inverse design of plasma metamaterial devices for optical
TRANSCRIPT
PHYSICAL REVIEW APPLIED 16, 014023 (2021)
Inverse Design of Plasma Metamaterial Devices for Optical Computing
Jesse A. Rodríguez,1,* Ahmed I. Abdalla,2 Benjamin Wang,1 Beicheng Lou,3 Shanhui Fan,3 andMark A. Cappelli1
1Department of Mechanical Engineering, Stanford University, Stanford, California 94305, USA
2Department of Physics, Stanford University, Stanford, California 94305, USA
3Department of Electrical Engineering, Stanford University, Stanford, California 94305, USA
(Received 14 February 2021; revised 25 April 2021; accepted 10 June 2021; published 9 July 2021)
We apply inverse-design methods to produce two-dimensional plasma metamaterial (PMM) devices.Backpropagated finite-difference frequency-domain (FDFD) simulations are used to design waveguidesand demultiplexers operating under both TE and TM modes. Demultiplexing and waveguiding aredemonstrated for devices composed of plasma elements with reasonable plasma densities of approxi-mately 7 GHz, allowing for future in situ training and experimental realization of these designs. We alsoexplore the possible applicability of PMMs to nonlinear boolean operations for use in optical computing.Functionally complete logical connectives (OR and AND) are achieved in the TM mode.
DOI: 10.1103/PhysRevApplied.16.014023
I. INTRODUCTION
Inverse-design methodology is an algorithmic techniqueby which optimal solutions are iteratively approximatedfrom an a priori set of performance metrics. When appliedto electromagnetically active systems [1–18], devices arebuilt by designating a design region wherein variousnumerical methods are used to solve Maxwell’s equationsfor a given distribution of sources and material domain.The constitutive properties (e.g., relative permittivity orpermeability) of the design region are then mapped totrainable parameters, which are modified to mold the prop-agation of source fields through the designated region in amanner that fulfills the user’s performance criteria. Typi-cally, the performance criteria is encoded as an objectivefunction that needs to be maximized or minimized. Inmany problems, the objective functions involve an innerproduct between the realized and the desired electromag-netic fields in regions of interest. Given Maxwell’s equa-tions as a constraint, automatic differentiation allows thecalculation of exact numerical gradients relating the cho-sen objective function to the parameters that encode thepermittivity structure of the training region. The algorithmthen iteratively adjusts these parameters to maximize theoverlap between the realized and desired modes of propa-gation [1–3]. The result is a device geometry specialized tothe user’s performance criteria.
Unfortunately, the set of physically realizable devicesfor inverse-design methods is a small subset of the config-uration space, leading to intensive investigation of devices,
which are amenable to these constraints such as photoniccrystal-style devices [1,4–7] and metasurfaces [8–11]. Ingeneral, these limitations require that certain restrictions beapplied during the training process. For example, utilizinga continuous range of permittivities throughout the simu-lation domain greatly improves the ability of the algorithmto extremize the objective function, but continuity is noteasily realizable with conventional manufacturing tech-niques and materials. To resolve this issue, a map from theparameterized design region to the set of actually manufac-turable device configurations is used. These maps encoderestrictions on the range and spatial distribution of thedomain permittivity. A common technique in the literatureis the use of nonlinear projection to design less versa-tile, but more easily manufacturable, binarized photonicdevices [12]. Such a device can be printed or created vialithography for uses in one-shot computing, but would beotherwise limited in general computing applications.
Rather than restricting the range of permittivities, wefocus instead on limiting the spatial configuration byparameterizing the design region as a plasma metama-terial (PMM). Our PMM is a periodic array of cylin-drical gaseous plasma elements (rods), which can beexperimentally realized via the use of discharge lampsor laser-generated plasmas. PMMs are ideal candidatesfor inversely designed devices since the plasma density,and hence the permittivity, of each element within thePMM can be dynamically and precisely tuned through acontinuous range of values [ε ∈ (−∞, 1) for a collison-less plasma]. This yields an infinite configuration spacefor training purposes [19] and allows a single device toserve multiple functions. Past work with inverse-design
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JESSE A. RODRÍGUEZ et al. PHYS. REV. APPLIED 16, 014023 (2021)
electromagnetic devices achieved reconfigurability viamethods such as refractive index changes [8] in mate-rials like liquid crystals [10]. In our case, the relativepermittivities of elements within the PMM device can bedynamically tuned by varying the gas discharge current. Inaddition, when the PMM is composed of elements that aresmall compared to the operating wavelength and the sourceis polarized properly, we can access localized surfaceplasmon (LSP) modes along the boundary between a pos-itive permittivity background and a negative permittivityplasma element. Such surface modes can yield more com-plex and efficient power transfer [20,21], allowing for aparticularly high degree of reconfigurability. Prior work byauthors Wang and Cappelli with plasma photonic crystals(PPCs) has already highlighted the richness of this geom-etry in waveguide [22] and band-gap [20,23,24] devices.The varied physics of PMMs is well matched to inverse-design methods, which promise to allow more holistic andefficient exploration of the configuration space.
