Evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity

Degree of paraxiality of a twist electromagnetic Gaussian Schell-model beam

The definition of the degree of paraxiality (DOP) for a stochastic electromagnetic field is applied to a twist stochastic electromagnetic field. As an illustrative example, DOP for a wide class of model stochastic fields, i.e., twist electromagnetic Gaussian Schell-model (TEGSM) fields, is discussed. The dependence of the DOP of the light source on its properties is also studied in detail. The numerical results show that the DOP of a TEGSM beam is determined by the rms widths of auto-correlation functions and the twist factor as well as by the degree of polarization. To explain the behavior of DOP, the far-field divergence angle of this beam source is also discussed.

Fast calculation of orbital angular momentum flux density of partially coherent Schell-model beams on propagation

Optical coherence has recently become a degree of freedom to modulate the orbital angular momentum (OAM) flux density of a partially coherent beam during propagation. However, the calculation of the OAM flux density for the partially coherent beam involves partial differential and four-dimensional integral operations, which poses drawbacks for its fast numerical calculations. In this paper, we present an efficient numerical protocol for calculating the OAM flux density of any partially coherent Schell-model beam propagating through a paraxial ABCD optical system by only adopting two-dimensional (2D) Fourier transforms. The general formalism is established in detail for the fast numerical calculation of the OAM flux density. It is found that the operation number in the developed algorithm is independent on the spatial coherence states of the beam. To demonstrate the validity of our algorithm, we calculate the OAM flux density of the partially coherent Laguerre-Gaussian beams during propagation with both the analytical and numerical methods. The obtained results are consistent well with each other. Moreover, the OAM flux density properties of two other classes of Schell-model beams, having no analytical solutions, are investigated as the specific examples. Our method provides a convenient way for studying the correlation-induced OAM density changes for any Schell-model beam propagation through a paraxial optical system.

Twisted electromagnetic elliptical multi-Gaussian Schell-model beams and their transmission in random media

We extend the scalar elliptical multi-Gaussian Schell-model (EMGSM) beams with twist phase to the electromagnetic domain and obtain the analytical expression for the propagation of the electromagnetic twisted EMGSM beams through random media. The twist phase-induced changes of the spectral density and degree of polarization of such beams on propagation are studied numerically. Results show that by adjusting the twist factor and the correlated parameters of the source, both the spectral density and degree of polarization not only rotate around the propagation axis but also exhibit diverse shapes. The flattopped ellipse-like and diamond-like shape maintain over a relatively long propagation distance and finally involve into Gaussian-like shape due to stronger atmospheric turbulence. The results will be useful in optical trapping and optical communication.

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.

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 C_n² 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.

Orbital angular moment of an electromagnetic Gaussian Schell-model beam with a twist phase

We derive the analytical formula for the orbital angular momentum (OAM) flux of a stochastic electromagnetic beam carrying twist phase [i.e., twisted electromagnetic Gaussian Schell-model (TEGSM) beam] in the source plane with the help of the Wigner distribution function. Furthermore, we derive the general expression of the OAM flux of a TEGSM beam on propagation with the help of a tensor method. As numerical examples, we explore the evolution properties of the OAM flux of a TEGSM beam propagating through a cylindrical thin lens or a uniaxial crystal. It is found that the OAM flux of a TEGSM beam closely depends on its twist factors and degree of polarization in the source plane, and one can modulate the OAM flux of a TEGSM beam by a cylindrical thin lens or a uniaxial crystal. Our results may be useful in some applications, such as particle manipulation and free-space optical communications, where light beam with OAM is preferred.

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.

Twist phase-induced changes of the statistical properties of a stochastic electromagnetic beam propagating in a uniaxial crystal

With the help of a tensor method, an analytical formula is derived for the cross-spectral density matrix of a twisted electromagnetic Gaussian Schell-model (EGSM) beam (i.e., EGSM beam with twist phase) propagating in a uniaxial crystal orthogonal to the optical axis. The twist phase-induced changes of the statistical properties, such as the spectral density, the degree of polarization and the degree of coherence, of an EGSM beam propagating in a uniaxial crystal are illustrated numerically. It is found that the distributions of the spectral density, the degree of polarization and the degree of coherence of a twisted EGSM beam in a uniaxial crystal all exhibit non-circular symmetries, which are quite different from those of a twisted EGSM beam in isotropic medium or in free space. One may use uniaxial crystal to determine whether an EGSM beam carries twist phase or not.

Experimental demonstration of ghost imaging with an electromagnetic Gaussian Schell-model beam

Recently, there has been a controversy about the dependence of the visibility of the ghost image on the degree of polarization (DOP) of a stochastic electromagnetic beam because of different definitions of the visibility. In this paper, we revisit ghost imaging with an electromagnetic Gaussian Schell-model (EGSM) beam. Through numerical examples based on the conventional definition of the visibility, we find that the visibility of the ghost image indeed increases or decreases with the increase of the DOP the beam source under certain conditions. We solve the controversy between literatures and the present paper through analyzing the r.m.s. widths of auto-correlation functions of the x component of the field and of the y component of the field. Furthermore, we carry out experimental demonstration of ghost imaging with an EGSM beam. Our experimental results verify the theoretical predictions.

