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Generation of high-dimensional caustic beams via phase holograms using angular spectral representation

Zhuo Sun, Juntao Hu, Yishu Wang, Wenni Ye, Yixian Qian, and Xinzhong Li

Open Access Open Access

Abstract

Using angular spectral representation, we demonstrate a generalized approach for generating high-dimensional elliptic umbilic and hyperbolic umbilic caustics by phase holograms. The wavefronts of such umbilic beams are investigated via the diffraction catastrophe theory determined by the potential function, which depends on the state and control parameters. We find that the hyperbolic umbilic beams degenerate into classical Airy beams when the two control parameters are simultaneously equal to zero, and elliptic umbilic beams possess an intriguing autofocusing property. Numerical results demonstrate that such beams exhibit clear umbilics in 3D caustic, which link the two separated parts. The dynamical evolutions verify that they both possess prominent self-healing properties. Moreover, we demonstrate that hyperbolic umbilic beams follow along a curve trajectory during propagation. As the numerical calculation of diffraction integral is relatively complex, we have developed an effective approach for successfully generating such beams by using phase hologram represented by angular spectrum. Our experimental results are in good agreement with the simulations. Such beams with intriguing properties are likely to be applied in emerging fields such as particle manipulation and optical micromachining.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Optical caustic is an intriguing phenomenon because it always manifests as a bright curve, surface, or hypersurface where light rays are focused, often appearing as patterns such as rainbows and teacups. Optical caustics have attracted significant attention since they can be customized into a variety of structured lights, including self-focusing beams [1–4], self-accelerating beams [5–7], and even diverse tailored beams [8,9]. Such beams have presented many excellent properties and have been applied in optical micromanipulation [10], optical micromachining [11], microscopic imaging [12], and spatio-temporal light bullets [13].

In general, stable optical caustics abide by the universal laws described by catastrophe theory [14–16]. In 1979, Berry proposed that when the dimension (d) of the control variable is less than four, there are seven stable catastrophe structures in nature, which are described by fold [17], cusp, swallowtail [18,19], butterfly [19,20], elliptic umbilic, hyperbolic umbilic, and parabolic umbilic catastrophe. The low-dimensional Airy beam, a typical example of a catastrophe beam (fold catastrophe) with non-diffracting, self-accelerating, and self-healing properties [5,6,21–28], has attracted considerable interest since it was first experimentally observed in 2007 [21,22]. Subsequently, Pearcey beams representing cusp catastrophe have also been extensively investigated [29] and exhibit interesting form-invariant and self-focusing properties. Later, high-dimensional swallowtail and butterfly beams were also observed experimentally [18,20]. These excellent works have contributed to the development of catastrophe optics and its applications. However, most findings mentioned above are only based on one state parameter. Thus far, such beams are infrequently observed because the diffraction integration becomes difficult to calculate owing to the coupling of two state parameters with the increasing dimensionality of control parameters.

In this work, we develop a general method for generating elliptic umbilic beams (EUBs) and hyperbolic umbilic beams (HUBs) with two state parameters by the diffraction catastrophe theory. Their caustics and wavefronts are discussed by mapping the high-dimensional catastrophe onto the low-dimensional space, and such beams also exhibit many interesting properties, including self-healing or following a curved trajectory [30]. Such beams may have applications in emerging fields such as designing caustic beams, particle manipulation, and optical micromachining. Note that the two catastrophe types exhibit similar potential function mathematically, we investigate them together.

2. Theory

According to catastrophe optic theory, EUBs and HUBs belong to high-dimension type of seven main catastrophe beams with two state parameters. The caustic field ${C_n}(a)$ can be obtained using the standard diffraction catastrophe integral:

$${C_n}(a) = \int_R {\exp [i{p_n}(a,s)]ds.}$$
Where ${p_n}(a,s)$ represents the canonical potential function related to control parameter ${a_j}$ and state parameter s. We define that d = n − 2 denotes the dimensionality (d) of the control parameters ${a_j}$ with $j = 1,2,\ldots ,n - 2$ ($n \ge 3$). The following discussions are limited to elliptic and hyperbolic umbilic catastrophe with n = 5. a = (a1, a2, a3) = (u, v, w) and s = (s1, s2) denote the control parameters and state parameters, which jointly determine the light field and the caustic properties, where u, v, and w denote the 3D control parameters, and s1, s2 are the two state parameters. R is the integral space. Specifically, for example, according to catastrophe theory, the potential function $p_5^e(u,v,w,{s_1},{s_2})$ for elliptic umbilic catastrophe is defined as follows:
$$p_5^e(u,v,w,{s_1},{s_2}) = \frac{1}{3}s_1^3 - {s_1}s_2^2 + w(s_1^2 + s_2^2) - v{s_2} - u{s_1}.$$

