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Abstract
An efficient scheme for the design of aperture fields (distributed sources) that radiate arbitrary trajectory curved (accelerating) beams, with enhanced controllability of various beam features, is presented. The scheme utilizes a frame-based phase-space representation of aperture fields to overcome the main hurdles in the design for large apertures: First, it uses the a-priory localization of caustic beams to significantly reduce the optimization problem’s variable space, to that of few Gaussian window coefficients accurately capturing those beams. Then, the optimization problem is solved in the reduced (local) spectral domain. We adopt a linearization approach that enables the solution by sequential application of conventional convex optimization tools, which are naturally compatible with the proposed phase-space representation. The localized nature of the Gaussian windows’ radiation is used also for fast field evaluation at a greatly reduced number of optimization constraint points. The significant enhancement in the controllability over the various beam parameters is demonstrated through a range of examples.
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1. Introduction
We are concerned with developing a methodology for the design and manipulation of curved (accelerating) beams [1–10]. The variety of applications include super-resolution imaging [11], light-sheet microscopy [12], optical coherent tomography [13], and particle manipulation [14], to name but a few. Several methods have been proposed for the generation of such field patterns, aiming to control various parameters of the emitted field distribution. Early attempts focused on control of the beam’s on-axis intensity profile. Analytic solutions of Maxwell’s equations, for wave objects that propagate in circular trajectories, in lossless and lossy media, were presented in [14,15]. These wave objects maintain their peak intensity along the beam-axis for large distances, even in an absorbing medium. Control of the on-axis intensity function for two-dimensional (2-D) surface-plasmon beams was demonstrated in [16].
Other works attempted to design the amplitude and phase of source field distributions on very wide apertures that give rise to desired patterns, using geometrical considerations of ray optics. For example, strong intensity profiles can be obtained at focal points of bundles of geometrical optics (GO) rays (forming Bessel-type beams) [17,18] or along smooth caustics of GO rays [19–23]. For these cases, the phase and amplitude of the corresponding aperture field distributions can be derived analytically. The principles, which were first used for cases that are of either simple curvature or high spatial symmetry, were extended in [24] for the design of 3-D caustic beams (CBs) that meander along complex trajectories with arbitrary curvature and torsion. The implementation was based on the “back-tracing” of rays from a predefined beam trajectory to the radiating aperture, to determine the source distribution’s phase. An iterative scheme for windowing and amplitude equalization of the aperture distribution enabled limited control over the field pattern width and along-axis amplitude. However, the strict caustic nature, imposed on the field by the GO-computed phase distribution, prohibited true control of these parameters. Advanced applications, such as in [25–28], require greater controlability over a broader range of field distribution parameters. Specifically, control of the beam’s cross-section intensity profile is of interest and motivates the presented work.
Enhanced field-pattern controlability can be obtained if the restrictions, imposed by the ray back-tracing strategy, are relaxed and the aperture distribution can be more freely optimized. This can be done, for example, by using techniques already explored for antenna array synthesis [29–31]. There, typically, the field intensity (or radiation pattern) is described on an infinite radius sphere, as a function of the angular coordinates $\theta$ and $\varphi$. The optimization often aims at concentrating the energy within a specified angular region (or a main lobe) while lowering the intensity below a certain upper-bound (termed the side-lobe level) outside of that region [29,32]. The main lobe’s width, often defined by the points where the intensity drops to $-3 \textrm {dB}$ of its maximal value, is termed the beam-width (BW). The optimization methods include linear programming techniques for convex optimization [29,32], as well as nature inspired heuristics [30,33], designed to address the non-convexity of the optimization goal functions and constraints, for high-complexity problems involving many array elements. They have successfully produced array pattern designs varying from directional pencil beams [34] to complex shaped patterns [32].
The design of a curved beam, however, poses additional challenges. While antenna arrays are typically composed of hundreds or thousands of sub-wavelength elements, placed at typical spacings of half a wavelength, we are interested in source arrays that mimic continuous aperture distributions. Thus, they are densely sampled, with deep sub-wavelength spacings. The CB design approach in [24] also suggests that electrically very large apertures are required for controlling the patterns away from the aperture. Typical aperture dimensions for caustic beam generation can be of hundreds of wavelengths in each axis, leading to a prohibitive number (millions) of optimization variables at the aperture. Also, unlike antenna design, where near-fields are considered mainly in the contexts of input impedance computation [35] or local field nullification for EM compatibility purposes [36], here, the design specifications are given in a large volume above the aperture, thus calling for the careful definition of the optimization problem and constraints.
