Structural stability of a nonparaxial singular mode beam

The exact explicit solution of the Maxwell equations for nonparaxial singular beams propagating in free space or in a homogeneous isotropic medium is considered. It is shown that, in the paraxial approximation, such solutions for mode beams of both lower and higher orders may turn into the solutions for guided modes or vortices of optical fibers. It is found that a variation of the Rayleigh length for a mode beam does not change the structure of phase and polarization singularities; it merely transforms their coordinates. In the paraxial limit, the singularities are shifted off the axis to regions with negligible light fluxes.     

Structure of a nonparaxial gaussian beam near the focus: III. Stability,eigenmodes, and vortices

Exact analytical structurally stable solutions of the Maxwell equations for singular mode beams propagating in free space or a uniform isotropic medium are obtained. Approximate boundary conditions are chosen in the form of the requirement that in the paraxial approximation the fields of nonparaxial mode beams in the waist plane are transformed into the fields of eigenmodes and vortices of weakly guiding optical fibers with the axial symmetry of refractive index. It is shown that optical vortices, in spite of a rather complex structure of field distribution, do not experience substantial changes in the beam form and reproduce, in general features, the field of paraxial vortices. Linear perturbations of the characteristic parameters of mode beams do not change the structure of their electromagnetic field. Nonparaxial singular beams have one more important property, in addition to the fact that the structure of these beams in the paraxial approximation is similar to the structure of the fields of eigenmodes in a fiber. The propagation constants of eigenmodes of a fiber exactly coincide (in the first approximation of perturbation theory) with the projection of the wave vector of a mode beam on the optical axis (an analog of the propagation constant). The possibility of the paraxial transition for nonparaxial mode beams with arbitrary values of azimuthal and radial indices is shown. The properties of nonparaxial modes are illustrated by numerous examples. The solutions obtained and the results of their analysis can be used for exact matching optical fibers and laser beams in various applications.     

Elegant Hermite-Laguerre-Gaussian beams

A novel family of paraxial laser beams called elegant Hermite-Laguerre-Gaussian beams (EHLGBs) are presented. The EHLGBs, a unity of elegant Hermite-Gaussian beams (EHGBs) and elegant Laguerre-Gaussian beams (ELGBs), constitute the exact and continuous transition modes between EHGBs and ELGBs when an additional parameter continuously changes. The virtual source for generation of the EHLGBs is identified. From the spectral representation of the elegant Hermite-Lauguerre-Gaussian waves, we derive the first three orders of nonparaxial corrections for the corresponding paraxial EHLGBs.     

Analytical solutions for laser modes in misaligned resonators

Laser modes of misaligned resonators transversally limited in gain are obtained analytically within the paraxial approximation. The eigenfunctions are shown to be complex Hermite-Gaussian functions, not centered either in intensity or in curvature center. Eigenvalues are also obtained, and the effect of resonator parameters and misalignment is analyzed, revealing the lack of selectivity of misalignment on order mode. The changes in the appearance of the modes are also shown.     

Hypergeometric modes

A new family of paraxial laser beams that form an orthogonal basis is discussed. When propagated in uniform space, these beams preserve their structure to scale. The intensity distribution profile for such beams is similar to that for the Bessel modes, representing a set of alternating bright and dark concentric rings. The complex amplitude of these beams is proportional to the degenerate (confluent) hypergeometric function, and therefore we term such beams hypergeometric (HyG) modes. The HyG modes are generated with a liquid-crystal microdisplay.     

Radial coherent and intelligent states of paraxial wave equation

Ladder operators for the radial index of the paraxial optical modes in the cylindrical coordinates are calculated. The operators obey the su(1,1) algebra commutation relations. Based on this Lie algebra, we found that coherent modes constructed as eigenstates of the destruction operator or resulting from the action of the displacement operator on the fundamental mode are different. Some properties of these two kinds of radial coherent modes are studied in detail.     