The design processes for binarized and PMM geome-tries are contrasted in Fig. 1. Binarized geometries[Fig. 1(a)] are challenging to optimize because of the req-uisite discontinuities. The jumps in permittivity from onematerial to another are difficult to handle in gradient-basedmethods that require the permittivity to vary continuously.Binarization can instead be approximated by any num-ber of techniques, including Gaussian blurring and highlynonlinear projections [12]. In these configurations, thetrainable parameters ρ (of dimension nxny) maintain aone-to-one correspondence with the individual pixels ofthe training region. In contrast, the number of trainingparameters in our PMM [Fig. 1(b)] is determined by thedimensions of the plasma rod array, where there is a one-to-one correspondence between each plasma element andthe nxny elements of ρ [Fig. 1(c)]. In the end, both param-eterization schemes result in a relative permittivity matrixεr (of size Nx × Ny) containing permittivity values at eachpixel in the simulation domain.
II. METHODS
Figure 2 provides a diagram of our PMM device in adirectional waveguide configuration. Subsequent config-urations exhibit the exact same array of tunable plasmaelements but with different entrance and exit waveguides.The device is simply a 10 × 10 array of tunable plasmarods suspended in air and spaced according to the limits ofexisting experimental facilites [22,23]. Modal sources areintroduced at the input waveguide(s) and allowed to scat-ter through the training region before again collecting atthe desired output waveguide(s). In this work, the fieldsare propagated through the domain via finite-differencefrequency-domain (FDFD) simulations computed withCeviche, an autograd-compliant electromagnetic simula-tion tool [2] that allows for calculation of the gradients of
(b)
(a)
(c)
ρ
ρ εr
εr E
EBlurNonlinearprojection
equations(FDFD)
ApplyPlasma
DispersionEquations
(FDFD)
Map toPlasma
Elements
J
J
Mask
Overlapintergral(s)
MaskOverlap
Intergral(s) ℒ
ℒ
Parameterization
Parameterization Physics
Physics
Objective calculation
Objective calculation
[Nx,Ny]
[nxny,1]
[nxny,1]
[Nx,Ny] [Nx,Ny]
[Nx,Ny]
Nx Nx
Nynxny
Scalar
Scalar
Penalty
→
→
FIG. 1. Flow chart describing the algorithmic design of either(a) binarized material with penalized material use [25] or (b) anarray of plasma columns either in the TM mode (Ez) or the TEmode (Ex). Though the parameterization stages are different, bothconfigurations can be given identical sources and objective cri-teria. J represents the modal source for the FDFD simulation.Examples of the training parameter vector ρ, permittivity matrixεr, and simulated E fields for a simple PMM device are providedin (c).
optimization objectives with respect to the input param-eters that encode the permittivity domain. Note that theuse of FDFD implies that all computed devices representthe steady-state solution achieved after some characteristictime. The simulation domain is discretized using a reso-lution of 50 pixels per lattice constant a and in each casepresented Nx = Ny = 1000 pixels. The only exceptions arethe logic gate devices, which are optimized at a resolutionof 30 pixels/a due to the high cost of the simulation. Inthose cases, the final device configurations are tested at thehigher resolution of 50 pixels/a and yield results that arequalitatively identical. A perfectly matched boundary layer(PML) 2a in width is applied along the domain boundaries.The polarization of the input source has a strong effect indevices of this nature, either Ez (E out of the page), whichwe call the TM case, or Ex (H out of the page), which wecall the TE case. Previous work with PMMs indicates thatboth the TM and TE responses are tunable, with the lat-ter benefiting from the presence of LSP modes [20,26],while the former makes more direct use of dispersive andrefractive effects [21–23]. The simulated Ez and Ex fieldsare masked to compute the field intensity and mode over-lap integrals along planes of interest within the problemgeometry. These integrals are used to calculate the objec-tive L(ρ). Our simulation tool, Ceviche, is then used tocompute numerical gradients of the objective with respect
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2r a
PML BC
Source
Probesra = 0.433
Metal waveguide
εrod
εbg = 1
ω = 1
FIG. 2. Schematic diagram for the PMM waveguide deviceconsisting of a 10 × 10 array of gaseous plasma rods suspendedin air with metal waveguides functioning as the entrance andexit(s). The rod radius is chosen to be commensurate with cur-rent experimental facilities. The “probe” slices represent wherethe mode overlap and intensity integrals that constitute the objec-tive function are calculated. εbg is the background permittivity. ω
is the nondimensionalized operating frequency ω/(2πc/a).
