Method for Computationally Efficient Design of Dielectric Laser Accelerators
Tyler Hughes, Georgios Veronis, Kent Wootton, R. Joel England, Shanhui Fan
MMethod for Computationally Efficient Design ofDielectric Laser Accelerator Structures T YLER H UGHES , G EORGIOS V ERONIS , K ENT
P. W
OOTTON , R. J
OEL E NGLAND , AND S HANHUI F AN Department of Applied Physics, Stanford University, 348 Via Pueblo, Stanford, CA, 94305, USA School of Electrical Engineering and Computer Science and Center for Computation and Technology(CCT), Louisiana State University, Baton Rouge, LA, 70803, USA SLAC National Accelerator Laboratory, 2575 Sand Hill Rd, Menlo Park, CA, 94025, USA Department of Electrical Engineering, Stanford University, 350 Serra Mall, Stanford, CA 94305, USA * [email protected] Abstract:
Dielectric microstructures have generated much interest in recent years as a means ofaccelerating charged particles when powered by solid state lasers. The acceleration gradient (orparticle energy gain per unit length) is an important figure of merit. To design structures withhigh acceleration gradients, we explore the adjoint variable method, a highly efficient techniqueused to compute the sensitivity of an objective with respect to a large number of parameters.With this formalism, the sensitivity of the acceleration gradient of a dielectric structure withrespect to its entire spatial permittivity distribution is calculated by the use of only two full-fieldelectromagnetic simulations, the original and ‘adjoint’. The adjoint simulation correspondsphysically to the reciprocal situation of a point charge moving through the accelerator gap andradiating. Using this formalism, we perform numerical optimizations aimed at maximizingacceleration gradients, which generate fabricable structures of greatly improved performance incomparison to previously examined geometries.
References and links
1. R. J. England, R. J. Noble, K. Bane, D. H. Dowell, C.-K. Ng, J. E. Spencer, S. Tantawi, Z. Wu, R. L. Byer, E. Peralta,and K. Soong, “Dielectric laser accelerators,” Reviews of Modern Physics , 1337 (2014).2. T. Plettner, P. Lu, and R. Byer, “Proposed few-optical cycle laser-driven particle accelerator structure,” PhysicalReview Special Topics-Accelerators and Beams , 111301 (2006).3. E. Peralta, K. Soong, R. England, E. Colby, Z. Wu, B. Montazeri, C. McGuinness, J. McNeur, K. Leedle, D. Walz,E. Sozer, B. Cowan, G. Travish, and R. Byer, “Demonstration of electron acceleration in a laser-driven dielectricmicrostructure,” Nature , 91–94 (2013).4. J. McNeur, M. Kozák, N. Schönenberger, K. J. Leedle, H. Deng, A. Ceballos, H. Hoogland, A. Ruehl, I. Hartl,O. Solgaard, J. S. Harris, R. L. Byer, and P. Hommelhof, “Elements of a dielectric laser accelerator,” arXiv preprintarXiv:1604.07684 (2016).5. K. J. Leedle, A. Ceballos, H. Deng, O. Solgaard, R. F. Pease, R. L. Byer, and J. S. Harris, “Dielectric laser accelerationof sub-100 kev electrons with silicon dual-pillar grating structures,” Optics letters , 4344–4347 (2015).6. C.-M. Chang and O. Solgaard, “Silicon buried gratings for dielectric laser electron accelerators,” Applied PhysicsLetters , 184102 (2014).7. J. Breuer, J. McNeur, and P. Hommelhoff, “Dielectric laser acceleration of electrons in the vicinity of single anddouble grating structures; theory and simulations,” Journal of Physics B: Atomic, Molecular and Optical Physics ,234004 (2014).8. J. Breuer, R. Graf, A. Apolonski, and P. Hommelhoff, “Dielectric laser acceleration of nonrelativistic electrons at asingle fused silica grating structure: Experimental part,” Physical Review Special Topics-Accelerators and Beams ,021301 (2014).9. M. Kozák, M. Förster, J. McNeur, N. Schönenberger, K. Leedle, H. Deng, J. Harris, R. Byer, and P. Hommelhoff,“Dielectric laser acceleration of sub-relativistic electrons by few-cycle laser pulses,” Nuclear Instruments and Methodsin Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment (2016).10. R. B. Palmer, “Acceleration theorems,” AIP Conf. Proc. , 90–100 (1995).11. C. Joshi, “The Los Alamos Laser Acceleration of Particles Workshop and beginning of the advanced acceleratorconcepts field,” AIP Conf. Proc. , 61–66 (2012).12. K. Soong, R. Byer, E. Colby, R. England, and E. Peralta, “Laser damage threshold measurements of optical materialsfor direct laser accelerators,” in “ADVANCED ACCELERATOR CONCEPTS: 15th Advanced Accelerator ConceptsWorkshop,” AIP Publishing, 2012, vol. 1507, pp. 511–515. a r X i v : . [ phy s i c s . op ti c s ] M a y
