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Airy transform of Laguerre-Gaussian beams

Guoquan Zhou, Fei Wang, and Shangshen Feng

Open Access Open Access

Abstract

Airy transform of Laguerre-Gaussian (LG) beams is investigated. As typical examples, the analytic expressions for the Airy transform of LG01, LG02, LG11, and LG12 modes are derived, which are special optical beams including the Airy and Airyprime functions. Based on these analytical expressions, the Airy transform of LG01, LG02, LG11, and LG12 modes are numerically and experimentally investigated, respectively. The effects of the control parameters α and β on the normalized intensity distribution of a Laguerre-Gaussian beam passing through Airy transform optical systems are investigated, respectively. It is found that the signs of the control parameters only affect the location of the beam spot, while the sizes of the control parameters will affect the characteristics of the beam spot. When the absolute values of the control parameters α and β decrease, the number of the side lobes in the beam spot, the beam spot size, and the Airy feature decrease, while the Laguerre-Gaussian characteristic is strengthened. By altering the control parameters α and β, the performance of these special optical beams is diversified. The experimental results are consistent with the theoretical simulations. The Airy transform of other Laguerre-Gaussian beams can be investigated in the same way. The properties of the Airy transform of Laguerre-Gaussian beams are well demonstrated. This research provides another approach to obtain special optical beams and expands the application of Laguerre-Gaussian beams.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Higher-order modes of axially symmetric laser cavities with spherical mirrors are Laguerre-Gaussian beams. The Laguerre-Gaussian (LG) mode is exactly the wave function of the common eigenvector of the orbital angular momentum and the total photon number operator [1]. The Laguerre-Gaussian beams can be generated by means of diffractive optics [2–4] or a liquid-crystal-on-silicon spatial light modulator [5] or a diode-pumped solid-state laser resonator [6] or a 4F spatial filtering system [7]. Propagation and diffraction properties of Laguerre-Gaussian beams have been fully studied. Propagation of Laguerre-Gaussian beams in apertured fractional Hankel transform systems [8], through a turbulent atmosphere have been derived [9, 10], through a slant non-Kolmogorov turbulence channel [11], reflected by a dielectric slab [12], diffracted by a soft-edge screen [13], and scattered by complicated shaped biological cells [14] have been investigated, respectively. The non-linear coaxial propagation of Laguerre-Gaussian and Gaussian beams has been explored in a homogeneous plasma [15]. The outcome of the coaxial superposition of two Laguerre-Gaussian beams is a new composite vortex beam [16]. Double-slit interference of Laguerre-Gaussian beams is verified to be controlled by the azimuthal phase [17]. The optical spin splitting of Laguerre-Gaussian beams can be enhanced by through graphene metamaterial slabs [18].

Superiority of Laguerre-Gaussian beams is that they possess orbital angular momentum. The orbital angular momentum of Laguerre-Gaussian beams can be determined by the diffraction patterns of a single slit [19] and the interference patterns of double-slit [20], respectively. Also, the orbital angular momentum of Laguerre-Gaussian beams has been treated within the non-paraxial framework [21]. Angular momentum densities of the transverse electric and the transverse magnetic terms of Laguerre-Gaussian beams have been depicted in free space [22]. The probabilities of the signal and the crosstalk orbital angular momentum states of a Laguerre-Gaussian beam through Kolmogorov and non-Kolmogorov turbulences have been examined [23]. The orbital angular momentum spectrum of a Laguerre-Gaussian beam through anisotropic non-Kolmogorov turbulence along the horizontal path has been modeled and calculated [24]. An orbital angular momentum encoding system with high-order radial indices of Laguerre-Gaussian beam has been used as the optical communication system [25]. The advanced mode demultiplexing of Laguerre-Gaussian beams based on the orbital angular momentum and the radial topological number has been exhibited by using the deep neural network flows [26].

The advantage of Laguerre-Gaussian beams is their capture ability. The off-axial optical capture by Laguerre-Gaussian beams has been examined in the Rayleigh and Mie domains [27]. Radiation force exerted on a sphere by Laguerre-Gaussian beams has been calculated by using the generalized Lorenz-Mie theory [28] and the far-field matching method [29], respectively. When the target atoms are deviated from the beam axis, the magnetic sublevel population of Ca+ ions in a Laguerre-Gaussian beam has been analyzed by first-order perturbation theory [30]. The dynamic polarizability of an atomic state in optical trapping by Laguerre-Gaussian beams has been proposed by using the state summation technology [31]. The spin of a dielectric spherical particle induced by a Laguerre-Gaussian beam with the orbital angular momentum has been demonstrated in a dielectric chiral medium [32]. The ultra-high precision and the spatial resolution atomic positioning in a double-Λ atomic system can be realized by using Laguerre-Gaussian beams [33]. The density distribution of trapped two-component Bose-Einstein condensates and their microscopic interaction with Laguerre-Gaussian beams have been investigated [34].

Other characteristics of Laguerre-Gaussian beams have been also well revealed. The focusing radially polarized Laguerre-Gaussian beam can be divided into a Bessel-like multi-ring and annular parts [35]. By using the binary phase plate, a Laguerre-Gaussian beam can be shaped as an illumination source [36]. The minimum effective beam width of a focused partially coherent Laguerre-Gaussian beam has been used to explore the focal shift [37]. High-order harmonic generation with a linearly polarized Laguerre-Gaussian beam has been suggested by different methods [38, 39]. The photon polarization tensor in pulsed Laguerre-Gaussian beams has been presented by means of the locally constant field approximation [40]. The radial and azimuthal modal decomposition of Laguerre-Gaussian beams has been explored by the optical correlation technique [41]. Vortex mode excitation by the Laguerre-Gaussian beams has been performed in a multimode fiber [42]. Also, the Laguerre-Gaussian beams have been extended to elegant Laguerre-Gaussian beams [43, 44] and cylindrical vector Laguerre-Gaussian beams [45, 46].

