Anisotropic Gaussian Schell-model Research Articles

Partially coherent sources with the combined quadratic phase

We construct a class of partially coherent sources with a new phase, which is consisted of separable and inseparable quadratic phases. As examples, the Gaussian distribution and rectangular flat-topped Gaussian distributions are selected as weight functions, respectively. The analytical propagation formulas of anisotropic Gaussian Schell-model (AGSM) beams and rectangular multi-Gaussian Schell-model (RMGSM) beams with the combined quadratic phase are derived. We investigate the propagation characteristics of the AGSM beam and the RMGSM beam carrying the combined quadratic phase. The twist effects of the combined quadratic phase on the spectral density (SD) and the degree of coherence (DOC) of the AGSM beam are quantitatively analyzed, and a specific method for calculating the critical value of the phase parameter is proposed. In addition, we qualitatively analyze the twist effects of the combined quadratic phase on the SD and the DOC of the RMGSM beam. We find that the propagation of the RMGSM beam is similar to that of the AGSM beam. These results may provide new inspiration for applications of synthesizing rotating beams in optics.

Statistical Characteristics of a Twisted Anisotropic Gaussian Schell-Model Beam in Turbulent Ocean

The analytical expression of the cross-spectral density function of a twisted anisotropic Gaussian Schell-model (TAGSM) beam transmitting in turbulent ocean is derived by applying a tensor method. The statistical properties, including spectral density, the strength of twist and beam width of the propagating beam are studied carefully through numerical examples. It is demonstrated that the turbulence of ocean has no effect on the rotation direction of the beam spot during propagation. However, the beam shape will degrade into a Gaussian profile under the action of oceanic turbulence with sufficiently long propagation distance, and a beam with a larger initial twist factor is more resistant to turbulence-induced degeneration. As oceanic turbulence becomes stronger, the beam spot spreads more quickly while the twist factor drops more rapidly upon propagation. The physical mechanisms of these phenomena are addressed in detail. The obtained results will be helpful in optical communication systems underwater.

Properties of off-axis hollow Gaussian-Schell model vortex beam propagating in turbulent atmosphere**Project supported by the Natural Science Foundation of Jilin Province, China (Grant No. 20180101031JC) and the Jilin Provincial Science Foundation for Basic Research, China (Grant No. 2019C040-7).

The analytical expression of off-axis hollow Gaussian–Schell model vortex beam (HGSMVB) generated by anisotropic Gaussian–Schell model source is first introduced. The evolution properties of off-axis HGSMVB propagating in turbulent atmosphere are analyzed. The results show that the off-axis HGSMVB with smaller coherence length or propagating in stronger turbulent atmosphere will evolve from dark hollow beam into Gaussian-like beam with a larger beam spot faster. The beams with different values of integer order N or the position for hollow and vortex factor R will have almost the same Gaussian-like spot distribution at the longer propagation distance.

Orbital angular moment of a partially coherent beam propagating through an astigmatic ABCD optical system with loss or gain

We derive the general expression for the orbital angular momentum (OAM) flux of an astigmatic partially coherent beam carrying twist phase [i.e., twisted anisotropic Gaussian-Schell model (TAGSM) beam] propagating through an astigmatic ABCD optical system with loss or gain. The evolution properties of the OAM flux of a TAGSM beam in a Gaussian cavity or propagating through a cylindrical thin lens are illustrated numerically with the help of the derived formula. It is found that we can modulate the OAM of a partially coherent beam by varying the parameters of the cavity or the orientation angle of the cylindrical thin lens, which will be useful in some applications, such as free-space optical communications and particle trapping.

