Generation of finite energy Airyprime beams by Airy transformation

Xiang Zang; Wensong Dan; Fei Wang; Fei Wang; Yimin Zhou; Yiqing Xu; Guoquan Zhou

doi:10.1364/OE.462295

1. Introduction

Optical transformation systems can transform one kind of beam into another with completely different characteristics. Among them, the most famous one is Airy transformation. Airy transformation is a useful tool that generates novel kinds of optical beams, and can modulate the amplitude and phase of a light field. For instance, a fundamental Gaussian beam can be transformed into an Airy beam by Airy transformation [1]. Therefore, the earliest beam model namely Gaussian beam is closely related to the popular Airy beam through Airy transformation. Firstly, the superposition of fundamental Gaussian beams, such as flat-topped Gaussian beams [2], hyperbolic-cosine Gaussian beams [3], double-half inverse Gaussian hollow beams [4], and controllable dark-hollow beams [5], is used to perform the Airy transformation, and the corresponding output beam is related to the Airy beams. Then, the Airy transformation of the higher-order modes, such as standard Laguerre-Gaussian beams [6], the standard Hermite-Gaussian beams [7], elegant Hermite-Gaussian beams [8], and Hermite-cosh-Gaussian beams [9], has been realized. Also, the Airy transformation of some special optical beams such as four-petal Gaussian beams [10], Lorentz-Gauss beams [11], and higher-order cosh-Gaussian beams [12] has been reported. Finally, the Airy transformation of the vortex beams such as Gaussian vortex beams has been implemented [13]. The common point of the Airy transformation of the latter three kinds of optical beams is that the output beam is a mixed beam, which includes the Airy beam and the Airy derivative beam. Due to the iterative relationship, the high order derivative of the Airy function can be expressed as the sum of the Airy and the Airyprime functions [14]. Therefore, the mixed beam must contain the Airy and the Airyprime beams.

The research on the characteristics of the Airy beam is quite thorough [15–31]. However, the characteristics of the Airyprime beam are not clear. Moreover, there will be a thinking inertia that the Airyprime beam has the same characteristics as the Airy beam. Is that thinking inertia true? The previously reported Airyprime beams are not actually Airyprime beams and can be called as 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 [32–35]. Recently, a new kind of abruptly autofocusing beam namely the circular Airyprime beam has been introduced theoretically and realized experimentally by us [36]. Up to now, the Airyprime beam with Cartesian symmetry has not been generated alone, and the properties of the Airyprime beam with Cartesian symmetry have not been reported. Our aim in this paper is to investigate the finite energy Airyprime (FEA) beam with Cartesian symmetry, hereafter, named the FEA beam. The Airy beam can be generated by the Airy transformation of a Gaussian beam. To generate the FEA beam, what kind of beam is needed to perform the Airy transformation, which is naturally what researchers are curious about? Based on the Airy transformation, therefore, a method to generate the FEA beam is introduced in this paper. The propagation characteristics including the intensity distribution, the centroid, the root-mean-square (r.m.s) beam width, the divergence angle and the beam propagation factor of the FEA beam are analyzed in detail both theoretically and experimentally. Also, the evolution characteristics of the FEA beam propagating in free space are examined. We hope to clarify the difference between the FEA and the Airy beams through this study.

2. Generation of the FEA beam by Airy transformation and its propagation characteristics

Let us start from an incident beam which is the coherent superposition of four elegant Hermite-Gaussian modes and propagates along the positive direction of z-axis. In the input plane, the electric field of such incident beam is expressed as

(1)$$\begin{aligned} E({x_0},{y_0})&\textrm{ = }E({x_0})E({y_0}) = 4{(\tau \gamma )^{3/2}}{E_{00}}({x_0},{y_0}) + 2{\tau ^{3/2}}{E_{01}}({x_0},{y_0}) + 2{\gamma ^{3/2}}{E_{10}}({x_0},{y_0}) + {E_{11}}({x_0},{y_0})\\ &= 4\exp \left( { - \frac{{x_0^2 + y_0^2}}{{w_0^2}}} \right)\left( {{\tau^{3/2}} + \frac{{{x_0}}}{{{w_0}}}} \right)\left( {{\gamma^{3/2}} + \frac{{{y_0}}}{{{w_0}}}} \right), \end{aligned}$$

with the elegant Hermite-Gaussian mode E_nm(x₀, y₀) being given by

(2)$${E_{nm}}({x_0},{y_0})\textrm{ = }{H_n}\left( {\frac{{{x_0}}}{{{w_0}}}} \right){H_m}\left( {\frac{{{y_0}}}{{{w_0}}}} \right)\exp \left( { - \frac{{x_0^2 + y_0^2}}{{w_0^2}}} \right),$$

where x₀ and y₀ are the transverse coordinates of the input plane. w₀ is the beam waist of a fundamental Gaussian beam. H_j is the Hermite polynomial of order j. n and m are the transverse mode numbers of elegant Hermite-Gaussian mode. The elegant Hermite-Gaussian mode described by Eq. (2) is called as EHG_nm mode. $\tau = w_0^2/(4{\alpha ^2})$ and $\gamma = w_0^2/(4{\beta ^2})$. α and β are Airy control parameters in two transverse directions of the Airy transform optical system. Moreover, α and β are assumed to be positive. The electric field of this incident beam passing through an Airy transform optical system is characterized by [1]:

(3)$$E(x,y) = E(x)E(y)\textrm{ = }\frac{1}{{\alpha \beta }}\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {E({x_0},{y_0})} } Ai\left( {\frac{{x - {x_0}}}{\alpha }} \right)Ai\left( {\frac{{y - {y_0}}}{\beta }} \right)d{x_0}d{y_0},$$

where x and y are two transverse coordinates of the output plane. Ai(·) is the Airy function. Because the electric field of the incident beam in the x- and y-directions can be separable, we only show the derivation of E(x) in the following analysis for simplicity. E(y) can be obtained by the same way for the derivation of E(x). By inserting the definition of the Airy function into Eq. (3) [37], E(x) can be expressed as

(4)$$E(x)\textrm{ = }\frac{{2\sqrt \tau }}{\pi }\int_{ - \infty }^\infty {({\tau ^{3/2}} + t)\exp \left( { - {t^2} - i2\sqrt \tau ut} \right)dt\int_{ - \infty }^\infty {\exp \left( {\frac{{i{u^3}}}{3} + \frac{{ixu}}{\alpha }} \right)} } du,$$

where t = x₀/w₀. By using the following integral formula [38]:

(5)$$\int_{ - \infty }^\infty {{x^s}\exp ( - {x^2} + 2px} )dx\textrm{ = }\frac{{\sqrt \pi }}{{{2^s}{i^s}}}\exp ({p^2}){H_s}(ip),$$

where s is an integer, E(x) reads as

(6)$$E(x)\textrm{ = }\frac{{2\tau }}{{\sqrt \pi }}\int_{ - \infty }^\infty {(\tau - iu)\exp \left( {\frac{{i{u^3}}}{3} - \tau {u^2} + \frac{{ixu}}{\alpha }} \right)} du.$$

Then using the the following integral formulae [38]:

(7)$$\int_{ - \infty }^\infty {\exp \left( {\frac{{i{u^3}}}{3} + ia{u^2} + ibu} \right)} du\textrm{ = }2\pi \exp \left( {\frac{{2i{a^3}}}{3} - iab} \right)Ai(b - {a^2}),$$

(8)$$\int_{ - \infty }^\infty {u\exp \left( {\frac{{i{u^3}}}{3} + ia{u^2} + ibu} \right)} du\textrm{ = }2\pi \exp \left( {\frac{{2i{a^3}}}{3} - iab} \right)[ - aAi(b - {a^2}) - iAi^{\prime}(b - {a^2})],$$

E(x) after the Airy transformation has an analytical form as

(9)$$E(x) ={-} 4\sqrt \pi \tau \exp \left( { - \frac{{{\tau^3}}}{3}} \right)\exp (\tau {x_1})Ai^{\prime}({x_1}),$$

where ${x_1} = (x + \alpha {\tau ^2})/\alpha$. Ai′(x) is the Airyprime function. The superscript “$^{\prime}$” means the derivation of Airy function with respect to x. Similarly, E(y) reads as

(10)$$E(y) ={-} 4\sqrt \pi \gamma \exp \left( { - \frac{{{\gamma^3}}}{3}} \right)\exp (\gamma {y_1})Ai^{\prime}({y_1}),$$

where ${y_1} = (y + \beta {\gamma ^2})/\beta$. We term the beam having the electric field shown in Eqs. (9) and (10) as the FEA beam. The FEA beam is acentric. Note that the derived FEA beam is acentric and their parameters correspond to the acentric Airy beam source which is exp[τ(x+ατ²)/α]Ai[(x+ατ²)/α]exp[γ(x+βγ²)/β]Ai[(x+βγ²)/β]. The transverse scales of the FEA beam are α and β. The modulation parameters in two transverse directions are τ and γ. The off-centre positions in two transverse directions are (−ατ², −βγ²). Therefore, the transverse scales, the modulation parameters and the off-centre positions of the FEA beam can be easily modulated by the Airy control parameters α and β. The light intensity of the FEA beam is then given by

(11)$$I(x,y) = I(x)I(y) = {|{E(x)} |^2}{|{E(y)} |^2}.$$

As the FEA beam has the same form in two transverse directions, only one transverse direction such as the x-direction is discussed. In order to further evaluate the properties of the FEA beam, we now investigate the first-order and the second-order moments of light intensity. The centroid in the x-direction of the FEA beam is defined as the first-order moment of light intensity and turns out to be [39–46]:

(12)$${X_c} = \frac{{\int_{ - \infty }^\infty {xI(x)dx} }}{{\int_{ - \infty }^\infty {I(x)dx} }}\textrm{ = }\frac{{\alpha \int_{ - \infty }^\infty {({x_1} - {\tau ^2})\exp (2\tau {x_1})A{{i^{\prime}}^2}({x_1})d{x_1}} }}{{\int_{ - \infty }^\infty {\exp (2\tau {x_1})A{{i^{\prime}}^2}({x_1})d{x_1}} }} = \frac{{ - 3\alpha (1 - 4{\tau ^3})}}{{4\tau (1 + 4{\tau ^3})}}.$$

