Variable-Width Beam Generation

• Variable-width beam generation algorithms are adaptive methods that replace a fixed beam width with a dynamic parameter governed by confidence, geometry, or physical constraints.
• They are applied across domains such as autoregressive decoding, wireless beamforming, optical beam shaping, and ion-beam material processing, enhancing performance and coverage.
• The underlying mechanisms include thresholding, gradient-based and genetic optimizations, and motion prediction to meet diverse operational requirements.

Searching arXiv for the cited papers and closely related "variable-width beam" usages across decoding, beamforming, and beam shaping. Variable-width beam generation algorithm denotes a family of procedures in which beam width is treated as an adaptive design variable rather than a fixed constant. In the cited literature, this notion appears in at least four distinct senses: the number of active hypotheses in autoregressive decoding, the spatial width or half-power beamwidth of antenna patterns, the transverse width of optical or nonlinear wavefields, and the effective spot size of a material-removal beam. Across these settings, the adjustable quantity is written as, for example, $w_i^t \le k$, $w_\ell$, beam width $b$, half-power beamwidth, $w(z)$, $W(s)$, or $w(t)=\Phi_1(t)$, and it is controlled by thresholding, curvature, phase-mask synthesis, motion prediction, or hardware-constrained optimization (Yang et al., 2020, Deutschmann et al., 2023, Roth et al., 2017, Zhang et al., 2024, Zhang et al., 15 Aug 2025, Zhalenchuck et al., 2022, Goutsoulas et al., 2018, Duque et al., 2018, Wu et al., 2015).

1. Domain-dependent meanings and common formal structure

A common source of ambiguity is that the word “beam” refers to different mathematical objects in different fields. In autoregressive generation it is a set of partial sequences; in wireless systems it is a radiated pattern synthesized by an array; in optics it is a propagating field whose trajectory, width, or orbital angular momentum may be prescribed; in ion-beam figuring it is a removal kernel whose spot size varies during scanning. The surveyed literature therefore does not describe a single canonical algorithm, but a recurrent design principle: replace a fixed-width parameter by an adaptive quantity tied to confidence, geometry, or physical constraints.

Domain	Variable-width quantity	Adaptation mechanism
Autoregressive decoding	$w_i^t$, $w_\ell$	pruning, refill, conformal thresholds
Wireless beamforming/tracking	beam width $b$, HPBW	weighted optimization, TIAM, GDCSA
Optical/nonlinear beams	$w(z)$, $w_\ell$0, $w_\ell$1	cone angle, curvature, variational parameters
Material processing	$w_\ell$2	variable diaphragm, pattern search

The mathematical pattern is similarly recurrent. A forward model maps control variables to a beam profile or candidate set; an objective then evaluates gain, ripple, sidelobes, coverage, spectral efficiency, sequence coverage, or removal error; and an update rule adjusts width subject to pruning, quantization, or machine-dynamics constraints. This structural similarity is explicit in beam-search pruning (Yang et al., 2020), weighted $w_\ell$3 beam synthesis (Roth et al., 2017), adaptive beam reconstruction for UAV-BSs (Zhang et al., 2024), AWBCD for XL-RIS (Zhang et al., 15 Aug 2025), Rayleigh–Ritz and Newton updates for self-trapped beams (Duque et al., 2018), and pattern-search trajectory optimization for variable-spot ion-beam figuring (Wu et al., 2015).

2. Autoregressive decoding and variable-width search beams

In "A Streaming Approach For Efficient Batched Beam Search" (Yang et al., 2020), variable-width beam generation is a decoding procedure on GPU architectures. The method maintains arrays of encoder-contexts $w_\ell$4, active beams $w_\ell$5, and finalized outputs $w_\ell$6, and introduces a refill mechanism controlled by a threshold fraction $w_\ell$7. When $w_\ell$8, the algorithm appends fresh inputs, initializes new beams with the start-of-sentence token, and then selects for expansion only those beams whose candidates have minimal length $w_\ell$9. Each candidate $b$0 at timestep $b$1 is assigned

$b$2

after which the pruning operator $b$3 applies absolute-threshold pruning,

$b$4

together with max-candidates-per-parent pruning. The resulting beam width is variable,

$b$5

The implementation groups active beams of equal length, concatenates candidate hidden states into a tensor of shape $b$6, and uses GPU-friendly scatter/gather operations. The paper states that Algorithm 1 matches exactly the output of a normally-batched variable-width beam search while refilling the batch to keep the GPU saturated.