3. J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E.Heebner, C. W. Siders, and C. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers tohigh average power,” Optics express , 13240–13266 (2008).14. T. Plettner and R. Byer, “Microstructure-based laser-driven free-electron laser,” Nuclear Instruments and Methods inPhysics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment , 63–66 (2008).15. J. Hebling, “Derivation of the pulse front tilt caused by angular dispersion,” Optical and Quantum Electronics ,1759–1763 (1996).16. S. Akturk, X. Gu, E. Zeek, and R. Trebino, “Pulse-front tilt caused by spatial and temporal chirp,” Optics Express ,4399–4410 (2004).17. N. K. Georgieva, S. Glavic, M. H. Bakr, and J. W. Bandler, “Feasible adjoint sensitivity technique for em designoptimization,” IEEE transactions on microwave theory and techniques , 2751–2758 (2002).18. M. Bakr and N. Nikolova, “An adjoint variable method for frequency domain tlm problems with conductingboundaries,” IEEE microwave and wireless components letters , 408–410 (2003).19. R.-E. Plessix, “A review of the adjoint-state method for computing the gradient of a functional with geophysicalapplications,” Geophysical Journal International , 495–503 (2006).20. M. B. Giles and N. A. Pierce, “An introduction to the adjoint approach to design,” Flow, turbulence and combustion , 393–415 (2000).21. G. Veronis, R. W. Dutton, and S. Fan, “Method for sensitivity analysis of photonic crystal devices,” Optics letters ,2288–2290 (2004).22. C. M. Lalau-Keraly, S. Bhargava, O. D. Miller, and E. Yablonovitch, “Adjoint shape optimization applied toelectromagnetic design,” Optics express , 21693–21701 (2013).23. A. Y. Piggott, J. Lu, K. G. Lagoudakis, J. Petykiewicz, T. M. Babinec, and J. Vučković, “Inverse design anddemonstration of a compact and broadband on-chip wavelength demultiplexer,” Nature Photonics , 374–377 (2015).24. P. Musumeci, S. Y. Tochitsky, S. Boucher, C. Clayton, A. Doyuran, R. England, C. Joshi, C. Pellegrini, J. Ralph,J. Rosenzweig, G. Sung, S. Tolmachev, A. Varfolomeev, A. J. Varfolomeev, T. Yarovoi, and R. Yoder, “High energygain of trapped electrons in a tapered, diffraction-dominated inverse-free-electron laser,” Physical review letters p.154801 (2005).25. E. Courant, C. Pellegrini, W. Zakowicz, M. Month, P. Dahl, and M. Dienes, “High-energy inverse free-electron laseraccelerator,” in “AIP Conference Proceedings,” , vol. 127 (AIP, 1985), vol. 127, pp. 849–874.26. W. Kimura, G. Kim, R. Romea, L. Steinhauer, I. Pogorelsky, K. Kusche, R. Fernow, X. Wang, and Y. Liu, “Laseracceleration of relativistic electrons using the inverse cherenkov effect,” Physical review letters , 546 (1995).27. J. Fontana and R. Pantell, “A high-energy, laser accelerator for electrons using the inverse cherenkov effect,” Journalof applied physics , 4285–4288 (1983).28. J. Bae, H. Shirai, T. Nishida, T. Nozokido, K. Furuya, and K. Mizuno, “Experimental verification of the theory on theinverse smith–purcell effect at a submillimeter wavelength,” Applied physics letters , 870–872 (1992).29. K. Mizuno, J. Pae, T. Nozokido, and K. Furuya, “Experimental evidence of the inverse smith–purcell effect,” Nature , 45–47 (1987).30. W. Shin and S. Fan, “Choice of the perfectly matched layer boundary condition for frequency-domain maxwellâĂŹsequations solvers,” Journal of Computational Physics , 3406–3431 (2012).31. A. Taflove and S. C. Hagness, Computational electrodynamics (Artech house publishers, 2000).32. M. Avriel,
Nonlinear programming: analysis and methods (Courier Corporation, 2003).33. M. Kozak, “Status report erlangen (hommelhoff group),” (2016). ACHIP 3rd Collaboration Meeting.34. J. Nocedal, “Updating quasi-newton matrices with limited storage,” Mathematics of computation , 773–782 (1980).35. B. M. Cowan, “Three-dimensional dielectric photonic crystal structures for laser-driven acceleration,” Phys. Rev. STAccel. and Beams , 011301 (2008).36. T. Zhang, J. Hirshfield, T. Marshall, and B. Hafizi, “Stimulated dielectric wake-field accelerator,” Physical Review E , 4647 (1997).
