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Interference enhancement effect in a single Airyprime beam propagating in free space

Wensong Dan, Xiang Zang, Fei Wang, Yimin Zhou, Yiqing Xu, Ruipin Chen, and Guoquan Zhou

Open Access Open Access

Abstract

An analytical expression of a single Airyprime beam propagating in free space is derived. Upon propagation in free space, a single Airyprime beam in arbitrary transverse direction is the coherent superposition of the Airyprime and the Airy-related modes, which results in the interference enhancement effect under the appropriate condition. The Airy-related mode is the conventional propagating Airy mode with an additional π/2 phase shift and a weight coefficient of half the normalized propagation distance. Due to the peak light intensity in the initial plane being set to be 1, the strength of interference enhancement effect is characterized by the maximum light intensity. The maximum light intensity of a single Airyprime beam propagating in free space is independent of the scaling factor and is only decided by the exponential decay factor. When the exponential decay factor is above the saturated value, the interference enhancement effect disappears. When the exponential decay factor decreases from the saturated value, the maximum light intensity of a single propagating Airyprime beam increases, and the position of maximum light intensity is getting farther away. With the increase of the scaling factor, the position of maximum light intensity of a single propagating Airyprime beam is extended. The intensity distribution and the transverse Poynting vector of a single propagating Airyprime beam are demonstrated in different observation planes of free space. The flow direction of transverse energy flux effectively supports the interference enhancement effect of a single propagating Airyprime beam. The Airyprime beam is experimentally generated, and the interference enhancement effect is experimentally confirmed. The interference enhancement effect is conducive to the practical application of a single Airyprime beam.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Due to peculiar and charming properties of non-diffraction [1], self-acceleration [2,3], and self-healing [4], an Airy beam is one of the superstar beams. The generation method of Airy beams has attracted much attention. Airy beams can be generated by using spatial light modulator [2], three-wave mixing processes [5], a binary phase element [6], in-plane diffraction [7], Airy laser [8], microfabricated continuous cubic phase plate [9], aberrations of a single lens [10], and diffraction of electrons through a nanoscale hologram [11], respectively. The physical instincts of Airy beams have been decoded by means of wave analysis and Wigner distribution function [12,13], respectively. In order to fully grasp the characteristics of Airy beams, the propagation of Airy beams in various media including free space [14,15], water [16], Kerr medium [17], photorefractive medium [18,19], turbulent atmosphere [20–22], uniaxial crystals orthogonal to the optical axis [23], chiral materials [24], saturated medium [25], nonlocal nonlinear medium [26,27], photonic lattice [28,29], nonlinear medium with nonlinear losses [30], medium with parabolic potential [31], nematic liquid crystals [32], the misaligned slab system with a rectangular aperture [33], turbulent ocean [34], and inhomogeneous medium [35], has been systematically investigated. The unique nature of an Airy beam makes it have extensive application prospects in the fields of optical removal of particles [36], curved plasma channel [37], spatiotemporal light bullets [38], femtosecond laser micromachining [39], super-resolution optical imaging [40], photonic interconnection [41], optical trapping/micromanipulation [42], and light sheet microscopy [43].

In this paper, the Airyprime beam whose form is similar to the form of the Airy beam is introduced in the Cartesian coordinate system. The pseudo Airyprime beams, whose electric field in the arbitrary transverse direction of the source plane is determined by the product of two Airyprime functions with opposite signs, have been reported [44–47]. In fact, the Airyprime beam is included in the output beam of the Airy transformation of the standard Laguerre-Gaussian beams [48], the standard Hermite-Gaussian beams [49], and the elegant Hermite-Gaussian beams [50]. Because the Airyprime beam is mixed with the Airy beam in the above Airy transformation, the propagation characteristics of the Airyprime beam have not been fully studied. In the remainder of this paper, we focus on the interference enhancement effect in a single Airyprime beam propagating in free space, which is the most significant difference between the Airyprime and the Airy beams.

2. Theoretical derivation of a single Airyprime beam propagating in free space

In the Cartesian coordinate system, x and y are two transverse coordinates, and z is the longitudinal coordinate, which is consistent with the direction of beam propagation. The electric field of an Airy beam in the initial plane z = 0 is characterized by

