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Evolutionary optimization of optical antennas
Thorsten Feichtner, Oleg Selig, Markus Kiunke, Bert Hecht
TL;DR
Optical antennas operate under material and excitation conditions that differ from the radio-frequency designs that have inspired them. This paper uses evolutionary optimization of gold-cube matrix antennas and finds a split-ring/two-wire hybrid with near-field enhancement about twice the reference, combining split-ring and dipole resonances in the visible regime.
Problem
Optical antenna materials and excitation conditions differ from the idealized assumptions behind radio-frequency designs, motivating alternative optical designs.
Source: p. 1Method
The authors use evolutionary optimization to search gold-cube matrix antennas for near-field intensity enhancement under focused Gaussian-beam illumination.
Source: p. 1Results
The optimized hybrid antenna outperforms the reference antennas by a factor of two in near-field intensity enhancement.
Source: p. 1Takeaways & Limitations
The split-ring/two-wire hybrid combines fundamental magnetic split-ring and electric dipole resonances in the visible wavelength regime.
Source: p. 1The fittest structure depends on the available primitive elements and boundary conditions, so changing either can yield a different optimum.
Source: p. 2
Abstract
from arXiv · showhide
The design of nano-antennas is so far mainly inspired by radio-frequency technology. However, material properties and experimental settings need to be reconsidered at optical frequencies, which entails the need for alternative optimal antenna designs. Here a checkerboard-type, initially random array of gold cubes is subjected to evolutionary optimization. To illustrate the power of the approach we demonstrate that by optimizing the near-field intensity enhancement the evolutionary algorithm finds a new antenna geometry, essentially a split-ring/two-wire antenna hybrid which surpasses by far the performance of a conventional gap antenna by shifting the n=1 split-ring resonance into the optical regime.
Evolutionary optimization of plasmonic nanostructures
The supplementary material compares the 21×21 matrix antenna with the 647 nm Gaussian focus and notes that the matrix and focal spot are comparable in size.
Evolutionary optimization of plasmonic nanostructures
- The supplementary material is titled “Evolutionary optimization of plasmonic nanostructures.”Source: Evolutionary optimization of plasmonic… · p. 7
SIZE RELATION OF MATRIX ANTENNA AND FOCAL AREA
- The matrix and gold-cube sizes were chosen so the excitation focus illuminates the whole structure, subjecting every cube to about the same light intensity.The figure compares the 21×21 matrix extension with the field profile of the 647 nm Gaussian focus.Source: SIZE RELATION OF MATRIX ANTENNA AND… · p. 7, Figure 1 · p. 7
- The size-relation figure concerns a Gaussian focus with NA = 1.Source: SIZE RELATION OF MATRIX ANTENNA AND… · p. 7, Figure 1 · p. 7
THE EXCITATION POLARIZATION
With linear excitation polarized along x, the field symmetries make many binary matrix structures physically redundant. The section describes counting structures with special symmetries and the areas needed to represent them.
THE EXCITATION POLARIZATION
- For linear x-polarized excitation, the fields are symmetric about the x-z plane and antisymmetric about the y-z plane.Source: THE EXCITATION POLARIZATION · p. 8
- As n grows, about three quarters of possible structures are physically redundant under the stated symmetry operations, so checking new structures can avoid simulations.Source: THE EXCITATION POLARIZATION · p. 8
- The 5×5-matrix figure concerns the areas needed to describe structures with specified properties, alongside the calculation of structures with special symmetries.Source: Figure 2 · p. 8, THE EXCITATION POLARIZATION · p. 8
Axial symmetry
The symmetry cases reduce how much of a binary matrix must be specified to describe a structure: axial symmetry uses a center-containing row and half-matrix, while point symmetry uses half a row and half-matrix.
Axial symmetry
- For axial symmetry, a center-containing row with n−1 elements and a half-matrix completely define the structure.Source: Axial symmetry · pp. 8–9
- Both symmetry cases describe structures using a half-matrix, with the accompanying row portion depending on the symmetry condition.Source: Axial symmetry · p. 8, Point symmetry · p. 9
Point symmetry
- For point symmetry, half a row with (n−1)/2 elements and a half-matrix describe the structure.Source: Point symmetry · p. 9
Point and axial symmetry
The symmetry sets overlap: matrices symmetric about both axes are also point symmetric, while origin symmetry combined with one axis implies symmetry about the other.
Point and axial symmetry
- Matrices symmetric about both axes are also point symmetric, so the three single-symmetry sets overlap.Source: Point and axial symmetry · p. 9
- Origin symmetry combined with symmetry about one axis also gives symmetry about the other axis.Source: Point and axial symmetry · p. 9
- The shared set is defined by a quarter-matrix including on-axis elements and excluding the center block.Source: Point and axial symmetry · p. 9
Counting
The count removes physical duplicates caused by mirror symmetries, correcting for structures that have both axis and point symmetry. For matrix sizes of five or larger, about 75% of structures are redundant.
