Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group
Journal Home About Issues in Progress Current Issue All Issues Feature Issues

Transformation of a Hermite-Gaussian beam by an Airy transform optical system

Guoquan Zhou, Fei Wang, Ruipin Chen, and Xia Li

Open Access Open Access

Abstract

Analytical expression of the Airy transform of an arbitrary Hermite-Gaussian beam is derived. The optical field in the x-direction of the Airy transform of Hermite-Gaussian beams with transverse mode number m is the sum of the zero-order derivative to mth-order derivative of the Airy function with different weight coefficients. The analytical expressions of the centre of gravity and the beam spot size of an arbitrary Hermite-Gaussian beam passing through an Airy transform optical system are also presented, which are very concise. Because the Airy transform of a Hermite-Gaussian beam has the same evolution law in the two transverse directions, only the effects of the control parameter α and the transverse mode number m on the normalized intensity distribution, the centre of gravity, and the beam spot size in the x-direction are theoretically investigated, respectively. The Airy transform of Hermite-Gaussian beams is also realized in the experiment. The influence of the control parameters on the normalized intensity distribution, the centre of gravity, and the beam spot size is experimentally investigated, respectively. The experimental results are consistent with the theoretical simulation results. When Hermite-Gaussian beams pass through an Airy transform optical system, the number of lobes may change, and the importance of lobes with the same status in the input plane may become different. By using the Airy transform of Hermite-Gaussian beams, the practical applications of Hermite-Gaussian beams can be extended.

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

1. Introduction

A Hermite-Gaussian beam is a higher-order mode with Cartesian symmetry [1]. Hermite-Gaussian beam can be excited by various methods, for instance, it has been excited in an end-pumped Nd: YVO4 laser by using an improved gain-shaping method [2]. By means of dual-off-axis pumping scheme, wavelength-tunable Hermite-Gaussian beams have been generated in Yb: CaGdAlO4 laser [3]. The properties of Hermite-Gaussian beams have been widely investigated. The diffraction of Hermite-Gaussian beams through a slit and a finite grating has been investigated [4,5]. The influence of the geometrical phase on the Hermite-Gaussian beams has been analyzed when they pass through transversal and parallel dielectric blocks [6]. The paraxial propagation of Hermite-Gaussian beams through the aligned and misaligned ABCD optical systems [7], atmospheric turbulence [8], complex optical system with apertures [9], and linear refractive index medium [10] have been investigated, respectively. The analytical distortion parameter of steady sate thermal blooming of Hermite-Gaussian beams in atmosphere has been derived [11]. The average intensity distribution of radial phased-locked partially coherent Hermite-Gaussian beams in oceanic turbulence has been evaluated [12]. The scattering properties of aerosol particles by a Hermite-Gaussian beam in marine atmosphere has been assessed [13]. The propagation properties of a partially coherent Hermite-Gaussian beam in uniaxial crystals orthogonal to the optical axis have been examined [14]. The Kerr effect on the propagation of Hermite-Gaussian beams has been weighed [15]. A group theoretical approach to the paraxial propagation of Hermite-Gaussian beams has been presented based on the factorization method [16]. Exact analytical expression of the beam propagation factor of a truncated Hermite-Gaussian beam has been derived [17]. In addition, the nonparaxial propagation of Hermite-Gaussian beams received attention [18]. Convolution neural networks were implemented to accurately detect the lowest 21 unique Hermite-Gaussian beams [19]. Atom and light-absorbing particles can be optically trapped and manipulated by a Hermite-Gaussian beam [20,21]. Second harmonics of a Hermite-Gaussian beam have been generated in collisionless and relativistic plasmas, respectively [22,23]. Electron acceleration by a circularly polarized Hermite-Gaussian beam has been realized in the plasma [24]. Based on a Wigner-function representation of Hermite-Gaussian beams, a new method of coherent mode decomposition has been proposed [25]. The inter-conversion between Hermite-Gaussian and Laguerre-Gaussian beams has been achieved [26]. The information capacity associated with measurements in Hermite-Gaussian and Laguerre-Gaussian modal basis in an optical system of finite aperture has been quantitatively compared [27]. Also, Hermite-Gaussian beams have been extended to elegant Hermite-Gaussian and Lorentz-Hermite-Gaussian beams [28,29].

To date, an Airy beam becomes a popular beam due to its advantages of non-diffraction, self-healing, and transverse acceleration [30–36]. The fascinating properties of Airy beams have led to widespread applications including optical particle clearing [37], curved plasma channel [38], spatiotemporal light bullets [39], laser microprocessing [40], and super-resolution imaging [41]. Although many methods for the generation of Airy beams have been proposed [42–49], the simplest and most effective method is the method of spatial light modulator [30]. The Airy transform is just based on the method of spatial light modulator. The famous Airy beam can be obtained by Airy transform of Gaussian beam [50]. The Airy transforms of flat-topped Gaussian beams, hyperbolic-cosine Gaussian beams, and double-half inverse Gaussian hollow beams have been reported [51–53], respectively. All these above beams used to perform the Airy transform can be obtained by superposition of Gaussian beams. Recently, the Airy transform of Laguerre-Gaussian beams, which are higher-order modes of axially symmetric laser cavities with spherical mirrors, has been realized theoretically and experimentally [54]. Unfortunately, the general analytic formula of Laguerre-Gaussian beams passing through an Airy transform optical system has not been derived. Moreover, only the effect of the control parameters of the Airy transform optical system on the normalized intensity distribution has been evaluated [54]. To our best knowledge, however, the Airy transform of a Hermite-Gaussian beam, which is another classical higher-order beam model, has not been reported yet. In this paper, therefore, the Airy transform of a Hermite-Gaussian beam is investigated. A general analytical formula for the Airy transform of a Hermite-Gaussian beam is derived. Moreover, the effect of the control parameters of the Airy transform optical system on the centre of gravity and the beam spot size of the Hermite-Gaussian beam passing through an Airy transform optical system is also investigated, which is not included in Ref. [54].

2. Transformation of a Hermite-Gaussian beam by an Airy transform optical system

The Airy transform optical system is schematized in Fig. 1. The optical system consists of two thin lenses with focal length f and a spatial light modulator (SLM). The SLM is used to impose to the input optical beam with a phase modulation. Therefore, the Airy transform has undergone two Fourier transforms, and there is a phase loading process after the first Fourier transform. The z-axis is the direction of beam propagation. The Hermite-Gaussian beam in the input plane z=0 is described by