The polynomial in Eq. (2) completely defines the properties and diffraction structure of EUBs [31]. The caustic field $Eub(X,Y,Z)$ for an EUB is expressed by

$$Eub(X,Y,Z) = \int\!\!\!\int_R {\exp [i(\frac{1}{3}{s_1}^3 - {s_1}{s_2}^2 + Z({s_1}^2 + {s_2}^2) - Y{s_2} - X{s_1})]d{s_1}d{s_2}} ,$$
where $X = {x / {{x_0}}}$, $Y = {y / {{y_0}}}$, and $Z = {z / {{z_0}}}$ correspond to the dimensionless 3D space coordinates, which match the control parameters u, v, and w, respectively, and ${x_0}$, ${y_0}$, and ${z_0}$ denote arbitrary scales. Note that Z no longer represents the traditional propagation axis. EUBs appear as a 3D wavefront: their caustics are hypersurfaces in space. To this end, we investigate and observe EUBs by projecting a 3D light field onto a 2D space. Here, one control parameter is held constant, and the residual two control parameters match the corresponding transverse coordinates. Consequently, EUBs (2D) are observed by mapping a 3D space onto a 2D space. Specifically, $Eub(X,Y,{a_3})$ is expressed by
$$Eub(X,Y,{a_3}) = \int\!\!\!\int_R {\exp [i(\frac{1}{3}{s_1}^3 - {s_1}{s_2}^2 + {a_3}({s_1}^2 + {s_2}^2) - Y{s_2} - X{s_1})]d{s_1}d{s_2}} ,$$
where a3 denotes an arbitrary constant. Figures 1(a1), 1(a2), and 1(a3) display the numerically obtained wavefront structures of EUBs, like a triangle in the case of a3 = −3, 0, and 3, respectively. EUBs exhibit various structures via varying the control parameter a3. We note that Ren et al. investigated triple-cusp beams [32], which were not generated by catastrophe and distinctly from the properties of our work.

 figure: Fig. 1.

Fig. 1. Numerical intensity distributions for $Eub(X,Y,{a_3})$ in the case of (a1) a3 = −3, (a2) a3 = 0, and (a3) a3 = 3. (b1), (b2) and (b3) $Hub(X,Y,{a_3})$ in case of a3 = −3, 0, and 3, respectively.

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Similarly, a 2D $Hub(X,Y,{a_3})$ can be written as

$$Hub(X,Y,{a_3}) = \int\!\!\!\int_R {\exp [i(s_1^3 + s_2^3 - {a_3}{s_1}{s_2} - Y{s_2} - X{s_1})]d{s_1}d{s_2}} ,$$
where the corresponding potential function $p_5^h(u,v,w,{s_1},{s_2}) = s_1^3 + s_2^3 - w{s_1}{s_2} - v{s_2} - u{s_1}$, where w is defined as the constant a3. Figures 1(b1), 1(b2), and 1(b3) exhibit numerically obtained wavefront distributions for HUBs when a3 = −3, 0, and 3, respectively. The hyperbolic umbilic beam evolves into an Airy beam when a3 = 0 since the corresponding potential function turns into $s_1^3 + s_2^3 - v{s_2} - u{s_1}$, which is the potential function for a 2D Airy beam. Consequently, an Airy beam is a special case of hyperbolic umbilic beams.

The potential function in optics is introduced to develop a three-dimensional caustic framework. In optics, it is not the individual rays that matter, but a series of rays in a region; the caustics are the singularities of these ray families. However, the singular mapping of the potential function defines the caustic, which manifests a geometrically stable diffraction pattern. In catastrophe theory, caustic is also defined as the abrupt transition of an optical system. For a system with two state parameters, caustic needs to satisfy two conditions: 1) The first derivative of the potential function with respect to the state variable is equal to zero; 2) the system’s corresponding Hess matrix [33] is equal to zero:

$$\left\{\begin{aligned} &\frac{{\partial {p_n}(u,v,w,{s_1},{s_2})}}{{\partial {s_1}}} = 0\\ &\frac{{\partial {p_n}(u,v,w,{s_1},{s_2})}}{{\partial {s_2}}} = 0 \end{aligned} \right.,$$
$$\begin{vmatrix} \frac{{{\partial^2}{p_n}}}{{\partial {s_1}^2}} & \frac{{{\partial^2}{p_n}}}{{\partial {s_1}\partial {s_2}}}\\ \frac{{{\partial^2}{p_n}}}{{\partial {s_2}\partial {s_1}}} &\frac{{{\partial^2}{p_n}}}{{\partial {s_2}^2}} \end{vmatrix} = 0,$$
where the three control variables u, v, and w match with the corresponding 3D spatial coordinates X, Y, and Z. Substituting the potential function $p_5^e(u,v,w,{s_1},{s_2})$ into Eqs. (6),(7), one can derive
$$\left\{ \begin{array}{l} {s_1}^2 - {s_2}^2 + 2w{s_1} - u = 0\\ - 2{s_1}{s_2} + 2w{s_2} - v = 0 \end{array} \right..$$

