Gaussian Schell-model Beams Research Articles

Twisting partially coherent light.

Twisted Gaussian Schell-model beams were introduced 25 years ago as a celebrated example of a "genuinely two-dimensional" partially coherent wavefield. Today, a definite answer about the effect that a twist phase should produce on an arbitrary cross-spectral density has not yet been reached. In the present Letter, the necessary and sufficient condition for a typical Schell-model partially coherent CSD endowed with axial symmetry to be successfully mapped onto a bonafide twisted CSD is addressed. In particular, it is proved that any shift-invariant degree of coherence of the form μ(|r1-r2|) is "twistable" if and only if the zeroth-order Hankel transform of the radial function μ(r)exp(ur2/2) (with u being the twist strength) turns out to be a well-defined, non-negative function.

Polarization properties of stochastic electromagnetic beams modulated by a wavefront-folding interferometer.

We consider the field generated by a wavefront-folding interferometer which is illuminated by a stochastic electromagnetic beam. The specular property and anti-specular property are discussed in the vector case. Take electromagnetic Gaussian Schell-model beam as an example, we investigate the spectral density, the spectral degree of coherence, the spectral degree of polarization as well as the state of polarization of the polarized portion of the field on propagation. Results show that the polarization properties including the degree of polarization, the orientation angle and the degree of ellipse can be adjusted by the phase difference of the interferometer and the phase retardation introduced by the prism. The results may be applied in free-space optical communication.

Fiber-coupling efficiency of Gaussian–Schell model beams through an ocean to fiber optical communication link

We theoretically study the fiber-coupling efficiency of Gaussian–Schell model beams propagating through oceanic turbulence. The expression of the fiber-coupling efficiency is derived based on the spatial power spectrum of oceanic turbulence and the cross-spectral density function. Our work shows that the salinity fluctuation has a greater impact on the fiber-coupling efficiency than temperature fluctuation does. We can select longer λ in the “ocean window” and higher spatial coherence of light source to improve the fiber-coupling efficiency of the communication link. We also can achieve the maximum fiber-coupling efficiency by choosing design parameter according specific oceanic turbulence condition. Our results are able to help the design of optical communication link for oceanic turbulence to fiber sensor.

The second–order moment statistics of a twisted electromagnetic Gaussian Schell-model propagating in a uniaxial crystal

Under the help of extended Huygens–Fresnel integral, we acquire the analytic expressions of second-order moments of the Wigner distribution function (WDF) for a twisted electromagnetic Gaussian Schell-model (TEGSM) beam propagating in a uniaxial crystal. Moreover, we obtain the formulae for propagation factor, the effective radius of curvature and Rayleigh range in which we studied the properties of the TEGSM beam numerically. It is found that the initial beam parameters and the uniaxial crystal parameters play a decisive role in the TEGSM beam propagating in a uniaxial crystal. The TEGSM beam with smaller absolute deviation of coherence width δxx and δyy is less affected by anisotropic diffraction in a uniaxial crystal, which is much different from the evolution properties of an EGSM beam in a uniaxial crystal. The Rayleigh range will decreases while the crystal parameter e increases. Thus, the propagation properties of a TEGSM beam could be controlled in a uniaxial crystal by changing the initial beam parameters and the uniaxial crystal parameters where it is necessary.

Partially coherent isodiffracting pulsed beams

We investigate a class of isodiffracting pulsed beams, which are superpositions of transverse modes supported by spherical-mirror laser resonators. By employing modal weights that, for stationary light, produce a Gaussian Schell-model beam, we extend this standard model to pulsed beams. We first construct the two-frequency cross-spectral density function that characterizes the spatial coherence in the space-frequency domain. By assuming a power-exponential spectral profile, we then employ the generalized Wiener-Khintchine theorem for nonstationary light to derive the two-time mutual coherence function that describes the space-time coherence of the ensuing beams. The isodiffracting nature of the laser resonator modes permits all (paraxial-domain) calculations at any propagation distance to be performed analytically. Significant spatiotemporal coupling is revealed in subcycle, single-cycle, and few-cycle domains, where the partial spatial coherence also leads to reduced temporal coherence even though full spectral coherence is assumed.

Entanglement and Complete Positivity: Relevance and Manifestations in Classical Scalar Wave Optics

AbstractEntanglement of states and Complete Positivity of maps are concepts that have achieved physical importance with the recent growth of quantum information science. They are however mathematically relevant whenever tensor products of complex linear (Hilbert) spaces are involved. We present such situations in classical scalar paraxial wave optics where these concepts play a role: propagation characteristics of coherent and partially coherent Gaussian beams; and the definition and separability of the family of Twisted Gaussian Schell Model (TGSM) beams. In the former, the evolution of the width of a projected one‐dimensional beam is shown to be a signature of entanglement in a two‐dimensional amplitude. In the latter, the partial transpose operation is seen to explain key properties of TGSM beams.