1. Introduction
Dielectric laser accelerators (DLAs) are periodic dielectric structures that, when illuminatedby laser light, create a near-field that may accelerate electrically charged particles such aselectrons [1]. A principal figure of merit for these DLA structures is the acceleration gradient,which signifies the amount of energy gain per unit length achieved by a particle that is phasedcorrectly with the driving field. DLAs may sustain acceleration gradients on the order of ∼ GV m − when operating using the high peak electric fields supplied by ultrafast (femtosecond) lasers.These acceleration gradients are several orders of magnitude higher than conventional particleaccelerators. As a result, the development of DLA can lead to compact particle accelerators thatenable new applications.In previous works, candidate DLA geometries were optimized for maximum accelerationradient by scanning through parameters of a specified structure geometry [2–9]. However, thisstrategy has limited potential to produce higher acceleration gradient structures because it onlysearches a small portion of the total design space.In this paper, we derive an analytical form for the sensitivity of the acceleration gradient ofa DLA structure with respect to its permittivity distribution using the adjoint-variable method(AVM). We may calculate this by use of only two full-field simulations. The first corresponds tothe typical accelerator setup, where the structure is illuminated with externally incident laser light.The second corresponds to the inverse process, where the same physical structure is simulated butnow with a charged particle traversing the structure as the source. Thus, this formalism explicitlymakes use of the reciprocal relationship between accelerators and radiators [10, 11]. We use thissensitivity information to perform optimizations, which generate DLA structures of much highergradients than previously explored geometries.This work is the first application of the AVM technique to the design of DLA structures andgives examples of fabricable structures that may improve the energy gain achievable with currentDLA technology. In addition, the optimized structures give insight into general design principlesfor DLAs, meaning that one may use the principle findings of this paper to design DLAs withouthaving to run optimizations directly. As an example, it was found that high gradient structuresoften include dielectric mirrors surrounding the particle gap, leading to higher field enhancement.This paper is organized as follows: We first outline the status of DLAs and basic designrequirements in section 2. We introduce AVM in section 3, where we derive the sensitivityof the acceleration gradient of a DLA with respect to its permittivity distribution. In section4, we show that the ‘adjoint’ solution corresponds to that of a radiating charge. In section 5,we describe and demonstrate algorithms for using the sensitivity information to design DLAstructures numerically.
2. A Brief Review of Dielectric Laser Accelerators
DLAs take advantage of the fact that dielectric materials have high damage thresholds at shortpulse durations and infrared wavelengths [1,4,12] when compared to metal surfaces at microwavefrequencies. This allows DLAs to sustain peak electromagnetic fields, and therefore accelerationgradients, that are 1 to 2 orders of magnitude higher than those found in conventional radiofrequency (RF) accelerators. Experimental demonstrations of these acceleration gradients havebeen made practical in recent years by the availability of robust nanofabrication techniquescombined with modern solid state laser systems [13]. By providing the potential for generatingrelativistic electron beams in relatively short length scales, DLA technology is projected to havenumerous applications where tabletop accelerators may be useful, including medical imaging,radiation therapy, and X-ray generation [1, 14]. To achieve high energy gain in a compact size, itis of principle interest to design structures that may produce the largest acceleration gradientspossible without exceeding their respective damage thresholds.Several recently demonstrated candidate DLA structures consist of a planar dielectric structurethat is periodic along the particle axis with either an semi-open geometry or a narrow (micronto sub-micron) vacuum gap in which the particles travel [2–9]. These structures are then side-illuminated by laser pulses. Fig. 1 shows a schematic of the setup, with a laser pulse incidentfrom the bottom.The laser field may also be treated with a pulse front tilt [15, 16] to enable group velocitymatching over a distance greater than the laser’s pulse length. For acceleration to occur, thedielectric structure must be designed such that the particle feels an electric field that is largelyparallel to its trajectory over many optical periods. In the following calculations, the geometryof the dielectric structure is represented by a spatially varying dielectric constant (cid:15) ( x , y ) . Weassume invariance in one coordinate ( ˆ z ) in keeping with the planar symmetry of most currentdesigns. However the methodology we present can be extended to include a third dimension. In - ε (x,y)xy β c βλλ Fig. 1. Diagram outlining the system setup for side-coupled DLA with an arbitrary dielectricstructure (cid:15) ( x , y ) (green). A charged particle moves through the vacuum gap with speed β c .The periodicity is set at βλ where λ is the central wavelength of the laser pulse. addition, our work approximates the incident laser pulse as a monochromatic plane wave at thecentral frequency, which is a valid approximation as long as the pulse duration is large comparedto the optical period.