The system’s Hess matrix (Eq. (7)) can be stated as

$$\left|\begin{array}{l} 2{s_1} + 2w\\ - 2{s_2} \end{array} \right.\left. \begin{array}{r} \textrm{ } - 2{s_2}\\ - 2{s_1} + 2w \end{array} \right|= 0.$$

Based on Eq. (9), one can obtain ${s_1} = w\cos \theta$ and ${s_2} = w\sin \theta$ by introducing the parameter $\theta$. Substituting these values into Eq. (8), we derive the caustic parameter equations for $Eub(X,Y,Z)$:

$$\begin{array}{l} u = {w^2}(\cos (2\theta ) + 2\cos \theta ),\\ v = {w^2}( - \sin (2\theta ) + 2\sin \theta ). \end{array}$$

Equation (10) visually describes the 3D caustic parameter equation of $Eub(X,Y,Z)$, where $\theta \in [0,2\pi ]$ is introduced. Figures 2(a1)–2(a2) show the caustic for $Eub(X,Y,Z)$ from two different perspectives, and it manifests as a 3D curved surface. The red and blue lines represent the 2D caustic lines by mapping the 3D caustic surfaces onto corresponding 2D planes when a3 = −5 and 5, respectively. Moreover, they also completely agree with the intensity patterns demonstrated in Figs. 1(a1) and 1(a3). We find that the 3D caustic consists of two symmetric and separated parts linked by a central umbilic [Fig. 2(a1)]. The 3D caustic includes three accelerating sheets intersecting in pairs to form three accelerating main lobes that follow curved trajectories and gradually shrink and focus on the umbilic with increasing w. Then they slowly separate and again form three accelerating and abducent sheets. The 3D caustic is symmetric at the central umbilic. Visualization 1 exhibits the dynamical 3D caustic.

 figure: Fig. 2.

Fig. 2. 3D caustics. (a1)–(a2) $Eub(X,Y,Z)$ and (b1)–(b2) $Hub(X,Y,Z)$.

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Similarly, the caustic parameter equations for $Hub(X,Y,Z)$ are derived via substituting $p_5^h(u,v,w,{s_1},{s_2}) = s_1^3 + s_2^3 - w{s_1}{s_2} - v{s_2} - u{s_1}$ into Eqs. (6) and (7):

$$\left\{ \begin{array}{l} 3{s_1}^2 - w{s_2} - u = 0\\ 3{s_2}^2 - w{s_1} - v = 0 \end{array} \right..$$

The system’s corresponding Hess matrix can be represented as

$$\left|\begin{array}{l} 6{s_1}\\ - w \end{array} \right.\left. \begin{array}{l} \textrm{ } - w\\ \textrm{ }6{s_2} \end{array} \right|= 0.$$

Similarly, we work out ${s_1} ={\pm} \frac{1}{6}w\exp (\tau )$ and ${s_2} ={\pm} \frac{1}{6}w\exp ( - \tau )$, which satisfies Eq. (12), where $\tau \in [ - \infty ,\infty ]$ is introduced for deriving and describing the following caustic parameter equations. We find that Eq. (11) remains invariant when replacing s1 with s2. Substituting s1 and s2 into Eq. (11), the caustic equations for Eub(X, Y, Z) are expressed as

$$\begin{array}{l} u = \frac{1}{{12}}{w^2}(\exp (2\tau ) \pm 2\exp ( - \tau )),\\ v = \frac{1}{{12}}{w^2}(\exp ( - 2\tau ) \pm 2\exp (\tau )). \end{array}$$

Both u and v have two solutions, expressed as u+, u- and v+, v-. However, there are only two combinations of equation sets owing to the structural equivalence of s1 and s2. i.e., (u+ and v+) (u- and v-). Figures 2(b1)–2(b2) show the 3D caustics for $Hub(X,Y,Z)$ from two different perspectives. The red, green, and blue lines display the 2D caustic lines by projecting the 3D caustic surface onto the corresponding 2D planes when a3 = −5, 0, and 5, respectively. They also fully correspond to those bright intensity patterns demonstrated in Figs. 1(b1)–1(b3). As demonstrated in Figs. 2(b1) and 2(b2), the 3D caustic consists of two intertwined sheets with cusp and smooth shapes, which slowly approach and coincide at the umbilic with increasing w. Interestingly, the umbilic evolves into a 2D dimensional Airy distribution when mapped onto the X-Y plane. Subsequently, the two sheets gradually separate. Visualization 2 displays its dynamical 3D caustic.

