Hermite Laguerre Gaussian Research Articles

Mapping structured Laguerre–Gaussian beam states onto the orbital Poincaré sphere in the form of controllable spatial trajectories

The structured Laguerre–Gaussian (LG) beam is a two-parameter superposition of 2n+ℓ+1 Hermite–Gaussian modes (where n and ℓ are a radial number and a topological charge of the initial LG beam) whose orbital angular momentum oscillations are controlled by phases and amplitude parameters. But we succeeded in reducing its representation to a simple sum of a standard LG mode and a hybrid Hermite–Laguerre–Gaussian (HLG) beam that is a key point in understanding a hidden geometry of the structured LG (sLG) beams and implementations of its unique prosperities. In assents, the hybrid HLG beam is mapped onto the orbital Poincaré sphere in the form of a plane trajectory along a main meridian of the sphere. However, the most intriguing thing is as follows. First, once we slightly perturb the HLG beam with a single LG mode, the flat trajectory turns into a complex multi-petalled tracery with multiple self-intersections due to cyclic variation of the phase parameter of the sLG beam. Moreover, the shape of the tracery as well as the birth and destruction of the self-intersection points can be controlled with the amplitude parameter. However, it is worth noting that when changing the beam parameters cyclically, the area outlined by the trajectory on the sphere is directly related to the geometric phase acquired by the sLG beam that can be treated as an additional degree of freedom for transmitting big data. In the article, we study the sLG beam properties and its mapping onto the orbital Poincarè sphere in the framework of a symplectic 4×4 matrix formalism while the orbital Stokes parameters are experimentally measured, and we have found good agreement between theory and experiment.

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.

Topological Charge of Propagation-Invariant Laser Beams

If a vortex propagation-invariant beam is given by all its intensity nulls, then its topological charge (TC) can be defined easily: its TC is equal to the sum of topological charges of all optical vortices in these intensity nulls. If, however, a propagation-invariant beam is given as a superposition of several light fields, then determining its TC is a complicated task. Here, we derive the topological charges of four different types of propagation-invariant beams, represented as axial superpositions of Hermite–Gaussian beams with different amplitudes and different phase delays. In particular, topological charges are obtained for such beam families as the Hermite–Laguerre–Gaussian (HLG) beams and two-parametric vortex Hermite beams. We show that the TC is a quantity resistant to changing certain beam parameters. For instance, when the parameters θ and α of the HLG beams are altered, the beam intensity also changes significantly, but the TC remains unchanged.

Propagation of the Hermite-Laguerre-Gaussian Schell-Model Beams Through Mode Converter

Mode converter is an astigmatic optical system that allows transforming the set of coherent Hermite-Gaussian modes with dislocation singularities into Laguerre-Gaussian one, which carries the OAM, and vice versa. The intermediate Hermite-Laguerre-Gaussian (HLG) modes are also obtained for a certain values of mode converter parameter. All these mode sets are used or can be used for information encoding in free space optical communication. Such mode transformation is well understood in the coherent case, however, the propagation of partially coherent HLG modes through the mode converter has not been addressed yet. Here, we derive a comprehensive closed-form expression for the evolution of the mutual intensity of HLG Schell-model beams (HLG-SMBs) during propagation through the mode converter. It is compared with the expression for HLG-SMBs propagation through rotationally symmetric system. We conclude that the mode converter also works for the partially coherent case, but only for intensity distribution. While the mutual intensity and, in particular, cross-correlation function are different in the output of the considered optical systems. This theoretical analysis is important for applications of the HLG sets in optical communication when the mode decoherence has to be taken into account.