3. Adjoint Variable Method
In a general DLA system, we may define the acceleration gradient ‘ G ’ over a time period ‘ T ’mathematically as follows: G = T ∫ T E | | ((cid:174) r ( t ) , t ) dt , (1)where (cid:174) r ( t ) is the position of the electron and E | | signifies the (real) electric field componentparallel to the electron trajectory at a given time.To maximize this quantity, we employ AVM [17, 18], which is a technique common to a widerange of fields, including seismology [19], aircraft design [20], and, recently, photonic devicedesign [21–23]. Many engineering systems can be described by a linear system of equations A ( γ ) z = b , where γ is a set of parameters describing the system. For a given set of parameters γ , solving this equation results in the solution ‘ z ’, from which an objective J = J ( z ) , whichis a function of the solution, can be constructed. The optimization of the engineering systemcorresponds to maximizing or minimizing J with respect to the parameters γ . For this purpose,AVM allows one to calculate the gradient of the objective function ∇ γ J for an arbitrary numberf parameters γ i with the only added computational cost of solving one additional linear systemˆ A T ¯ z = − dJdz T , which is often called the ‘adjoint’ problem. For a more comprehensive overviewof the method, we refer the reader to [17].Here we provide the derivation of AVM specifically for the optimization of the acceleratorstructures. Since the structure is invariant in the ˆ z direction, we work in two dimensions, examiningonly the H z , E x and E y field components. For an approximately monochromatic input lasersource with angular frequency ω , the electric fields are, in general, of the form (cid:174) E ((cid:174) r , t ) = Re (cid:110) (cid:174) E ((cid:174) r ) exp ( i ω t ) (cid:111) , (2)where now (cid:174) E is complex.Let us assume the particle we wish to accelerate is moving on the line y = (cid:174) v = β c ˆ x , where c is the speed of light in vacuum and β ≤
1. The x position of the particleas a function of time is given by x ( t ) = x + β c t , where x represents an arbitrary choice ofinitial starting position. For normal incidence of the laser (laser propagating in the + ˆ y direction),phase velocity matching between the particle and the electromagnetic fields is established byintroducing a spatial periodicity in our structure of period βλ along ˆ x , where λ is the laserwavelength. In the limit of an infinitely long structure (or equivalently, T → ∞ ) we may rewriteour expression for the gradient in Eq. (1) as an integral over one spatial period, given by G = βλ Re (cid:26) exp (− i φ ) ∫ βλ dx E x ( x , ) exp (cid:16) i πβλ x (cid:17)(cid:27) . (3)Here the quantity φ = π x βλ is representative of the phase of the particle as it enters thespatial period. In further calculations, we set φ =
0, only examining the acceleration gradientsexperienced by particles entering the accelerator with this specific phase. Since we have arbitrarilycontrol over our input laser phase, this does not impose any constraint on the acceleration gradientattainable.To simplify the following derivations, we define the following inner product operation involvingthe integral over two vector quantities (cid:174) a and (cid:174) b over a single period volume V (cid:48) (cid:104) (cid:174) a , (cid:174) b (cid:105) = (cid:104)(cid:174) b , (cid:174) a (cid:105) = ∫ V (cid:48) d v (cid:16) (cid:174) a · (cid:174) b (cid:17) = ∫ βλ dx ∫ ∞−∞ d y (cid:16) (cid:174) a · (cid:174) b (cid:17) . (4)With this definition, we then have the gradient G = Re {(cid:104) (cid:174) E , (cid:174) η (cid:105)} , (5)where (cid:174) η = (cid:174) η ( x , y ) = βλ exp (cid:16) i πβλ x (cid:17) δ ( y ) ˆ x . (6)Now, we wish to examine the sensitivity of G with respect to an arbitrary parameter, γ , whichmay represent a shifting of material boundary, changing of dielectric constant at a point, or anyother change to the system. Differentiating Eq. (5) gives dGd γ = Re (cid:40)(cid:42) d (cid:174) Ed γ , (cid:174) η (cid:43)(cid:41) . (7)Here we have made use of the fact that (cid:174) η does not depend on γ .From Maxwell’s equations in the frequency domain, we may express our electromagneticproblem in terms of a linear operator ˆ A as × ∇ × (cid:174) E ((cid:174) r ) − k (cid:15) r ((cid:174) r ) (cid:174) E ((cid:174) r ) ≡ ˆ A (cid:174) E ((cid:174) r ) = − i µ ω (cid:174) J ((cid:174) r ) . (8)Here, k = ω / c , (cid:15) r is the relative permittivity, (cid:174) J represents a current density source, and anon-magnetic material is assumed ( µ = µ ). Differentiating Eq. (8) with respect to γ , andassuming that the current source ( (cid:174) J ) does not depend on γ , we see that d (cid:174) Ed γ = − ˆ A − d ˆ Ad γ (cid:174) E . (9)ˆ A is self-adjoint under our inner product, (cid:104) ˆ A (cid:174) a , (cid:174) b (cid:105) = (cid:104) (cid:174) a , ˆ A (cid:174) b (cid:105) , and the same is true for ˆ A − and d ˆ Ad γ .Using these facts and combining Eq. (7) with Eq. (9), we find