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Three-dimensional dielectric photonic crystal structures for laser-driven acceleration

Benjamin M. Cowan*Tech-X Corporation, 5621 Arapahoe Avenue, Boulder, Colorado 80303, USA

and Stanford Linear Accelerator Center, 2575 Sand Hill Road, Menlo Park, California 94025, USA(Received 3 November 2007; published 17 January 2008)

We present the design and simulation of a three-dimensional photonic crystal waveguide for linearlaser-driven acceleration in vacuum. The structure confines a synchronous speed-of-light acceleratingmode in both transverse dimensions. We report the properties of this mode, including sustainable gradientand optical-to-beam efficiency. We present a novel method for confining a particle beam using opticalfields as focusing elements. This technique, combined with careful structure design, is shown to have alarge dynamic aperture and minimal emittance growth, even over millions of optical wavelengths.

DOI: 10.1103/PhysRevSTAB.11.011301 PACS numbers: 41.75.Jv, 42.70.Qs, 41.85.�p

I. INTRODUCTION

The extraordinary electric fields available from lasersystems make laser-driven charged particle accelerationan exciting possibility. Because of the maturity of rf accel-erator technology, it is tempting to try to scale down tradi-tional microwave accelerators to optical wavelengths.However, this proposal encounters several obstacles inthe structure design. First, because of the significant lossof metals at optical frequencies, we wish instead to usedielectric materials for a structure. Second, manufacturinga circular disk-loaded waveguide on such small scalesposes a significant challenge. For these reasons, we mustconsider structures which differ significantly from thoseused in conventional accelerators.

So far, macroscopic, far-field structures have been in-vestigated experimentally as a means to exploit high laserintensities for linear particle acceleration. In the laserelectron acceleration program (LEAP), a free-spacemode was used to modulate an unbunched electron beam[1]. The experiment demonstrated the expected linear scal-ing of energy modulation with laser electric field as well asthe expected polarization dependence. The next stage ofthe experimental program, to be performed at the E163facility at SLAC, seeks to demonstrate net energy gain byfirst optically bunching the electron beam using an IFEL[2,3]. Beyond that, the desire is to demonstrate a scalableacceleration mechanism. For that purpose, a near-field,guided-mode structure would be more suitable.

Photonic crystals [4] provide a means of guiding aspeed-of-light optical mode in an all-dielectric structure.They have been investigated for some time for metallic rfaccelerator structures because of their potential for elimi-nating a major source of beam breakup instability [5,6]. Inthe optical regime, a synchronous mode has been shown toexist in a photonic crystal fiber [7]. Several years ago, astudy was conducted of two-dimensional planar structures[8]. While the study was informative, such two-

dimensional planar structures are ultimately impracticalbecause they only confine the accelerating mode in onetransverse direction. In this paper we present a three-dimensional structure which overcomes this deficiency,confining the mode in both transverse dimensions.

In Sec. II we present the structure geometry and accel-erating mode. Then in Sec. III we analyze the performanceof the structure in terms of accelerating gradient andoptical-to-beam efficiency. In Sec. IV we address single-particle beam dynamics in the structure, presenting afocusing-defocusing lattice and demonstrating stablebeam propagation.

II. STRUCTURE GEOMETRY ANDACCELERATING MODE

The accelerator structure is based on the so-called‘‘woodpile’’ geometry, a well-established three-dimensional photonic crystal lattice designed to provide acomplete photonic band gap in a structure with a straight-forward fabrication process [9]. The lattice consists oflayers of dielectric rods in vacuum, with the rods in eachlayer rotated 90� relative to the layer below and offset halfa lattice period from the layer two below, as shown with thecoordinate system in Fig. 1. We consider laser accelerationusing a wavelength of 1550 nm, in the telecommunicationsband where many promising sources exist [10]. At thiswavelength silicon has a normalized permittivity of �r ��=�0 � 12:1 [11]. We let a be the horizontal lattice con-stant, and take c � a

��2p

. As described in [9] the lattice hasa face-centered cubic structure and exhibits an omnidirec-tional band gap. Based on those results, we take the rodwidth to be w � 0:28a. The ratio of the width of the bandgap to its center frequency is 18.7%.