Direct generation of orthogonally polarized dual-wavelength orbital Poincaré modes at 639 nm and 721 nm in a blue-diode pumped Pr:YLF laser

We report on the first, to the best of our knowledge, direct generation of orthogonally polarized dual-wavelength laser with orbital Poincaré (OP) modes at 639 nm and 721 nm in a blue-diode pumped Pr:YLF laser. Beyond the single-wavelength emission, dual-wavelength laser can be approached by adjusting the tilt angle of the output mirror in the critical cavity. In addition, either monochromatic or bi-chromatic orbital Poincaré modes, including Hermite-Gaussian (HG) modes, Laguerre-Gaussian (LG) modes and Hermite-Laguerre Gaussian (HLG) modes can be directly generated by simple off-axis pump scheme and displacement of the output mirror. Moreover, hybrid eigenmodes with S-shape beam profile resulting from dual singularities at the wavefront are observed, as well as the mode switching along with the increase of the pump power. Meanwhile, these composite modes are replicated by simulations using the coherent superposition of higher order HG modes. This work may shed new light on the dual-wavelength laser at visible regime with various modes for potential applications.

Direct generation of 523 nm orbital Poincaré mode from a diode-pumped Pr3+:LiYF4 laser with an off-axis optical needle pumping geometry.

We demonstrate an ultra-compact (8 mm-long) green orbital Poincaré Pr3+:LiYF4 (YLF) laser, in which the intracavity lasing modes are represented as eigenmodes on an equivalent orbital Poincaré sphere (eOPS). This is achieved through the simple application of an 'optical-needle' pumping geometry in conjunction with off-axis laser mode-manipulation. Optical vortices that are produced and mapped on north and south poles of the eOPS exhibit a maximum output power of 304 mW with a pump-to-laser-output slope efficiency of 12%. In addition, higher-order structured modes, including Ince Gaussian (IG), helical IG (HIG), Hermite-Laguerre Gaussian (HLG) and vortex lattices, can be directly generated from the laser cavity.

Laser transverse modes of spherical resonators: a review [Invited

The study of structured laser beams has been one of the most active fields of research for decades, particularly in exploring laser beams with orbital angular momentum. The direct generation of structured beams from laser resonators is deeply associated with the formation of transverse modes. The wave representations of transverse modes of spherical cavities are usually categorized into Hermite–Gaussian (HG) and Laguerre–Gaussian (LG) modes for a long time. Enormous experimental results have revealed that the generalized representation for the transverse modes is the Hermite–LG (HLG) modes. We make a detailed overview for the theoretical description of the HLG modes from the representation of the spectral unitary group of order 2 in the Jordan–Schwinger map. Furthermore, we overview how to derive the integral formula for the elliptical modes based on the Gaussian wave-packet state and the inverse Fourier transform. The relationship between the HLG modes and elliptical modes is linked by the quantum Fourier transform. The most striking result is that the HLG modes can be exactly derived as the superposition of the elliptical modes without involving Hermite and Laguerre polynomials. Finally, we discuss the application of the HLG modes in characterizing the propagation evolution of the vortex structures of HG beams transformed by an astigmatic mode converter. This overview certainly provides not only a novel formula for transverse modes, but also a pedagogical insight into quantum physics.

Generating High-Order Transverse Patterns in Optically Pumped Semiconductor Lasers

High-order pattern formation in an optically pumped semiconductor laser (OPSL) under a selective pumping with varying spatial overlap between the pump beam and transverse modes is explored. In contrast to transverse pattern generation by off-axis pumped solid-state lasers where the mode order can be flexibly increased, experimental results reveal that the selective pumping fails to realize high-order mode operation in OPSLs when the pump-to-mode size ratio and pump power are insufficiently large. On the other hand, several high-order patterns belonging to the Hermite-Laguerre-Gaussian (HLG) modes are observed when scanning a large-ratio pump beam to specific positions on the gain chip. These HLG modes are experimentally confirmed to mainly originate from the transverse non-uniformity of present OPSL chip, while their structural features cannot be simply correlated with the pump scanning positions. Nevertheless, it is believed that the first-time observation of pure HLG modes under the large-ratio pumping can offer useful insights into the high-order pattern manipulation in OPSLs.