The same paper gives a rough complexity comparison. Fixed-width beam search requires $b$7 expansions per input, whereas Var-Batch replaces $b$8 by an average variable beam size $b$9, and Var-Stream further reduces idle warps by maintaining $w(z)$0. On WMT’19 De→En with beam size $w(z)$1 on a 32 GB V100, the reported results are: Fixed BLEU $w(z)$2, time $w(z)$3 s; Var-Batch BLEU $w(z)$4, time $w(z)$5 s; Var-Stream BLEU $w(z)$6, time $w(z)$7 s. The paper summarizes this as 71 % faster than Fixed and 17 % faster than Var-Batch, with no BLEU loss. Additional results report 8–10 % speedups at $w(z)$8, a similar 6 % speedup over Var-Batch on a 16 GB V100 at $w(z)$9, an increase in average expansions per step from $W(s)$0 on ATIS with decode time reduced from $W(s)$1 s to $W(s)$2 s and unchanged F1, and near-matching of Fixed decoding on Penn Treebank shift-reduce parsing with $W(s)$3 expansions per step.

"Conformal Autoregressive Generation: Beam Search with Coverage Guarantees" (Deutschmann et al., 2023) recasts variable width as a set-prediction problem. Given an autoregressive model $W(s)$4, the goal is to output a set $W(s)$5 such that

$W(s)$6

Calibration computes per-step thresholds $W(s)$7 from exchangeable calibration data by sorting non-conformity or confidence scores $W(s)$8, setting

$W(s)$9

and discarding the lowest $w(t)=\Phi_1(t)$0 samples at each step. In inference, every continuation $w(t)=\Phi_1(t)$1 is retained iff $w(t)=\Phi_1(t)$2, so the beam width $w(t)=\Phi_1(t)$3 is determined by uncertainty rather than by a fixed $w(t)=\Phi_1(t)$4. Under exchangeability and continuous score distributions, Proposition 3 gives

$w(t)=\Phi_1(t)$5

The paper notes worst-case growth up to $w(t)=\Phi_1(t)$6 in degenerate models, but reports typical complexity comparable to beam search with an average width $w(t)=\Phi_1(t)$7. Empirically, on integer addition with $w(t)=\Phi_1(t)$8, per-step $w(t)=\Phi_1(t)$9 yields true coverage approximately $w_i^t$0, against guarantees $w_i^t$1, with mean beam size adapting from approximately $w_i^t$2 to approximately $w_i^t$3; on chemical reaction product prediction with $w_i^t$4, per-step $w_i^t$5 gives marginal coverage approximately $w_i^t$6 and $w_i^t$7 versus guarantees approximately $w_i^t$8 and $w_i^t$9, with mean beam widths adapting from approximately $w_\ell$0 to approximately $w_\ell$1.

3. Hybrid beamforming synthesis and adaptive UAV beam tracking

"Arbitrary Beam Synthesis of Different Hybrid Beamforming Systems" (Roth et al., 2017) formulates variable-width beam generation as a constrained optimization problem over the beamforming vector $w_\ell$2. The array-factor magnitude is

$w_\ell$3

with desired pattern $w_\ell$4 and weighting function $w_\ell$5. The objective is a weighted $w_\ell$6 error,

$w_\ell$7

subject to hardware constraints $w_\ell$8, $w_\ell$9, including sub-array and fully-connected hybrid RF structures, per-element amplitude or total-power constraints, and quantized phase-shifter constraints. The paper explicitly states that one often chooses $b$0 such as $b$1 to balance ripple versus main-lobe flatness. It also states that the method is not constrained to a certain antenna array geometry and can handle 1D, 2D, and even 3D geometries like cylindric arrays. The algorithmic recipe is: discretize the spatial domain into a fine grid, formulate $b$2 as a finite sum, initialize $b$3 from several random or heuristic starts, use a gradient-based NLP or, with quantized phases, an MINLP solver, exploit FFT/IFFT on uniform grids, and compare local minima from different initializations. The paper adds that no full closed-form $b$4 exists and describes the method as purely numerical.