$${E_A}(x,y,0)\textrm{ = }C\exp \left( {\frac{{ax}}{{{w_0}}}} \right)Ai\left( {\frac{x}{{{w_0}}}} \right)\exp \left( {\frac{{ay}}{{{w_0}}}} \right)Ai\left( {\frac{y}{{{w_0}}}} \right),$$
where a is an exponential decay factor, and w0 is a scaling factor. C is a control parameter of the light intensity and ensures that the peak light intensity in the initial plane is always 1. According to the form of the above Airy beam, an Airyprime beam is introduced. The electric field of an Airyprime beam in the initial plane z = 0 is described by
$$E(x,y,0) = C\exp \left( {\frac{{ax}}{{{w_0}}}} \right)Ai^{\prime}\left( {\frac{x}{{{w_0}}}} \right)\exp \left( {\frac{{ay}}{{{w_0}}}} \right)Ai^{\prime}\left( {\frac{y}{{{w_0}}}} \right),$$
where Ai′(·) is the Airyprime function. The paraxial propagation of the Airyprime beam in free space is governed by [51]:
$$E(x,y,z) = \frac{k}{{2\pi iz}}\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {E(x^{\prime},y^{\prime},0)} \exp \left\{ {\frac{{ik}}{{2z}}[{{(x - x^{\prime})}^2} + {{(y - y^{\prime})}^2}]} \right\}dx^{\prime}} dy^{\prime},$$
where k = 2π/λ with λ being the wavelength. As the electric field in the initial plane can be separable in the x- and y-directions, we first derive the electric field in the x-direction. By using the following integral formulae [52,53]:
$$Ai^{\prime}(x)\textrm{ = }\frac{i}{{2\pi }}\int_{ - \infty }^\infty {u\exp \left( {\frac{{i{u^3}}}{3} + ixu} \right)} du,$$
$$\int_{ - \infty }^\infty {\exp ( - {p^2}{x^2} + qx} )dx\textrm{ = }\frac{{\sqrt \pi }}{p}\exp \left( {\frac{{{q^2}}}{{4{p^2}}}} \right),$$
$$\int_{ - \infty }^\infty {u\exp \left( {\frac{{i{u^3}}}{3} + ib{u^2} + icu} \right)} du\textrm{ = } - 2\pi \exp \left( {\frac{{2i{b^3}}}{3} - ibc} \right)[bAi(c - {b^2}) + iAi^{\prime}(c - {b^2})],$$
the electric field in the x-direction of the Airyprime beam propagating in free space can be analytically expressed as
$$E(x,z) = {E_{Ap}}(x,z) + {E_{Ar}}(x,z),$$
with EAp(x, z) and EAr(x, z) being given by
$${E_{Ap}}(x,z) = \sqrt C \exp \left( {\frac{{ax}}{{{w_0}}} - \frac{{a{z^2}}}{{2z_0^2}} + \frac{{i{a^2}z}}{{2{z_0}}} + \frac{{ixz}}{{2{w_0}{z_0}}} - \frac{{i{z^3}}}{{12z_0^3}}} \right)Ai^{\prime}\left( {\frac{x}{{{w_0}}} - \frac{{{z^2}}}{{4z_0^2}} + \frac{{iaz}}{{{z_0}}}} \right),$$
$${E_{Ar}}(x,z) = \sqrt C \frac{{iz}}{{2{z_0}}}\exp \left( {\frac{{ax}}{{{w_0}}} - \frac{{a{z^2}}}{{2z_0^2}} + \frac{{i{a^2}z}}{{2{z_0}}} + \frac{{ixz}}{{2{w_0}{z_0}}} - \frac{{i{z^3}}}{{12z_0^3}}} \right)Ai\left( {\frac{x}{{{w_0}}} - \frac{{{z^2}}}{{4z_0^2}} + \frac{{iaz}}{{{z_0}}}} \right),$$
where ${z_0} = kw_0^2$. The subscripts Ap and Ar denote the Airyprime and the Airy-related modes, respectively. Equation (7) manifests that a single Airyprime beam propagating in free space doesn't keep the Airyprime form. Upon propagation in free space, an Airy-related mode, which is the conventional propagating Airy mode with an additional π/2 phase shift and a weight coefficient of half the normalized propagation distance, will emerge in the propagating Airyprime beam. Therefore, a single propagating Airyprime beam in the transverse direction is the coherent superposition of the Airyprime and the Airy-related modes, which will result in the interference enhancement effect under the appropriate condition. The light intensity in the x-direction of the Airyprime beam propagating in free space is found to be
$$I(x,z) = {I_{Ap}}(x,z) + {I_{Ar}}(x,z) + {I_{cr}}(x,z),$$
with IAp(x, z), IAr(x, z) and Icr(x, z) being given by
$${I_{Ap}}(x,z) = C\exp \left( {\frac{{2ax}}{{{w_0}}} - \frac{{a{z^2}}}{{z_0^2}}} \right){\left|{Ai^{\prime}\left( {\frac{x}{{{w_0}}} - \frac{{{z^2}}}{{4z_0^2}} + \frac{{iaz}}{{{z_0}}}} \right)} \right|^2},$$
$${I_{Ar}}(x,z) = \frac{{C{z^2}}}{{4z_0^2}}\exp \left( {\frac{{2ax}}{{{w_0}}} - \frac{{a{z^2}}}{{z_0^2}}} \right){\left|{Ai\left( {\frac{x}{{{w_0}}} - \frac{{{z^2}}}{{4z_0^2}} + \frac{{iaz}}{{{z_0}}}} \right)} \right|^2},$$
$$\begin{array}{l} {I_{cr}}(x,z) = \frac{{iCz}}{{2{z_0}}}\exp \left( {\frac{{2ax}}{{{w_0}}} - \frac{{a{z^2}}}{{z_0^2}}} \right)\left[ {A{{i^{\prime}}^ \ast }\left( {\frac{x}{{{w_0}}} - \frac{{{z^2}}}{{4z_0^2}} + \frac{{iaz}}{{{z_0}}}} \right)Ai\left( {\frac{x}{{{w_0}}} - \frac{{{z^2}}}{{4z_0^2}} + \frac{{iaz}}{{{z_0}}}} \right)} \right.\\ \begin{array}{ccc} {}&{}&{} \end{array}\left. { - Ai^{\prime}\left( {\frac{x}{{{w_0}}} - \frac{{{z^2}}}{{4z_0^2}} + \frac{{iaz}}{{{z_0}}}} \right)A{i^ \ast }\left( {\frac{x}{{{w_0}}} - \frac{{{z^2}}}{{4z_0^2}} + \frac{{iaz}}{{{z_0}}}} \right)} \right], \end{array}$$
where the asterisk indicates the complex conjugation. IAp(x, z) and IAr(x, z) correspond to the light intensity in the x-direction of the Airyprime and the Airy-related modes, respectively. Icr(x, z) means the light intensity in the x-direction of the cross term. Similarly, the analytically electric field in the y-direction of the Airyprime beam propagating in free space is obtained by
$$\begin{array}{l} E(y,z) = {E_{Ap}}(y,z) + {E_{Ar}}(y,z) = \sqrt C \exp \left( {\frac{{ay}}{{{w_0}}} - \frac{{a{z^2}}}{{2z_0^2}} + \frac{{i{a^2}z}}{{2{z_0}}} + \frac{{iyz}}{{2{w_0}{z_0}}} - \frac{{i{z^3}}}{{12z_0^3}}} \right)\\ \begin{array}{ccc} {}&{}&{\begin{array}{cc} {}& \times \end{array}} \end{array}\left[ {Ai^{\prime}\left( {\frac{y}{{{w_0}}} - \frac{{{z^2}}}{{4z_0^2}} + \frac{{iaz}}{{{z_0}}}} \right) + \frac{{iz}}{{2{z_0}}}Ai\left( {\frac{y}{{{w_0}}} - \frac{{{z^2}}}{{4z_0^2}} + \frac{{iaz}}{{{z_0}}}} \right)} \right], \end{array}$$
The light intensity in the y-direction of the Airyprime beam propagating in free space is found to be I(y, z) = |E(y, z)|2. The electric field and the light intensity of a single Airyprime beam propagating in free space read as
$$E(x,y,z) = E(x,z)E(y,z),$$
$$I(x,y,z) = I(x,z)I(y,z),$$
Now, we prove that the electric field of a single Airyprime beam propagating in free space satisfies the paraxial wave equation:
$$\frac{{{\partial ^2}E(x,y,z)}}{{\partial {x^2}}} + \frac{{{\partial ^2}E(x,y,z)}}{{\partial {y^2}}} + 2ik\frac{{\partial E(x,y,z)}}{{\partial z}} = 0.$$

