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Abstract
This work introduces and demonstrates a method for tailoring multi-focus abruptly autofocusing beams (MFAABs) in free space. Since MFAABs are formed by superposing the wavefronts of abruptly autofocusing beams (AABs), the number, positions, and relative intensity of foci are directly and independently adjusted during the pre-design of AABs. Also, we show that the size of the light field involves the characteristics of focus, such as peak intensity, focal size, and subsequent oscillations. We apply the double-phase hologram method to generate the required complex field with a phase-only spatial light modulator in experiments. With flexible control on foci characteristics, this work may broaden potential applications of autofocusing beams.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Since optical Airy beam was demonstrated in 2007 [1], accelerating beams have attracted extensive attention due to their peculiar propagation properties [2]. By constructing circular Airy beams, a new type of beams, abruptly autofocusing beams (AABs), was theoretically proposed [3] and experimentally observed [4], whose energy is initially distributed away from the center and converging to the focus in a parabolic manner as the beam propagates, so that the intensity at the focus increases by several orders of magnitude. This kind of beams has been used in the regions requiring an abruptly changed beam intensity, such as material micromachining [5], particle manipulation [6] and super-resolution imaging [7].
Later, methods were developed to improve the focus features or to introduce new characteristics to AABs, including adjustment of the phase chirp rate [8], modification of Fourier spectrum profiles [9], combination with a vortex phase [10–13], introduction of a scaling factor to the phase [14], and extension to nonparaxial scheme [15]. Based on similar principles, various beams with novel focusing properties have been proposed subsequently [16–23]. Besides, some ways to generate multi-focus autofocusing beams have also been studied. For example, a complex amplitude modulation on polarized Airy beams could induce various multi-focus phenomena [24]. With frequency spectrum modified, a pair of symmetric Airy beams would evolve into a multi-focus autofocusing Airy beam (MAAB) with four off-axis foci at its focal plane [25]. Recently, on-axis multi-focus autofocusing optical beams were also reported by superposing two one-dimensional (1D) symmetric cosine accelerating beams [26]. However, these methods can only control either the number or positions of foci, leaving other critical characteristics unadjustable.
In this work, a method for a full control on multi-focus abruptly autofocusing beams (MFAABs) is explored and verified. We first derive paraxial parabolic AABs from the Fourier space with desired peak intensity by scaling the light field window. Then MFAABs are realized by superposing the wavefronts of pre-designed AABs. This designing method involves more degrees of freedom, including the number of foci and each focus’ relative intensity and position in free space. To perform MFAABs with phase-only elements, we employ the double-phase hologram (DPH) method [27] for complex field encoding. Finally, we demonstrate three types of MFAABs (including on-axis and off-axis cases) and observe the unique multi-focus phenomena in experiments using a spatial light modulator (SLM).
2. Focal characteristics control
The MFAAB proposed here is composed of several single-focus circular AABs. Following the bottom-up clue, the key to our method lies in constructing individual components, then the coordination between each other. A parabolic trajectory is chosen as the primary demonstration of single-focus circular AABs for its non-diffraction [28] and ease of implementation. We use the caustic method [29] and define the trajectory function in a cylindrical coordinate system as:
where $r_0$ is radius of the central dark region, and $z_c$ is the intersection of the caustic at the central axis, which could be regarded as the distance of focus approximately.Under the paraxial approximation, by using the Fourier transform effect of a lens, the spatial spectrum of the initial field in the real plane ($z=0$) can be directly tailored in the phase plane. We write the complex wavefront in the phase plane as $\psi (r)=a(r)\exp {[\mathrm {i}\phi (r)]}$, where $a(r)$ and $\phi (r)$ are the amplitude and phase distributions, respectively. The phase term under the paraxial approximation can be expressed as [17]:
where $k$ is the wave number and $f$ is focal length of Fourier lens. Figure 1(a) shows the design principle and how light rays form AABs , and Fig. 1(b) the corresponding numerical propagating profile. Note that the simulations of beam dynamics in this work are based on the angular spectrum method [30].
Fig. 1. (a) Ray-optics schematics of a section of AABs. (b) Corresponding light intensity distribution in numerical simulation (logarithmic scale).