A Laguerre-Gaussian beam is a classical beam model, and the corresponding fundamental mode is a Gaussian beam. However, the novel special optical beams are also emerging continuously, one of which is an Airy beam. Due to three distinctive features of non-diffraction, self-healing, and transverse acceleration, Airy beams have been attracted much attention [47–60]. The connection between Airy beams and classical Gaussian beams can be realized by Airy transform. The Airy transform of a Gaussian beam is an Airy beam, and likewise the Airy transform of an Airy beam is a Gaussian beam [61]. The Airy transform of a flat-topped Gaussian beam, a hyperbolic-cosine Gaussian beam, and a double-half inverse Gaussian hollow beam becomes different kinds of Airy-related beams [62–64]. At present, all the beams used to perform the Airy transform are Gaussian beams or optical beams described by superposition of Gaussian beams. In particular, the Airy transform of high-order Gaussian beams such as Laguerre-Gaussian beams is more concerned by optical researchers, which is also the reason why this paper studies the Airy transform of these beams.

2. Transformation of Laguerre-Gaussian beams by an Airy transform optical system

A Cartesian coordinate system is established, and the z-axis is taken as the beam propagation direction. A Laguerre-Gaussian beam in the source plane z = 0 is characterized by

$$E_{nm}^{}({x_0},{y_0},0) = {\left( {\frac{{\sqrt 2 {\rho_0}}}{{{w_0}}}} \right)^m}L_n^m\left( {\frac{{2\rho_0^2}}{{w_0^2}}} \right)\exp \left( { - \frac{{\rho_0^2}}{{w_0^2}}\textrm{ + }im{\varphi_0}} \right),$$
where ${\rho _0} = {(x_0^2\textrm{ + }y_0^2)^{1/2}}$ and ${\varphi _0} = {\tan ^{ - 1}}({y_0}/{x_0})$. x0 and y0 are two transverse coordinates in the source plane. w0 is the Gaussian waist. $L_n^m(.)$ is the associated Laguerre polynomial. n and m are the radial and the angular mode numbers. The Laguerre-Gaussian beam represented by Eq. (1) is called as LGnm mode. The optical field of a Laguerre-Gaussian beam passing through an Airy transform optical system is given by [61]:
$$E_{nm}^{}(x,y) = \frac{1}{{|{\alpha \beta } |}}\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {E_{nm}^{}({x_0},{y_0},0)} } Ai\left( {\frac{{x - {x_0}}}{\alpha }} \right)Ai\left( {\frac{{y - {y_0}}}{\beta }} \right)d{x_0}d{y_0},$$
where x and y are two transverse coordinates in the output plane. α and β are the control parameters of the Airy transform optical system in the x- and y-directions. Ai(·) is the Airy function and is defined by
$$Ai(x) = \frac{1}{{2\pi }}\int_{ - \infty }^\infty {\exp \left( {\frac{{i{u^3}}}{3} + ixu} \right)} du.$$
It seems to be impossible to obtain the generally analytic formula of Laguerre-Gaussian beams passing through an Airy transform optical system. However, the analytical expression of the Airy transform of each Laguerre-Gaussian beam can be presented. Here, we choose to derive the analytic formulas of the Airy transform of LG01, LG02, LG11, and LG12 modes. By using the following mathematical integral formulae [65, 66]:
$$L_n^m(x) = \sum\limits_{l = 0}^n {\frac{{{{( - 1)}^l}(n + m)!{x^l}}}{{(m + l)!(n - l)!l!}}} ,$$
$$\int_{ - \infty }^\infty {\exp ( - {b^2}{x^2} - sx)} dx = \frac{{\sqrt \pi }}{b}\exp \left( {\frac{{{s^2}}}{{4{b^2}}}} \right),$$
$$\int_{ - \infty }^\infty {\exp \left( {\frac{{i{u^3}}}{3} + ip{u^2} + iqu} \right)} du\textrm{ = }2\pi \exp \left[ {ip\left( {\frac{{2{p^2}}}{3} - q} \right)} \right]Ai(q - {p^2}),$$
the analytic expressions of the Airy transform of LG01, LG02, LG11, and LG12 modes are found to be
$$\begin{aligned}E_{01}(x, y)=& \frac{-\sqrt{2} \pi w_{0}^{3}}{2|\alpha \| \beta|} \exp \left(\frac{w_{0}^{6}}{96 \alpha^{6}}+\frac{w_{0}^{6}}{96 \beta^{6}}+\frac{w_{0}^{2} x}{4 \alpha^{3}}+\frac{w_{0}^{2} y}{4 \beta^{3}}\right)\left[\left(\frac{w_{0}^{2}}{4 \alpha^{3}}+\frac{i w_{0}^{2}}{4 \beta^{3}}\right)\right.\\ & \times A i\left(\frac{w_{0}^{4}}{16 \alpha^{4}}+\frac{x}{\alpha}\right) Ai \left(\frac{w_{0}^{4}}{16 \beta^{4}}+\frac{y}{\beta}\right)+\frac{1}{\alpha} A i^{\prime}\left(\frac{w_{0}^{4}}{16 \alpha^{4}}+\frac{x}{\alpha}\right) \\ &\left.\times A i\left(\frac{w_{0}^{4}}{16 \beta^{4}}+\frac{y}{\beta}\right)+\frac{i}{\beta} A i\left(\frac{w_{0}^{4}}{16 \alpha^{4}}+\frac{x}{\alpha}\right) A i^{\prime}\left(\frac{w_{0}^{4}}{16 \beta^{4}}+\frac{y}{\beta}\right)\right], \end{aligned}$$
