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Abstract
We propose and demonstrate a type of multi-focus autofocusing beams, circular hyperbolic umbilic beams (CHUBs), based on the double-active variable caustics in catastrophe theory. The mathematical form is more general compared to circular Airy, Pearcey and swallowtail beams. The CHUBs can generate multi-focus at its optical axis, while the on-axis intensity fluctuates up to two orders of magnitude that of the maximum intensity in the initial plane. Using the concept of topographic prominence, we quantify the autofocusing ability. We construct the criteria for selecting the effective foci, and then explore the influence of related parameters. Our findings suggest that the CHUBs could be a suitable tool for multi-particle manipulation, optical tweezers, optical lattices and related applications.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
There has been a tremendous development of interest in the area now known as catastrophe theory in mathematics since the first studies in the mid-1960s of Rene Thom's Stabilite Structurelle et Morphogenese, which finally appeared in 1974 [1]. Caustics typically appear as points, lines, surfaces, or hypersurfaces as a result of catastrophes. Each catastrophe creates a stable diffractive structure with rapid shifts in the diffraction structure's critical points. Diffraction sudden changes exhibit themselves as optical caustics. As the dimension of its control parameter space increases, there are fold (A2), cusp (A3), swallowtail (A4), butterfly (A5), and wigwam (A6) potential functions, as well as elliptical umbilic (D4-), hyperbolic umbilic (D4+), parabolic umbilic (D5), and symbolic umbilic (E6) diffraction catastrophe of the potential functions of two active variables [2,3]. Caustics can be sculpted into propagation-invariant light [4], and it can be used to interpret the abruptly autofocusing vortex beams [5], to present a modern viewpoint on the classes of structured light [6], and other physics phenomenon such as nonequilibrium quantum multi-body processes [7]. The hyperbolic umbilic catastrophes, according to Thom, model the breaking of a wave. They are examples of co-rank 2 catastrophes closely related to the geometry of nearly spherical surfaces, and can be seen in the focal surfaces formed by light reflecting off a surface in three dimensions: umbilical point. The hyperbolic umbilic beams were first observed experimentally in spheroidal drops with rainbow scattering by Nye [8] and Marston et al. [9] in 1984, who performed an analysis based on geometric optics. Subsequently, Kaduchak et al. [10] and Nye [11] mathematically analyzed the properties of the hyperbolic umbilic function.
Beam focusing has long been a popular research area. For many applications, focusing the energy on the wavefront onto the target spot abruptly while retaining a low intensity profile until then is advantageous. Since the theoretical proposal [12] and experimental demonstration [13] of Airy beams, there has been an increasing interest in abruptly autofocusing (AAF) beams. The circular Airy beams (CABs) [14,15], based on circular symmetry and the combination of the properties of self-bending and non-diffractive of the 2 + 1-D Airy beams, have tight-focusing characteristics [16,17]. Properties of circular autofocusing beams can be changed with certain modifications such as off-axial vortex singularities [18] and cross-phase [19–21]. We have previously studied CABs with optical vortices, blocked rings, radial polarization and partially coherent properties [22–29]. Liu et al. used diffractive optical elements (DOEs) to achieve autofocus multiple times of 1 + 1-D beams in free space [30]. Wu et al. introduced and demonstrated a method for tailoring multi-focus abruptly autofocusing beams (MFAAB) in free space using double-phase hologram method [31]. Similarly, the abruptly autofocusing ability of circular Pearcey beams (CPBs) can reach three orders of magnitude in comparison to the maximum intensity in the initial plane [32–34] and can be controlled in nonlocally defocusing nonlinear media [35]. The swallowtail [36–38] and butterfly [39] beams corresponding to the higher two examples of single active variable caustics in catastrophe theory, have also been investigated.
Although extensive research has been carried out on fold and cuspoid catastrophes, to the best of our knowledge, optical beams with circular symmetry and umbilic catastrophes have not been investigated. The theoretical analysis of the integral of the higher-order canonical potential function is more complicated, and the oscillation phase factor is difficult to converge in numerical simulations [40].
In this work, we report the theoretical and experimental demonstration of a type of multi-focusing beams, namely circular hyperbolic umbilic beams (CHUBs), based on the hyperbolic umbilic caustics in the standard diffraction catastrophe integral. This type of beam tends to focus automatically multiple times. We find that the intensity fluctuations can reach two orders of magnitude at axis, compared to the highest intensity in its initial plane. Certain parameters can be modified to tailor the multifocal characteristics, making it a good candidate for applications such as optical trapping, micromanipulation and driven optical lattices in quantum simulations, etc.