In an MFAAB, the number and positions of foci are determined by the trajectories of AABs previously designed in phase space. For an AAB with a given trajectory, its phase distribution $\phi (r)$ is unique and the remaining degrees of freedom to control the characteristics of the focus lies in the amplitude profile $a(r)$. To maximize the energy efficiency [8], we define $a(r)$ as:
where $R$ is the radius of initial light field. Because there must be light rays passing through the focus to produce the auto-focusing effect, $R$ needs to meet:Next, we will discuss the influence of parameter $R$ on the focus characteristics. The complex amplitude $E(z)$ along the central axis can be obtained by calculating the angular spectrum diffraction integral in a paraxial situation:
We use a laser with a wavelength of 1.55 $\mathrm{\mu}$m and a lens with a focal length of $f=75$ mm in this work. Figure 2(a) shows an AAB with parameters set as: {$r_0=0.3$ mm, $z_c=30$ mm, $R=2$ mm}, and the numerical and theoretical results of its intensity on the central axis are shown in Fig. 2(b). Except for a slight position offset of the intensity peak, their intensity profiles are basically consistent. Thus the prediction of focus feature by Eq. (5) is proved to be accurate, which can guide and simplify our designing process. Derived from Eq. (5), the intensity profiles along $z$ axis and peak intensities with varying $R$ are as shown in Fig. 2(c). A higher peak intensity and narrower full width at half maximum (FWHM) can be obtained with $R$ increasing within limits, while too large $R$ will decrease the maximum intensity because a part of the energy is transferred to subsequent oscillations. The result reveals a remarkable association between peak intensity and the value of $R$ in a considerable range. A target peak intensity can be directly matched to a suitable $R$ value, enabling us to regulate the intensity of each focus in an MFAAB independently.
Fig. 2. (a) Intensity distribution of designed AAB (scaled logarithmically to show the trajectories of AABs). (b) Theoretical predicted and numerical result of intensity distribution along the central axis. (c) Theoretical intensity distribution and peak intensities with different R values. The ordinate represents the ratio of the light intensity ($I$) to the maximum value on the initial plane ($I_0$).
3. Construction of MFAABs using phase-only elements
An MFAAB can be regarded as the superposition of several AABs with different trajectories and focus positions, which is technically realized by linearly superimposing the complex wavefronts corresponding to different trajectories. If all the foci are on the central axis, the wavefront of such an MFAAB can be written as:
Figures 2(b) and 2(c) show that an AAB has almost no intensity distribution before the focus, but its subsequent oscillations might pose a perturbation to the following foci. The interaction effect among AABs depends on their relative phase, which varies with the values of additional phase $c_j$ in Eq. (6). A group of well-selected $c_j$ can minimize the inter-focus disturbance to ensure that superposed foci have the same relative intensities as they were separately designed. In this work, the additional phase $c_j$ is optimized by sweeping its value, with steps of 0.01$\mathrm{\pi}$, in a range of $[0, 2\mathrm{\pi} )$. We optimize the $c_j$ term in the complex amplitude $\psi _j(r)$ of the AABs in the order of the distance of their focus to the initial plane, from the nearest ($j = 1$) to the farthest ($j = n$). We first set $c_1$ as 0 and calculate the central light field distribution $E_1(z)$ of the first AAB by the angular spectrum diffraction integral. The next step is the optimization of $c_2$: the central light field distribution $E_2(z)$ of the second AAB is calculated from $\psi _2(r)$, and then superimposed to $E_1(z)$ to select the optimal value of $c_2$, with the criterion that the peak intensity of the second AAB should remain least affected by $E_1(z)$. Repeating such a strategy, the optimal values of all the $c_j$ can be determined in succession.
Our method also adapts to generate multiple foci on a focal plane. The construction process is similar to the former case, except that the focal point of each AAB component is off-axis. An off-axis focus is obtained by simply multiplying the expression of AAB in phase space by a linear phase along the offset direction, equivalent to a translation in real space:
The key to generating high-quality MFAABs is to control the complex amplitude of the wavefront precisely. Several approaches have been proposed to directly control the complex amplitude of the light field, such as optical metasurfaces [31,32], and diffractive optical elements (DOEs) with varying opaque areas on pixel units for modulating transmittance [33]. However, these elements require complicated design and fabrication and also lack dynamic adjustability. An alternative way to control the complex amplitude is to encode it into phase terms. One of such techniques is DPH coding method [34]. The approach is based on the apparent fact that a complex field, with its amplitude function $a(x,y)$ not greater than $1$, can be decomposed into two phase components:
We follow the method developed by Arrizón and his collaborators [27] to integrate the two phase components of DPH coding, which use four adjacent pixels ($2\times 2$) to form a complex amplitude modulation unit, namely the macro-pixel. The complex light field is discretely sampled at the interval of macro-pixel $D=2d$, where $d$ is the period of sub-pixel, equal to the pixel pitch of the SLM here. Then we interlace two phase components calculated from Eq. (9) in the four sub-pixels of such a modulation unit, by taking the diagonal elements as $\theta _1$ and anti-diagonal elements as $\theta _2$. In this way, the modulation effect of the four sub-pixels together is nearly equivalent to the complex amplitude modulation of the macro-pixel. Further description of the method and analysis of the light field reconstruction results are given in [27].
