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

Asymmetric cosine-Gaussian Schell-model sources

We introduce a new class of Schell-model source whose spatial correlation function carries a finite series of nontrivial phases and is modulated by cosine function. The propagating formula for the cross-spectral density (CSD) function of the beams generated by this new source in free space is derived and used to investigate the characteristics of the light fields for different modulation parameters. The results show that the appearance of the modulating function and the nontrivial phase sequence cause the radiated fields with extraordinary characteristics during propagation, such as self-splitting and self-interfering in an asymmetric manner. This method of jointly controlling the magnitude and phase of the source complex coherence state provides a new approach for achieving beam shaping.

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.

Multi-hyperbolic sine-correlated beams and their statistical properties in turbulent atmosphere

We introduce a partially coherent beam, called a multi-hyperbolic sine-correlated (MHSC) beam, by employing a multi-hyperbolic sine function to modulate the spectral degree of coherence. Based on the extended Huygens-Fresnel principle and second-order moments of the Wigner distribution function, we derive the analytical expressions for the spectral intensity, the root-mean-square (rms) angular width and the M² factor in turbulent atmosphere. Numerical results show that the intensity profile, which keeps the dark-hollow invariant in free space, will be gradually destroyed by the turbulence along the propagation distance. We believe that the MHSC beams have significant advantage over the hyperbolic sine-correlated beams for reducing the turbulence-induced degradation, especially for the MHSC beams with a higher beam order N. The effects of the beams parameters and the turbulent atmosphere on the beam quality are analyzed in detail.

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.

Experimental synthesis of partially coherent sources

A flexible pseudo-mode sampling superposition method for synthesizing partially coherent sources has been introduced that can be thought of as an approximate discrete representation of Gori's nonnegative definiteness criterion for designing spatial correlation functions. Importantly, without performing formidable mode analysis, this method enables us to develop a convenient and efficient experimental technology to customize partially coherent sources without sacrificing theoretical accuracy. As an example, we experimentally generate a new, to the best of our knowledge, class of nontrivial pseudo-Schell model sources recently proposed by de Sande et al. Our approach opens up a useful avenue for manipulating nontrivial partially coherent beams and promotes applications for optical tweezers and photolithography.

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.

Statistical properties of partially coherent radially and azimuthally polarized rotating elliptical Gaussian beams in oceanic turbulence with anisotropy

A new class of partially coherent radially and azimuthally polarized rotating elliptical Gaussian (PCRPREG and PCAPREG) beams is introduced. The analytical expressions of the PCRPREG and PCAPREG beams propagating through anisotropy oceanic turbulence are derived based on the extended Huygens-Fresnel principle and the spatial power spectrum of oceanic turbulence. The effects of beam waist size w₀, coherence width σ₀, propagation distance z and oceanic turbulence parameters on the evolution statistics properties of PCRPREG and PCAPREG beams are studied in detail by numerical simulation. Our results indicate that with the increase of the propagation distance in the far field region, the normalized initial profile with a doughnut-like distribution of PCRPREG and PCAPREG beams gradually converts into a flat-topped one, and finally evolves into a Gaussian-like beam profile. We also find that the salinity-induced turbulence fluctuation makes a greater contribution to the decrease of beam quality compared with the temperature-induced turbulence fluctuation. Furthermore, the full width at half maximum becomes wider for the larger propagation distance z and wavelength λ or the smaller dissipation rate ε. Our work will pave the way for the development of underwater optical communication and underwater laser radar in oceanic environment.

Twisted partially coherent array sources and their transmission in anisotropic turbulence

A new class of twisted Schell-model array correlated sources are introduced based on Mercer's expansion. It turns out that such sources can be expressed as superposition of fully coherent Laguerre-Gaussian modes, and the twistable condition is established. Furthermore, on the basis of a stretched coordinate system and a quadratic approximation, analytical expressions for the mutual coherence function of an anisotropic non-Kolmogorov turbulence and the cross-spectral density of a twisted Gaussian Schell-model array beam are rigorously derived. Due to the presence of the twist phase, the beam spot and the degree of coherence rotate as they propagate, but their rotation centers are different. It is shown that the anisotropy of turbulence causes an anisotropic beam spreading in the horizontal and vertical directions. However, impressing a twist phase on source beams can significantly inhibit this effect. For an anticipated atmospheric channel condition, a comprehensive selection of initial optical signal parameters, receiver aperture size and receiver capability, etc., is necessary. Our work is helpful for exploring new forms of twistable sources, and promotes guidance on optimization of partial coherent beam applications.

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.

Vectorial optical fields: recent advances and future prospects

Driven by their potential applications, vectorial optical fields with spatially inhomogeneous states of polarization within the cross section have drawn significant attention recently. This work intends to review some of the latest development of this rapidly growing field of optics and offer a general overview of the current status of this field in a few areas. Mathematical descriptions of generalized vectorial optical fields are provided along with several special examples. A time-reversal methodology for the creation of a wide variety of exotic optical focal fields with prescribed characteristics within the focal volume is presented. Recently developed methods for the generation of vectorial optical fields that utilize fiber lasers, digital lasers, vectorial optical field generator, metasurfaces or photoalignment liquid crystals are summarized. The interactions of these vectorial optical fields with various micro- and nano-structures are presented and the prospects of their potential applications are discussed. The connection of vectorial optical fields with higher dimensionality in quantum information is summarized.

