Extinction of coherent backscattering by a disordered photonic crystal with a Dirac spectrum

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by  R. A. Sepkhanov

Extinction of coherent backscattering by a disordered photonic crystal with a Dirac
spectrum
R. A. Sepkhanov,1 A. Ossipov,2 and C. W. J. Beenakker1
1Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands
2School of Mathematical Sciences, University of Nottingham,
University Park, Nottingham, NG7 2RD, United Kingdom
(Dated: October2008)
Photoniccrystalswithatwo-dimensionaltriangularlatticehaveaconicalsingularityinthespec-
9
trum. Close to this so-called Dirac point, Maxwell’s equations reduce to the Dirac equation for an
0
ultrarelativistic spin-1/2particle. Hereweshowthatthehalf-integer spinandtheassociated Berry
0
2 phase remain observable in the presence of disorder in the crystal. While constructive interference
ofascalar(spin-zero)waveproducesacoherentbackscatteringpeak,consistingofadoublingofthe
n disorder-averagedreflectedphotonflux,thedestructiveinterferencecausedbytheBerryphasesup-
a
presses thereflectedintensityatananglewhichisrelatedtotheangleofincidencebytime-reversal
J
symmetry. We demonstrate this extinction of coherent backscattering by a numerical solution of
7 Maxwell’s equations and compare with analytical predictions from the Dirac equation.

]
PACSnumbers: 42.25.Dd,42.25.Hz,42.70.Qs,03.65.Vf
l
l
a
h
Coherent backscattering is a rare example of an opti- tial and final wave vectors k and k ). This absence of
- i f
s cal interference effect that is systematically constructive backscattering can be understood either in terms of the
e
in a random medium [1]. A reciprocal pair of waves (re- Berry phase difference of π between time reversed scat-
m
lated by time reversal symmetry) arrive in phase at the tering sequences [10] or in terms of the antisymmetry
t. observer regardless of the path length or scattering se- S(ki kf)= S( kf ki) of the scattering matrix
a → − − →−
quence. By measuring the angular profile of the diffu- of the Dirac equation [11]. Absence of backscattering
m
sively reflected intensity for plane wave illumination and in graphene might be measurable if scanning probe mi-
- averagingovertherandomscattering,apeakisobserved croscopy can provide the required angular resolution of
d
n ataspecificreflectionangle[2,3,4]. Underoptimalcon- the electron flow [12].
o ditions(samepolarizationofincidentandreflectedwave, Hereweinvestigateanalternativerealizationoftheex-
c transportmeanfreepathl muchlargerthanwavelength tinctionofcoherentbackscatteringthatreliesonphotons
[
λ), the average peak and background intensity have ra- rather than ultrarelativistic fermions. The half-integer

2 tio =2 [5, 6]. This factor of two enhancement follows spinrequiredfortheBerryphaseofπ isproducedbythe
v direRctly by comparing the coherent addition of intensi- Dirac-typebandstructureofatriangular-latticephotonic
4 ties (first sum the two wave amplitudes, then square to crystal [13, 14]. By using photons rather than electrons
2 obtain the total intensity) with the incoherent addition thedifficultyofangularresolveddetectionisavoided. We
1
(first square, then sum). first discuss the effect at the level of the Dirac equation,
0
Because coherent backscattering does not depend on and then test the predictions with a numerical solution
.
0 the random scattering phase shifts, it is a sensitive tool of the full Maxwell’s equations.
1
toprobesystematicphaseshiftsthatcontaininformation Weconsideraphotoniccrystalwithatwo-dimensional
8
about internal degrees of freedom of the scatterers and (2D) triangular lattice structure. The hexagonal first
0
: the photons. One example in light scattering from cold Brillouin zone is shown in Fig. 1. Haldane and Raghu
v atoms is the coupling of the photon polarization to the [13] showed that a pair of almost degenerate envelope
i
X collective spinof the atomic ensemble, whichcan change Bloch waves (Ψ1,Ψ2) Ψ near a corner of the Brillouin
≡
the constructive into a destructive interference [7]. This zone can be representedby a pseudospin, coupled to the
r
a effect is similar to the change from =2 to =1/2 in orbital motion. On length scales large compared to the
R R
electronic systems with strong spin-orbit coupling [8]. latticeconstantaandforfrequenciesnearthedegeneracy
Afewyearsago,Bliokh[9]discussedanaltogetherdif- frequency ωD, the wave equation reduces to
ferentandmoredramaticswitchfromconstructivetode- ∂ ∂ ω ω
D
structive interference: ultrarelativistic fermions (with an iσx iσy Ψ= − Ψ, (1)
(cid:18)− ∂x − ∂y(cid:19) v
energy much greater than the rest energy) would have a D
completeextinctionofcoherentbackscattering,so =0, withPaulimatricesσx,σy. Thisisthe2DDiracequation
as a consequence of the Berry phase of π accumRulated ofaspin-21 particlewithzeromassandgroupvelocityvD
along a closed trajectory by a half-integer spin pointing (of order aωD). The dispersion relation
in the direction of motion. Indeed, it was previously no- (ω ω )2 =v2(k2+k2) (2)
ticed by Ando et al. in the context of graphene [10] that − D D x y
thescatteringamplitudes(φ)formasslessDiracfermions has a double cone with a degeneracy at frequency ω
D
vanishes when φ π (with φ the angle between the ini- (the so-called Dirac point).
