Brillouin Scattering Self-Cancellation

1.7 MB·

by  Omar Florez

Brillouin Scattering Self-Cancellation
O. Florez,1 P.F. Jarschel,1 Y.A.V. Espinel,1 C.M.B. Cordeiro,1
T.P. Mayer Alegre,1 G.S. Wiederhecker,1 and P. Dainese1,∗
1Gleb Wataghin Physics Institute, University of Campinas, 13083-970 Campinas, SP, Brazil
(Dated: January 21, 2016)
Theinteractionbetweenlightandacousticphononsisstronglymodifiedinsub-wavelengthconfine-
ment, and has led to the demonstration and control of Brillouin scattering in photonic structures
such as nano-scale optical waveguides and cavities. Besides the small optical mode volume, two
physical mechanisms come into play simultaneously: a volume effect caused by the strain induced
refractive index perturbation (known as photo-elasticity), and a surface effect caused by the shift
of the optical boundaries due to mechanical vibrations. As a result proper material and structure
6 engineering allows one to control each contribution individually. In this paper, we experimentally
1 demonstrate the perfect cancellation of Brillouin scattering by engineering a silica nanowire with
0 exactlyopposingphoto-elasticandmoving-boundaryeffects. Thisdemonstrationprovidesclearex-
2 perimentalevidencethattheinterplaybetweenthetwomechanismsisapromisingtooltoprecisely
control the photon-phonon interaction, enhancing or suppressing it.
n
a
J
I. INTRODUCTION mechanisms in the case of a cylindrical-wire waveguide
0
under a purely radial acoustic expansion.
2
Brillouin scattering arises from the interaction be-

] tween an electromagnetic wave and an acoustic wave.
cs An incident photon at frequency ω is scattered into (a) us
i an up- or down-shifted photon with frequencies ω ±Ω
t Photo-
p due to the absorption or creation of a phonon with fre- elastic Moving-
o
quency Ω, respectively. The ability to strongly confine boundary
.
s both optical and acoustic modes in photonic structures
c
has opened new opportunities to control their interac-
i
s tion [1, 2], and several new or enhanced applications
y have been explored, such as in photonic signal process- E0 ∆nmb
h
ing [1–7], demonstration and tailoring of slow light de-
p
vices[8–11],actuationandcontrolofphotonicstructures E
[ s
using optical forces [12–14], observation of Raman-like

1
scattering [15, 16], on-chip Brillouin scattering [17–19],
v generation of frequency combs [20, 21] and new sensing -2 -1 0 1 2 2x/d
8
applications [22]. This interaction has also been a plat- ∆n
4 pe
2 formforfundamentalphysicsexperimentssuchascoher- (b)
5 ent phonon generation [23], cooling mechanical modes Sz (W/µm2)
0 to ground state [24, 25], generation of squeezed light 3 0.7 12
1. states [26], and synchronization of micromechanical os- 00..35 10 m)
160 cniilslTmartaodbrsietti[ow2n7e]ae.lnly,thteheopmtiocsatl caonmdmaocnouisnttiecraficetlidosn imneclohwa-- E/Es02 0.1 68 1410 V/
v: contrastwaveguidesisthephoto-elasticeffect(pe-effect), 1 4 E (s
i a well-known volume-effect in which the acoustic strain- 2
X
fields perturb the material’s refractive index [28]. For
0 0
r example, this effect dominates Brillouin scattering in 0.5 1.0 1.5 2.0
a
standard optical fibers [29]. However, as the surface- Diameter (µm)
to-volume ratio becomes larger and the index contrast
higher, such as in nano-optical waveguides and cavities, FIG. 1. Illustration of the Photo-elastic and Moving-
the optical field experiences an additional surface effect Boundary effects: (a) refractive index perturbation due
due to the vibrating boundary [30–33], referred to as to the pe-effect (red) and due to the mb-effect (blue) in a
moving-boundary effect (or mb-effect). The schematic nanowire under a linear radial expansion, overlaid with the
in Figure 1a illustrates the interplay between these two electric field profile (d: wire diameter, x: horizontal coordi-
nate, and u : surface displacement); (b) surface field ampli-
s
tude E as a function of wire diameter (blue: absolute value
s
for 1 W optical power; red: amplitude relative to the center
∗ dainese@ifi.unicamp.br field E0). Inset: Poynting vector for 0.55 µm diameter.