Inserting Eq. (15) into Eq. (17), one can obtain

$$E(y,z)\left[ {\frac{{{\partial^2}E(x,z)}}{{\partial {x^2}}}\textrm{ + }2ik\frac{{\partial E(x,z)}}{{\partial z}}} \right] + E(x,z)\left[ {\frac{{{\partial^2}E(y,z)}}{{\partial {y^2}}} + 2ik\frac{{\partial E(y,z)}}{{\partial z}}} \right] = 0.$$
By using ∂Ai′(x)/∂x = xAi(x), the following equations are found to be zero:
$$\frac{{{\partial ^2}E(x,z)}}{{\partial {x^2}}}\textrm{ + }2ik\frac{{\partial E(x,z)}}{{\partial z}} = 0,$$
$$\frac{{{\partial ^2}E(y,z)}}{{\partial {y^2}}} + 2ik\frac{{\partial E(y,z)}}{{\partial z}} = 0.$$
Therefore, the electric field of a single Airyprime beam propagating in free space is the solution of the paraxial wave equation.

For the convenience of comparison, a single Airy beam propagating in free space is listed as follow:

$$\begin{array}{l} {E_A}(x,y,z) = C\exp \left( {\frac{{ax}}{{{w_0}}} + \frac{{ay}}{{{w_0}}} - \frac{{a{z^2}}}{{z_0^2}} + \frac{{i{a^2}z}}{{{z_0}}} + \frac{{ixz}}{{2{w_0}{z_0}}} + \frac{{iyz}}{{2{w_0}{z_0}}} - \frac{{i{z^3}}}{{6z_0^3}}} \right)\\ \begin{array}{ccc} {}&{}&{\begin{array}{cc} {}&{} \end{array}} \end{array} \times Ai\left( {\frac{x}{{{w_0}}} - \frac{{{z^2}}}{{4z_0^2}} + \frac{{iaz}}{{{z_0}}}} \right)Ai\left( {\frac{y}{{{w_0}}} - \frac{{{z^2}}}{{4z_0^2}} + \frac{{iaz}}{{{z_0}}}} \right), \end{array}$$
Equation (21) shows that the Airy beam propagating in free space keeps the Airy form unchanged.

To insight the evolution of the beam profile, we now investigate the transverse energy flux of a single Airyprime beam during the free space propagation. The transverse energy flux of a single Airyprime beam propagating in free space is described by the transverse Poynting vector. The transverse Poynting vector S⊥ is defined by [54]

$${{\boldsymbol S}_ \bot } = \frac{1}{{2k}}{\mathop{\rm Im}\nolimits} \left[ {{E^ \ast }(x,y,z)\frac{{\partial E(x,y,z)}}{{\partial x}}{{\boldsymbol e}_x} + {E^ \ast }(x,y,z)\frac{{\partial E(x,y,z)}}{{\partial y}}{{\boldsymbol e}_y}} \right],$$
where Im denotes taking the imaginary part. ex and ey are unit vectors along the x- and y-directions of the Cartesian coordinate system. Inserting Eq. (15) into Eq. (22), one can obtain the transverse Poynting vector of a single Airyprime beam propagating in free space.

3. Theoretical calculations and analyses

In this section, the interference enhancement effect of a single Airyprime beam propagating in free space is explored by using theoretical calculations. The wavelength is fixed to be λ=532 nm in the following calculations.

