Coherence and polarization properties of a radially polarized beam with variable spatial coherence

Propagation of radially polarized beams with a Hermite non-uniformly correlated array in free space and turbulent atmosphere

We introduce what we believe to be a novel class of radially polarized partially coherent beams in which the correlation function possesses a Hermite non-uniformly correlated array. The source parameter conditions required to generate a physical beam are derived. The statistical properties of such beam propagating in free space and turbulent atmosphere are thoroughly examined using the extended Huygens-Fresnel principle. It is shown that the intensity profile of such beams presents a controllable periodic grid distribution due to its multi-self-focusing propagation property and can keep the shape in free space while propagating in turbulent atmosphere, it exhibits self-combining properties over a long-ranges. Owing to the interaction between the non-uniform correlation structure and the non-uniform polarization, this beam can locally self-recover the polarization state after propagating a long distance in a turbulent atmosphere. Furthermore, the source parameters play essential roles in determining the distribution of spectral intensity, the state of polarization, and the degree of polarization of the RPHNUCA beam. Our results may benefit multi-particle manipulation and free-space optical communication applications.

Nonparaxial Propagation of Bessel Correlated Vortex Beams in Free Space

The nonparaxial propagation of partially coherent beams carrying vortices in free space is investigated using the method of decomposition of the incident field into coherent diffraction-free modes. Modified Bessel correlated vortex beams with the wavefront curvature are introduced. Analytical expressions are presented to describe the intensity distribution and the degree of coherence at different distances. The evolution of the intensity distribution during beam propagation for various source parameters is analyzed. The effects of nonparaxiality in the propagation of tightly focused coherent vortex beams are analyzed.

Effect of optical spatial coherence on localized spin angular momentum density in tightly focused light [Invited]

Optical coherence is one of the most fundamental characteristics of light and has been viewed as a powerful tool for governing the spatial, spectral, and temporal statistical properties of optical fields during light-matter interactions. In this work, we use the optical coherence theory developed by Emil Wolf as well as the Richards-Wolf's vectorial diffraction method to numerically study the effect of optical coherence on the localized spin density of a tightly focused partially coherent vector beam. We find that both the transverse spin and longitudinal spin, with the former induced by the out-of-phase longitudinal field generated during strong light focusing and the latter induced by the vortex phase in the incident beam, are closely related to the optical coherence of the incident beam, i.e., with the decrease of the transverse spatial coherence width of the incident beam, the magnitude of the spin density components decreases as well. The numerical findings are interpreted well with the two-dimensional degrees of polarization between any two of the three orthogonal field components of the tightly focused field. We also explore the roles of the topological charge of the vortex phase on enhancing the spin density for the partially coherent tightly focused field. The effect of the incident beam's initial polarization state is also discussed.

Radially polarized cosine non-uniformly correlated beams and their propagation properties

We introduce a kind of radially polarized partially coherent (RPPC) beam with a prescribed non-uniform correlation function, called a radially polarized cosine non-uniformly correlated (RPCNUC) beam. Based on the extended Huygens-Fresnel principle, we study the propagation properties in free space and in a turbulent atmosphere. Unlike RPPC beams with uniform coherence, RPCNUC beams possess the invariance of dark hollow cores and radial polarization, and exhibit self-focusing properties. In a turbulent atmosphere, the intensity distribution demonstrates self-healing properties over a certain propagation distance. We also investigate how to adjust the beam parameters to reduce the turbulence-induced degradation in detail.

Statistical properties of a partially coherent vector beam with controllable spatial coherence, vortex phase, and polarization

We report on a partially coherent radially polarized power-exponent-phase vortex (PC-RP-PEPV) beam with various distributions of intensity, controllable coherence width, vortex phase, and polarization. The statistical properties of the PC-RP-PEPV beam depend on topological charge, power order, polarization states, and coherence width, which differ from those of conventional radially polarized beams. Here, the initial radial polarization state will transform to complex ellipse polarization state during propagation. By modulating the topological charge of the PC-RP-PEPV beam, the intensity structure of the beam can be adjusted from circular to polygonal. Finally, PC-RP-PEPV beams were experimentally generated, and were consistent with numerical simulation results. This work has applications in optical manipulation, optical measurements, and optical information processing.