2. Theory
2.1 Hyperbolic umbilic function
In catastrophe theory, the cuspoid caustic field ${C_n}(x )$ is defined in terms of standard diffraction integrals, which can be represented as [2]:
The canonical potential function ${p_n}({x;s} )$ is referred to as a polynomial in the above equation, and x is a vector or a scalar that parametrizes the potential. For the hyperbolic umbilic caustics belonging to umbilical catastrophes, we use Thom's standard form of potential function [1]
Hereafter the corresponding caustic field will be abbreviated as HU function
It is clear from Eq. (3) that $\textrm {HU} ({x,y,z} )= \textrm {HU} ({y,x,z} )$. Note that when $z = 0$, the integral degenerates into the multiplication of two Airy functions
Equation (4) means that the 2 + 1-D Airy beams is a special case of the HU beams except for a scaling factor.
In contrast to the cusp caustics, it is difficult to get the numerical computation to converge if Eq. (3) is employed directly for double integration - the integrand oscillates rapidly for larger values of u or v. A contour rotation in the complex s plane can be utilized to transform the highest-order phase factor $\textrm{exp} ({\textrm{i}{s^{n + 2}}} )$ into a decaying factor and speed up integration convergence for $n = 1\sim 4$ cases in Eq. (1). Fortunately, we discover that with the help of the well-studied airy function [41]
The common factor v has been extracted outside the bracket in Eq. (6) to speed up the calculation, namely the Qin Jiushao’s algorithm or Horner's method [42].
2.2 2 + 1-D circular hyperbolic umbilic beams
To define the 2 + 1-D circular hyperbolic umbilic beams, two of the three variables need to be chosen as beam coordinates, with the left variable $\xi $ acting as a constant parameter. Obviously, considering the symmetry of the state variables $(x\& y)$ in Eq. (3), there are two options, namely HU12 and HU23 functions, which are defined as:
Note that x, y, and $\xi $ in Eq. (7) are all dimensionless independent variables of the HU function.
Figure 1 visualizes the modulus of the HU12 [Figs. 1(a1)–1(a5)] and HU23 [Figs. 1(b1)–1(b5)] functions taking five different $\xi $ values. Note the similarity of Fig. 1(a1) to the 2 + 1-D Airy function.
Fig. 1. Numerical calculations of the modulus of the (a1)-(a4) HU12 and (b1)-(b4) HU23 functions of five different $\xi $ values. The coordinates in the x and y directions both range from -60 to +60.
Thereafter, we focus on the HU12 function, which is more closely related to the Airy function. The definitions of CPBs and CSBs in their initial planes are simply mapping of the beam coordinate along the radial direction of cylindrical coordinates to the x axis direction corresponding to the caustics coordinates. In contrast, without loss of generality, the corresponding 2 + 1-D circular hyperbolic umbilic beams can be defined in the $z = 0$ plane as
The parameters in Eq. (8) are listed in Table 1.
Table 1. Parameter explanations in Eq. (8)
The interpolated coordinates ${\rho _1}$ and ${\varphi _{\textrm{intp}}}$ are from the initial plane of CHUB to the HU function. The parameter ${C_0}$ is related to the optical power in the bucket (PIB). In the next simulations, we modify the value of ${C_0}$ to adjust the optical power to 1mW while keeping the scaling factor ${\rho _0} = 0.1\textrm{mm}$ constant.
Figure 2 depicts the intensity distributions of the CHUBs in $z = 0$ plane when $a = 0.002$, ${\rho _1} = 10{\rho _0}$ and ${\varphi _{\textrm{intp}}} = 0$, while $\xi $ takes 0, 1, 5 and 15. Figure 2(a) is the reversed ($\cos {\varphi _{\textrm{intp}}} = 1 > 0$) circular Airy beams. It can be seen from Figs. 2(b)–2(d) that when $\xi $ increases, the number of rings in the initial planes will increase accordingly.
Fig. 2. Numerical intensity distributions of the CHUBs for different $\xi $ values in initial plane: (a)$\xi = 0$; (b)$\xi = 1$; (c)$\xi = 5$ and (d) $\xi = 15$. Other parameters are $a = 0.002$, ${\rho _0} = 0.1\textrm{mm}$, ${\rho _1} = 10{\rho _0}$, ${\varphi _{\textrm{intp}}} = 0$ and $\lambda = 1064\textrm{nm}$. The unit of intensity in the color bar is W/m2.