$$\begin{aligned}E_{02}(x, y)=& \frac{\pi w_{0}^{2}}{2|\alpha||\beta|} \exp \left(\frac{w_{0}^{6}}{96 \alpha^{6}}+\frac{w_{0}^{6}}{96 \beta^{6}}+\frac{w_{0}^{2} x}{4 \alpha^{3}}+\frac{w_{0}^{2} y}{4 \beta^{3}}\right)\left[\left(\frac{w_{0}^{6}}{8 \alpha^{6}}-\frac{w_{0}^{6}}{8 \beta^{6}}+\frac{i w_{0}^{6}}{8 \alpha^{3} \beta^{3}}\right.\right.\\ &\left.+\frac{w_{0}^{2} x}{\alpha^{3}}-\frac{w_{0}^{2} y}{\beta^{3}}\right) A i\left(\frac{w_{0}^{4}}{16 \alpha^{4}}+\frac{x}{\alpha}\right) A i\left(\frac{w_{0}^{4}}{16 \beta^{4}}+\frac{y}{\beta}\right)+\left(\frac{w_{0}^{4}}{2 \alpha^{4}}+\frac{i w_{0}^{4}}{2 \alpha \beta^{3}}\right) \\ & \times A i^{\prime}\left(\frac{w_{0}^{4}}{16 \alpha^{4}}+\frac{x}{\alpha}\right) A i\left(\frac{w_{0}^{4}}{16 \beta^{4}}+\frac{y}{\beta}\right)-\left(\frac{w_{0}^{4}}{2 \beta^{4}}-\frac{i w_{0}^{4}}{2 \alpha^{3} \beta}\right) A i\left(\frac{w_{0}^{4}}{16 \alpha^{4}}+\frac{x}{\alpha}\right) \\ &\left.\times A i^{\prime}\left(\frac{w_{0}^{4}}{16 \beta^{4}}+\frac{y}{\beta}\right)+\frac{2 i w_{0}^{2}}{\alpha \beta} A i^{\prime}\left(\frac{w_{0}^{4}}{16 \alpha^{4}}+\frac{x}{\alpha}\right) A i^{\prime}\left(\frac{w_{0}^{4}}{16 \beta^{4}}+\frac{y}{\beta}\right)\right], \end{aligned}$$
$$\begin{aligned}E_{11}(x, y)=& \frac{\sqrt{2} \pi w_{0}^{3}}{4|\alpha||\beta|} \exp \left(\frac{w_{0}^{6}}{96 \alpha^{6}}+\frac{w_{0}^{6}}{96 \beta^{6}}+\frac{w_{0}^{2} x}{4 \alpha^{3}}+\frac{w_{0}^{2} y}{4 \beta^{3}}\right)\left[\left(\frac{2 w_{0}^{2}}{\alpha^{3}}+\frac{w_{0}^{8}}{16 \alpha^{9}}+\frac{w_{0}^{8}}{32 \alpha^{3} \beta^{6}}\right.\right.\\ &\left.+\frac{2 i w_{0}^{2}}{\beta^{3}}+\frac{i w_{0}^{8}}{16 \beta^{9}}+\frac{i w_{0}^{8}}{32 \alpha^{6} \beta^{3}}+\frac{3 w_{0}^{4} x}{4 \alpha^{6}}+\frac{w_{0}^{4} y}{4 \alpha^{3} \beta^{3}}+\frac{i w_{0}^{4} x}{4 \alpha^{3} \beta^{3}}+\frac{3 i w_{0}^{4} y}{4 \beta^{6}}\right) \\ & \times A i\left(\frac{w_{0}^{4}}{16 \alpha^{4}}+\frac{x}{\alpha}\right) A i\left(\frac{w_{0}^{4}}{16 \beta^{4}}+\frac{y}{\beta}\right)+\left(\frac{4}{\alpha}+\frac{w_{0}^{6}}{4 \alpha^{7}}+\frac{w_{0}^{6}}{8 \alpha \beta^{6}}+\frac{i w_{0}^{6}}{8 \alpha^{4} \beta^{3}}\right.\\ &\left.+\frac{w_{0}^{2} x}{\alpha^{4}}+\frac{w_{0}^{2} y}{\alpha \beta^{3}}\right) A i^{\prime}\left(\frac{w_{0}^{4}}{16 \alpha^{4}}+\frac{x}{\alpha}\right) A i\left(\frac{w_{0}^{4}}{16 \beta^{4}}+\frac{y}{\beta}\right)+\left(\frac{w_{0}^{6}}{8 \alpha^{3} \beta^{4}}+\frac{4 i}{\beta}+\frac{i w_{0}^{6}}{8 \alpha^{6} \beta}\right.\\ &\left.+\frac{i w_{0}^{6}}{4 \beta^{7}}+\frac{i w_{0}^{2} x}{\alpha^{3} \beta}+\frac{i w_{0}^{2} y}{\beta^{4}}\right) A i\left(\frac{w_{0}^{4}}{16 \alpha^{4}}+\frac{x}{\alpha}\right) A i^{\prime}\left(\frac{w_{0}^{4}}{16 \beta^{4}}+\frac{y}{\beta}\right) \\ &\left.+\left(\frac{w_{0}^{4}}{2 \alpha \beta^{4}}+\frac{i w_{0}^{4}}{2 \alpha^{4} \beta}\right) A i^{\prime}\left(\frac{w_{0}^{4}}{16 \alpha^{4}}+\frac{x}{\alpha}\right) A i^{\prime}\left(\frac{w_{0}^{4}}{16 \beta^{4}}+\frac{y}{\beta}\right)\right], \end{aligned}$$
$$\begin{aligned}E_{12}(x, y)=& \frac{\pi w_{0}^{2}}{|\alpha||\beta|} \exp \left(\frac{w_{0}^{6}}{96 \alpha^{6}}+\frac{w_{0}^{6}}{96 \beta^{6}}+\frac{w_{0}^{2} x}{4 \alpha^{3}}+\frac{w_{0}^{2} y}{4 \beta^{3}}\right)\left\{\left[\left[-\frac{7 w_{0}^{6}}{16 \alpha^{6}}-\frac{w_{0}^{12}}{128 \alpha^{12}}+\frac{7 w_{0}^{6}}{16 \beta^{6}}\right.\right.\right.\\ &+\frac{w_{0}^{12}}{128 \beta^{12}}-\frac{3 w_{0}^{2} x}{2 \alpha^{3}}-\frac{w_{0}^{8} x}{8 \alpha^{9}}+\frac{3 w_{0}^{2} y}{2 \beta^{3}}+\frac{w_{0}^{8} y}{8 \beta^{9}}-\frac{w_{0}^{4} x^{2}}{4 \alpha^{6}}+\frac{w_{0}^{4} y^{2}}{4 \beta^{6}}-\frac{i w_{0}^{2}}{2 \alpha \beta}\left(\frac{7 w_{0}^{4}}{8 \alpha^{2} \beta^{2}}\right.\\ &\left.\left.+\frac{w_{0}^{10}}{64 \alpha^{8} \beta^{2}}+\frac{w_{0}^{10}}{64 \alpha^{2} \beta^{8}}+\frac{3 w_{0}^{6} x}{16 \alpha^{5} \beta^{2}}+\frac{3 w_{0}^{6} y}{16 \alpha^{2} \beta^{5}}\right)\right] A i\left(\frac{w_{0}^{4}}{16 \alpha^{4}}+\frac{x}{\alpha}\right) A i\left(\frac{w_{0}^{4}}{16 \beta^{4}}+\frac{y}{\beta}\right) \\ &-\left[\frac{5 w_{0}^{4}}{4 \alpha^{4}}+\frac{w_{0}^{10}}{32 \alpha^{10}}+\frac{w_{0}^{6} x}{4 \alpha^{7}}+\frac{i w_{0}^{2}}{2 \alpha \beta}\left(\frac{5 w_{0}^{2}}{2 \beta^{2}}+\frac{w_{0}^{8}}{16 \alpha^{6} \beta^{2}}+\frac{w_{0}^{8}}{16 \beta^{8}}+\frac{w_{0}^{4} x}{4 \alpha^{3} \beta^{2}}+\frac{3 w_{0}^{4} y}{4 \beta^{5}}\right)\right] \\ & \times A i^{\prime}\left(\frac{w_{0}^{4}}{16 \alpha^{4}}+\frac{x}{\alpha}\right) A i\left(\frac{w_{0}^{4}}{16 \beta^{4}}+\frac{y}{\beta}\right)+\left[\frac{5 w_{0}^{4}}{4 \beta^{4}}+\frac{w_{0}^{10}}{32 \beta^{10}}+\frac{w_{0}^{6} y}{4 \beta^{7}}-\frac{i w_{0}^{2}}{2 \alpha \beta}\left(\frac{5 w_{0}^{2}}{2 \alpha^{2}}\right.\right.\\ &\left.\left.+\frac{w_{0}^{8}}{16 \alpha^{8}}+\frac{w_{0}^{8}}{16 \alpha^{2} \beta^{6}}+\frac{3 w_{0}^{4} x}{4 \alpha^{5}}+\frac{w_{0}^{4} y}{4 \alpha^{2} \beta^{3}}\right)\right] A i\left(\frac{w_{0}^{4}}{16 \alpha^{4}}+\frac{x}{\alpha}\right) A i^{\prime}\left(\frac{w_{0}^{4}}{16 \beta^{4}}+\frac{y}{\beta}\right) \\ &\left.-\frac{i w_{0}^{2}}{2 \alpha \beta}\left(6+\frac{w_{0}^{6}}{4 \alpha^{6}}+\frac{w_{0}^{6}}{4 \beta^{6}}+\frac{w_{0}^{2} x}{\alpha^{3}}+\frac{w_{0}^{2} y}{\beta^{3}}\right) A i^{\prime}\left(\frac{w_{0}^{4}}{16 \alpha^{4}}+\frac{x}{\alpha}\right) A i^{\prime}\left(\frac{w_{0}^{4}}{16 \beta^{4}}+\frac{y}{\beta}\right)\right\}, \end{aligned}$$
where Ai′(·) is the Airyprime function and is given by
$$Ai^{\prime}(x) = \frac{i}{{2\pi }}\int_{ - \infty }^\infty {u\exp \left( {\frac{{i{u^3}}}{3} + ixu} \right)} du.$$