$${E_{mn}}({x_0},{y_0}) = {E_m}({x_0},0){E_n}({y_0},0) = {H_m}\left( {\frac{{\sqrt 2 {x_0}}}{{{w_0}}}} \right){H_n}\left( {\frac{{\sqrt 2 {y_0}}}{{{w_0}}}} \right)\exp \left( { - \frac{{x_0^2 + y_0^2}}{{w_0^2}}} \right),$$
where x0 and y0 are the transverse coordinates in the input plane. w0 is the Gaussian waist. Hj is the jth-order Hermite polynomial. m and n are the transverse mode numbers in the x0- and y0-directions, respectively. The optical field of a Hermite-Gaussian beam passing through an Airy transform optical system is given by [50]:
$${E_{mn}}(x,y) = {E_m}(x){E_n}(y) = \frac{1}{{|{\alpha \beta } |}}\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {{E_{mn}}({x_0},{y_0})} } Ai\left( {\frac{{x - {x_0}}}{\alpha }} \right)Ai\left( {\frac{{y - {y_0}}}{\beta }} \right)d{x_0}d{y_0},$$
where α and β are the control parameters of the Airy transform optical system in the x- and y-directions, respectively. x and y are two transverse coordinates in the output plane. Ai(·) is the Airy function. Since the optical field in the x- and y-directions is separable, we first derive the optical field in the x-direction. For the optical field in the y-direction, it can be obtained by replacing the parameter of the optical field in the x-direction. In the process of derivation, the following mathematical integral formulas are used [55,56]:
$${H_m}(x) = \sum\limits_{l = 0}^{[m/2]} {\frac{{{{( - 1)}^l}{2^{m - 2l}}m!}}{{l!(m - 2l)!}}} {x^{m - 2l}},$$
$$\int_{ - \infty }^\infty {\exp \left( {\frac{{i{u^3}}}{3} + ip{u^2} + iqu} \right)} du\textrm{ = }2\pi \exp \left[ {ip\left( {\frac{{2{p^2}}}{3} - q} \right)} \right]Ai(q - {p^2}),$$
where [m/2] means taking the integer portion of m/2. Therefore, the analytical expression in the x-direction of the Airy transform of a Hermite-Gaussian beam is given by
$$\begin{array}{l} {E_m}(x) = \frac{{{i^m}{w_0}\alpha }}{{2\sqrt {\pi |\alpha |} }}\int_{ - \infty }^\infty {{H_m}\left( {\frac{{ - {w_0}\xi }}{{\sqrt 2 }}} \right)} \exp\left( {\frac{{i{\alpha^3}{\xi^3}}}{3} - \frac{{w_0^2{\xi^2}}}{4} + i\xi x} \right)d\xi \\ \begin{array}{ccc} {}&{}&\textrm{ = } \end{array}\frac{{{i^m}\sqrt \tau }}{{\sqrt \pi }}\int_{ - \infty }^\infty {\sum\limits_{l = 0}^{[m/2]} {\frac{{{{( - 1)}^{m - l}}m!}}{{l!(m - 2l)!}}} {{\left( {\frac{\alpha }{{|\alpha |}}2\sqrt {2\tau } } \right)}^{m - 2l}}} {u^{m - 2l}}\exp\left( {\frac{{i{u^3}}}{3} - \tau {u^2} + i\frac{x}{\alpha }u} \right)du\\ \begin{array}{ccc} {}&{}&\textrm{ = } \end{array}{i^m}2\sqrt {\pi \tau } \exp \left( {\frac{{2{\tau^3}}}{3} + \frac{{\tau x}}{\alpha }} \right)\sum\limits_{l = 0}^{[m/2]} {\frac{{{{( - 1)}^{m - l}}m!}}{{l!(m - 2l)!}}} {\left( {\frac{\alpha }{{|\alpha |}}2\sqrt {2\tau } } \right)^{m - 2l}}\sum\limits_{s = 0}^{m - 2l} {{C_{m - 2l,s}}} A{i^{(s)}}({x_1}), \end{array}$$
where $\tau = w_0^2/(4{\alpha ^2})$, ${x_1} = (x + \alpha {\tau ^2})/\alpha$, and Ai(s)(·) denotes the sth-order derivative of the Airy function. Here, the zero-order derivative of the Airy function is stipulated to be itself. The weight coefficient ${C_{m - 2l,s}}$ is determined by
$${C_{0,0}}\textrm{ = }1,$$
$${C_{1,0}}\textrm{ = } - i\tau {C_{0,0}},{C_{1,1}}\textrm{ = } - i{C_{0,0}},$$
$${C_{t,0}} ={-} i\tau {C_{t - 1,0}},{C_{t,s}} = ( - i\tau {C_{t - 1,s}} - i{C_{t - 1,s - 1}}),{C_{t,t}} ={-} i{C_{t - 1,t - 1}},t > 1,0 < s < t - 1.$$

 figure: Fig. 1.

Fig. 1. A schematic diagram of the Airy transform optical system.

Download Full Size | PDF

According to Eq. (5), the analytical optical fields in the x-direction of the first five Hermite-Gaussian beams passing through an Airy transform optical system are found to be