Figure 1 represents the contour graph of the intensity distribution in the x-z plane of different Airyprime beams. To facilitate comparative analysis, the intensity distribution in the x-z plane of different Airy beams is also plotted in Fig. 1. We note that the light intensity of a single propagating Airyprime beam in some observation planes of free space will exceed 1, which is caused by the interference enhancement of the Airyprime and the Airy-related modes. The range between the two vertical dotted lines is the range of interference enhancement. Here, the phenomenon of light intensity being greater than 1 is called as the interference enhancement effect. Moreover, the strength of interference enhancement effect is described by the maximum light intensity. The greater the maximum light intensity, the stronger the interference enhancement effect. When a = 0.1 and w0 = 0.1 mm, the maximum light intensity of a single Airyprime beam propagating in free space is 1.19, and the position of maximum light intensity is z = 0.579 m, as well as the range of interference enhancement is 0.411 m ≤ z≤0.762 m. When a = 0.12 and w0 = 0.1 mm, the maximum light intensity of a single Airyprime beam propagating in free space is 1.03, and the position of maximum light intensity is z = 0.537 m, as well as the range of interference enhancement is 0.469 m ≤ z≤0.604 m. When a = 0.1 and w0 = 0.05 mm, the maximum light intensity of a single Airyprime beam propagating in free space is same as that in the case of a = 0.1 and w0 = 0.1 mm, and the position of maximum light intensity is z = 0.145 m, as well as the range of interference enhancement is 0.103 m ≤ z≤0.191 m. Under the same maximum light intensity, the expansion of the range of interference enhancement effect can be realized by increasing the scaling factor w0. No matter how the exponential decay factor a and the scaling factor w0 are changed, however, the maximum light intensity of a single Airy beam propagating in free space always remains 1, and the position of the maximum light intensity must be in the initial plane. The white dashed curves marked in Fig. 1 denote the trajectory of peak intensity. The trajectory of peak intensity of a single Airy beam first moves upward and then trembles. While the trajectory of peak intensity of a single Airyprime beam consists of two segment curves, and the first segment trajectory of peak intensity is lower than the corresponding trajectory of peak intensity of a single Airy beam. However, the second segment trajectory of peak intensity of a single Airyprime beam is more upward than the corresponding trajectory of peak intensity of a single Airy beam.

 figure: Fig. 1.

Fig. 1. Contour graph of the intensity distribution in the x-z plane of the Airyprime (the left column) and the Airy (the right column) beams propagating in free space. (a) and (b) a = 0.1 and w0 = 0.1 mm; (c) and (d) a = 0.12 and w0 = 0.1 mm; (e) and (f) a = 0.1 and w0 = 0.05 mm.

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To better illustrate Fig. 1, the intensity distribution in the x-direction of the two components for the Airyprime beam in the initial plane z = 0 and the observation plane z = 0.5 m is displayed in Fig. 2, and the intensity distribution in the x-direction of the Airy beam in different observation planes is demonstrated in Fig. 3. The parameters a = 0.1 and w0 = 0.1 mm are used for calculations in Figs. 2 and 3. When z = 0, the peak intensity of a single Airyprime beam appears on the second peak from the right. When z = 0.5 m, the peak intensity of a single Airyprime beam emerges on the first peak from the right. The phenomenon that the trajectory of peak intensity of a single Airyprime beam jumps from down to up corresponds to the process of the peak light intensity migrating from the second peak to the first peak. When the single Airyprime beam leaves the initial plane, the Airy-related mode appears. The peak intensities in the x-direction of the Airyprime mode and the Airy beams both decrease upon free space propagation. As shown in Figs. 2(c) and 3(b), the peak intensities in the x-direction of the observation plane z = 0.5 m for the Airyprime mode and the Airy beams are both smaller than 1. The weight coefficient of the Airy-related mode is proportional to the propagation distance. When the Airy-related mode propagates to a certain observation plane, the superposition effect caused by the positive action of the weight coefficient and the negative action of the peak intensity of the Airy beam makes the peak intensity of the Airy-related mode greater than 1. When z = 0.5 m, the peak intensity in the x-direction of the Airy-related mode is far greater than 1. The tail variation in the trajectory of peak intensity of a single Airy beam in Fig. 1(b) can interpreted as follow. As z0 = 0.118 m, the tail variation in the trajectory of peak intensity of a single Airy beam corresponds to the far field. A single Airy beam in the far field will evolve into a Gaussian-like beam. During the evolution of a single Airy beam into a Gaussian-like beam, the trajectory of peak intensity moves toward the negative direction of the x-axis, which is shown in Figs. 3(c) and 3(d).

 figure: Fig. 2.

Fig. 2. The intensity distribution in the x-direction of the two components for the Airyprime beam in the initial plane z = 0 and the observation plane z = 0.5 m. a = 0.1 and w0 = 0.1 mm. (a) I(x, 0); (b) I(x, 0.5 m); (c) IAp(x, 0.5 m); (d) IAr(x, 0.5 m).

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 figure: Fig. 3.

Fig. 3. The intensity distribution in the x-direction of the Airy beam in different observation planes. a = 0.1 and w0 = 0.1 mm. (a) z = 0; (b) z = 0.5 m; (c) z = 0.8 m; (d) z = 1 m.

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The intensity distributions in the x-z plane of the Airyprime beam, the Airyprime mode, the Airy-related mode and the cross term are demonstrated in Figs. 4 and 5. a = 0.1 and w0 = 0.1 mm in Fig. 4, and a = 0.15 and w0 = 0.1 mm in Fig. 5. Figure 4 shows that the region of interference enhancement effect stems from the Airy-related mode. The region of interference enhancement effect originates from the joint action of the conventional propagating Airy mode and the weight coefficient proportional to the propagating distance. Icr(x, z) is always less than or equal to zero in Figs. 4 and 5. Figure 5 manifests that there is no interference enhancement effect in the case of a = 0.15 and w0 = 0.1 mm. With the increase of the exponential decay factor a, the maximum light intensity of IAp(x, z) increases, while the maximum light intensity of IAr(x, z) decreases. More importantly, the maximum negative value of Icr(x, z) increases rapidly by increasing the exponential decay factor a.

 figure: Fig. 4.

Fig. 4. Contour graph of the intensity distribution in the x-z plane of the Airyprime beam with a = 0.1 and w0 = 0.1 mm. (a) I(x, z); (b) IAp(x, z); (c) IAr(x, z); (d) Icr(x, z).

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 figure: Fig. 5.

Fig. 5. Contour graph of the intensity distribution in the x-z plane of the Airyprime beam with a = 0.15 and w0 = 0.1 mm. (a) I(x, z); (b) IAp(x, z); (c) IAr(x, z); (d) Icr(x, z).