to the training parameters via forward-mode differentia-tion [2]. Ceviche uses the Adam optimization algorithm[27] (gradient ascent, in essence) to iteratively adjust ρ
and thereby maximize L. Optimization is conducted withlearning rates ranging from 0.001–0.005, and the defaultAdam hyperparameters β1 = 0.9, and β2 = 0.999.
In summary, our optimization problem is
maxρ
L(E)
given ∇ × 1μ0
∇ × E − ω2ε(ω, ρ)E = −iωJ ,
where L is the objective composed of a set of mode overlapintegrals (L2 inner product of simulated field with desiredpropagation mode), ρ is an nxny-dimensional vector thatcontains the permittivity values of each of the PMM ele-ments, μ0 is the vacuum permeability, E is the electricfield, J is a current density used to define a fundamentalmodal source at the input waveguide, ω is the field fre-quency, and ε(ω, ρ) is the spatially dependent permittivitythat is encoded by ρ. In practice, this permittivity distri-bution is achieved by varying the plasma density throughcontrol of the discharge current in each of the PMM ele-ments according to the Drude model, which is where thedependence on ω arises. The nondimensionalized plasmafrequencies in ρ are mapped to element permittivities viathe Drude model with no collisionality (loss):
ε = 1 − ω2p
ω2 ,
where ω2p = nee2/ε0me is the plasma frequency squared,
ne is the electron density, e is the electron charge, me is theelectron mass, and εo is the free-space permittivity.
Of course, real gas discharge plasmas will be affectedby some degree of collisional damping (represented bythe loss parameter γ in the Drude model). Such damp-ing can be brought to very low levels in practice byreducing the discharge pressure, but this can come at theexpense of plasma density (and correspondingly, plasmafrequency), which affects the plasma dielectric constant.Furthermore, the neglect of collisional damping can leadto inaccurate results, particularly in the TE polarizationwhere LSP modes can occur. Experimental noise and errorin discharge current can also have an effect on device func-tionality; see the Supplemental Material for a sensitivitystudy that is conducted on the optimal devices presented inthe following sections [28]. We choose to ignore collisionsand other nonideal factors in this particular study for sev-eral reasons. First, experimental results in our prior studiesof PPCs where we operate primarily with TE-polarizedincident fields have agreed reasonably well with the col-lisionless Drude model [22,23]. In a study focused on aPPC band-gap device that utilized the LSP resonance, weshowed experimentally that the resonance is present andstrong for a relatively high collisionality where γ /ωp =0.12 [26]. Second, the optimization algorithm used here isless stable and more time consuming when the permittivityin the domain is of a complex data type, testing the limits ofour available computational resources. Finally, this studyis focused primarily on demonstrating that inverse designis readily applied to PMMs and its application representsa fruitful line of research. In future work, collisional-ity along with other nonidealities such as experimentalnoise and plasma nonuniformity within the dischargeswill be accounted for as we move closer to experimentalrealization.
III. RESULTS AND DISCUSSION
A. Directional waveguide
Electromagnetic waveguides are fairly common ininverse-design devices either as the primary function or asbuilding blocks to more complicated devices [5,18]. Pre-sented first in Fig. 3 are field simulations and permittivitiesof straight waveguide devices designed for either the TMor TE polarization. Operating at ω = 1 × (2πc/a), or anondimensionalized source frequency ω = ω/(2πc/a) =1, these represent the simplest type of optimization prob-lem for this configuration. In both polarization cases, thedomain is initialized with r/a = 0.433, background per-mittivity εbg = 1, and εrod = 0.75 for every plasma ele-ment in the array. The plasma frequency of each of the rodsis unconstrained during training (ωp ∈ [0, ∞]). The rodpermittivities are initialized at 0.75 to allow the source toreach the exit waveguides in the first training epoch, which
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TEEw
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Rel
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→
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FIG. 3. (a) TE FDFD simulation showing (a.i) the field magni-tude |Hz|2 (z out of the page) for an optimized straight waveguideand (a.ii) the relative permittivity domain that gives rise to thisbehavior, where the maximum plasma frequency among theplasma elements is ωp = 1.17. (b) FDFD simulation for the TMcase showing (b.i) |Ez|2 and (b.ii) the relative permittivity with amaximum plasma frequency of ωp = 1.68.