2
We can simplistically characterize the strength of the trol their interaction, i.e. enhancing or suppressing it.
mb-andpe-effectsbyevaluatingtheirindexperturbation In [32], an analogous argument in terms of optical forces
depth·area products. As the wire boundary expands out (electrostriction and radiation pressure) has been theo-
by a small acoustic displacement u , the refractive in- retically explored for a variety of structures and materi-
s
dex in a small region near the surface abruptly increases als,demonstratingtherichnessofsuchinterplaybetween
by ∆n = n −n = 0.45 (where n =1.45 and the two mechanisms. In this paper, we demonstrate ex-
mb glass air glass
n = 1.0 are the refractive indices of the silica wire perimentally that we can achieve exact cancellation of
air
and air cladding, respectively). Therefore the strength backwardBrillouinscatteringbysimplyvaryingthewire
of the moving-boundary effect is: η = ∆n A = diameter. It is interesting to note that physically all
mb mb mb
(n −n )πdu , where A = πdu is the area in conditions favor strong photon-phonon interaction: (i)
glass air s mb s
which the mb-effect effect takes place (here d is the wire phase-matching condition is satisfied, (ii) both optical
diameter). Although ∆n is a relatively large pertur- andacousticfieldsarestronglyconfinedandhighlyover-
mb
bation, it affects only a small area (for thermally excited lappingspatiallyand,mostimportantly,(iii)individually
acoustic waves in the GHz frequency range, the bound- each interaction mechanism is strong. However, due to a
arydisplacementisoftheorderof10−16 m). Clearly,the precise control of the acoustic and optical mode profiles,
mb-effect is larger the higher the index contrast is. The the surface vibrations produces a dielectric perturbation
photo-elastic effect, on the other hand, induces a pertur- thatcancelsoutexactlytheperturbationcausedbyinter-
bation throughout the entire wire cross-section, and so nalbodystrainvibrations. WerefertothiseffectasBril-
A = πd2/4. Despite its tensorial nature, the order of louin scattering Self-Cancellation (BSC), since the can-
pe
magnitude of the pe-effect can be estimated by consid- cellationarisesfromthesameacousticmodethatcreates
ering only the radial strain component S (any other each effect individually. Experimentally demonstrating
rr
deformation in the longitudinal or azimuthal direction the BSC-effect validates our fundamental understanding
is ignored here, however a rigorous analysis is presented ofthephoton-phononinteractioninsub-wavelengthcon-
furtherinthisarticle). Assumingtheradialacousticdis- finementregimes,openingupthepossibilitytoselectively
placementincreaseslinearlywithr asu =u 2r/d(with suppress or further enhance the interaction by exploring
r s
rbeingtheradiuscoordinate),thestrainbecomessimply both effects simultaneously.
S =∂ u =2u /d. This then results in a index change The exact contribution from each mechanism can be
rr r r s
of∆npe =−n3glassp11uds [28]. Incontrasttothemb-effect, calculated using standard perturbation theory (see Sup-
a radial expansion leads to a reduction in the refractive plementary material for details). The strength of the
index (this is true since silica glass has a positive photo- photon-phonon coupling through photo-elastic effect is
elastic coefficient p = 0.121 [34]). The strength of the expressed mathematically as [34]:
11
peset-iemffaectet tishatthetnheηrpaeti=o η−pne/3gηlamssbp1=1π−duns3g.lassWp1i1th/4t∆hnatm,bw≈e κpe = ω8ε0 (cid:90) E∗s·∆(cid:15)∗pe·EpdA, (1)
−0.2. This order of magnitude estimate indicates that
both effects are indeed comparable in silica nanowires where∆(cid:15)pe =−n4p·Sistherelativepermittivitypertur-
and, moreover, can have opposite signs. bation caused by the acoustic strain tensor (S), n and p
are the material’s refractive index and photo-elastic ten-
Obviously, this simple analysis overlooks the crucial
sor, while E and E are the power-normalized electric
role of the optical fields. Not only the field strength of p s
field profiles for the pump and scattered waves. Physi-
bothincidentandscatteredwavesmustbenon-negligible
cally, the overlap integral simply represents a spatial av-
in the region where the index perturbation occurs, but
erage of dielectric perturbation weighted by the optical
also its spatial profile plays as a weighing function for
fields’ profiles. As discussed before, the same acoustic
each perturbation in a first-order spatial average. Fig-
wavethatcausesthepe-effect,alsocausesadisplacement
ure 1b shows the evolution of the field on the surface as
ontheboundariesthatdefinethewaveguidestructure(in
a function of wire diameter for an optical wavelength of