The numerical examples use a ULA with $b$5, half-$b$6 spacing, and $b$7, with beam widths $b$8, $b$9, and $w(z)$0, comparing sub-array and fully-connected architectures with and without 2-bit phase quantization. For the objective, the paper specifies $w(z)$1, 512 angle samples, and either sum-power $w(z)$2 or per-element $w(z)$3. Representative metrics include, for fully-connected designs, average gain $w(z)$4, $w(z)$5, and $w(z)$6 dB with max sidelobe levels $w(z)$7, $w(z)$8, and $w(z)$9 dB for $w_\ell$00, respectively, while fully-connected plus 2 bit phase quantization gives average gain $w_\ell$01, $w_\ell$02, and $w_\ell$03 dB with max sidelobes $w_\ell$04, $w_\ell$05, and $w_\ell$06 dB. The design parameters are explicitly summarized in the paper: $w_\ell$07 sets target gain and width, $w_\ell$08 trades passband flatness and sidelobe attenuation, transition-band width sets the roll-off region, $w_\ell$09 sets gain penalty versus feasibility, and larger phase quantization $w_\ell$10 moves the design closer to the unquantized pattern.

"An Accurate Beam-Tracking Algorithm with Adaptive Beam Reconstruction via UAV-BSs for Mobile Users" (Zhang et al., 2024) treats variable width as an online control variable in beam tracking. For a uniform planar array with $w_\ell$11 elements and inter-element spacing $w_\ell$12, the half-power beamwidths in the azimuth and elevation cuts satisfy

$w_\ell$13

The paper then couples beam reconstruction to target motion through a time-interval adjustment mechanism (TIAM). With angular drift modeled as uniformly accelerated motion,

$w_\ell$14

the next rebuild time is set as

$w_\ell$15

so that fast-moving users trigger shorter intervals and slow users incur fewer rebuilds. The full BAB-AR pipeline combines three elements: a Gaussian process regression model trained by position-known assisted UAVs to predict azimuth and elevation angles; a global dynamic crow search algorithm to localize position-unknown assisted UAVs without historical trajectory; and beam steering at the U-UAV side using predicted angles and TIAM interval control. The paper reports angle-relative error of GDCSA-estimated angles $w_\ell$16 rad across CTRV, CTRA, and random motions, stable SNR $w_\ell$17 dB for worst-case motions at $w_\ell$18 transmit power, the data-rate model $w_\ell$19, and an energy-efficiency expression

$w_\ell$20

Its stated adaptation constraint is that TIAM enforces the MU’s angular drift never to exceed half the instantaneous HPBW in either cut, guaranteeing continuous coverage.

4. Near-field variable-width coverage and codebook design for XL-RIS

"Near-Field Variable-Width Beam Coverage and Codebook Design for XL-RIS" (Zhang et al., 15 Aug 2025) moves the variable-width problem to near-field reconfigurable intelligent surfaces. The setting is an indoor downlink with $w_\ell$21 transmissive XL-RIS panels flush-mounted on the YOZ-plane of a room, each RIS being a uniform planar array with typically $w_\ell$22 sub-wavelength elements and spacing $w_\ell$23. Because the aperture is “extremely large,” the paper adopts a spherical-wave near-field model rather than a plane-wave approximation. The design region $w_\ell$24 may be a rectangle, sector, or arbitrary shape and is partitioned into sample points $w_\ell$25. For each codeword, the paper defines spectral efficiency

$w_\ell$26

and frames a bi-objective problem: maximize average spectral efficiency over $w_\ell$27 and minimize outage probability

$w_\ell$28

subject to phase quantization $w_\ell$29.