→
2
FIG. 1: Hexagonal first Brillouin zone of a two-dimensional
FIG. 2: A pair of reciprocal waves that interfere destruc-
triangular lattice photonic crystal, with equifrequency con-
tively, resulting in the extinction of coherent backscattering.
tours centered at the corners. The three solid circles are re-
Arrowsattheair-crystalinterfaceindicatetheincidentplane
lated by translation over a reciprocal lattice vector, so they
wave(redsolidlines),thediffusivelyreflectedwave(bluesolid
areequivalent,whilescatteringfromasolidtoadottedcircle
lines),andthespecularlyreflectedwave(blackdashedlines).
issuppressedifthescatteringpotentialissmoothonthescale Thetwoinitialwaves(wavevectorkiair,refractedtoki)follow
of the lattice constant a. The large dashed circle at the cen-
time-reversedsequencesofscatteringevents(darkcircles), to
ter is the equifrequency contour in air, included to indicate end up in two final waves (wave vector k , refracted to kair)
f f
therefractionattheair-crystalinterface(asexpressedbyEq. withaphasedifferenceofπ+(ki+kf)·δr. Forkf =−kionly
(4)).
the Berry phase difference of π remains. As a consequence
of this destructive interference, the intensity of the reflected
wave indicated in blue is suppressed. By measuring the an-
TheBerryphaseassociatedwiththepseudospindegree
gular profile of the average reflected intensity a minimum of
of freedom was calculated in Ref. [14]. The solution of nearly zero intensity will result at thisangle.
Eq. (1) with a definite wave vector k=(k ,k ) is
x y
Ψ= −1/2 (ω−ωD)/vD cos(θ/2) , (3) isspecularlyreflectedatanangleθspec =π θ. Theinter-
C (cid:18) kx+iky (cid:19)≡(cid:18)eiφsin(θ/2)(cid:19) face reflectivity R (the fraction of the in−cident photon
int
fluxthatisspecularlyreflected)followsfromthetransfer
with a normalization constant. The angles φ,θ define
C matrix of the air-crystalinterface calculatedin Ref. [15].
the Bloch vector (cosφsinθ,sinφsinθ,cosθ), represent-
In the approximationof maximal coupling we find
ing the direction of the pseudospin on the Bloch sphere.
Because of the dispersion relation(2) the angle θ =π/2,
1 1 (k /k)2
so the Bloch vector lies in the x y plane, pointing in R (k )= − − y , (6)
the direction of k. The Berry ph−ase φB is one half the int y 1+p1−(ky/k)2
solidanglesubtendedatthe originbythe rotatingBloch p
vector. A rotation of k by 360◦ in the x y plane thus hencetheinterfacereflectivityiszeroforky =0. Numer-
produces a Berry phase φ =π. − ical solutions of Maxwell’s equations [15] give Rint(0)
B ≃
Becauseofrefractionattheinterfacesx=0andx=L 0.05 for ω near ωD, so this is a reasonably accurate ap-
between the photonic crystal and air, we need to distin- proximation.
guish the initial and final wave vectors k , k of enve- Disorder inside the photonic crystal produces a back-
i f
lope Bloch waves inside the crystal (velocity v ) from groundof diffusively reflected waves in an angular open-
D
the correspondingvalueskair,kair ofplanewavesoutside ing δθ k/kair 1 around θ0 = π arcsin(2πc/3ωa).
i f ≃ ≪ −
(velocity c). For the crystallographic orientation shown The reciprocity angle θ∗ for coherent backscattering is
related to the incident angle θ of Eq. (5) by
inFig. 1,thewavevectorsbeforeandafterrefraction(at
a given frequency ω) are related by [15]
θ∗ =π arcsin(2πc/3ωa cki,y/ω). (7)
− −
kair =k +2π/3a, (4a)
y y
We will choose k small ( k) but nonzero, so that
c2(kair)2 ω2 =0=v2k2 (ω ω )2, (4b) i,y ≪
− D − − D Rint 1 while still coherent backscattering at angle θ∗
≪
can be resolved from specular reflection at angle θ .
with a the lattice constant and ω the frequency of the spec
D
Dirac point. We denote kair = kair and k = k. The extinction of coherent backscattering by destruc-
| | | | tive interference of reciprocal waves is illustrated in Fig.