$${E_0}(x) = 2\sqrt {\pi \tau } \exp \left( {\frac{{2{\tau^3}}}{3} + \frac{{\tau x}}{\alpha }} \right)Ai({x_1}),$$
$${E_1}(x) = \frac{{ - 4\sqrt {2\pi } \alpha \tau }}{{|\alpha |}} \exp \left( {\frac{{2{\tau^3}}}{3} + \frac{{\tau x}}{\alpha }} \right)[\tau Ai({x_1}) + Ai^{\prime}({x_1})],$$
$$\begin{array}{l} {E_2}(x) = 4\sqrt {\pi \tau } \exp \left( {\frac{{2{\tau^3}}}{3} + \frac{{\tau x}}{\alpha }} \right)[(1 + 4{\tau ^3})Ai({x_1})\textrm{ + }8{\tau ^2}Ai^{\prime}({x_1}) + 4\tau A{i^{(2)}}({x_1})]\\ \begin{array}{{cc}} {}&{} \end{array} = 4\sqrt {\pi \tau } \exp \left( {\frac{{2{\tau^3}}}{3} + \frac{{\tau x}}{\alpha }} \right)\left[ {\left( {1 + 8{\tau^3} + \frac{{4\tau x}}{\alpha }} \right)Ai({x_1})\textrm{ + }8{\tau^2}Ai^{\prime}({x_1})} \right], \end{array}$$
$$\begin{array}{l} {E_3}(x)\textrm{ = }\frac{{ - 4\sqrt {2\pi } \alpha \tau }}{{|\alpha |}} \exp \left( {\frac{{2{\tau^3}}}{3} + \frac{{\tau x}}{\alpha }} \right)[(6\tau + 8{\tau ^4})Ai({x_1}) + (6 + 24{\tau ^3})Ai^{\prime}({x_1})\\ \begin{array}{{cc}} {}&{} \end{array} + 24{\tau ^2}A{i^{(2)}}({x_1}) + 8\tau A{i^{(3)}}({x_1})]\\ \begin{array}{{cc}} {}&{} \end{array}\textrm{ = }\frac{{ - 4\sqrt {2\pi } \alpha \tau }}{{|\alpha |}} \exp \left( {\frac{{2{\tau^3}}}{3} + \frac{{\tau x}}{\alpha }} \right)\left[ {\left( {14\tau + 32{\tau^4} + \frac{{24{\tau^2}x}}{\alpha }} \right)Ai({x_1})} \right.\\ \begin{array}{{cc}} {}&{} \end{array} + \left. {\left( {6 + 32{\tau^3} + \frac{{8\tau x}}{\alpha }} \right)Ai^{\prime}({x_1})} \right], \end{array}$$
$$\begin{array}{l} {E_4}(x) = 2\sqrt {\pi \tau } \exp \left( {\frac{{2{\tau^3}}}{3} + \frac{{\tau x}}{\alpha }} \right)[(12\textrm{ + }96{\tau ^3} + 64{\tau ^6})Ai({x_1}) + (192{\tau ^2} + 256{\tau ^5})Ai^{\prime}({x_1})\\ \begin{array}{ccc} {}&{}& + \end{array}(96\tau + 384{\tau ^4})A{i^{(2)}}({x_1}) + 256{\tau ^3}A{i^{(3)}}({x_1}) + 64{\tau ^2}A{i^{(4)}}({x_1})]\\ \begin{array}{cc} {}&{} \end{array} = 2\sqrt {\pi \tau } \exp \left( {\frac{{2{\tau^3}}}{3} + \frac{{\tau x}}{\alpha }} \right)[(12\textrm{ + }448{\tau^3} + 512{\tau^6}\textrm{ + }\frac{{96\tau x}}{\alpha } + \frac{{512{\tau^4}x}}{\alpha } + \frac{{64{\tau^2}{x^2}}}{{{\alpha^2}}})Ai({x_1})\\ \begin{array}{ccc} {}&{}& + \end{array}({320{\tau^2} + 512{\tau^5} + \frac{{256{\tau^3}x}}{\alpha }}) Ai^{\prime}({x_1})], \end{array}$$
where Ai′(·) is the Airyprime function. Therefore, the optical field in the x-direction of the Airy transform of Hermite-Gaussian beams with transverse mode number m is the sum of the zero-order derivative to mth-order derivative of the Airy function with different weight coefficients. The light intensity of a Hermite-Gaussian beam passing through an Airy transform optical system is defined by
$${I_{mn}}(x,y) = {I_m}(x){I_n}(y) = {|{{E_m}(x)} |^2}{|{{E_n}(y)} |^2}.$$
The centre of gravity in the x-direction of a Hermite-Gaussian beam passing through an Airy transform optical system is given by [57–61]:
$${X_{m,c}} = \frac{{\int_{ - \infty }^\infty {x{{|{{E_m}(x)} |}^2}dx} }}{{\int_{ - \infty }^\infty {{{|{{E_m}(x)} |}^2}dx} }}.$$
Therefore, the centre of gravity in the x-direction of a Hermite-Gaussian beam with m=1 passing through an Airy transform optical system can be calculated by
$${X_{1,c}} = \alpha \left( {\frac{{\int_{ - \infty }^\infty {{x_1} \exp (2\tau {x_1}){{[\tau Ai({x_1}) + Ai^{\prime}({x_1})]}^2}d{x_1}} }}{{\int_{ - \infty }^\infty {\exp (2\tau {x_1}){{[\tau Ai({x_1}) + Ai^{\prime}({x_1})]}^2}d{x_1}} }} - {\tau^2}} \right) ={-} \frac{{3{\alpha ^3}}}{{w_0^2}}.$$
Similarly, one can obtain
$$X_{0,c}^{} ={-} \frac{{{\alpha ^3}}}{{w_0^2}},X_{2,c}^{} ={-} \frac{{5{\alpha ^3}}}{{w_0^2}},X_{3,c}^{} ={-} \frac{{7{\alpha ^3}}}{{w_0^2}},X_{4,c}^{} ={-} \frac{{9{\alpha ^3}}}{{w_0^2}}.$$
The beam spot size in the x-directions of a Hermite-Gaussian beam passing through an Airy transform optical system yields [57–61]:
$${W_{m,x}} = {\left[ {\frac{{\int_{ - \infty }^\infty {{x^2}{{|{{E_m}(x)} |}^2}dx} }}{{\int_{ - \infty }^\infty {{{|{{E_m}(x)} |}^2}dx} }} - X_{m,c}^2} \right]^{1/2}}.$$
Accordingly, the beam spot size in the x-directions of a Hermite-Gaussian beam with m=1 passing through an Airy transform optical system is found to be
$$\begin{array}{l} {W_{1,x}} = {\left\{ {{\alpha^2}\frac{{\int_{ - \infty }^\infty {x_1^2\exp (2\tau {x_1}){{[\tau Ai({x_1}) + Ai^{\prime}({x_1})]}^2}d{x_1}} }}{{\int_{ - \infty }^\infty {\exp (2\tau {x_1}){{[\tau Ai({x_1}) + Ai^{\prime}({x_1})]}^2}d{x_1}} }} - 2\alpha {\tau^2}{X_{1,c}} - {\alpha^2}{\tau^4} - X_{1,c}^2} \right\}^{1/2}}\\ \begin{array}{cc} {}& = \end{array}\frac{{|\alpha |}}{{2\sqrt 2 \tau }}{(24{\tau ^3} + 3)^{1/2}}. \end{array}$$
Similarly, one can have
$$W_{0,x}^{} = \frac{{|\alpha |}}{{2\sqrt 2 \tau }}{(8{\tau ^3} + 1)^{1/2}},W_{2,x}^{} = \frac{{|\alpha |}}{{2\sqrt 2 \tau }}{(40{\tau ^3} + 7)^{1/2}},$$
$$W_{3,x}^{} = \frac{{|\alpha |}}{{2\sqrt 2 \tau }}{(56{\tau ^3} + 13)^{1/2}},W_{4,x}^{} = \frac{{|\alpha |}}{{2\sqrt 2 \tau }}{(72{\tau ^3} + 21)^{1/2}}.$$
According to the above rules, it can be concluded that the centre of gravity and the beam spot size of a Hermite-Gaussian beam passing through an Airy transform optical system are determined by
$$X_{m,c}^{} ={-} (2m + 1)\frac{{{\alpha ^3}}}{{w_0^2}},Y_{n,c}^{} ={-} (2n + 1)\frac{{{\beta ^3}}}{{w_0^2}},$$
$$W_{m,x}^{} = \frac{{|\alpha |}}{{2\sqrt 2 \tau }}{[8(2m + 1){\tau ^3} + m(m + 1) + 1]^{1/2}},W_{n,y}^{} = \frac{{|\beta |}}{{2\sqrt 2 \gamma }}{[8(2n + 1){\gamma ^3} + n(n + 1) + 1]^{1/2}},$$
where $\gamma = w_0^2/(4{\beta ^2})$. Equations (22) and (23) can be proved to be correct by calculating the centre of gravity and the beam spot size of the Airy transform of an arbitrary Hermite-Gaussian beam.

3. Numerical simulations and results

The Airy transform properties of Hermite-Gaussian beams are demonstrated by using the obtained expressions. The Gaussian waist is fixed as w0=0.6 mm in the following calculations. Since the optical field of the Airy transform of a Hermite-Gaussian beam has the same evolution law in the x- and y-directions, only the transverse direction such as x-direction is selected as the research object. Figure 2 shows the normalized intensity distribution in the x0-direction of different Hermite-Gaussian beams in the input plane z=0. It is only when m is odd that the on-axis intensity is equal to zero. When m is an even number, the on-axis intensity is ranked as the maximum of (m/2 + 1)th priority. A Hermite-Gaussian beam has m valleys with zero intensity in the x0-direction. The beam spot size in the x0-direction of an initial Hermite-Gaussian beam is (2m+1)1/2w0/2. Therefore, the beam spot size in the x0-direction of the input plane increases with the increase of transverse mode number m. Normalized intensity distribution in the x-direction of Hermite-Gaussian beams passing through an Airy transform optical system with α=0.7 mm is demonstrated in Fig. 3. The first peak of normalized intensity is not the maximum. The number of side lobes in the beam spot of Hermite-Gaussian beams after Airy transform increases sharply with the increase of transverse mode number m. When the control parameter α is positive, the side lobes are located in the negative direction of x-axis. Figure 4 represents the normalized intensity distribution in the x-direction of Hermite-Gaussian beams passing through an Airy transform optical system with α=0.4 mm. The comparison between Figs. 3 and 4 denotes that the number of side lobes in the beam spot of Airy transform of a Hermite-Gaussian beam decreases with the decrease of control parameter α. When the control parameter α tends to zero, the Airy transform of a Hermite-Gaussian beam also tends to the initial Hermite-Gaussian beam itself, which is not displayed graphically to save space.

 figure: Fig. 2.

Fig. 2. Normalized intensity distribution in the x0-direction of Hermite-Gaussian beams in the input plane z=0. (a) m=1. (b) m=2. (c) m=3. (d) m=4.

Download Full Size | PDF

 figure: Fig. 3.

Fig. 3. Normalized intensity distribution in the x-direction of Hermite-Gaussian beams passing through an Airy transform optical system with α=0.7 mm. (a) m=1. (b) m=2. (c) m=3. (d) m=4.

Download Full Size | PDF

 figure: Fig. 4.

Fig. 4. Normalized intensity distribution in the x-direction of Hermite-Gaussian beams passing through an Airy transform optical system with α=0.4 mm. (a) m=1. (b) m=2. (c) m=3. (d) m=4.

