Hermite Laguerre Gaussian Beams Research Articles

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.

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.

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.

Hermite–Laguerre–Gaussian beams in strongly nonlocal nonlinear media

We obtain an exact Hermite–Laguerre–Gaussian (HLG) beam solution of the Snyder–Mitchell mode, which is confirmed by the simulation of the nonlocal nonlinear Schrödinger equation. HLG breathers and solitons can be obtained by taking different input power. The HLG beams are a unity of Hermite–Gaussian beams (HGBs) and Laguerre–Gaussian beams (LGBs), which form the exact and continuous transition modes between HGBs and LGBs when an additional parameter changes continuously.

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.

Vectorial Hermite–Laguerre–Gaussian beams beyond the paraxial approximation

Starting from the vectorial Rayleigh–Sommerfeld integrals, the free-space propagation expressions for vectorial Hermite–Laguerre–Gaussian (HLG) beams beyond the paraxial approximation are derived. The far-field expressions and the scalar paraxial results are given as special cases of our general expressions. The intensity distributions of vectorial nonparaxial HLG beams are studied and illustrated with numerical examples.

Far-field properties and beam quality of vectorial Hermite–Laguerre–Gaussian beams beyond the paraxial approximation

The far-field properties and beam quality of vectorial nonparaxial Hermite–Laguerre–Gaussian (HLG) beams are studied in detail, where, instead of the second-order-moments-based M 2 factor, the extended power in the bucket (PIB) and βparameter are used to characterize the beam quality in the far field and the intensity in the formulae is replaced by the z component of the time-averaged Poynting vector 〈 S z 〉 . It is found that the 〈 S z 〉 PIB and βparameter of vectorial nonparaxial HLG beams depend on the mode indices n, m, αparameter and waist-width-to-wavelength ratio w 0/ λ and the PIB and βparameter are additionally dependent on the bucket's size taken.

The beam quality of vectorial nonparaxial Hermite-Laguerre-Gaussian beams

Based on the vectorial Rayleigh-Sommerfeld diffraction integrals, the closed-form expressions of vectorial nonparaxial circularly polarized Hermite-Laguerre-Gaussian (HLG) beams in the far-field are derived. The vectorial nonparaxial circularly polarized Gaussian beam is treated as a special case of our general formulae. The power in the bucket (PIB) is extended to the nonparaxial regime, where the intensity is replaced by the z component of the time-averaged Poynting vector, and used to characterize the beam quality of nonparaxial beams in the far field. Numerical calculation and analysis show the dependence of the PIB of vectorial nonparaxial HLG beams on the waist width-to-wavelength ratio w0/λ, as well as on the parameter α, mode indexes n and m, and bucket's size chosen.

Propagation of Hermite–Laguerre–Gaussian beams through a paraxial optical ABCD system with rectangular hard-edged aperture

Based on a recurrence approach the analytical equations of the propagation of Hermite–Laguerre–Gaussian (HLG) beams through a paraxial optical ABCD system with rectangular hard-edged aperture are derived. The results are strict and can be used to study the propagation of HLG beams of any orders. The effect of the aperture on the propagation of HLG beams and the focal shifts are analyzed with numerical examples.

Generalized Gaussian beams

A unity of Hermite–Gaussian (HG) and Laguerre–Gaussian (LG) beam families is proposed by introducing an additional parameter. Continuous changing of the introduced parameter allows one to transform HG beams into LG beams in a continuous way, keeping some important properties of both families, for example, structural stability under propagation. The generalized beams (called Hermite–Laguerre–Gaussian beams) are investigated by theoretical and experimental means.