Next, we discuss the propagation properties of the CHUBs by numerical calculations using the angular spectrum representation method [43]:
In order to quantitatively describe the autofocusing capability, we use the ratio of the intensity on the optical axis to the maximum intensity in the initial plane to represent the autofocusing behavior: $\eta (z )= I({0,0,z} )/\max [{I({x,y,z} )} ]$.
Figure 3 shows the simulations of a CHUB propagation in the case of $a = 0.002$, $\xi = 12$, ${\rho _1} = 50{\rho _0}$ and ${\varphi _{\textrm{intp}}} = 0$. Figure 3(a) plots the autofocusing capability $\eta $ with the propagation distance z. Figure 3(b) depicts the side-view intensity in $x - o - z$ plane ($y = 0$). Figures 3(c1)–3(c4) depict the intensity distributions at the cross-sectional locations of the highest two maximum and adjacent minimum positions. It can be seen that the on-axis intensity undergoes several fluctuations as the propagation distance increases, which means that the beam has several on-axis autofocusing point.
Fig. 3. Simulation of propagation of a CHUB when $a = 0.002$, $\xi = 12$, ${\rho _1} = 50{\rho _0}$ and ${\varphi _{\textrm{intp}}} = 0$: (a) autofocusing capability $\eta $ as a function of propagation distance z; (b) side view of intensity in $x - o - z$ plane within 0.7 m; (c1) and (c3) cross-sectional intensity distributions in the highest two planes marked in (a) and (b); (c2) and (c4) distributions of the valley bottom adjacent to the highest peak of the curve in (a). The intensity is displayed in kW/m2.
2.3 Influence of parameters on the progression process
It can be seen from Eq. (8) that there are two main parameters describing the beam: interpolation slope ${\varphi _{\textrm{intp}}}$ and interpolation intercept ${\rho _1}$. Other parameters, Gaussian decaying factor $a$, HU function parameter $\xi $, scale factor ${\rho _0} = 0.1\textrm{mm}$ and optical power should be regarded as constants.
The influence of ${\rho _1}$ on autofocusing ability is shown in Fig. 4. The intensity distributions in $z = 0$ planes of five different values (${\rho _1}/{\rho _0} ={-} 10,0,10,20,50$) are shown in Figs. 4(a1)–4(a5) when $a = 0.002$, $\xi = 10$ and ${\varphi _{\textrm{intp}}} = 0$, and the related $\eta $ curves are shown in Figs. 4(b1)–4(b5). The number of foci on the z axis will rise as the interpolation intercept ${\rho _1}$ grows larger within a given range, as will the depth of focus (fluctuations of intensity). Please note that all color bars are not normalized in order to quantitatively depict the difference in their autofocusing abilities.
Fig. 4. Propagation simulations of CHUBs with different interpolation intercept ${\rho _1}$ when $a = 0.002$, $\xi = 10$ and ${\varphi _{\textrm{intp}}} = 0$: (a1)-(a5) intensity distributions in the $z = 0$ planes; (b1)-(b5) fluctuations of autofocusing ability $\eta $ with propagation distance z (${\rho _1}$ values marked in the pictures) corresponding to (a1)-(a5).
Next, we focus on quantifying the description of the $\eta (z )$ curve in order to systematically explore the effects of these parameters.
Although interferometric visibility is widely used to characterize the contrast of interference fringes, its definition depends primarily on the highest and lowest intensities and does not adequately captures the fluctuations in unequal amplitude oscillations. Inspired by the topographic prominence (TP) widely used to describe the relative heights of curve peaks in topography and signal processing [44,45], we also use this concept in the following discussion. The TP of a focal point of the curve is the least drop required from this peak's top to the summit of an adjacent higher peak. If there is no higher peak (i.e. the highest peak), find the lowest points on the left and right sides of the main peak, and TP is the height difference between the main peak and the higher of the two lowest points. Since $\eta $ is dimensionless, TP is a dimensionless value for each peak. Figure 5(a) illustrates a schematic diagram of the TP concept mentioned above. The TP value of each peak is indicated by the red vertical double-arrow line. Please notice that the peaks do not include the curve's terminals.