Propagation of a Vector Cosine-Gaussian Correlated Beam Through an Active GRIN Medium

This paper presents a convenient approach that uses the tensor method to study the propagation of a nonuniformly correlated electromagnetic Cosine-Gaussian Schell-model (ECGSM) beam through an active gradient-index (GRIN) medium. It is shown that the initial correlation structure has a significant influence on the intensity gain; an ECGSM beam ( n = 1) experiences more gain than a conventional electromagnetic Gaussian Schell-model (EGSM) beam ( n = 0). The inhomogeneous gain induced by the medium leads to a nonlinear modulation of beam parameters such as the spectral intensity, coherence, and polarization. Although the gain leads to a broadening of the beam width, the coherence gradually decreases, which yields a useful guideline for manipulating active GRIN materials to generate numerous partially coherent beams. It is found that the polarization ellipse in the center of a conventional EGSM beam gradually evolves into circular polarization. However, the state of polarization of a nonuniformly correlated ECGSM beam gradually changes into linear polarization. The ability to manipulate light beams using active GRIN materials opens an alternative avenue for the generation of complex vector beams, which promises important supports in the fields of waveguide amplifiers, beam shaping, optical sensors, and beam transformer devices.

Temporal spectrum of beam wander for Gaussian Schell-model beams propagating in atmospheric turbulence with finite outer scale: publisher's note.

This publisher's note corrects a typo in the title in Opt. Lett.38, 1887 (2013)OPLEDP0146-959210.1364/OL.38.001887.

Shaping the intensity and degree of coherence of a partially coherent beam by a 4f optical system with an amplitude filter

It is known that a 4f optical system with an amplitude filter can be adopted to shape the intensity of a coherent Gaussian beam. In this paper, we investigate the propagation of a typical kind of partially coherent beam called a Gaussian Schell-model (GSM) beam through a 4f optical system with an amplitude filter in the frequency plane. It is demonstrated both numerically and experimentally that the 4f optical system with an amplitude filter can be used to shape the intensity and degree of coherence of a GSM beam simultaneously. The shaped beam displays interesting properties and forms controllable far-field intensity lattices, which may be useful for particle trapping and information transfer.

Propagation of radially polarized multi-cosine Gaussian Schell-model beams in non-Kolmogorov turbulence

Recently, we introduced a new class of radially polarized beams with multi-cosine Gaussian Schell-model(MCGSM) correlation function based on the partially coherent theory (Tang et al., 2017). In this manuscript, we extend the work to study the statistical properties such as the spectral density, the degree of coherence, the degree of polarization, and the state of polarization of the beam propagating in isotropic turbulence with a non-Kolmogorov power spectrum. Analytical formulas for the cross-spectral density matrix elements of a radially polarized MCGSM beam in non-Kolmogorov turbulence are derived. Numerical results show that lattice-like intensity pattern of the beam, which keeps propagation-invariant in free space, is destroyed by the turbulence when it passes at sufficiently large distances from the source. It is also shown that the polarization properties are mainly affected by the source correlation functions, and change in the turbulent statistics plays a relatively small effect. In addition, the polarization state exhibits self-splitting property and each beamlet evolves into radially polarized structure upon propagation.

Propagation Factors of Partially Coherent Model Beams in Oceanic Turbulence

Based on the extended Huygens–Fresnel principle and second-order moments of the Wigner distribution function, we have derived the analytical expressions of the propagation factor of four kinds of partially coherent model beams in oceanic turbulence, including Gaussian Schell-model (GSM), Bessel–Gaussian Schell-model, Laguerre–Gaussian Schell-model, and cosine-Gaussian Schell-model (CGSM). It is found that the modulation of the spectral degree of coherence takes advantage over the GSM beams for reducing the turbulence-induced degradation. The beams with larger index ( $n$ , $\beta $ , $m$ ), initial beamwidth, and the wavelength or the smaller coherent length will be less affected by the oceanic turbulence. In addition, the influences of the oceanic parameters on the beam quality are discussed in detail, and the degradation of beams can be enhanced with the decrease of the rate of dissipation of turbulence kinetic energy per unit mass of fluid and the increase of the rate of dissipation of mean-square temperature, also the salinity fluctuation has greater contribution to the decrease of the beam quality than that of the temperature fluctuation. Moreover, the comparison of the normalized propagation factor of the four kinds of beams demonstrates that the CGSM is more robust against the oceanic turbulence than other partially coherent model beams. Our results can be helpful in the design of an optical communication system in an oceanic environment.