The Airyprime function can be used directly in MATLAB. The outcomes of the Airy transform of LG01, LG02, LG11, and LG12 modes are composite beams including the Airy and Airyprime functions. As shown in Eq. (1), the optical field of the Laguerre-Gaussian beam has the structure of exp(imφ0) in the source plane. After the Airy transform, however, this structure of exp(imφ) disappears, which cannot be found in Eqs. (7)-(10). For ease of understanding, let’s discuss Eqs. (7) and (8). The optical field of LG01 and LG02 modes in the source plane can be rewritten as

$$E_{01}^{}({x_0},{y_0},0) = \frac{{ - \sqrt 2 {w_0}}}{2}\left[ {\frac{{\partial G({x_0},{y_0})}}{{\partial {x_0}}} + i\frac{{\partial G({x_0},{y_0})}}{{\partial {y_0}}}} \right],$$
$$E_{02}^{}({x_0},{y_0},0) = \frac{{w_0^2}}{2}\left[ {\frac{{{\partial^2}G({x_0},{y_0})}}{{\partial x_0^2}} - \frac{{{\partial^2}G({x_0},{y_0})}}{{\partial y_0^2}} + 2i\frac{{{\partial^2}G({x_0},{y_0})}}{{\partial {x_0}\partial {y_0}}}} \right],$$
where$G({x_0},{y_0}) = \exp ( - \rho _0^2/w_0^2)$. We note that $Ai^{\prime\prime}(x) = xAi(x)$. Therefore, the Airy transform of a Gaussian function is an Airy function. The Airy transform of the first derivative of a Gaussian function is the sum of Airy and Airyprime functions with different weights. The Airy transform of the second derivative of a Gaussian function is the sum of Airy function, Airyprime function, and the second derivative of Airy function with different weights. Equations (7) and (8) are the concrete presentations of the above conclusions. In order to save space, moreover, Eqs. (7) and (8) are written to be very compact. Similarly, Eqs. (9) and (10) can be analyzed by the same procedure. When the control parameters α and β tend to zero, Eq. (2) reduces to be
$$\begin{array}{l} E_{nm}^{}(x,y) = \int_{ - \infty }^\infty {\int_{ - \infty }^\infty {E_{nm}^{}({x_0},{y_0},0)} } \delta ({x_0} - x)\delta ({y_0} - y)d{x_0}d{y_0}\\ \begin{array}{{ccc}} {}&{}&{\begin{array}{{cc}} {}&{ = {{\left( {\frac{{\sqrt 2 \rho }}{{{w_0}}}} \right)}^m}L_n^m\left( {\frac{{2\rho_{}^2}}{{w_0^2}}} \right)\exp \left( { - \frac{{\rho_{}^2}}{{w_0^2}}\textrm{ + }im\varphi } \right),} \end{array}} \end{array} \end{array}$$
where δ(.) is a Dirac function, $\rho = {(x_{}^2\textrm{ + }y_{}^2)^{1/2}}$, and $\varphi = {\tan ^{ - 1}}(y/x)$. The light intensity of a Laguerre-Gaussian beam passing through an Airy transform optical system is given by
$${I_{nm}}(x,y) = {|{{E_{nm}}(x,y)} |^2}.$$
The Airy transform of other Laguerre-Gaussian beams can be obtained by using the similar derivation.

