Standard Hermite-Gaussian Beam Research Articles

Modulated high-order Hermite-Gaussian beams with uniform intensity distribution

Hermite-Gaussian beams have been studied since the early years of laser modes. With the recent development of structured light, different modulation methods have been demonstrated to control the energy, polarization and propagation behavior of Hermite-Gaussian beams. In this work we introduce an inverse design approach to modulate the energy distribution among different lobes of Hermite-Gaussian beams. In this approach, the desired intensity profiles of Hermite-Gaussian beams are the starting point. The required spatial phase masks are calculated through inverse Fourier transform, then projected on a spatial light modulator to convert conventional Gaussian beams into modulated Hermite-Gaussian beams. The experimental modulated high-order (m = 10, n = 10) Hermite-Gaussian beams show much improved energy distribution among different lobes over standard Hermite-Gaussian beams. These modulated Hermite-Gaussian beams also have improved beam quality factor M2 after modulation. These results demonstrated the feasibility to fine-tune the intensity profiles of Hermite-Gaussian beams, which have applications in structured illumination, particle manipulation and optical communications.

Experimental realization of modulated Hermite-Gaussian laser modes: a maximum number of highly intense lobes.

Here, we present an experimental method that redistributes the optical energy among the lobes of high-order standard Hermite-Gaussian (SHG) laser modes in a controlled manner. We numerically designed diffractive optical elements, displayed over a spatial light modulator for redistribution of optical energy that converts low and moderate intense lobes into all highly intense lobes and vice versa at the Fourier plane. Such precise generation of modulated HG (MHG) laser modes offers a maximum number of highly intense lobes compared to SHG modes. Hence, we envisage that MHG beams may surpass SHG beams in many applications, such as particle manipulation and optical lithography, where highly intense lobes play a significant role.

Properties of Airy transform of elegant Hermite-Gaussian beams

Airy transform of elegant Hermite-Gaussian beams is theoretically and experimentally investigated. The analytically general formula of the Airy transform of elegant Hermite-Gaussian beams is presented. The analytically general expressions of the centroid and the beam spot size of an elegant Hermite-Gaussian beam passing through an Airy transform optical system are derived. The effects of the Airy control parameters and the transverse mode numbers on the intensity distribution, the centroid, and the beam spot size of elegant Hermite-Gaussian beams passing through Airy transform optical systems are investigated, respectively. The experimental results coincide with the theoretical results. Also, the Airy transform of the elegant and the standard Hermite-Gaussian beams is compared. The analytical expression of the Airy transform of the elegant Hermite-Gaussian beam is more concise than that of the standard Hermite-Gaussian beam. Although the optical fields in the x-direction of the Airy transform of the elegant and the standard Hermite-Gaussian beams with transverse mode number m > 1 are both the sum of the zero-order derivative up to mth-order derivative of the Airy function, the weight coefficient of the same order derivative of the Airy function is different for the elegant and the standard transformed Hermite-Gaussian beams. The centroids of the elegant and the standard Hermite-Gaussian beams passing through the same Airy transform optical system are the same. However, the beam spot size of the elegant transformed Hermite-Gaussian beam is less than or equal to that of the standard transformed Hermite-Gaussian beam. The meaning of elegance is also embodied in the Airy transform of the elegant optical beams. This research reveals the difference between the Airy transform of the elegant and the standard Hermite-Gaussian beams.

Evolution properties of partially coherent standard and elegant Hermite-Gaussian beams in uniaxial crystals.

The propagation properties of a partially coherent elegant Hermite-Gaussian beam (EHGB) and a partially coherent standard Hermite-Gaussian beam (SHGB) in uniaxial crystals orthogonal to the optical axis are studied. The propagation expression for the cross-spectral density of the partially coherent EHGB and SHGB propagating in a uniaxial crystal is derived. With the help of the derived formula, some analyses are illustrated by numerical examples related to the evolution properties of the spectral density, the spectral degree of coherence, and the effective beam width of the partially coherent EHGB and SHGB in a uniaxial crystal. It is shown that propagation properties of the partially coherent EHGB and SHGB in an isotropic crystal are much different from the evolution properties in an anisotropic crystal. The evolution behavior of spectral density and the spectral degree of coherence depends on the parameters of the uniaxial crystals (the ratio of extraordinary and ordinary refractive indices), and beam parameters including the coherence length and the beam order, as well as the propagation distance.

Comparative study of propagation properties of partially coherent standard and elegant Hermite-Gaussian beams in inhomogeneous atmospheric turbulence

Based on the extended Huygens-Fresnel principle (EHFP) and the definition of the Wigner distribution function (WDF), the propagation properties of partially coherent standard Hermite-Gaussian beams (SHGBs) and elegant Hermite-Gaussian beams (EHGBs) in inhomogeneous atmospheric turbulence have been investigated. The M2-factor, the curvature radius and the root-mean-square (rms) angular width of partially coherent SHGBs and EHGBs have been explored numerically and comparatively. It can be found that the saturation propagation distances (SPDs) of the M2-factor and the rms angular width of EHGB and SHGB in inhomogeneous turbulence with a small zenith angle of π/6 are about 20km and 0.3km, respectively. What’s more, the M2-factor and the curvature radius of the partially coherent EHGB are smaller than that of the partially coherent SHGB in inhomogeneous turbulence. It can be shown that the M2-factor and curvature radius depend on the zenith angle, coherence width, inner scale of the turbulence, beam order, wave number and waist width in inhomogeneous turbulence. Furthermore, it is found that the partially coherent SHGB and EHGB have the same rms angular width in inhomogeneous turbulence.