Download Full Size | PDF

The variation of the centre of gravity and the beam spot size of the Airy transform of Hermite-Gaussian beams in the x-direction with the control parameter α is shown in Fig. 5. The centre of gravity of the Airy transform of different Hermite-Gaussian beams in the x-direction versus the control parameter α has the same variation rule. The sign of the centre of gravity in the x-direction is opposite to that of the control parameter α. With the increase of control parameter α, the centre of gravity in x-direction decreases. In other words, the larger the absolute value of the control parameter α, the greater the deviation of the centre of gravity from the origin. The absolute value of the centre of gravity in the x-direction increases with the increase of transverse mode number m. The beam spot size in the x-direction is independent of the sign of the control parameter α. When the control parameter α increases from a negative value to zero, the beam spot size in the x-direction decreases first, and then nearly keeps unvaried. When the control parameter α continuously increases from zero to a positive value, the beam spot size in the x-direction remains almost unchanged at first, and then increases. Therefore, the curve of the beam spot size in the x-direction versus the control parameter α has a flat bottom. The beam spot size in the x-direction increases with the increase of transverse mode number m.

 figure: Fig. 5.

Fig. 5. The centre of gravity (a) and the beam spot size (b) in the x-direction of the Airy transform of Hermite-Gaussian beams as a function of the control parameter α.

Download Full Size | PDF

Finally, the complete image of the Airy transform of Hermite-Gaussian beams with two transverse directions is presented. Figure 6 shows the normalized intensity distribution of different Hermite-Gaussian beams in the input plane z=0. The concepts of row and column are used to describe the beam spot of Hermite-Gaussian beams. The beam spot of Hermite-Gaussian beams in the input plane has lobes of n+1 rows and m+1 columns. Therefore, the number of lobes in Fig. 6(d) is the largest and that in Fig. 6(a) is the smallest. It can be seen that the marginal lobes are the strongest and the central lobes are the weakest. Figures 7–10 show the normalized intensity distribution of Hermite-Gaussian beams passing through different Airy transform optical systems with α=β=0.5 mm, 0.4 mm, 0.3 mm, and 0.2 mm, respectively. When the control parameters α and β are comparable to the Gaussian waist, the beam spot of the Airy transform of Hermite-Gaussian beams reflects the Airy feature. The larger the transverse mode number m, the more lobes in the x-direction. The larger the transverse mode number n, the greater the number of lobes in the y-direction. The Airy and the Hermite-Gaussian features are complementary. With the decrease of control parameters α and β, the Airy feature weakens, while the Hermite-Gaussian feature enhances. When the control parameters α and β are reduced to 0.3 mm, the number of lobes in the beam spot of the Airy transform of Hermite-Gaussian beams decreases significantly. When α=β=0.2 mm, the number of lobes in the beam spot of the Airy transform of Hermite-Gaussian beams is the same as that in the input plane. However, the status and the importance of these lobes have changed. After the Airy transform, the lobes with the same status in the input plane become of different importance. The closer the control parameters are to zero, the closer the beam spot of the Airy transform of Hermite-Gaussian beams is to the input beam spot. Figures 7–10 indicate that the beam spot of Airy transform of Hermite-Gaussian beams depends on the control parameters α and β.

 figure: Fig. 6.

Fig. 6. Normalized intensity distribution of Hermite-Gaussian beams in the input plane z=0. (a) m=0 and n=4. (b) m=1 and n=3. (c) m = n=2. (d) m=3 and n=2.

Download Full Size | PDF

 figure: Fig. 7.

Fig. 7. Normalized intensity distribution of Hermite-Gaussian beams passing through an Airy transform optical system with α=β=0.5 mm. (a) m=0 and n=4. (b) m=1 and n=3. (c) m = n=2. (d) m=3 and n=2.

Download Full Size | PDF

 figure: Fig. 8.

Fig. 8. Normalized intensity distribution of Hermite-Gaussian beams passing through an Airy transform optical system with α=β=0.4 mm. (a) m=0 and n=4. (b) m=1 and n=3. (c) m = n=2. (d) m=3 and n=2.

Download Full Size | PDF

 figure: Fig. 9.

Fig. 9. Normalized intensity distribution of Hermite-Gaussian beams passing through an Airy transform optical system with α=β=0.3 mm. (a) m=0 and n=4. (b) m=1 and n=3. (c) m = n=2. (d) m=3 and n=2.

Download Full Size | PDF

 figure: Fig. 10.

Fig. 10. Normalized intensity distribution of Hermite-Gaussian beams passing through an Airy transform optical system with α=β=0.2 mm. (a) m=0 and n=4. (b) m=1 and n=3. (c) m = n=2. (d) m=3 and n=2.

Download Full Size | PDF

4. Experimental results

In this section, we carry out the experiment to generate Hermite-Gaussian beams, and then implement the Airy transform of the generated beams. The experimental setup is illustrated in Fig. 11. A fundamental Gaussian laser beam with a wavelength of 532 nm generated by a diode pumped solid state (DPSS) laser is first expanded by a beam expander, and then goes towards a reflective mode spatial light modulator (SLM1, Holoeye Pluto-VIS, pixel size: 8µm×8µm). The SLM1 acts as a phase-only screen to convert the incident Gaussian beam into the Hermite-Gaussian beam. In order to generate high quality Hermite-Gaussian beams, the phase plate synthesis method introduced in Ref. [62] is adopted. The inset Fig. 11(a) shows an example of the phase plate for producing a Hermite-Gaussian beam with mode indices of m = n=2. After the SLM1, the reflective diffraction orders pass through a 4f optical systems consisting of two lenses L1 and L2 with focal lengths f1=f2=250 mm. An iris is placed in the rear focal plane of L1 to block other unwanted diffraction orders, and only the first diffraction order is allowed to pass through. In the rear focal plane of L2, the optical beam can be regarded as the Hermite-Gaussian source with specified mode indices at z=0. Figure 12 presents the normalized intensity distributions of the generated Hermite-Gaussian sources with four different mode indices. It can be seen that the experimental results are in good agreement with the theoretical results shown in Fig. 6, except that there are slightly inhomogeneous intensities between the corner lobes. The waist size w0 of the generated beams obtained from theoretical fitting is about 0.6 mm.

 figure: Fig. 11.

Fig. 11. Schematic diagram of the experimental setup for generation of the Hermite-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; BS: beam splitter; L1, L2, L3, L4: thin lenses; BPA: beam profile analysis.

Download Full Size | PDF

 figure: Fig. 12.

Fig. 12. Experimental distribution of normalized intensity distribution of Hermite-Gaussian beams in the input plane z=0. (a) m=0 and n=4. (b) m=1 and n=3. (c) m = n=2. (d) m=3 and n=2.

Download Full Size | PDF

Once the Hermite-Gaussian source is generated, it passes through another 4f optical system composed of two lenses L3 and L4, performing the Airy transform of the Hermite-Gaussian beam. The focal lengths of L3 and L4 are f3=400 mm and f4=150 mm, respectively. A SLM2 (Holoeye LETO, pixel size: 6.4 µm×6.4 µm) on which a cubic phase $\Phi ({x_1},{y_1}) = ({{\alpha^3}{k^3}x_1^3 + {\beta^3}{k^3}y_1^3} )/3{f_3} - (2k{f_3} + 2k{f_4} + \pi )$ where k is a wavenumber is loaded, is just located in the rear/front focal plane of L3/L4. The inset Fig. 11(b) shows a typical loaded phase on the SLM2 with α=β=0.3 mm. A beam profile analysis (BPA) is placed in the rear focal (output) plane of L4 to record the intensity distribution of Airy transformed beam. Note that in our 4f system for L3 and L4, the focal lengths of L3 and L4 are not equal to each other. This setting only changes the magnification of the output beam, i.e., the magnification is M = f4/f3 without changing its amplitude and phase distributions.

Figures 13–16 show the experimental results of normalized intensity distributions of output beams with four different mode indices when the control parameters are α=β=0.5 mm, 0.4 mm, 0.3 mm, and 0.2 mm, respectively. The results correspond to the theoretical calculations shown in Figs. 7–10. It can be seen from Figs. 13–16 that the intensity distributions are in good agreement with the theoretical results, indicating that our experimental system is applicable for the Airy transform of Hermite-Gaussian beams or other laser beam modes.

 figure: Fig. 13.