3. Numerical results

The characteristics of the Airy transform of four Laguerre-Gaussian modes, which are LG01, LG02, LG11, and LG12 modes, are studied numerically in detail in this section. First, the density plots of the normalized intensity distributions of four modes in the source plane z = 0 are shown in Fig. 1. The Gaussian waist w0 is fixed at 3.6 mm in all subsequent numerical calculations. The beam spot of Laguerre-Gaussian beams is dark hollow and has n + 1 layers, and each layer takes on annular distribution. Moreover, there is a dark ring between each layer. Figure 2 shows the effect of the signs of the control parameters α and β on the normalized intensity distribution of LG01 mode passing through Airy transform optical systems. The locations in the x- and y-directions of a transformed Laguerre-Gaussian beam are determined by the signs of the control parameters α and β, respectively. When the control parameter α is positive, the beam spot is located at the left half of the symmetrical coordinate system. Otherwise, the beam spot is located at the right half of the symmetrical coordinate system. When the control parameter β is positive, the beam spot is located at the lower half of the symmetrical coordinate system. If the parameter control β is negative, the beam spot is located at the upper half of the symmetrical coordinate system. Therefore, the location of the beam spot in quadrants depending on the combination of signs of the control parameters α and β obviously follows from the influence of the signs separately, which is described in the previous lines. Moreover, the signs of the control parameters α and β don’t affect the intensity, which is a priori clear from Eq. (2).

 figure: Fig. 1.

Fig. 1. Distribution of normalized intensity of Laguerre-Gaussian beams in the initial source plane z = 0. (a) LG01 mode; (b) LG02 mode; (c) LG11 mode; (d) LG12 mode.

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 figure: Fig. 2.

Fig. 2. Distribution of normalized intensity of the LG01 mode passing through Airy transform optical systems with different signs of α and β. (a) α=β=6 mm; (b) α=-6 mm and β=6 mm; (c) α=6 mm and β=-6 mm; (d) α=β=-6 mm.

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 figure: Fig. 3.

Fig. 3. Distribution of normalized intensity of the LG01 mode passing through different Airy transform optical systems. (a) α=β=6 mm; (b) α=β=4 mm; (c) α=β=2.5 mm; (d) α=β=1 mm.

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 figure: Fig. 4.

Fig. 4. Distribution of normalized intensity of the LG02 mode passing through different Airy transform optical systems. (a) α=β=6 mm; (b) α=β=4 mm; (c) α=β=2.5 mm; (d) α=β=1 mm.

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 figure: Fig. 5.

Fig. 5. Distribution of normalized intensity of the LG11 mode passing through different Airy transform optical systems. (a) α=β=6 mm; (b) α=β=4 mm; (c) α=β=2.5 mm; (d) α=β=1 mm.

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 figure: Fig. 6.

Fig. 6. Distribution of normalized intensity of the LG12 mode passing through different Airy transform optical systems. (a) α=β=6 mm; (b) α=β=4 mm; (c) α=β=2.5 mm; (d) α=β=1 mm.

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Now, we focus on the influence of the sizes of the control parameters α and β on the normalized intensity distribution of Laguerre-Gaussian beams passing through Airy transform optical systems. Figures 3–6 illustrate the Airy transform of LG01, LG02, LG11, and LG12 modes with different values of α and β, respectively. Here the symmetric case of α=β is considered. Although the Laguerre-Gaussian modes are different, the influence of the sizes of the control parameters α and β on the normalized intensity distributions of Laguerre-Gaussian beams passing through Airy transform optical systems is the same. When the absolute values of the control parameters α and β decrease, the number of side lobes in the beam spot diminishes, resulting in a decrease in the beam spot size. When α and β decrease to 2.5 mm, the number of the side lobes in the beam spot is apparently less. When α andβ are reduced to 1 mm, the beam spot is close to but still different from its initial intensity distribution. When the control parameters α and β tend to zero, the beam spot is almost the same as the corresponding initial intensity distribution. When the absolute values of the control parameters α and β are much larger than the Gaussian waist w0, the Airy transform of LG01, LG02, LG11, and LG12 modes shows much Airy features. When the absolute values of the control parameters α and β are much smaller than the Gaussian waist w0, such as α=β=1 mm, the Airy feature decreases with the decrease of the absolute values of the control parameters α and β until the Airy feature almost disappears and the Laguerre-Gaussian feature appears. By changing the control parameters α and β, therefore, the Airy transform of a Laguerre-Gaussian beam can become a special optical beam with rich performance. The physical explanation of the influence of the control parameters α and β on intensity distribution is as follow. The Airy transform has twice Fourier transforms, and there is a phase loading process after the first Fourier transform. Moreover, the control parameters α and β determine the loaded phase. As a result, the control parameters α and β affect the intensity distribution in the output plane.

The influences of the radial and the angular mode numbers n and m on the normalized intensity distributions of Laguerre-Gaussian beams passing through an Airy transform optical system can also be elucidated by Figs. 3–6. The effect of the radial mode number n on the normalized intensity distributions of Laguerre-Gaussian beams passing through the same Airy transform optical system is more significant, while the influence of the angular mode number m is relatively slight. When the radial mode number n is fixed, the beam spots of different Laguerre-Gaussian beams passing through the same Airy transform optical system are similar to some extent. However, the angular mode number m affects the details of the beam spots, which also lead to the different beam spots for the different angular mode number.