Super cosh-Gauss nonuniformly correlated radially polarized beam and its propagation characteristics

In this paper, a new kind of partially coherent vector beam termed as super cosh-Gauss nonuniformly correlated radially polarized (SCNRP) beam is introduced. Such beam source exhibits almost perfect coherence between two points that are within the beam center region or located on a ring concentric with the beam center. However, the coherence drops or even vanishes when the two points leave the central region and are located on the concentric rings with different radii. The second-order statistical properties, such as the spectral density, the state of polarization (SOP), and the degree of polarization (DOP) of such beam upon free-space propagation are studied through numerical examples. Our results reveal that the beam displays a self-focusing property during propagation. The focusing ability can be enhanced with increasing the beam index and decreasing the beam's spatial coherence width, whereas the DOP and SOP remain unchanged on propagation. Meanwhile, we establish an experimental system with the use of a radial polarization converter and a digital micro-mirror device to synthesize the SCNRP beam with controllable beam index and spatial coherence width. The spectral density and polarization properties of the synthesized beam during propagation are measured and analyzed in the experiment. The experimental results agree well with our theoretical predictions.

Self-focusing propagation characteristics of a radially-polarized beam in nonlinear media

In this study, an analytical formula for the self-focusing length of a radially polarized beam (RPB) is first derived, which has a similar behavior to the semi-empirical Marburger formula of a Gaussian beam, and is beneficial to quantitatively and qualitatively analyze practical experimental scenarios. However, the relation of the self-focusing length with the states of polarization (SoPs) was evaluated, and it was found that RPB with spatially inhomogeneous SoP at the field cross-section can retain a further self-focusing length compared to a beam with a spatially homogeneous one. The influence of the topological charge on the self-focusing length is explored, which shows that RPB with a low topological charge can achieve a high-power density at a relatively further receiver plane. Therefore, it is demonstrated that the RPB as a laser source not only extends the self-focusing length, but also improves the power density of the target. With the help of RPB, it is possible to realize a controllable self-focusing length and a high target optical power density, which may have potential applications in fine optical manipulation, optical communication, high-power long-range laser atmospheric propagation, and related areas.

Depolarization of Vector Light Beams on Propagation in Free Space

Nonparaxial propagation of the vector vortex light beams in free space was investigated theoretically. Propagation-induced polarization changes in vector light beams with different spatial intensity distributions were analyzed. It is shown that the hybrid vector Bessel modes with polarization-OAM (orbital angular momentum) entanglement are the exact solutions of the vector Helmholtz equation. Decomposition of arbitrary vector beams in the initial plane z = 0 into these polarization-invariant beams with phase and polarization singularities was used to analyze the evolution of the polarization of light within the framework of the 2 × 2 coherency matrix formalism. It is shown that the 2D degree of polarization decreases with distance if the incident vector beam is not the modal solution. The close relationship of the degree of polarization with the quantum-mechanical purity parameter is emphasized.

Radially polarized twisted partially coherent vortex beams

We introduce a new type of partially coherent vector beam, named the radially polarized twisted partially coherent vortex (RPTPCV) beam. Such a beam carries the twist phase and the vortex phase simultaneously, and the initial state of polarization (SOP) is radially polarized. On the basis of the pseudo-modal expansion and fast Fourier transform algorithm, the second-order statistics such as the spectral density, the degree of polarization (DOP) and the SOP, propagation through a paraxial ABCD optical system are investigated in detail through numerical examples. The results reveal that the propagation properties of the RPTPCV beam closely depends on the handedness of the twist phase and the vortex phase. When the handedness of the two phases is same, the beam profile is easier to remain a dark hollow shape and the beam spot rotates faster during propagation, compared to the partially coherent vortex beam or the RPTPCV beam with the opposite handedness of the two phases. In addition, the same handedness of two phases resists the coherence induced de-polarization of the beam upon propagation, and the SOP is also closely related to the handedness, topological charge of the vortex phase and the twist factor of the twist phase, providing an efficient way to modulate the beam's DOP and SOP in the output plane. Moreover, we establish an experiment setup to generate the RPTPCV beam. The average spectral density and the polarization properties are examined in the experiment. The experimental results agree reasonable well with the theoretical predictions. Our results will be useful for particle manipulating, free-space optical communications, and polarization lidar systems.