The r.m.s beam width in the x-directions is defined as the second-order moment of light intensity and is found to be [39–46]:

(13)$$\begin{aligned} {W_x} &= {\left[ {\frac{{\int_{ - \infty }^\infty {{x^2}I(x)dx} }}{{\int_{ - \infty }^\infty {I(x)dx} }} - X_c^2} \right]^{1/2}} = {\left[ {\frac{{{\alpha^2}\int_{ - \infty }^\infty {{{({x_1} - {\tau^2})}^2}\exp (2\tau {x_1})A{{i^{\prime}}^2}({x_1})d{x_1}} }}{{\int_{ - \infty }^\infty {\exp (2\tau {x_1})A{{i^{\prime}}^2}({x_1})d{x_1}} }} - X_c^2} \right]^{1/2}}\\ &=\alpha \frac{{{{(3 + 80{\tau ^3} + 16{\tau ^6} + 128{\tau ^9})}^{1/2}}}}{{2\sqrt 2 (\tau + 4{\tau ^4})}}. \end{aligned}$$

The divergence angle in the x-direction is given by [39–46]:

(14)$$\begin{aligned} {\theta _x} &= \frac{1}{k}{\left\{ {\frac{1}{{\int_{ - \infty }^\infty {I(x)dx} }}\int_{ - \infty }^\infty {{{\left|{\frac{{\partial E(x)}}{{\partial x}}} \right|}^2}} dx} \right\}^{1/2}} = \frac{1}{{k\alpha }}{\left\{ {\frac{{\int_{ - \infty }^\infty {\exp (2\tau {x_1}){{[{x_1}Ai({x_1}) + \tau Ai^{\prime}({x_1})]}^2}d{x_1}} }}{{\int_{ - \infty }^\infty {\exp (2\tau {x_1})A{{i^{\prime}}^2}({x_1})d{x_1}} }}} \right\}^{1/2}}\\ & =\frac{1}{{k{w_0}}}\frac{{{{(3 + 4{\tau ^3})}^{1/2}}}}{{{{(1 + 4{\tau ^3})}^{1/2}}}}, \end{aligned}$$

where k = 2π/λ with λ being the optical wavelength. The cross second-order moment in the x-direction reads as

(15)$$< x{\theta _x} > = \frac{\pi }{{ik}}\frac{{\int_{ - \infty }^\infty {x\left\{ {{{\left[ {\frac{{\partial E(x)}}{{\partial x}}} \right]}^ \ast }E(x) - \frac{{\partial E(x)}}{{\partial x}}{E^ \ast }(x)} \right\}dx} }}{{\int_{ - \infty }^\infty {I(x)} dx}} = 0,$$

where the asterisk denotes the complex conjugation. Since E(x) is a real number, the cross second-order moment in the x-direction always equals to zero. The beam propagation factor in the x-direction yields [39–46]:

(16)$$M_x^2 = 2k{(W_x^2\theta _x^2 - < x{\theta _x}{ > ^2})^{1/2}} = 2k{W_x}{\theta _x} = \frac{{{{(3 + 4{\tau ^3})}^{1/2}}{{(3 + 80{\tau ^3} + 16{\tau ^6} + 128{\tau ^9})}^{1/2}}}}{{2\sqrt 2 {\tau ^{3/2}}{{(1 + 4{\tau ^3})}^{3/2}}}}.$$

Similarly, the centroid, the r.m.s beam width, the divergence angle and the beam propagation factor in the y-direction of the FEA beam can be obtained by parameter substitution. For the sake of comparison, the centroid, the r.m.s beam width, the divergence angle and the beam propagation factor in the x-direction of the incident beam are listed as follows:

(17)$${X_c} = \frac{{2{w_0}{\tau ^{3/2}}}}{{1 + 4{\tau ^3}}},{W_x} = \frac{{{w_0}{{(3 + 16{\tau ^6})}^{1/2}}}}{{2(1 + 4{\tau ^3})}},$$

(18)$${\theta _x} = \frac{1}{{k{w_0}}}{\left( {\frac{{3 + 4{\tau^3}}}{{1 + 4{\tau^3}}}} \right)^{1/2}},M_x^2 = \frac{{{{(3 + 4{\tau ^3})}^{1/2}}{{(3 + 16{\tau ^6})}^{1/2}}}}{{{{(1 + 4{\tau ^3})}^{3/2}}}}.$$

Comparing Eqs. (12)–(16) to Eqs. (17) and (18), it is found that only the divergence angles of the incident beam and the FEA beam are the same, and other parameters are modulated by the Airy transformation.