Beam spreading and M2-factor of electromagnetic Gaussian Schell-model beam propagating in inhomogeneous atmospheric turbulence

The main purpose of this paper is to study the beam spreading and M2-factor of electromagnetic Gaussian Schell-model (EGSM) beam through inhomogeneous atmospheric turbulence. The analytical formulas for the root-mean-square (rms) spatial width, rms angular width, M2-factor of EGSM beam in turbulence are derived by using the second-order moments of the Wigner distribution function and extended Huygens-Fresnel integral. Our results show that the relative M2-factor of EGSM beam decreases with decreasing of zenith angle, initial coherence lengths and initial degree of polarization and with increasing of beam widths and inner scale. It is also shown that as the zenith angle is equal to π/6, the saturated propagation distances of the relative rms angular width, the relative rms spatial width, and the relative M2-factor are about 0.3km, 3km and 20km, respectively. And it also can be found that the relative M2-factor becomes small when the initial degree of polarization closes to 0.

Propagation properties of an orthogonal cosine-Gaussian Schell-model beam in uniaxial crystals orthogonal to the optical axis

The analytical expressions for an orthogonal cosine-Gaussian Schell-model (OCGSM) beam propagating in uniaxial crystals orthogonal to the optical axis are derived. Based on the formulas derived, the propagation properties, such as the normalized spectral density, the spectral degree of coherence (SDOC), and the effective beam width of an OCGSM beam in uniaxial crystals are illustrated. Numerical results show that, by changing the beam mode order n and m, the OCGSM beam can split into two- or four-beamlet elliptical beams and keep invariant in uniaxial crystals during propagation, which is different from the conventional Gaussian Schell-model beam. The SDOC of the OCGSM beam will evolve into an elliptical shape and the effective beam width of the OCGSM beam will spread faster along the x direction than the y direction in uniaxial crystals. In addition, the significant influence of the beam mode order, the spatial coherence length and the ratio of the extraordinary refractive index to the ordinary refractive index of the uniaxial crystals on the evolution properties of an OCGSM beam is discussed in detail.

Strehl ratio and spread rate of Gaussian–Schell model beams in the marine–atmosphere link of weak to strong scintillations

We present the models of the Strehl ratio (SR) and the spread rate of Gaussian–Schell model (GSM) beams in the long slant path of marine–atmosphere turbulence. The models are developed based on the extended Rytov theory which permits us extending the results of weak scintillation into the moderate-to-strong scintillation regimes. Our results show that there exists a minimum of SR for selected waist radiuses and the minimum SR is dependent on the transverse coherent width of light source. The spread rate increases with increasing the light wavelength but it decreases with increasing the waist radius and the inner scale. The effects of turbulence on the peak received irradiance can be reduced by selecting larger waist radius of transmitting aperture and longer light wavelength.

Kurtosis parameter K of arbitrary electromagnetic beams propagating through non-Kolmogorov turbulence

General analytical formulae for the kurtosis parameters K (K parameters) of the arbitrary electromagnetic (AE) beams propagating through non-Kolmogorov turbulence are derived, and according to the unified theory of polarization and coherence, the effect of degree of polarization (DOP) of an electromagnetic beam on the K parameter is studied. The analytical formulae can be given by the second-order moments and fourth-order moments of the Wigner distribution function for AE beams at source plane, the two turbulence quantities relating to the spatial power spectrum, and the propagation distance. Our results can also be extended to the arbitrary beams and the arbitrary spatial power spectra of Kolmogorov turbulence or non-Kolmogorov turbulence. Taking the stochastic electromagnetic Gaussian Schell-model (SEGSM) beam as an example, the numerical examples indicate that the K parameters of a SEGSM beam in non-Kolmogorov turbulence depend on propagation distance, the beam parameters and turbulence parameters. The K parameter of a SEGM beam is more sensitive to effect of turbulence with smaller inner scale and generalized exponent parameter. A non-polarized light has the strongest ability of resisting turbulence (ART), however, a fully polarized SEGSM beam has the poorest ART.

Coherence and polarization properties of laser propagating through biological tissues

Based on the extended Huygens-Fresnel principle, the analytical expressions of the cross-spectral density matrix elements for random electromagnetic Gaussian Schell-model (GSM) beam propagating in biological tissues are derived, and used to study the changes in spectral degree of coherence μ and spectral degree of polarization P of random electromagnetic GSM beams with the propagation distance z propagating through the different biological tissues. It is shown that the changes closely depend on the species of the biological tissues, beam wave length, the interval between two field points and propagation distance. The spectral degree of coherence μ and the spectral degree of polarization P of the ultraviolet ray (λ=0.325μm) will quickly decrease during the propagation process, which implies that the damage of the ultraviolet ray to biological tissues is strong. The bigger structure constant of the refractive-index Cn2 corresponds to the smaller change of μ and P. There exists the obvious effect of the interval between two field points on the spectral degree of coherence and the spectral degree of polarization of random electromagnetic GSM beams passing biological tissues. The obtained results can provide the theoretical and experimental basis for the analysis to the coherence and polarization properties of random electromagnetic beams propagating through the complex biological tissues.