consistently led to higher field intensities at the desired exitafter training. In this and all other simulations, the permit-tivity of the input and output waveguide walls is set toε = −1000 to serve as a lossless metal. The objective issimply the L2 inner product of the simulated field with anm = 1 propagation mode at the location of the probe in thedesired exit waveguide,
Lwvg =∫
E · E∗m=1dldesired exit.
No penalty is needed to achieve a decent standard ofobjective functionality. We can see from Fig. 3 that boththe TE and TM modes yield good performance, with aslightly better result in the TM-mode straight waveguide.The optimized distribution of plasma densities (hence rel-ative permittivities) in the plasma rods for the TM case isnot surprising, representing a rectangular bridge of lowerrefractive index than its surroundings. However, the TEcase settled on a distribution that was unintuitive.
Next, we define an objective intended to produce awaveguide with a 90◦ bend. The algorithm is initializedalmost identically to the straight waveguide case, the onlydifference being that the objective overlap integral is com-puted at the bottom exit waveguide. No penalty is imposedat the incorrect exit waveguide as it is deemed unnecessaryfor satisfactory performance. The results for this case arefound in Fig. 4. Again we see that, while both cases resultin good device performance, the TM case again slightlyoutperforms the TE case, which had some field leakageinto the undesired output waveguide.
The devices presented in Figs. 3 and 4 appear to makeexpert use of the underlying electromagnetics. Inverse
TEE w
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FIG. 4. (a) TE FDFD simulation showing (a.i) the field mag-nitude |Hz|2 (z out of the page) for an optimized bent waveguideand (a.ii) the relative permittivity domain that gives rise to thisbehavior, where the maximum plasma frequency among theplasma elements is ωp = 1.27. (b) FDFD simulation for the TMcase showing (b.i) |Ez|2 and (b.ii) the relative permittivity with amaximum plasma frequency of ωp = 1.97.
design allows a nuanced search of the configuration space,yielding more effective designs that preserve the inputmode. Again, we emphasize that the final training resultsare not intuitive configurations. A human-reasoned designfor this device may more closely resemble that of Wanget al. in Ref. [22], where the elements along the desiredpath of propagation are made equivalent to the back-ground permittivity (i.e., the plasma is turned off) andthe remainder of the rods are activated at the lowest per-mittivity (highest plasma frequency) possible to approxi-mate a metallic waveguide. Instead, the algorithm settledon a configuration where a high contrast in permittiv-ity of neighboring plasma columns is only present nearthe edges of the device to prevent leaking. The remain-ing elements are given a seemingly random distributionof plasma frequencies, keeping in line with typical resultsof inverse-design schemes where strange structures oftenarise [12,15]. Figure 5 graphs the evolution of the objec-tive while training for these waveguide devices. The curvessuggest that a local maximum is achieved in each opti-mization scheme. The spurious changes in the TE objec-tives are likely due to the excitation of LSPs. Becausewe use a collisionless plasma model, once the permittiv-ity of the plasma elements is pushed to negative valuesby the parameter evolution, small changes in plasma fre-quency can excite localized surface plasmon resonancesthat strongly affect the overall wave propagation.
Overall, the results depicted in Figs. 3 and 4 are encour-aging. Unlike binarized photonic devices, which are static,the PMM geometry allows a single device to achieve bothpropagation modes because the permittivity of the deviceelements can be tuned electronically. Quickly switchingbetween the straight and bent waveguide behavior, which
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TE
TM
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FIG. 5. Evolution of the objective functions for (a) the TEwaveguides in the (a.i) straight waveguide case and (a.ii) thebent waveguide case. The same is presented for (b) the TMwaveguides. The y values of these plots are arbitrary as they arenormalized by the initial (epoch 0) value.
is easily enabled by the controllable plasma current, opensup the possibility of transistorlike optical switches.
