Changes in the spectrum of twist anisotropic Gaussian Schell-model beams propagating through turbulent atmosphere

Based on the extended Huygens-Fresnel principle, the spectrum of twist anisotropic Gaussian Schell-model (TAGSM) beams propagating through turbulent atmosphere is derived analytically by using the partially coherent complex curvature tensor. The relative spectral shift of TAGSM beams propagating through turbulent atmosphere is closely related with the strength of atmospheric turbulence, the beam’s parameter and the radial coordinate. The on-axis spectral shift of TAGSM beams propagating through turbulent atmosphere changes from blue shift to red shift with the increasing of the propagation distance z, and at a certain propagation distance z, there exists a rapid transition of the spectrum at the critical position r_c.     

Spectral properties of apertured stochastic electromagnetic twist anisotropic Gaussian Schell-model beam propagating in turbulent atmosphere

Based on the extended Huygens-Fresnel principle, analytical formulas are derived for the cross-spectral density matrix of an apertured stochastic electromagnetic twist anisotropic Gaussian Schell-model (ETAGSM) beam propagating in a turbulent atmosphere by use of a tensor method. Spectral properties of apertured ETAGSM beam are closely related with the strength of atmospheric turbulence, the aperture widths and the beam's parameters, etc. Our main attention was focused on the influence of the aperture widths, atmospheric turbulence, twist parameters and partial coherence on the spectral properties (including spectral degree of polarization, the spectral degree of coherence and the spectral density) of apertured ETAGSM beam propagating in turbulent atmosphere. Numerical calculation results and analysis are given.     

Changes in the spectrum of Gaussian Schell-model beams propagating through turbulent atmosphere

Based on the extended Huygens-Fresnel principle, the spectrum of Gaussian Schell-model (GSM) beams propagating through turbulent atmosphere is derived analytically. It is shown that, if the scaling law is valid, the normalized spectrum S(ω) of GSM beams propagating through turbulent atmosphere is the same as the normalized source spectrum S⁽⁰⁾(ω), whether GSM source is quasi-homogenous or not. On the other hand, if the scaling law fails, S(ω) of GSM is different from S⁽⁰⁾(ω). The structure constant of the refractive index, transverse coordinate of observation point and spatial correlation length of the source affect the spectrum, which is illustrated numerically.     

Polarization properties of nonparaxial partially polarized Gaussian Schell-model beams

The polarization properties of nonparaxial partially polarized Gaussian Schell-model (PGSM) beams are studied. Based on the beam coherence-polarization matrix, the propagation formula of the degree of polarization for the nonparaxial PGSM beams is derived. The paraxial approximation is dealt with as special cases of our general results. Some conditions that limit the choice of the parameters are established. The sufficiency condition of polarization invariance in propagation for the nonparaxial PGSM beams is derived. By using the derived formulae, the propagation properties of the degree of polarization for the nonparaxial PGSM beam in free space are illustrated and analyzed with numerical examples. Some detailed comparisons of the obtained results with the paraxial results are made.     

Propagation of partially polarized Gaussian Schell-model beams in anomalously dispersive media

By use of a tensor method, analytical transform formulae for isotropic partially polarized GSM beams propagating in anomalously dispersive media are derived. Based on the derived formula, the propagation properties of isotropic partially polarized GSM beams in anomalously dispersive media are studied in detail. The results show that the polarization and the irradiance distribution and the coherence degree properties are influenced by the anomalously dispersive media and beam's initial coherence. In addition, the evolution of the spectrum shows that spectrum splitting and spectral shift occur during propagation in anomalously dispersive media.     

Slanting propagation of stochastic electromagnetic twist anisotropic Gaussian–Schell model beams in turbulent atmosphere

Based on the extended Huygens–Fresnel diffraction integral and the unified theory of coherence and polarization, the slanting propagation properties of stochastic electromagnetic twist anisotropic Gaussian–Schell model (ETAGSM) beams in turbulent atmosphere have been investigated. Both numerical calculation and physical interpretation are obtained. The influence of the atmospheric turbulence and the source parameters on the polarization distribution, intensity distribution and coherence distribution has been studied in great detail. The investigation reveals that slanting propagation can weaken the influence of atmospheric turbulence on the spectral properties of stochastic ETAGSM beams more obviously. It is also shown that we can modulate the spectral degree of polarization in the output plane especially in the far field by simply controlling the coherence, the spot width or the twisty properties of the source beams. The distribution of the spectral degree of polarization and cross-polarization of the beams in the output plane is also given out.     