In this work, we present an optimization scheme that enables the efficient design of aperture distributions and provides more comprehensive control of the various beam features. The optimization approach relies on the localized nature of the caustic field pattern, but does not restrict the phase distribution to that associated with the back-propagated rays. Instead, it uses the heuristic approach in [24] as the initial step in the optimization and, more importantly, for the reduction of the optimization problem’s variable space. This is carried out using the frame-based phase-space method [37,38], in which the aperture distribution is presented as a superposition of Gaussian windows (GWs) over a discrete spatial-directional (spectral) lattice. In this representation, the field produced by the aperture is decomposed into a discrete set of tilted and shifted Gaussian beams propagators (GBs) that are emanating from the aperture. This representation extracts local radiation properties of the aperture field that are mapped into the beam-domain data (phase-space coefficients), thereby leading to compact spectral representations [39–41]. Therefore, it can efficiently and accurately describe fields that are locally attributed to contributions by few ray paths, such as the caustic beams under consideration.
Using the GO-based aperture distribution in [24], the beam-domain components with the strongest coefficients, which are the most significant for the field reconstruction, are kept, while the majority are discarded. The reduced set of GWs is then used within the selected optimization framework [32]. The representation of the aperture distribution using a limited number of GWs also enables the fast repetitive computation of radiated fields in the volume, as part of the optimization process: an index of radiated fields, corresponding to each of the windows, can be pre-computed once. Then, for each aperture distribution, only multiplications by the pertinent weights and summations are carried. Moreover, the inherently-localized nature of the contributions enables the reduction of the region in which constraints on the radiated field pattern are imposed to the beam axis’ vicinity. There, the Gaussian beam (GB)-approximation of each GW’s contribution is valid and can be computed analytically at a significantly reduced complexity. To avoid additional computational burdens, in the form of solution multiplicity common to global optimization methods, e.g., [30,31] and [32], we adopt a linearization approach [42] that enables the solution by sequential application [43] of conventional convex optimization tools that can be integrated naturally with the proposed phase-space representation.
The remainder of the paper is organized as follows: Section 2. formulates the optimization problem. Section 3. describes the phase-space representation of aperture fields, its usage for reduction of the number of optimization variables, and the approximate GB-representation of radiated fields. Section 4. describes the optimization scheme, used in conjunction with the GB field representation. Section 5. showcases the method’s capabilities via representative examples. Section 6. concludes the work and discusses avenues for further research.
2. Problem formulation
Consider the trajectory ${{\textbf {r}}_{\textrm {b}}(\sigma )}$, defined in a 3-D domain as
Fig. 1. Illustration of the desired field pattern in 3-D. The beam-axis trajectory, ${\textbf {r}}_\text {b}(\sigma )$, is marked red, with the local beam coordinates indicated. The ONA is painted blue, for the case of an ellipse, in the normal–binormal plane, whose radii are the BWs.
Curved beam design, unlike that of an antenna pattern, implies that specifications should be defined in three dimensions. In this work, we define the on-axis area (ONA) for each point along the beam-axis, ${\sigma }$, as a convex region in the plane perpendicular to ${{\textbf {r}}_{\textrm {b}}(\sigma )}$, i.e., in the normal–binormal plane in (4). For an elliptical ONA, the ellipse’s radii can be considered the desired BWs in these local coordinates (see Fig. 1). Accordingly, the off-axis area (OFA) is defined as the infinite area in that plane complementary to the ONA. The design problem can be formulated as an optimization one, in a similar manner to that used for antenna array synthesis: minimize a functional on the aperture field $u_0$: $f(u_0):\mathbb {C}^{2}\xrightarrow {} \mathbb {R}_{+}$, subject to certain constraints (e.g., upper bounds, exact values, etc., in the ONA or OFA) on either $I(u_0)$, $u_0$, or both.
