference law using a modified version of the Young's interferometer

We present a method to verify the electromagnetic spectral interference law experimentally, which connects the combined two-beam usual Stokes parameters with the corresponding single beam usual Stokes parameters. Using polarizers and phase retarders in a modified version of the Young's interferometer, the usual Stokes parameters and the generalized Stokes parameters are determined for a laser beam. The experimentally determined single point (usual) Stokes parameters at an axial point in the observation plane for the combined beam are compared with their theoretically expected values obtained by putting the experimental data in the electromagnetic spectral interference law. A fairly good agreement is found between the two values. Not only does the proposed method provides a comparison between the two kinds of Stokes parameters but also validates the role of the generalized Stokes parameters as modulation parameters for the usual Stokes parameters.     

Changes of polarization of light beams on propagation through tissue

The values of the degree of polarization and the coherence of the light beam are of great importance in many areas. In this paper, we study the effects of tissue turbulence on the degree of polarization of a partially coherent electromagnetic beam. Based on the unified theory of coherence and polarization for random electromagnetic beams and the spectral density of the index of refraction fluctuations of tissue, we have established the detailed formula for calculating the change of the degree of polarization of the beam when propagating through tissue. Compared with the light propagation through atmosphere, the result shows that although the propagation distance in tissue imaging is very short, the degree of polarization of the beam may change greatly. An example was given which clearly shows the effect of the correction properties and the tissue turbulence on the degree of polarization of the light beam propagating within the tissue.     

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.     

Generalized Stokes parameters of random electromagnetic vortex beams propagating through atmospheric turbulence

By using the extended Huygens–Fresnel principle, the analytical expressions for the generalized Stokes parameters of random electromagnetic Gaussian Schell-model (GSM) vortex beams propagating through atmospheric turbulence are derived, and used to study the changes in spectral Stokes parameters of random electromagnetic GSM vortex beams in atmospheric turbulence and to compare the results of random electromagnetic GSM vortex-free beams. The influence of atmospheric turbulence on the spectral Stokes parameters is analyzed. The validity of our results is interpreted physically.     

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.     

Elliptic incoherent accessible solitons in strongly nonlocal media

We study the propagation of elliptic incoherent accessible solitons in strongly nonlocal media with noninstantaneous Kerr nonlinearity. For this soliton to exist, the coherence properties of the incoherent beam should be anisotropic. The total power of the incident beam should also equal to a critical value which depend on the beam width as well as the coherence properties. When initial parameters of the beam do not satisfy the existence conditions, the elliptic incoherent accessible solitons will undergo linear harmonic oscillation in different states. Corresponding properties are studied in detail.