Four-beamlets laser array and its propagation

A new type of laser array named four-beamlets laser array is proposed. Based on the diffraction integrals, analytical propagation formulae for the four-beamlets laser array passing through paraxial aligned and misaligned optical systems are derived, respectively. The four-beamlets laser array can also be expressed as a superposition of a series of Hermite Gaussian modes by using polynomial expansion. As a numerical example, the propagation properties of a four-beamlets laser array in free space are illustrated graphically. The fractional Fourier transform of the four-beamlets laser array is also studied. Our model provides a convenient way to describe laser array with four-beamlets.     

Hypergeometric-Gaussian modes

We studied a novel family of paraxial laser beams forming an overcomplete yet nonorthogonal set of modes. These modes have a singular phase profile and are eigenfunctions of the photon orbital angular momentum. The intensity profile is characterized by a single brilliant ring with the singularity at its center, where the field amplitude vanishes. The complex amplitude is proportional to the degenerate (confluent) hypergeometric function, and therefore we term such beams hypergeometric-Gaussian (HyGG) modes. Unlike the recently introduced hypergeometric modes [Opt. Lett. 32, 742 (2007)], the HyGG modes carry a finite power and have been generated in this work with a liquid-crystal spatial light modulator. We briefly consider some subfamilies of the HyGG modes as the modified Bessel Gaussian modes, the modified exponential Gaussian modes, and the modified Laguerre-Gaussian modes.     

单模光纤波导模的傍轴传输特性

将LP01模表示成用拉盖尔-高斯模线性叠加的形式,利用柯林斯公式推导出单模光纤波导模通过傍轴ABCD光学系统的解析传输公式。用所得的解析公式对LP01模在自由空间的传输和聚焦特性作了研究,并和高斯基模进行了比较。数值计算结果表明,用高斯基模描述LP01模会产生一定误差。LP01模经过透镜聚焦后存在焦移,且焦移随着菲涅尔数传输特性的减小而增大。选择合适的菲涅尔数聚焦后,LP01模会出现长焦深,这是与聚焦的高斯模所不同的。     

Universal description of geometric phases in higher-order optical modes bearing orbital angular momentum

We study geometric phases that arise from (cyclic) transformations of the transverse spatial structure of paraxial optical modes. Our approach involves bosonic ladder operators that, in the spirit of the quantum-mechanical harmonic oscillator, generate sets of transverse optical modes. It applies to modes of all orders in a very natural way and provides a universal geometric interpretation of the phase shifts acquired by nonastigmatic modes under typical experimental conditions.     

Complex modes and instability of full-vectorial beam propagation methods

Full-vectorial beam propagation methods (FVBPMs) are widely used to model light waves propagating in high-index-contrast optical waveguides. We show that the paraxial FVBPM and wide-angle FVBPMs based on diagonal Padé approximants are analytically unstable for waveguides with complex modes. The instability cannot be removed by enlarging the computational domain, increasing the numerical resolution, or using perfectly matched layers, because the complex modes are highly confined around the waveguide core.     

强非局域非线性介质中光束传输的Ince-Gauss解

利用强非局域非线性介质中傍轴光束传输的线性模型（Snyder-Mitchell模型）讨论了椭圆坐标系下光束传输过程,通过设立Ince多项式对Gauss函数的调制解得到了强非局域非线性介质中光束稳定传输的Ince-Gauss解.当Ince-Gauss光束的入射功率为临界功率时,光束保持孤子形式传输,否则传输光束的束宽呈现周期性波动,即为呼吸子形式.同时还数值模拟了呼吸子的传输过程.Ince-Gauss光在一定条件下可以连续转换为Hermite-Gauss光或Laguerre-Gauss光,图示展现了几个低阶Ince型光孤子及其转换情况. 关键词： 强非局域非线性介质 Ince-Gauss光 Laguerre-Gauss光 Hermite-Gauss光     

Tracking spatial modes in nearly hemispherical microcavities

We measure experimentally the spatial intensity profiles and resonant frequencies of the transverse modes of nearly hemispherical microcavities with cavity length and mirror curvatures below 10 microm. These resonators possess axially symmetric Gauss-Laguerre-like modes, but do not display the frequency degeneracies typical of large-scale optical cavities. It is possible to interpret these results using a paraxial model of cavity propagation that includes nonparabolic optical elements.     