A comparative study of standard and elegant Hermite–Gaussian beams propagating through turbulent atmosphere

Based on the extended Huygens–Fresnel principle and the definition of the second-order moments of the Wigner distribution function (WDF), the analytical expressions for the root-mean-square (rms) beam width and far-field divergence angle, curvature radius and M 2-factor of standard Hermite–Gaussian (SHG) and elegant Hermite–Gaussian (EHG) beams passing through turbulent atmosphere are derived and compared. It is shown that in turbulent atmosphere the far-field divergence angle of SHG and EHG beams is equal under the same conditions, but the rms beam width, curvature radius and the M 2-factor of SHG and EHG beams are different except for beam orders m = 0 and m = 1. The relative rms beam width, relative curvature radius and relative M 2-factor of SHG beams are less than those of EHG beams. Therefore, the conclusion that SHG beams are less influenced by turbulence than EHG beams can be drawn if we examine one of the above three relative beam parameters.

Vectorial structure of the far field of an elegant Hermite–Gaussian beam

Based on the vector angular spectrum method and the method of stationary phase, an analytical expression for the vectorial structure of the far field of an elegant Hermite–Gaussian beam is derived. The analytical formulae of the energy flux distributions of the TE term, the TM term, and the whole beam are presented in the far field. Analytical expressions for the ratios of the powers of the TE and TM terms to those of the elegant Hermite–Gaussian beam are obtained without any approximation. The physical pictures of the far field of an elegant Hermite–Gaussian beam are demonstrated and compared with those of the far field of the corresponding standard Hermite–Gaussian beam. This research reveals the internal vectorial structure of the far field of an elegant Hermite–Gaussian beam from an alternative viewpoint.

Partially coherent elegant Hermite–Gaussian beams

Elegant Hermite–Gaussian beams (EHGBs) are extended to the partially coherent case. An explicit and analytical formula is derived for the cross-spectral density of a partially coherent EHGB propagating through an aligned or misaligned paraxial ABCD optical system. The propagation properties of a partially coherent EHGB in free space and its focusing properties through a thin lens are studied numerically, and are compared to those of a partially coherent standard Hermite–Gaussian beam (SHGB). It is found that the propagation and focusing properties of a partially coherent EHGB are closely related to its initial coherence. A partially coherent EHGB spreads slower than a partially coherent SHGB in free-space propagation. A partially coherent EHGB can be focused more tightly than a partially coherent SHGB.

Average intensity and spreading of an elegant Hermite–Gaussian beam in turbulent atmosphere

The propagation of an elegant Hermite-Gaussian beam (EHGB) in turbulent atmosphere is investigated. Analytical propagation formulae for the average intensity and effective beam size of an EHGB in turbulent atmosphere are derived based on the extended Huygens-Fresnel integral. The corresponding results of a standard Hermite-Gaussian beam (SHGB) in turbulent atmosphere are also derived for the convenience of comparison. The intensity and spreading properties of EHGBs and SHGBs in turbulent atmosphere are studied numerically and comparatively. It is found that the propagation properties of EHGBs and SHGBs are much different from their properties in free space, and the EHGB and SHGB with higher orders are less affected by the turbulence. What's more, the SHGB spreads more rapidly than the EHGB in turbulent atmosphere under the same conditions. Our results will be useful in long-distance free-space optical communications.

Fractionalization of optical beams: I Planar analysis

We apply the tools of the fractional calculus to introduce new fractional-order beam solutions to the paraxial wave equation that can be regarded as intermediate solutions between the known integral-order solutions. We restrict our attention to the fractionalization of the elegant and standard Hermite-Gaussian beams.

The kurtosis parametric characterization of the passage of a standard Hermite–Gaussian beam and an elegant Hermite–Gaussian beam through a fractional Fourier transformation system with a spherically aberrated lens

The propagation characteristics of the kurtosis parameters of a standard Hermite–Gaussian (SHG) beam and of an elegant Hermite–Gaussian (EHG) beam, each passing through a fractional Fourier transformation (FRFT) system with a spherically aberrated lens, are studied in detail. Some numerical calculations are made by introducing an efficient algorithm, based on the Collins diffraction integral formula. The resulting graphs illustrate the striking difference between ideal FRFT systems and those with a spherically aberrated lens. The kurtosis parameters of both SHG and EHG beams passing through a type I Lohmann system with a spherically aberrated lens are seen to change with the fractional order periodically and the fundamental period is 4, but for type II the fundamental period is 2. Different values of spherical aberration coefficients affect the kurtosis parameters in greatly different ways. The values of the kurtosis parameters of a SHG beam passing through either type of Lohmann system with a spherically aberrated lens are no longer equal to those of an EHG beam, even when they have the same fractional orders and the same spherical aberration coefficients.

A comparative study of elegant and standard Hermite–Gaussian beams

The closed-form expressions for the propagation of elegant Hermite–Gaussian (H–G) beams passing though a first-order optical ABCD system are derived by means of which a comparative study of the propagation properties of standard and elegant H–G beams is performed. Furthermore, the beam propagation factor and/or power in the bucket (PIB) are chosen as the criteria for characterizing beam quality, and the beam quality of standard and elegant H–G beams is compared.