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.     

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.     

Quantum state for the two-variable Hermite polynomial–Gaussian laser modes of the electromagnetic field

In Ref. [7] by using the plane-wave representation of the fundamental Gaussian mode as a seed function, Enderlein and Pampaloni derived higher-order beam modes (Laguerre–Gaussian laser beams) of the electromagnetic field by acting with differential operators on this fundamental solution. In this paper we show that their approach can be improved by directly performing the integration with the aid of the two-variable Hermite polynomials, in so doing these modes can exhibit some new good characters. Their relationship to the eigenmodes of light propagation in the quadratic graded-index (GRIN) medium is pointed out. By introducing the two-mode entangled state representation the quantum states corresponding to these modes are also found which turns out to be the Hermite-polynomial excitation on the two-mode squeezed vacuum state.     

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.     

Propagation of an Airy–Gaussian beam in uniaxial crystals

Under the paraxial approximation, the analytical propagation expression of an Airy–Gaussian beam(Ai GB) in uniaxial crystals orthogonal to the optical axis is investigated. The propagation dynamics of the Ai GB is given for different ratios of the extraordinary index to the ordinary refractive index. It has been found that the continuity and the self-bending effect of Ai GB become weaker when the ratio increases. From the figure of the maximum intensity of Ai GB, one can see that the maximum intensity is not monotone decreasing due to the anisotropic effect of the crystals. The intensity distribution of Ai GB in different distribution factors is shown. The Ai GB converges toward a Gaussian beam as the distribution factor increases.     

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.     

Effect of turbulent atmosphere on the on-axis average intensity of Pearcey–Gaussian beam

The propagation characteristics of the Pearcey–Gaussian(PG) beam in turbulent atmosphere are investigated in this paper.The Pearcey beam is a new kind of paraxial beam,based on the Pearcey function of catastrophe theory,which describes diffraction about a cusp caustic.By using the extended Huygens–Fresnel integral formula in the paraxial approximation and the Rytov theory,an analytical expression of axial intensity for the considered beam family is derived.Some numerical results for PG beam propagating in atmospheric turbulence are given by studying the influences of some factors,including incident beam parameters and turbulence strengths.     

Vortex and anti-vortex compositions of exact elegant Laguerre–Gaussian vector beams

Reformulation of conventional beam definitions into their bidirectional versions and use of Hertz potentials make beam fields exact vector solutions to Maxwell’s equations. This procedure is applied to higher-order elegant Laguerre–Gaussian beams of transverse magnetic and transverse electric polarization. Their vortex and anti-vortex co-axial compositions of equal and opposite topological charges are given in a closed analytic form. Polarization components of the composed beams are specified by their radial and azimuthal indices. The longitudinal components are common for beam compositions of both types; meanwhile, their transverse components are different and comprise two—nonparaxial and paraxial—separate parts distinguished by a paraxial parameter and its inverse, respectively. The new solutions may appear useful in modeling and tailoring of arbitrary vector beams.     

Quantitative comparison of self-healing ability between Bessel–Gaussian beam and Airy beam

The self-healing ability during propagation process is one of the most important properties of non-diffracting beams. This ability has crucial advantages to light sheet-based microscopy to reduce scattering artefacts, increase the quality of the image and enhance the resolution of microscopy. Based on similarity between two infinite-dimensional complex vectors in Hilbert space, the ability to a Bessel–Gaussian beam and an Airy beam have been studied and compared. Comparing the evolution of the similarity of Bessel–Gaussian beam with Airy beam under the same conditions, we find that Bessel–Gaussian beam has stronger self-healing ability and is more stable than that of Airy beam. To confirm this result, the intensity profiles of Bessel–Gaussian beam and Airy beam with different similarities are numerically calculated and compared.     

Tight focusing properties of radially polarized Lorentz–Gaussian beam

Focusing properties of radially polarized Lorentz–Gauss beam with one on-axis optical vortex was investigated by vector diffraction theory. Results show that intensity distribution in the focal region can be altered considerably by charge number of the optical vortex and the beam parameters. Many novel focal patterns may occur, Such as Peak-centered, donut focal shapes which is potentially useful in optical tweezers, material processing and laser printing.     