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

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

Second-order moments of an electromagnetic Gaussian Schell-model beam in a uniaxial crystal

We derive the analytical expressions for the second-order moments of an electromagnetic Gaussian Schell-model (EGSM) beam propagating in a uniaxial crystal. With the help of the derived formulas, we study the evolution properties of the propagation factor, the effective radius of curvature and the Rayleigh range of an EGSM beam in a uniaxial crystal. It is found that the evolution properties of an EGSM beam in a uniaxial crystal are much different from its evolution properties in free space and are closely determined by the initial beam parameters and the parameters of the uniaxial crystal. The uniaxial crystal provides one way for modulating the properties of an EGSM beam.

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.

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

In a recent publication [Appl. Phys. Lett, 100, 051108 (2012)], a radially polarized (RP) beam with variable spatial coherence (i.e., partially coherent RP beam) was generated experimentally. In this paper, we derive the realizability conditions for a partially coherent RP beam, and we carry out theoretical and experimental study of the coherence and polarization properties of a partially coherent RP beam. It is found that after passing through a thin lens, both the degree of coherence and the degree of polarization of a partially coherent RP beam varies on propagation, while the state of polarization of the completely polarized part of such beam remains invariant. The variations of the degree of coherence and the degree of polarization depend closely on the initial spatial coherence. Our experimental results agree well with the theoretical predictions.

Statistical properties of a nonparaxial cylindrical vector partially coherent field in free space

Analytical nonparaxial propagation formula for the cross-spectral density matrix of a cylindrical vector partially coherent beam in free space is derived based on the generalized Raleigh-Sommerfeld diffraction integrals. Statistical properties, such as average intensity, degree of polarization and degree of coherence, of a nonparaxial cylindrical vector partially coherent field are illustrated numerically, and compared with that of a paraxial cylindrical vector partially coherent beam. It is found that the statistical properties of a nonparaxial cylindrical vector partially coherent field are much different from that of a paraxial cylindrical vector partially coherent beam, and are closely determined by the initial beam width and correlation coefficients. Our results will be useful for modulating the properties of a nonparaxial cylindrical vector partially coherent field.

Scattering-induced changes in the degree of polarization of a stochastic electromagnetic plane-wave pulse

The scattering of a stochastic electromagnetic plane-wave pulse on a deterministic spherical medium is investigated. An analytical formula for the degree of polarization (DOP) of the scattered field in the far zone is derived. Letting pulse duration T(0) → ∞, our formula can be applied to study the scattering of a stationary stochastic electromagnetic light wave. Numerical results show that the DOP of the far zone field is closely determined by the size of the spherical medium when the incident field is a stochastic electromagnetic plane-wave pulse. This is much different from the case when the incident field is a stationary stochastic electromagnetic light wave, where the DOP of the far zone field is independent of the size of the medium. One may obtain the information of the spherical medium by measuring the scattering-induced changes in the DOP of a stochastic electromagnetic plane-wave pulse.

Experimental measurement of the beam parameters of an electromagnetic Gaussian Schell-model source

We report the experimental measurement of the beam parameters of an electromagnetic Gaussian Schell-model (EGSM) source based on the technique for measuring the fourth-order correlation function (i.e., intensity correlation function). Furthermore, we carry out experimental measurement of the intensity distribution of a focused EGSM beam, and we carry out theoretical simulation using the measured beam parameters. Our experimental results agree reasonably well with the theoretical predictions.

Modulation of spectral intensity, polarization and coherence of a stochastic electromagnetic beam

Analytical formula for the cross-spectral density matrix of a stochastic electromagnetic Gaussian Schell-model (EGSM) beam truncated by a circular phase aperture propagating in free space is derived with the help of a tensor method, which provides a reliable and fast way for studying the propagation and transformation of a truncated EGSM beam. Statistics properties, such as the spectral intensity, the degree of coherence, the degree of polarization and the polarization ellipse of a truncated EGSM beam in free space are studied numerically. The propagation factor of a truncated EGSM beam is also analyzed. Our numerical results show that we can modulate the spectral intensity, the polarization, the coherence and the propagation factor of an EGSM beam by a circular phase aperture. It is found that the phase aperture can be used to shape the beam profile of an EGSM beam and generate electromagnetic partially coherent dark hollow or flat-topped beam, which is useful in some applications, such as optical trapping, material processing, free-space optical communications.