From the optimization problem’s description follow the various challenges involved with its solution: (i) The cost of the optimization is directly related to the problems’s variables space dimensionality, i.e, to the aperture’s electrical length squared, if described using its samples, which can be of millions of unknowns. (ii) Any optimization technique will involve repeated numerical computations of $I(u_0)$ via (2), at a cost dictated by the aperture’s area. (iii) The number of observation points at which $I(u_0)$ should be evaluated for computing $f(u_0)$ and imposing the constraints can be excessive. (iv) Finally, straightforward formulation of the constraints typically gives rise to non-convexity of the optimization problem, with respect to the variables describing $u_0$. Challenges (i)-(iii) can be tackled by using a phase-space representation of the field aperture, as described in Section 3. Challenge (iv) will be addressed in Section 4.
3. Phase-space and Gaussian beam representations for efficient optimization
In this Section, we lay the foundation for addressing challenges (i)-(iii) outlined in Section 2., toward the efficient optimization of the aperture distribution, by adopting a phase-space representation of aperture fields. We begin by formulating the phase-space decomposition of aperture fields (Section 3.1). Then, in Section 3.2, we describe its usage, in conjunction with the method in [24], for reducing the optimization variable space to that of a smaller sub-set of phase-space coefficients, thus addressing challenge (i). Section 3.3 discusses the representation’s advantages for the efficient computation of radiated fields, both by exact computation and by means of approximate GB propagation, thus addressing challenge (ii).
3.1 Phase-space representation of aperture fields
Let us first define the frame representation used for efficiently describing the aperture fields. In this representation, the aperture field is described by beam-domain weights, $a_{{\textbf {n}}}$, of field components associated with points on the spatial-directional four-dimensional frame lattice
Fig. 2. Phase-space representation of aperture fields. Each beam represents the field produced by a spectral component corresponding to a point on the directional lattice, emanating from a grid point on the spatial lattice.
We proceed by choosing a proper synthesis ("mother") window $\psi ({{{{\textbf {r}}}_{\textrm {a}}}})$. The frame representation of the aperture field, $u_0({{{{\textbf {r}}}_{\textrm {a}}}})$, in the $z=0$ plane is given by
3.2 Optimization variable-space dimensionality reduction
The beam-domain data extracts local (directional) radiation properties of the aperture field about the spatial lattice points ${{{{\textbf {r}}}_{\textrm {a}}}}$. For CBs, this field is a-priory localized, since the rays that form the caustic surface emanate from a specific curve in the aperture in space-dependent directions [24]. Denoting $S$ the phase of the aperture distribution, $u_0^{\textrm {bt}}({{{{\textbf {r}}}_{\textrm {a}}}})$, computed via the ray back-tracing technique, the aperture field is localized about the so-called Lagrange manifold:
3.3 Efficient computation of radiated fields
The GW representation of the aperture distribution enables us to significantly reduce the number of optimization variables. Yet, at each stage of the optimization process, it is necessary to compute $u({\textbf {r}})$, via the integral (2) over a distribution $u_0({\textbf {r}})$, at a number of observation point that is dictated by the optimization problem’s formulation. Let us assume first that the optimization constraints are imposed in the entire volume. Accurate numerical representation of $u_0({\textbf {r}})$ requires its sampling at sub-wavelength spacings. Computation of $u({\textbf {r}})$ at a given $z$-plane could be performed, via fast Fourier transform (FFT), as in [24], at a cost of $N_xN_y \textrm {log}(N_xN_y)$, $N_x$ and $N_y$ being the numbers of uniformly spaced samples in the $x$ and $y$ axes. However, this quasi-linear complexity is still excessive, especially if constraints are imposed at many different $z$-planes, when many ($N_\textrm {it}$) optimization iterations are required.
