Gaussian Schell-model Source Research Articles

Study on the construction of twisted cosine partially coherent beams and their propagation characteristics

We propose a novel Schell model source for generating twisted partially coherent beams with an initial radius of curvature, which is called a twisted flat-topped cosine Gaussian Schell-model (TFCGSM) source. The TFCGSM beam comprises a wavefront phase and a flat-top structure, with the source degree of coherence determined by two cosine functions. Based on the Huygens–Fresnel principle, the general analytical expression of the cross-spectral density function of the TFCGSM beam propagating through the paraxial ABCD optical system is derived, and then its propagation properties are studied. The results show that the conversion of the array of the beam and the non-uniform structure can be realized by adjusting the parameters in the source plane. As the propagation distance of the TFCGSM beam increases, it rotates around the axis and increases the intensity of the array distribution. Surprisingly, the initial radius of curvature can cause the beam to rotate. The unique shape and properties of the TFCGSM beam create new possibilities for optical communication and enhanced optical functions.

Propagation Properties of Generalized Schell-Model Pulse Sources in Dispersive Media

A model of a generalized pulse source, whose complex degree of temporal coherence is described by a function of the nth power difference of two instants, was constructed. As examples, we consider the generalized Gaussian and multi-Gaussian Schell-model pulse sources and study their propagation in dispersive media. It is indicated that such pulse beams present unique self-focusing, off-axis self-shifting and asymmetric self-splitting characteristics by adjusting the power exponent and phase parameters. Further, we explicitly discuss how the coherence time, summation factor as well as the dispersive coefficient significantly affect the self-focusing and self-shifting behaviors of the pulse beam. The results will benefit some applications involving pulse shaping, optical trapping and remote sensing.

Changes in the statistical properties of electromagnetic double multi-Gaussian Schell-model beams on propagation in free space.

A class of electromagnetic random sources with a multi-Gaussian functional form in the spectral density and in the correlations part of the cross-spectral density matrix is introduced based on the genuine cross-spectral density-matrix theory. The analytic propagation formulas of the cross-spectral density matrix of such beams propagating in free space are derived by use of Collins' diffraction integral. With the help of analytic formulas, the evolution of statistical characteristics, i.e.,the spectraldensity, the spectral degree of polarization, and the spectral degree of coherence for such beams in free space are analyzed numerically. Employing the multi-Gaussian functional form in the cross-spectral density matrix introduces one more freedom in the modeling of Gaussian Schell-model sources.

Lensless ghost imaging of a partially coherent vortex source in atmospheric turbulence

The image quality of ghost imaging was mainly degraded because of the beam wander and scintillation caused by atmospheric turbulence. References show that the vortex phase can effectively deduce the negative effects and improve the critical resolution and image quality of ghost imaging in free space which can be used to improve its quality through turbulence. In this study, we investigate the quality of the ghost imaging of a Gaussian Schell-model vortex source with different topological charges and coherent widths during propagation through atmospheric turbulence with different strengths using numerical simulations. The results show that the image quality is improved with increasing topological charge in turbulence, which was further proved via experimental measurements. Our study can be applied to free space optical communication and information encryption based on ghost imaging systems to reduce the influence of turbulence.

Partially coherent conical refraction promises new counter-intuitive phenomena

In this paper, we extend the paraxial conical refraction model to the case of the partially coherent light using the unified optical coherence theory. We demonstrate the decomposition of conical refraction correlation functions into well-known conical refraction coherent modes for a Gaussian Schell-model source. Assuming randomness of the electrical field phase of the input beam, we reformulated and significantly simplified the rigorous conical refraction theory. This approach allows us to consider the propagation of light through a conical refraction crystal in exactly the same way as in the classical case of coherent radiation. Having this in hand, we derive analytically the conical refraction intensity both in the focal plane and in the far field, which allows us to explain and rigorously justify earlier experimental findings and predict new phenomena. The last include the counterintuitive effect of narrowing of the conical refraction ring width, disappearance of the dark Poggendorff’s ring in the Lloyd’s plane, and shift of Raman spots for the low-coherent conical refraction light. We also demonstrate a universal power-law dependence of conical refraction cones coherence degree on the input correlation length and diffraction-free propagation of the low-coherent conical refraction light in the far field.