Beforeconsideringthediffusereflectionfromthedisor-
2. The two series of time reversed scattering events
deredphotoniccrystal,weaddressthespecularreflection
fromtheair-crystalinterfacethatispresentevenwithout S+ = ki k1 k2 kn−1 kn ki and
any disorder. A plane wave incident at an angle S− =ki → kn→ k→n−·1·· → →k2 →k1− ki
→− →− →···→− →− →−
have the same scattering amplitude up to a Berry phase
θ =arcsin(ckair/ω)=arcsin(2πc/3ωa+ck /ω) (5) difference φ = π [9, 10]. This destructive interference
i,y i,y B
3
suppresses the reflected intensity at angle θ∗. If the final Thesuppression(12)ofthereflectedintensitywithina
wavevectork deviatesfromtheexactbackscatteringdi- narrowangularopeningδθ 1/klaroundthereciprocity
f
rection ki,thephasedifference∆φ=φB+(ki+kf) δr angleθ∗ iscompensatedby≃anexcessintensityδR 1/kl
depends−on the separation δr of the first and last sc·at- of the diffusively reflected wave at angles away fr≃om θ∗.
teringevents [5]. The Berryphase difference φ remains This compensation is required by current conservation
B
equal to π, because both the Berry phases accumulated [17],butdifficulttoobserveforkl 1. Wewilltherefore
≫
along S+ and S− are incremented by the same amount not consider it in what follows.
(half the angle between k and k ). We now compare the analytical predictions from the
f i
−
By including the Berry phase in the theory [1, 5, 6] Dirac equation with a numerical solution of Maxwell’s
of coherent backscattering of scalar waves in the weak equations. As in earlier work[15], we use the meep soft-
scattering regime (l λ), it follows that the incoherent ware package [18] to solve Maxwell’s equations in the
≫
partR ofthereflectivityRremainsunaffectedwhilethe time domain [19] for a continuous plane wave source
0
interferencepartδR acquiresanoverallfactorcosφ [9]: (timedependence eiωt)switchedongradually. Wecal-
B
∝
culate the reflected intensity, projected onto transverse
R=R0+δRcosφB. (8) modes,asafunctionoftimeandtakethelargetimelimit
toobtainthestationaryreflectivityR. Wetookthe time
The entire angular profile of coherent backscattering in
sufficiently large that the sum of the total transmission
the Dirac equation (where φ = π) may therefore be
B and total reflection differs from unity by less than 0.03.
obtained, for l λ, by simply changing the sign of the
The photonic crystal consists of a triangular lattice
≫
known results for δR for scalar waves (where φ = 0).
B of parallel dielectric rods (dielectric constant 8.9, radius
Inparticular,fork k andδk =k +k 1/lone
i,y f,y i,y ρ = 0.3a) in air. The orientation of the lattice is as
≪ ≪ 0
has
shown in Fig. 2. The magnetic field is taken parallel
to the rods (TE polarization). The conical singularity
1 l+z
0
R0 = πN(1+z0/l)(cid:18)1− L+2z (cid:19), (9) in the band structure is at frequency ωD = 3.03c/a,
0 with a slope dω/dk v = 0.432c. We discretize
D
1 ≡
δR= (1+z /l) the transverse wave vector by means of periodic bound-
0
πN ary conditions: k = 2πn/W, n = 0, 1, 2,..., with
y
± ±
1 (l+z0)δkcoth[δk(L+2z0)] . (10) W = 1200a. The longitudinal dimension of the lattice
× −
(cid:8) (cid:9) is taken at L = 60√3a (corresponding to 121 rows of
The normalization factor N = kW/2π is chosen such dielectric rods). The frequency ω of the incident plane
thatRisthefractionoftheincidentphotonfluxwhichis waveischosensuchthatω ω =0.206c/aissufficiently
D
reflectedinasingletransversemodewhenkf,y =2πn/W large that N = 88 1, b−ut sufficiently small that the
(n = 0, 1, 2,...) is discretized by periodic boundary trigonaldistortiono≫fthe circularequifrequencycontours
± ±
conditions at y = 0 and y = W. (This normalization is insignificant.
