Ince-Gaussian Modes Research Articles

Transition from Ince-Gaussian beams to nondiffractive Mathieu beams.

We show that under the appropriate conditions, the Ince-Gaussian modes (IGBs) of stable resonators display a behavior very similar to that of the Mathieu beams (MBs), exhibiting nondiffracting propagation and self-healing properties. We show that the high-order IGB propagates in a quasi-nondiffractive manner within the same conical region as any nondiffractive beam, even when their profiles do not match exactly. Our results indicate new, to our knowledge, methods to generate a quasi-nondiffractive MB from spherical resonators and provide more efficient ways to generate them in the Fourier space. These high-order IGBs are an excellent option for applications where a quasi-nondiffractive, but not exact, behavior is required.

Ince-Gaussian laser beams as superposition of Hermite-Gaussian or Laguerre-Gaussian beams

Abstract: We obtain explicit analytic expressions for the Ince-Gaussian (IG) beams for several first indices p = 3, 4, 5, 6. Earlier, explicit expressions have been derived for amplitudes of the IG beams with p = 0, 1, 2 and without regard for the ellipticity parameter. Here, we give expressions for the amplitudes of 24 IG beams written as superpositions of the Laguerre-Gaussian (LG) or Hermite-Gaussian (HG) beams, with the superposition coefficients explicitly depending on the ellipticity parameter. Simultaneously expressing the IG modes both via the LG and HG modes allows easily obtaining the IG modes in the extreme cases when the ellipticity parameter is zero or infinite. Explicit dependence of the obtained expressions for the IG modes on the ellipticity allows the intensity pattern at the beam cross-section to be varied by continuously varying the parameter value. For the first time, intensity distributions are obtained for the IG beams with negative ellipticity parameter.

Talbot-like pattern evolution in complex structured light from a unitary transformation.

Astigmatic unitary transformations allow for the adiabatic connections of all feasible states of paraxial Gaussian beams on the same modal sphere, i.e., Hermite-Laguerre-Gaussian (HLG) modes. Here, we present a comprehensive investigation into the unitary modal evolution of complex structured Gaussian beams, comprised of HLG modes from disparate modal spheres, via astigmatic transformation. The non-synchronized higher-order geometric phases in cyclic transformations originate a Talbot-effect-like modal evolution in the superposition state of these HLG modes, resulting in pattern variations and revivals in transformations with specific geodesic loops. Using Ince-Gaussian modes as an illustrative example, we systematically analyze and experimentally corroborate the beamforming mechanism behind the pattern evolution. Our results outline a generic modal conversion theory of structured Gaussian beams via astigmatic unitary transformation, offering a new approach for shaping spatial modal structure. These findings may inspire a wide variety of applications based on structured light.

Observation of Boyer-Wolf Gaussian modes

Stable laser resonators support three fundamental families of transverse modes: the Hermite, Laguerre, and Ince Gaussian modes. These modes are crucial for understanding complex resonators, beam propagation, and structured light. We experimentally observe a new family of fundamental laser modes in stable resonators: Boyer-Wolf Gaussian modes. By studying the isomorphism between laser cavities and quadratic Hamiltonians, we design a laser resonator equivalent to a quantum two-dimensional anisotropic harmonic oscillator with a 2:1 frequency ratio. The generated Boyer-Wolf Gaussian modes exhibit a parabolic structure and show remarkable agreement with our theoretical predictions. These modes are also eigenmodes of a 2:1 anisotropic gradient refractive index medium, suggesting their presence in any physical system with a 2:1 anisotropic quadratic potential. We identify a transition connecting Boyer-Wolf Gaussian modes to Weber nondiffractive parabolic beams. These new modes are foundational for structured light, and open exciting possibilities for applications in laser micromachining, particle micromanipulation, and optical communications.

Deformable polarization singularity array by superimposed Ince–Gaussian beams

We present a numerical and experimental study of polarization singularity arrays, which are generated by the coherent superposition of orthogonal, linearly polarized, even and odd Ince–Gaussian (IG) beams. The generated array consists of multiple polarization singularities of different types. The type of the polarization singularities can be adjusted effectively by switching the linear polarization states of two IG modes and by modifying the relative phase factor. In addition, we study the evolution of the arrays with the ellipticity parameter, which leads to the splitting and spatial rearrangement of polarization singularities. This work may have implications for material processing, optical trapping, and manipulation.