Twist phase and classical entanglement of partially coherent light.

We demonstrate that the presence of a twist phase in a random light beam leads to classical entanglement between phase space degrees of freedom of the beam. We find analytically the bi-orthogonal decomposition of the Wigner function of a twisted Gaussian Schell-model (TGSM) source and quantify its entanglement by evaluating the Schmidt number of the decomposition. We show that (i) classical entanglement of a TGSM source vanishes concurrently with the twist in the fully coherent limit and (ii) entanglement dramatically increases as the source coherence level decreases. We also show that the discovered type of classical entanglement of a Gaussian Wigner function does not degrade on beam propagation in free space.

Variant-Coherence Gaussian Sources

The celebrated Gaussian Schell model source with its shift-invariant degree of coherence may be the basis for devising sources with space-variant properties in the spirit of structured coherence. Starting from superpositions of Gaussian Schell model sources, we present two classes of genuine cross-spectral densities whose degree of coherence varies across the source area. The first class is based on the use of the Laplace transform while the second deals with cross-spectral densities that are shape-invariant upon paraxial propagation. For the latter, we present a set of shape-invariant cross-spectral densities for which the modal expansion can be explicitly found. We finally solve the problem of ascertain whether an assigned cross-spectral density is shape-invariant by checking if it satisfies a simple differential constraint.

Three modal decompositions of Gaussian Schell-model sources: comparative analysis.

Representation of the cross-spectral density (CSD) function of an optical source or beam as the incoherent superposition of mutually uncorrelated modes are widely used in imaging systems and in free space optical communication systems for simplification of the analysis and reduction of the time-consuming integral calculations. In this paper, we examine the equivalence and the differences among three modal representation methods: coherent-mode representation (CMR), pseudo-mode representation (PMR) and random mode representation (RMR) for the Gaussian Schell-model (GSM) source class. Our results reveal that for the accurate reconstruction of the CSD of a generic GSM source, the CMR method requires superposition of the least number of optical modes, followed by PMR and then by RMR. The three methods become equivalent if a sufficiently large number of optical modes are involved. However, such an equivalence is limited to the second-order statistics of the source, e.g., the spectral density (average intensity) and the degree of coherence, while the fourth-order statistics, e.g., intensity-intensity correlations, obtained by the three methods are quite different. Furthermore, the second- and the fourth- order statistics of the GSM beam propagating through a deterministic screen and dynamic random screens with fast and slow time cycling are investigated through numerical examples. It is found that the properties of the second-order statistics of the beams obtained by the three methods are the same, irrespectively of the characteristics of the screens, whereas those of the fourth-order statistics remain different.

Spectral polarization of Gaussian Schell-model beams.

We present a class of broadband electromagnetic Gaussian Schell-model sources whose state of polarization is both uniform and identical for all frequencies, but whose far-zone polarization properties strongly depend on wavelength. Also, these sources can produce beams whose polarized portion is always linearly polarized but with a polarization angle that evolves on propagation. Our results offer new insights into the behavior of broadband partially coherent sources.

Effects of source spatial partial coherence on intensity statistics of optical beams in mono-static turbulent channels.