our case the wire boundary), and again using first order
1.55 µm (the inset shows the intensity profile for a 0.55
perturbation, the moving-boundary coupling coefficient
µmdiameter,clearlydistributedoveranarealargerthan
is given by [31]:
the wire itself). The surface field increases as the diam-
eter is reduced, reaches a maximum and then decreases κ = ω(cid:15)0 (cid:73) u (∆(cid:15) E∗ ·E +∆(cid:15)−1E∗ ·E )dl,
for very small diameter. This defines a region where the mb 8 ⊥ mb s,(cid:107) p,(cid:107) mb s,⊥ p,⊥
mb-effect might become relevant in the photon-phonon (2)
interaction, somewhere between 0.5 and 2.0 µm in di- where the perturbation coefficients are ∆(cid:15) = n2 −
mb glass
ameter. This behavior allows us to scan the nanowire n2 and ∆(cid:15)−1 =(n−2−n−2 )n4 , and u is the acous-
air mb air glass air ⊥
diameter until we find a point in which the optical field tic displacement normal to the wire surface. In Equa-
provides just the right weighing, so that the elasto-optic tion 2, the normal component of the electric field is eval-
and the moving-boundary effects cancel out exactly. uated in the outer region (air cladding) to correctly take
With these two effects simultaneously perturbing the into account the field discontinuity (this is equivalent to
dielectricconstant,preciseengineeringoftheopticaland the definition in [31] using the continuous normal dis-
acousticmodesinacertainstructurecanbeusedtocon- placement field). Physically this line integral represents
3
(a) TR21 -1m)1250 (c) R a yleig h limit -1/2m) 01 0.(5e) 1.0 1.5 2.0
µβ (a10 Phaseβ am=a2tβching: 2 -5(10--21 Diameter (µm)
5 1/L
1
2
κ-3
0
0 2 4 6 8 10 12
-4
Frequency (GHz)
moving-boundary (mb)
photo-elastic (pe)
10-4
Radial
(d) R pe + mb
(b) R displacement 01 8
01 10-5 (f)
1.0
TR 6
0.5 -1/2m)10-6 TR22 23 5-1/2m) 4
( -0
0 1/2 κL-10-64 6 8 10 12 1/2 (1L 02
-0.5 1
κ0 0.5 1.0 1.5 2.0
-10-5 R -2
-1.0 TR21 R 03 Diameter (µm)
-10-4 02
Frequency (GHz)
FIG.2. AcousticmodesandPerturbationstrength: nanowirecross-sectionunderdeformationduetothetwofundamental
Rayleighacousticmodes: axiallyasymmetrictorsional-radialTR modein(a)andaxiallysymmetricradialR modein(b).
21 01
Solidanddashedlinesrepresentrespectivelythedeformedandun-deformedparametriclines,andthecolorprofilerepresentsthe
normalizedradialacousticdisplacementamplitude;(c)dispersionrelationforbothfamilies(R andTR )forananowirewith
0m 2m
0.55 µm diameter. The horizontal continuous line represents the phase-matching condition for backward Brillouin scattering;
(d) coupling coefficients calculated for each phase-matched acoustic mode also for a nanowire with 0.55 µm diameter(green:
moving-boundarycontribution;orange: photo-elasticcontribution; andblack: totalpe +mb);(e)and(f)couplingcoefficients
as a function of the nanowire diameter. The coupling coefficients were calculated for acoustic modes normalized to thermal
energy (at 300 K), and the factor L1/2 scales the normalization to any waveguide length.
an average of the dielectric perturbation in an infinites- calpropagationconstants,respectively),andthecrossing
imal area along the waveguide perimeter, weighted by points with the dispersion curves determine the frequen-
the optical fields. To first-order, the overall interaction cies of the acoustic modes involved in the interaction.
strengthisdeterminedbysumofthecouplingcoefficients Due to such small wire diameter (from 0.5 and 2.0 µm),
κ=κ +κ andtheeffectofBrillouinSelf-Cancellation there are only a few acoustic modes per family, which
pe mb
is achieved when κ =−κ . leads to multi-peaked Brillouin spectrum [2].
mb pe
In order to evaluate the coupling coefficients in Equa- The fundamental modes in each families, R and
01
tions 1 and 2, one must obtain the acoustic mode pro- TR , are of particular interest because they have the
21
files. Each mode creates a different perturbation pro- largest acoustic displacement near the surface (thus en-
file, both on the surface and throughout the wire cross- hancing mb-effect). In the high frequency limit – when
section. The acoustic modes in a cylindrical rod geom- theacousticfrequencyismuchlargerthanthemodecut-
etry can be categorized in symmetry-based modal fami- off frequency – the phase velocity of these two modes
lies [35], and can be calculated analytically. The funda- (R and TR ) approaches a certain limit, so-called
01 21
mental optical mode interacts most efficiently with two Rayleigh speed (and for that these modes are referred
acoustic mode families: axially symmetric radial (R ) to as Rayleigh modes) [37, 38]. Brillouin scattering due
0m
modes and axially asymmetric torsional-radial (TR ) to such Rayleigh modes has been observed recently [39].