that dGd γ = Re (cid:26)(cid:28) − ˆ A − d ˆ Ad γ (cid:174) E , (cid:174) η (cid:29)(cid:27) = Re (cid:26)(cid:28) (cid:174) E , − d ˆ Ad γ ˆ A − (cid:174) η (cid:29)(cid:27) . (10)Thus, if we define a second simulation with a source of − (cid:174) η and fields (cid:174) E aj ,ˆ A (cid:174) E aj = − i µ ω (cid:174) J aj = − (cid:174) η, (11)then the field solution, (cid:174) E aj = − ˆ A − (cid:174) η , can be easily identified in Eq. (10). The sensitivity of theacceleration gradient can finally be expressed as dGd γ = Re (cid:26)(cid:28) (cid:174) E , d ˆ Ad γ (cid:174) E aj (cid:29)(cid:27) . (12)The only quantity in this expression that depends on the parameter γ is d ˆ Ad γ . As we will soondiscuss, this quantity will generally be trivial to compute. On the other hand, the full fieldcalculations of (cid:174) E and (cid:174) E aj are computationally expensive, but may be computed once and usedfor an arbitrarily large set of parameters γ i . This gives the AVM approach a significant scalingadvantage with respect to traditional direct sensitivity methods, which require a separate full-fieldcalculation for each parameter being investigated. It is this fact that we leverage with AVM to doefficient optimizations over a large design space.To confirm that this derivation matches the results obtained by direct sensitivity analysis, weexamine a simple accelerator geometry composed of two opposing dielectric squares each ofrelative permittivity (cid:15) . We take a single γ parameter to be the relative permittivity of the entiresquare region. Because we only change the region inside the dielectric square, we may identifythe d ˆ Ad γ operator for this parameter by examining Eq. (8), giving d ˆ Ad (cid:15) ((cid:174) r ) = (cid:40) − k if (cid:174) r in square0 otherwise . (13)Thus, given the form of the acceleration gradient sensitivity given in Eq. (12), combinedwith Eq. (13), the change in acceleration gradient with respect to changing the entire squarepermittivity is simply given by the integral of the two field solutions over the square region,labeled ‘ sq ’ dGd (cid:15) sq = − k Re (cid:26)∫ sq d (cid:174) r (cid:174) E ((cid:174) r ) · (cid:174) E aj ((cid:174) r ) (cid:27) . (14)In Fig. 2 we compare this result with the direct sensitivity calculation where the system ismanually changed and simulated again. The two methods agree with excellent precision. )b) Relative Permittivity ( ε ) G r a d i en t ( E )
2 3 4 5 6 d G / d ε εε Fig. 2. Demonstration of AVM in calculating sensitivities. (a) The acceleration gradient ( G )of a square accelerator structure (inset) as a function of the square’s relative permittivity.The particle traverses along the dotted line and a plane wave is incident from the bottomof the structure. (b) The sensitivity dGd (cid:15) of the gradient with respect to changing the squarerelative permittivity for direct central difference (solid line) dGd (cid:15) = G ( (cid:15) + ∆ (cid:15) )− G ( (cid:15) − ∆ (cid:15) ) ∆ (cid:15) andusing AVM (circles). The two calculations agree with excellent precision. The dotted line at dGd (cid:15) =
0, corresponds to local minima and maxima of G ( (cid:15) ) above. Extending this example to the general case of perturbing the permittivity at an arbitrary position (cid:174) r (cid:48) , we see that dGd (cid:15) ((cid:174) r (cid:48) ) = − k Re (cid:26)∫ d (cid:174) r (cid:174) E ((cid:174) r ) · (cid:174) E aj ((cid:174) r ) δ ((cid:174) r − (cid:174) r (cid:48) ) (cid:27) (15) = − k Re (cid:110) (cid:174) E ((cid:174) r (cid:48) ) · (cid:174) E aj ((cid:174) r (cid:48) ) (cid:111) . (16)
4. Reciprocity
With the AVM form derived, we now wish to re-examine the adjoint source term from Eq. (11)in another interpretation. Let us now consider the fields radiated by a point particle of charge q flowing through our domain at y = (cid:174) v = β c ˆ x . In the time domain, we canepresent the current density of this particle as (cid:174) J rad ( x , y ; t ) = q β c δ ( x − x − c β t ) δ ( y ) ˆ x . (17)We may take the Fourier transform of (cid:174) J rad with respect to time to examine the current densityin the frequency domain, giving (cid:174) J rad ( x , y ; ω ) = q β c δ ( y ) ˆ x ∫ ∞−∞ dt exp (− i ω t ) δ ( x − x − c β t ) (18) = q exp (cid:16) i ω ( x − x ) c β (cid:17) δ ( y ) ˆ x (19) = q exp (cid:16) i πβλ x (cid:17) exp (− i φ ) δ ( y ) ˆ x . (20)Comparing with the source of our adjoint problem, (cid:174) J aj = − i ωµ (cid:174) η , we can see that (cid:174) J aj = − i exp ( i φ ) π q β c µ (cid:174) J rad . (21)This finding shows that the adjoint field solution ( (cid:174) E aj ) corresponds (up to a complex constant)to the field radiating from a test particle flowing through the accelerator structure. To put thisanother way, in order to calculate the acceleration gradient sensitivity with AVM, we mustsimulate the same structure operating both as an accelerator ( ˆ A (cid:174) E = − i ωµ (cid:174) J acc ) and as a radiator( ˆ A (cid:174) E aj = − i ωµ (cid:174) J aj ).It is understood that one way to create acceleration is to run a radiative process in reverse. Indeed,this is the working principle behind accelerator schemes such as inverse free electron lasers [24,25],inverse Cherenkov accelerators [26, 27], and inverse Smith-Purcell accelerators [28, 29]. Here,we see that this relationship can be expressed in an elegant fashion using AVM.