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The maximum light intensity and the corresponding position of a single Airyprime beam propagating in free space depend on the exponential decay factor a and the scaling factor w0. First, the effect of the exponential decay factor a on the maximum light intensity and the corresponding position of a single Airyprime beam propagating in free space is investigated. The maximum light intensity and the corresponding position of a single Airyprime beam propagating in free space as a function of the exponential decay factor a is shown in Fig. 6. Imax(x, y, z) and zm denote the maximum light intensity and the corresponding position, respectively. The colored lines in Fig. 6 are added for illustration. w0 is fixed to be 0.1 mm and the exponential decay factor a varies from 0.02 to 0.16 in Fig. 6. When the exponential decay factor a is greater than or equal to 0.125, the interference enhancement effect disappears, and the maximum light intensity of a single propagating Airyprime beam is always 1, as well as the position of maximum light intensity appears in the initial plane. Therefore, a = 0.125 is the saturated value of the interference enhancement effect. Note that this saturated value is independent of the value of w0. In Eqs. (8) and (9), one can replace x/w0, y/w0 and z/z0 with three dimensionless variables. As a result, the only remaining beam parameter is a. With the decrease of the exponential decay factor a, the maximum light intensity of a single propagating Airyprime beam increases, and the position of maximum light intensity is getting farther away.

 figure: Fig. 6.

Fig. 6. The maximum light intensity (a) and the corresponding position (b) of a single propagating Airyprime beam with w0 = 0.1 mm as a function of the exponential decay factor a.

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Then, the influence of the scaling factor w0 on the maximum light intensity and the corresponding position of a single propagating Airyprime beam is examined, which is displayed in Fig. 7. The exponential decay factor a is fixed to be 0.1 and the scaling factor w0 varies from 0.02 mm to 0.16 mm in Fig. 7. With the increase of the scaling factor w0, the position of maximum light intensity of a single propagating Airyprime beam is extended, which is caused by the increase of z0. However, the maximum light intensity of a single propagating Airyprime beam is independent of the scaling factor w0.

 figure: Fig. 7.

Fig. 7. The maximum light intensity (a) and the corresponding position (b) of a single propagating Airyprime beam with a = 0.1 as a function of the scaling factor w0.

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Figure 8(a) shows the transverse coordinates of maximum light intensity of a single propagating Airyprime beam with w0 = 0.1 mm as a function of the exponential decay factor a. The transverse coordinates of maximum light intensity are indicated by xm and ym. Moreover, xm = ym. When the exponential decay factor a increases and doesn't exceed the saturated value 0.125, the transverse coordinates of maximum light intensity decrease. When the exponential decay factor a further increases from the saturated value 0.125, the change of xm and ym is too small to be detected. When a < 0.125, xm and ym are positive. When a≥0.125, xm and ym are negative. The transverse coordinates of maximum light intensity of a single propagating Airyprime beam with a = 0.1 as a function of the scaling factor w0 is displayed in Fig. 8(b). With the increase of the scaling factor w0, the transverse coordinates of maximum light intensity also increase. xm and ym are proportional to w0. The curve in Fig. 8(b) can be described approximately by xm = ym = 5.431w0−0.005 mm.

 figure: Fig. 8.

Fig. 8. (a) The transverse coordinates of maximum light intensity of a single propagating Airyprime beam with w0 = 0.1 mm as a function of the exponential decay factor a; (b) The transverse coordinates of maximum light intensity of a single propagating Airyprime beam with a = 0.1 as a function of the scaling factor w0.

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Finally, the intensity distribution of a single propagating Airyprime beam in different observation planes of free space is demonstrated in Figs. 9–11. w0 is fixed to be 0.1 mm, and a = 0.1, 0.04, and 0.02 in Figs. 9–11, respectively. Upon propagation in free space, the first lobe along the diagonal will evolve from the weak lobe to the dominant lobe, and the number of the lobes decreases. The magnitude of light intensity in the process of free space propagation first decreases, then increases, and finally decreases, which is not shown to save space. Figures 9(d), 10(d), and 11(d) correspond to the intensity profile in the observation plane of the strongest interference enhancement effect. When the interference enhancement effect is the strongest, the light intensity reaches the maximum, and the number of side lobes is the least, as well as two weak tails along the two transverse directions appears in the intensity pattern. The maximum light intensity in the case of a = 0.1, 0.04, and 0.02 is 1.42, 5.49, and 12.82, respectively. When w0 = 0.1 mm, the strongest interference enhancement effect requires a long propagation distance to occur. To shorten the position of the maximum light intensity zm, the effective approach is to reduce the scaling factor w0, which is shown in Fig. 12. a = 0.1 and w0 = 0.07 mm in Fig. 12. Although w0 decreases only a little, the position of the maximum light intensity is nearly half of that in the case of w0 = 0.1 mm.

 figure: Fig. 9.

Fig. 9. The intensity distribution of a single propagating Airyprime beam in different observation planes of free space. a = 0.1 and w0 = 0.1 mm. (a) z = 0. (b) z = 0.35 m. (c) z = 0.4 m. (d) z = 0.579 m.

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 figure: Fig. 10.

Fig. 10. The intensity distribution of a single propagating Airyprime beam in different observation planes of free space. a = 0.04 and w0 = 0.1 mm. (a) z = 0. (b) z = 0.37 m. (c) z = 0.6 m. (d) z = 0.869 m.

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 figure: Fig. 11.

Fig. 11. The intensity distribution of a single propagating Airyprime beam in different observation planes of free space. a = 0.02 and w0 = 0.1 mm. (a) z = 0. (b) z = 0.4 m. (c) z = 0.8 m. (d) z = 1.203 m.

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 figure: Fig. 12.