Potentials of radial partially coherent beams in free-space optical communication: a numerical investigation.

Nonuniformly correlated partially coherent beams (PCBs) have extraordinary propagation properties, making it possible to further improve the performance of free-space optical communications. In this paper, a series of PCBs with varying degrees of coherence in the radial direction, academically called radial partially coherent beams (RPCBs), are considered. RPCBs with arbitrary coherence distributions can be created by adjusting the amplitude profile of a spatial modulation function imposed on a uniformly correlated phase screen. Since RPCBs cannot be well characterized by the coherence length, a modulation depth factor is introduced as an indicator of the overall distribution of coherence. By wave optics simulation, free-space and atmospheric propagation properties of RPCBs with (inverse) Gaussian and super-Gaussian coherence distributions are examined in comparison with conventional Gaussian Schell-model beams. Furthermore, the impacts of varying central coherent areas are studied. Simulation results reveal that under comparable overall coherence, beams with a highly coherent core and a less coherent margin exhibit a smaller beam spread and greater on-axis intensity, which is mainly due to the self-focusing phenomenon right after the beam exits the transmitter. Particularly, those RPCBs with super-Gaussian coherence distributions will repeatedly focus during propagation, resulting in even greater intensities. Additionally, RPCBs also have a considerable ability to reduce scintillation. And it is demonstrated that those properties have made RPCBs very effective in improving the mean signal-to-noise ratio of small optical receivers, especially in relatively short, weakly fluctuating links.

Circularly symmetric cusped random beams in free space and atmospheric turbulence.

A class of random stationary, scalar sources producing cusped average intensity profiles (i.e. profiles with concave curvature) in the far field is introduced by modeling the source degree of coherence as a Fractional Multi-Gaussian-correlated Schell-Model (FMGSM) function with rotational symmetry. The average intensity (spectral density) generated by such sources is investigated on propagation in free space and isotropic and homogeneous atmospheric turbulence. It is found that the FMGSM beam can retain the cusped shape on propagation at least in weak or moderate turbulence regimes; however, strong turbulence completely suppresses the cusped intensity profile. Under the same atmospheric conditions the spectral density of the FMGSM beam at the receiver is found to be much higher than that of the conventional Gaussian Schell-model (GSM) beam within the narrow central area, implying that for relatively small collecting apertures the power-in-bucket of the FMGSM beam is higher than that of the GSM beam. Our results are of importance to energy delivery, Free-Space Optical communications and imaging in the atmosphere.

Polarization properties of Gaussian-Schell model beams propagating in a space-to-ground optical communication downlink.

Based on the extended Huygens-Fresnel principle, the element expression of a cross-spectral matrix of Gaussian-Schell-model (GSM) laser beams propagating in a space-to-ground optical communication downlink is derived analytically. With full consideration of both the source parameters and the zenith angle, the changes in the degree of polarization (DOP) and the state of polarization (SOP) are investigated. The simulation results indicate that the source parameters play a crucial role in determining both DOP and SOP in space-to-ground optical communication downlinks. Compared to the source parameters, the zenith angle has no obvious influence on the polarization properties when it is smaller than 88°. The results of this work provide a design basis for a polarization shift keying optical communication system with a GSM laser beam.

Propagation factor of electromagnetic concentric rings Schell-model beams in non-Kolmogorov turbulence**Project supported by the National Basic Research Program of China (Grant No. 2013CBA01702), the National Natural Science Foundation of China (Grant Nos. 61377016, 11104049, 10974039, 61575055, and 61575053), the Fundamental Research Funds for the Central Universities, China (Grant No. HIT.BRETIII.201406), and the Program for New Century Excellent Talents in University,

We derive an analytical expression for the propagation factor (known as M2-factor) of electromagnetic concentric rings Schell-model (EM CRSM) beams in non-Kolmogorov turbulence by utilizing the extended Huygens–Fresnel diffraction integral formula and the second-order moments of the Wigner distribution function (WDF). Our results show that the EM CRSM beam has advantage over the scalar CRSM beam for reducing the turbulence-induced degradation under suitable conditions. The EM CRSM beam with multi-rings far-fields in free space is less affected by the turbulence than the one with dark-hollow far-fields or the electromagnetic Gaussian Schell-model (EGSM) beam. The dependence of the M2-factor on the beam parameters and the turbulence are investigated in detail.