3. Experimental results and discussions

We discuss the generation and evolution of a light field from the viewpoint of the angular spectrum and experimentally investigate the propagation dynamics. The angular spectrum is considered the essential parameter of a light field since an arbitrary optical field can be obtained by Fourier transforming its angular spectrum. As a result, the angular spectrum defines the optical light structure and its propagation dynamics. The angular spectrum $\widetilde {Eub}({K_X},{K_Y},a_3^{})$ of $Eub(X,Y,{a_3})$ can be written as

$$\begin{aligned} \widetilde {Eub}({K_X},{K_Y},{a_3}) &= \int\!\!\!\int_R {Eub(X,Y,{a_3}){e^{ - i{K_X}X}}{e^{ - i{K_Y}Y}}} dXdY\\ &\textrm{ = }\exp [i( - \frac{1}{3}{K_X}^3 + {K_X}{K_Y}^2 + {a_3}({K_X}^2 + {K_Y}^2))], \end{aligned}$$
where $\widetilde {Eub}({K_X},{K_Y},{a_3})$ denotes the angular spectrum of the initial optical field $Eub(X,Y,{a_3},\xi = 0)$ with the dimensionless propagation distance $\xi$; ${K_X}$ and ${K_Y}$ are the corresponding dimensionless spatial frequencies to X and Y, respectively. Here, for simplicity, the scaling factor $1/{(2\pi )^2}$ is neglected in the angular spectrum. Similarly, the angular spectrum $\widetilde {Hub}({K_X},{K_Y},a_3^{})$ for a hyperbolic umbilic beam $Hub(X,Y,{a_3})$ can be written as follows:
$$\begin{aligned} \widetilde {Hub}({K_X},{K_Y},a_3^{}) &= \int\!\!\!\int_R {Hub(X,Y,a_3^{}){e^{ - i{K_X}X}}{e^{ - i{K_Y}Y}}dXdY} \\ &{\kern 1cm} = \exp [i( - {K_X}^3 - {K_Y}^3 - {a_3}{K_X}{K_Y})]. \end{aligned}$$
We find that the spectral phases are expressed by polynomials, consequently, such umbilic beams can be generated by imposing the spectral phase onto a plane wave. The experimental setup to observe such umbilic beams with a computer-controlled reflective spatial light modulator (SLM, Holoeye, $1920 \times 1080$ pixels) is shown in Fig. 3. A He-Ne laser was used to generate an expanded and collimated beam with $\lambda = 632\; \textrm{nm}$, which was reflected by the SLM, where the corresponding spectral phase was loaded. Subsequently, this reflected and modulated beam was Fourier transformed by a Fourier lens with the focal length $f^{\prime} = 150\; \textrm{mm}$. Finally, the wavefront’s intensity was recorded using a movable CCD ($1600 \times 1200$ pixels; pixel size of 4 µm).

 figure: Fig. 3.

Fig. 3. Experimental setup. BE: beam expander; BS: beam splitter; SLM: spatial light modulator; and L: lens.

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Figures 4(a1), 4(a2), and 4(a3) display the spectral phases for elliptic umbilic beams $Eub(X,Y,{a_3},\xi = 0)$ in the case of a3 = −3, 0, and 3, respectively, and the corresponding experimental intensity distributions are exhibited in Figs. 4(b1)–4(b3), which completely agree with those numerical results demonstrated in Figs. 1(a1)–1(a3). Figures 5(a1)–5(a3) and Figs. 5(b1)–5(b3) depict those similar numerical and experimental results for $Hub(X,Y,{a_3},\xi = 0)$, and both coincide highly. Additionally, the spectral phase depicted in Eq. (15) turns into a cubic phase ${K_x}^3 + K_y^3$ when a3 = 0; consequently, the induced experimental intensity distribution is identical to that of Airy beams.

 figure: Fig. 4.

Fig. 4. The spectral phases of $Eub(X,Y,{a_3},\xi = 0)$ in the case of (a1) a3 = −3, (a2) a3 = 0, and (a3) a3 = 3. (b1)–(b3) Corresponding experimental results.

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 figure: Fig. 5.

Fig. 5. The spectral phases of $Hub(X,Y,{a_3},\xi = 0)$ in the case of (a1) a3 = −3, (a2) a3 = 0, and (a3) a3 = 3. (b1)–(b3) Corresponding experimental results.