B. Demultiplexer
Next, we present a frequency demultiplexer designed todistinguish between frequencies of ω1 = 1 and ω2 = 1.1.Inverse-design methods have been used before to optimizewavelength demultiplexers in optical photonic structures[9,12]. The optimization problem here is fundamentallydifferent from the waveguiding case for three reasons:(1) a single device must now adhere to an objective basedon two simulations, one for each operating frequency, (2)the permittivity distribution is slightly different for thetwo frequencies because of the Drude dispersion relation,and (3) a penalty is required to minimize leakage into thewrong output waveguide. The third difference is especiallysubtle since we want to discourage any leakage, not justa spurious m = 1 mode. This means we cannot use thesame mode integral as before. Instead we simply penalizethe simulated field intensity at the incorrect exits:
∫ |E|2dl.With all this considered, the demultiplexer objective, Ldmp,is
Ldmp =(∫
Eω1 · E∗m=1dlω1 exit
)
×(∫
Eω1.1 · E∗m=1dlω1.1 exit
)
−(∫
|Eω1|2dlω1.1 exit
) (∫|Eω1.1|2dlω1 exit
),
where Eω1 is the simulated field for the ω = 1 source andEω1.1 is the simulated field for the ω = 1.1 source.
ω = 1 ω = 1.1
(a)
(b.ii)(b.i) (b.iii)
(a.ii)(a.i) (a.iii)
ω = 1
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1
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TEEw
TMBw
→
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FIG. 6. (a) TE FDFD simulation showing (a.i) the field mag-nitude |Hz|2 (z out of the page) for ω1 = 1 and (a.ii) ω1 = 1.1,along with (a.iii) the relative permittivity of the plasma rodswhen ω1 = 1 where the maximum plasma frequency among theplasma elements is ωp = 1.17. (b) FDFD simulations and ω1 =1 permittivities for the TM case where the maximum plasmafrequency is ωp = 1.95.
After the 1250 epochs of training, the TE and TMdevices presented in Fig. 6 are obtained. Though thesedevices represent a jump in complexity, the objective func-tionalities are still accomplished quite well. The largestdiscrepancy is the minor leakage into the incorrect exitwaveguide in the TE case operating at ω1 = 1.1. It isinteresting to note that, despite the increased frequencysensitivity of the device in the TE mode when its con-stituent plasma elements have ωp > ωsource and thereforeεrod < 0, the algorithm did not push rod permittivities deepinto negative values. Perhaps bringing the source frequen-cies closer together might allow more effective usage ofLSP excitations to differentiate the two frequencies. Figure7 shows the evolution of the objective for the two devices.It appears as before that a suitable local maximum isobtained in both cases. Once again, the final distributionof plasma frequency among the PMM elements for thedemultiplexers provides little suggestion of their intendedfunction.
(a) (b)TE TM
Training epoch0 250 500 750 0 250 500 750
0
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Training epoch1000 12501000 1250
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FIG. 7. Evolution of the objective functions for (a) the TEdemultiplexer and (b) the TM demultiplexer. Once again, they axes on these plots are effectively arbitrary since they arenormalized by the initial (epoch 0) value.
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C. Low plasma frequency designs
Now, while these cases do exhibit compelling results,there is much to be said about their experimental realiz-ability. Using the dimensions of plasma discharge tubeslike those in Refs. [22,23], the dimensionalized lattice fre-quency is estimated to be approximately 20 GHz. Thusthese devices call for plasma frequencies as high asapproximately 39 GHz, or, equivalently, plasma densitiesof approximately 2 × 1019 m−3. These conditions are onlypossible through pulsed operation, which introduces com-plex transient behavior. In an effort to examine how thesedevices may be realized with currently existing plasmasources, we consider the case where the lowest frequencysource allowed by the waveguides (width 2a yieldingω = 0.25) is used and the plasma frequency is limited to7 GHz (an attainable quasi-steady-state operation condi-tion). These limits are enforced via a reparameterizationof the optimization algorithm where an arctan barrier isemployed. The results for these waveguide cases along
(a.iv)(a.iii)
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(a.iii)
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TEEw
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(b)
TMBw
→
→
FIG. 8. (a) TE FDFD simulation showing (a.i) the field magni-tude |Hz|2 (z out of the page) for an optimized straight waveguideand (a.ii) the relative permittivity domain that gives rise to thisbehavior, along with (a.iii) the field magnitude for the bentwaveguide and (a.iv) its permittivity domain. (b) FDFD simu-lations and domains for the TM polarization. Among all of thesecases, the highest plasma frequency is ωp = 0.338, which corre-sponds to approximately 6.75 GHz within existing experimentalfacilities.