Second-order statistics of a radially polarized partially coherent twisted beam in a uniaxial crystal

The Wigner function (WDF) has been used to study the second-order moments for a radially polarized partially coherent twisted (RPPCT) beam (i.e., a RPPC beam with a twist phase) propagating in a uniaxial crystal. With the help of the extended Huygens-Fresnel integral formula and definition of the WDF, the analytical formulas for propagation factor (M²-factor), effective radius of curvature (ERC), and Rayleigh range of a RPPCT beam propagating in a uniaxial crystal have been derived. Our numerical results show that the M²-factor of a RPPCT beam with larger absolute value of the twist factor or lower coherence is less affected by anisotropic diffraction in a uniaxial crystal. The dependence of the ERC and Rayleigh range of a RPPCT beam propagating in a uniaxial crystal on the parameter e along the x direction is much different from that along the y direction due to anisotropic diffraction. We can judge how much a RPPCT beam carries the twist factor by measuring the deviation percentage of the M²-factor of a RPPCT beam propagating in a uniaxial crystal and even can modulate the properties of a RPPCT beam by varying the beam parameters, which will be useful in some applications where a RPPCT beam is required.

Propagation of multi-cosine-Laguerre-Gaussian correlated Schell-model beams in free space and atmospheric turbulence

We introduce a class of random stationary, scalar source named as multi-cosine-Laguerre-Gaussian-correlated Schell-model (McLGCSM) source whose spectral degree of coherence (SDOC) is a combination of the Laguerre-Gaussian correlated Schell-model (LGCSM) and multi-cosine-Gaussian correlated Schell-model (McGCSM) sources. The analytical expressions for the spectral density function and the propagation factor of a McLGCSM beam propagating in turbulent atmosphere are derived. The statistical properties, such as the spectral intensity and the propagation factor, of a McLGCSM beam are illustrated numerically. It is shown that a McLGCSM beam exhibits a robust ring-shaped beam array with adjustable number and positions in the far field by directly modulating the spatial structure of its SDOC in the source plane. Moreover, we provide a detailed insight into the theoretical origin and characteristics of such a ring-shaped beam array. It is demonstrated that these peculiar shaping properties are the concentrated manifestation of the individual merits respectively associated with the Laguerre- and multi-cosine-related factors of the whole SDOC. Our results provide a novel scheme to generate robust and controllable ring-shaped beam arrays over large distances, and will widen the potentials for manipulation of multiple particles, free-space optical communications and imaging in the atmosphere.

Random sources for cusped beams

We introduce two novel classes of partially coherent sources whose degrees of coherence are described by the rectangular Lorentz-correlated Schell-model (LSM) and rectangular fractional multi-Gaussian-correlated Schell-model (FMGSM) functions. Based on the generalized Collins formula, analytical expressions are derived for the spectral density distributions of these beams propagating through a stigmatic ABCD optical system. It is shown that beams belonging to both classes form the spectral density apex that is much higher and sharper than that generated by the Gaussian Schell-model (GSM) beam with a comparable coherence state. We experimentally generate these beams by using a nematic, transmissive spatial light modulator (SLM) that serves as a random phase screen controlled by a computer. The experimental data is consistent with theoretical predictions. Moreover, it is illustrated that the FMGSM beam generated in our experiments has a better focusing capacity than the GSM beam with the same coherence state. The applications that can potentially benefit from the use of novel beams range from material surface processing, to communications and sensing through random media.

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.

Second-order statistics of a radially polarized cosine-Gaussian correlated Schell-model beam in anisotropic turbulence

Recently, we introduced a new class of radially polarized cosine-Gaussian correlated Schell-model (CGCSM) beams of rectangular symmetry based on the partially coherent electromagnetic theory [Opt. Express23, 33099 (2015)]. In this paper, we extend the work to study the second-order statistics such as the average intensity, the spectral degree of coherence, the spectral degree of polarization and the state of polarization in anisotropic turbulence based on an extended von Karman power spectrum with a non-Kolmogorov power law α and an effective anisotropic parameter. Analytical formulas for the cross-spectral density matrix elements of a radially polarized CGCSM beam in anisotropic turbulence are derived. It is found that the second-order statistics are greatly affected by the source correlation function, and the change in the turbulent statistics induces relatively small effect. The significant effect of anisotropic turbulence on the beam parameters mainly appears nearα=3.1, and decreases with the increase of the anisotropic parameter. Furthermore, the polarization state exhibits self-splitting property and each beamlet evolves into a radially polarized structure in the far field. Our work enriches the classical coherence theory and may be important for free-space optical communications.