4. Experimental results

In this section, we carry out the experiment for the Airy transform of Laguerre-Gaussian beams. The experimental setup is illustrated in Fig. 7. A linearly polarized Gaussian beam (wavelength λ=633 nm) generated by a He-Ne laser is first expanded by a beam expander (BE), and then goes toward a reflective-mode spatial light modulator (SLM1, Holoeye, Pluto-VIS) which acts as a phase screen controlled by the computer program. In order to generate LGnm mode, the phase pattern $\theta (x,y) = m\varphi + H[L_n^m(x,y)]$, where H denotes a unit step-function, is first calculated. Then, a blazed phase grating with phase shift k0x with k0 being the spatial frequency is imparted the phase pattern θ(x, y). As a result, the generated phase grating is loaded to SLM1. The inset Fig. 7(a) shows the typical phase grating for generation of the LG11 mode. When the incident beam is reflected from SLM1, the first diffraction order after SLM1 is regarded as the Laguerre-Gaussian beam with prescribed mode number n and m, and is selected out by a circular aperture (not shown in the schematics of Fig. 7). The generated Laguerre-Gaussian beam then arrives at SLM2 (Holoeye 2008), which is use to impose a cubic phase of the incident beam. The phase screen is computed as the interference of the cubic phase exp[ik3(α3x3 +β3y3)/3f3] with a plane wave exp(ik1x) where k1 is the spatial frequency. The phase pattern is shown in the inset Fig. 7(b). After the SLM2, the first diffraction order whose electric field is Eout(r)=Ein(r)exp[ik3(α3x3 +β3y3)/3f3] is selected out. A lens with focal length f = 250 mm is placed at the distance f of SLM2 to perform the Fourier transform of the first diffraction order. Finally, a beam profile analysis (BPA) is located at the rear focal plane of the lens to measure the spectral density of the output beam.

 figure: Fig. 7.

Fig. 7. Schematic diagram of the experimental setup for generation of the Laguerre-Gaussian beam as well as for the measurement of its spectral density after the Airy transform. BE: beam expander; SLM1 and SLM2: spatial light modulator; BPA: beam profile analysis.

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The difference between the experimental setup and the theoretical calculation is that there is no Fourier transform lens placed between SLM1 and SLM2 in the experiment. In theory, the first step is the Fourier transform of a LG beam, and then the cubic phase is loaded on the Fourier transform surface, and finally another Fourier transform is performed. If a Fourier transform lens is placed between SLM1 and SLM2 in the experiment, the beam spot on SLM2 will be very small. If the beam spot size on SLM2 is small, the error of loading the cubic phase will increase due to the limitation of the pixel size of SLM2. Fortunately, it is known that the LG modes are exact solutions of the paraxial wave equation, which means that the Fourier transform of the LG modes is still the LG modes, i.e., the mathematical expression of the electric field of the LG modes remains unchanged after Fourier transform, except for the reduction of the beam spot size. Therefore, the above experimental setup can still perform Airy transform of the LG beams, whether the Fourier transform lens between SLM1 and SLM2 is placed or not. It should be emphasized that the Airy transform using the above experimental setup is only applicable to optical beams which are exact solutions of the paraxial wave equation.

In our experiment, the Gaussian waist w0 is chosen as 0.5 mm. Figures 8–13 correspond to the experimental results of Figs. 1–6, respectively. It is obvious that the experimental results are in good agreement with the theoretical simulation results in the above section. As shown in Fig. 8, the quality of the formed initial Laguerre-Gaussian mode is slightly poor. This is due to the fact that only the mode phase is used on SLM1 without taking into account the amplitude distribution. In order to improve the quality of mode formation, various coding methods including the method of encoded binary diffractive element [67] and the superpixel-based spatial amplitude and phase modulation method [68] will be used in our future experiments.

 figure: Fig. 8.

Fig. 8. Experimental distribution of normalized intensity of Laguerre-Gaussian beams in the initial source plane z = 0. (a) LG01 mode; (b) LG02 mode; (c) LG11 mode; (d) LG12 mode.

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 figure: Fig. 9.

Fig. 9. Experimental distribution of normalized intensity of the LG01 mode passing through different Airy transform optical systems. (a) α=β=6 mm; (b) α=-6 mm and β=6 mm; (c) α=6 mm and β=-6 mm; (d) α=β=-6 mm.

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 figure: Fig. 10.

Fig. 10. Experimental distribution of normalized intensity of the LG01 mode passing through different Airy transform optical systems. (a) α=β=6 mm; (b) α=β=4 mm; (c) α=β=2.5 mm; (d) α=β=1 mm.

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 figure: Fig. 11.

Fig. 11. Experimental distribution of normalized intensity of the LG02 mode passing through different Airy transform optical systems. (a) α=β=6 mm; (b) α=β=4 mm; (c) α=β=2.5 mm; (d) α=β=1 mm.

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 figure: Fig. 12.

Fig. 12. Experimental distribution of normalized intensity of the LG11 mode passing through different Airy transform optical systems. (a) α=β=6 mm; (b) α=β=4 mm; (c) α=β=2.5 mm; (d) α=β=1 mm.

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 figure: Fig. 13.

Fig. 13. Experimental distribution of normalized intensity of the LG12 mode passing through different Airy transform optical systems. (a) α=β=6 mm; (b) α=β=4 mm; (c) α=β=2.5 mm; (d) α=β=1 mm.

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5. Summary

The Airy transform of Laguerre-Gaussian beams is investigated. Though generally analytic formula of Laguerre-Gaussian beams passing through an Airy transform optical system cannot be presented, the analytical expression of the Airy transform of each Laguerre-Gauss beam can be obtained. As an example, the analytic formulas of the Airy transform of LG01, LG02, LG11, and LG12 modes are derived, respectively.

The normalized intensity distributions of LG01, LG02, LG11, and LG12 modes are depicted in the initial source plane z = 0. The beam spot of Laguerre-Gaussian beams is dark hollow and has n + 1 layers. Moreover, each layer has a ring distribution, and there is a dark ring between each layer. The Airy transform of LG01, LG02, LG11, and LG12 modes is numerically and experimentally investigated, respectively.

The effects of the signs of the control parameters α and β on the normalized intensity distribution of a Laguerre-Gaussian beam passing through Airy transform optical systems are first investigated. The signs of the control parameters α and β only affect the location of the beam spot. Then, the effects of the sizes of the control parameters α and β on the normalized intensity distribution of a Laguerre-Gaussian beam passing through Airy transform optical systems are examined. As the absolute values of the control parameters α and β decrease, the number of the side lobes in the beam spot diminishes, and the beam spot size also reduces. When the absolute values of the control parameters α and β are much larger than the Gaussian waist w0, the Airy transform of Laguerre-Gaussian beams shows much Airy feature. With decreasing the absolute values of the control parameters α and β, the Airy feature decreases, and the Laguerre-Gaussian characteristic is strengthened. When the control parameters α and β tend to zero, therefore, the beam spot tends to the corresponding initial intensity distribution. The experimental results are consistent with the theoretical simulations.