We form a waveguide by removing all dielectric mate-rial in a region which is rectangular in the transverse x andy dimensions, and extends infinitely in the e-beam propa-gation direction z. In addition, there are two modifications:In order to avoid deflecting fields in the accelerating mode,we make the structure vertically symmetric by inverting*[email protected]

PHYSICAL REVIEW SPECIAL TOPICS - ACCELERATORS AND BEAMS 11, 011301 (2008)

1098-4402=08=11(1)=011301(9) 011301-1 © 2008 The American Physical Society

http://dx.doi.org/10.1103/PhysRevSTAB.11.011301

the upper half of the lattice so it is a vertical reflection ofthe lower half. Second, we extend the central bars into thewaveguide by 0:124a on each side in order to suppressquadrupole fields. The consequences of this perturbation

will be discussed further in Sec. IV. The geometry, with adefect waveguide introduced, is shown schematically inFig. 2 and visually in Fig. 3.

The inversion of half the lattice introduces a planardefect where the two halves meet, but this waveguide stillsupports a confined accelerating mode. Indeed, the mode islossless to within the tolerance of the calculation, placingan upper bound on the loss of 0:48 dB=cm. Its fields areshown in Fig. 4. For this mode a=� � 0:364, so using a1550 nm source determines a � 565 nm. The individualrods are then 158 nm wide by 200 nm tall. The mode wascomputed using the finite-difference time-domain method,including a uniaxial perfectly matched layer to includelosses [12].

guide planar defect

rod extendsinfinitely in z

crossbar centeredat z = nacrossbar centeredat z = (n + 1/2)a z

y

x

FIG. 2. The geometry of a vertically symmetric waveguidestructure.

FIG. 3. A symmetric waveguide.

FIG. 1. A diagram of 8 layers (2 vertical periods) of thewoodpile lattice.

FIG. 4. The accelerating field seen by a speed-of-light particle,averaged over a lattice period, normalized to the acceleratingfield on axis, shown with structure contours for a transverse sliceat z � 0. The inner rectangle denotes the interface between freespace and the absorbing boundary in the simulation.

BENJAMIN M. COWAN Phys. Rev. ST Accel. Beams 11, 011301 (2008)

011301-2

III. PERFORMANCE OF THE ACCELERATINGMODE

We can now explore the performance of this structurefrom the point of view of accelerating gradient and optical-to-beam efficiency. The gradient is limited by opticalbreakdown to the structure material. We can define aparameter, which is a property of the mode itself andindependent of the material, which relates the acceleratinggradient to the breakdown threshold of the dielectric.While the mechanism of optical breakdown is not fullyunderstood for most materials, let us suppose that break-down occurs when the electromagnetic energy densityexceeds a certain threshold uth in the material. We thendefine the damage impedance of an accelerating mode by

Zd �E2

acc

2umaxc;

where Eacc is the accelerating gradient on axis and umax isthe maximum electromagnetic energy density anywhere inthe material. Then we can write the sustainable accelerat-ing gradient as

Eacc ��2Zduthc

p: (1)

For the mode in the woodpile waveguide, we have Zd �11:34 �. Optical breakdown studies in silicon have showna breakdown threshold of uth � 13:3 J=cm3 at � �1550 nm and 1 ps FWHM pulse width [13], resulting inan unloaded accelerating gradient of 301 MV=m.

With designs for future high-energy colliders calling forbeam power exceeding 10 MW, optical-to-beam efficiencyis a critical parameter for a photonic accelerator. Here wecompute the efficiency, and the characteristics of the par-ticle beam required to optimize the efficiency. Unlikeconventional metallic disk-loaded waveguide structures,the waveguide we consider here has a group velocity asignificant fraction of the speed of light; in this case vg �0:269c. As the particle bunch and laser pulse traverse thestructure, the bunch will generate wakefields in the wave-guide which destructively interfere with the incident laserpulse. Because the beam-driven wakefield has amplitudeproportional to the bunch charge, wakefields limit thepractical amount of charge one can accelerate: If q is thebunch charge, then the energy gained by the bunch scalesas q, but the energy loss due to wakefields scales as q2.Increasing the charge too much ultimately decreases theenergy gain, and thus the efficiency, of the structure. Toimprove the efficiency, we can embed the structure withinan optical cavity in order to recycle the remaining laserpulse energy for acceleration of subsequent optical bunchtrains.