Fig. 12. The intensity distribution of a single propagating Airyprime beam in different observation planes of free space. a = 0.1 and w0 = 0.07 mm. (a) z = 0. (b) z = 0.18 m. (c) z = 0.2 m. (d) z = 0.2832 m.

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Figure 13 shows the transverse Poynting vector of a single propagating Airyprime beam with a = 0.04 and w0 = 0.1 mm in different observation planes of free space. When 0 < z < zm, the transverse energy flux flows out from the other lobes, and flows into the first lobe along the diagonal. When z = zm, the energy is mainly concentrated on the dominant lobe, which is the first lobe along the diagonal and evolved from the weak lobe. When z ≥ zm, and the transverse energy flux flows out from the dominant lobe. The flow direction of transverse energy flux is consistent with the conclusion on the maximum light intensity and the corresponding position.

 figure: Fig. 13.

Fig. 13. The transverse Poynting vector of a single propagating Airyprime beam in different observation planes of free space. a = 0.04 and w0 = 0.1 mm. (a) z = 0. (b) z = 0.37 m. (c) z = 0.6 m. (d) z = 0.869 m.

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This above interference enhancement effect can resist the light intensity loss caused by atmospheric disturbance, and it is very useful for atmospheric optical communication. Long distance atmospheric optical communication can be realized by reducing the exponential decay factor a and increasing the scaling factor w0. Moreover, the increase of the scaling factor w0 can expand the range of interference enhancement effect.

4. Experimental results

In this section, we carry out the experiment to generate the Airyprime beam and to measure the intensity profile of a single propagating Airyprime beam. Figure 14 shows the corresponding schematic diagram of the experimental setup. A Gaussian beam with λ=532 nm produced by a solid-stated laser (Laser Quantum, model ventus) is expanded by a 50× beam expander (BE), and then is reflected by a reflecting mirror (RM), and finally is split into two identical portions by a 50: 50 intensity beam splitter (BS). One portion is incident into a reflective mode spatial light modulator (SLM, Holoeye LETO-3, pixel size: 6.4 µm×6.4 µm) which is uploaded with predesigned holograms that contain the essential information of the Airyprime beam. The method of phase plate synthesis we used in SLM can be found in Ref. [55]. After reflection from the SLM, the modulated light passes through a 4f system which contains two identical lenses L1 and L2 with the focal length f1 = f2 = 250 mm. A circular aperture (CA) placed in the rear focal plane of L1 is used to select out the first diffraction order and to filter out other unwanted diffraction orders. In the rear focal plane of L2 namely the initial plane z = 0, one can obtain the expected Airyprime beam. A beam profile analysis (BPA, Ophir, model BGS-USB3-LT665, pixel pitch: 4.4 µm; pixel resolution: 2752 × 2192) mounted on the electric translation stage along the propagation z-axis is used to recorded the intensity distribution of a single Airyprime beam propagating in free space.

 figure: Fig. 14.

Fig. 14. The experimental setup of the generation of the Airyprime beam and the measurement of the intensity profile of a single propagating Airyprime beam.

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Figures 15–18 represent the corresponding experimental results of intensity distribution of a single propagating Airyprime beam in different observation planes of free space. w0 = 0.1 mm, and a = 0.1, 0.04, and 0.02 in Figs. 15–17, respectively. w0 = 0.07 mm and a = 0.1 in Fig. 18. The experimental beam profiles in Figs. 15–18 are well consistent with the theoretical beam profiles as shown in Figs. 9–12. The deviation between the experimental and the theoretical light intensities is no more than 5%. As long as a is less than the saturation value 0.125, the phenomenon of the interference enhancement effect must be observed. The smaller a is, moreover, the stronger the interference enhancement effect is. Without affecting the strength of interference enhancement effect, the position of the maximum light intensity can be regulated by w0.

 figure: Fig. 15.

Fig. 15. Experimental result of the intensity distribution of a single propagating Airyprime beam in different observation planes of free space. a = 0.1 and w0 = 0.1 mm. (a) z = 0. (b) z = 0.35 m. (c) z = 0.4 m. (d) z = 0.579 m.

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 figure: Fig. 16.

Fig. 16. Experimental result of the intensity distribution of a single propagating Airyprime beam in different observation planes of free space. a = 0.04 and w0 = 0.1 mm. (a) z = 0. (b) z = 0.37 m. (c) z = 0.6 m. (d) z = 0.869 m.

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 figure: Fig. 17.

Fig. 17. Experimental result of the intensity distribution of a single propagating Airyprime beam in different observation planes of free space. a = 0.02 and w0 = 0.1 mm. (a) z = 0. (b) z = 0.4 m. (c) z = 0.8 m. (d) z = 1.203 m.

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 figure: Fig. 18.

Fig. 18. Experimental result of the intensity distribution of a single propagating Airyprime beam in different observation planes of free space. a = 0.1 and w0 = 0.07 mm. (a) z = 0. (b) z = 0.18 m. (c) z = 0.2 m. (d) z = 0.2832 m.

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5. Summary

An analytical expression of a single Airyprime beam propagating in free space is derived by means of mathematical techniques. As we all know, the Airy beam propagating in free space still keeps the Airy form. However, the Airyprime beam propagating in free space doesn't keep the Airyprime form. Upon propagation in free space, a single Airyprime beam in arbitrary transverse direction is the coherent superposition of the Airyprime and the Airy-related modes. The Airy-related mode is the conventional propagating Airy mode with an additional π/2 phase shift and a weight coefficient of half the normalized propagation distance. These two modes will produce the interference enhancement effect under the appropriate condition. As the peak light intensity of a single Airyprime beam in the initial plane is set to be 1, the strength of interference enhancement effect is described by the maximum light intensity.

