Realization and measurement of Airy transform of Gaussian vortex beams
Introduction
Airy beams have three fascinating features: non-diffraction, self-healing and transverse acceleration [1], [2], [3], [4], [5], [6], [7], [8], [9], making them ideal light sources in broad application prospects such as optical micro-manipulation [10], optical trapping [11], plasma channel [12], light bullets [13], [14], optical micro-imaging [15], laser micro-processing [16], and so on. Many methods have been proposed to generate Airy beams successfully [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], but the most basic method is based on spatial light modulator (SLM) [1]. Among the generation methods based on SLM, the simplest one is the Airy transform. The Airy transform of a Gaussian beam is a well-known Airy beam [29]. When performing the Airy transforms of a flat-topped Gaussian beam, a hyperbolic-cosine Gaussian beam and a double-half inverse Gaussian hollow beam, the output beams are an Airy-related beam [30], a novel finite Airy-related beam [31] and a superposition of finite Airy beams [32], respectively. In the past, the attention is concentrated on the Airy transform of vortex-free beams. Recently, the Airy transform of Laguerre-Gaussian beams, which are the familiar vortex beams, has been realized theoretically and experimentally by us [33]. Unfortunately, the universal analytical formula for Laguerre-Gaussian beams passing through an Airy transform optical system has not been derived yet. Moreover, only the effect of the Airy control parameters on the normalized intensity distribution has been investigated [33]. Therefore, more characteristics of the Airy transform of vortex beams are expected to be explored.
A Gaussian vortex beam is a typical vortex beam. Propagation properties of the Gaussian vortex beam with initial radial polarization in the nonparaxial framework have been investigated [34]. The singularity of the polarization in the diffracted Gaussian vortex beams by a half-plane screen and an annular aperture has been examined [35], [36]. The Riemann-Silberstein vortices of Gaussian vortex beams with one topological charge (TC) show their dynamic evolution in free space [37]. Gaussian vortex beams have been proved to have superiority in power coupling of optical systems with the Cassegrain-telescope receivers in the turbulent atmosphere [38]. The property of vectorial structure of the Gaussian vortex beam has been demonstrated in the far field [39]. Propagation characteristic of elliptical Gaussian vortex beams in uniaxial crystal orthogonal to the optical axis has been exhibited [40]. The evolution of the transverse electric term, the transverse magnetic term and the whole Gaussian vortex beam has been displayed in the source region [41]. The distribution of angular momentum density of a Gaussian vortex beam has been depicted in the source region and in the far field, respectively [42], [43]. The Characteristics related to intensity distribution of the Gaussian vortex beam in atmospheric turbulence has been numerically simulated [44]. The transformation law of axial light intensity of a partially coherent Gaussian vortex beam has been deduced [45]. In this paper, the Airy transform of the Gaussian vortex beams is studied. The beam propagation factor defined by the second-order moments is also known as the beam quality factor. The beam propagation factor allows one to access the “quality” of the output beam. Besides the intensity information, the phase information is another important property of an optical beam. Therefore, not only the effects of the Airy control parameters and the TC on the normalized intensity distribution, the centroid and the beam spot size of the output beam are to be investigated, but also the influences of the Airy control parameters and the TC on the phase distribution and the beam propagation factor of the output beam are to be discussed, which cannot be found in the previous literature of the Airy transform.
Section snippets
Derivation of Airy transform of Gaussian vortex beams
Fig. 1 illustrates a typical optical system used to realize the Airy transform. The optical system is composed of two thin lenses with focal length f, forming 4f optical system, and a SLM located in the rear focal plane of the first lens. By using the SLM, a phase modulation is imposed to the optical beam after the first Fourier transform. The electric field of a Gaussian vortex beam with TC m in the input plane is described by [46], [47]
where x0 and y0
Numerical simulations and results
By using the formulae derived in the previous section, the characteristics of the Airy transform of Gaussian vortex beams with TC m = ± 1 and ± 2 are analyzed through numerical examples. Without loss of generality, the Gaussian waist w0 = 0.5 mm is fixed in the following analysis. It was reported in the known literature that the signs of the Airy control parameters α and β only affect the location of the output beams, and the positive and the negative signs of the Airy control parameters α and β
Experimental results
In this section, we carry out the experiment to implement the Airy transform of Gaussian vortex beams with TC m = ± 1 and ± 2, and to measure the characteristics of the output beams including the intensity distribution, phase distribution, beam centroid, beam spot size and beam propagation factor. The influences of the Airy control parameters and TC on the characteristics of the output beam are experimentally studied in detail.
Fig. 9 shows the experimental setup for the generation of Gaussian
Summary
As a summary, we have derived analytical expressions for the electric fields of Gaussian vortex beams with arbitrary TC m passing through an Airy transform optical system. The output electric field with TC m is the sum of the zero-order derivative up to the mth-order derivative of the Airy function with different weight coefficients. Since the TC m = ± 1 and ± 2 are the most common cases, the attention is focused on the Airy transform of Gaussian vortex beams with m = ± 1 and ± 2. The closed
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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