Propagation of the chirped-Airy–Gaussian–Hermite–Laguerre–Gaussian wave packets in free space

We investigate the propagation properties of the chirped-Airy–Gaussian–Hermite–Laguerre–Gaussian (CAGHLG) wave packets analytically and numerically. We mainly discuss the features of the wave packets with various chirped conditions, modes and angles. The results reveal that the intensity distributions and the phase patterns of the CAGHLG wave packets change with different modes. Meantime, the CAGHLG wave packets can realize the transformation of correlation wave packets by altering angle. Additionally, we obtain two kinds of inversions of the CAGHLG wave packets by choosing appropriate chirp factors in the course of transmission.

Hybrid topological evolution of multi-singularity vortex beams: generalized nature for helical-Ince-Gaussian and Hermite-Laguerre-Gaussian modes.

A generalized family of scalar structured Gaussian modes including helical-Ince-Gaussian (HIG) and Hermite-Laguerre-Gaussian (HLG) beams is presented with physical insight upon the hybrid topological evolution nature of multi-singularity vortex beams carrying orbital angular momentum. Considering the physical origins of intrinsic coordinates aberration and the Gouy phase shift, a closed-form expression is derived to characterize the general modes in astigmatic optical systems. Moreover, a graphical representation, singularities hybrid evolution nature (SHEN) sphere, is proposed to visualize the topological evolution of the multi-singularity beams, accommodating HLG, HIG, and other typical subfamilies as characteristic curves on the sphere surface. The salient properties of SHEN sphere for describing the precise singularities splitting phenomena, exotic structured light fields, and Gouy phase shift are illustrated with adequate experimental verifications.

Gaussian mode families from systems of rays

Gaussian mode families, including Laguerre–Gaussian, Hermite–Gaussian and generalized Hermite–Laguerre–Gaussian beams, can be described via a geometric optics construction. Ray families crossing the focal plane are represented as one-parameter families of ellipses, parametrized by curves on an analog Poincaré sphere for rays. We derive the optical path length weighting the rays, and find it to be related to the Pancharatnam–Berry connection on the Poincaré sphere. Dressing the rays with Gaussian beams, the approximation returns the Laguerre-, Hermite- and generalized Hermite–Laguerre–Gaussian beams exactly. The approach strengthens the connection between structured light and Hamiltonian optics, opening the possibility to new structured Gaussian beams.

Originating an integral formula and using the quantum Fourier transform to decompose the Hermite-Laguerre-Gaussian modes into elliptical orbital modes

We exploit the SU(2) representation of the Hermite-Laguerre-Gaussian (HLG) mode to manifest the successive transformation between Hermite-Gaussian (HG) and Laguerre-Gaussian (LG) modes. We theoretically confirmed that the time-dependent coherent state for the HLG modes can be simplified as a closed form of Gaussian wave packet. We further employ the explicit closed form to originate an integral of the Gaussian wave-packet state over the elliptical orbit to represent the elliptical orbital mode with fractional orbital angular momentum. On the other hand, we also derive the elliptical orbital mode as the superposition of the degenerate HLG modes. By using the derived formulae and the quantum Fourier transform, the HLG mode is inversely expressed as the superposition of the elliptical orbital modes. The derived representation unambiguously reveals the connection between HLG modes and bundles of elliptical orbits.

Swings and roundabouts: optical Poincaré spheres for polarization and Gaussian beams.

The connection between Poincaré spheres for polarization and Gaussian beams is explored, focusing on the interpretation of elliptic polarization in terms of the isotropic two-dimensional harmonic oscillator in Hamiltonian mechanics, its canonical quantization and semiclassical interpretation. This leads to the interpretation of structured Gaussian modes, the Hermite-Gaussian, Laguerre-Gaussian and generalized Hermite-Laguerre-Gaussian modes as eigenfunctions of operators corresponding to the classical constants of motion of the two-dimensional oscillator, which acquire an extra significance as families of classical ellipses upon semiclassical quantization.This article is part of the themed issue 'Optical orbital angular momentum'.