If $N_\textrm {it}$ is of the order of $N$, it becomes beneficial to numerically compute (2) for each uniformly sampled $\varphi _{\textbf {n}}({{{{\textbf {r}}}_{\textrm {a}}}})$, to form a vocabulary of field values that should only be weighted and summed in each iteration. The potential savings in computation time come at the additional cost of storing the $N$ radiated fields. For GWs that are highly localized in the $z=0$ plane and produce highly localized fields in any other $z$-plane, FFT-based field evaluation can be applied to smaller, truncated, source and observer regions, near the centers of the GW and corresponding ONA. This compromise of accuracy away from the ONA implies that one could also use, instead of the exact computation of (2), for each $\varphi _{\textbf {n}}({{{{\textbf {r}}}_{\textrm {a}}}})$, its analytically calculated GB, $B_{{\textbf {n}}}({\textbf {r}})$, which is valid near its axis. The latter choice, which will be used in this work, reduces the cost of the vocabulary computation even further, regardless of the choice of GWs. It also allows for the more natural future extension of the method to cases of non-planar apertures.
Limiting the observation region, for each $z$-plane, aligns also with the need in reducing the number of optimization constraints, for feasibility purposes, by not imposing them in the far OFA, where the fields are, inherently, of low intensity. The optimization technique described in the next Section is designed in accordance with these observations, thus addressing challenge (iii).
4. Optimization technique
In this Section, we describe the proposed optimization method. We first formulate the optimization problem used in this work, in general terms of the minimized functional and imposed constraints. Then, we specify it for the GWs and GBs framework described in Section 3. This enables addressing challenges (i) and (ii), highlighted in Section 2., as well as challenge (iii), for the specific formulation of the optimization problem. Finally, we describe the selected optimization algorithm, addressing challenge (iv).
Ideally, one would like to dictate the intensity $I(\textbf {r}_i)$ in the ONA, to a prescribed tolerance $\varepsilon$, and reduce it to minimum in the OFA. These design specifications can be translated, in a manner similar to that in [47], to the following optimization problem:
We would like to formulate the problem such that fewer field samples can be used, without under-sampling. The proposed algorithm sets constraints on the intensity only in the ONA and part of the OFA near the ONA. To eliminates solutions of high intensity outside of the sampled/constrained regions, the aperture energy is minimized, rather than the side-lobe level. The problem can be written as
The optimization problem in (20) is non-convex in $\textbf {a}$, due to the quadratic form in the ONA constraint. Yet, it can be effectively solved by repeated application of convex optimization tools, with gradual convergence to a satisfactory aperture design. To that end, the problem is reformulated, for each step, in an approximate linearized manner, by defining the vector $\textbf {c}_i = \textbf {a}^{H}\textbf {b}_i^{H}\textbf {b}_i$, where $\textbf {a}$ is the solution from the previous step, as
Algorithm 1. Beam forming sequential optimizations
5. Numerical examples
In this section, the optimization method is showcased via the synthesis of a curved beam. The beam axis is a right-handed helical curve, described by
where $R=P=45\lambda$. The design specifications include the target value for the peak intensity along the beam-axis and the requirement for a uniform intensity within the beam’s ONA. In the OFA, the intensity should be generally low. In all the examples, the constraints are defined in terms of the points at which they are imposed, ${\textbf {r}}_i$ and ${\textbf {r}}_j$, in the ONA and OFA, respectively, and the values assigned to those points. The usage of GWs and corresponding GBs enables, as explained in 3.3, the reduction of the constrained region of the ONA to the vicinity of the beam axis. As the solution is sought in the subspace of dominant GWs in $u_0^{\textrm {bt}}({{{{\textbf {r}}}_{\textrm {a}}}})$, which corresponds to an Airy function profile associated with the caustic, a reasonable choice for the ONA width in the normal direction is of the order of the Airy function’s main lobe’s width. In the binormal direction, the BW can be made smaller than that achieved via the back-tracing technique. With these two widths being the radii of an elliptical ONA, the constrained part of the OFA can be an ellipse of somewhat larger radii. Figure 3 shows the constraint structure for a single cross-section of the beam. The solid line marks the edge of the ONA. A certain transition area exists between this line and the closest to axis OFA constraint points, where no constraints are imposed.Fig. 3. Setting the constraints. A blue ellipse marks the ONA. Constraint points in the ONA and OFA are located on black and red marked dashed contours, respectively
Throughout this section, we quantify the performance of the designed aperture distributions using various measures. These include the standard and the average deviation (AD) of $I_{\textrm {b}}$ and the BW from their target values. As the proposed procedure enables enhanced control of the ONA intensity pattern, it is important to quantify also the difference between the desired and synthesized ONA intensities. For a uniform ONA intensity, this index can be thought of as the beam homogeneity (BH), defined as
In what follows, we demonstrate the proposed method’s capability to improve the design, compared $u_0^{\textrm {bt}}({{{{\textbf {r}}}_{\textrm {a}}}})$ from [24], in terms of the various measures and, specifically, the BH and OEP. This will be shown for the cases of a large bi-normal BW beam and a narrow BW beam with a modulated on-axis intensity profile.