To avoid overlap of different diffraction orders, the reconstructed light field should be limited to a size smaller than the spacing of adjacent diffraction orders. Since the light field of a single AAB on the real plane has a radially symmetric form, we define its maximum radius as $R_0$, which can be estimated by the ray-optics model shown in Fig. 1(a):
Compared with another commonly used method proposed by Davis, et al. [35], which attaches a linear phase grating (typically no less than 8 pixels/period), the macro-pixel coding technique uses only two pixels in a sampling period, providing a much larger spacing between diffraction orders for the light field reconstruction. This feature will bring significant benefits for generating MFAABs. First, it directly offers a larger lateral offset space for off-axis foci. On the other hand, it helps to enhance the intensity of the focus, because a greater trajectory curvature [28] and a larger $R$ value [Eq. (5)] can both contribute to higher focus intensity and a larger $R_0$. Equality (5) also indicates that a more distant focal point requires a larger $R_0$ to maintain a specific intensity, i.e., a larger $R_0$ enables us to tailor a more distant focal point.
4. Numerical and experimental demonstrations of MFAABs
The experimental setup for generating and observing MFAABs is illustrated in Fig. 3. The laser with a wavelength of 1.55 $\mathrm{\mu}$m is first expanded by a beam expander to about 12 mm in diameter and then horizontally polarized. Before reaching the SLM (Santec SLM-200, 10-bit gray level, $1920 \times 1080$ pixel resolution, 8.0 $\mathrm{\mu}$m pixel pitch), the light beam is cut off by a 5-mm diameter aperture to approximate a plane wave within limits. Light modulated by the SLM undergoes the Fourier transform of the lens, then evolves into the required field distribution on the back focal plane of the lens. A camera (AVT Goldeye P-008 SWIR, 14-bit dynamic range, 320 $\times$ 256 pixel resolution, 30 $\mathrm{\mu}$m $\times$ 30 $\mathrm{\mu}$m pixel size) is installed on a motorized rail to record the profiles of MFAABs at different positions over the propagation range (15 cm), with 1 mm of each step. To show the feasibility of our method, we construct three kinds of MFAABs: three foci on the central axis, three and four foci on the focal plane. The foci in each case have close peak intensities.
Fig. 3. Schematic of the experimental setup for generation and detection of MFAABs. BE: beam expander; LP: linear polarizer; M: mirror; BS: beam splitter; SLM: spatial light modulator; L: lens. The focal length of the lens is 75 mm.
To construct three foci with similar peak intensities along the central axis, we design three AABs in a group focusing on the central axis at the same angle, with their trajectory parameters set as: {$r_{01}=0.6$ mm, $z_{c1}=60$ mm; $r_{02}=0.9$ mm, $z_{c2}=90$ mm; $r_{03}=1.2$ mm, $z_{c3}=120$ mm}. Their trajectories are similar, and they have been scaled in different proportions. From a dynamical perspective, it can be assumed that the rays through each focal point are all emitted from a circle with a radius of 1.5 mm. An AAB with a farther focus is calculated with a larger initial light field area to offset the diffraction loss during propagation. Using the results of Eq. (5) for prediction, we set the parameters of light field area and additional phase as: {$R_1=2.15$ mm, $c_1=0$; $R_2=2.19$ mm, $c_2=0.6\mathrm{\pi}$; $R_3=2.25$ mm, $c_3=0.17\mathrm{\pi}$}, which can keep the relative intensities (set to be identical in this example) of AABs unchanged after the superposition. The corresponding amplitude and phase distributions are shown in Figs. 4(e) and 4(f).
Fig. 4. The numerical simulation and experiment results of MFAAB with three foci on the central axis. (a) (b) Side-view of the beam in the X-Z plane in simulation and experiment. Green dash lines represent the caustic, and withe dash lines mark the position of the central axis. (c) (d) Intensity distribution on the central axis in simulation and experiment. The results in the experiment are normalized. (e) (f) The amplitude and phase distribution of the wavefront used in the simulation. (g) The complex field encoding result and phase diagram loaded on SLM.
The numerical and experimental results shows that the light converges along different trajectories behind the image plane (Figs. 4(a), 4(b)), and produces three focal points with the virtual intensity on the central axis simultaneously (Figs. 4(c), 4(d)).
