Polarization properties of partially coherent electromagnetic elegant Laguerre–Gaussian beams in turbulent atmosphere

Based on the extended Huygens–Fresnel Principle and the unified theory of coherence and polarization, the analytical formulas for the polarization degree and spectral coherence degree of partially coherent electromagnetic elegant Laguerre–Gaussian (PCEELG) beams through turbulent atmosphere are obtained theoretically in detail. It is found that the polarization degree of PCEELG beams tends to the value of its source plane after a sufficiently long propagation distance in turbulent atmosphere. Furthermore, this value is independent of the beam orders, the correlation length in the source plane and the structure constant of the turbulent atmosphere ( $C_{n}^{2}$ ). The polarization degree of PCEELG beams also acquires a particular value at a certain distance in free space, which is different from the value in the source plane. The spectral coherence degree of PCEELG beams has an oscillatory behavior for different propagation distance, beam orders and  $C_{n}^{2}$ .     

Partially coherent elegant Hermite–Gaussian beams

Elegant Hermite–Gaussian beams (EHGBs) are extended to the partially coherent case. An explicit and analytical formula is derived for the cross-spectral density of a partially coherent EHGB propagating through an aligned or misaligned paraxial ABCD optical system. The propagation properties of a partially coherent EHGB in free space and its focusing properties through a thin lens are studied numerically, and are compared to those of a partially coherent standard Hermite–Gaussian beam (SHGB). It is found that the propagation and focusing properties of a partially coherent EHGB are closely related to its initial coherence. A partially coherent EHGB spreads slower than a partially coherent SHGB in free-space propagation. A partially coherent EHGB can be focused more tightly than a partially coherent SHGB.     

Propagation of truncated modified Laguerre–Gaussian beams

By expanding the circ function into a finite sum of complex Gaussian functions and applying the Collins formula, the propagation of hard-edge diffracted modified Laguerre–Gaussian beams (MLGBs) through a paraxial ABCD system is studied, and the approximate closed-form propagation expression of hard-edge diffracted MLGBs is obtained. The transverse intensity distribution of the MLGB carrying finite power can be characterized by a single bright and symmetric ring during propagation when the aperture radius is very large. Starting from the definition of the generalized truncated second-order moments, the beam quality factor of MLGBs through a hard-edged circular aperture is investigated in a cylindrical coordinate system, which turns out to be dependent on the truncated radius and the beam orders.     

Composite vortex beams by coaxial superposition of Laguerre–Gaussian beams

We propose the generation of novel composite vortex beams by coaxial superposition of Laguerre–Gaussian (LG) beams with common waist position and waist parameter. Computer-generated holography by conjugate-symmetric extension is applied to produce the holograms of several composite vortex beams. Utilizing the holograms, fantastic light modes including optical ring lattice, double dark-ring and double bright-ring composite vortex beams etc. are numerically reconstructed. The generated composite vortex beams show diffraction broadening with some of them showing dynamic rotation around beam centers while propagating. Optical experiments based on a computer-controlled spatial light modulator (SLM) verify the numerical results. These novel composite vortex beams possess more complicated distribution and more controllable parameters for their potential application in comparison to conventional optical ring lattice.     

Rectangular hard-edged aperture diffracted Laguerre–Gaussian beams

Based on the relations between Laguerre–Gaussian (LG) and Hermite–Gaussian (HG) modes and by introduced the complex Gaussian expansion method for two dimensional rectangular aperture, the approximate analytical propagation expressions of the rotational symmetrical LG beams along with their even and odd modes through a paraxial ABCD optical system with rectangular hard-edged aperture are derived. As special cases of the results, the corresponding closed-forms of the circular aperture diffracted LG beams and non-truncated LG beams are also given. Numerical examples are given to prove the validity of this approximate analytical method and illustrate the propagation properties of the rectangular hard-edged aperture diffracted LG beams.     

Fresnel tomography and interferometric technique for characterizing Laguerre–Gaussian beams

The Fresnel tomograms of Laguerre–Gaussian beams are found and the relation of the two-point spatial correlation function with the tomographic map is discussed. The spatial distribution and signature of vorticity of Laguerre–Gaussian modes is analyzed by measuring the two-point spatial correlation function of the one-dimensional projection of the vortex field. An interferometric approach, based on a Fourier projection algorithm, for the experimental determination of the two-point correlation function of the full complex vortex field at the cross-section plane of the recorded interference pattern is also described; this approach allows one to characterize the vortex structure and spatial coherence along the direction of propagation.     

Dynamics of collinear Laguerre–Gaussian beams in nonlocal nonlinear media

We study the optical fields that are produced by two collinear Laguerre–Gaussian (LG) beams in strongly nonlocal nonlinear media. Various novel kinds of rotating breathers and solitons, such as quadrangle, pentacle, pentagon, etc. can be obtained by combining different mode-index LG beams with the different relative amplitude. The rotating and translating asymmetric breathers and solitons can be obtained when the difference of the azimuthal charge between the two combining LG beams is 1.     