Mueller Matrix Polarimetry with Invariant Polarization Pattern Beams

A wide class of nonuniformly totally polarized beams that preserve their transverse polarization pattern during paraxial propagation was studied. Beams of this type are of interest, in particular, in polarimetric techniques that use a single input beam for the determination of the Mueller matrix of a homogeneous sample. In these cases, in fact, it is possible to test the sample response to several polarization states at once. The propagation invariance of the transverse polarization pattern is an interesting feature for beams used in these techniques, because the polarization state of the output beam can be detected at any transverse plane after the sample, without the use of any imaging/magnifying optical system. Furthermore, exploiting the great variety of the beams of this class, the ones that better fit specific experimental constrains can be chosen. In particular, the class also includes beams that present all possible polarization states across their transverse section (the full Poincaré beams (FPB)). The use of the latter has recently been proposed to increase the accuracy of the recovered Mueller matrix elements. Examples of FPBs with propagation-invariant polarization profiles and its use in polarimetry are discussed in detail. The requirement of invariance of the polarization pattern can be limited to the propagation in the far field. In such a case, less restrictive conditions are derived, and a wider class of beams is found.

Statistical properties of a partially coherent radially polarized vortex beam propagating in a uniaxial crystal

Free-space propagation and experimental generation of a partially coherent radially polarized (PCRP) vortex beam were studied recently [Opt. Express24, 13714 (2016)OPEXFF1094-408710.1364/OE.24.013714]. In this work, we explore the statistical properties of such a PCRP vortex beam propagating in a uniaxial crystal. We show that the anisotropy of the refractive index of the uniaxial crystal induces the asymmetrical distribution of the intensity, the degree and the state of polarization, as well as the degree of coherence of the beam during propagation. Further, by comparing the asymmetrical distribution of the statistical properties of the PRCP vortex beam with those of a PRCP beam without a vortex phase, we find that the asymmetrical features can be used for determining whether a PCRP beam carries the vortex phase. Further, we show that from the far-field distribution of the degree of coherence, we could quantify the topological charge and distinguish the handedness of the vortex phase. Our findings provide a novel approach for measuring the phase information of the partially coherent vortex beams.

Partially coherent radially polarized circular Airy beam

We propose and demonstrate a new, to the best of our knowledge, kind of partially coherent vector beam called the partially coherent radially polarized circular Airy beam (PCRPCAB). The PCRPCAB inherits the autofocusing ability of the radially polarized circular Airy beam (RPCAB) and can create an optical potential well at the center of the beam, whose depth can be adjusted by changing the coherent width. We find that, as coherent width decreases, the intensity becomes higher in the dark notch caused by the polarization singularity, and the singularity of the degree of polarization (DOP) remains along propagation, with its waist controllable by the coherent width. Our results make the PCRPCAB a good candidate for optical micromanipulation, disordered optical lattices, etc.

Statistical properties of an electromagnetic Gaussian Schell-model beam propagating in uniaxial crystals along the optical axis

Within the angular-spectrum representation, we study the partially coherent beam propagating in uniaxial crystals along the optical axis. By a method of vortex expansion, we derive the analytical solution for the cross-spectral density (CSD) function of an electromagnetic Gaussian Schell-model (EGSM) beam. We demonstrate that the analytical expression of CSD function can be written into a quasi-coherent-mode representation, whose basis vectors are constructed by the elegant Laguerre-Gaussian (eLG) functions. Several limits of the analytical solution are examined and good agreements with previous theories are obtained. Moreover, we calculate the energy density and degree of polarization (DOP) of the EGSM beam, from which the effects of coherent degree on the propagating properties are revealed. It is found that the energy conversion between circularly polarized components becomes rapid when the degree of coherence is decreasing. For all degree of coherence, the energy density and DOP exhibit a similar saturated behavior in the far field.

Vector partially coherent beams with prescribed non-uniform correlation structure

We introduce a general strategy for the synthesis of vector partially coherent beams (PCBs) with a prescribed non-uniform correlation structure. With it, we characterize a specific family of such beams, termed radially polarized Hermite non-uniformly correlated (RPHNUC) beams. These beams possess unusual propagation properties compared to vector PCBs with uniform correlation structure; for example, they maintain their dark hollow core and evolve multi-ring structures. These beams may prove useful in free-space optical communications, optical trapping, and polarization-sensitive imaging.

Partially coherent radially polarized fractional vortex beam

A new kind of partially coherent vector beam, named a partially coherent radially polarized fractional vortex (PCRPFV) beam, is introduced as a natural extension of the recently introduced scalar partially coherent fractional vortex beams [Zeng et al., Opt. Express26, 26830 (2018)10.1364/OE.26.026830]. Realizability conditions and propagation formulas for a PCRPFV beam are derived. Statistical properties of a focused PCRPFV beam, such as average intensity, degree of polarization, state of polarization and cross-spectral density matrix, are illustrated in detail and compared with that of a partially coherent radially polarized integer vortex beam and a scalar partially coherent fractional vortex beam. It is found that the statistical properties of a PCRPFV beam are qualitatively different from these simpler beam classes and are strongly determined by the vortex phase (i.e., fractional topological charge) and initial coherence width. We demonstrate experimental generation of PCRPFV beams and confirm their behavior. Our results will be useful for the rotating and trapping of particles, the detection of phase objects, and polarization lidar systems.