Although only the Airy transform of LG01, LG02, LG11, and LG12 modes is involved here, the Airy transform of other Laguerre-Gaussian beams can be investigated in the same way. The properties of the Airy transform of Laguerre-Gaussian beams are well demonstrated by this research. By performing the Airy transform of Laguerre-Gaussian beams, special optical beams which are combination of several Airy and Airyprime beams can be obtained. By altering the control parameters α and β, the performance of these special optical beams is diverse, which may have potential applications. Due to the mixture of Laguerre-Gaussian and Airy features, these special optical beams can be used not only in the application field of Laguerre-Gaussian beams, but also in the application field of Airy beams. Therefore, this research provides another approach to obtain special optical beams and expands the application of Laguerre-Gaussian beams. Moreover, the Airy transform whose kernel is an Airy function can be extended to the fractional Airy transform whose kernel is a generalized Airy function [69], which will be discussed elsewhere.

Funding

National Natural Science Foundation of China (11974313, 11874046).

Disclosures

The authors declare no conflicts of interest.

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Figures (13)
Fig. 1.
Fig. 1. Distribution of normalized intensity of Laguerre-Gaussian beams in the initial source plane z = 0. (a) LG01 mode; (b) LG02 mode; (c) LG11 mode; (d) LG12 mode.

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Fig. 2.
Fig. 2. Distribution of normalized intensity of the LG01 mode passing through Airy transform optical systems with different signs of α and β. (a) α=β=6 mm; (b) α=-6 mm and β=6 mm; (c) α=6 mm and β=-6 mm; (d) α=β=-6 mm.

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Fig. 3.
Fig. 3. Distribution of normalized intensity of the LG01 mode passing through different Airy transform optical systems. (a) α=β=6 mm; (b) α=β=4 mm; (c) α=β=2.5 mm; (d) α=β=1 mm.

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Fig. 4.
Fig. 4. Distribution of normalized intensity of the LG02 mode passing through different Airy transform optical systems. (a) α=β=6 mm; (b) α=β=4 mm; (c) α=β=2.5 mm; (d) α=β=1 mm.

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Fig. 5.
Fig. 5. Distribution of normalized intensity of the LG11 mode passing through different Airy transform optical systems. (a) α=β=6 mm; (b) α=β=4 mm; (c) α=β=2.5 mm; (d) α=β=1 mm.

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Fig. 6.
Fig. 6. Distribution of normalized intensity of the LG12 mode passing through different Airy transform optical systems. (a) α=β=6 mm; (b) α=β=4 mm; (c) α=β=2.5 mm; (d) α=β=1 mm.

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Fig. 7.
Fig. 7. Schematic diagram of the experimental setup for generation of the Laguerre-Gaussian beam as well as for the measurement of its spectral density after the Airy transform. BE: beam expander; SLM1 and SLM2: spatial light modulator; BPA: beam profile analysis.

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Fig. 8.
Fig. 8. Experimental distribution of normalized intensity of Laguerre-Gaussian beams in the initial source plane z = 0. (a) LG01 mode; (b) LG02 mode; (c) LG11 mode; (d) LG12 mode.

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Fig. 9.
Fig. 9. Experimental distribution of normalized intensity of the LG01 mode passing through different Airy transform optical systems. (a) α=β=6 mm; (b) α=-6 mm and β=6 mm; (c) α=6 mm and β=-6 mm; (d) α=β=-6 mm.

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Fig. 10.
Fig. 10. Experimental distribution of normalized intensity of the LG01 mode passing through different Airy transform optical systems. (a) α=β=6 mm; (b) α=β=4 mm; (c) α=β=2.5 mm; (d) α=β=1 mm.

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Fig. 11.
Fig. 11. Experimental distribution of normalized intensity of the LG02 mode passing through different Airy transform optical systems. (a) α=β=6 mm; (b) α=β=4 mm; (c) α=β=2.5 mm; (d) α=β=1 mm.

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Fig. 12.
Fig. 12. Experimental distribution of normalized intensity of the LG11 mode passing through different Airy transform optical systems. (a) α=β=6 mm; (b) α=β=4 mm; (c) α=β=2.5 mm; (d) α=β=1 mm.

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Fig. 13.
Fig. 13. Experimental distribution of normalized intensity of the LG12 mode passing through different Airy transform optical systems. (a) α=β=6 mm; (b) α=β=4 mm; (c) α=β=2.5 mm; (d) α=β=1 mm.

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Equations (15)

Equations on this page are rendered with MathJax. Learn more.