Generation of multi-axis Laguerre–Gaussian beams from?geometric modes of a hemiconfocal cavity

We study the transformation of multi-axis Laguerre–Gaussian beams from optical beams comprising Hermite–Gaussian modes with ray–wave duality. By use of cylindrical lenses, Hermite–Gaussian modes can be transformed into Laguerre–Gaussian modes possessing optical orbital angular momentum. The superposed Hermite–Gaussian modes localized on geometric trajectories are generated from a degenerate hemiconfocal cavity and transformed into multi-axis Laguerre–Gaussian beams. Experimental results of the structured beams are systematically manipulated and in good agreement with theoretical analyses.     

Propagation properties of Airy–Gaussian beams

The propagation of the Airy–Gaussian beams is studied in strongly nonlocal nonlinear media analytically and numerically. The linear momentum of the analytical Airy–Gaussian beam solution of the Snyder–Mitchell model is not conservational, which is the reason that results in the disagreement between the analytical Airy–Gaussian beam solution and the numerical simulations of the nonlocal nonlinear Schr?dinger equation in the case of strong nonlocality. The quasi-Airy–Gaussian soliton in the Gaussian-shaped response material can be obtained when the parameter χ ₀ is large enough, and the patterns of Airy–Gaussian beams are variable periodically in liquid crystal material during propagation.     

Effective radius of curvature of Hermite–Gaussian array beams

Analytical expressions for the effective radius of curvature, R, of Hermite–Gaussian (H–G) array beams propagating in free space for both coherent and incoherent combinations are derived. It is shown that for the two types of beam combination a minimum of the effective radius of curvature, R_min, appears as the propagation distance z increases. For the coherent combination, R is larger than that for the incoherent combination. The position z_min where the effective radius of curvature reaches its minimum is further away from the source plane for the coherent combination than that for the incoherent combination. For the two types of beam combination, R and z_min increase with increasing beam number, increasing beam separation distance, increasing waist width, and decreasing beam order and wavelength. In particular, the R of single H–G beams is always smaller than that of H–G array beams; the R of Gaussian array beams is always larger than that of H–G array beams.     

Propagation aspects of Mathieu–Gaussian beams in turbulence

Based on our recent source plane formulation, the propagation characteristics of Mathieu–Gaussian beams in turbulent atmosphere are investigated. In this connection, the average receiver plane intensity expression is deduced using the Huygens–Fresnel integral. Our results offered in the form of graphical illustrations reveal that, for some settings of source and propagation parameters, the center of the source beam is evacuated after propagation, while the initially smaller side lobes begin to grow. In a parallel development, the angular distribution of the beam also changes. At small Gaussian source sizes and transverse components of the wave vector, the source beam profile remains almost invariant throughout the propagation. The larger refractive index structure constant values cause the final Gaussian beam profile to be attained at earlier propagation distances. Smaller refractive index structure constants, on the other hand, do not change the beam profile substantially from that of free space.     

Interaction of Airy–Gaussian beams in saturable media

Based on the nonlinear Schr o¨dinger equation, the interactions of the two Airy–Gaussian components in the incidence are analyzed in saturable media, under the circumstances of the same amplitude and different amplitudes, respectively. It is found that the interaction can be both attractive and repulsive depending on the relative phase. The smaller the interval between two Airy–Gaussian components in the incidence is, the stronger the intensity of the interaction. However, with the equal amplitude, the symmetry is shown and the change of quasi-breathers is opposite in the in-phase case and out-of-phase case. As the distribution factor is increased, the phenomena of the quasi-breather and the self-accelerating of the two Airy–Gaussian components are weakened. When the amplitude is not equal, the image does not have symmetry. The obvious phenomenon of the interaction always arises on the side of larger input power in the incidence. The maximum intensity image is also simulated. Many of the characteristics which are contained within other images can also be concluded in this figure.     

Scintillation advantages of lowest order Bessel–Gaussian beams

For a weak turbulence propagation environment, the scintillation index of the lowest order Bessel–Gaussian beams is formulated. Its triple and single integral versions are presented. Numerical evaluations show that at large source sizes and large width parameters, when compared at the same source size, Bessel–Gaussian beams tend to exhibit lower scintillations than the Gaussian beam scintillations. This advantage is lost however for excessively large width parameters and beyond certain propagation lengths. Large width parameters also cause rises and falls in the scintillation index of off-axis positions toward the edges of the received beam. Comparisons against the fundamental Gaussian beam are made on equal source size and equal power basis. PACS  42.25.Dd; 42.25.Bs; 42.68.Bz; 42.68.-w     

Soliton guidance and nonlinear coupling for polarized vector spiraling elliptic Hermite–Gaussian beams in nonlocal nonlinear media

We investigate the incoherent beams with two orthogonal polarizations in nonlocal nonlinear media, one of which is a fundamental Gaussian beam and the other is spiraling elliptic Hermite–Gaussian beam carrying the orbital angular momentum(OAM).Using the variational approach, we obtain the critical power and the critical OAM required for the vector spiraling elliptic Hermite–Gaussian solitons.In the strong nonlocality region, two components of the vector beam contribute to the nonlinear refractive index in a linear manner by the sum of their respective power.The nonlinear refractive index exhibits a circularly symmetrical profile in despite of the elliptic shapes for spiraling Hermite–Gaussian beams.We find that in the strong nonlocality region, the critical power and the rotational velocity are the same regardless of the relative ratio of the constituent powers.The nonlinear refractive index loses its circular symmetry in weak nonlocality region, and the nonlinear coupling effect is observed.Due to the radiation of the OAM, the damping of the rotation is predicted, and can be suppressed by decreasing the proportion of the spiraling elliptic component of the vector beam.     