Counting
- Axis- and point-symmetric matrices require separate counting adjustments because mirroring can produce physically redundant structures, while matrices with both symmetries are counted completely.The count corrects for overlap between the symmetry classes before determining the number of physically unique matrices.Source: Counting · pp. 9–10
- About 75% of structures are redundant for matrix sizes of five and larger.The fraction of nonredundant square matrices approaches 0.25 as matrix size increases.Source: Counting · p. 10, Figure 3 · p. 10
SPEED AND CONVERGENCE
Simulations stop at 35 fs to reduce computational effort while preserving the relative fitness ranking. The best individuals are then simulated again to full convergence for accurate results.
SPEED AND CONVERGENCE
- 35 fs simulations leave more than 98% of initial power outside the simulation volume and cut simulation time by about half, but do not provide very high absolute accuracy.A full FDTD simulation generally takes 70–80 fs to converge; the best individuals are re-simulated until full convergence.Source: SPEED AND CONVERGENCE · pp. 10–11
Methods of creating new individuals
The algorithm creates descendants through random generation, mutation, or crossing genomes from two parents. These methods are applied with equal probability, and duplicate structures are replaced.
Methods of creating new individuals
- Random descendants independently assign each block a 50% chance of being gold or void, introducing diversity into the genetic material.Source: Methods of creating new individuals · p. 11
- Mutation toggles each block of parent A between gold and void with a 10% chance, allowing a good structure to develop without mixing genomes.Source: Methods of creating new individuals · p. 12
- Linear and spiral crossing combine the first half of one parent genome with the second half of another, with a randomly chosen split point.The method combines the left or inner region of one antenna with the right or outer region of another.Source: Methods of creating new individuals · p. 12
- Each generation method is used with equal probability; crossing selects a second parent from the remaining four fittest individuals, and simulated duplicates are replaced.Source: Methods of creating new individuals · p. 12
Development of higher-fitness individuals by means of evolutionary optimization
The evolutionary search raises near-field enhancement above the reference antenna by generation 42, while structural features emerge alongside fitness changes. Opening a gap produces the first large fitness jump, and mutation can introduce features after temporary declines.
Development of higher-fitness individuals by means of evolutionary optimization
- By generation 42, near-field enhancement surpasses the reference antenna, although the 35 fs fitness estimates are typically 10–20% low.The first generation had 20 random individuals with enhancements from 0.2 to 43.8.Source: Development of higher-fitness… · pp. 12–13, Figure 5 · p. 13
- The probability plots show a 40×20 nm arm developing left of the gap and a more triangular structure emerging on the right.The plots track the probability of gold occupancy at each matrix position among the ten best structures in each generation; the gap remains empty.Source: Development of higher-fitness… · pp. 12–13
- Opening the gap by removing the block directly above the center produces the first large jump in maximum fitness.The gap places charge carriers at the minimum possible distance from the optimization position; solidifying nearby areas supports currents and higher surface charge density at the gap.Source: Development of higher-fitness… · p. 13
- Fitness can fall when offspring lack a new key feature, while increased mutation may initially lower fitness but generate features that lead to better structures.Source: Development of higher-fitness… · pp. 13–14
QUALITY FACTOR
The optimized antenna has a quality factor of about 20. The paper relates higher Q and near-field enhancement to thinner antenna cross sections.
QUALITY FACTOR
- Q ≈20 is reported for the optimal antenna, with reduced radiation resistance from thinner cross sections explaining the increased quality factor.The resonance length also decreases with smaller cross section d.Source: QUALITY FACTOR · p. 14
- Table I compares Q-factor, resonance width ∆λ, and maximum near-field intensity enhancement Emax across antennas with varying cross section at constant resonance wavelength.Source: Table I · p. 14, QUALITY FACTOR · p. 14
- Thinner antennas show higher maximum near-field intensity enhancement through stronger gap confinement and the lightning rod effect.Source: QUALITY FACTOR · p. 14
RADIATION PATTERN
The fittest matrix antenna’s radiation pattern differs only slightly from the reference, while its directivity is higher. The comparison evaluates a dipole emitter at the highest-enhancement position.
RADIATION PATTERN
- The radiation comparison places a dipole emitter at the highest near-field-enhancement position and evaluates its far-field power flow at λres = 650 nm.The structures’ z symmetry means only upper-half-space radiation power is evaluated; the figure uses θ and ϕ coordinates and cylindrical projections.Source: RADIATION PATTERN · pp. 14–15
- The reference antenna has a symmetrical dipolar pattern and directivity Dref = 2.09 dBi, slightly below the ideal thin-wire half-wave dipole value Dλ/2 = 2.15 dBi.Source: RADIATION PATTERN · p. 16
- The fittest antenna has directivity DEA = 2.31 dBi versus Dref = 2.09 dBi, with only a slight pattern deviation and more pronounced radiation in the z-direction.The paper relates the enhanced directivity, through reciprocity, to optimization for Gaussian-beam irradiation.Source: RADIATION PATTERN · p. 16