Propagation properties of partially polarized Gaussian Schell-model beams through an astigmatic lens

Based on the beam coherent-polarization (BCP) matrix approach and propagation law of partially coherent beams, analytical propagation equations of partially polarized Gaussian Schell-model (PGSM) beams through an astigmatic lens are derived, which enables us to study the propagation-induced polarization changes and irradiance distributions at any propagation distance of PGSM beams through an astigmatic lens within the framework of the paraxial approximation. Detailed numerical results for a PGSM beam passing through an astigmatic lens are presented. A comparison with the aberration-free case is made, and shows that the astigmatism affects the propagation properties of PGSM beams.     

部分相干电磁光束在湍流介质中传输的偏振变化

 从推广的惠更斯-菲涅尔原理出发,推导出了部分相干电磁光束的偏振态在湍流介质中传输的表达式。并以电磁高斯-谢尔模型(EGSM)光束为例,研究了湍流对电磁高斯-谢尔模型光束偏振态的影响。研究结果表明,对于轴上点,湍流介质的折射率结构常数越大,偏振度趋于最大值的速度越快,达到的最大值越小;光斑越大,偏振度达到最大值的位置离光源越远,在光斑增大的过程中,偏振度所达到的极大值会先增大后减小,最后保持与光源相同的偏振度不变。对于轴外点,一个固定的z,光的偏振度随着离轴距离的增大而逐渐下降,并最终等于零。折射率结构常数越大,偏振度随离轴距离的增大而下降得越缓慢;光斑越大,偏振度随离轴距离的增大下降得越快。     

Focal shift of partially polarized Gaussian Schell-model beams

A detailed study of the focal shift of partially polarized Gaussian Schell-model (PGSM) beams is made on the basis of the beam coherence-polarization (BCP) matrix approach. It is shown that, apart from the Fresnel number and truncation parameter, the focal shift of PGSM beams depends on the angle of polarizer and the ratio of degree of spatial coherence.     

Propagation of partially coherent controllable dark hollow beams with various symmetries in turbulent atmosphere

Normalized intensity distribution, the complex degree of coherence and power in the bucket for partially coherent controllable dark hollow beams (DHBs) with various symmetries propagating in atmospheric turbulence are derived using tensor method and investigated in detail. Analytical results show that, after sufficient propagation distance, partially coherent DHBs with various symmetries eventually become circular Gaussian beam (without dark hollow) in turbulent atmosphere, which is different from its propagation properties in free space. The partially coherent DHBs return to a circular Gaussian beam rapidly for stronger turbulence, higher coherence, lower beam order, smaller p or smaller beam waist width. Another interesting observation is that the profile of the complex degree of coherence attains a similar profile to that of the average intensity of the related beam propagating in a turbulent atmosphere. Besides the laser power focusablity of DHBs are better than that of Gaussian beam propagating in turbulent atmosphere.     

Propagation properties of partially coherent Laguerre-Gaussian beams in turbulent atmosphere

To study the propagation properties of partially coherent Laguerre-Gaussian (PLG) beams through turbulent atmosphere, the analytical formulas are derived for the angular width and the beam-propagation factor (M²-factor) of PLG beams by using the extended Huygens-Fresnel principle and the second-order moments of the Wigner distribution function (WDF). The corresponding numerical results are also calculated. When propagation distance increases, the angular width is found to spread faster for PLG beams with higher beam order, smaller correlation length and bigger structure constant The angular width of PLG beams decreases with increase in waist width (w₀).The M²-factor of PLG beams with higher beam order and smaller correlation length is less affected by turbulence with increase in propagation distance. The propagation properties of the M²-factor for PLG beams with the smaller structure constant are better than that with bigger structure constant . The M²-factor of PLG beams decreases with increase in the wavelength λ, and it is also less affected by turbulence for beams with higher order and smaller correlation length. Furthermore, for the PLG beams with the same beam order, the angular width and the M²-factor keep invariable in free space.     