Within the validity of paraxial approximation, the electric field E(x, z) of the FEA beam propagating in free space can be treated by the Huygens-Fresnel integral and is found to be after integral calculation [47]:

(19)$$\begin{aligned} E(x,z) &={-} 4\sqrt \pi \tau \exp \left( {\frac{{2{\tau^3}}}{3}} \right)\sqrt {\frac{k}{{2\pi iz}}} \int_{ - \infty }^\infty {\exp \left( {\frac{{\tau x^{\prime}}}{\alpha }} \right)Ai^{\prime}\left( {\frac{{x^{\prime}}}{\alpha } + {\tau^2}} \right)\exp \left\{ {\frac{{ik}}{{2z}}[{{(x - x^{\prime})}^2}]} \right\}dx^{\prime}} \\ & ={-} 4\sqrt \pi \tau \exp \left( {\frac{{2{\tau^3}}}{3}\textrm{ + }\frac{{\tau x}}{\alpha } - \frac{{8{\tau^3}{z^2}}}{{z_0^2}} + \frac{{4i{\tau^3}z}}{{{z_0}}} + \frac{{2i\tau xz}}{{\alpha {z_0}}} - \frac{{16i{\tau^3}{z^3}}}{{3z_0^3}}} \right)\\ & \times \left[ {\frac{{2i\tau z}}{{{z_0}}}Ai\left( {{\tau^2} + \frac{x}{\alpha } - \frac{{4{\tau^2}{z^2}}}{{z_0^2}} + \frac{{4i{\tau^2}z}}{{{z_0}}}} \right) + Ai^{\prime}\left( {{\tau^2} + \frac{x}{\alpha } - \frac{{4{\tau^2}{z^2}}}{{z_0^2}} + \frac{{4i{\tau^2}z}}{{{z_0}}}} \right)} \right]\\ & = \frac{{2i\tau z}}{{{z_0}}}{E_{Ai}}(x,z) + {E_{Ap}}(x,z), \end{aligned}$$

where ${z_0} = kw_0^2$. The subscript Ai and Ap denote the Airy and the Airyprime modes, respectively. It is known that the Airy beam in free-space propagation is still the Airy function with its argument depending on propagation distance [15]. While for the FEA beam, one can see from Eq. (19) that it will be associated with an Airy mode, and therefore, it becomes the coherent superposition of an Airyprime mode and an Airy mode during free space propagation. This feature will lead to the particular propagation characteristics compared to that of the finite energy Airy beam under the same beam parameter, especially for the evolution of main lobe which we will investigate in the following section through numerical examples.

3. Numerical calculations and analysis

In this section, the properties of the FEA beam are examined through numerical examples. w₀ is set to be 0.5 mm in the following calculations. As the electric field of the FEA beam has the same evolution law in the two transverse directions, first only the x-direction is considered. Figure 1 presents the normalized intensity distribution in the x₀-direction of different incident beams in the input plane. When the Airy control parameter is small such as α=0.1 mm, the intensity distribution in the x₀-direction is a Gaussian distribution. With the increase of the Airy control parameter α, a side peak appears on the left side, and the height of the side peak increases gradually, as well as the main peak moves to the right gradually. The normalized intensity distribution in the x-direction of FEA beams generated by the Airy transformation is shown in Fig. 2. When α=0.1 mm, the intensity distribution of the FEA beam is similar with that of the incident beam in the input plane. When α=0.3 mm, two side peaks emerge in the left side, and the position of the main peak has shifted to the right. As the Airy control parameter α increases further, the number of side peaks increases, and the size of beam profile broadens. When α=0.4 mm, the original main peak has evolved into the largest side peak, and the second peak from the right side has become the main peak. When α=0.5 mm, the first peak from the right side further degenerates into the peak with the third largest intensity.

Fig. 1. Normalized intensity distribution in the x₀-direction of different incident beams in the input plane. (a) α=0.1 mm; (b) α=0.3 mm; (c) α=0.4 mm; (d) α=0.5 mm.

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Fig. 2. Normalized intensity distribution in the x-direction of FEA beams generated by different Airy transform optical systems. (a) α=0.1 mm; (b) α=0.3 mm; (c) α=0.4 mm; (d) α=0.5 mm.

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Figure 3 illustrates the centroid, the r.m.s beam width, the divergence angle and the beam propagation factor in the x₀-direction of incident beams as a function of the Airy control parameter α. With the increase of the Airy control parameter α, the centroid first increases from zero to 0.25 mm and then decreases. When α=0.32 mm, the centroid reaches the maximum value 0.25 mm. When the Airy control parameter α increases from zero to 1 mm, the r.m.s beam width first keeps unvaried at 0.25 mm, then decreases, thirdly increases, and finally tends to the saturated value 0.43 mm. When α=0.26 mm, the r.m.s beam width reaches the minimum value 0.21 mm. With the increase of the Airy control parameter α, the divergence angle first keeps unvaried, then increases, and finally tends to the saturated value. The minimum divergence angle is the divergence angle of the Gaussian beam. When the Airy control parameter α increases from zero to 1 mm, the beam propagation factor first remains unchanged at 1, then increases, and finally tends to the saturated value 3. The reason for the saturated value 3 is that when the Airy control parameter increases, the electric field of such incident beam in the x₀-direction approximately reduces to an EHG₁₁ mode whose beam propagation factor is just 3. The centroid, the r.m.s beam width, the divergence angle and the beam propagation factor in the x-direction of the FEA beam as a function of the Airy control parameter α is shown in Fig. 4. When α≤0.31 mm, the centroid is positive and is slightly greater than zero. When the Airy control parameter α increases from zero to 0.31 mm, the centroid first increases and then decreases. When α=0.25 mm, the centroid reaches the maximum value 0.11 mm. When α>0.31 mm, the centroid becomes negative, and the absolute value of the centroid increases with increasing the Airy control parameter α. With the increase of the Airy control parameter α, the r.m.s beam width first keeps unvaried at 0.25 mm and then increases. The divergence angle in the x-direction of the FEA beam is the same with that of the input beam, implying that the divergence angle is independent of the Airy transformation. When the Airy control parameter α increases from zero to 1 mm, the beam propagation factor first remains unvaried at 1 and then increases, which is unilaterally caused by the r.m.s beam width.

