Twisted Gaussian Schell-model Research Articles

Propagation of the COAM matrix of a twisted Gaussian Schell-model beam

The Coherence-Orbital Angular Momentum (COAM) matrix elements of an optical beam describe its mode-to-mode radial correlations. An analytic expression for the COAM matrix of a Twisted Gaussian Schell-Model (TGSM) beam propagating in a vacuum is derived and analyzed in detail, with the help of the Huygens–Fresnel integral. Additionally, we show how to recover the cross-spectral density of the TGSM beam from its COAM matrix, expressing the result in an elegant symmetric form. Due to the unique properties of the TGSM beam, the analysis of its individual OAM correlations will deliver further insight for their applications to imaging, laser communication, optical tweezers, and other modalities.

Coherence-OAM matrix properties of a twisted Gaussian Schell model beam.

Based on the cross-spectral density (CSD) function, the coherence-orbital angular momentum (COAM) matrix of twisted Gaussian Schell-model (TGSM) beam is proposed, and the COAM matrix is used to describe the correlation between OAM modes in TGSM optical field. The COAM matrix characteristics of TGSM beam are analyzed by numerical simulation. The results show that the COAM matrix characteristics of TGSM beam depend on the initial parameters of the beam. In addition, a method of generating TGSM beam by superposition of COAM matrix element modes is described, and the influence of different initial parameters on the superposition characteristics is studied. The results reveal the internal relationship between the coherent structure of the optical field, the twist phase and the OAM modes. Our work helps to explore new expressions of partially coherent beams and promote the practical application of optimizing partially coherent beams.

Classical entanglement of twisted random light propagating through atmospheric turbulence [Invited

We examine the impact of the atmospheric turbulence on a recently discovered type of classical entanglement of partially coherent beams endowed with a twist phase. We derive a compact analytical expression for the Schmidt number of a bi-orthogonal decomposition of the Wigner function of a twisted Gaussian Schell-model (TGSM) beam propagating through the turbulent atmosphere. We elucidate conditions for a TGSM source to generate a strongly classically entangled paraxial field over a desired propagation distance in the turbulent atmosphere. Our results will find applications to free-space optical communications and motivate further research into classical entanglement with random light.

Twisted Gaussian Schell-model breathers and solitons in strongly nonlocal nonlinear media.

Based on the Snyder-Mitchell linear model and the cross-spectral density (CSD) function, the analytical propagation formula of twisted Gaussian Schell-model (TGSM) beams in strongly nonlocal nonlinear medium (SNNM) is derived. Then the propagation characteristics of TGSM beam are studied. It is found that the soliton radius is jointly determined by the initial power, coherence length, and twist factor; the degree of spatial coherence is adjusted by changing the twist factor without affecting the soliton intensity. In the case of non-soliton properties, there is a threshold of coherence length which makes partially coherent beams have the same evolution law as completely coherent beams. Furthermore, increasing the twist factor, decreasing the coherence length and initial power can improve the beam quality of the beam propagating in SNNM.

Evaluation of twisted Gaussian Schell model beams produced with phase randomized coherent fields

The twisted Gaussian Schell Model describes a family of partially coherent beams that present several interesting characteristics, and as such have attracted attention in classical and quantum optics. Recent techniques have been demonstrated to synthesize these beams from a coherent source using a discrete set of ‘pseudo-modes’, where the phase of each mode is randomized so that they are mutually incoherent. Here we investigate this technique and evaluate the resulting beam parameters, such as divergence, coherence length and twist phase. We show that for a finite set of modes there is also some residual coherence, which can have an observable effect. A theoretical model is developed for the output field that includes residual coherence and agrees very well with experimental data. In addition, we demonstrate a simple method to measure the twist phase using double slit interference.

Properties of the Rotation and Mergence of Twisted Gaussian Schell Model Array Beams Propagating in Turbulent Biological Tissues

In this paper, an analytical expression for describing propagation properties of twisted Gaussian Schell model array (TGSMA) beams through turbulent biological tissues is derived based on the extended Huygens Fresnel integral. With the help of the formulae, properties of the rotation and mergence for the TGSMA beams in turbulent biological tissues are researched in detail. It is found that the TGSMA beams go through the distinct mergence period in the far field besides phenomena of abruption and rotation in the near field, and turbulent biological tissues play a dominated role in mergence of the TGSMA beams. These novel results may be helpful in optical trapping.

Rotation of degree of coherence and redistribution of transverse energy flux induced by non-circular degree of coherence of twisted partially coherent sources.