Propagation properties of radially polarized multi-Gaussian Schell-model beams in oceanic turbulence

By utilizing the extended Huygens-Fresnel principle, we derive the analytical formulas for the cross-spectral density matrix elements of a radially polarized multi-Gaussian Schell-model (RPMGSM) beam propagating in oceanic turbulence. Effects of beam parameters and oceanic turbulence parameters on the propagation properties of RPMGSM beams are investigated in detail by numerical simulation. Our results show that the RPMGSM beam with larger beam order M has an advantage over the radially polarized Gaussian Schell-model beam for reducing turbulence-induced degradation. Compared with temperature-induced turbulence fluctuation, the salinity-induced turbulence fluctuation makes a greater contribution to the influence on propagation properties of RPMGSM beams.

Scintillation properties of a partially coherent vector beam with vortex phase in turbulent atmosphere

We undertake a computational and experimental study of an advanced class of structured beams, partially coherent radially polarized vortex (PCRPV) beams, on propagation through atmospheric turbulence. A computational propagation model is established to simulate this class of beams, and it is used to calculate the average intensity and on-axis scintillation index of PCRPV beams. On comparison with other classes of structured beams, such as partially coherent vortex beams and partially coherent radially polarized beams, it is found that the PCRPV beams, which structure phase, coherence and polarization simultaneously, show marked improvements in atmospheric propagation. The simulation results agree reasonably well with the experimental results. These beams will be useful in free-space optical communications and remote sensing.

Propagation Characteristics of a Twisted Cosine-Gaussian Correlated Radially Polarized Beam

Recently, partially coherent beams with twist phases have attracted growing interest due to their nontrivial dynamic characteristics. In this work, the propagation characteristics of a twisted cosine-Gaussian correlated radially polarized beam such as the spectral intensity, the spectral degree of coherence, the degree of polarization, the state of polarization, and the spectral change are investigated in detail. Due to the presence of the twisted phase, the beam spot, the degree of coherence, and the state of polarization experience rotation during transmission, but the degree of polarization is not twisted. Meanwhile, although their rotation speeds closely depend on the value of the twist factor, they all undergo a rotation of π / 2 when they reach the focal plane. Furthermore, the effect of the twist phase on the spectral change is similar to the coherence, which is achieved by modulating the spectral density distribution during transmission. The twist phase opens up a useful guideline for manipulation of novel vector structure beams and enriches potential applications in the field of beam shaping, optical tweezers, optical imaging, and free space optical communications.

Coupling Efficiency of a Partially Coherent Radially Polarized Vortex Beam into a Single-Mode Fiber

We study the problem of coupling partially coherent radially polarized (PCRP) vortex beams into a single-mode optical fiber. Using the well-known concept of the cross-spectral density (CSD) matrix, we derive a general expression for the coupling efficiency of the partially coherent beam into a single-mode fiber. We adopt PCRP vortex beams for incident beams and use our general results to discuss the effects of the coherence, topological charge, and wavelength on the coupling efficiency of an optical beam focused onto a single-mode fiber with a lens. Our results should be useful for any application that requires coupling of partially coherent beams into optical fibers.

Focus shaping of the radially polarized Laguerre-Gaussian-correlated Schell-model vortex beams

In this paper, we introduce a new kind of partially coherent vector beam with special correlation function and vortex phase named radially polarized Laguerre-Gaussian-correlated Schell-model (LGCSM) vortex beam as a natural extension of scalar LGCSM vortex beam. The realizability conditions for such beam are derived. The tight focusing properties of a radially polarized LGCSM vortex beam passing through a high numerical aperture (NA) objective lens are investigated numerically based on the vectorial diffraction theory. We find that not only the transverse component but also the longitudinal component of the focal field distributions can be shaped by regulating the structures of the correlation functions, which is quite different from that of the conventional radially polarized partially coherent beam. Moreover, a series of wildly used focal field with novel structure, e.g., focal spot, flat-topped or doughnut beam profiles, needle-like focal field and controllable three-dimensional (3D) optical cage, were obtained. These results indicate that the focus shaping can be achieved by combining the regulation of the structures of the correlation functions with the regulation of beam parameters effectively. Our results may be useful for potential applications in optical trapping, optical high-resolution microscopy and optical data storage.