Fig. 13. Experimental distribution of normalized intensity distribution of Hermite-Gaussian beams passing through an Airy transform optical system with α=β=0.5 mm. (a) m=0 and n=4. (b) m=1 and n=3. (c) m = n=2. (d) m=3 and n=2.

Download Full Size | PDF

 figure: Fig. 14.

Fig. 14. Experimental distribution of normalized intensity distribution of Hermite-Gaussian beams passing through an Airy transform optical system with α=β=0.4 mm. (a) m=0 and n=4. (b) m=1 and n=3. (c) m = n=2. (d) m=3 and n=2.

Download Full Size | PDF

 figure: Fig. 15.

Fig. 15. Experimental distribution of normalized intensity distribution of Hermite-Gaussian beams passing through an Airy transform optical system with α=β=0.3 mm. (a) m=0 and n=4. (b) m=1 and n=3. (c) m = n=2. (d) m=3 and n=2.

Download Full Size | PDF

 figure: Fig. 16.

Fig. 16. Experimental distribution of normalized intensity distribution of Hermite-Gaussian beams passing through an Airy transform optical system with α=β=0.2 mm. (a) m=0 and n=4. (b) m=1 and n=3. (c) m = n=2. (d) m=3 and n=2.

Download Full Size | PDF

In order to investigate the effect of the control parameters on the center of gravity and the r.m.s (root mean square) beam spot size of the output beams, Hermite-Gaussian beams with mode indices m=1, 2, 3, and 4 as well as n=0 are experimentally measured. For each mode, the range of control parameter α is from 0 to 0.55 mm with step 0.05 mm, and β=0. The BPA first captures all the intensity distributions of the Airy transformed beams with different α. Each graph can be represented as an intensity matrix I(xi, yj), where (xi, yj) denotes the spatial coordinates. Therefore, the centre of gravity and the r.m.s beam spot size can be evaluated by the following formulae

$${X_{m,c}} = \sum\limits_i^{{N_1}} {\sum\limits_j^{{N_2}} {{x_i}} } I({x_i},{y_j})/\sum\limits_i^{{N_1}} {\sum\limits_j^{{N_2}} {I({x_i},{y_j})} } ,$$
$$W_{m,x}^2 = \sum\limits_i^{{N_1}} {\sum\limits_j^{{N_2}} {{{({x_i} - {X_{m,c}})}^2}} } I({x_i},{y_j})/\sum\limits_i^{{N_1}} {\sum\limits_j^{{N_2}} {I({x_i},{y_j})} } ,$$
where N1=1928 and N2=1448. The dependence of the centre of gravity and the r.m.s beam spot size of four Hermite-Gaussian modes on the control parameter α are illustrated in Fig. 17(a) and 17(b), respectively. For comparison, the corresponding theoretical results (solid curves) calculated from Eqs. (22) and (23) with magnification M = f4/f3 are also plotted. The experimental results show that with the increase of control parameter, the centre of gravity moves towards the negative x-coordinate direction, while the beam spot size almost remains unchanged for α<0.3 mm, and then increases suddenly for α>0.3 mm. Moreover, the experimental results are consistent with the theoretical results.

 figure: Fig. 17.

Fig. 17. Experimental measurement of the centre of gravity (a) and the beam spot size (b) in the x-direction of the Airy transform of Hermite-Gaussian beams as a function of the control parameter α.

Download Full Size | PDF

5. Summary

The Airy transform of Hermite-Gaussian beams is investigated theoretically and experimentally. The analytical expression of the optical field of the Airy transform of an arbitrary Hermite-Gaussian beam in the x-direction is derived. The optical field in the x-direction of the Airy transform of Hermite-Gaussian beams with transverse mode number m is the sum of the zero-order derivative to mth-order derivative of the Airy function with different weight coefficients. For the optical field in the y-direction, it can be obtained by replacing the parameter of the optical field in the x-direction. The general analytic formulae for the centre of gravity and the beam spot size of a Hermite-Gaussian beam passing through an Airy transform optical system are also derived.

Because the Airy transform of a Hermite-Gaussian beam has the same evolution law in two transverse directions, the effects of the control parameter α and the transverse mode number m on the normalized intensity distribution, the centre of gravity, and the beam spot size in the x-direction are theoretically investigated, respectively. The number of the side lobes in the beam spot decreases with the decrease of control parameter α. When the control parameter α approach zero, the Airy transform of a Hermite-Gaussian beam also tends to the initial Hermite-Gaussian distribution. The centre of gravity or the beam spot size in the x-direction of the Airy transform of different Hermite-Gaussian beams have the same evolution law versus the control parameter α. The sign of the centre of gravity in the x-direction is opposite to that of the control parameter α. The larger the absolute value of the control parameter α is, the greater the deviation of the center of gravity from the origin. The beam spot size in the x-direction is independent of the sign of the control parameter α. When the control parameter α increases from zero, the beam spot size in the x-direction first remains unchanged, and then increases. The number of the side lobes in the beam spot, the absolute value of the centre of gravity in the x-direction, and the beam spot size in the x-direction all increase with the increase of transverse mode number m.

The complete image of the Airy transform of Hermite-Gaussian beams with two transverse directions is also demonstrated. The beam spot of the Airy transform of Hermite-Gaussian beams depends on the control parameters α and β. When the control parameters α and β are comparable to the Gaussian waist, the beam spot of the Airy transform of Hermite-Gaussian beams shows Airy feature. With the decrease of control parameters α and β, the number of lobes in the beam spot decreases, and the Airy feature weakens until it disappears completely, while the Hermite-Gaussian feature enhances. With the change of the control parameters α and β, the beam spot is diverse and interesting. When a Hermite-Gaussian beam passes through an Airy transform optical system, the number of lobes in the beam spot may change, and the lobes with the same status in the input plane may become of different importance.

Finally, the Airy transform of Hermite-Gaussian beams is realized in the experiment. The influence of the control parameters on the normalized intensity distribution, the centre of gravity, and the beam spot size is experimentally investigated, respectively. The experimental results are consistent with the theoretical simulation results. The properties of Airy transform of Hermite-Gaussian beams are well displayed in this research. By using the Airy transform of Hermite-Gaussian beams, special optical beams containing the Airy function and the different higher order derivatives of the Airy function can be obtained, which may have novel potential applications. Therefore, this research extends the practical applications of Hermite-Gaussian beams.

Funding

National Natural Science Foundation of China (11974313, 11874046, 11874323).

Disclosures

The authors declare no conflicts of interest.

References

1. K. M. Luk and P. K. Yu, “Generation of Hermite-Gaussian beam modes by multipoles with complex source points,” J. Opt. Soc. Am. A 2(11), 1818–1820 (1985). [CrossRef]  

2. F. Schepers, T. Bexter, T. Hellwig, and C. Fallnich, “Selective Hermite-Gaussian mode excitation in a laser cavity by external pump beam shaping,” Appl. Phys. B 125(5), 75 (2019). [CrossRef]  

3. Y. Shen, Y. Meng, X. Fu, and M. Gong, “Wavelength-tunable Hermite-Gaussian modes and an orbital-angular-momentum-tunable vortex beam in a dual-off-axis pumped Yb: CALGO laser,” Opt. Lett. 43(2), 291–294 (2018). [CrossRef]  

4. O. Mata-Mendez and F. Chavez-Rivas, “Diffraction of Hermite-Gaussian beams by a slit,” J. Opt. Soc. Am. A 12(11), 2440–2445 (1995). [CrossRef]  

5. O. Mata-Mendez and F. Chavez-Rivas, “Diffraction of Gaussian and Hermite-Gaussian beams by finite gratings,” J. Opt. Soc. Am. A 18(3), 537–545 (2001). [CrossRef]  

6. S. A. Carvalho and S. De Leo, “The effect of the geometrical optical phase on the propagation of Hermite-Gaussian beams through transversal and parallel dielectric blocks,” J. Mod. Opt. 66(5), 548–556 (2019). [CrossRef]  