Training Epoch Training Epoch
Training Epoch
TE
TM
(a)
(b)
(a.i) (a.ii)
(b.ii)(b.i)
Straight Bent
Training epoch
Training epoch
Training epoch
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0 200 400 600
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FIG. 9. Evolution of the objective functions for (a) the low-frequency TE waveguides in the (a.i) straight waveguide caseand (a.ii) the bent waveguide case. The same is presented for (b)the TM waveguides. The y axes on these plots are again arbitrary.
with the evolution of the objective function are presentedin Figs. 8 and 9, respectively.
We observe that for both objective functionalities, theTM-polarization case exhibits very strong performancewhile the TE case struggles to match its higher-frequencyalternatives. The reason for this is likely enhanced cou-pling to LSP modes. Since the plasma rods are now muchsmaller in radius than the wavelength of the source, LSPsare much more likely to appear and have a strong effecton the overall wave propagation. This is evident in theTE objective training curves, which are far more erraticthan their TM counterparts. Since the algorithm adjuststhe permittivities of many rods at each step after calcu-lating the gradients with respect to each parameter, thepresence of LSP excitation can cause unexpected spuriouslosses that cause the objective to abruptly drop. In addition,the Adam optimization algorithm includes “momentum”as part of its gradient ascent routine [27], so the objec-tive is not guaranteed to increase monotonically. We seethe same performance discrepancies in the demultiplexer,shown in Fig. 10, and the demultiplexer objective, plot-ted in Fig. 11. Despite the losses in the TE-polarizationcases, these lower-frequency simulations suggest that areconfigurable device operating in the TM mode couldbe constructed with experimental discharge tubes such asthose described in our prior studies [22,23]. More workneeds to be done to identify objective functionalities thatcan reap the benefits of coupling into TE LSPs, per-haps with devices, which require an extremely sensitivefrequency response.
D. Optical logic gates
The success of these preliminary devices encouragesexperimentation with more complicated objectives and
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ω = 0.25 ω = 0.27(b.ii)(b.i) (b.iii)
(a.ii)(a.i) (a.iii)
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FIG. 10. (a) TE FDFD simulation showing (a.i) the field mag-nitude |Hz|2 (z out of the page) for ω = 0.25 and (a.ii) ω =0.27, along with (a.iii) the relative permittivity of the plasmarods when ω = 0.25 where the maximum plasma frequencyamong the plasma elements is ωp = 0.288, which corresponds toapproximately 5.75 GHz within our existing experimental facil-ities. (b) FDFD simulations and ω = 0.25 permittivities for theTM case where the maximum plasma frequency is ωp = 0.358or approximately 7.13 GHz experimentally.
device geometries. Past work has established a connectionbetween the wave dynamics inherent to electromagneticphenomena and the complicated computations performedby recurrent neural networks (RNNs) [29]. This type ofanalog computing appears to be the natural applicationof these PMM devices. The reconfigurability of gaseousplasmas means that the complicated classification tasks,which are the staple of RNNs can be readily trained andimplemented in situ. As a natural departure from tradi-tional von Neumann architectures, these optical computingplatforms might yield significant advantages in bandwidthand throughput in an age where electronic computingappears to be reaching physical limits. Here we explore theapplication of inverse design to the fabrication of devicescrucial to the development of more general optical comput-ing platforms. Specifically, we consider the basic buildingblock of traditional computing platforms: the electroniclogic circuit. As physical realizations of boolean algebra,logic gates serve as the backbone for the core arithmetic,
Training Epoch Training Epoch
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Training epoch0 100 200 300 0 100 200 300
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40
–25
0
25
–50Obj
ectiv
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Training epoch400 500400 500
Obj
ectiv
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(b)
FIG. 11. Evolution of the objective functions in the low-frequency case for (a) the TE demultiplexer and (b) the TMdemultiplexer. Once again, the y axes on these plots are arbitrary.
Training Epoch
Q
P
Q NAND P
Q AND P
Const. src.