To compute the efficiency, we follow the treatmentdescribed in [14]. According to that description, the effi-ciency of the structure depends upon several parameters.First, the characteristic impedance of the mode, which

describes the relationship between input laser power andaccelerating gradient [15], is

Zc �E2

acc�2

P� 484 �;

where P is the laser power. Second, the group velocity ofthe mode affects the efficiency, as modes with groupvelocity closer to the speed of light couple better to arelativistic particle beam. This is quantified by the lossfactor, which is given by [16]

k �1

4

c�g1� �g

Zc�2 ;

where �g � vg=c. An optical bunch with charge q willradiate fields in the accelerating mode with deceleratinggradient equal to kq. We can use the loss factor as a point ofcomparison with conventional accelerators. We find thatthe woodpile structure couples much more strongly to aparticle beam than an rf accelerator, with a loss factor of5:56� 109 �V=m�=pC versus 19:7 �V=m�=pC for theSLAC linac [17]. However, this is entirely due to thesmaller wavelength. If we compare the quantity k�2, wefind values of 0:013 V m=pC for the woodpile structure and0:22 V m=pC for the SLAC linac.

Finally, the Cerenkov impedance ZH parametrizes theenergy loss due to wideband Cerenkov radiation.Following [18], we can estimate ZH from its value for abulk dielectric with a circular hole of radius R, which is

ZH �Z0

2��R=��2;

where Z0 � 376:73 � is the impedance of free space. Forarbitrary accelerating structures, we can let R be a lengthcharacterizing the radius of the vacuum waveguide, even ifthe guide is not circular. In our case the aperture is rectan-gular, so we define the parameter R by R �

��Ap

=2 �0:451�, where A is the aperture area. This yields ZH �295 �.

The accelerator, particle beam, and laser fields operateon several length scales. First, the particles must be opti-cally bunched in order to sample only a small range ofoptical phases; thus each optical microbunch must havelength � � or equivalently have duration on the order oftens of attoseconds. We consider particles to come in trainsof N optical bunches, with the optical bunches beingspaced by �. The length of the train is then N�. We alsowish to use ultrafast laser pulses of duration �� on theorder of 1 ps. Because the group velocity of the waveguideis less than c, a relativistic particle beam will outrun thelaser pulse. We must therefore construct the accelerator inshort segments, and recouple laser pulses at every segment.In a segment of length L, the laser pulse will slip withrespect to the particle bunch train by time �� L=c��1� �g�=�g. Following [14], we assume an acceleratorsegment is inside an optical cavity with a round-trip loss of

THREE-DIMENSIONAL DIELECTRIC PHOTONIC . . . Phys. Rev. ST Accel. Beams 11, 011301 (2008)

011301-3

5%, and the laser pulses are coupled into the cavity througha beam splitter with reflectivity r. We consider each bunchin our train to have the same charge, and assume that theduration of the train is much less than the slippage time ��.We also assume that �� 3��. For example, if wechoose a laser pulse duration of 1 ps, then �� 3 ps,which gives a structure length of L � 331 �m. Then N �L=� � 214.