Nonparaxial propagation of phase-flipped Gaussian beams

This paper derives the closed-form expressions for nonparaxial phase flipped Gaussian （PFG） beams propagating in free space, through a knife edge and an aperture, which enable us to study nonparaxial propagation properties of PFG beams and to compare nonparaxial results with paraxial ones. It is found that the f parameter, offsetting distance of the knife edge and truncation parameter affect the nonparaxial beam propagation properties. Only under certain conditions the paraxial approximation is applicable. The results are illustrated by numerical examples.     

Vectorial structure of the far field of an elegant Hermite–Gaussian beam

Based on the vector angular spectrum method and the method of stationary phase, an analytical expression for the vectorial structure of the far field of an elegant Hermite–Gaussian beam is derived. The analytical formulae of the energy flux distributions of the TE term, the TM term, and the whole beam are presented in the far field. Analytical expressions for the ratios of the powers of the TE and TM terms to those of the elegant Hermite–Gaussian beam are obtained without any approximation. The physical pictures of the far field of an elegant Hermite–Gaussian beam are demonstrated and compared with those of the far field of the corresponding standard Hermite–Gaussian beam. This research reveals the internal vectorial structure of the far field of an elegant Hermite–Gaussian beam from an alternative viewpoint.     

Propagation of a Hermite–Laguerre–Gaussian beam in a turbulent atmosphere

The propagation and spreading of a Hermite–Laguerre–Gaussian (HLG) beam in a turbulent atmosphere has been investigated. Based on the extended Huygens–Fresnel integral and some mathematical techniques, analytical expressions for the average intensity, the effective beam size, and the kurtosis parameter of an HLG beam in a turbulent atmosphere are derived, respectively. The average intensity distribution and the spreading properties of HLG beams in a turbulent atmosphere are numerically demonstrated. Upon propagation in a turbulent atmosphere, the central lobes in the beam spot of the HLG beam will evolve into the dominant lobes, and the peripheral lobes around the central lobes will evolve into the subdominant lobes. The influences of the additional angle parameter and the transversal mode numbers on the propagation of HLG beams in a turbulent atmosphere are also discussed. As the coherence length of the turbulence is determined by the propagation distance, the effect of the additional angle parameter on the effective beam size is related to the propagation distance. The kurtosis parameter generally increases with increasing the additional angle parameter. The influence of the transversal mode numbers on the kurtosis parameter is related to the additional angle parameter and the propagation distance. According to the practical need of free-space optical communications and remote sensing, the HLG beam in a turbulent atmosphere can be controlled by choice of the additional angle parameter and the transversal mode numbers.     

Dynamic evolution of Riemann–Silberstein vortices for Gaussian vortex beams

The dynamic evolution of Riemann–Silberstein (RS) vortices for Gaussian vortex beams with topological charges m = ± 1 in free space is studied. It is shown that for Gaussian on-axis vortex beams there exist both RS vortex with m = + 2 and circular edge dislocation. For Gaussian off-axis vortex beams the circular edge dislocation splits into two RS vortices with opposite topological charges m = ± 1 and the RS vortex with m = + 2 decays into two vortices with same topological charges m = + 1. The motion of RS vortices takes place by varying the propagation distance, waist width, off-axis parameter, or topological charge. RS vortices for Gaussian vortex-free beams can be treated as a special case. The results are illustrated analytically and numerically.     

Partially coherent elegant Hermite–Gaussian beam in turbulent atmosphere

Based on the extended Huygens–Fresnel integral, analytical formulas for the cross-spectral density, mean-squared beam width and angular spread of a partially coherent elegant Hermite–Gaussian (HG) beam in turbulent atmosphere are derived. The evolution properties of the average intensity, spreading and directionality of a partially coherent elegant HG beam in turbulent atmosphere are studied numerically. It is found that the partially coherent elegant HG beam with smaller initial coherence width, larger beam order and longer wavelength is less affected by the atmospheric turbulence. Compared to the partially coherent standard HG beam, the partially coherent elegant HG beam is less affected by turbulence under the same condition. Furthermore, it is found that there exist equivalent partially coherent standard and elegant HG beams, equivalent fully coherent standard and elegant HG beams, and an equivalent Gaussian–Schell-model beam may have the same directionality as a fully coherent Gaussian beam whether in free space or in turbulent atmosphere. Our results can be utilized in short and long atmospheric optical communication systems.