The effects of the exponential decay factor a and the scaling factor w0 on the maximum light intensity and the corresponding position of a single Airyprime beam propagating in free space are investigated, respectively. The maximum light intensity of a single Airyprime beam propagating in free space is independent of the scaling factor w0 and is only decided by the exponential decay factor a. When the exponential decay factor a is above the saturated value 0.125, the interference enhancement effect disappears. When the exponential decay factor a decreases from the saturated value, the maximum light intensity of a single propagating Airyprime beam increases, and the position of maximum light intensity is getting farther away. When a = 0.02, the maximum light intensity can be theoretically up to 12.82. With the increase of the scaling factor w0, the position of maximum light intensity of a single propagating Airyprime beam is extended. In order to better understand the interference enhancement effect, the intensity distribution and the transverse Poynting vector of a single propagating Airyprime beam are demonstrated in different observation planes of free space. The flow direction of transverse energy flux effectively supports the interference enhancement effect of a single propagating Airyprime beam.

Also, the Airyprime beam is experimentally generated, and the interference enhancement effect of a single propagating Airyprime beam is experimentally confirmed. By selecting the smaller a and w0, a strong interference enhancement effect appears on the observation plane close to the initial plane. By selecting the smaller a and the larger w0, a strong interference enhancement effect appears on the observation plane away from the initial plane. In this research, the interference enhancement effect of a single Airyprime beam propagating in free space is discovered, which is not possessed by a single propagating Airy beam. The interference enhancement effect is conducive to the practical application of a single Airyprime beam.

Funding

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

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (18)
Fig. 1.
Fig. 1. Contour graph of the intensity distribution in the x-z plane of the Airyprime (the left column) and the Airy (the right column) beams propagating in free space. (a) and (b) a = 0.1 and w0 = 0.1 mm; (c) and (d) a = 0.12 and w0 = 0.1 mm; (e) and (f) a = 0.1 and w0 = 0.05 mm.

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Fig. 2.
Fig. 2. The intensity distribution in the x-direction of the two components for the Airyprime beam in the initial plane z = 0 and the observation plane z = 0.5 m. a = 0.1 and w0 = 0.1 mm. (a) I(x, 0); (b) I(x, 0.5 m); (c) IAp(x, 0.5 m); (d) IAr(x, 0.5 m).

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Fig. 3.
Fig. 3. The intensity distribution in the x-direction of the Airy beam in different observation planes. a = 0.1 and w0 = 0.1 mm. (a) z = 0; (b) z = 0.5 m; (c) z = 0.8 m; (d) z = 1 m.

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Fig. 4.
Fig. 4. Contour graph of the intensity distribution in the x-z plane of the Airyprime beam with a = 0.1 and w0 = 0.1 mm. (a) I(x, z); (b) IAp(x, z); (c) IAr(x, z); (d) Icr(x, z).

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Fig. 5.
Fig. 5. Contour graph of the intensity distribution in the x-z plane of the Airyprime beam with a = 0.15 and w0 = 0.1 mm. (a) I(x, z); (b) IAp(x, z); (c) IAr(x, z); (d) Icr(x, z).

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Fig. 6.
Fig. 6. The maximum light intensity (a) and the corresponding position (b) of a single propagating Airyprime beam with w0 = 0.1 mm as a function of the exponential decay factor a.

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Fig. 7.
Fig. 7. The maximum light intensity (a) and the corresponding position (b) of a single propagating Airyprime beam with a = 0.1 as a function of the scaling factor w0.

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Fig. 8.
Fig. 8. (a) The transverse coordinates of maximum light intensity of a single propagating Airyprime beam with w0 = 0.1 mm as a function of the exponential decay factor a; (b) The transverse coordinates of maximum light intensity of a single propagating Airyprime beam with a = 0.1 as a function of the scaling factor w0.

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Fig. 9.
Fig. 9. The intensity distribution of a single propagating Airyprime beam in different observation planes of free space. a = 0.1 and w0 = 0.1 mm. (a) z = 0. (b) z = 0.35 m. (c) z = 0.4 m. (d) z = 0.579 m.

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Fig. 10.
Fig. 10. The intensity distribution of a single propagating Airyprime beam in different observation planes of free space. a = 0.04 and w0 = 0.1 mm. (a) z = 0. (b) z = 0.37 m. (c) z = 0.6 m. (d) z = 0.869 m.

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Fig. 11.
Fig. 11. The intensity distribution of a single propagating Airyprime beam in different observation planes of free space. a = 0.02 and w0 = 0.1 mm. (a) z = 0. (b) z = 0.4 m. (c) z = 0.8 m. (d) z = 1.203 m.

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Fig. 12.
Fig. 12. The intensity distribution of a single propagating Airyprime beam in different observation planes of free space. a = 0.1 and w0 = 0.07 mm. (a) z = 0. (b) z = 0.18 m. (c) z = 0.2 m. (d) z = 0.2832 m.

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Fig. 13.
Fig. 13. The transverse Poynting vector of a single propagating Airyprime beam in different observation planes of free space. a = 0.04 and w0 = 0.1 mm. (a) z = 0. (b) z = 0.37 m. (c) z = 0.6 m. (d) z = 0.869 m.

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Fig. 14.
Fig. 14. The experimental setup of the generation of the Airyprime beam and the measurement of the intensity profile of a single propagating Airyprime beam.

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Fig. 15.
Fig. 15. Experimental result of the intensity distribution of a single propagating Airyprime beam in different observation planes of free space. a = 0.1 and w0 = 0.1 mm. (a) z = 0. (b) z = 0.35 m. (c) z = 0.4 m. (d) z = 0.579 m.

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Fig. 16.
Fig. 16. Experimental result of the intensity distribution of a single propagating Airyprime beam in different observation planes of free space. a = 0.04 and w0 = 0.1 mm. (a) z = 0. (b) z = 0.37 m. (c) z = 0.6 m. (d) z = 0.869 m.