5. Applications
Now that we have shown how to use AVM to compute the sensitivity of the acceleration gradientwith respect to the permittivity distribution, we will show practical applications of these results.First, for computational modeling, the problem must be transitioned from a continuous space to adiscrete space. Here we make the transition using a finite-difference frequency-domain (FDFD)formalism [30,31]. The electromagnetic fields now exist on a Yee lattice and the linear operator ˆ A becomes a sparse, complex symmetric matrix, A , relating the vector of electric field components, e , to the input current source components b as A e = b . (22)To solve for the field components, this system must be solved numerically for e . In two-dimensions, this is usually done directly by use of “lower-upper” (LU) decomposition methods forsparse matrices. Only the right hand side of Eq. (22) is different between the original and adjointsimulations. Therefore after the A matrix is factored to solve the original simulation, its factoredform may be saved and reused for the adjoint calculation, which cuts the total computationalrunning time roughly in half.Written in terms of this discrete system, the acceleration gradient is = Re { e T η } , (23)where η is now a discretized version of (cid:174) η . Similarly, the sensitivity of the gradient with respect tochanging the permittivity at pixel ‘ i ’ is given by dGd (cid:15) i = − k Re { e i ¯ e i } , (24)where, as before, ¯ e is the solution of the adjoint problem A ¯ e = − η . (25)For all simulations, we use an FDFD program developed specifically for this work, although acommercial package would also be sufficient. We have chosen a grid spacing that correspondsto 200 grid points per free space wavelength in each dimension. Perfectly matched layers areimplemented as absorbing regions on the edges parallel to the electron trajectory, with periodicboundary conditions employed on boundaries perpendicular to the electron trajectory. A total-fieldscattered-field [31] formalism is used to create a perfect plane wave input for the accelerationmode. Since we now have a highly efficient method to calculate dGd (cid:15) i , we proceed to use this informationto maximize the acceleration gradient with respect to the permittivity distribution. We use aniterative algorithm based on batch gradient ascent [32]. During each iteration, we first calculate dGd (cid:15) i for all pixels ‘ i ’ within some specified design region. Then, we update each (cid:15) i grid as follows (cid:15) i : = (cid:15) i + α dGd (cid:15) i . (26)Here, α is a step parameter that we can tune. We need α to be small enough to find local maxima,but large enough to have the optimization run in reasonable amount of time. This process is theniterated until convergence on G . During the course of optimization, the permittivity distributionis considered as a continuous variable, which is not realistic in physical devices. To address thisissue, we employ a permittivity capping scheme during optimization. We define a maximumpermittivity ‘ (cid:15) m ’ corresponding to a material of interest. During the iterative process, if therelative permittivity of any cell becomes either less than 1 (vacuum) or greater than (cid:15) m , that cell ispushed back into the acceptable range. It was found that with this capping scheme, the structuresconverged to binary (each pixel being either vacuum or material with a permittivity of (cid:15) m ) after anumber of iterations without specifying this choice of binary materials as a requirement of theoptimization. Therefore, only minimal post-processing of the structures was required.The results of this optimization scheme are shown in Fig. 3(b-d) for three different (cid:15) m valuescorresponding to commonly explored DLA materials. The design region was taken to be arectangle fully surrounding but not including the particle gap. The design region was madesmaller for higher index materials, since making it too large led to divergence during the iteration.We found that a totally vacuum initial structure worked well for these optimizations. However,initially random values between 1 and (cid:15) m for each pixel within the design region also gavereasonable results.This optimization scheme seems to favor geometries consisting of a staggered array of field-reversing pillars surrounding the vacuum gap, which is already a popular geometry for DLA.However, these optimal designs also include reflective mirrors on either side of the pillar array,which suggests that for strictly higher acceleration gradients, it is useful to use dielectric mirrorsto resonantly enhance the fields in the gap.