Fig. 3. Different parameters in the x₀-direction of incident beams as a function of the Airy control parameter α. (a) Centroid; (b) r.m.s beam width; (c) Divergence angle; (d) Beam propagation factor.

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Fig. 4. Different parameters in the x-direction of FEA beams as a function of the Airy control parameter α. (a) Centroid; (b) r.m.s beam width; (c) Divergence angle; (d) Beam propagation factor.

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Figures 5 and 6 show the normalized intensity distribution in the x-direction of the FEA beam at different propagation distance z. The Airy control parameter is α=0.3 mm in Fig. 5 and α=0.4 mm in Fig. 6. When α=0.3 mm, the first peak from the right side is always the main peak and shifts to the right upon propagation in free space. When α=0.4 mm, the first peak from the right side which is not the main peak in the plane z = 0 evolves into the main peak upon free space propagation, and the first peak from the right side in the observation plane z = 0.5z₀ happens to be the main peak. Upon propagation in free space, the number of side peaks will decrease to 1. Moreover, the last side peak covers a wide range, and the light intensity at the junction of the main peak and the last side peak is not zero. When the FEA beam propagates in free space, the smaller the control parameter is, the faster the whole evolution process is. Figure 7 shows the contour graph of the normalized intensity distribution in the x-z plane of the FEA beam, which is like a jet of flame. As the Airy control parameter α increases, the trajectory of the main lobe is elongated and is deflected upward. Particularly, there appears a secondary appreciable lobe whose trajectory is downward upon propagation when α =0.4mm. In the case of α = 0.5mm, the third appreciable side lobe appears. The different trajectories of lobes may have potential application in separation of micro particles and cells. In addition, compared to the Airy beam under the same parameters, the intensity of main lobe of the FEA beam decays much slower during free space propagation, and even is slightly enhanced because of the interference of two mixed modes shown in Eq. (19).

Fig. 5. Normalized intensity distribution in the x-direction of the FEA beam propagating in free space. α=0.3 mm. (a) z = 0.125z₀; (b) z = 0.25z₀; (c) z = 0.5z₀; (d) z = z₀.

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Fig. 6. Normalized intensity distribution in the x-direction of the FEA beam propagating in free space. α=0.4 mm. (a) z = 0.25z₀; (b) z = 0.5z₀; (c) z = z₀; (d) z = 2z₀.

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Fig. 7. Contour graph of the normalized intensity distribution in the x-z plane of FEA beams propagating in free space. (a) α=0.1 mm; (b) α=0.3 mm; (c) α=0.4 mm; (d) α=0.5 mm.

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Finally, the incident beams and the FEA beams with the two transverse directions are considered. Figure 8 shows normalized intensity distribution of incident beams in the input plane. When α=β=0.1 mm, there is only one lobe. With the increase of the Airy control parameter α, the number of lobes increases. When α=β=0.3 mm, the number of lobes is 3 including one dominant lobe and two minor lobes of the same level. When α=β=0.4 mm, the number of lobes reaches the maximum value 4 including one dominant lobe, two secondary important lobes of the same level, and one weakest lobe. When the Airy control parameters increase further from 0.4 mm, the number of lobes keeps unvaried. Normalized intensity distribution of FEA beams generated by the Airy transform optical system is demonstrated in Fig. 9. With the increase of the Airy control parameter α, the Airyprime feature is enhanced, which results in the increase of number of lobes. In this process, however, the position of the dominant lobe moves diagonally from the upper right corner to the next. Due to the differential operation on the Airy function, the beam profile of the FEA beam in the source plane exhibits more visible side lobes and the brightest lobe may shift to the second lobe from the corner (determined by the beam parameters), which is quite different from that of the finite energy Airy beam source. Figures 10 and 11 show normalized intensity distribution of the FEA beams in different observation planes of free space. α=β=0.3 mm in Fig. 10 and α=β=0.4 mm in Fig. 11. Upon propagation in free space, each lobe tends to be connected together, and the number of lobes reduces, which is caused by the self-acceleration and diffraction. Figure 10(a) and 10(b) manifest that the FEA beam possesses short non-diffraction. With the increase of α and β, the non-diffraction characteristic is enhanced, indicating that the beam profile can keep invariant for longer propagation distance, as shown in Fig. 11(a) and 11(b). In the case of α=β=0.3mm, the beam pattern at z = z₀ evolves into a solid beam spot with two tails along the x- and the y-directions, and keeps its beam profile unchanged for further propagation. While in the case of α=β=0.4mm, the beam profile turns into a solid beam spot accompanied by two elongated tails at z = 2z₀, then, the beam profile keeps invariant for further propagation. We may come to a conclusion for the intensity evolution of the FEA beam in free-space propagation. The beam pattern will eventually evolve into a dominant Gaussian type spot with two tails along two transverse directions, but the tails elongate and the transition distance becomes long as the Airy control parameters increase. It is well known that the finite energy Airy beam will go through a short non-diffraction distance during free space propagation. The energy of the side lobes will flow to the main lobe to keep the non-diffraction state. Then the profile degenerates into Gaussian type eventually as it propagates continuously. However, the beam pattern of the FEA beam propagating in free space will eventually evolve into a dominant Gaussian type spot accompanied by two tails along two transverse directions [see Fig. 9(d)], and keeps the profile unchanged for further propagation except for the beam expanding (not shown here). Hence, the significant difference between the far-field intensity patterns of the FEA beam and the Airy beam is these two tails.