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The control of structural waves by modulating parameters has recently been an innovative scene in radiation propagation and advanced imaging [34]. Next, the the propagation field $Eub(X,Y,{a_3},\xi )$ of elliptic umbilic beams can be approximately described by the angular spectrum integral

$$\scalebox{0.88}{$\begin{aligned} Eub(X,Y,{a_3},\xi ) = &\int {\int_{ - \infty }^{ + \infty } {\widetilde {Eub}({K_X},{K_Y})} \exp [i({K_X}X + {K_Y}Y - ({K_X}^2 + {K_Y}^2)\xi )]d{K_X}d{K_Y}} \\ &\;\;\;\;\;= \int {\int_{ - \infty }^{ + \infty } {\exp [i( - \frac{1}{3}{K_X}^3 + {K_X}{K_Y}^2 + ({a_3} - \xi )({K_X}^2 + {K_Y}^2) + X{K_X} + Y{K_Y})]} d{K_X}d{K_Y}} , \end{aligned}$}$$
where $\xi = {\xi _b}/(2K\xi _0^{})$ represents the dimensionless propagation distance with ${\xi _b}$ being the real propagation distance. ${\xi _0}$ is an arbitrary scaling factor. $K = \sqrt {{K_X}^2 + {K_Y}^2 + {K_\xi }^2}$ represents the dimensionless wavenumber. Figures 6(a1), 6(a2), and 6(a3) exhibit the numerical wavefront distribution of $Eub(X,Y,{a_3},\xi )$ at the propagation distances $\xi$ = 0, 3, and 8, respectively. Figures 6(a1)–6(a3) show that the initial field presents a symmetrical triangular distribution, which gradually shrinks inward to form a focusing point like an umbilic with increasing propagation distance that subsequently expands outward to form a triangular distribution again. Consequently, the autofocusing property for elliptic umbilic beams facilitates its application in particle manipulation. The corresponding experimental results at the ${\xi _b}$ = 0, 300, and 800 mm planes are displayed in Figs. 6(b1), 6(b2), and 6(b3), respectively, completely coinciding with those numerical ones, where ${a_3} = 4$ and ${x_0} = {y_0} = {z_0} = {\xi _\textrm{0}}\textrm{ = 5} \mathrm{\mu m}$. Visualization 3 exhibits the propagation dynamics. Note that the propagation fields are not be solved in closed form, but derived by fast Fourier Transform method.

 figure: Fig. 6.

Fig. 6. Elliptic umbilic beams $Eub(X,Y,{a_3},\xi )$. (a1)–(a3) Numerical wavefront distribution for $\xi = 0,3,and\textrm{ 8}$ planes. (b1)–(b3) Corresponding experimental results.

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For a better representation of the propagation characteristics, the propagation caustic can also be derived by Eqs. (6) and (7):

$$\begin{array}{l} X = {w^2}( - \sin (2\theta ) - 2\sin \theta ),\\ Y = {w^2}(\cos (2\theta ) - 2\cos \theta ),\\ \xi = {a_3} - w, \end{array}$$
where $\theta \in [0,2\pi ]$. Figure 7(a) exhibits the 3D propagation caustic, where the red, yellow, and blue lines represent the caustic lines in the $\xi $ = 0, 3, and 8 planes, respectively. Moreover, they coincide with those intensity distribution patterns demonstrated in Figs. 6(a1)–6(a3). As displayed in Fig. 7(a), the caustic consists of three accelerating sheets, which form a shape like a triangular pyramid. The three sheets propagate along curved trajectories and slowly shrink to an umbilic like the 3D caustic depicted in Fig. 2. This can be explained by the fact that the propagation field described by Eq. (16) exhibits the same form as that of Eq. (4) except that there is an extra propagation factor $\xi$ in the quadratic term. This means the propagation caustic possesses the same structure as the 3D elliptic umbilic beam. To visualize the propagation caustic, Visualization 4 exhibits the dynamics related to Fig. 6. The caustic exhibits a curved property that would be beneficial to design accelerating beams manually.

 figure: Fig. 7.

Fig. 7. The 3D propagation caustic. (a) $Eub(X,Y,{a_3},\xi )$ and (b) $Hub(X,Y,{a_3},\xi )$

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The discussion for $Hub(X,Y,{a_3},\xi )$ is similar to that of $Eub(X,Y,{a_3},\xi )$. The propagation $Hub(X,Y,{a_3},\xi )$ is expressed by

$$Hub(X,Y,{a_3},\xi ) = \int {\int_{ - \infty }^{ + \infty } {\exp [i( - {K_X}^3 - {K_Y}^3 - {a_3}{K_X}{K_Y} + X{K_X} + Y{K_Y} - {K_X}^2\xi - {K_Y}^2\xi )]} d{K_X}d{K_Y}}. $$