100 200 300 400 500
Iteration Number G r a d i en t ( E ) Y P o s i t i on ( λ ) X Position ( λ ) c)b) d)a) Si:Si N : ε m = 11.8 ε m = 3.9 ε m = 2.1SiO : Fig. 3. Demonstration of the structure optimization for β = .
5, laser wavelength λ = µ m,and a gap size of 400 nm. A plane wave is incident from the bottom in all cases. (a)Acceleration gradient as a function of iteration number for different maximum relativepermittivity values, corresponding to those of Si, Si N , and SiO at the laser wavelength. Theoptimizations converge after about five-hundred iterations. (b-d) Final structure permittivitydistributions (white = vacuum, black = (cid:15) m ) corresponding to the three curves in (a). Eightperiods are shown, corresponding to four laser wavelengths. For each (b-d), design regionwidths on each side of the particle gap were given by 1, 2, and 4 µ m for Si, Si N , and SiO ,respectively. It was observed that for random initial starting permittivity distributions, the same structures asshown in Fig. 3 are generated every time. Furthermore, these geometries are remarkably similarto those recently proposed [33], although these designs do not include the reflective front mirror.These findings together suggest that the proposed structures may be close to the globally optimalstructure for maximizing G .It was further found that convergence could be achieved faster by a factor of about ten byincluding a ‘momentum’ term in the update equation. This term corresponds to the sensitivitycalculated at the last iteration multiplied by a constant, α (cid:48) <
1. Explicitly, for iteration number ‘ j ’and pixel ‘ i ’ (cid:15) ( j + ) i : = (cid:15) ( j ) i + α (cid:34) dGd (cid:15) i ( j ) + α (cid:48) dGd (cid:15) i ( j − ) (cid:35) . (27) .3. Acceleration Factor Maximization DLAs are often driven with the highest input field possible before damage occurs. Therefore,another highly relevant quantity to maximize is the ‘acceleration factor’ given by the accelerationgradient divided by the maximum electric field amplitude in the structure. This quantity willultimately limit the amount of acceleration gradient we can achieve when running at damagethreshold. Explicitly, the acceleration factor is given by f A = G max {| (cid:174) E |} . (28)Here, | (cid:174) E | is a vector of electric field magnitudes in our structure, and the max {} function isdesigned to pick out the highest value of this vector in either our optimization or material region,depending on the context. We would like to use the same basic formalism to maximize f A .However, since the max {} function is not differentiable, this is not possible directly. Instead wemay use a ‘smooth-max’ function to approximate max {} as a weighted sum of vector componentsmax {| (cid:174) E |} ≈ (cid:205) i | (cid:174) E i | exp (cid:0) a | (cid:174) E i | (cid:1)(cid:205) i exp (cid:0) a | (cid:174) E i | (cid:1) . (29)Here, the parameter a ≥ a = a and examiningthe acceleration factors of the resulting optimized structures, we determined that a = f A . If a is too large, the calculation may induce floating point overflow orrounding error issues.Using this smooth-max function, one may calculate d f A d (cid:15) i analytically and perform structureoptimizations in the same way that was discussed previously. The derivation of the adjoint sourceterm is especially complicated and omitted for brevity, although the end result is expressed solelyin terms of the original fields, the adjoint fields, and the d ˆ Ad γ operator, as before. Two structureswith identical parameters but optimized, respectively, for maximum G and f A are shown in Fig.4. On the left, we see that the G maximized structure shows the characteristic dielectric mirrors,giving resonant field enhancement. On the right is the structure optimized for f A , which haseliminated most of its dielectric mirrors and also introduces interesting pillar shapes. In Table 1the main DLA performance quantities of interest are compared between these two structures.Whereas the acceleration gradient is greatly reduced when maximizing for f A , the f A value itselfis improved by about 25% or 23% depending on whether one measures the maximum field inthe design region or the material-only region, respectively. These findings suggest that the AVMstrategy is effective in designing not only resonant, high acceleration gradient structures, but alsonon-resonant structures that are more damage resistant. In the future, when more components ofDLA are moved on-chip (such as the optical power delivery), it will be important to have controlover the resonance characteristics of the DLA structures to prevent damage breakdown at theinput facet. Our technique may be invaluable in designing structures with tailor-made qualityfactors for this application.