7. Y. Cai and C. Chen, “Paraxial propagation of a partially coherent Hermite-Gaussian beam through aligned and misaligned ABCD optical systems,” J. Opt. Soc. Am. A 24(8), 2394–2401 (2007). [CrossRef]  

8. X. Ji, X. Chen, and B. Lü, “Spreading and directionality of partially coherent Hermite-Gaussian beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 25(1), 21–28 (2008). [CrossRef]  

9. P. Sun, J. Guo, and T. Wang, “Propagation equation of Hermite-Gauss beams through a complex optical system with apertures and its application to focal shift,” J. Opt. Soc. Am. A 30(7), 1381–1386 (2013). [CrossRef]  

10. A. A. Kovalev, V. V. Kotlyar, and S. G. Zaskanov, “Diffraction integral and propagation of Hermite-Gaussian modes in a linear refractive index medium,” J. Opt. Soc. Am. A 31(5), 914–919 (2014). [CrossRef]  

11. Z. Ding, X. Li, J. Cao, and X. Ji, “Thermal blooming effect of Hermite-Gaussian beams propagating through the atmosphere,” J. Opt. Soc. Am. A 36(7), 1152–1160 (2019). [CrossRef]  

12. D. Liu, Y. Wang, and H. Zhong, “Average intensity of radial phase-locked partially coherent standard Hermite-Gaussian beam in oceanic turbulence,” Opt. Laser Technol. 106, 495–505 (2018). [CrossRef]  

13. Q. Huang, M. Cheng, L. Guo, J. Li, X. Yan, and S. Liu, “Scattering of aerosol particles by a Hermite-Gaussian beam in marine atmosphere,” Appl. Opt. 56(19), 5329–5335 (2017). [CrossRef]  

14. H. Liu, J. Xia, and Y. Lu, “Evolution properties of partially coherent standard and elegant Hermite-Gaussian beams in uniaxial crystals,” J. Opt. Soc. Am. A 34(12), 2102–2109 (2017). [CrossRef]  

15. X. Fan, X. Ji, H. Yu, H. Wang, Y. Deng, and L. Chen, “Kerr effect on propagation characteristics of Hermite-Gaussian beams,” Opt. Express 27(16), 23112–23123 (2019). [CrossRef]  

16. S. C. Y. Cruz and Z. Gress, “Group approach to the paraxial propagation of Hermite-Gaussian modes in a parabolic medium,” Ann. Phys. 383, 257–277 (2017). [CrossRef]  

17. K. Mihoubi, A. Bencheikh, and A. Manallah, “The beam propagation factor M2 of truncated standard and elegant-Hermite-Gaussian beams,” Opt. Laser Technol. 99, 191–196 (2018). [CrossRef]  

18. K. Duan, B. Wang, and B. Lü, “Propagation of Hermite-Gaussian and Laguerre-Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 22(9), 1976–1980 (2005). [CrossRef]  

19. L. R. Hofer, L. W. Jones, J. L. Goedert, and R. V. Dragone, “Hermite-Gaussian mode detection via convolution neural networks,” J. Opt. Soc. Am. A 36(6), 936–943 (2019). [CrossRef]  

20. T. P. Meyrath, F. Schreck, J. L. Hanssen, C.-S. Chuu, and M. G. Raizen, “A high frequency optical trap for atoms using Hermite-Gaussian beams,” Opt. Express 13(8), 2843–2851 (2005). [CrossRef]  

21. A. P. Porfirev and R. V. Skidanov, “Optical trapping and manipulation of light-absorbing particles by means of a Hermite-Gaussian laser beam,” J. Opt. Technol. 82(9), 587–591 (2015). [CrossRef]  

22. J. Wadhwa and A. Singh, “Generation of second harmonics by a self-focused Hermite-Gaussian laser beam in collisionless plasma,” Phys. Plasmas 26(6), 062118 (2019). [CrossRef]  

23. J. Wadhwa and A. Singh, “Generation of second harmonics of intense Hermite-Gaussian laser beam in relativistic plasma,” Laser Part. Beams 37(01), 79–85 (2019). [CrossRef]  

24. H. S. Ghotra and N. Kant, “TEM modes influenced electron acceleration by Hermite-Gaussian laser beam in plasma,” Laser Part. Beams 34(3), 385–393 (2016). [CrossRef]  

25. T. Tanaka, “Coherent mode decomposition using mixed Wigner functions of Hermite-Gaussian beams,” Opt. Lett. 42(8), 1576–1579 (2017). [CrossRef]  

26. G. Liang and Q. Wang, “Controllable conversion between Hermite Gaussian and Laguerre Gaussian modes due to cross phase,” Opt. Express 27(8), 10684–10691 (2019). [CrossRef]  

27. S. Restuccia, D. Giovannini, G. Gibson, and M. Padgett, “Comparing the information capacity of Laguerre-Gaussian and Hermite-Gaussian modal sets in a finite-aperture system,” Opt. Express 24(24), 27127–27136 (2016). [CrossRef]  

28. G. Zhou, “Vectorial structure of the far field of an elegant Hermite-Gaussian beam,” Opt. Laser Technol. 44(1), 218–225 (2012). [CrossRef]  

29. G. Zhou, Z. Ji, and G. Ru, “Complete analytical expression of Lorentz-Hermite-Gauss laser beams,” Laser Eng. 40(1-3), 127–147 (2018).

30. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007). [CrossRef]  

31. J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008). [CrossRef]  

32. G. Zhou, R. Chen, and X. Chu, “Propagation of Airy beams in uniaxial crystals orthogonal to the optical axis,” Opt. Express 20(3), 2196–2205 (2012). [CrossRef]  

33. J. Rogel-Salazar, H. Jiménez-Romero, and A. S. Chávez-Cerda, “Full characterization of Airy beams under physical principles,” Phys. Rev. A 89(2), 023807 (2014). [CrossRef]  

34. G. Zhou, R. Chen, and G. Ru, “Propagation of an Airy beam in a strongly nonlocal nonlinear media,” Laser Phys. Lett. 11(10), 105001 (2014). [CrossRef]  

35. G. Zhou, R. Chen, and X. Chu, “Propagation of cosh-Airy beams in uniaxial crystals orthogonal to the optical axis,” Opt. Laser Technol. 116, 72–82 (2019). [CrossRef]  

36. Y. Zhou, Y. Xu, X. Chu, and G. Zhou, “Propagation of cosh-Airy and cos-Airy beams in parabolic potential,” Appl. Sci. 9(24), 5530 (2019). [CrossRef]  

37. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008). [CrossRef]  

38. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009). [CrossRef]  

39. D. Abdollahpour, S. Suntsov, D. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105(25), 253901 (2010). [CrossRef]  

40. A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101(7), 071110 (2012). [CrossRef]  

41. S. Jia, J. C. Vaughan, and X. Zhuang, “Isotropic three-dimensional super-resolution imaging with a self- bending point spread function,” Nat. Photonics 8(4), 302–306 (2014). [CrossRef]  

42. T. Ellenbogen, N. Voloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009). [CrossRef]  

43. H. Dai, X. Sun, D. Luo, and Y. Liu, “Airy beams generated by a binary phase element made of polymer-dispersed liquid crystals,” Opt. Express 17(22), 19365–19370 (2009). [CrossRef]  

44. L. Li, T. Li, S. Wang, and S. Zhu, “Plasmonic Airy beam generated by in-plane diffraction,” Phys. Rev. Lett. 107(12), 126804 (2011). [CrossRef]  

45. G. Porat, I. Dolev, O. Barlev, and A. Arie, “Airy beam laser,” Opt. Lett. 36(20), 4119–4121 (2011). [CrossRef]  

46. R. Cao, Y. Yang, J. Wang, J. Bu, and M. Wang, “Microfabricated continuous cubic phase plate induced Airy beams for optical manipulation with high power efficiency,” Appl. Phys. Lett. 99(26), 261106 (2011). [CrossRef]  