Structurally Invariant Higher-Order Ince-Gaussian Beams and Their Expansions into Hermite-Gaussian or Laguerre-Gaussian Beams

Paraxial beam modes, which propagate in space and focus without changing their transverse intensity pattern, are of great value for multiplexing transmitted data in optical communications, both in waveguides and in free space. The best-known paraxial modes are the Hermite-Gaussian and Laguerre-Gaussian beams. Here, we derive explicit analytical expressions for Ince-Gaussian (IG) beams for several first values of the indices p = 3, 4, 5, and 6. In total, we obtain expressions for the amplitudes of 24 IG beams. These formulae are written as superpositions of the Laguerre-Gaussian (LG) or Hermite-Gaussian (HG) beams, with the superposition coefficients explicitly depending on the ellipticity parameter. Due to simultaneous representation of the IG modes via the LG and HG modes, it is easy to obtain the IG modes in the limiting cases wherein the ellipticity parameter is zero or approaches infinity. The explicit dependence of the obtained expressions for the IG modes on the ellipticity parameter makes it possible to change the intensity pattern at the beam cross-section by continuously varying the parameter values. For the first time, the intensity distributions of the IG beams are obtained for negative values of the ellipticity parameter. The obtained expressions could facilitate a theoretical analysis of properties of the IG modes and could find practical applications in the numerical simulation or generation of such beams with a liquid-crystal spatial light modulator.

Controlled experimental generation of perturbed high-order Ince-Gaussian laser modes.

We present an experimental approach for generating perturbed high-order Ince-Gaussian laser modes by transforming the low and moderate-intensity lobes of high-order Ince-Gaussian (IG) modes into high-intensity lobes and vice versa. This perturbation reshuffles optical energy among the different lobes and generates new, to the best of our knowledge, modulated Ince-Gaussian (MIG) modes. Computer-generated holograms displayed over spatial light modulators were used to modulate the IGMs. Compared to IG modes, MIG modes are generated precisely in a sense that enhances the IG modes and provides a maximum number of highly intense lobes in a particular mode. That enables the newly generated MIG modes to be utilized more efficiently than IG modes in applications such as particle manipulation and optical trapping of microparticles, which exploit highly intense lobes.

Study on 1.9 [formula omitted]m structured lasers based on Ince–Gaussian modes superposition with multi-modulation by different directions off-axis dual-end-pump

The double IGp,pe modes multi-modulation superposition evolution structured lasers output is realized, based on the proposed different directions off-axis dual-end-pumping Tm:YLF laser method. Under the modes orthogonal superposition, the theoretical simulation and experimental measurement results have a high degree of agreement, and the orthogonal superposition of double IGp,pe modes within 5 orders is realized, which proves that different order modes and optical field intensity ratio have the capability to modulate the formation of vortex array structured beams. When the low-order IGp,pe modes of the same order are orthogonally superimposed, the phenomenon of superposition of different phase differences occurs, which proves the possibility of modulating the superposition of IGp,pe modes by the phase difference. At the same time, it is found that the regular pattern output of high-order IGp,m modes (odd mode m ≤ 3 or even mode m ≤ 2) can be realized at different angles, which proves the ability of rotation angle to modulate the superposition of double IGp,pe modes. Finally, the 1.9μm Tm:YLF structure laser output with double IGp,pe modes multi-factor composite modulation superposition evolution within 10 orders is realized. When the 10-order double IG10,10e modes are superimposed, the output power exceeds 5.7 W, and the conversion efficiency is higher than 17.5%.

Exploring the ellipticity dependency on vector helical Ince-Gaussian beams and their focusing properties.

We present a numerical study of the intensity and polarization structure of vector helical Ince-Gaussian (VHIG) modes, which present a distinct subclass of vector Ince-Gaussian modes with defined parameter settings. The intensity profile of VHIG beams has an elliptic hollow structure, while the polarization distribution shows multiple single-charge polarization vortices arranged along a line. By selecting the mode order, phase factor and ellipticity of the VHIG beams, we can control the number of elliptic rings, the number of polarization vortices, and the topology of the vector singularity. Furthermore, we simulate the focusing properties of VHIG beams based on vector diffraction theory. Our results indicate that the ellipticity parameter of VHIG beams could be a valuable degree of freedom to generate attractive transverse profiles and longitudinal distributions under focusing, which may have implications for lithography, material processing, optical communication, and even optical trapping and manipulation.