Figure 8(a) illustrates the numerical side views of a HUB in the $Y - \xi$ plane, where the green line denotes the propagation trajectory. Figures 8(b1), 8(b2), 8(b3), and 8(b4) exhibit numerical intensity distributions at the propagation planes of $\xi$ = 0, 1, 2, and 3, respectively. Such beams exhibit accelerating properties that coincide with the curved caustic lines described in Fig. 8(a). The corresponding experimental results at ${\xi _b}$ = 0, 200, 400, and 600 mm planes are shown in Figs. 8(c1), 8(c2), 8(c3), and 8(c4), respectively, which are in excellent agreement with the numerical results, where ${a_3} = 4$ and ${x_0} = {y_0} = {z_0} = {\xi _\textrm{0}}\textrm{ = 10} \mathrm{\mu m}$. Visualization 5 exhibits the evolution dynamics.

 figure: Fig. 8.

Fig. 8. Hyperbolic umbilic beams $Hub(X,Y,{a_3},\xi )$. (a) The numerical side views in the $Y - \xi$ plane. (b1), (b2), (b3), and (b4) numerical intensity distributions at the propagation planes of $\xi$ = 0, 1, 2, and 3, respectively. (c1)–(c4) Corresponding experimental results.

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Likewise, the propagation caustic parameter equations can be derived by using Eq. (14):

$$\begin{array}{l} X = 2{K_X}\xi + \frac{{{a_3}^3}}{{36{K_X} + 12\xi }} - \frac{{{a_3}}}{3}\xi + 3{K_X}^2,\\ Y = \frac{{2{a_3}^2\xi }}{{36{K_X} + 12\xi }} - \frac{2}{3}{\xi ^2} + {a_3}{K_X} + 3{(\frac{{{a_3}^2}}{{36{K_X} + 12\xi }} - \frac{\xi }{3})^2}. \end{array}$$

According to the Hess matrix, we can obtain ${K_Y} = \frac{{{a_3}^2}}{{36{K_X} + 12\xi }} - \frac{\xi }{3}$. Figure 7(b) demonstrates the 3D propagation caustics, where the red, green, blue, and black lines denote the caustic lines in the $\xi $ = 0, 1, 2, and 3 planes, respectively, which agree with the maximum intensity distributions depicted in Figs. 8(b1)–8(b4). The propagation caustic includes two sheets that manifest as smooth and cusp shapes. Additionally, as demonstrated in Fig. 7(b), two main lobes propagate along two different accelerating trajectories. Visualization 6 illustrates the dynamical propagation caustic associated with Fig. 8. The animation displays an accelerating property that can also be employed to create waveguide structures along curved trajectories.

Additionally, we continue to investigate the self-construction property for elliptic umbilic beams. In simulations, we make the intensity of a cusp zone zero and discuss the propagation fields. Figures 9(a1), 9(a2), and 9(a3) exhibit the damaged intensity distributions in the $\xi $ = 0, 0.7, and 8 planes, respectively—the damaged cusp reforms with increasing propagation distance. Experimentally, the intensity for the cusp in the initial field is blocked using an opaque object. Figures 9(b1)–9(b3) display the corresponding blocked intensity distributions of Figs. 9(a1)–9(a3) at the ${\xi _b}$ = 0, 100, and 800 mm planes, which coincide with the numerical results. Visibly, the elliptic umbilic beams exhibit a strong self-construction ability. Similarly, Fig. 10 depicts similar results as Fig. 9 for hyperbolic umbilic beams, where the peak area is blocked. EUBs and HUBs exhibit a forceful self-healing ability, which can effectively improve any anti-interference ability. This intriguing property is especially beneficial to depth microscopic imaging.

 figure: Fig. 9.

Fig. 9. The self-construction ability for EUBs. Numerical damaged intensity distributions at $\xi =$ (a1) 0, (a2) 0.7, and (a3) 8 planes. (b1-b3) Corresponding experimental results at ${\xi _b}$ = 0, 100 and 800 mm planes, respectively.

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 figure: Fig. 10.

Fig. 10. The self-construction ability of HUBs. Numerical damaged intensity distributions at $\xi =$ (a1) 0, (a2) 1, and (a3) 2 planes. (b1-b3) Corresponding experimental results at ${\xi _b}$ = 0, 200, and 400 mm planes, respectively.

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4. Conclusions

Using diffraction catastrophe, we demonstrate an effective approach for generating high-dimensional caustic beams with two state parameters. The 3D caustics exhibit their intensity profiles and shapes. The dynamics markedly display their evolutions, and numerical and experimental results verify that such intriguing beams possess interesting accelerating and self-reconstruction abilities, especially the self-autofocusing properties of EUBs. Such exotic properties may give rise to new applications in depth microscopic imaging, wavefront control, and optical micromanipulation.