Fig. 8. Normalized intensity distribution of incident beams in the input plane. (a) α=β=0.1 mm; (b) α=β=0.3 mm; (c) α=β=0.4 mm; (d) α=β=0.5 mm.

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Fig. 9. Normalized intensity distribution of FEA beams generated by different Airy transform optical systems. (a) α=β=0.1 mm; (b) α=β=0.3 mm; (c) α=β=0.4 mm; (d) α=β=0.5 mm.

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Fig. 10. Normalized intensity distribution of the FEA beam propagating in free space. α=β=0.3 mm. (a) z = 0.125z₀; (b) z = 0.25z₀; (c) z = 0.5z₀; (d) z = z₀.

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Fig. 11. Normalized intensity distribution of the FEA beam propagating in free space. α=β=0.4 mm. (a) z = 0.25z₀; (b) z = 0.5z₀; (c) z = z₀; (d) z = 2z₀.

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4. Experimental results

In this section, a experimental setup is established to generate the FEA beam via Airy transformation of four mixed elegant Hermite-Gaussian modes, and the propagation characteristics including intensity distribution, r.m.s beam width and beam propagation factor are measured experimentally. The schematic diagram of the experimental setup is displayed in Fig. 12. The first step is the generation of four mixed elegant Hermite-Gaussian modes in the input plane. A Gaussian laser beam with λ=532 nm produced by a solid-stated laser (Laser Quantum, model ventus) is first expanded by a 50× beam expander (BE), then is reflected by a reflecting mirror (RM), and finally is split into two portions by a 50: 50 intensity beam splitter (BS). The transmitted beam impinges on a phase-only spatial light modulator (SLM₁, Holoeye LETO-3, pixel size: 6.4µm×6.4µm) working at reflective mode. The SLM’s screen acts as a phase hologram to simultaneously modulate the amplitude and phase of the incident beam. The method for encoding a prescribe complex electric filed into the hologram is described in Ref. [48]. Figure 13(a) shows a typical phase hologram loaded on the SLM₁ to generate an incident beam with α=β=0.1mm. After reflected by the SLM₁, the modulated light is reflected by the BS again and enters a 4f optical system composed of lens L₁ and L₂ with the focal length f₁= f₂= 250mm. A circular aperture (CA) which is placed in the rear focal plane of L₁ is used to select the first diffraction order and filter out other unwanted diffraction orders. In the rear focal plane of L₂, which is the input plane, the four mixed elegant Hermite-Gaussian modes with w₀= 0.5 mm is generated. Figure 14 shows the experimental results of the normalized intensity distribution of the generated beam in the input plane captured by the beam profile analysis (BPA). The BPA (model: BGS-USB3-LT665) produced by Ophir company has a 2752×2192 pixel array with a 4.4µm pitch. The experimental result is well consistent with the theoretical result shown in Fig. 8.

Fig. 12. The experimental setup of the generation of the incident beam, the realization of the Airy transformation of the incident beam, and the measurement of the intensity characteristics for the FEA beam.

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Fig. 13. (a) Phase hologram loaded on SLM₁ for the generation of the incident beam with α=β=0.1 mm; (b) Phase hologram loaded on SLM₂ for the generation of the FEA beam with α=β=0.1 mm.

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Fig. 14. Experimental result of normalized intensity distribution of incident beams in the input plane. (a) α=β=0.1 mm; (b) α=β=0.3 mm; (c) α=β=0.4 mm; (d) α=β=0.5 mm.

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The next step is the implementation of the Airy transformation of the incident beam. Another 4f optical system, which contains two lenses L₃ and L₄ with the focal length f₃= f₄= 250 mm, and a spatial light modulator (SLM₂, Holoeye Pluto-2, pixel size: 8µm×8µm) are used to perform the Airy transformation. The SLM₂ is located in the rear/front focal plane of L₃/L₄. A cubic phase ψ(x, y) = (α³k³×³+β³k³y³)/(3f₃³)–(2kf₃+2kf₄+π) is loaded on the SLM₂. Figure 13(b) displays the cubic phase hologram with α=β=0.1mm loaded on SLM₂. After the Airy transformation of the incident beam, the FEA beam is generated in the rear focal plane of L₄ (see the output plane in Fig. 12).