As described above, trying to accelerate too muchcharge can reduce the efficiency of an accelerator due towakefield effects. Therefore, for a given r and N, thestructure has a maximum efficiency �max. For each N,we choose r to maximize �max. These optimum valuesare plotted in Fig. 5. From this we see that the efficiencycan be made quite high. Even with just a single bunch, theefficiency reaches 38%, while for a train of 100 bunches,the efficiency is 76%. Once the optimum r is computed foreach N, the total train charge qt and external laser pulseamplitude are chosen so that they together satisfy twoconditions: First, that the peak unloaded gradient is equalto the maximum sustainable gradient in the structure,which we take to be 301 MV=m from the above discus-sion. Second, we require that the charge maximize theefficiency. We plot the optimum charge as a function ofthe number of bunches in Fig. 6. As expected from theresults in [14], the optimum total charge is low, only1.41 fC for a single bunch and 12.3 fC for 100 bunches.With the charge having been computed, we can then findthe average initial gradient (the gradient experienced bythe first optical bunch in the train) for each N. We can alsocompute the induced energy spread on the beam as thedifference between the average accelerating gradient ex-perienced by the first and last bunches in the train, relativeto the average initial gradient. These quantities are plottedin Fig. 7. From this plot we see that wakefields have a

serious effect on the acceleration. For just a single bunch,the average initial gradient is reduced from 218 MV=m(which is reduced from 301 MV=m due to the advance-ment of the electrons with respect to the Gaussian laserpulse envelope) to 197 MV=m. The minimum at around 7optical bunches is due to two competing effects. First, asthe number of bunches increases from the single-bunchcase, the total charge increases, generating more wake-fields which are recycled within the cavity. But we seefrom Fig. 5 that, as the number of bunches increases, thereflectivity decreases, so a smaller fraction of these wake-fields is recycled. More concerning is the effect on theenergy spread of the beam. Even with only two bunches,the spread in gradient is 6.2%, and with 5 bunches, thespread is 16.5%.

0 20 40 60 80 100

Number of optical bunches

0.3

0.4

0.5

0.6

0.7

0.8

0.9

ropt

ηmax

FIG. 5. The optimum reflectivity and the corresponding maxi-mum efficiency as a function of the number of optical micro-bunches in the bunch train.

0 20 40 60 80 100


0

2

4

6

8

10

12

14

Tota

lCha

rge

(fC

)

FIG. 6. The optimum total charge of a bunch train or maxi-mum efficiency as a function of the number of optical micro-bunches.

0 20 40 60 80 100


190

192

194

196

198

200

202

204

Ave

rage

initi

algr

adie

nt(M

V/m

)

0 20 40 60 80 1000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Rel

ativ

een

ergy

spre

ad

FIG. 7. The average initial gradient and induced energy spreadas a function of the number of optical microbunches.


011301-4

IV. PARTICLE BEAM DYNAMICS

As remarked above, we adjusted the geometry of thestructure because of beam dynamics considerations,namely, to make the structure symmetric in order to sup-press deflecting fields. Here we examine the particle beamdynamics in this structure in more detail. Two concernsimmediately arise. First, the structure has an extremelysmall aperture, approximately � in each transverse dimen-sion, so confining the beam within the waveguide presentsa serious challenge. Second, since the structure is notazimuthally symmetric, particles out of phase with theaccelerating field will experience transverse forces. As itturns out, an investigation of the second problem will yieldinsight into the first: The optical fields in the structureprovide focusing forces which are quite strong comparedto conventional quadrupole magnets and can be used toconfine a beam within the waveguide. Therefore, we beginwith a description of the transverse forces in the woodpilestructure. We then propose a focusing scheme, and com-pute the requirements on the phase-space extent of theparticle beam.

In our analysis of the beam dynamics in this structure,we use the average of the fields over one longitudinalperiod of the photonic crystal waveguide. This is justifiedby two effects. First, we consider the particles to be rela-tivistic, and the longitudinal velocity deviates from thespeed of light by only O�1=2�, where is the Lorentzfactor. Second, the submicron period together with fieldintensities limited by damage threshold prohibits particlesfrom acquiring relativistic momentum on the scale of asingle period. To see this, one need only examine a nor-malized vector potential of the fields, defined by