Generation of Ince-Gaussian Beams Using Azocarbazole Polymer CGH.

Ince–Gaussian beams, defined as a solution to a wave equation in elliptical coordinates, have shown great advantages in applications such as optical communication, optical trapping and optical computation. However, to ingress these applications, a compact and scalable method for generating these beams is required. Here, we present a simple method that satisfies the above requirement, and is capable of generating arbitrary Ince–Gaussian beams and their superposed states through a computer-generated hologram of size 1 mm2, fabricated on an azocarbazole polymer film. Other structural beams that can be derived from the Ince–Gaussian beam were also successfully generated by changing the elliptical parameters of the Ince–Gaussian beam. The orthogonality relations between different Ince–Gaussian modes were investigated in order to verify applicability in an optical communication regime. The complete python source code for computing the Ince–Gaussian beams and their holograms are also provided.

High-peak-power structured beams by an Nd:YAG/Cr4+:YAG laser in a near-hemispherical cavity.

An Nd:YAG/Cr4+:YAG passively Q-switched (PQS) laser in a near-hemispherical cavity is exploited to generate high-order structured pulsed fields. Under tightly focused on-axis pumping, radial-order Laguerre-Gaussian (LG) modes with controllable mode orders by the input pump power are realized to exhibit quite stable temporal behavior. The pulse repetition rates of radial-LG modes can reach up to 78 kHz with an average output power of 0.57 W and peak power beyond 300 W under a 5-W pump level. Furthermore, by introducing 1D off-axis pumping into the PQS laser, various structured pulsed fields with transverse morphologies as high-order Ince-Gaussian (IG) modes are further created. With clean and well-defined beam structures, the IG pulsed fields can be nicely reconstructed by the resonant modes of the inhomogeneous Helmholtz equation for spherical cavities. More importantly, these high-order PQS IG modes reveal highly regular pulse trains with the maximum pulse repetition rate beyond 20 kHz and overall peak power higher than 1.5 kW.

Tailoring Ince-Gaussian arrays based on a computer-generated hologram.

The continually expanding interest in Ince-Gaussian modes has arisen due to their highly complex and widely varying spatial structures, which allow these fascinating diverse light modes to be significant to various topics. In this paper, we propose a technique for generating beam arrays that contain four Ince-Gaussian modes. By utilizing a single digital hologram, the intensity distributions of each mode in the array can be manipulated independently and flexibly. This work provides a feasible method to realize the controllable multiplexing of Ince-Gaussian modes, which has potential application in numerous important areas.

Optical storage of Ince-Gaussian modes in warm atomic vapor.

We report on the optical storage of Ince-Gaussian modes in a warm rubidium vapor cell based on electromagnetically induced transparency protocol, and we also qualitatively analyze how atomic diffusion affects the retrieved beams after storage. Ince-Gaussian modes possess very complex and abundant spatial structures and form a complete infinite-dimensional Hilbert space. Successfully storing such modes could open up possibilities for fundamental high-dimensional optical communication experiments.

Mixing Ince–Gaussian modes through sum-frequency generation

We present a nonlinear mixing study of Ince–Gaussian (IG) beams in non-collinear sum-frequency generation, completing the fundamental analysis of all three families of paraxial wave equation solutions under this nonlinear process. Through theoretical and experimental results, we consider the nonlinear mixing of even and odd IG modes, and transition between the paraxial modes by controlling the eccentricity parameter ϵ and helical IG modes possessing arbitrary orbital angular momentum. Additionally, the selection rule of IG modes in the nonlinear mixing processes is determined. This work is important for optical communications and quantum protocols.

Classically entangled Ince–Gaussian modes

Complex vector light modes, classically entangled in their spatial and polarization degrees of freedom (DoF), have become ubiquitous in a vast diversity of research fields. Crucially, while polarization is limited to a bi-dimensional space, the spatial mode is unbounded, and it can be specified by any of the sets of solutions the wave equation can support in different coordinate systems. Here, we report on a class of vector beams with elliptical symmetry where the spatial DoF is encoded in the Ince–Gaussian modes of the cylindrical elliptical coordinates. We outline their geometric representation on the higher-order Poincaré sphere, demonstrate their experimental generation, and analyze the quality of the generated modes via Stokes polarimetry. We anticipate that such vector modes will be of great relevance in applications, such as optical manipulations, laser material processing, and optical communications among others.

