Three different types of astigmatic Hermite-Gaussian beams with orbital angular momentum

Abstract

In this work, we study three different types of astigmatic (anisotropic) Hermite-Gaussian (aHG) beams whose complex amplitude in the Fresnel diffraction zone is described by the complex argument of a Hermite polynomial of degree (n, 0). The first-type beam is a circularly symmetric optical vortex with topological charge n that has passed through a cylindrical lens. The outgoing optical vortex ‘splits’ into n first-order optical vortices, carrying an orbital angular momentum (OAM) per photon of n. The second-type beam is a Hermite-Gaussian (HG) beam of order (n, 0) generated by passing an elliptic HG beam through a cylindrical lens, which acts by imprinting an OAM into the original HG beam. The OAM of such a beam is a sum of vortex and astigmatic components and can reach large values (tens and hundreds of thousands per photon). We derive an exact formula to describe the OAM of these aHG beams. Under certain conditions, the zero intensity lines of the aHG beam ‘merge’ into an n-fold degenerate intensity null on the optical axis, with the OAM of the beam becoming equal to n. The third type is an elliptical optical vortex with topological charge n that has passed through a cylindrical lens. With a special choice of the ellipticity degree (1:3), the beam retains its structure upon propagation, with the on-axis degenerate intensity null not ‘splitting’ into n optical vortices. The beam is shown to carry a fractional OAM not equal to n. Using intensity distributions of the aHG beam in the foci of two cylindrical lenses, we measure a normalized OAM of the elliptic aHG beam, with the deviation from a theoretical estimate being as low as 7% (experimental OAM-13.62 and theoretical OAM-14.76).

Simplifying the Experimental Detection of the Vortex Topological Charge Based on the Simultaneous Astigmatic Transformation of Several Types and Levels in the Same Focal Plane

“…It can be seen from the intensity distribution in the focal plane after the astigmatic transformation of the vortex Laguerre–Gaussian modes with a TC value l into the Hermite–Gaussian modes with orders ( p , l ) = (0, l ) [ 67 , 68 , 69 ] that the TC of the vortex beam was visualized in the intensity of the astigmatically transformed beam.…”

An astigmatic transform of a fractional-order edge dislocation

“…Roots of the Hermite polynomial have symmetric pairs relative to the origin, with the Hermite polynomial roots H n alternating with the roots H n+1 . Hence, considering that n < ν < (n + 1), the roots (10) need to be located between the maximum-number zeros of the Hermite polynomial, H n and H n+1 . At large-order fractional edge dislocation, ν ≫ 1, the Tricomi function in ( 5) is described by a different asymptotic relation on the horizontal axis [27]:…”

“…Also, focusing of a high-order optical vortex (OV) with an astigmatic lens [5,6], conversion of an astigmatic sin-Gaussian beam in a nonlinear medium [7], and an astigmatic mode conversion in a laser cavity [8,9] have been reported. Elliptic optical Gaussian beams with astigmatic phase were investigated in [10,11], at first with the conversion of an HG beam (0, n) by means of a tilted cylindrical lens and later by studying a mode beam in which a conventional OV with the charge n. In the last case, the charge embedded into an elliptic astigmatic Gaussian beam remains unchanged upon propagation, not splitting into a bunch of simple OVs. Other topics relating to the astigmatic transform are the propagation of elliptic OVs [12], the use of an astigmatic transform for measuring the topological charge (TC) of a single OV [13], and analyzing an astigmatic vortical HG beam [14].…”

Astigmatic transformation of a fractional-order edge dislocation

“…В [8,9] исследовалось астигматическое модовое преобразование внутри лазерного резонатора. Оптические эллиптические Гауссовы вихри с астигматической фазой рассматривались ранее в [10,11]. В [10] рассматривалось преобразование пучка Эрмита-Гаусса порядка (0, n) с помощью повернутой цилиндрической линзы.…”

“…Оптические эллиптические Гауссовы вихри с астигматической фазой рассматривались ранее в [10,11]. В [10] рассматривалось преобразование пучка Эрмита-Гаусса порядка (0, n) с помощью повернутой цилиндрической линзы. В [11] рассмотрен модовый пучок, у которого канонический оптический вихрь с топологическим зарядом (ТЗ), равным n, внедренный в эллиптический астигматический Гауссов пучок, сохраняется при распространении и не расщепляется на простые оптические вихри.…”