5.1 Large binormal width beam
Our first example demonstrates the proposed design procedure. In it, the helical beam in (22) is designed with a desired binormal BW of $20\lambda$ and a uniform intensity $I_{\textrm {b}}=1$. The procedure begins by computing, via back-tracing, the distribution $u_0^{\textrm {bt}}({{{{\textbf {r}}}_{\textrm {a}}}})$, for which the magnitude is shown in Fig. 4(a). Next, following Section 3.1, its phase-space representation coefficients, $a_{\textbf {n}}^{\textrm {bt}}$, are computed. The variable space is then truncated, according to Section 3.2, before the iterative optimization process of Section 4. is applied.
Fig. 4. (a) Magnitude of the initial aperture distribution designed using ray back-tracing. Corresponding phase-space coefficient magnitude (in dB) for ${{\bar {{\textbf {r}}}_{\textrm {a}}}}/\lambda =(65.27,27.14)$ (b) before and (c) after truncation. The Lagrange manifold is marked by a black cross.
For the aperture field distribution, $u_0^{\textrm {bt}}({{{{\textbf {r}}}_{\textrm {a}}}})$, $b=3z_{\textrm {max} }$ was used, where $z_{\textrm {max}} = 270\lambda$ is the maximal height of interest above the aperture. This choice guarantees the collimation of the radiated GBs up to the distance of interest. The phase-space grid was designed with an over-completeness parameter $\nu =1/3$ and a matching parameter of 1 (see [37]), such that $\Delta _x = \Delta _y = 12.8\lambda$ and $\Delta _{\xi _x} = \Delta _{\xi _y} = 0.02$. This resulted in roughly $11$ million nonzero phase-space components, in the visible spectrum $\sqrt {\bar \xi ^{2}_x+\bar \xi ^{2}_y} < 1$. The truncation of the coefficients set was carried with the threshold conveniently set to $-32$ dB with respect to the largest coefficient, leading to $N \approx 50,000$ GWs, corresponding to beams emanating from 405 spatial phase-space grid points in the aperture. The truncation is visualized in (Fig. 4(b)) and (Fig. 4(c)). These show a cross-section of $a_{\textbf {n}}^{\textrm {bt}}$, i.e., the visible spectrum coefficients’ absolute value, on the phase grid, for a specific spatial point in the frame lattice, before (Fig. 4(b)) and after (Fig. 4(c)) truncation. The strongest coefficients are concentrated around the Lagrange manifold (defined in Section 3.2), marked by a black cross. For comparison, the number of non-zero aperture sampling points using the method in [24] is roughly 5 million.
For the optimization, the beam axis was sampled every $7\lambda$ in the $z$-axis, and the constraints were imposed on constant radii ellipses, sampled uniformly with respect to the angular coordinate, at spacings of $\Delta \varphi =30^{\circ }$, in the ONA, and $\Delta \varphi =18^{\circ }$, in the OFA (see Fig. 3). We used $4$ ellipses per cross-section in the ONA, and $3$ ellipses in the constrained part of the OFA, defined between 1.5 and 1.8 times the BW. These parameters were chosen to reduce the number of constraints. In Algorithm 1, we set $\varepsilon =0.2$ $\delta =0.5I_{\textrm {b}}(\sigma )$, corresponding to the half-power BW definition. We set $\gamma = 0.02$ so that the number of iterations is sufficient for convergence. Figures 5(a) and 5(b) show the field intensity cross-section at $\sigma =161.3\lambda$, for the back-traced and optimized radiated fields, respectively, where the black curve indicates the ONA’s boundary. The computed BH index in (23) were $0.75$ and $0.44$, for the back-traced and optimized beams, respectively, indicating the homogeneity enhancement. The computed OEP in (24) dropped from $10^{-4}$ to $6.2\cdot 10^{-5}$ after the optimization. Both measures indicate the enhanced beam synthesis accuracy.