The proposed solver is adaptive-weight block-coordinate descent. After approximating the continuous region by a weighted sum, the optimization becomes

$w_\ell$30

For element $w_\ell$31, the paper defines $w_\ell$32, $w_\ell$33, $w_\ell$34, and $w_\ell$35, and under the practical condition $w_\ell$36 reduces the subproblem to maximizing

$w_\ell$37

With

$w_\ell$38

the paper gives a unique optimal phase as the nearest quantized value to either $w_\ell$39 or $w_\ell$40, depending on the sign variable $w_\ell$41. The outer loop then updates the weights by increasing $w_\ell$42 whenever $w_\ell$43, which the paper identifies as improving fairness and minimizing outage. The output codewords are assembled into a joint multi-RIS codebook.

The reported performance emphasizes geometric flexibility and robustness. AWBCD converges in approximately 20–30 outer iterations, with monotonically increasing average spectral efficiency and decreasing outage. The paper states that rectangles of varying size, shifts, multiple disjoint rectangles, and even “T”- or “L”-shaped regions are successfully covered. Increasing RIS size from $w_\ell$44 to $w_\ell$45 yields a 2–3 bps/Hz spectral-efficiency gain and halved outage. One-bit quantization suffers approximately 1 bps/Hz loss, whereas $w_\ell$46 bits is described as virtually identical to continuous. Against ring, sector, and fresnel codebooks designed for single RIS, AWBCD and even the simpler BCD achieve higher median spectral efficiency and substantially smaller tail outage, especially at large angles off the RIS normal, while the paper notes that AWBCD incurs a penalty at the beam center in favor of “edge-boosting,” producing more uniform coverage and lower outage (Zhang et al., 15 Aug 2025).

5. Optical beam shaping, accelerating Bessel beams, and self-trapped beams

"A simple algorithm for the design of accelerating Bessel beams with adjustable features along their propagation" (Zhalenchuck et al., 2022) describes variable-width beam generation through radius-dependent holographic design. Starting from the angular-spectrum representation of a scalar field and the standard Bessel-beam solution

$w_\ell$47

the paper allows the local cone angle to vary with radius so that a prescribed trajectory $w_\ell$48, beam radius $w_\ell$49, intensity $w_\ell$50, and local topological charge $w_\ell$51 can all be imposed. The stationary-phase mapping yields, in the simplest paraxial axicon,

$w_\ell$52

while beam width is set through the first zero of the Bessel function,

$w_\ell$53

The local grating period then satisfies

$w_\ell$54

with $w_\ell$55. The input-plane field on the SLM is chosen as

$w_\ell$56

The step-by-step procedure is to sample $w_\ell$57, compute $w_\ell$58, $w_\ell$59, $w_\ell$60, and $w_\ell$61, invert the stationary-phase mapping to obtain ring radii, determine tilt angles $w_\ell$62 and $w_\ell$63, set amplitudes from $w_\ell$64, interpolate these quantities over $w_\ell$65, and encode the final phase–amplitude hologram. The paper further states that the same algorithm extends to the non-paraxial regime by replacing the Fresnel mapping with a non-paraxial analog.

"Independent amplitude and trajectory/beam-width control of nonparaxial beams" (Goutsoulas et al., 2018) gives a local relation between beam width and curvature for caustic beams. If the trajectory is parametrized by arc length as $w_\ell$66, its curvature is

$w_\ell$67

The asymptotic Airy-type analysis of the Rayleigh–Sommerfeld integral yields

$w_\ell$68

from which the local main-lobe width is identified as

$w_\ell$69

This is the paper’s central statement that beam width depends solely on the local curvature of the trajectory. The corresponding phase mask satisfies

$w_\ell$70

and the amplitude mask for a prescribed maximum caustic amplitude $w_\ell$71 is

$w_\ell$72

The paper’s algorithm tabulates the trajectory and its derivatives, computes $w_\ell$73, $w_\ell$74, $w_\ell$75, and $w_\ell$76, integrates to obtain $w_\ell$77, and then constructs $w_\ell$78. Numerical validation covers circular, elliptic, parabolic, and cubic trajectories; constant, Gaussian-shaped bump, and sigmoid or sinusoidal amplitude profiles; caustic paths that coincide with theoretical $w_\ell$79 to within numerical precision; and beam widths that agree with $w_\ell$80 within a few percent.