47. P. Acebal, L. Carretero, S. Blaya, and A. Murciano, “Generation of high-quality tunable one-dimensional Airy beams using the aberrations of a single lens,” IEEE Photonics J. 4(5), 1273–1280 (2012). [CrossRef]  

48. N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Aire, “Generation of electron Airy beams,” Nature 494(7437), 331–335 (2013). [CrossRef]  

49. J. D. Ring, C. J. Howls, and M. R. Dennis, “Incomplete Airy beams: finite energy from a sharp spectral cutoff,” Opt. Lett. 38(10), 1639–1641 (2013). [CrossRef]  

50. Y. Jiang, K. Huang, and X. Lu, “The optical Airy transform and its application in generating and controlling the Airy beam,” Opt. Commun. 285(24), 4840–4843 (2012). [CrossRef]  

51. Y. Jiang, K. Huang, and X. Lu, “Airy related beam generated from flat-topped Gaussian beams,” J. Opt. Soc. Am. A 29(7), 1412–1416 (2012). [CrossRef]  

52. L. Ez-zariy, F. Boufalah, L. Dalil-Essakali, and A. Belafhal, “A conversion of the hyperbolic-cosine Gaussian beam to a novel finite Airy-related beam using an optical Airy transform system,” Optik 171, 501–506 (2018). [CrossRef]  

53. M. Yaalou, E. M. El Halba, Z. Hricha, and A. Belafhal, “Transformation of double–half inverse Gaussian hollow beams into superposition of finite Airy beams using an optical Airy transform,” Opt. Quant. Electron. 51(3), 64–75 (2019). [CrossRef]  

54. G. Zhou, F. Wang, and S. Feng, “Airy transform of Laguerre-Gaussian beams,” Opt. Express 28(13), 19683–19699 (2020). [CrossRef]  

55. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).

56. O. Vallée and S. Manuel, Airy Functions and Applications to Physics (Imperial College Press, 2010).

57. R. Martínez-Herrero and P. M. Mejías, “Second-order spatial characterization of hard-edge diffracted beams,” Opt. Lett. 18(19), 1669–1671 (1993). [CrossRef]  

58. G. Zhou, “Generalized beam propagation factors of truncated partially coherent cosine-Gaussian and cosh-Gaussian beams,” Opt. Laser Technol. 42(3), 489–496 (2010). [CrossRef]  

59. G. Zhou, “Far field structural properties of a Gaussian vortex beam,” Laser Eng. 26(1-2), 1–17 (2013).

60. Y. Ni, Z. Ji, G. Ru, and G. Zhou, “Propagation properties of controllable dark hollow laser beams in uniaxial crystals along the optical axis,” Laser Eng. 31(3-6), 0826001 (2011). [CrossRef]  

61. Y. Zhou, Y. Xu, and G. Zhou, “Beam propagation factor of a cosh-Airy beam,” Appl. Sci. 9(9), 1817 (2019). [CrossRef]  

62. V. Arrizon, U. Ruiz, R. Carrada, and L. A. Gonzalez, “Pixelated phase computer holograms for the accurate encoding of scalar complex fields,” J. Opt. Soc. Am. A 24(11), 3500–3507 (2007). [CrossRef]  

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (17)
Fig. 1.
Fig. 1. A schematic diagram of the Airy transform optical system.

View in Article | Download Full Size | PDF

Fig. 2.
Fig. 2. Normalized intensity distribution in the x0-direction of Hermite-Gaussian beams in the input plane z=0. (a) m=1. (b) m=2. (c) m=3. (d) m=4.

View in Article | Download Full Size | PDF

Fig. 3.
Fig. 3. Normalized intensity distribution in the x-direction of Hermite-Gaussian beams passing through an Airy transform optical system with α=0.7 mm. (a) m=1. (b) m=2. (c) m=3. (d) m=4.

View in Article | Download Full Size | PDF

Fig. 4.
Fig. 4. Normalized intensity distribution in the x-direction of Hermite-Gaussian beams passing through an Airy transform optical system with α=0.4 mm. (a) m=1. (b) m=2. (c) m=3. (d) m=4.

View in Article | Download Full Size | PDF

Fig. 5.
Fig. 5. The centre of gravity (a) and the beam spot size (b) in the x-direction of the Airy transform of Hermite-Gaussian beams as a function of the control parameter α.

View in Article | Download Full Size | PDF

Fig. 6.
Fig. 6. Normalized intensity distribution of Hermite-Gaussian beams in the input plane z=0. (a) m=0 and n=4. (b) m=1 and n=3. (c) m = n=2. (d) m=3 and n=2.

View in Article | Download Full Size | PDF

Fig. 7.
Fig. 7. Normalized intensity distribution of Hermite-Gaussian beams passing through an Airy transform optical system with α=β=0.5 mm. (a) m=0 and n=4. (b) m=1 and n=3. (c) m = n=2. (d) m=3 and n=2.

View in Article | Download Full Size | PDF

Fig. 8.
Fig. 8. Normalized intensity distribution of Hermite-Gaussian beams passing through an Airy transform optical system with α=β=0.4 mm. (a) m=0 and n=4. (b) m=1 and n=3. (c) m = n=2. (d) m=3 and n=2.

View in Article | Download Full Size | PDF

Fig. 9.
Fig. 9. Normalized intensity distribution of Hermite-Gaussian beams passing through an Airy transform optical system with α=β=0.3 mm. (a) m=0 and n=4. (b) m=1 and n=3. (c) m = n=2. (d) m=3 and n=2.

View in Article | Download Full Size | PDF

Fig. 10.
Fig. 10. Normalized intensity distribution of Hermite-Gaussian beams passing through an Airy transform optical system with α=β=0.2 mm. (a) m=0 and n=4. (b) m=1 and n=3. (c) m = n=2. (d) m=3 and n=2.

View in Article | Download Full Size | PDF

Fig. 11.
Fig. 11. Schematic diagram of the experimental setup for generation of the Hermite-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; BS: beam splitter; L1, L2, L3, L4: thin lenses; BPA: beam profile analysis.

View in Article | Download Full Size | PDF

Fig. 12.
Fig. 12. Experimental distribution of normalized intensity distribution of Hermite-Gaussian beams in the input plane z=0. (a) m=0 and n=4. (b) m=1 and n=3. (c) m = n=2. (d) m=3 and n=2.

View in Article | Download Full Size | PDF

Fig. 13.
Fig. 13. Experimental distribution of normalized intensity distribution of Hermite-Gaussian beams passing through an Airy transform optical system with α=β=0.5 mm. (a) m=0 and n=4. (b) m=1 and n=3. (c) m = n=2. (d) m=3 and n=2.

View in Article | Download Full Size | PDF

Fig. 14.
Fig. 14. Experimental distribution of normalized intensity distribution of Hermite-Gaussian beams passing through an Airy transform optical system with α=β=0.4 mm. (a) m=0 and n=4. (b) m=1 and n=3. (c) m = n=2. (d) m=3 and n=2.

View in Article | Download Full Size | PDF

Fig. 15.
Fig. 15. Experimental distribution of normalized intensity distribution of Hermite-Gaussian beams passing through an Airy transform optical system with α=β=0.3 mm. (a) m=0 and n=4. (b) m=1 and n=3. (c) m = n=2. (d) m=3 and n=2.

View in Article | Download Full Size | PDF

Fig. 16.
Fig. 16. Experimental distribution of normalized intensity distribution of Hermite-Gaussian beams passing through an Airy transform optical system with α=β=0.2 mm. (a) m=0 and n=4. (b) m=1 and n=3. (c) m = n=2. (d) m=3 and n=2.

View in Article | Download Full Size | PDF

Fig. 17.
Fig. 17. Experimental measurement of the centre of gravity (a) and the beam spot size (b) in the x-direction of the Airy transform of Hermite-Gaussian beams as a function of the control parameter α.