Fig. 5. Intensity cross-sections for the (a) back-traced and (b) optimized designs. The black ellipses indicate the on-axis area.
5.2 OEP and binormal beam-width reduction
In the next example, using the proposed technique, we address the back-tracing scheme’s inability to produce field patterns of small controllable BWs [24]. The phase-space representation parameters were set as in the previous example. Furthermore, as the desired widths are not necessarily achievable by using the method in [24], the current example also uses the same reduced set of GWs.
Figure 6(a) shows the AD of the beam’s binormal BW from the desired one, as a function of the desired BW, for the back-traced and optimized cases, for the trajectory in (22). It can be seen that, with mere back-tracing, the AD increases with the decrease in the desired BW. However, with the proposed method, it is smaller and remains constant across all studied BW values. Fig. 6(b) and Fig. 6(c), present the $-3\textrm {dB}$ iso-surface plots of the field intensities, for the back-tracing and optimized designs, respectively, for a desired BW of $10\lambda$. The black lines indicate the desired BW. Unlike the back-tracing’s field, the optimized one meets the BW requirements almost perfectly along the entire beam-axis. This is quantified by the BH and OEP, plotted in Figs. 6(d) and Fig. 6(e), respectively, as a function of the desired BWs, for the back-tracing (black) and optimized (red) designs. In both parameters, the optimized distributions outperform the back-tracing’s ones, in the entire range of studied BW values.
Fig. 6. (a) AD of the BW from its desired value, for the back-tracing (black) and optimized (red) designs, as a function of the desired BW. $-3\textrm {dB}$ iso-surface plots for the (b) back-tracing and (c) optimized beams. The black lines mark the desired $-3\textrm {dB}$ width. (d) BH and (e) OEP indices for the back-tracing (black) and optimized (red) designs, as a function of the desired BWs.
Fig. 7. Beam intensity profiles in the cross-section $\sigma =287.46\lambda$, for desired BWs of $10\lambda$ (a,d), $5\lambda$ (b,e), and $2.5\lambda$ (c,f) for back-tracing (a,b,c) and optimized (d,e,f) designs. The black ellipses mark the ONA.
Some insight into these results, can be gained from examining the field intensity cross-section, plotted in Fig. 7 for $\sigma =287.46\lambda$, for the back-tracing’s (top) and optimized (bottom) beams, for several desired BWs: $10\lambda$ (left), $5\lambda$ (middle), and $2.5\lambda$ (right). The caustic trajectory’s shape is inherited by the back-tracing design’s field. This translates into a "spill-over" of the power beyond the bounds of the ONA, which worsens with the decrease in the desired BW. The effect is avoided almost entirely by the optimized design, with very mild spill-over at the smallest BW value.
6. Conclusion
A systematic method for the design and manipulation of curved (accelerating) beams, via an optimization scheme, was presented. By employing a local spectrum representation of the aperture fields, we have significantly reduced the number of optimization variables, while providing sufficient degrees of freedom for the design. This enabled the crucial speed up of the various stages of the computation. The proposed representation is general and can be used in conjunction with various optimization methods. Using it to accelerate the sequential convex optimization method [32] resulted in enhanced controllability over various beam parameters that could not be reached using back-tracing techniques.
The proposed approach can be extended to cases of non-planar apertures and for the control of more features of complicated profile CBs, e.g., [23]. Its further development can include more advanced and unconventional strategies for designing the phase-space lattice and GW parameters. A natural follow-up to this work is its extension to the design of time-varying apertures for the generation of time-depended (pulsed) curved beam fields. This can be carried out in a similar manner, by using the pulsed-beam spectral representation of the fields, relying on the additional pre-optimized localization in the temporal phase-space spectral variable [50] for reduction of the optimization variable space. Other avenues of research include the design of aperture distributions for curved beams in media that contain mild inhomogeneity, as well as extensions to the cases of stochastic and uncertain media.
Funding
Israel Science Foundation (531/19).
Disclosures
The authors declare no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research. The presented simulated results can be reproduced using the reported parameters.
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