Fig. 5. Schematic diagram of TP: (a) illustration of TP concept; (b) scatter of simulations of total TP summed over all effective foci using our inclusion criteria when $a = 0.002$ and $\xi = 15$; (c) simulation of the $\eta $ curve with nearly largest total TP in (b) when ${\rho _1} = 112{\rho _0}$ and ${\varphi _{\textrm{intp}}} ={-} 0.005\mathrm{\pi }$.
Trapping depth and the number of foci in an optical field are important characteristics in applications such as optical lattices, multi-particle optical trapping, etc. The TP indicates the relative intensity fluctuations of each focus, and the base of the $\eta $ curve should also be considered to evaluate the focusing capability of each focus thoroughly. This implies that whether a focal point can form an effective trap needs to be judged according to certain criteria. Therefore, we define the inclusion criteria for selecting effective focus in the following discussion. After sorting the peaks from highest to lowest, select the foci whose TP exceeds 50% of their peak and is greater than 30% in height compared to the highest peak. The first requirement excludes peaks with insufficient focus depth (higher valley bottom), while the second rule filters out small fluctuations in the curve's tail.
The autofocusing ability $\eta $ is better when the interpolation direction is near $+ x$ direction (${\varphi _{\textrm{intp}}} \approx 0$) and ${\rho _1}$ gets larger. The dependencies of $\eta $ on ${\rho _1}$ and ${\varphi _{\textrm{intp}}}$ is presented in Fig. 5(b) when $a = 0.002$ and $\xi = 15$. Each dot represents the total TP summed over all the effective foci as the result of two parameters ${\rho _1}$ and ${\varphi _{\textrm{intp}}}$. Figure 5(c) shows a simulation of the autofocusing ability $\eta (z )$ under circumstances ${\rho _1} = 112{\rho _0}$ and ${\varphi _{\textrm{intp}}} ={-} 0.005\mathrm{\pi }$ corresponding to the highest point in Fig. 5(b).
The total TP summed over 17 effective foci can reach 2.2k. In contrast, the circular Airy beams (CABs), circular Pearcey beams (CPBs) and circular swallowtail beams (CSBs) are single-shot autofocusing beams. The unique properties of CHUBs - multiple on-axis foci with intensities of up to two orders of magnitude - makes them suitable candidates for constructing many-body optical traps.
3. Experimental results and discussions
Figure 6 shows our experimental setup for generating the CHUBs. We use the single-pixel checkerboard method [46] for simultaneous encoding of amplitude and phase on a single reflective spatial light modulator (BNS, P512-1064-PCIe, 512 × 512 pixels, pixel spacing of 15 µm). The 1064nm laser shaped by a polarization maintaining fiber is collimated by a beam expander. A neutral density filter, two half-wave plates and a sandwiched polarizing beam splitter are used to generate linearly polarized beam with tunable power that meets the requirements of the SLM modulation. According to the translation property of the Fourier transform, we add a linear phase in the hologram loaded on the SLM to separate zero diffraction order from the unmodulated spot. Then, in the position of the $2f$ plane, the zero-order component of the spectrum is selected by an aperture, finally resulting in a spatially distributed CHUB in the $4f$ plane. A beam profiler (Spiricon, SP620U, 1600 × 1200 pixels, pixel spacing of 4.4 µm) is used to observe the beam profile and measure the intensity distributions behind the $4f$ plane. In Fig. 6, the CCD is intentionally drawn behind the $4f$ plane, which is typical in practical measurements.
Fig. 6. Experimental setup. 1064, 1064nm laser; PMF, polarization maintaining fiber; NDF, neutral density filter; BE, beam expander; PBS, polarizing beam splitter; BT, beam trap; HWP, half-wave plates; SLM, spatial light modulator; L, $f = 25\textrm{cm}$ lens; AP, aperture; M, mirrors; CCD, charge coupled device.
Figure 7 depicts the experimental results of one standard group [Fig. 7(a)] and four control groups [Fig. 7(b)–7(e)], compared with the corresponding theoretical simulations. The optical power is reduced by scattering in the neighboring air media at each focal point, therefore the higher the autofocusing ability and the more focal points, the greater the power loss. The experimental results support our ideas: for short propagation distances, the agreement with simulations is strong, but as the distance rises, the deviation of $\eta $ increases. Nonetheless, our experiments reveal that increasing the Gaussian decaying factor a reduces the distance between foci. Parameter $\xi $ of HU function primarily impacts the number of foci. Interpolation intercept ${\rho _1}$ controls the depth of focus, while the interpolation direction ${\varphi _{\textrm{intp}}}$ has a major influence on autofocusing behavior.
