Shaping soliton properties in Mathieu lattices

Study on acoustic radiation force of a rigid sphere arbitrarily positioned in a zero-order Mathieu beam

The expressions of the axial and transverse acoustic radiation forces of a rigid sphere arbitrarily positioned in a zero-order Mathieu beam are derived in this paper. The expansion coefficients of the off-axis zero-order Mathieu beam are obtained using the addition theorem of the Bessel functions, and numerical experiments are conducted to verify the theory. The three-dimensional acoustic radiation forces on a rigid sphere are studied when the beam is set at different ellipticity parameters, half-cone angles, and offsets of the incident wave relative to the particle center. Simulation results show that the axial acoustic radiation forces of the rigid sphere are always positive, but the transverse forces vary with the positions of the particle and the beam parameters. Also, by changing the frequency, half-cone angle, and offset of the zero-order Mathieu beam, the value and direction of the transverse forces can be adjusted, which has applications in controlling the rigid sphere to be close to or away from the beam axis. Furthermore, the finite element model is set up to verify the theoretical model, and the results obtained by the two methods are in good agreement. This work may contribute to a better understanding of the underlying mechanisms of the particle manipulation with different acoustic beams.

Generation of on-demand quasi-Mathieu beams with a controlled generation of spatial spectrum of angular Mathieu-Gauss functions with a digital laser

This study provided an intra-cavity method for the selective generation of all kinds of quasi-Mathieu beams. The method employed L-type digital lasers to selectively generate the Fourier spectrum of the gaussian-modulated angular Mathieu function. The lasing field then underwent a Fourier-transform with an extra-cavity lens, and was converted into quasi-Mathieu beams after passing through an axicon. The selection of the lasing quasi-Mathieu beams was controlled by the projection phase of the intra-cavity spatial light modulator (SLM) of digital lasers, which provided flexibility in dynamically generating on-demand quasi-Mathieu beams. The formalism of the resulting quasi-Mathieu beams is detailed in this paper. The nondiffracting characteristics of the resulting quasi-Mathieu beams were verified both numerically and experimentally. The capability of dynamically controlled generation and manipulation of lasing quasi-Mathieu beams by the proposed method is beneficial to practical applications of Mathieu beams.

Morphing discrete diffraction in nonlinear Mathieu lattices

Discrete optical gratings are essential components to customize structured light waves, determined by the band structure of the periodic potential. Beyond fabricating static devices, light-driven diffraction management requires nonlinear materials. Up to now, nonlinear self-action has been limited mainly to discrete spatial solitons. Discrete solitons, however, are restricted to the eigenstates of the photonic lattice. Here, we control light formation by nonlinear discrete diffraction, allowing for versatile output diffraction states. We observe morphing of diffraction structures for discrete Mathieu beams propagating nonlinearly in photosensitive media. The self-action of a zero-order Mathieu beam in a nonlinear medium shows characteristics similar to discrete diffraction in one-dimensional waveguide arrays. Mathieu beams of higher orders show discrete diffraction along curved paths, showing the fingerprint of respective two-dimensional photonic lattices.

Wavefronts and caustics associated with Mathieu beams

In this work we compute the wavefronts and the caustics associated with the solutions to the scalar wave equation introduced by Durnin in elliptical cylindrical coordinates generated by the function A(ϕ)=ce_ν(ϕ,q)+ise_ν(ϕ,q), with ν being an integral or nonintegral number. We show that the wavefronts and the caustic are invariant under translations along the direction of evolution of the beam. We remark that the wavefronts of the separable Mathieu beams generated by A(ϕ)=ce_ν(ϕ,q) and A(ϕ)=se_ν(ϕ,q) are cones and their caustic is the z axis; thus, they are not structurally stable. However, in general, the Mathieu beam generated by A(ϕ)=ce_ν(ϕ,q)+ise_ν(ϕ,q) is stable because locally its caustic has singularities of the fold and cusp types. To show this property, we present the wavefronts and the caustics for the Mathieu beams with characteristic value a_ν=0 and q=0,0.2,0.3,0.5. For q=0, we obtain the Bessel beam of order zero; in this case, the wavefronts are cones and the caustic coincides with the z axis. For q≠0, the wavefronts are deformations of conical ones, and the caustic surface, for some values of q, has singularities of the cusp ridge type. Furthermore, we remark that the set of Mathieu beams with characteristic value a_ν=0 and 0≤q<1 has associated a caustic with singularities of the swallowtail type, which is structurally stable. Therefore, we conclude that this type of Mathieu beam is more stable than plane waves, Bessel beams, parabolic beams, and those generated by A(ϕ)=ce_ν(ϕ,q) and A(ϕ)=se_ν(ϕ,q). To support this conclusion, we present experimental results showing the pattern obtained after obstructing a plane wave, the Bessel beam of order m=5, and the Mathieu beam of order m=5 and q=50 with complex transversal amplitude given by Ce₅(ξ,50)ce₅(η,50)+iSe₅(ξ,50)se₅(η,50), where (ξ, η) are the elliptical coordinates on the plane.