The last step is the measurement of the intensity distribution, the r.m.s beam width and the beam propagation factor of the generated FEA beam. A BS located between L₄ and BPA₁ is used to split the generated FEA beam into two identical portions. A BPA₁ mounted on the electric translation stage is used to record the beam pattern of the FEA beam during free space propagation. Also, the intensity distribution, the centroid and the r.m.s beam width of the generated FEA beam in the output plane are measured by the BPA₁. In order to measure the beam propagation factor, a focal lens L₅ with the focal length f₅= 400mm is placed in the equivalent output plane, and a BPA₂ which is mounted on another translation stage along the propagation axis (z-direction) is used to measure the variation of the r.m.s beam width near the focal plane of L₅. The use of L₅ is to produce the secondary beam waist with small size near the focal plane of L₅. Thereby, the variation of the r.m.s beam width with propagation distance near the focal plane is appreciable, which is convenient for experimental measurement. It is worthy pointing out that the BPA used in this paper is the same BPA, and we have numbered the BPA in different measurement positions for the convenience of description.

Figure 15 displays the corresponding experimental results of normalized intensity distribution of the generated FEA beams with different Airy control parameters. The characteristics of the beam profile are closely related to the Airy control parameters. When the Airy control parameters increase, the number of side lobes also increases, while the intensity of symmetrical side lobes is a little inconsistent. By comparing with the theoretical intensity profile shown in Fig. 9, it can be clearly found that the experimental results are basically consistent with the numerical simulation results.

Fig. 15. Experimental result of normalized intensity distribution of FEA beams generated by different Airy transform optical systems. (a) α=β=0.1 mm; (b) α=β=0.3 mm; (c) α=β=0.4 mm; (d) α=β=0.5 mm.

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The measurement process of the centroid, the r.m.s beam width and the beam propagation factor in the x-direction of the FEA beam is interpreted as follow. First, the pictures of intensity distribution at different propagation distances are recorded by the BPA. For each picture, it can be represented as a N×M intensity matrix, and the intensity in each pixel is expressed as I(x_i, y_j), i = 1, 2…, j = 1, 2…, where (x_i, y_j) is the coordinate at that pixel. Then, the centroid and the r.m.s beam width in the x-direction are calculated by the following formulae:

(20)$${X_c} = \sum\limits_{i = 1}^N {\sum\limits_{j = 1}^M {{x_i}I({x_i},{y_j})} } /\sum\limits_{i = 1}^N {\sum\limits_{j = 1}^M {I({x_i},{y_j})} } ,$$

(21)$${W_x} = {\left[ {\sum\limits_{i = 1}^N {\sum\limits_{j = 1}^M {x_i^2I({x_i},{y_j})} } /\sum\limits_{i = 1}^N {\sum\limits_{j = 1}^M {I({x_i},{y_j})} } - X_c^2} \right]^{1/2}},$$

On the basis of Eqs. (20)–(21), one can obtain the centroid and the r.m.s beam width in the x-direction. To measure the beam propagation factor, the r.m.s beam width W_x(z) after the lens L₅ at different propagation distances z is first calculated using Eqs. (20)–(21). The variation of the r.m.s beam width versus the propagation distance z follows a parabola law:

(22)$$W_x^2(z) = J{z^2} + Kz + L.$$

From the above equation, the parameter J, K and L can be obtained by fitting the curve of W_x²(z) as a function of z. Then, the beam waist and the divergence angle in the x-direction after the lens L₅ are given by

(23)$${w_x} = {(4JL - {K^2})^{1/2}}/2{J^{1/2}},{\theta _x} = \mathop {\lim }\limits_{z \to \infty } \frac{{{W_x}(z)}}{z} = {J^{1/2}},$$

where w_x is the beam waist namely the minimum r.m.s beam width. Therefore, the beam propagation factor in the x-direction of the FEA beam is found to be

(24)$$M_x^2 = 2k{w_x}{\theta _x}\textrm{ = }k{(4JL - {K^2})^{1/2}}.$$

The dependence of the centroid and the r.m.s beam width in the x-direction of the FEA beam on the Airy control parameter α is experimentally measured and illustrated in Fig. 16(a) and Fig. 16(b). The circular dots and the solid curve correspond to the experimental and the theoretical values, respectively. When the Airy control parameter α increases from zero to 0.3 mm, the centroid in the x-direction first increases and then decreases. When α>0.3 mm, the centroid in the x-direction is negative, and it drops with increasing the Airy control parameter α. This is because that the number of side lobes obviously increase when the Airy control parameter α>0.3mm. The r.m.s beam width in the x-direction equals 0.25 mm and remains unchanged for α<0.2 mm. Nevertheless, the r.m.s beam width rises rapidly as the Airy control parameter increases further ifα>0.2 mm. The experimental result of the beam propagation factor in the x-direction of the FEA beam as a function of the Airy control parameter α is shown in Fig. 16(c). The variation of the beam propagation factor with the Airy control parameter is similar to that of the r.m.s beam width. Figure 16(d) shows our experimental results of the r.m.s beam width at different propagation distances. Blue dots correspond to the case of α=0.2mm, and purple dots denotes the case of α=0.4mm. By curve fitting, the parameters J, K and L are 5.17×10⁻⁷, −3.89 ×10⁻⁴ mm and 7.84×10⁻²mm² for α=0.2mm. As to α=0.4mm, the parameters J, K and L are 3.73×10⁻⁶, −2.9×10⁻³mm and 5.8×10⁻¹mm², respectively. The experimental beam propagation factor in the case of α=0.2mm is 1.24, which is slightly larger than the theoretical value 1.03. The experimental beam propagation factor in the case of α=0.4mm is 5.83, and the corresponding theoretical value is 5.30, which is close to each other. The difference between the experimental and the theoretical results is mainly caused by the imperfect incident beam and the background noise.