Wavefront spacing and Gouy phase in strongly focused fields: the role of polarization

The wavefront spacings and the Gouy phases are investigated in the strongly focused fields with four differently polarized incident waves. It is shown that these four incident polarizations will lead to nine axial field components, which are found actually to only have five kinds of wavefront spacings and Gouy phases. The four analytical and one numerical expressions are derived for these five kinds of wavefront spacings in the immediate neighborhood of the focus. Our results show that the polarization state, the semiaperture angle, and beam size influence the wavefront spacings and the Gouy phases in different ways. The difference between two very near wavefront spacings can be bigger than a half wavelength. It is also found that the axial field components with the polarization states orthogonal to that of the incident wave always have larger wavefront spacings.

Vector Hermite-Gaussian correlated Schell-model beam

A new kind of partially coherent vector beam named vector Hermite-Gaussian correlated Schell-model (HGCSM) beam is introduced as a natural extension of recently introduced scalar HGCSM beam. The realizability and beam conditions for a vector HGCSM beam with uniform state of polarization (SOP) or non-uniform SOP are derived, respectively. Furthermore, analytical formulae for a vector HGCSM beam propagating in free space are derived, and the propagation properties of a vector HGCSM beam with uniform SOP or non-uniform SOP in free space are studied and analyzed in detail. We find that the behaviors of a vector HGCSM beam on propagation are quite different from those of a conventional vector partially coherent beam with uniform SOP or non-uniform SOP, and modulating the structures of the correlation functions cannot only modulate the intensity distribution, but also the state of polarization, the degree of polarization and the polarization singularities of a partially coherent vector beam on propagation. Furthermore, we report experimental generation of a radially polarized HGCSM beam for the first time. Our results provide a novel way for polarization modulation.

Vortex phase-induced changes of the statistical properties of a partially coherent radially polarized beam

Partially coherent radially polarized (PCRP) beam was introduced and generated in recent years. In this paper, we investigate the statistical properties of a PCRP beam embedded with a vortex phase (i.e., PCRP vortex beam). We derive the analytical formula for the cross-spectral density matrix of a PCRP vortex beam propagating through a paraxial ABCD optical system and analyze the statistical properties of a PCRP vortex beam focused by a thin lens. It is found that the statistical properties of a PCRP vortex beam on propagation are much different from those of a PCRP beam. The vortex phase induces not only the rotation of the beam spot, but also the changes of the beam shape, the degree of polarization and the state of polarization. We also find that the vortex phase plays a role of resisting the coherence-induced degradation of the intensity distribution and the coherence-induced depolarization. Furthermore, we report experimental generation of a PCRP vortex beam for the first time. Our results will be useful for trapping and rotating particles, free-space optical communications and detection of phase object.

Propagation properties of a radially polarized partially coherent twisted beam in free space

We introduce a radially polarized partially coherent twisted beam based on the unified theory of coherence and polarization. We also derive expressions of this beam propagating in free space and examine the influence of twist factor and coherence of the source plane on its propagation properties. It is found that both twist factor and coherence of the source plane affect its average intensity, polarization, and coherence. Those results also show that the beam is rotating with beam propagation, and the beam's rotation is independent of coherence of the source plane.

Generation and propagation of a vector cosine-Gaussian correlated beam with radial polarization

Scalar cosine-Gaussian-correlated Schell-model (CGCSM) beams of circular or rectangular symmetry were introduced just recently. In this paper, a new kind of partially coherent vector beam named vector CGCSM beam with radial polarization (i.e., radially polarized CGCSM beam) is introduced. The realizability conditions for a radially polarized CGCSM source and the beam condition for radiation generated by such source are derived. The statistical properties, such as the average intensity, the degree of coherence, the degree of polarization and the state of polarization, of a radially polarized CGCSM beam focused by a thin lens are analyzed in detail. It is found that the statistical properties of a radially polarized CGCSM beam are quite different from those of a conventional radial polarized partially coherent beam with Gaussian correlated Schell-model function. Furthermore, we first report experimental generation of a radially polarized CGCSM beam and measure its focusing properties. Our experimental results are consistent with the theoretical predictions.