Funding

Natural Science Foundation of Zhejiang Province (LXZ22A040001); National Natural Science Foundation of China (11974102, 11974314); Jinhua Science and Technology Bureau (20211043).

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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Supplementary Material (6)

NameDescription
Visualization 1       elliptic umbilic beams of caustic
Visualization 2       hyperbolic umbilic beams of caustic
Visualization 3       elliptic umbilic beams of propagation dynamics
Visualization 4       elliptic umbilic beams of propagation caustic
Visualization 5       hyperbolic umbilic beams of propagation dynamics
Visualization 6       hyperbolic umbilic beams of propagation caustic

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 (10)
Fig. 1.
Fig. 1. Numerical intensity distributions for $Eub(X,Y,{a_3})$ in the case of (a1) a3 = −3, (a2) a3 = 0, and (a3) a3 = 3. (b1), (b2) and (b3) $Hub(X,Y,{a_3})$ in case of a3 = −3, 0, and 3, respectively.

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Fig. 2.
Fig. 2. 3D caustics. (a1)–(a2) $Eub(X,Y,Z)$ and (b1)–(b2) $Hub(X,Y,Z)$.

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Fig. 3.
Fig. 3. Experimental setup. BE: beam expander; BS: beam splitter; SLM: spatial light modulator; and L: lens.

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Fig. 4.
Fig. 4. The spectral phases of $Eub(X,Y,{a_3},\xi = 0)$ in the case of (a1) a3 = −3, (a2) a3 = 0, and (a3) a3 = 3. (b1)–(b3) Corresponding experimental results.

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Fig. 5.
Fig. 5. The spectral phases of $Hub(X,Y,{a_3},\xi = 0)$ in the case of (a1) a3 = −3, (a2) a3 = 0, and (a3) a3 = 3. (b1)–(b3) Corresponding experimental results.

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Fig. 6.
Fig. 6. Elliptic umbilic beams $Eub(X,Y,{a_3},\xi )$. (a1)–(a3) Numerical wavefront distribution for $\xi = 0,3,and\textrm{ 8}$ planes. (b1)–(b3) Corresponding experimental results.

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Fig. 7.
Fig. 7. The 3D propagation caustic. (a) $Eub(X,Y,{a_3},\xi )$ and (b) $Hub(X,Y,{a_3},\xi )$

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Fig. 8.
Fig. 8. Hyperbolic umbilic beams $Hub(X,Y,{a_3},\xi )$. (a) The numerical side views in the $Y - \xi$ plane. (b1), (b2), (b3), and (b4) numerical intensity distributions at the propagation planes of $\xi$ = 0, 1, 2, and 3, respectively. (c1)–(c4) Corresponding experimental results.

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Fig. 9.
Fig. 9. The self-construction ability for EUBs. Numerical damaged intensity distributions at $\xi =$ (a1) 0, (a2) 0.7, and (a3) 8 planes. (b1-b3) Corresponding experimental results at ${\xi _b}$ = 0, 100 and 800 mm planes, respectively.

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Fig. 10.
Fig. 10. The self-construction ability of HUBs. Numerical damaged intensity distributions at $\xi =$ (a1) 0, (a2) 1, and (a3) 2 planes. (b1-b3) Corresponding experimental results at ${\xi _b}$ = 0, 200, and 400 mm planes, respectively.

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Equations (19)