Effects of a rectangular aperture on the vectorial structure of a Gaussian beam in the far-field regime

With the help of the angular spectrum representation and the Gaussian function expansions of the hard-edge aperture function, the vectorial structure of a linearly polarized Gaussian beam (GB) diffracted by a rectangular aperture is analyzed in detail. It is found that the sizes of the energy flux density spots and the energy fluxes of the TE and TM terms depend on the aperture configuration and the polarization direction of the incident GB. The far fields may have smaller spots and larger energy fluxes for a GB diffracted by a rectangular aperture compared to that by a square aperture with the same beam intensity. And another potential application in information encoding and transmission for free-space communications is also proposed in addition to re-focusing to enhance the optical storage density. This encoding scheme has the benefit of easy implementation without modulating any properties of the light source.     

Tunable gradient force of hyperbolic-cosine–Gaussian beam with vortices

Gradient force plays an important role in optical tweezers technique. In this paper, the tunable gradient force in focal plane of the hyperbolic-cosine–Gaussian (ChG) beam is investigated numerically. The ChG beam contains one spiral vortex and one non-spiral vortex. Simulation results show that the gradient force distribution can be altered considerably by decentered parameters of ChG beam, topological number of the spiral vortex, and vortex parameter of the non-spiral vortex. Many novel gradient force patterns can occur, which means corresponding optical traps may come into being, including ring optical trap, multiple-point trap pattern, line optical trap, rectangle trap pattern, and rhombus trap pattern. In addition, force pattern evolution principle may also differ significantly.     

Effect of Non-Kolmogorov Turbulence on the Spreading of Polychromatic Partially Coherent Hermite–Gaussian Array Beams

Based on the extended Huygens–Fresnel principle, we derive an analytical expression for the beam width of polychromatic partially coherent Hermite-Gaussian array (PPCHGA) beams propagating through non-Kolmogorov turbulence and study in detail the effect of bandwidth, array parameters, and non-Kolmogorov turbulence on the beam-width spreading. We show that the beam width of PPCHGA beams increases with increase in the bandwidth, beam number, and relative distance of beam separation. The spreading of polychromatic array beams with increasing generalized exponent parameter is smaller than that of monochromatic array beams under the same conditions. The results are illustrated by numerical examples.     

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  $C_{n}^{2}$ .     

Propagation of Gaussian beam array through an optical system in?turbulent atmosphere

The propagation of Gaussian beam array through an optical system in turbulent atmosphere is studied in this paper. Analytical propagation formulas for the propagation of a Gaussian beam array through an optical system in turbulent atmosphere are derived based on the extended Huygens–Fresnel diffraction principle. Numerical examples are provided to illustrate the propagation property. It is shown that the intensity profile at the receiving plane is significantly affected by the optical system, turbulence intensity and the propagation distance.     

Gaussian beam decomposition of high frequency wave fields using expectation–maximization

A new numerical method for approximating highly oscillatory wave fields as a superposition of Gaussian beams is presented. The method estimates the number of beams and their parameters automatically. This is achieved by an expectation–maximization algorithm that fits real, positive Gaussians to the energy of the highly oscillatory wave fields and its Fourier transform. Beam parameters are further refined by an optimization procedure that minimizes the difference between the Gaussian beam superposition and the highly oscillatory wave field in the energy norm.     

Influence of the in-plane magnetic field on the Andreev conductance of a superconductor–insulator–normal metal structure

Pronounced conductance due to electrons experiencing Andreev reflection from a superconducting condensate has been observed in superconductor (aluminum)–insulator (aluminum oxide)–normal metal (copper) tunnel junctions at low voltages, along with single-electron tunneling. It has been discovered experimentally that the collective current is suppressed in the magnetic field parallel to the tunnel junction plane and the Andreev conductance decreases nearly twofold in a field of ～20–30 mT.     

On the structure of open–closed topological field theory in two dimensions

I discuss the general formalism of two-dimensional topological field theories defined on open–closed oriented Riemann surfaces, starting from an extension of Segal's geometric axioms. Exploiting the topological sewing constraints allows for the identification of the algebraic structure governing such systems. I give a careful treatment of bulk-boundary and boundary-bulk correspondences, which are responsible for the relation between the closed and open sectors. The fact that these correspondences need not be injective nor surjective has interesting implications for the problem of classifying ‘boundary conditions’. In particular, I give a clear geometric derivation of the (topological) boundary state formalism and point out some of its limitations. Finally, I formulate the problem of classifying (on-shell) boundary extensions of a given closed topological field theory in purely algebraic terms and discuss reducibility of boundary extensions.