Correlations between intensity fluctuations in apertured stochastic electromagnetic twist anisotropic Gaussian–Schell model beam propagating in turbulent atmosphere

Based on the extended Huygens–Fresnel principle, a formula has been derived for the correlations between intensity fluctuations at two points in the cross-section of an apertured stochastic electromagnetic twist anisotropic Gaussian Schell-model (ETAGSM) beam propagating in turbulent atmosphere by use of tensor method. Our main attention was concentrated on the influence of the aperture widths, atmospheric turbulence, twist parameters and partial coherence on the correlations between intensity fluctuations of apertured stochastic ETAGSM beam propagating in turbulent atmosphere. Numerical calculation results and analysis show that strong atmospheric turbulence, beam's twist properties and high coherence weaken the influence of the aperture on the correlations between intensity fluctuations of ETAGSM beam propagating in turbulent atmosphere.

Far-zone polarization distributionproperties of partially coherent beams with non-uniform source polarization distributions inturbulent atmosphere

It has already been found that, in turbulent atmosphere, many partially coherent electromagnetic beams with uniform source polarization distributions can regain these distribution patterns in the far field. However, the far-zone polarization properties of beams with non-uniform source polarization distributions are not sufficiently studied and the condition for an electromagnetic beam to reconstruct its source polarization distribution in the far zone is not established. Using a type of electromagnetic anisotropic Gaussian Schell-model (GSM) beams which can have non-uniform polarization distributions on the source plane, we find that, under the influence of turbulent atmosphere, the transverse polarization distribution will finally become uniform starting with a non-uniform source polarization distribution, and the far-field degree of polarization is affected by the source intensity parameters, but not by the source spatial coherence parameters. We also find that, electromagnetic anisotropic GSM beams can regain their source polarization patterns in the far field when propagating through atmospheric turbulence if and only if, the two intensity distributions corresponding to the two orthogonal field components on the source plane are the same, or are different only for a constant parameter. The validity of this condition is unaffected by the intensity and spatial coherence profiles on the source plane.

Evolution properties of the intensity distribution of an anisotropic Gaussian Schell-model beam in a turbulent atmosphere

An analytical expression for the cross-spectral density of an anisotropic Gaussian Schell-model (AGSM) beam propagating in a turbulent atmosphere is derived, which is featured by its clear physical meaning. The evolution properties of the intensity distribution of an AGSM beam in a turbulent atmosphere are studied thoroughly. It is found that owing to the anisotropy of the source coherence, the intensity distribution can have many different evolution processes; but under the influence of the turbulent atmosphere, it will finally take on a circular shape. The effects of lowering the source coherence on the circularization speed of the intensity distribution are also investigated. It is found that, when the anisotropy of the source coherence is taken into account, lowering the source coherence can accelerate or decelerate the circularization of the intensity distribution. We propose and demonstrate five kinds of conditions, and, under each condition, lowering the source coherence has a unique and definite effect on the circularization speed of the intensity distribution. In particular, we give an analysis about the equivalence between the problem studied in one of our five conditions and that in the work of Cai and He [Appl. Phys. Lett. 89, 041117 (2006)] and show that their relevant conclusion is incorrect.

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 of a twisted anisotropic Gaussian Schell-model beam beyond the paraxial approximation

With the help of a tensor method, an analytical nonparaxial propagation formula for a twisted anisotropic Gaussian Schell-model (GSM) beam in free space is derived based on the generalized Raleigh-Sommerfeld diffraction integral. The far-field nonparaxial propagation expression for a twisted anisotropic GSM beam is also derived. The paraxial approximation is dealt with as a special case of our general result. Our numerical results show that the nonparaxial propagation properties of a twisted anisotropic GSM beam are closely related to the initial beam parameters (i.e., twist factor, transverse spot width matrix, coherence width matrix and wavelength) and the propagation distance. Our formulae provide a convenient and powerful way for studying the paraxial and nonparaxial propagation of an isotropic or anisotropic GSM beam with or without twist phase in free space.

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.

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.

Z-scan experiment with anisotropic Gaussian Schell-model beams

We analyze the z-scan experiment with anisotropic Gaussian Schell-model (AGSM) beams. The expression for the cross-spectral density of the AGSM beam passing through the lens and onto the nonlinear thin sample is derived. Based on the expression, we simulate the results of the z-scan experiment theoretically and analyze the effects of the e factor (e=w(0x)/w(0y)) and the spatial degree of coherence in the x and y orientations on the on-axis z-scan transmittance. It is found that DeltaTp(-v) becomes larger with an increment of the e factor and the spatial degree of coherence. So we can improve the sensitivity of the z-scan experiment by increasing the e factor and the spatial degree of coherence. The results are helpful for improving the sensitivity of the z-scan experiment.