is chosen to simplify the comparison with the numerical Disorderisintroducedbyrandomlyvaryingthe radius
calculations described later on.) ρ(r) of the dielectric rods at position r, according to
Eqs. (9) and (10) are approximate results from the ρ(r) = ρ +δρ(r). The variation δρ(r) is spatially cor-
0
radiativetransferequation,accurateforadisorderedslab related over a length ξ in a Gaussian manner,
ofthicknessLnotmuchsmallerthanthetransportmean
freepathl. Theparameterz ,theso-calledextrapolation N
0
length, depends on the reflectivity R of the interface δρ(r)= A exp( r r 2/2ξ2), (13)
int i i
−| − |
between the photonic crystal and vacuum, according to Xi=1
[16]
where r (i = 1,2,... ) is a randomly chosen point in
i
N
z0 = 14πl 11+−CC12, C1 = k1 Z0kRint(ky)dky, (11a) ttξhh=ee i2cnraty.esrItvtaalilsa(en−sds∆enA,t∆iia)il.sthaWatreattnhodeookcmoNrlyrel=cahtoi3os2en0n0le,anm∆gtph=liξtu0od.f0e4thiane,
C = 4 k 1 (k /k)2R (k )dk . (11b) disorderis largerthanthe lattice constanta to minimize
2 y int y y
π Z0 q − scattering between the two inequivalent corners of the
Brillouin zone (solid and dashed circles in Fig. 1). For
SubstitutionofEq.(6)givesthesimpleanswerz =l for
0 these disorder parameters we found that only about 2%
the extrapolation length of the air-crystal interface.
of the incident flux is reflected into the opposite corner.
Collectingresults,wearriveatthefollowinglineshape
The numerical data in Fig. 3 is for the incident wave
of the reflectivity near the reciprocity angle:
vector k = 32π/W, averaged over 80 realizations of
i,y
−
the disorder. The angle of incidence (measured relative
4 l
R= lδkcoth[δk(L+2l)] to the positive x-axis) is θ = 0.67rad = 38.4◦. The
πN (cid:18) − L+2l(cid:19)
specularly reflected wave at an angle θ = π θ =
spec
→ 3π4N(Ll+2l2)δk2 for δk →0. (12) 2T.h4e7rraedcip=ro1c4i1ty.5◦anhgalset(r7a)nissveθr∗se=m2o.4d0ernaudm=be1r37n.5=−◦,−c1o6r-.
4
responding to the mode number n=+16.
Boththespecularlyreflectedwaveatn= 16andthe
−
extinctionofthe coherentbackscatteringatn=+16are
clearlyvisibleinFig.3. Theextinctionatthereciprocity
angle is not complete, presumably because of scattering
between inequivalent corners of the Brillouin zone. The
analyticaltheorypredictsaparaboliclineshapeofthere-
flectivity around the reciprocity angle, given by Eq. (7),
the width of which depends on the value of the trans-
port mean free path l. A fit to the numerical data gives
l = 10a (dotted curve in Fig. 3), which then implies
a background of incoherent diffuse reflection at a value
R 0.006whichis somewhatlargerthanthe numerical
0
≈
data (dashed horizontal line).
In conclusion, we have shown that coherent backscat-
FIG. 3: Plot of the reflectivity R (fraction of incident flux tering of radiation from a disordered triangular lattice
reflected in a single transverse mode) as a function of the photonic crystal is extinguished at an angle that is re-
transverse modeindex n(related tothetransverse wavevec- ciprocal to the angle of incidence. This effect has an
tor by k = 2πn/W). The specularly reflected flux is at
f,y analogue for electrons [9, 10], but it should be easier to
n = −16 and the extinction of coherent backscattering is at
observe for photons because angular resolved detection
n = +16 (indicated by arrows). The curve is calculated nu-
is more feasible. The observationof the extinction of co-
merically from Maxwell’s equations. The dashed and dotted
herentbackscatteringwouldbe a strikingdemonstration
lines are, respectively, theanalytical predictions (9) and (12)
for the incoherent background and the line shape near the ofaspin-1/2Berryphase inadisorderedopticalsystem.
extinction angle, for a transport mean free path of l=10a. We acknowledgediscussions with J. H. Bardarsonand
C.W.Groth. This researchwassupportedbythe Dutch
Science Foundation NWO/FOM.
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