Inpu
t
Training Epoch
Q = 0, P = 0 Q = 0, P = 1 Q = 1, P = 0 Q = 1, P = 1(a)
(b) (c)
Training EpochTraining epoch0 250 500 750
10
20
30
0
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ectiv
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1000
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ield
mag
nitu
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0.6
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FIG. 12. (a) E-field magnitudes for the OR gate in each of theinput cases. (b) The evolution of the objective throughout train-ing. (c) The relative permittivity domain along with labels for thedevice input and exit waveguides where the maximum plasmafrequency is ωp = 1.738.
storage, and processing operations used in modern com-puting architectures. Optical logic gates could serve asthe basic components of optical computing platforms [30],and have been realized using photonic crystal systems[31–33] as well as other techniques like semiconduc-tor optical amplifiers [34]. We propose the applicationof inverse-design PMM devices as an avenue for thefabrication of reconfigurable optical logic devices.
These devices represent the most complicated objectivesconsidered thus far, particularly since the boolean arith-metic is an inherently nonlinear algebra. Because of this,we remove the plasma-frequency constraints imposed forthe simpler devices and use a TM-polarized source fre-quency of ω = 2 to allow for more complex propagationstructures. As with the demultiplexers, the same devicemust be optimized to handle multiple cases depending onthe source. Figures 12 and 13 illustrate our platform for therealization of boolean logic through plasma arrays. Logi-cal 1s and 0s are captured by the presence or absence of amodal source at each of the two input data waveguides. Toaccount for situations where a logical 1 is required whileboth databits are off (as is the case for a NOR or NANDsignal), a constant field source is provided via the bot-tom waveguide. This in turn necessitates the addition ofa “ground” sink in the case when a logical 0 is required.The inclusion of this sink has the added benefit of dissuad-ing reflection back into the input waveguides, which wouldhinder any efforts to link connectives and perform morecomplicated computations.
While the previous cases were run entirely by thealgorithm, the complex nature of this problem requires theaddition of many hyperparameters in the form of a set ofvariable weights (w1–w8 and q1–q8) that could be manuallytuned after each optimization run. With additional comput-ing power, one could design another shell of optimization
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LAND = w1
∫EQ0,P0 · E∗
m=1dlNAND − w2
∫|EQ0,P0|2dlAND + w3
∫EQ1,P0 · E∗
m=1dlNAND
− w4
∫|EQ1,P0|2dlAND + w5
∫EQ0,P1 · E∗
m=1dlNAND − w6
∫|EQ0,P1|2dlAND
+ w7
∫EQ1,P1 · E∗
m=1dlAND − w8
∫|EQ1,P1|2dlNAND,
LOR = q1
∫EQ0,P0 · E∗
m=1dlNOR − q2
∫|EQ0,P0|2dlOR + q3
∫EQ1,P0 · E∗
m=1dlOR
− q4
∫|EQ1,P0|2dlNOR + q5
∫EQ0,P1 · E∗
m=1dlOR − q6
∫|EQ0,P1|2dlNOR
+ q7
∫EQ1,P1 · E∗
m=1dlOR − q8
∫|EQ1,P1|2dlNOR,
where the weights are (5, 1, 5, 10, 5, 10, 10.9, 8.5) for theAND case and (4, 3, 1, 1.5, 1, 1.5, 1, 1) for the OR case. Theresults are found to be highly dependent on the choice ofthese weighting constants, suggesting that some form ofautomatic method for obtaining them might lead to bet-ter performance. In our case we manually tune the weightsfollowing each run of optimization with Ceviche, where,for example, w8 in LAND would be increased following arun where the algorithm converged on a design with toomuch field intensity in the NAND exit for the Q = 1, P = 1case. When using machine learning algorithms, it is com-mon that hyperparameters such as the weights introducedhere can have a major impact on the convergence of thealgorithm to an effective design, and many methods existfor tuning hyperparameters toward that end [35]. In ourcase, we did not have the computational resources to do so.Experimenting with different configurations of the objec-tive determined that the simple linear combination of the
Q
P
Q NAND P
Q AND P
Const. src.
Inpu
t
Training Epoch
Q = 0, P = 0 Q = 0, P = 1 Q = 1, P = 0 Q = 1, P = 1(a)
(b) (c)
Training EpochTraining epoch0 250 500 750
20
40
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0
Obj
ectiv
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1000
80
E-f
ield
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FIG. 13. (a) E-field magnitudes for the AND gate in each ofthe input cases. (b) The evolution of the objective throughouttraining. (c) The relative permittivity domain along with labelsfor the device input and exit waveguides where the maximumplasma frequency is ωp = 1.728.
field integrals above is the easiest to coax into good per-formance. Attempting to maximize the ratio between fieldintensities at the output and ground waveguides results inan optimization domain with many singularities, makingconvergence exceedingly difficult, and an objective con-sisting of a product of differences would yield the sameobjective for both the AND and OR gate since the parity ofthese objectives is the same. Nevertheless, these productsand ratios of field integrals may be useful when employedin the optimization of the hyperparameter weights wi andqi should they be determined automatically as mentionedabove.