A �eEa

mc2 ;

where E is the electric field magnitude. The angular de-viation of a particle trajectory within a period isO�A=�. Inour case, E � 301 MV=m, so A � 3:3� 10�4. Thus, thevariation of a particle’s trajectory within a period is negli-gible, so the time-averaged force on the particle is wellapproximated by the force due to the longitudinally aver-aged fields. We therefore proceed to define the longitudi-nally averaged fields, as seen by a speed-of-light particlebeam. Taking the fields to have ei!t time dependence, wecan define the averaged electric field amplitude by

�E�x; y� �1

a

Z a

0E�x; y; z�eikzzdz;

where E is the complex electric field amplitude and kz �k0 � !=c is the Bloch wave number in the periodic struc-ture. It follows that the average amplitude of the transverseforce on a particle is given by

F? �iek0r? �Ez:

Thus, Ez essentially forms a ‘‘potential’’ for the transverseforce. We note that the factor of i implies that particlesexperience a transverse force to the extent that they are outof phase with the crest of the accelerating wave.

Since the mode has speed-of-light phase velocity, Ezsatisfies r2

?�Ez � 0 in the vacuum region. This implies

that r? � F? � 0, so that focusing in both transverse di-rections simultaneously is impossible. It also means thatwe can expand �Ez as a polynomial in the complex coor-dinate w � �x iy�=�. Using the symmetries of the struc-ture, we find we can write �Ez as

�E z �X1m�0

A2m Re�w2m�: (2)

We then have that

@Fx@x

��x�0;y�0�

ie��

A2:

Thus, a particle ahead in phase by (kz�!t � ) willexperience focusing gradients Kx and Ky, in the x and ydirections, respectively, given by

Kx � �Ky � �Re�ieA2

��Ee�i

��

eA2

��Esin: (3)

This is equivalent to a quadrupole magnet with focusinggradient �A2=��c� sin. The coefficients A2m for m> 1correspond to nonlinear forces.

To examine the implications of these forces for ourstructure, we extract the coefficients A2m from the com-puted fields. We do so by fitting polynomials of a formsimilar to that in Eq. (2) for m � 0; . . . ; 6 to the computedEx and Ey fields; these are sufficient to compute the co-efficients of the other components using the Maxwellequations. As noted in Sec. II, we have modified thestructure geometry by extending the central bars into thewaveguide in order to suppress the quadrupole fields. Thisinsertion distance was determined by computing the fittedA2 coefficient as a function of insertion distance, andfinding the value for which A2 � 0. This prevents particlesfrom being driven out of the waveguide by the linearfocusing force, but does not yet provide a mechanism forstable propagation within such a small waveguide.

To address the question of stable propagation, we notefrom Eq. (3) that, because of the optical scale of �, thefocusing gradient can be quite large. This suggests usingthese strong focusing forces to confine the particle beam byrunning a similar structure with the laser field �=2 out ofphase with the particle beam. We explore this possibility byagain modifying our structure by inserting the central bars,this time to optimize the fields for focusing. For thispurpose we wish to suppress the lowest order nonlinearfield component, in this case the octupole field given by theA4 coefficient, for maximum stability. We find that, byinserting the central bars by 0:188a, we can suppress theoctupole field, while the focusing gradient for fields at


011301-5

damage threshold is equivalent to an 831 kT=m magnet.The accelerating field for this structure geometry is shownin Fig. 8.

Having found suitable optical modes for both accelera-tion and focusing, we can now describe a system of focus-ing elements designed to confine the particle beam. Weconsider a focusing-defocusing (F0D0) lattice,1 in whicheach cell consists of a focusing segment of length Lf run�=2 behind in phase for focusing in the x direction, fol-lowed by an accelerating segment of length La, a focusingsegment of length Lf run ��=2 behind in phase fordefocusing in the x direction, and then another acceleratingsegment of length La [19]. We are free to choose theparameters Lf and La. Let K be the peak focusing gradientfor the focusing segment, given by Eq. (3) with E theenergy of the ideal particle. Assuming an initial idealparticle energy of 1 GeV, we find that, if we run thefocusing structure at damage threshold (with an on-axisaccelerating gradient of 133 MV=m), we haveK � 2:49�105 m�2. For our lattice we choose Lf � 0:2=

��Kp�

401 �m, which gives a focal length of f � 10:0 mm.We also choose the betatron phase advance per half-cellto be �=4; then La �

��2pf � 14:2 mm. The maximum

beta function value is then �F � 48:4 mm, the length ofthe unit cell is Lc � 2�La Lf� � 29:1 mm, and the be-tatron period is 4Lc � 117 mm. As the particle energyincreases, we consider these lengths to scale as 1=

��Kp/��

Ep

.