6. Discussion
We found that AVM is a reliable method for optimizing DLA structures for both maximumacceleration gradient and also acceleration factor. The optimization algorithm discussed showsgood convergence and rarely requires further post-processing of structures to create binarypermittivity distributions. Therefore, it is a simple and effective method for designing DLAs.Whereas most structure optimization in this field uses parameter sweeps to search the designspace, the efficiency of our method allows us to more intelligently find optimal geometries without ) b)X Position ( λ )
0 1 2 3 4 0 1 2 3 4 Y P o s i t i on ( λ ) Fig. 4. Demonstration of the final structures after optimization for (a) maximizing gradientonly, (b) maximizing the acceleration factor. β = .
5, laser wavelength λ = µ m, gap sizeof 400 nm. (cid:15) m = .
1, corresponding to SiO . In (a), the high gradients are achieved usingreflective dielectric mirrors to confine and enhance the fields in the center region. In (b),these dielectric mirrors are removed and the pillar structures are augmented. The structure in(b) shows a 23% increase in the acceleration factor in the material region when compared to(a). Table 1. Acceleration factor ( f A ) before and after maximization. Quantity Value (max G ) Value (max f A ) ChangeGradient ( E ) 0.1774 0.0970 -45.32%max {| (cid:174) E |} in design region 4.1263 1.7940 -56.52%max {| (cid:174) E |} in material region 2.7923 1.2385 -55.84% f A in design region 0.0430 0.0541 +25.81% f A in material region 0.0635 0.0783 +23.31%shape parameterization. Furthermore, the structures that we design are fabricable. Although noDLA structures have been tested at the proposed wavelength of 2 µ m, both simulations [2] andexperimental results from other wavelengths [5] show gradients far below those presented here.We had limited success designing DLA structures in the relativistic ( β ≈
1) regime, especially forhigher index materials, such as Si. We believe this is largely due to the stronger coupling betweenelectron beam and incident plane wave at this energy. The characteristics of the adjoint sourcechange dramatically at the β = β ≥
1, the adjointfields become propagating by process of Cherenkov radiation. Upon using the above describedalgorithm, the gradients diverge before returning to low values, no matter the step size α . Theonly way to mitigate this problem is to choose prohibitively small design regions or low indexmaterials, such as SiO .The AVM formalism presented in this work may also be extended to calculate higher orderderivatives of G . For each higher order, the form of the derivative of G can be derived in aashion very similar to the one outlined for first order. Given the full Hessian H i , j = d Gd (cid:15) i d (cid:15) j , ascalculated by AVM, one could use Newton’s method to do optimization. However, to performexactly, this calculation would require as many additional simulations as there are grid pointswithin the design region. Therefore, these higher order methods are inconvenient for our purposeswhere there are generally tens of thousands of design pixels. This limitation may be averted byusing approximate methods for finding the inverse Hessian [34], which may provide substantialimprovement to optimization results and convergence speeds in certain cases. However, in ourcase there was no need to explore beyond first order due to the relative success and speed of thealgorithm presented.As future works, our goal is to fabricate and test these structures experimentally, as well asinclude further metrics into the optimization if necessary, such as favoring larger feature sizesand incorporating focusing effects. Furthermore, this method is of great interest in designingwaveguide-coupled accelerator structures, where typical designs optimized for plane wave inputare likely suboptimal. This will be of critical importance when moving the optical power deliverysource on-chip.In addition to the side-incident geometry explored, this technique is applicable to designing otherdielectric-based accelerator structures. This includes particle-laser co-propagating schemes [35]and perhaps dielectric wakefield acceleration [36], among others. Therefore, we expect that ourresults may find use in the larger advanced accelerator community.
7. Conclusion
We have introduced the adjoint variable method as a powerful tool for designing dielectric laseraccelerators for high gradient acceleration and high acceleration factor. We have further shownthat the adjoint simulation is sourced by a point charge flowing through the accelerator, whichquantifies the reciprocal relationship between an accelerator and a radiator.Optimization algorithms built on this approach allow us to search a substantially larger designspace and generate structures that give gradients far above those normally used for each material.Furthermore, the structures designed by AVM are fundamentally not constrained by shapeparameterization, allowing never-before-seen geometries to be generated and tested.
Funding
This work was supported by the Gordon and Betty Moore Foundation under grant GBMF4744(Accelerator on a Chip), and by the U.S. Department of Energy under Contract DE-AC02-76SF00515.