Fig. 16. Experimental result of the centroid (a), the r.m.s beam width (b), and the beam propagation factor (c) in the x-direction of the FEA beam as a function of the Airy control parameterα; (d) The experimental r.m.s beam width in the x-direction as a function of propagation distance z at the cases of α=0.2 mm and α=0.4 mm.

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Figures 17 and 18 present the experimental results of normalized intensity distribution of the FEA beam withα=β=0.3 mm and α=β=0.4 mm propagating in free space, respectively. When propagating in free space, each lobe in the case of different Airy control parameters is gradually connected together, and the number of lobes decreases, which is the common feature. With the increase of the Airy control parameters, the beam profile during free space propagation is more likely to remain the beam profile in the plane z = 0, exhibiting a certain non-diffractive feature. The experimental intensity profiles are basically consistent with the theoretical intensity profiles as shown in Figs. 10 and 11. The difference between the experimental and the theoretical intensity profiles may be caused by the large pixel size of SLM₂ and the imperfect incident beam.

Fig. 17. Experimental result of normalized intensity distribution of the FEA beam propagating in free space. α=β=0.3 mm. (a) z = 0.125z₀; (b) z = 0.25z₀; (c) z = 0.5z₀; (d) z = z₀.

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Fig. 18. Experimental result of normalized intensity distribution of the FEA beam propagating in free space. α=β=0.4 mm. (a) z = 0.125z₀; (b) z = 0.25z₀; (c) z = 0.5z₀; (d) z = z₀.

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5. Summary

Based on the Airy transformation of the coherent superposition of EHG₀₀, EHG₀₁, EHG₁₀, and EHG₁₁ modes with specific weight coefficients, a FEA beam is generated alone for the first time. Moreover, the transverse scales, the modulation parameters and the off-centre positions of the generated FEA beam can be easily modulated by the Airy control parameters. Analytical expressions of the centroid, the r.m.s beam width, the divergence angle and the beam propagation factor of the FEA beam are derived in the output plane of the Airy transformation, respectively. As a comparison, analytical expressions of the centroid, the r.m.s beam width, the divergence angle and the beam propagation factor of the incident beam namely the coherent superposition of four elegant Hermite-Gaussian modes are also presented. Analytical electric field of the FEA beam propagating in free space is derived. Particularly, the FEA beam upon free space propagation will be associated with an additional Airy mode, which will result in the different propagation characteristics from those of the corresponding finite energy Airy beam.

The effects of the Airy control parameters on the intensity distribution, the centroid, the r.m.s beam width and the beam propagation factor of the FEA beam are theoretically and experimentally investigated. With the increase of the Airy control parameters, the Airyprime feature is enhanced, which results in the increase of number of lobes, and the position of the dominant lobe moves diagonally from the upper right corner to the next. When the Airy control parameters don't exceed 0.31mm, the positive centroid first increases and then decreases with increasing the Airy control parameters. When the Airy control parameters are above 0.31mm, the absolute value of the negative centroid increases with increasing the Airy control parameters. With the increase of the Airy control parameters, the r.m.s beam width first keeps unvaried at 0.25mm and then increases, and the beam propagation factor first remains unvaried at 1 and then increases.

The properties of the finite energy Airyprime beam propagating in free space are also theoretically and experimentally studied. Upon propagation in free space, each lobe of the FEA beam is gradually connected together, and the number of lobes reduces. The beam pattern of FEA beams propagating in free space will evolve into a solid beam spot accompanied by two tails, which disappears in the pattern of Airy beams. With the increase of the Airy control parameters, the non-diffraction characteristic of the FEA beam is enhanced, and the trajectory of the main lobe is elongated and deflected upward, while the trajectory of multi appreciable side lobes is downward. The intensity of main lobe of the FEA beam propagating in free space decays much slowly and even is slightly enhanced.

This research affords an effective and novel approach to generate FEA beams. Moreover, the properties of FEA beams are well demonstrated by this research and are proved to be completely different from those of finite energy Airy beams. This research is also beneficial to expand the potential application of FEA beams. When extended to the atmospheric propagation, the single FEA beam is also the coherent superposition of the Airyprime and the Airy modes. The interference enhancement effect caused by the Airyprime and the Airy modes can resist the light intensity loss caused by atmospheric disturbance. However, a single Airy beam does not have interference enhancement effect. Therefore, a single FEA beam is more suitable for atmospheric optical communication than a single Airy beam.

Funding

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

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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Generation of finite energy Airyprime beams by Airy transformation

Abstract

1. Introduction

2. Generation of the FEA beam by Airy transformation and its propagation characteristics

3. Numerical calculations and analysis

4. Experimental results

5. Summary

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (18)

Equations (24)

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