The lattice described in the previous paragraph is stableunder linear betatron motion. However, nonlinear trans-verse forces in the accelerating and focusing modes maycause instabilities which limit the dynamic aperture of thelattice and constrain the allowable emittance of the particlebeam. To determine the effect of these forces and computethe required emittance of a particle beam, we perform fullsimulations of particle trajectories. These simulations in-clude the dynamics of the particles in all six phase-spacecoordinates. They were performed by using the polynomialfits as analytic expressions for the fields, and propagatingparticle trajectories within each segment using an ordinarydifferential equation solver based on an explicit Runge-Kutta (4, 5) formula.

To compute the dynamic aperture of the lattice, wesimulate particles with initial positions uniformly distrib-uted throughout the waveguide aperture. The initial trans-verse momenta of the particles are taken to be 0, as are theinitial deviations of the particle energies from the idealcase. We first simulate particles which are exactly on crestinitially. We propagate the particles for 3 m through thelattice and record whether or not each particle collides withthe waveguide edge. The initial particle positions, colorcoded as to the particle’s survival, is shown in Fig. 9. In thatfigure, a red marker indicates that a particle initially at thatposition exited the waveguide aperture within 1 m ofpropagation. A green marker indicates that it remainedwithin the waveguide through 1 m, but exited before 3 m,while a blue marker indicates that the particle remained inthe waveguide through all 3 m of propagation. We see from

−4 −2 0 2 4

x (µm)

−4

−2

0

2

4y

(µm

)

−2.4

−1.8

−1.2

−0.6

0.0

0.6

1.2

1.8

2.4

FIG. 8. The accelerating field in the focusing structure modi-fied to suppress the octupole moment. As in Fig. 4, the quantityplotted here is the accelerating field seen by a speed-of-lightparticle, averaged over a photonic crystal lattice period.

−0.6 −0.4 −0.2 0.0 0.2 0.4 0.6

x (µm)

−0.6

−0.4

−0.2

0.0

0.2

0.4

0.6

y(µ

m)

Lost by 1 mSurvived after 1 m, lost by 3 mSurvived after 3 m

FIG. 9. Initial particle positions, with colors indicatingwhether, and when, each particle exited the waveguide aperture.The bounds of the plot correspond to the physical aperture of thestructure.

1For the remainder of this section, we refer to the F0D0 latticeas simply the ‘‘lattice,’’ not to be confused with the photoniccrystal lattice.


011301-6

the plot that most particles that are driven out of thewaveguide exited within the first meter. The figure alsoindicates that the dynamic aperture extends to almost theentire physical aperture of the structure (the dynamic ap-erture being wider in x than y because the initial longitu-dinal particle positions are at the high-�x point of thelattice). The initial energy of the particles is 1 GeV, andafter 3 m the energy of the ideal particle is 1.87 GeV. Theellipse indicated in the figure is the ellipse of largest areawhich still contains only particles which remain in thewaveguide. If we take this ellipse to be the 3� boundaryof the particle distribution, we can use the relation �2 ��" to obtain the emittance requirement of the lattice. Wefind normalized emittances of

"�n�x � 9:2� 10�10 m; "�n�y � 1:09� 10�9 m: (4)

While these emittances are several orders of magnitudesmaller than those produced by conventional rf injectors,because of the small bunch charge as discussed in Sec. III,

−0.6 −0.4 −0.2 0.0 0.2 0.4 0.6

s (µm)

−0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6A

vera

ged

Ez

Ez(s, 0)Ez(0, s)Ez(s, s)

FIG. 10. Lineouts of average accelerating field seen by aspeed-of-light particle, normalized to the accelerating field onaxis. The fields are shown along the x axis (blue), the y axis(green), and the x � y diagonal (red).