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$${C_n}(a) = \int_R {\exp [i{p_n}(a,s)]ds.}$$
$$p_5^e(u,v,w,{s_1},{s_2}) = \frac{1}{3}s_1^3 - {s_1}s_2^2 + w(s_1^2 + s_2^2) - v{s_2} - u{s_1}.$$
$$Eub(X,Y,Z) = \int\!\!\!\int_R {\exp [i(\frac{1}{3}{s_1}^3 - {s_1}{s_2}^2 + Z({s_1}^2 + {s_2}^2) - Y{s_2} - X{s_1})]d{s_1}d{s_2}} ,$$
$$Eub(X,Y,{a_3}) = \int\!\!\!\int_R {\exp [i(\frac{1}{3}{s_1}^3 - {s_1}{s_2}^2 + {a_3}({s_1}^2 + {s_2}^2) - Y{s_2} - X{s_1})]d{s_1}d{s_2}} ,$$
$$Hub(X,Y,{a_3}) = \int\!\!\!\int_R {\exp [i(s_1^3 + s_2^3 - {a_3}{s_1}{s_2} - Y{s_2} - X{s_1})]d{s_1}d{s_2}} ,$$
$$\left\{\begin{aligned} &\frac{{\partial {p_n}(u,v,w,{s_1},{s_2})}}{{\partial {s_1}}} = 0\\ &\frac{{\partial {p_n}(u,v,w,{s_1},{s_2})}}{{\partial {s_2}}} = 0 \end{aligned} \right.,$$
$$\begin{vmatrix} \frac{{{\partial^2}{p_n}}}{{\partial {s_1}^2}} & \frac{{{\partial^2}{p_n}}}{{\partial {s_1}\partial {s_2}}}\\ \frac{{{\partial^2}{p_n}}}{{\partial {s_2}\partial {s_1}}} &\frac{{{\partial^2}{p_n}}}{{\partial {s_2}^2}} \end{vmatrix} = 0,$$
$$\left\{ \begin{array}{l} {s_1}^2 - {s_2}^2 + 2w{s_1} - u = 0\\ - 2{s_1}{s_2} + 2w{s_2} - v = 0 \end{array} \right..$$
$$\left|\begin{array}{l} 2{s_1} + 2w\\ - 2{s_2} \end{array} \right.\left. \begin{array}{r} \textrm{ } - 2{s_2}\\ - 2{s_1} + 2w \end{array} \right|= 0.$$
$$\begin{array}{l} u = {w^2}(\cos (2\theta ) + 2\cos \theta ),\\ v = {w^2}( - \sin (2\theta ) + 2\sin \theta ). \end{array}$$
$$\left\{ \begin{array}{l} 3{s_1}^2 - w{s_2} - u = 0\\ 3{s_2}^2 - w{s_1} - v = 0 \end{array} \right..$$
$$\left|\begin{array}{l} 6{s_1}\\ - w \end{array} \right.\left. \begin{array}{l} \textrm{ } - w\\ \textrm{ }6{s_2} \end{array} \right|= 0.$$
$$\begin{array}{l} u = \frac{1}{{12}}{w^2}(\exp (2\tau ) \pm 2\exp ( - \tau )),\\ v = \frac{1}{{12}}{w^2}(\exp ( - 2\tau ) \pm 2\exp (\tau )). \end{array}$$
$$\begin{aligned} \widetilde {Eub}({K_X},{K_Y},{a_3}) &= \int\!\!\!\int_R {Eub(X,Y,{a_3}){e^{ - i{K_X}X}}{e^{ - i{K_Y}Y}}} dXdY\\ &\textrm{ = }\exp [i( - \frac{1}{3}{K_X}^3 + {K_X}{K_Y}^2 + {a_3}({K_X}^2 + {K_Y}^2))], \end{aligned}$$
$$\begin{aligned} \widetilde {Hub}({K_X},{K_Y},a_3^{}) &= \int\!\!\!\int_R {Hub(X,Y,a_3^{}){e^{ - i{K_X}X}}{e^{ - i{K_Y}Y}}dXdY} \\ &{\kern 1cm} = \exp [i( - {K_X}^3 - {K_Y}^3 - {a_3}{K_X}{K_Y})]. \end{aligned}$$
$$\scalebox{0.88}{$\begin{aligned} Eub(X,Y,{a_3},\xi ) = &\int {\int_{ - \infty }^{ + \infty } {\widetilde {Eub}({K_X},{K_Y})} \exp [i({K_X}X + {K_Y}Y - ({K_X}^2 + {K_Y}^2)\xi )]d{K_X}d{K_Y}} \\ &\;\;\;\;\;= \int {\int_{ - \infty }^{ + \infty } {\exp [i( - \frac{1}{3}{K_X}^3 + {K_X}{K_Y}^2 + ({a_3} - \xi )({K_X}^2 + {K_Y}^2) + X{K_X} + Y{K_Y})]} d{K_X}d{K_Y}} , \end{aligned}$}$$
$$\begin{array}{l} X = {w^2}( - \sin (2\theta ) - 2\sin \theta ),\\ Y = {w^2}(\cos (2\theta ) - 2\cos \theta ),\\ \xi = {a_3} - w, \end{array}$$
$$Hub(X,Y,{a_3},\xi ) = \int {\int_{ - \infty }^{ + \infty } {\exp [i( - {K_X}^3 - {K_Y}^3 - {a_3}{K_X}{K_Y} + X{K_X} + Y{K_Y} - {K_X}^2\xi - {K_Y}^2\xi )]} d{K_X}d{K_Y}}. $$
$$\begin{array}{l} X = 2{K_X}\xi + \frac{{{a_3}^3}}{{36{K_X} + 12\xi }} - \frac{{{a_3}}}{3}\xi + 3{K_X}^2,\\ Y = \frac{{2{a_3}^2\xi }}{{36{K_X} + 12\xi }} - \frac{2}{3}{\xi ^2} + {a_3}{K_X} + 3{(\frac{{{a_3}^2}}{{36{K_X} + 12\xi }} - \frac{\xi }{3})^2}. \end{array}$$
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