E n m ( x 0 , y 0 , 0 ) = ( 2 ρ 0 w 0 ) m L n m ( 2 ρ 0 2 w 0 2 ) exp ⁡ ( − ρ 0 2 w 0 2  +  i m φ 0 ) ,
E n m ( x , y ) = 1 | α β | ∫ − ∞ ∞ ∫ − ∞ ∞ E n m ( x 0 , y 0 , 0 ) A i ( x − x 0 α ) A i ( y − y 0 β ) d x 0 d y 0 ,
A i ( x ) = 1 2 π ∫ − ∞ ∞ exp ⁡ ( i u 3 3 + i x u ) d u .
L n m ( x ) = ∑ l = 0 n ( − 1 ) l ( n + m ) ! x l ( m + l ) ! ( n − l ) ! l ! ,
∫ − ∞ ∞ exp ⁡ ( − b 2 x 2 − s x ) d x = π b exp ⁡ ( s 2 4 b 2 ) ,
∫ − ∞ ∞ exp ⁡ ( i u 3 3 + i p u 2 + i q u ) d u  =  2 π exp ⁡ [ i p ( 2 p 2 3 − q ) ] A i ( q − p 2 ) ,
E 01 ( x , y ) = − 2 π w 0 3 2 | α ‖ β | exp ⁡ ( w 0 6 96 α 6 + w 0 6 96 β 6 + w 0 2 x 4 α 3 + w 0 2 y 4 β 3 ) [ ( w 0 2 4 α 3 + i w 0 2 4 β 3 ) × A i ( w 0 4 16 α 4 + x α ) A i ( w 0 4 16 β 4 + y β ) + 1 α A i ′ ( w 0 4 16 α 4 + x α ) × A i ( w 0 4 16 β 4 + y β ) + i β A i ( w 0 4 16 α 4 + x α ) A i ′ ( w 0 4 16 β 4 + y β ) ] ,
E 02 ( x , y ) = π w 0 2 2 | α | | β | exp ⁡ ( w 0 6 96 α 6 + w 0 6 96 β 6 + w 0 2 x 4 α 3 + w 0 2 y 4 β 3 ) [ ( w 0 6 8 α 6 − w 0 6 8 β 6 + i w 0 6 8 α 3 β 3 + w 0 2 x α 3 − w 0 2 y β 3 ) A i ( w 0 4 16 α 4 + x α ) A i ( w 0 4 16 β 4 + y β ) + ( w 0 4 2 α 4 + i w 0 4 2 α β 3 ) × A i ′ ( w 0 4 16 α 4 + x α ) A i ( w 0 4 16 β 4 + y β ) − ( w 0 4 2 β 4 − i w 0 4 2 α 3 β ) A i ( w 0 4 16 α 4 + x α ) × A i ′ ( w 0 4 16 β 4 + y β ) + 2 i w 0 2 α β A i ′ ( w 0 4 16 α 4 + x α ) A i ′ ( w 0 4 16 β 4 + y β ) ] ,
E 11 ( x , y ) = 2 π w 0 3 4 | α | | β | exp ⁡ ( w 0 6 96 α 6 + w 0 6 96 β 6 + w 0 2 x 4 α 3 + w 0 2 y 4 β 3 ) [ ( 2 w 0 2 α 3 + w 0 8 16 α 9 + w 0 8 32 α 3 β 6 + 2 i w 0 2 β 3 + i w 0 8 16 β 9 + i w 0 8 32 α 6 β 3 + 3 w 0 4 x 4 α 6 + w 0 4 y 4 α 3 β 3 + i w 0 4 x 4 α 3 β 3 + 3 i w 0 4 y 4 β 6 ) × A i ( w 0 4 16 α 4 + x α ) A i ( w 0 4 16 β 4 + y β ) + ( 4 α + w 0 6 4 α 7 + w 0 6 8 α β 6 + i w 0 6 8 α 4 β 3 + w 0 2 x α 4 + w 0 2 y α β 3 ) A i ′ ( w 0 4 16 α 4 + x α ) A i ( w 0 4 16 β 4 + y β ) + ( w 0 6 8 α 3 β 4 + 4 i β + i w 0 6 8 α 6 β + i w 0 6 4 β 7 + i w 0 2 x α 3 β + i w 0 2 y β 4 ) A i ( w 0 4 16 α 4 + x α ) A i ′ ( w 0 4 16 β 4 + y β ) + ( w 0 4 2 α β 4 + i w 0 4 2 α 4 β ) A i ′ ( w 0 4 16 α 4 + x α ) A i ′ ( w 0 4 16 β 4 + y β ) ] ,
E 12 ( x , y ) = π w 0 2 | α | | β | exp ⁡ ( w 0 6 96 α 6 + w 0 6 96 β 6 + w 0 2 x 4 α 3 + w 0 2 y 4 β 3 ) { [ [ − 7 w 0 6 16 α 6 − w 0 12 128 α 12 + 7 w 0 6 16 β 6 + w 0 12 128 β 12 − 3 w 0 2 x 2 α 3 − w 0 8 x 8 α 9 + 3 w 0 2 y 2 β 3 + w 0 8 y 8 β 9 − w 0 4 x 2 4 α 6 + w 0 4 y 2 4 β 6 − i w 0 2 2 α β ( 7 w 0 4 8 α 2 β 2 + w 0 10 64 α 8 β 2 + w 0 10 64 α 2 β 8 + 3 w 0 6 x 16 α 5 β 2 + 3 w 0 6 y 16 α 2 β 5 ) ] A i ( w 0 4 16 α 4 + x α ) A i ( w 0 4 16 β 4 + y β ) − [ 5 w 0 4 4 α 4 + w 0 10 32 α 10 + w 0 6 x 4 α 7 + i w 0 2 2 α β ( 5 w 0 2 2 β 2 + w 0 8 16 α 6 β 2 + w 0 8 16 β 8 + w 0 4 x 4 α 3 β 2 + 3 w 0 4 y 4 β 5 ) ] × A i ′ ( w 0 4 16 α 4 + x α ) A i ( w 0 4 16 β 4 + y β ) + [ 5 w 0 4 4 β 4 + w 0 10 32 β 10 + w 0 6 y 4 β 7 − i w 0 2 2 α β ( 5 w 0 2 2 α 2 + w 0 8 16 α 8 + w 0 8 16 α 2 β 6 + 3 w 0 4 x 4 α 5 + w 0 4 y 4 α 2 β 3 ) ] A i ( w 0 4 16 α 4 + x α ) A i ′ ( w 0 4 16 β 4 + y β ) − i w 0 2 2 α β ( 6 + w 0 6 4 α 6 + w 0 6 4 β 6 + w 0 2 x α 3 + w 0 2 y β 3 ) A i ′ ( w 0 4 16 α 4 + x α ) A i ′ ( w 0 4 16 β 4 + y β ) } ,
A i ′ ( x ) = i 2 π ∫ − ∞ ∞ u exp ⁡ ( i u 3 3 + i x u ) d u .
E 01 ( x 0 , y 0 , 0 ) = − 2 w 0 2 [ ∂ G ( x 0 , y 0 ) ∂ x 0 + i ∂ G ( x 0 , y 0 ) ∂ y 0 ] ,
E 02 ( x 0 , y 0 , 0 ) = w 0 2 2 [ ∂ 2 G ( x 0 , y 0 ) ∂ x 0 2 − ∂ 2 G ( x 0 , y 0 ) ∂ y 0 2 + 2 i ∂ 2 G ( x 0 , y 0 ) ∂ x 0 ∂ y 0 ] ,
E n m ( x , y ) = ∫ − ∞ ∞ ∫ − ∞ ∞ E n m ( x 0 , y 0 , 0 ) δ ( x 0 − x ) δ ( y 0 − y ) d x 0 d y 0 = ( 2 ρ w 0 ) m L n m ( 2 ρ 2 w 0 2 ) exp ⁡ ( − ρ 2 w 0 2  +  i m φ ) ,
I n m ( x , y ) = | E n m ( x , y ) | 2 .
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