FIG. 11. Initial particle positions, with colors indicating whether, and when, each particle exited the waveguide aperture, for initialphases ranging from �90 to 90 mrad.


011301-7

the transverse brightness (charge per phase-space area)required in the optical structure is similar to that of con-ventional accelerators.

Another consequence of suppressing the quadrupolefields in the structure is low induced energy spread dueto nonuniformity of the accelerating field. Figure 10 showsthe average accelerating field seen by a speed-of-lightparticle—the same quantity plotted in Fig. 4—along thex axis, the y axis, and the x � y diagonal. We see a regionof uniform field near the center of the waveguide, consis-tent with an r4 dependence. While the field decreases alongthe diagonal, few particles within the dynamic aperture canbe significantly displaced along the diagonal, since �x and�y are not simultaneously large. The uniformity of theaccelerating field is borne out directly in the simulation:If we weight the particles within the dynamic aperture sothat their initial transverse spatial distribution is Gaussianwith the 3� ellipse shown in Fig. 9, we find that theyacquire a relative energy spread of only 7� 10�4 after3 m of propagation.

We can also examine the beam dynamics for particleswhich are slightly out of phase with the crest of the accel-erating laser field. Running the same simulation as above,but with the particles given an initial phase offset, we findthat the dynamic aperture is reduced as particles becomefurther out of phase. This is shown in Fig. 11. For particlesahead in phase (< 0), we can see in the plots a patterncharacteristic of a fourth-order resonance, indicating thatthe octupole field in the accelerating structure becomessignificant as particles become off crest. The difference inaperture shape between positive and negative phase shiftscould be attributed to the sign difference of that octupolefield.

With simulations of the dynamic aperture of the latticefor both on- and off-crest particles, we can now computethe dynamics of a realistic particle bunch, with variation inall six phase-space coordinates. We take the initial normal-ized transverse emittances to be those computed above foron-crest particles and given in Eq. (4). We also take the rmsphase spread and relative beam energy deviation to be� � 10 mrad and �� 10�3, respectively. We trackthe ensemble of particles for 3 m of propagation, as before,and record the phase-space coordinates every 10 cm. Wefind that 261 of the 13 228 particles simulated, or 2.0%,exited the waveguide before completing the 3 m propaga-tion. From the recorded phase-space coordinates we cantrack the normalized emittances as a function of propa-gated distance. The transverse emittances are shown inFig. 12. We see that the emittances increase initially, butthat this increase slows as the propagation continues. Forthe full 3 m, "�n�x increases by 10.1%, while "�n�y grows by7.3%. Thus, the growth in invariant emittance is muchsmaller than the energy gain, so that the geometric emit-tance of the beam will still be adiabatically damped. Thisshows that particle bunches can be propagated stably in a

photonic crystal accelerator without an extraordinary en-hancement to beam brightness.

V. CONCLUSION

We have found numerically an accelerating mode in athree-dimensional dielectric photonic crystal structure.This mode propagates with speed-of-light phase velocityand is confined in both transverse directions with low loss,and it is therefore suitable for acceleration of relativisticparticles. Based on breakdown measurements for bulkdielectric, we estimate that the structure can sustain un-loaded gradients in excess of 300 MV=m. We computedthat the structure can operate with high optical-to-beamefficiency using the mechanism of recycling laser energy inan optical cavity. However, the energy spread among opti-cal bunches in a train remains a concern; some a mecha-nism must be found to counter this effect or such anaccelerator would need to operate with less than optimumcharge.

We also showed a novel method for beam confinementin a small aperture which uses optical fields in a modifiedstructure to focus the particle beam. We found that thismechanism only required brightness comparable to con-ventional accelerators, and resulted in minimal emittancegrowth even for significant propagation distance.