We investigate, via both experimental measurements and wave-optics computer simulations, the statistical characteristics of the fluctuating intensity, such as the average intensity, the beam wander and the scintillation index, of the Gaussian Schell-model (GSM) beams on passing through double-pass, monostatic turbulence channels with either a retro-reflector (RR) or a flat mirror (FM). Our experimental results reveal that the enhanced backscatter (EBS) gradually weakens as the spatial coherence of the GSM source decreases, and eventually disappears for the sufficiently low source spatial coherence states. The r.m.s beam wander remains practically invariant with the variation of the source coherence width in the range from 0.2 to 6.0 mm both in the case of the RR and the FM, which formed the RR case being much smaller. In addition, it is found that the long-term scintillation index of the untracked beam with the RR is smaller than that with the FM, while the situation is reversed for the short-term scintillation index of the tracked beam. In both cases, the scintillation index decreases as the spatial coherence of the GSM source decreases. The obtained computer simulation results agree reasonably well with the experimental results. In addition, the effects of spatial coherence on statistical characteristics of the GSM beams along a 1 km propagation distance through the double-pass monostatic turbulence are also investigated using wave-optics simulation. We also carry out evaluation and comparison of the intensity probability density functions for the RR and FM cases and for various source coherence states that are of utmost importance for free-space optical communications in retro-reflection modulation regime. In addition, our findings will be beneficial for the development of remote sensing and directed energy laser applications in the presence of air turbulence.

Spectral invariance of Gaussian Schell-model beams.

It is well known that in general the spectrum of a beam that is generated by a partially coherent source will change on propagation. Here we derive necessary and sufficient conditions under which the often-used Gaussian Schell-model sources can produce beams whose normalized spectrum is invariant everywhere, or is invariant just along the beam axis. These sources are not necessarily quasi-homogeneous or obeying the scaling law.

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.

Reconstruction of an electromagnetic Gaussian Schell-model source from far-zone intensity measurements.

An electromagnetic Gaussian Schell-model source that produces a random beam may be characterized by eight independent quantities. We show how far-zone measurements of the Stokes parameters, together with the Hanbury Brown-Twiss coefficient, allow one to determine all the source parameters. This method provides, to the best of our knowledge, a new tool to identify distant sources.

Propagation of Gaussian Schell-model beams through a jet engine exhaust.

Theoretical predictions of light beam interactions with jet engine exhaust are of importance for optimization of various optical systems, including LIDARs, imagers and communication links operating in the vicinity of aircrafts and marine vessels. Here we extend the analysis previously carried out for coherent laser beams propagating in jet engine exhaust, to the broad class of Gaussian Schell-Model (GSM) beams, being capable of treating any degree of coherence in addition to size and radius of curvature. The analytical formulas for the spectral density (SD) and the spectral degree of coherence (DOC) of the GSM beam are obtained and analyzed on passage through a typical jet engine exhaust region. It is shown that for sources with high coherence, the transverse profiles of the SD and the DOC of the GSM beams gradually transition from initially circular to elliptical shape upon propagation at very short ranges. However, such transition is suppressed for sources with lower coherence and disappears in the incoherent source limit, implying that the GSM source with low source coherence is an excellent tool for mitigation of the jet engine exhaust-induced anisotropy of turbulence. The physical interpretation and the illustration are included.

Twisted elliptical multi-Gaussian Schell-model beams and their propagation properties.

We introduce a new kind of partially coherent source whose cross-spectral density (CSD) function is described as the incoherent superposition of elliptical twisted Gaussian Schell-model sources with different beam widths and transverse coherence widths, named twisted elliptical multi-Gaussian Schell-model (TEMGSM) beams. Analytical expression for the CSD function propagating through a paraxial ABCD optical system is derived with the help of the generalized Collins formula. Our results show that the TEMGSM beam is capable of generating a flat-topped elliptical beam profile in the far field, and the beam spot during propagation exhibits clockwise/anti-clockwise rotation with respect to its propagation axis. In addition, the analytical expressions for the orbital angular momentum (OAM) and the propagation factor are also derived by means of the Wigner distribution function. The influences of the twisted factor and the beam index on the OAM and the propagation factor are studied and discussed in detail.