Propagation properties of a partially polarized electromagnetic twist anisotropic Gaussian Schell-model beam in turbulent atmosphere

Based on the extended Huygens–Fresnel integral formula, analytical formulae for the elements of cross-spectral density matrix of partially polarized electromagnetic twist anisotropic Gaussian Schell-model (TAGSM) beam propagating in turbulent atmosphere can been derived by a tensor method. Our main attention was focus on the effect of the atmospheric turbulence, twist parameters and partial coherence on the spectral degree of polarization, the spectral degree of coherence and the spectral density. Numerical calculation results and analysis are given.

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 rc.

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.

Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere

Analytical formulas are derived for the propagation of a partially coherent twisted anisotropic Gaussian Schell-model (TAGSM) beam in a turbulent atmosphere. Propagation properties of a TAGSM beam in a turbulent atmosphere are investigated in detail. It is found that a TAGSM beam will become a circular beam in a turbulent atmosphere, and low coherence and larger twist have an effect of anticircularization of the beam spot. The beam spot spreads more rapidly for lower coherence, larger twist, or stronger turbulence.

Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system

By expanding the hard-aperture function into a finite sum of complex Gaussian functions, we derived an approximate analytical formula for a partially coherent twisted anisotropic Gaussian Schell-model (AGSM) beam propagating through an apertured paraxial general astigmatic (GA) optical system by use of a tensor method. The results obtained by using the approximate analytical formula are in good agreement with those obtained by using the numerical integral calculation. Our formulas avoid time-consuming numerical integration and provide a convenient and effective way for studying the propagation and transformation of a partially coherent twisted AGSM beam through an apertured paraxial GA optical system.

Fractional Fourier transform for partially coherent beam in spatial-frequency domain

By using Fourier transform and the tensor analysis method, the fractional Fourier transform (FRT) in the spatial-frequency domain for partially coherent beams is derived. Based on the FRT in the spatial-frequency domain, an analytical transform formula is derived for a partially coherent twisted anisotropic Gaussian–Schell model (GSM) beam passing through the FRT system. The connections between the FRT formula and the generalized diffraction integral formulae for partially coherent beams through an aligned optical system and a misaligned optical system in the spatial-frequency domain are discussed, separately. By using the derived formula, the intensity distribution of partially coherent twisted anisotropic GSM beams in the FRT plane are studied in detail. The formula derived provide a convenient tool for analysing and calculating the FRTs of the partially coherent beams in spatial-frequency domain.

Application of the Wigner distribution function to studying spectral changes of twisted anisotropic Gaussian Schell-Model beams

Summary By using the Wigner distribution function (WDF), a general closed-form expression for the spectrum of twisted anisotropic Gaussian Schell-model (AGSM) beams in passage through a first-order ABCD system is derived. The spectral changes of twisted AGSM beams propagating in free space and through a thin lens, and the spectral changes of conversional Gaussian-Schell model (GSM) beams are treated as special cases of our general expression. Our theoretical and numerical results show that the WDF provides a powerful and simple tool in analyzing propagation properties of general AGSM beams. Specifically, the spectral behavior of twisted AGSM beams and conventional GSM beams can be treated in a unified and analytical way.

Diagonalization and symmetrization of anisotropic twisted Gaussian Schell-model beams

On the basis of the Wigner distribution function and second-order moments matrix methods, the diagonalization and symmetrization of anisotropic twisted Gaussian Schell-model (TGSM) beams are studied. The synthesis procedure for anisotropic TGSM beams is described and illustrated with numerical examples. It is shown that an anisotropic TGSM beam can be transformed into an aligned simple astigmatic (ASA) beam by four steps and then transformed into a stigmatic beam with twist by subsequent two steps.