Quasi-one-dimensional optical lattices for soliton manipulation

Based on angular spectrum engineering, we report the generation of optical lattices whose two-dimensional transverse nondiffracting pattern can be reduced to a quasi-one-dimensional intensity structure formed by either a single or multiple parallel channels. Remarkably, many features for each channel such as its maximum intensity, modulation, width, or separation among channels, can be controlled and modified in order to meet the requirements of particular applications. In particular, we demonstrate that these lattices can provide useful schemes for soliton routing and steering. We demonstrate the existence domain of ground-state solitons for the single quasi-one-dimensional lattice, and we show that these nondiffracting beams allow "push and pull" dynamics among the neighbor solitons propagated along the nondiffracting channels generated.

Polarization holograms allow highly efficient generation of complex light beams

We report a viable method to generate complex beams, such as the non-diffracting Bessel and Weber beams, which relies on the encoding of amplitude information, in addition to phase and polarization, using polarization holography. The holograms are recorded in polarization sensitive films by the interference of a reference plane wave with a tailored complex beam, having orthogonal circular polarizations. The high efficiency, the intrinsic achromaticity and the simplicity of use of the polarization holograms make them competitive with respect to existing methods and attractive for several applications. Theoretical analysis, based on the Jones formalism, and experimental results are shown.

Quantitative characterization of the energy circulation in helical beams by means of near-field diffraction

We present a method to measure the skew angle of the wave-fronts in an optical vortex, which is directly related with the energy flux. It is based on the analysis of the evolution on propagation of the near-field diffraction pattern generated by a single-slit, consisting of two main lobes that shift in opposite directions depending on the vortex helicity. The transverse displacement of each lobe as a function of the propagation distance allows to quantify the energy circulation. Analytical, numerical and experimental results are compared, showing good agreement. We illustrate the method for the case of Bessel beams, although we also discuss other types of helical beams, such as Laguerre-Gauss and Mathieu beams.

Solitons in spiraling Vogel lattices

We address light propagation in Vogel optical lattices and show that such lattices support a variety of stable soliton solutions in both self-focusing and self-defocusing media, whose propagation constants belong to domains resembling gaps in the spectrum of a truly periodic lattice. The azimuthally rich structure of Vogel lattices allows generation of spiraling soliton motion.

Stripe-like quasi-nondiffracting optical lattices

We introduce stripe-like quasi-nondiffracting lattices that can be generated via spatial spectrum engineering. The complexity of the spatial shapes of such lattices and the distance of their almost diffractionless propagation depend on the width of their ring-like spatial spectrum. Stripe-like lattices are extended in one direction and are localized in the orthogonal one, thereby creating either straight or curved in any desired fashion optically-induced channels that may be used for optical trapping, optical manipulation, or optical lattices for quantum and nonlinear optics applications. As an illustrative example, here we show their potential for spatial soliton control. Complex networks consisting of several intersecting or joining stripe-like lattices suited to a particular application may also be constructed.

Numerical study for selective excitation of Mathieu-Gauss modes in end-pumped solid-state laser systems

This study reports a possible first systematic approach to the selective excitations of all Mathieu-Gauss modes (MGMs) in end-pumped solid-state lasers with a new kind of axicon-based stable laser resonator. The study classifies MGMs into two categories, and explores and verifies the approach to excite each MGM category using numerical simulations. Controlling both the "cavity mode gain" and the "cavity conical asymmetry" of the axicon-based stable laser resonator achieves the proposed selective MGM-excitation approach.

Experimental generation of Mathieu-Gauss beams with a phase-only spatial light modulator

We present a novel method for the efficient generation of even, odd, and helical Mathieu-Gauss beams of arbitrary order and ellipticity by means of a phase-only spatial light modulator (SLM). Our method consists of displaying the phase of the desired beam in the SLM; the reconstructed field is obtained on-axis following a spatial filtering process with an annular aperture. The propagation invariance and topological properties of the generated beams are investigated numerically and experimentally.

Stable solitons in elliptical photonic lattices

We introduce the basic properties of solitons in elliptical photonic lattices induced optically by a superposition of Mathieu beams. Owing to the modulation of the intensity along its elliptical rings, these lattices allow novel dynamics of propagation, being possible, for the first time to our knowledge, to propagate solitons in an elliptic motion with varying rotation rate.

Generation of Mathieu-Gauss modes with an axicon-based laser resonator

We report what is to our best knowledge the first observation of Mathieu-Gauss modes directly generated in an axicon-based stable resonator. By slightly breaking the symmetry of the cavity we were able to generate single lowest and high-order Mathieu-Gauss modes of high quality. The observed transverse modes have an inherent elliptic structure and exhibit remarkable agreement with theoretical predictions.

Focal shift in vector Mathieu-Gauss beams

The three dimensional distribution of focused vector Mathieu-Gauss beams (vMG) is studied in the vicinity of the geometrical focus of an unapertured thin lens. We adopt two different intensity based criteria for defining the actual focus. Our analysis confirms the existence of a focal shift towards the lens for this type of beams. The dependence of the focal shift on the different parameters of the beams is discussed in detail. Beams with different states of polarization are studied as well, and it is shown that the focal shift is independent of the polarization.

Normalization of the Mathieu-Gauss optical beams

A series scheme is discussed for the determination of the normalization constants of the even and odd Mathieu-Gauss (MG) optical beams. We apply a suitable expansion in terms of Bessel-Gauss (BG) beams and also answer the question of how many BG beams should be used to synthesize a MG beam within a tolerance. The structure of the normalization factors ensures that MG beams will always be normalized independently of the particular normalization adopted for the Mathieu functions. In this scheme, the normalization constants are expressed as rapidly convergent series that can be calculated to an arbitrary precision.