Properties of partially coherent elegant Laguerre-Gaussian beam in free space and oceanic turbulence

The analytical expression of partially coherent elegant Laguerre-Gaussian (PCELG) beam generated by anisotropic Gaussian Schell-model sources is given. The analytical formulas of CSD for a PCELG beam propagating in free space and oceanic turbulence are derived. The characteristics of average intensity and spectral degree of coherence of a PCELG beam propagating in free space and oceanic turbulence are investigated by using numerical simulations. It is shown that the stronger oceanic turbulence will accelerate the evolution of a PCELG beam propagating in oceanic turbulence, and the influences of coherence length on the spectral degree of coherence will disappear as the propagation distance increases. The findings may be useful in underwater optical communications.

Phase distribution reconstruction in reflective ghost diffraction

The commonly used ghost imaging technique can be used to obtain the reflectivity of a reflective object, but leaves the phase distribution of the reflection coefficient not retrieved. To reconstruct the phase of a reflective object, we propose a kind of generalized reflective ghost diffraction (GD) scheme, in which the reference screens are inserted into the reference path of the standard reflective GD setup. With the help of the three-step phase-shifting method, we provide an algorithm to reconstruct the phase distribution of the reflective object. The applicability of this theoretical proposal is demonstrated via numerical simulations. The effects of the transverse coherence of the Gaussian Schell-model source on the phase reconstruction are also discussed.

Propagation properties of a partially coherent four-petal Lorentz–Gauss beam in non-Kolmogorov turbulence

The model of a partially coherent four-petal Lorentz–Gauss beam generated by the Gaussian Schell-model source was first introduced. The cross-spectral density function of a partially coherent four-petal Lorentz–Gauss beam propagating in non-Kolmogorov turbulence was derived by using the extended Huygens–Fresnel integral. The spreading and coherence properties of a partially coherent four-petal Lorentz–Gauss beam propagating in non-Kolmogorov turbulence were studied. The results show that such a beam propagating in non-Kolmogorov turbulence with either larger , smaller , larger , or smaller will lose the four-petal profile distribution and evolve into a solid beam with Gaussian-like distribution more rapidly as the propagation distance increases. The results also show that the spectral degree of coherence of a partially coherent four-petal Lorentz–Gauss beam between two points and will decrease as the distance of two points increases in the far field.

Generalized Hanbury Brown–Twiss effect in partially coherent electromagnetic beams

The recently introduced concept of Stokes fluctuations generalizes both the Hanbury Brown--Twiss effect and the notion of scintillation. Here we apply this new framework to the specific example of a Gaussian Schell-model (GSM) beam. We derive formulas for Stokes scintillations and Stokes fluctuation correlations, which explicitly express the dependence of these quantities on the GSM source parameters. It is found that the normalized Stokes scintillations vary significantly with position. Also, they can be either positively or negatively correlated.

Higher-order twisted/astigmatic Gaussian Schell-model cross-spectral densities and their separability features.

Adding a twist phase term to the cross-spectral density (CSD) function of a partially coherent source can be done if and only if the resulting function remains nonnegative definite. Constraints on the twist term that guarantee the validity of the resulting CSD have been derived only for Twisted Gaussian Schell-model (TGSM) sources. Here, an infinite family of higher-order TGSM sources is introduced, whose CSDs are expressed as products of the CSD of a standard TGSM source times Hermite polynomials of arbitrary orders and suitable arguments. All the members present the same twist term and for all of them the twist-coherence constraint keeps obeying the form valid for a standard TGSM source. They can be used as building blocks for constructing an endless number of valid twisted CSDs, with an assigned value of the twist parameter and intensity and/or coherence features that can be very different from those of a standard TGSM source. Through partial transposition, higher-order TGSM CSDs are converted into higher-order Astigmatic Gaussian Schell-model (AGSM) CSDs. The problem of the separability of higher-order TGSM and AGSM CSDs is addressed, and conditions ensuring their separability are derived.