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Fig. 17.
Fig. 17. Experimental result of the intensity distribution of a single propagating Airyprime beam in different observation planes of free space. a = 0.02 and w0 = 0.1 mm. (a) z = 0. (b) z = 0.4 m. (c) z = 0.8 m. (d) z = 1.203 m.

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Fig. 18.
Fig. 18. Experimental result of the intensity distribution of a single propagating Airyprime beam in different observation planes of free space. a = 0.1 and w0 = 0.07 mm. (a) z = 0. (b) z = 0.18 m. (c) z = 0.2 m. (d) z = 0.2832 m.

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Equations (22)

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

E A ( x , y , 0 )  =  C exp ⁡ ( a x w 0 ) A i ( x w 0 ) exp ⁡ ( a y w 0 ) A i ( y w 0 ) ,
E ( x , y , 0 ) = C exp ⁡ ( a x w 0 ) A i ′ ( x w 0 ) exp ⁡ ( a y w 0 ) A i ′ ( y w 0 ) ,
E ( x , y , z ) = k 2 π i z ∫ − ∞ ∞ ∫ − ∞ ∞ E ( x ′ , y ′ , 0 ) exp ⁡ { i k 2 z [ ( x − x ′ ) 2 + ( y − y ′ ) 2 ] } d x ′ d y ′ ,
A i ′ ( x )  =  i 2 π ∫ − ∞ ∞ u exp ⁡ ( i u 3 3 + i x u ) d u ,
∫ − ∞ ∞ exp ⁡ ( − p 2 x 2 + q x ) d x  =  π p exp ⁡ ( q 2 4 p 2 ) ,
∫ − ∞ ∞ u exp ⁡ ( i u 3 3 + i b u 2 + i c u ) d u  =  − 2 π exp ⁡ ( 2 i b 3 3 − i b c ) [ b A i ( c − b 2 ) + i A i ′ ( c − b 2 ) ] ,
E ( x , z ) = E A p ( x , z ) + E A r ( x , z ) ,
E A p ( x , z ) = C exp ⁡ ( a x w 0 − a z 2 2 z 0 2 + i a 2 z 2 z 0 + i x z 2 w 0 z 0 − i z 3 12 z 0 3 ) A i ′ ( x w 0 − z 2 4 z 0 2 + i a z z 0 ) ,
E A r ( x , z ) = C i z 2 z 0 exp ⁡ ( a x w 0 − a z 2 2 z 0 2 + i a 2 z 2 z 0 + i x z 2 w 0 z 0 − i z 3 12 z 0 3 ) A i ( x w 0 − z 2 4 z 0 2 + i a z z 0 ) ,
I ( x , z ) = I A p ( x , z ) + I A r ( x , z ) + I c r ( x , z ) ,
I A p ( x , z ) = C exp ⁡ ( 2 a x w 0 − a z 2 z 0 2 ) | A i ′ ( x w 0 − z 2 4 z 0 2 + i a z z 0 ) | 2 ,
I A r ( x , z ) = C z 2 4 z 0 2 exp ⁡ ( 2 a x w 0 − a z 2 z 0 2 ) | A i ( x w 0 − z 2 4 z 0 2 + i a z z 0 ) | 2 ,
I c r ( x , z ) = i C z 2 z 0 exp ⁡ ( 2 a x w 0 − a z 2 z 0 2 ) [ A i ′ ∗ ( x w 0 − z 2 4 z 0 2 + i a z z 0 ) A i ( x w 0 − z 2 4 z 0 2 + i a z z 0 ) − A i ′ ( x w 0 − z 2 4 z 0 2 + i a z z 0 ) A i ∗ ( x w 0 − z 2 4 z 0 2 + i a z z 0 ) ] ,
E ( y , z ) = E A p ( y , z ) + E A r ( y , z ) = C exp ⁡ ( a y w 0 − a z 2 2 z 0 2 + i a 2 z 2 z 0 + i y z 2 w 0 z 0 − i z 3 12 z 0 3 ) × [ A i ′ ( y w 0 − z 2 4 z 0 2 + i a z z 0 ) + i z 2 z 0 A i ( y w 0 − z 2 4 z 0 2 + i a z z 0 ) ] ,
E ( x , y , z ) = E ( x , z ) E ( y , z ) ,
I ( x , y , z ) = I ( x , z ) I ( y , z ) ,
∂ 2 E ( x , y , z ) ∂ x 2 + ∂ 2 E ( x , y , z ) ∂ y 2 + 2 i k ∂ E ( x , y , z ) ∂ z = 0.
E ( y , z ) [ ∂ 2 E ( x , z ) ∂ x 2  +  2 i k ∂ E ( x , z ) ∂ z ] + E ( x , z ) [ ∂ 2 E ( y , z ) ∂ y 2 + 2 i k ∂ E ( y , z ) ∂ z ] = 0.
∂ 2 E ( x , z ) ∂ x 2  +  2 i k ∂ E ( x , z ) ∂ z = 0 ,
∂ 2 E ( y , z ) ∂ y 2 + 2 i k ∂ E ( y , z ) ∂ z = 0.
E A ( x , y , z ) = C exp ⁡ ( a x w 0 + a y w 0 − a z 2 z 0 2 + i a 2 z z 0 + i x z 2 w 0 z 0 + i y z 2 w 0 z 0 − i z 3 6 z 0 3 ) × A i ( x w 0 − z 2 4 z 0 2 + i a z z 0 ) A i ( y w 0 − z 2 4 z 0 2 + i a z z 0 ) ,
S ⊥ = 1 2 k Im [ E ∗ ( x , y , z ) ∂ E ( x , y , z ) ∂ x e x + E ∗ ( x , y , z ) ∂ E ( x , y , z ) ∂ y e y ] ,
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