Generation and propagation of an electromagnetic Gaussian Schell-model vortex beam

We outline the propagation of an electromagnetic Gaussian Schell-model (EGSM) vortex beam through a paraxial ABCD optical system and analyze the vortex phase-induced changes of the statistical properties, such as average intensity, state of polarization, and degree of polarization (DOP), of a focused EGSM beam. It is found that one can shape the beam profile of an EGSM vortex beam in the focal plane through varying its initial topological charge, DOP, and coherence widths. Furthermore, we first report experimental generation of an EGSM vortex beam and measure its focusing properties in experiments. Our experimental results are consistent with the numerical results and may be useful in material thermal processing and particle trapping.

Statistical properties of a partially coherent cylindrical vector beam in oceanic turbulence

Propagation of a partially coherent cylindrical vector Laguerre-Gaussian (PCCVLG) beam passing through oceanic turbulence is studied with the help of the extended Huygens-Fresnel integral formula and unified theory of coherence and polarization of light. Analytical formula for the cross-spectral density matrix of a PCCVLG beam propagating in oceanic turbulence is derived, and the statistical properties, such as intensity distribution and degree of polarization, of a PCCVLG beam on propagation in oceanic turbulence are illustrated in detail. It is found that the statistical properties of a PCCVLG beam in oceanic turbulence vary as the sea-related parameters, initial coherence length, and mode orders vary, and such beam is depolarized on propagation. Our results will be useful in optical underwater communications, imaging, and sensing.

Statistical properties in Young's interference pattern formed with a radially polarized beam with controllable spatial coherence

Experimental generation of a radially polarized (RP) beam with controllable spatial coherence (i.e., partially coherent RP beam) was reported recently [Appl. Phys. Lett. 100, 051108 (2012)]. In this paper, we carry out theoretical and experimental studies of the statistical properties in Young's two-slit interference pattern formed with a partially coherent RP beam. An approximate analytical expression for the cross-spectral density matrix of a partially coherent RP beam in the observation plane is obtained, and it is found that the statistical properties, such as the intensity, the degree of coherence and the degree of polarization, are strongly affected by the spatial coherence of the incident beam. Our experimental results are consistent with the theoretical predictions, and may be useful in some applications, where light field with special statistical properties are required.

Generation and propagation of partially coherent beams with nonconventional correlation functions: a review [invited]

Partially coherent beams with nonconventional correlation functions have displayed many extraordinary properties, such as self-focusing and self-splitting, which are totally different from those of partially coherent beams with conventional Gaussian correlated Schell-model functions and are useful in many applications, such as optical trapping, free-space optical communications, and material thermal processing. In this paper, we present a review of recent developments on generation and propagation of partially coherent beams with nonconventional correlation functions.

Theoretical and experimental studies of the spectral changes of a polychromatic partially coherent radially polarized beam

In a recent publication (Appl. Phys. Lett. 100 (2012) 051108), a monochromatic partially coherent radially polarized (RP) beam was generated experimentally. In this paper, we analyze the spectral changes of a polychromatic partially coherent RP beam focused by a thin lens for the first time, and compare with that of a focused scalar polychromatic GSM beam. Furthermore, we report experimental generation of a polychromatic partially coherent RP beam and carry out experimental measurement of the spectral changes of such beam focused by a thin lens. Our results show that the behavior of the spectral changes of a focused polychromatic partially coherent RP beam is different from that of a focused scalar polychromatic GSM beam. Our experimental results are consistent with the theoretical predictions.

State of polarization and propagation factor of a stochastic electromagnetic beam in a gradient-index fiber

With the help of a tensor method, we investigate the evolution properties of the state of polarization of an electromagnetic Gaussian Schell-model beam propagating through a gradient-index (GRIN) fiber. We find that the Stokes parameters and the polarization ellipse exhibit periodicity. The initial beam parameters affect the values of the Stokes parameters and the parameters of the polarization ellipse. Furthermore, based on the second-order moments of the Wigner distribution function, the explicit expression for the propagation factor (known as the M(2) factor) in the GRIN fiber is derived. It is shown that the M(2) factor remains invariant on propagation and is determined only by the initial beam parameters.

Nonparaxial propagation properties of a vector partially coherent dark hollow beam

Based on the generalized Raleigh-Sommerfeld diffraction integrals, analytical nonparaxial propagation formulas for the elements of the cross-spectral density matrix of a vector partially coherent dark hollow beam (DHB) in free space are derived. The effect of spatial coherence and beam waist sizes on the statistical properties of a nonparaxial vector DHB is studied numerically. It is found that one can modulate the statistical properties of a nonparaxial vector DHB by varying its initial spatial coherence, which will be useful in some applications where nonparaxial beams are commonly encountered.