Statistics properties of a cylindrical vector partially coherent beam

Cylindrical vector partially coherent beam is introduced as a natural extension of cylindrical vector coherent beam based on the unified theory of coherence and polarization. Analytical propagation formula for the cross-spectral density matrix of a cylindrical vector partially coherent beam passing through paraxial ABCD optical system is derived based on the generalized Collins integral formula. As an application example, the statistics properties, such as the average intensity, spreading and the degree of polarization, of a cylindrical vector partially coherent beam propagating in free space are studied in detail. It is found that the statistics properties of a cylindrical vector partially coherent beam are much different from a cylindrical vector coherent beam. Our results may find applications in connection with laser beam shaping and optical trapping.

M2-factor of a stochastic electromagnetic beam in a Gaussian cavity

With the help of a tensor method, an explicit expression for the M2-factor of a stochastic electromagnetic Gaussian Schell-model (EGSM) beam in a Gaussian cavity is derived. Evolution properties of the M2-factor of an EGSM beam in a Gaussian cavity are studied numerically in detail. It is found that the behavior of the M2-factor of an EGSM beam in a Gaussian cavity is determined by the statistical properties of the source beam and the parameters of the cavity. Thermal lens effect induced changes of the M2-factor of an EGSM beam in a Gaussian cavity is also investigated. Our results will be useful in many applications, such as free-space optical communications, laser radar system, optical trapping and optical imaging, where stochastic electromagnetic beams are required.

Second-order statistics of a twisted gaussian Schell-model beam in turbulent atmosphere

We present a detailed investigation of the second-order statistics of a twisted gaussian Schell-model (TGSM) beam propagating in turbulent atmosphere. Based on the extended Huygens-Fresnel integral, analytical expressions for the second-order moments of the Wigner distribution function of a TGSM beam in turbulent atmosphere are derived. Evolution properties of the second-order statistics, such as the propagation factor, the effective radius of curvature (ERC) and the Rayleigh range, of a TGSM beam in turbulent atmosphere are explored in detail. Our results show that a TGSM beam is less affected by the turbulence than a GSM beam without twist phase. In turbulent atmosphere the Rayleigh range doesn't equal to the distance where the ERC takes a minimum value, which is much different from the result in free space. The second-order statistics are closely determined by the parameters of the turbulent atmosphere and the initial beam parameters. Our results will be useful in long-distance free-space optical communications.

Spatio-temporal coupling of random electromagnetic pulses interacting with reflecting gratings

Matrix optics is applied to a class of random, in time and space, electromagnetic pulsed beam-like (REMPB) radiation interacting with linear optical elements. A 6×6 order matrix describing transformation of a six-dimensional state vector including four spatial and two temporal positions within the field is used to derive conditions for spatio-temporal coupling. An example is included which deals with a spatio-temporal coupling in a typical REMPB on reflection from a reflecting grating. Electromagnetic nature of such interaction is explored via considering dependence of the degree of polarization of the reflected REMPB on its source and on the structure of the grating.

Propagation factor of a stochastic electromagnetic Gaussian Schell-model beam

Analytical formula is derived for the propagation factor (known asM(2)-factor) of a stochastic electromagnetic Gaussian Schell-model (EGSM) beam in free space and in turbulent atmosphere. In free space, the M(2)-factor of an EGSM beam is mainly determined by its initial degree of polarization, r.m.s. widths of the spectral densities and correlation coefficients, and its value remains invariant on propagation. In turbulent atmosphere, the M(2)-factor of an EGSM beam is also determined by the parameters of the turbulent atmosphere, and its value increases on propagation. The relative M(2)-factor of an EGSM beam with lower correlation factors, larger r.m.s. widths of the spectral densities and longer wavelength is less affected by the atmospheric turbulence. Under suitable conditions, an EGSM beam is less affected by the atmospheric turbulence than a scalar GSM beam (i.e. fully polarized GSM beam). Our results will be useful in long-distance free-space optical communications.

Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams

Radiation force of a focused scalar twisted Gaussian Schell-model (TGSM) beam on a Rayleigh dielectric sphere is investigated. It is found that the twist phase affects the radiation force and by raising the absolute value of the twist factor it is possible to increase both transverse and longitudinal trapping ranges at the real focus where the maximum on-axis intensity is located. Numerical calculations of radiation forces induced by a focused electromagnetic TGSM beam on a Rayleigh dielectric sphere are carried out. It is found that radiation force is closely related to the twist phase, degree of polarization and correlation factors of the initial beam. The trapping stability is also discussed.

Effects of astigmatism on spectra, coherence and polarization of stochastic electromagnetic beams passing through an astigmatic optical system

Analytical formulas for the cross-spectral density matrix of stochastic electromagnetic Gaussian Schell-model (EGSM) beams passing through an astigmatic optical system are derived. We show both analytically and by numerical examples the effects of astigmatism on spectra, coherence and polarization of stochastic electromagnetic EGSM beams propagating through an astigmatic lens. A comparison with the aberration-free case is made, and shows that the astigmatism has significant effect on the spectra, coherence and polarization.