Elegant Ince-Gaussian beams

The existence of elegant Ince-Gaussian beams that constitute a third complete family of exact and biorthogonal elegant solutions of the paraxial wave equation is demonstrated. Their transverse structure is described by Ince polynomials with a complex argument. Elegant Ince-Gaussian beams constitute exact and continuous transition modes between elegant Laguerre-Gaussian and elegant Hermite-Gaussian beams. The expansion formulas among the three elegant families are derived.     

Hollow Gaussian beams and their propagation properties

A new mathematical model, described as hollow Gaussian beams (HGBs), is proposed to describe a dark hollow laser beam (DHB). The area of the dark region across the HGBs can easily be controlled by proper choice of the beam parameters. Based on the Collins integral, an analytical propagation formula for the HGBs through a paraxial optical system is derived. The HGBs also can be expressed as a superposition of a series of Lagurerre-Gaussian modes by use of a polynomial expansion. As a numerical example, the propagation properties of a DHB in free space are illustrated graphically. The HGBs provide a convenient and powerful way to describe and treat the propagation of DHBs and can be used conveniently to analyze atoms manipulated with a DHB.     

All-Optical Spin–Orbit Coupling of Light in Coherent Media Using Rotating Image

The possibility of all-optical spin–orbit coupling (SOC) of light is investigated based on a rotating spinor image traveling through an electromagnetically induced transparency (EIT) medium. It is shown that the paraxial evolution of the spinor image composed of two Laguerre–Gaussian (LG) modes with different frequencies can be analogous with the quantum dynamics of a spin-1/2 particle with strong and tunable SOC governed by the Pauli equation, where the spin-up and -down states have different effective masses. Using realistic EIT parameters with cold atoms, both the radial inhomogeneity of the strong control field and the atomic density distribution with comparable size are considered. The results confirm that the large group refractive index varying in the radial dimension mimicking the central potential can greatly enhance the spin–orbit interaction, leading to visible spatial quantization of the oppositely oriented spin states, equivalent to the two LG modes.     

Ray theory of microbending

A ray theory is given for randomly bent (two-dimensional) optical fibers that have arbitrary index profiles and arbitrary curvature spectra. Simple closed form results are given for power-law profiles and spectra. No approximation is made besides the small bending approximation and the paraxial ray optics approximation. In particular, the coupling between all modes is effectively taken into account.     

Self-imaging with finite energy

General solutions and conditions are presented for paraxial waves that image themselves with different scales through free propagation. These waves, represented as superpositions of Gauss-Laguerre modes, have finite energy and thus finite effective width. The self-imaging wave fields described by Montgomery [J. Opt. Soc. Am. 57, 772 (1967)], which possess a Fourier transform that is confined to a ring structure, are obtained as a specific limiting case of an infinite aperture.     

Focus wave modes in optical parametric generators

It is shown that in the paraxial approximation the conditions for noncollinear phase matching for three-wave parametric interaction in a nonlinear crystal coincide with the conditions for generation of focus wave modes. As a result, the creation of broadband localized optical fields (pulsed beams without diffraction and dispersion spreading) inside the crystals of parametric generators is feasible.     

Ince-Gaussian beams

We demonstrate the existence of the Ince-Gaussian beams that constitute the third complete family of exact and orthogonal solutions of the paraxial wave equation. Their transverse structure is described by the Ince polynomials and has an inherent elliptical symmetry. Ince-Gaussian beams constitute the exact and continuous transition modes between Laguerre and Hermite-Gaussian beams. The propagating characteristics are discussed as well.