Propagation of partially coherent Bessel-Gaussian beams in turbulent atmosphere

The characteristics of partially coherent Bessel-Gaussian beams propagating in turbulent atmosphere are investigated. Based on the extended Huygens–Fresnel principle, the influence of topological charges and coherence of the source on the intensity and the degree of coherence in the received plane are considered. The influence of atmospheric turbulence on beam profile and coherence in the received plane is also analyzed. It is found that a Bessel-Gaussian shaped intensity distribution will eventually transform into a Gaussian distribution after propagating in turbulent atmosphere. Meanwhile, topological charges, coherence of the source and atmospheric turbulence will also influence the propagation characterizations of the beams.     

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.     

Effect of polarization on the evolution of electromagnetic hollow Gaussian Schell-model beam

Based on the theory of coherence, an analytical propagation formula for partially polarized and partially coherent hollow Gaussian Schell-model beams (HGSMBs) passing through a paraxial optical system is derived. Furthermore, we show that the degree of polarization of source may affect the evolution of HGSMBs and a tunable dark region may exist. For two special cases of fully coherent and partially coherent δ_xx = δ_yy, normalized intensity distributions are independent of the polarization of source.     

Propagation of non-uniformly polarized beams in a turbulent atmosphere

Based on the extended Huygens-Fresnel diffraction integral, the propagation of non-uniformly polarized (NUP) beam in a turbulent atmosphere has been investigated. The influence of the source parameters and the atmospheric turbulence on the intensity distribution and polarization distribution has been studied in great details. It is shown that the non-uniform polarization of the beam influences the intensity distribution and makes the polarization distribution oscillate in the output plane. It is also found that the atmospheric turbulence can reduce the influence of non-uniform polarization of the beam on the intensity distribution and polarization distribution in the output plane.     

Changes in polarization of Gaussian Schell-model beams and partially polarized Gaussian Schell-model beams through a polarization grating

The polarization properties of Gaussian Schell-model (GSM) beams and partially polarized Gaussian Schell-model (PGSM) beams passing through a polarization grating (PG) are studied based on the beam coherence-polarization (BCP) matrix formulism, where the finite size of the PG is considered. Detailed numerical calculation results are given and compared with the previous work.     

The aperture effect on the propagation of partially polarized Gaussian Schell-model beams

Based on the beam coherence-polarization (BCP) matrix approach and propagation law of partially coherent beams, the propagation of partially polarized Gaussian Schell-model (PGSM) beams through an aperture is studied. Numerical calculation results are given to illustrate the propagation-induced polarization changes and irradiance distribution of PGSM beams for the apertured case.     

Spectral shifts and spectral switches of elliptical Gaussian beam propagating through turbulent atmosphere

Based on the extended Huygens-Fresnel principle, the spectrum of elliptical Gaussian beam (EGB) propagating through turbulent atmosphere can be derived analytically by tensor method. It can avoid time-consuming numerical integral that was commonly used in the previous study of spectral changes. Analytical results show that the on-axis normalized spectrum S(ω) of EGB propagating through turbulent atmosphere is different from the original spectrum S₀(ω) and there exist spectral shifts and spectral switches for EGB propagating through turbulent atmosphere. Besides, spectral shifts and spectral switches of EGB are closely related with the structure constant of the turbulent atmosphere, the beam parameters and the coordinate of observation point. Compared with the Gaussian beam, there are two spectral switches for EGB propagating through turbulent atmosphere.     

Focusing of stochastic electromagnetic Gaussian Schell-model beams through a high numerical aperture objective

Based on vectorial Debye diffraction theory, the focusing properties of stochastic electromagnetic Gaussian Schell-model beams in the focal region of high numerical aperture objective are investigated. Expressions for the intensity distribution and the degree of polarization are derived near the focus. Numerical calculations are performed to analyze the influences of varying corresponding parameters on the changes in the intensity distribution and in the degree of polarization in the focal region.     

Propagation of a decentered astigmatic partially coherent beam in a turbulent atmosphere

Propagation properties of a decentered general astigmatic partially coherent beam (i.e., decentered twisted anisotropic Gaussian Schell-model beam) in a turbulent atmosphere are investigated. Analytical formulas for the cross-spectral density and decentered parameter of a decentered astigmatic partially coherent beam propagating in a turbulent atmosphere are derived. Irradiance properties of a decentered astigmatic partially coherent beam in a turbulent atmosphere are studied graphically, and are found to be quite different from its properties in free space.