View in Article | Download Full Size | PDF

Equations (25)

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

E m n ( x 0 , y 0 ) = E m ( x 0 , 0 ) E n ( y 0 , 0 ) = H m ( 2 x 0 w 0 ) H n ( 2 y 0 w 0 ) exp ⁡ ( − x 0 2 + y 0 2 w 0 2 ) ,
E m n ( x , y ) = E m ( x ) E n ( y ) = 1 | α β | ∫ − ∞ ∞ ∫ − ∞ ∞ E m n ( x 0 , y 0 ) A i ( x − x 0 α ) A i ( y − y 0 β ) d x 0 d y 0 ,
H m ( x ) = ∑ l = 0 [ m / 2 ] ( − 1 ) l 2 m − 2 l m ! l ! ( m − 2 l ) ! x m − 2 l ,
∫ − ∞ ∞ 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 m ( x ) = i m w 0 α 2 π | α | ∫ − ∞ ∞ H m ( − w 0 ξ 2 ) exp ⁡ ( i α 3 ξ 3 3 − w 0 2 ξ 2 4 + i ξ x ) d ξ  =  i m τ π ∫ − ∞ ∞ ∑ l = 0 [ m / 2 ] ( − 1 ) m − l m ! l ! ( m − 2 l ) ! ( α | α | 2 2 τ ) m − 2 l u m − 2 l exp ⁡ ( i u 3 3 − τ u 2 + i x α u ) d u  =  i m 2 π τ exp ⁡ ( 2 τ 3 3 + τ x α ) ∑ l = 0 [ m / 2 ] ( − 1 ) m − l m ! l ! ( m − 2 l ) ! ( α | α | 2 2 τ ) m − 2 l ∑ s = 0 m − 2 l C m − 2 l , s A i ( s ) ( x 1 ) ,
C 0 , 0  =  1 ,
C 1 , 0  =  − i τ C 0 , 0 , C 1 , 1  =  − i C 0 , 0 ,
C t , 0 = − i τ C t − 1 , 0 , C t , s = ( − i τ C t − 1 , s − i C t − 1 , s − 1 ) , C t , t = − i C t − 1 , t − 1 , t > 1 , 0 < s < t − 1.
E 0 ( x ) = 2 π τ exp ⁡ ( 2 τ 3 3 + τ x α ) A i ( x 1 ) ,
E 1 ( x ) = − 4 2 π α τ | α | exp ⁡ ( 2 τ 3 3 + τ x α ) [ τ A i ( x 1 ) + A i ′ ( x 1 ) ] ,
E 2 ( x ) = 4 π τ exp ⁡ ( 2 τ 3 3 + τ x α ) [ ( 1 + 4 τ 3 ) A i ( x 1 )  +  8 τ 2 A i ′ ( x 1 ) + 4 τ A i ( 2 ) ( x 1 ) ] = 4 π τ exp ⁡ ( 2 τ 3 3 + τ x α ) [ ( 1 + 8 τ 3 + 4 τ x α ) A i ( x 1 )  +  8 τ 2 A i ′ ( x 1 ) ] ,
E 3 ( x )  =  − 4 2 π α τ | α | exp ⁡ ( 2 τ 3 3 + τ x α ) [ ( 6 τ + 8 τ 4 ) A i ( x 1 ) + ( 6 + 24 τ 3 ) A i ′ ( x 1 ) + 24 τ 2 A i ( 2 ) ( x 1 ) + 8 τ A i ( 3 ) ( x 1 ) ]  =  − 4 2 π α τ | α | exp ⁡ ( 2 τ 3 3 + τ x α ) [ ( 14 τ + 32 τ 4 + 24 τ 2 x α ) A i ( x 1 ) + ( 6 + 32 τ 3 + 8 τ x α ) A i ′ ( x 1 ) ] ,
E 4 ( x ) = 2 π τ exp ⁡ ( 2 τ 3 3 + τ x α ) [ ( 12  +  96 τ 3 + 64 τ 6 ) A i ( x 1 ) + ( 192 τ 2 + 256 τ 5 ) A i ′ ( x 1 ) + ( 96 τ + 384 τ 4 ) A i ( 2 ) ( x 1 ) + 256 τ 3 A i ( 3 ) ( x 1 ) + 64 τ 2 A i ( 4 ) ( x 1 ) ] = 2 π τ exp ⁡ ( 2 τ 3 3 + τ x α ) [ ( 12  +  448 τ 3 + 512 τ 6  +  96 τ x α + 512 τ 4 x α + 64 τ 2 x 2 α 2 ) A i ( x 1 ) + ( 320 τ 2 + 512 τ 5 + 256 τ 3 x α ) A i ′ ( x 1 ) ] ,
I m n ( x , y ) = I m ( x ) I n ( y ) = | E m ( x ) | 2 | E n ( y ) | 2 .
X m , c = ∫ − ∞ ∞ x | E m ( x ) | 2 d x ∫ − ∞ ∞ | E m ( x ) | 2 d x .
X 1 , c = α ( ∫ − ∞ ∞ x 1 exp ⁡ ( 2 τ x 1 ) [ τ A i ( x 1 ) + A i ′ ( x 1 ) ] 2 d x 1 ∫ − ∞ ∞ exp ⁡ ( 2 τ x 1 ) [ τ A i ( x 1 ) + A i ′ ( x 1 ) ] 2 d x 1 − τ 2 ) = − 3 α 3 w 0 2 .
X 0 , c = − α 3 w 0 2 , X 2 , c = − 5 α 3 w 0 2 , X 3 , c = − 7 α 3 w 0 2 , X 4 , c = − 9 α 3 w 0 2 .
W m , x = [ ∫ − ∞ ∞ x 2 | E m ( x ) | 2 d x ∫ − ∞ ∞ | E m ( x ) | 2 d x − X m , c 2 ] 1 / 2 .
W 1 , x = { α 2 ∫ − ∞ ∞ x 1 2 exp ⁡ ( 2 τ x 1 ) [ τ A i ( x 1 ) + A i ′ ( x 1 ) ] 2 d x 1 ∫ − ∞ ∞ exp ⁡ ( 2 τ x 1 ) [ τ A i ( x 1 ) + A i ′ ( x 1 ) ] 2 d x 1 − 2 α τ 2 X 1 , c − α 2 τ 4 − X 1 , c 2 } 1 / 2 = | α | 2 2 τ ( 24 τ 3 + 3 ) 1 / 2 .
W 0 , x = | α | 2 2 τ ( 8 τ 3 + 1 ) 1 / 2 , W 2 , x = | α | 2 2 τ ( 40 τ 3 + 7 ) 1 / 2 ,
W 3 , x = | α | 2 2 τ ( 56 τ 3 + 13 ) 1 / 2 , W 4 , x = | α | 2 2 τ ( 72 τ 3 + 21 ) 1 / 2 .
X m , c = − ( 2 m + 1 ) α 3 w 0 2 , Y n , c = − ( 2 n + 1 ) β 3 w 0 2 ,
W m , x = | α | 2 2 τ [ 8 ( 2 m + 1 ) τ 3 + m ( m + 1 ) + 1 ] 1 / 2 , W n , y = | β | 2 2 γ [ 8 ( 2 n + 1 ) γ 3 + n ( n + 1 ) + 1 ] 1 / 2 ,
X m , c = ∑ i N 1 ∑ j N 2 x i I ( x i , y j ) / ∑ i N 1 ∑ j N 2 I ( x i , y j ) ,
W m , x 2 = ∑ i N 1 ∑ j N 2 ( x i − X m , c ) 2 I ( x i , y j ) / ∑ i N 1 ∑ j N 2 I ( x i , y j ) ,
Select as filters
Select Topics Cancel
Top
       
© Copyright 2024 | Optica Publishing Group. All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies.
Privacy | Terms of Use