Fig. 7. Comparison of experimental results (black scatter) and theoretical simulations (blue curve) of autofocusing ability $\eta $ versus propagation distance z under multiple sets of parameters: (a) standard parameter set - $a = 0.002$, $\xi = 15$, ${\rho _1} = 100{\rho _0}$ and ${\varphi _{\textrm{intp}}} = 0$; (b) increase a to 0.02; (c) reduce $\xi $ to 8; (d) reduce ${\rho _1}$ to 20${\rho _0}$; (e) reduce ${\varphi _{\textrm{intp}}}$ to -0.1$\mathrm{\pi }$. The rest of the parameters in (b)-(e) are the same as in (a).
Figure 8(a) plots our experimental results corresponding to the simulation with a larger total TP value in Fig. 5(b). Figure 8(b) shows the hologram loaded on the SLM encoded using the checkerboard method. Figures 8(c1)–8(c4) and 8(d1)–8(d4) are the simulations and experimental results in the two intensity maxima and adjacent minima planes marked by red vertical lines in Fig. 8(a), respectively. Note that the optical power in the experiment differs from the simulation value (1mW) but remains constant, and that the $\eta $ curve is independent of absolute power, implying that the theoretical and experimental color bar values are not comparable - it is the relative changes in each group that need to be compared.
Fig. 8. Comparison of experimental results with theoretical simulations of parameter set listed in Fig. 5(c): (a) on-axis intensity profile $\eta (z )$; (b) computer-generated hologram; (c1)-(c4) numerical intensity distributions in the four extreme planes marked by red vertical lines in (a); (d1)-(d4) experimental results in the corresponding planes in (c1)-(c4). The light intensity in (c1)-(c4) is in kW/m2. Note that the color bar of (d1)-(d4) are the relative intensity recorded by the CCD, which should be proportional to the theoretical value in (c1)-(c4).
The experimental results show that when the CCD is moved to the autofocusing planes, the outer rings darken and the on-axis intensity approaches a maximum value; when moved to the minimum intensity positions, the surrounding ring is bright and the center is dark. Given the limitations of the experimentally generated beam size and the resolution of our SLM, and the fact that the beam will be scattered by air turbulence to reduce the power especially at focal points, the measured ratio is smaller than the theoretical expectation as the propagation distance increases. Our simulations are in good agreement with the experimental results.
4. Conclusions
We propose and investigate the circular hyperbolic umbilic beams corresponding to the hyperbolic umbilic caustics in catastrophe theory, and find theoretically and experimentally that multiple foci can be generated on optical axis without relying on other optical components. In reality, given constant optical power, the autofocusing ability of all circular autofocusing beams is mainly determined by the maximum intensity in their initial planes. The three parameters $\xi$, ${\rho _1}$ and ${\varphi _{\textrm{intp}}}$ primarily influence the multi-focusing capabilities. In view of the fact that the normalization scale ${\rho _0}$ and Gaussian factor a are involved in many studies, their influence on the transmission is relatively simple. The higher the Gaussian decaying factor a, that is, the higher the intensity of the inner main rings, the weaker the autofocusing capacity $\eta (z )$; the larger the HU function parameter $\xi $ - the more number of outer rings, the lower the energy of the main ring, and the stronger the corresponding autofocusing ability. The HU function coordinates get larger when the interpolation intercept ${\rho _1}$ is larger, and the corresponding function value is relatively modest, which also leads to a drop in the initial main ring energy, enhancing the autofocusing ability. By smoothly changing these parameters, the properties of the focusing field can be continuously adjusted. These beams can be utilized to build one-dimensional single-beam particle traps, or to drive the optical lattice to achieve adiabatic control of the quantum simulation system.
This study enriches the understanding of the autofocusing beams and provides references for future research of high-order catastrophe beams. As an interesting candidate for various applications, we hope that this multi-focusing properties could open a door for multi-particle trapping, optical lattices, and broaden more applications of the catastrophe theory.
Funding
Fundamental Research Funds for the Central Universities (2016XZZX004-01, 2017QN81005); National Natural Science Foundation of China (11474254, 11804298); National Key Research and Development Program of China (2017YFA0304202).
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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