"Numerical realization of the variational method for generating self-trapped beams" (Duque et al., 2018) addresses controllable beam width in nonlinear media via a numerical Rayleigh–Ritz procedure. The field $w_\ell$81 satisfies a generalized nonlinear Schrödinger equation,

$w_\ell$82

and stationary modes are sought with $w_\ell$83. The trial function depends on real variational parameters $w_\ell$84; in the explicit vortex/azimuthon/multipole ansatz given in the paper, $w_\ell$85 and $w_\ell$86 control the beam widths along $w_\ell$87 and $w_\ell$88. The effective Lagrangian is evaluated numerically on a grid,

$w_\ell$89

and stationary conditions are

$w_\ell$90

The solver is a multidimensional Newton–Raphson iteration,

$w_\ell$91

with gradients and Hessians obtained by centered finite differences. The paper states that each iteration evaluates $w_\ell$92 roughly $w_\ell$93 times, each evaluation costing $w_\ell$94 operations; that tolerance $w_\ell$95–$w_\ell$96 on parameter changes is typical; that ill-conditioned Hessians near bifurcations may require damped Newton or quasi-Newton steps; and that the resulting profiles can be launched in split-step simulations to verify robustness and azimuthal stability. The examples include fundamental solitons, vortices, multipoles, and azimuthons, with widths emerging from the solved values of $w_\ell$97.

6. Variable-spot ion beams, photonic-crystal shaping, and recurrent trade-offs

"Variable-spot ion beam figuring" (Wu et al., 2015) is a beam-generation algorithm in the sense of material removal rather than wave propagation. The method scans at constant velocity $w_\ell$98 while varying the spot width $w_\ell$99 by a variable diaphragm, with fixed transverse width $b$00. The continuous removal model is

$b$01

and the paper introduces the concept of integral etching time,

$b$02

The 2D surface map is reduced to multiple 1D belt-like profiles by rastering, grouping, and superposition. For each belt, the optimization minimizes

$b$03

subject to machine-dynamics constraints

$b$04

where $b$05. The paper uses generalized pattern search over piecewise-trapezoidal diaphragm velocity profiles. Reported simulations give, for 1D optimization, convergence in $b$06 pattern-search iterations under $b$07, $b$08, $b$09, and $b$10, with RMS error below 1 % of the peak target etching time. For a $b$11 surface with random Gaussian roughness, the initial map PV $b$12 nm and RMS $b$13 nm are reduced after one iteration to PV $b$14 nm and RMS $b$15 nm overall, and to PV $b$16 nm and RMS $b$17 nm in the central $b$18 region. The same section explicitly contrasts the method with conventional dwell-time IBF, which varies speed instead of spot size.

"Beam shaping using genetically optimized two-dimensional photonic crystals" (Gagnon et al., 2012) presents a discrete-structure beam-generation algorithm based on multiple scattering and a genetic algorithm. The incident field is a complex-source beam

$b$19

expanded around each cylinder as

$b$20

with closed-form coefficients

$b$21

The scattered field is computed by the multiple-scattering method, and the photonic-crystal design is encoded as a bit string indicating which candidate cylinders are present. The fitness compares irradiance profiles at a target plane,

$b$22

with higher fitness $b$23. The paper specifies a population size $b$24, roulette-wheel selection, uniform crossover with probability $b$25, mutation with probability $b$26 per bit, elitism, and termination after either a fixed maximum generation of approximately 5000 or stagnation of best fitness. For a square lattice of holes with diameter $b$27 in index $b$28, incident TM-polarized CSB with $b$29 and $b$30, and target Hermite–Gaussian orders $b$31 at $b$32, the paper reports $b$33 and insertion efficiency $b$34 for $b$35, and $b$36 with $b$37 for $b$38. Tolerance tests varying $b$39 by $b$40 or $b$41 by $b$42 retain $b$43.

Taken together, these works show that “variable-width beam generation algorithm” is best understood as a cross-domain design pattern rather than a single method. The width variable may encode uncertainty, passband support, curvature, codeword-region coverage, or spot size; the update may be a pruning rule, a block-coordinate phase update, a Newton step, a genetic mutation, or a direct-search move. A plausible implication is that comparisons across subfields should focus less on the word “beam” and more on the role played by the adaptive width parameter in the forward model, objective, and constraints.