It is known that a twisted Gaussian Schell-model (TGSM) beam with elliptical Gaussian amplitude will rotate its beam spot upon propagation because of the vortex structure of the transverse energy flux. In this paper, we study a special kind of twisted partially coherent beams named twisted Hermite-Gaussian correlated Schell model (HGCSM) beam whose degree of coherence (DOC) is non-circularly symmetric but the source amplitude is of the circular Gaussian profile. Our results reveal that the beam spot (average intensity distribution) does not rotate during propagation even if the circular symmetry of the beam spot is broken. However, the DOC pattern shows the rotation under propagation. From the investigation of the transverse energy flux and OAM density flux, we attribute the nontrivial rotation phenomenon to the redistribution of the transverse energy flux by non-circular DOC. Furthermore, based on Hyde's approach [J. Opt. Soc. Am. A37, 257 (2020)10.1364/JOSAA.381772], we introduce a method for the generation of this class of twisted partially coherent sources. The non-rotation of the beam spot and rotation of the DOC are demonstrated in experiment.

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.

Experimental synthesis of partially coherent beam with controllable twist phase and measuring its orbital angular momentum

Abstract Twist phase is a nontrivial second-order phase that only exists in a partially coherent beam. Such twist phase endows the partially coherent beam with orbital angular momentum (OAM) and has unique applications such as in super-resolution imaging. However, the manipulation and the detection of the twist phase are still far from easy tasks in experiment. In this work, we present a flexible approach to generate a famous class of twisted Gaussian Schell-model (TGSM) beam with controllable twist phase by the superposition of the complex field realizations using a single phase-only spatial light modulator. The precise control of the amplitude and phase of the field realizations allows one to manipulate the strength of the twist phase easily. In addition, we show that the twist factor, a key factor that determines the strength of twist phase and the amount of OAM, can be measured by extracting the real part of the complex degree of coherence of the TGSM beam. The experiment is carried out with the help of the generalized Hanbury Brown and Twiss experiment as the generated TGSM beam obeys Gaussian statistics. The flexible control and detection of the twist phase are expected to find applications in coherence and OAM-based ghost imaging.

Influence of Kerr nonlinearity on propagation characteristics of twisted Gaussian Schell-model beams.

The analytical propagation formulae of twisted Gaussian Schell-model (TGSM) beams through nonlinear Kerr media are derived. It is found that a TGSM beam is less sensitive to Kerr nonlinearity than a Gaussian Schell-model (GSM) beam. Furthermore, the propagation characteristics of TGSM beams with stronger twist and worse spatial coherence are less affected by Kerr nonlinearity. The self-focusing effect enhances the beam twist, but degrades the beam spatial coherence. In the atmosphere (one kind of self-focusing media), a TGSM beam has greater resistance to self-focusing effects and atmospheric turbulence effects than a GSM beam or an ideal Gaussian beam.

Evolution properties of the orbital angular momentum spectrum of twisted Gaussian Schell-model beams in turbulent atmosphere.

We derive the analytical formula of the energy weight of each orbital angular momentum (OAM) mode of twisted Gaussian Schell-model (TGSM) beams propagating in weak turbulent atmosphere. The evolution of its OAM spectrum is studied by numerical calculation. Our results show that the OAM spectrum of a TGSM beam changes with the beam propagating in turbulent atmosphere, which is completely different from that of the TGSM beam propagating in free space. Furthermore, influences of the source parameters and the turbulence parameters on the OAM spectrum of a TGSM beam in turbulent atmosphere are analyzed. It is found that the source parameters and turbulence parameters, such as twist factor, coherence length, beam waist size, and structure constant, have a significant influence on the OAM spectrum, but the value of the wavelength and inner scale have little influence. Increasing the beam waist size or decreasing the coherence length would lead to the OAM spectrum broadened in the source plane, but would be robust for the OAM modes of the TGSM beam in the turbulent atmosphere. It is clear that the bigger the value of the twist factor, the more asymmetric the OAM mode of the TGSM beam is, and the better mode distribution can be maintained when it propagates in turbulent atmosphere. Our results have potential applications in reducing the error rate of free-space optical communication and detecting the atmospheric parameters.

Ghost Imaging with a Partially Coherent Beam Carrying Twist Phase in a Turbulent Ocean: A Numerical Approach

Ghost imaging (GI) is an indirect imaging approach that can retrieve an object’s image even in a harsh environment through measuring the fourth-order correlation function (FOCF) between the signal and idle optical paths. In this paper, we study lensless GI with a partially coherent beam carrying twist phase, i.e., twisted Gaussian Schell-model (TGSM) beam, in the presence of oceanic turbulence. Explicit expression of the FOCF is derived based on the optical coherence theory and Rytov approximation, and the effects of the twist phase and the oceanic turbulence on the quality and visibility of image are investigated in detail through numerical examples. Our results show that the simulated oceanic turbulence strongly affects the GI. The quality of image decreases monotonously with an increase of the strength of turbulence whereas the visibility increases. When the illumination light carries a twist phase, the visibility of the image is improved while the quality of the image is reduced in contrast to those without a twist phase. By properly selecting the strength of the twist phase, the image can still be maintained at an acceptable level of quality with high visibility. Furthermore, it is found that the quality and visibility of the ghost image are less affected by the oceanic turbulence using a TGSM beam with larger twist factor. Our findings will be useful for the application of GI in an oceanic turbulent environment.