Entanglement of photons with complex spatial structure in Hermite-Laguerre-Gaussian modes

Photons with complicated spatial mode structures can be applied for different quantum information tasks. Here, we show the entanglement of photons with complex singularity patterns called Hermite-Laguerre-Gaussian (HLG) modes. Measuring one photon of the entangled pairs by HLG mode basis to define its singularity pattern, we can steer the singularity structure of its partner, while the initial singularity structure of the photons is undefined. We also calculate the HLG specific quantum-correlation function. It can be used to extend the quantum key distribution protocols and to tune experiments dealing with high-order transverse modes. In addition we discuss orbital angular momentum properties of the HLG modes and summarize some features of the singularity pattern of the HLG modes with varying angle parameter.

An interpretation and guide to single-pass beam shaping methods using SLMs and DMDs

Exquisite manipulations of light can be performed with devices such as spatial light modulators (SLMs) and digital micromirror devices (DMDs). These devices can be used to simulate transverse paraxial beam wavefunction eigenstates such as the Hermite–Laguerre–Gaussian mode families. We investigate several beam shaping methods in terms of the wavefunctions of scattered light. Our analysis of the efficiency, behaviour and limitations of beam shaping methods is applied to both theory and experiment. The deviation from the ideal output from a valid beam shaping method is shown to be due to experimental factors which are not necessarily being accounted for. Incident beam mode shape, aberration, and the amplitude/phase transfer functions of the DMD and SLM impact the distribution of scattered light and hence the effectiveness and efficiency of a beam shaping method. Correcting for these particular details of the optical system accounts for all differences in efficiency and mode fidelity between experiment and theory. We explicitly show the impact of experimental parameter variations so that these problems may be diagnosed and corrected in an experimental beam shaping apparatus. We show that several beam shaping methods can be used for the production of beam modes in a single pass and the choice is based on the particular experimental conditions.

Non-integer OAM beam shifts of Hermite–Laguerre–Gaussian beams

We have studied the effect of non-integer Orbital Angular Momentum (OAM) on OAM enhanced beam shifts, for in-plane (Goos–Hänchen) and out-of-plane (Imbert–Fedorov) shifts, using Hermite–Laguerre–Gaussian beams. Contrary to naive expectation we find, theoretically and experimentally, that the non-integer OAM beam shifts do not interpolate linearly between the integer OAM beam shifts.

Nonparaxial propagation of Hermite—Laguerre—Gaussian beams in uniaxial crystal orthogonal to the optical axis

Analytical expressions for the three components of the nonparaxial propagation of a Hermite—Laguerre—Gaussian (HLG) beam in uniaxial crystal orthogonal to the optical axis are derived. The intensity distribution of an HLG beam and its three components propagating in a uniaxial crystal orthogonal to the optical axis are demonstrated by numerical examples. Although the y and z components of an HLG beam in the incident plane are both equal to zero, they emerge upon propagation inside the uniaxial crystal. Moreover, the beam profile of the x component is relatively stable and the beam profiles of the y and z components have the same evolution law. If the ratio of the extraordinary refractive index to the ordinary refractive index is larger than unity, the beam profile of the HLG beam is elongated in the x direction and generally rotates clockwise. Otherwise, the beam profile of the HLG beam is elongated in the y direction and generally rotates anticlockwise. This research is beneficial to the optical trapping and nonlinear optics involved in the rotation of a beam profile.

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.

Position measurement of non-integer OAM beams with structurally invariant propagation

We present a design to generate structurally propagation invariant light beams carrying non-integer orbital angular momentum (OAM) using Hermite-Laguerre-Gaussian (HLG) modes. Different from previous techniques, the symmetry axes of our beams are fixed when varying the OAM; this simplifies the calibration technique for beam positional measurement using a quadrant detector. We have also demonstrated analytically and experimentally that both the OAM value and the HLG mode orientation play an important role in the quadrant detector response. The assumption that a quadrant detector is most sensitive at the beam center does not always hold for anisotropic beam profiles, such as HLG beams.

General astigmatic transform of Hermite–Laguerre–Gaussian beams

The general astigmatic transform, or two-dimensional non-separable linear canonical transform of a Hermite-Laguerre-Gaussian beam, is investigated by theoretical means. Some corollaries that apply to Hermite-Gaussian and Laguerre-Gaussian beam propagation are presented and discussed.