While we have described the central features of a pho-tonic accelerator here, many issues remain to be addressedin order to develop a practical accelerator design. Onecritical issue is a greater understanding of the structurebreakdown threshold. While the studies in [13] on bulksilicon can provide an estimate for sustainable gradient, theeffects of the geometric structure on the breakdown thresh-old must be investigated. Ultimately, the sustainable gra-dient in the structure will have to be determined directly byexperimentation on manufactured prototypes.

FIG. 12. Transverse emittances of the particle bunch as afunction of propagated distance.


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Another issue to be addressed is the presence of higher-order modes in the waveguide. As remarked in theIntroduction, photonic crystals have been attractive asaccelerator structures because of their potential for elimi-nating a major source of beam breakup instability. This isbecause a photonic crystal waveguides only confinesmodes with frequencies in the band gap of the lattice.Early indications from simulation indicate that the wave-guide described here supports only a small number (lessthan 10) of higher-order modes. However, this issue needsto be examined in more detail to determine the propertiesof each individual mode and their effects on acceleratorperformance.

Finally, while fabrication of these structures at opticallength scales is likely to be challenging, much work onfabricating the woodpile lattice is underway in the pho-tonics community. Woodpile lattices with a small numberof layers have been constructed using a layer-by-layertechnique [20] and via wafer fusion [21]. Other experi-mental techniques, such as interferometric alignment [22]and stacking by micromanipulation [23], hold promise forefficient, cost-effective fabrication. Preliminary studies[24] on the fabrication tolerance of the accelerator struc-tures discussed here indicated an rms layer-to-layer hori-zontal misalignment tolerance of 25–30 nm. This figurewas based on the criterion that the wave number error besmall enough that the accumulated phase error is less than� over a distance of 100 �m (the distance in which thelaser pulse envelope slips by roughly 1 ps with respect tothe particle beam). Such tolerances are encouraging, sincecurrent semiconductor processing technology can meetthat requirement [25]. However, much more detailed stud-ies are necessary to determine the effects of fabricationerror on other accelerator properties, including loss, effi-ciency, and beam confinement.

ACKNOWLEDGMENTS

The author would like to thank R. Siemann, S. Fan, andA. Chao for helpful advice. This work was supported byDepartment of Energy Contracts No. DE-AC02-76SF00515 (SLAC) and No. DE-FG03-97ER41043-II(LEAP).

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[2] C. D. Barnes, E. R. Colby, and T. Plettner, in AdvancedAccelerator Concepts: Tenth Workshop, Mandalay Beach,CA, 2002, edited by C. E. Clayton and P. Muggli(American Institute of Physics, Melville, NY, 2002),p. 294.

[3] C. M. S. Sears, E. R. Colby, R. Ischebeck, C. M.McGuinness, R. H. Siemann, J. E. Spencer, D. Walz,R. L. Byer, and T. Plettner, in Proceedings of the 2007Particle Accelerator Conference, Albuquerque, NM, USA,2007, edited by C. Petit-Jean-Genaz (IEEE, Piscataway,NJ, 2007), pp. 1894–1898.

[4] J. D. Joannopoulos, R. D. Meade, and J. N. Winn, PhotonicCrystals: Molding the Flow of Light (Princeton UniversityPress, Princeton, N.J., 1995).

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[12] A. Taflove and S. C. Hagness, ComputationalElectrodynamics: The Finite-Difference Time-DomainMethod (Artech House, Boston, 2000), 2nd ed.

[13] B. Cowan, in Twelfth Advanced Accelerator Concepts,Lake Geneva, WI, 2006, edited by M. Conde andC. Eyberger (American Institute of Physics, Melville,NY, 2006), pp. 837–843.

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[19] S. Y. Lee, Accelerator Physics (World Scientific,Singapore, 1999), pp. 48–50.

[20] J. G. Fleming and S. Y. Lin, Opt. Lett. 24, 49 (1999).[21] S. Noda, K. Tomoda, N. Yamamoto, and A. Chutinan,

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[24] B. Cowan, in Twelfth Advanced Accelerator Concepts,Lake Geneva, WI, 2006 (Ref. [13]), pp. 844–850.

[25] International technology road map for semiconductors(2005).


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