We also tailor MFAABs with three and four foci on the focal plane. We choose the following parameters to construct an initial AAB: {$r_0=0.8$ mm, $z_c=80$ mm, $R=2.25$ mm}. In the case of three foci, the trajectory is superimposed with its two copies made at the positions of $\pm$0.4 mm translation along the X-axis (Fig. 5), while the case of four foci is composed of four trajectories respectively translated by ($\pm 0.4$ mm, $\pm 0.4$ mm) in the X-Y plane (Fig. 6). The corresponding amplitude and phase distributions of the above cases are shown in Figs. 5(e) and 5(f) and Figs. 6(e) and 6(f), respectively. In both cases, the foci have uniform peak intensities on the focal planes set at $z=81$ mm, consistent with our simulation results.
Fig. 5. The numerical simulation and experiment results of MFAAB with three foci on the focal plane. (a) (b) Side-view of the beam in the X-Z plane in simulation and experiment. Green dash lines represent the caustic. (c) (d) Intensity profile on the center-line of the focal plane in simulation and experiment ($z=81$ mm, $y=0$). (e) (f) The amplitude and phase distribution of the wavefront used in the simulation. (g) The complex field encoding result and phase diagram loaded on SLM. (h1)-(h3) (i1)-(i3) Normalized intensity distribution at different distance in simulation and experiment, corresponding to the white dash lines in (a) and (b) ($z=62$ mm, $z=76$ mm, $z=81$ mm, respectively). Each picture displays an area of 2.4 mm $\times$ 2.4 mm.
Fig. 6. The numerical simulation and experiment results of MFAAB with four foci on the focal plane. (a) (b) Side-view of the beam in the X-Z plane in simulation and experiment. Green dash lines represent the caustic. (c) (d) Intensity profile on the center-line of the focal plane in simulation and experiment ($z=81$ mm, $y=0$). (e) (f) The amplitude and phase distribution of the wavefront used in the simulation. (g) The complex field encoding result and phase diagram loaded on SLM. (h1)-(h3) (i1)-(i3) Normalized intensity distribution at different distance in simulation and experiment, corresponding to the white dash lines in (a) and (b) ($z=62$ mm, $z=76$ mm, $z=81$ mm, respectively). Each picture displays an area of 2.4 mm $\times$ 2.4 mm.
The phase patterns for MFAABs generations of the above three cases are as shown in Fig. 4(g), Fig. 5(g), and Fig. 6(g), respectively. All the patterns have an effective circular area, out of which light intensity is considered to be 0 in the calculation. In experiments, the patterns are loaded at the center of SLM. The part of incident light outside this effective circle will be attributed to noise after modulations. Therefore, it is necessary to use a diaphragm to control the size of the light, and its aperture should be only slightly larger than the phase modulation area. In addition, due to the limited modulation efficiency of SLM, a part of the light reflected without being modulated converges at the rear focal plane of the lens (image plane), then diverges rapidly. Moreover, the difference between the experiment and the simulation results may also be caused by the following reasons: misalignment of the lens placement, deviation of the light field from the ideal plane wave, interference of the remaining noise light, etc.
The method of wavefront superposition can also be combined with other means for focus and trajectory characteristics control, such as introducing the vortex phase, and designing the trajectory of other curves, etc. If there are devices that can directly control the complex amplitude of the light field, such as metasurfaces, MFAAB can also be designed in real space to obtain a more compact device (without a lens to do the Fourier transform).
5. Conclusion
This work proposes and demonstrates a new type of autofocusing beams, with multiple foci adjustable in their number, positions, and peak intensities. We first construct a parabolic type AAB in Fourier space based on the caustics method. Through theoretical calculations and numerical simulations, we verify that the characteristics of the focus, especially the peak intensity, can be adjusted by scaling the initial light field size. MFAABs with on-axis and off-axis foci are both constructed by superimposing multiple AAB wavefronts pre-designed with expected trajectories and focal intensity. Attaching an additional phase to each AAB can counteract the interference effect and maintain relative intensity during the superposition process. Because the realization of MFAABs requires a complex modulation of the light field, we employ the double-phase hologram method to encode the complex wavefront in a phase-only element using an SLM. Finally, we observe MFAABs with expected focusing phenomena. The novel beams may be applied to optical trapping, particle manipulating, micromachining, and medical treatment.
Funding
National Key Research and Development Program of China (2019YFA0706302); National Natural Science Foundation of China (11774437); Basic and Applied Basic Research Foundation of Guangdong Province (2021B1515020093); Science and Technology Planning Project of Guangdong Province (2018B010114002); Local Innovative and Research Teams Project of Guangdong Pearl River Talents Program (2017BT01X121).
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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