The results shown in Figs. 12 and 13 are compelling. Inall cases, the field intensity at the correct output waveguideis substantially higher than at the incorrect output. Thoughsome operating cases exhibit substantial loss, we under-stand that these losses are necessary when trying to achievea nonlinear objective through linear dynamics. For exam-ple, the AND gate exhibits a large amount of leakage in theupper-right corner of the device. This odd mode of propa-gation (halfway between the two exit guides) allows smallchanges in the sources to change which output waveguidesees a higher field intensity. Overall, these results show usthat simple PMM device architectures can be reconfiguredto achieve a wide array of highly nontrivial functions wheninverse design is utilized.
IV. CONCLUSION
There are several opportunities for further work basedon the results discussed so far. The waveguide and demul-tiplexer devices designed here can be readily implementedand compared against simulated functionality using exist-ing two-dimensional (2D) PPC devices. Those parameterswhich are here taken as initial conditions (r/a, εrod) canthemselves be used as training parameters. The lattice con-stant in particular is noted to have a large impact on thetrade-off between transmission to the output waveguide
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and control of propagation in the design region. Thoughin this study we choose the lattice constant to be com-mensurate with existing experimental equipment, moretightly packed rods would likely lead to better perfor-mance. Consider also the prospect of a hexagonal or a3D PMM arrangement. The nondimensionalized operationfrequency (ω in terms of c/a) may also prove to be a use-ful optimization parameter as the dispersive physics, whichare the operational basis for these devices, are highly fre-quency dependent. In this study, aside from choosing thelowest frequency possible to reduce the plasma densityrequirements, other operating frequencies are determinedwithout any substantial amount of reasoning. Of particu-lar interest in the future are the effects of magnetizationon the landscape of possible PMM devices. When used inconjunction with circularly polarized light, gaseous plas-mas are capable of achieving εr > 1, opening the doorto even more complex logical operations. While severaltools exist for the simulation and inverse design of pho-tonic devices [36], more development is required on theapplication of inverse design to magnetized PMMs toincorporate such physics. Ceviche also includes a finite-difference time-domain solver that can be used to computenumerical gradients for optimization. Operating in the timedomain would allow for a number of interesting designsto be pursued, such as analog recurrent neural networks,which take advantage of nonlinearities in the plasma per-mittivity. The simulations in this study can also be furtherrefined by adding collisionality (loss) and nonuniformplasma density profiles to the simulation. If the objec-tive functionalities shown in this report can be achievedwith these two nonideal factors considered, then experi-mental realization would likely prove to be much moreattainable. Finally, should experimental realization directlyfrom simulation results prove difficult (we consider thisto be a possible outcome), the device training could actu-ally be carried out in situ, with plasma element currentsadjusted while the objective (transmission at a certainpoint, for example) is determined in real time. In effect, theCeviche forward-mode differentiation scheme that we usehere could be carried out in the laboratory with real fieldmeasurements instead of simulated fields. This methodwould automatically take all nonideal factors into account.
In conclusion, we consider the application of inverse-design methods to the creation of highly optimized2D PMM devices—including waveguides, demultiplexers,and photonic logic gates—for both TE and TM propaga-tion modes. Objective functions of varying complexitiesare considered depending on the desired device. The opti-mization algorithm is shown to take advantage of the richphysics inherent to PMMs, including tunable reflection,refraction, and the presence of localized surface plas-mon modes. In the waveguiding and demultiplexer cases,simulated devices are shown with experimentally realiz-able parameters with few necessary changes to existing
experimental infrastructure. However, further work isneeded, both in simulations and experimental develop-ment, to enable device development for more complexobjectives like boolean logic as well as other devices withcomplex wave-propagation structures.
ACKNOWLEDGMENTS
A.A. thanks the Tomkat Center for support via theEnergy Impact Fellowship. This research is also par-tially supported by the Air Force Office of ScientificResearch through a Multi-University Research Initiative(MURI) with Dr. Mitat Birkan as Program Manager.J.A.R. acknowledges support by the U.S. Department ofEnergy, Office of Science, Office of Advanced ScientificComputing Research, Department of Energy Computa-tional Science Graduate Fellowship under Award No.[DE-SC0019323].
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