Generating bona fide twisted Gaussian Schell-model beams.

We experimentally realize bona fide twisted Gaussian Schell-model (TGSM) beams by converting an anisotropic GSM beam into a TGSM beam with a set of just three cylindrical lenses. In contrast to the previously reported experimental technique for generating TGSM beams, we are able to explicitly generate anisotropic GSM beams with simultaneously controlled beam widths and transverse coherence lengths along two mutually orthogonal directions, prior to converting them to genuine TGSM beams.

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.

Second-order moments of a twisted Gaussian Schell-model beam in anisotropic turbulence

Analytical formulas for the Wigner distribution functions and the second-order moments (SOMs) of a twisted Gaussian Schell-model (TGSM) beam propagating through anisotropic turbulence have been derived by means of a tensor method. It is found that the propagation law for the SOMs of a TGSM beam spreading in turbulence can be described as a first-order optical systems propagation law with an additional turbulence-induced effect. Based on the SOMs, second-order statistics in terms of the effective beam width, the M2-factor, the orbital angular momentum flux, and the effective radius curvature are analyzed in detail. One finds that the anisotropic turbulence leads to an anisotropic spreading of light beams, and a TGSM beam is less affected by turbulence than a Gaussian Schell-model beam without the twist phase. For short distance propagation, the light beam is more sensitive to the initial beam parameters than turbulence parameters, while for sufficiently long transmission distance, the beam characteristics are determined at most by turbulence statistics. The method can be extended to study the propagation characteristics of various coherent and partially coherent complex Gaussian beams, such as flat-topped beams, dark hollow beams, and array beams in turbulence, and promotes important supports in free-space optical communications and remote sensing.

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.

Statistical properties of a radially polarized twisted Gaussian Schell-model beam in an underwater turbulent medium.

Average intensity and the normalized powers of the completely polarized and the completely unpolarized portions of a radially polarized twisted Gaussian Schell-model (TGSM) beam propagating in underwater turbulence are examined. In our formulation, our previously obtained atmospheric turbulence solution for the same radially polarized TGSM beam using the extended Huygens-Fresnel principle is utilized, with the inclusion of our recently derived expression for the atmospheric turbulence structure constant in terms of underwater turbulence parameters. Effects of the rate of dissipation of mean-squared temperature, rate of dissipation of kinetic energy per unit mass of fluid, kinematic viscosity, and the contribution of the temperature-to-salinity ratio to the refractive index spectrum on the average intensity, and the normalized powers of the completely polarized and completely unpolarized portions of a radially polarized TGSM beam propagating in underwater turbulence are presented.

Propagation of a radially polarized twisted Gaussian Schell-model beam in turbulent atmosphere

Propagation properties of a scalar twisted Gaussian Schell-model (TGSM) beam in turbulent atmosphere has been studied in detail. In this paper, we explore the propagation properties of a radially polarized TGSM beam in turbulent atmosphere. Analytical formulae for the elements of the cross-spectral density matrix of a radially polarized TGSM beam propagating in turbulent atmosphere are derived based on the extended Huygens–Fresnel integral. We find that the atmospheric turbulence induces degeneration of the intensity distribution and depolarization of a radially polarized TGSM beam on propagation, while the twist phase plays a role in anti-degradation and anti-depolarization. Furthermore, we also find that the twist phase induces not only the rotation of the beam spot but also changes of the state of polarization on propagation, which are caused by the fact that the twist phase imposes orbital angular momentum on the beam. Thus, the twist phase can be used to control the propagation properties of radially polarized partially coherent beam. Our results will be useful in some applications, such as optical trapping and free-space optical communications.

Twisted Gaussian Schell-model beams as series of partially coherent modified Bessel-Gauss beams.

We show that twisted Gaussian Schell-model (TGSM) beams can be represented through an incoherent superposition of partially coherent beams carrying optical vortices and whose cross-spectral densities are expressed in terms of modified Bessel functions. Moreover, starting from this result, we show that the modal expansion of the cross-spectral density of a TGSM source can be directly obtained through simple mathematics.

Apodized design of diffractive axicons for twisted partially coherent light

Within the framework of inverse diffractive optics, we present a design for diffractive axicons in twisted, spatially partially coherent fields, in particular twisted Gaussian Schell-model (TGSM) fields. The design is based on the method of stationary phase. A general modification is introduced to the inverse diffractive optics approach for improving the synthesized optical element to produce the desired intensity distribution. Both the design and modification are demonstrated with annular-aperture axicons generating uniform-intensity axial line segments in partially coherent TGSM illumination.