Parametric upconversion of Ince-Gaussian modes.

The Ince-Gaussian (IG) mode, a recently discovered type of structured Gaussian beam, corresponds to eigenfunctions of the paraxial wave equation in elliptical coordinates. This propagation-invariant mode is of significance in various domains, in particular, its nonlinear transformation; however, there have been few relevant studies to date. In this Letter, we report the parametric upconversion of IG modes and associated full-field selection rule for the first time, to the best of our knowledge. We demonstrate that IG signals can be perfectly upconverted by a flattop-beam pump; in contrast, significant mode distortion occurred when using the most common Gaussian pump. Particular attention was given to the origin of the distortion, i.e., radial-mode degeneration induced by the sum-frequency generation excited by a Gaussian pump. This proof-of-principle demonstration has great significance in relevant areas, such as high-dimensional quantum frequency interfacing and upconversion imaging.

Optical mode conversion through nonlinear two-wave mixing

Hermite\penalty1000-\hskip0ptGaussian (HG), Laguerre\penalty1000-\hskip0ptGaussian (LG), and Ince-Gaussian (IG) modes have been of considerable importance for photonics. They have revealed a unique way to optical manipulation and are particularly promising for optical communications. This paper combines HG modes, as a basis, inside the nonlinear crystal and the generated second harmonic field turns to LG and IG modes. Here we present a way to use second order nonlinear media to convert optical beam modes as we wish.

Shaping optical spin flow topologies by the translation of tailored orbital phase flow

Beneath being a well-established tool in optical manipulation for spinning and eventually sorting trapped objects, transverse energy flow (TEF) of light represents a versatile, illustrative and instructive approach to understand singular light fields and their properties. Up to now, these transverse flows were mainly investigated in its orbital part, for example, as in optical vortices, whereas the study of complex spin flow density (SFD) topologies has not yet reached the same mature state. Especially, the defined customization of energy flow and respective SFD fields has not yet been initiated. By the application of higher-order Laguerre– as well as Ince–Gaussian modes, being well-studied tools in optical manipulation, we demonstrate for the first time to our knowledge the development of sophisticated SFD configurations. We show shaping the flow field and controlling the type, number, spatial location and order of formed singular points in these TEFs. Our approach facilitates on-demand sculpting energy flow fields and their complex topologies for future applications in, for example, optical manipulation based on tailored polarization structures.

Effects of Cr4+ ions on forming Ince–Gaussian modes in passively Q-switched microchip solid-state lasers

The effects of Cr4+ ions on forming Ince–Gaussian (IG) laser modes in a tilted pumped Cr4+:YAG passively Q-switched (PQS) Nd:YAG microchip lasers has been studied experimentally and theoretically. The formation of Cr4+-ion domains in Cr4+:YAG crystal has been proposed based on the spatial distribution of Cr4+ ions in Cr4+:YAG crystal. The spatial modulation effect of Cr4+-ion domains on selecting a transverse mode has been investigated based on their periodic distribution and nonlinear absorption of the Cr4+ saturable absorber. The formation of IG modes in the Nd:YAG/Cr4+:YAG PQS microchip laser under tilted pumping has been demonstrated theoretically with the spatial modulation of the Cr4+-ion domains in Cr4+:YAG crystal and the inversion population distribution, by taking account of the thermal lens effect. The IG modes selected theoretically in the Nd:YAG/Cr4+:YAG PQS microchip laser are in good agreement with experimentally obtained IG modes. This work provides a theoretical model of the selection of IG modes with Cr4+-ion domains in a tilted beam pumped Nd:YAG/Cr4+:YAG PQS microchip laser to produce efficient, stable and controllable IG laser modes.

Sculpting complex polarization singularity networks.

Polarization singularities in vectorial light fields have become an important tool for different cutting-edge applications such as information processing with integer information units. However, even though vectorial singularities naturally form configurations of multiple singular points, up to now only rather simple, mostly cylindrical vector beams including single central singularities, have been considered. Here we demonstrate the customization of extended singularity networks embedded in a class of complex polarization structures based on general Ince-Gaussian modes, namely, Ince-Gaussian vector modes. Contributing to fundamental singular optics, our investigations evince the conservation of tailored singularity arrangements upon 3D propagation, whereby respective modes enlarge the range of stable vectorial fields, paving the way to information technologies with a significantly enhanced number of degrees of freedom.