2m
modes[36]. Figures2aand2bshowthewirecross-section In this limit, the Rayleigh speed is lower than the bulk
underthedeformationcausedbythefundamentalmodes, longitudinal or transverse speeds, which in turn means
TR and R , respectively. The acoustic dispersion re- that the transverse wavevector becomes imaginary and
21 01
lation calculated for a silica wire with 0.55 µm diameter the acoustic field profile decays exponentially from the
is shown in Figure 2c. The horizontal line represents the surface inwards [37]. In other words, in the limit of high
phasematchingconditionforbackwardBrillouinscatter- frequency, these modes are pure surface waves. Even be-
ing β = 2β (where β and β are the acoustic and opti- lowtheRayleighlimit,alargesurfacedisplacementisex-
a a
4
pected, as can be seen from the profile in Figures 2a and 5
2b. The calculated coupling coefficients for this 0.55 µm (a) Experiment
wire diameter are shown in Figure 2d, in which the indi- )4 Theory
vidual contribution from each effect is shown separately u.
a.
feoffrecmtocdaensnuopttboe1n2egGleHczte.dCalenadr,lyi,ntfhaects,hiitftdinogm-binoautnedsafroyr sity (3 3 µm
some modes. Specifically, Figures 2e and 2f show the n2
e
calculated mb- and pe-coupling coefficients as a function nt
of the wire diameter for the two Rayleigh modes (R01 I1
and TR ). The pe-effect dominates the interaction for
21
theTR21 modethroughouttheentirediameterrangeex- 0
plored here. There is no diameter in which mb-effect 5 6 7 8 9 10 11 12
Frequency (GHz)
compensates the pe-effect, and therefore we do not ex-
pecttoobserveself-cancellationeffectforTR . Thiscan
21 Nanowire
be easily understood based on the azimuthal cos2φ de- 5 (b)
SMF SMF
pendence,whichmeansthatthewiresurfaceisdeformed
4 input ouput
toasomewhatellipticalshape. Thesignofthedielectric )
u. R
pplearcteumrbeanttiosnignduaelotnogstuhrefawceireshpifetrifmolelotewrs(it.hee. praodsiitailvediisn- y (a.3 R01 02 Transitions
the regions the wire expands out and negative where it sit
n2
is contracted). As a result, the average mb-perturbation e
(i.e. the line integral in κmb) is reduced. For the R01 Int1 TR21 R03
mode, on the other hand, the mb- and pe-effects are of
thesamemagnitudeandhaveoppositesigns,asoursim-
0
ple analysis indicated. Clearly, at a diameter equals to 5 6 7 8 9 10 11 12
1.1 µm, they exactly cancel out and no backward Bril- Frequency (GHz)
louin scattering should be observed for this mode.
FIG.3. TheBrillouinspectrum: (a)experimental(green)
and theoretical (black) Brillouin backscattering spectra for
a sample with a diameter of 0.55 µm. The shaded region
II. EXPERIMENTAL RESULTS
represents the scattering peak due to the single-mode fiber
(SMF)pigtails;(b)contributionstothetotalspectrumarising
Samples of silica nanowires were fabricated by heating from the transition regions (red) and from the actual center
andstretchingstandardtelecommunicationsfiber[40]. A nanowire (blue). The theoretical spectrum in (a) is the sum
geometric schematic is shown in the inset of Figure 3b. of these contributions. Insets: scanning electron microscope
Twotransitionregionsconnecttheactualnanowire(cen- image of a nanowire and schematic of the sample structure,
tral region) to single-mode fibers at both input and out- indicating the SMF input/output pigtails, the transition re-
gions and the center nanowire.
put ends. All nanowires are approximately 8 cm long
and have transition regions of approximately 5 cm on
each side. We report results on samples with diameter
ranging from ∼0.51 to 1.34 µm. The precise wire diame- showthecalculatedcontributionsfromthenanowirewith
terwasdeterminedusinganon-destructiveexperimental uniformdiameter(inblue)andfromthetransitionregion
technique based on forward Brillouin scattering. Both with varying diameter along its length[40] (in red). The
experimental setups for forward and backward Brillouin transition region leads to broad scattering bands, which
scattering characterization are described in the Supple- arises because the phase-matching frequency sweeps the
mentary materials. acousticdispersioncurveasthediametervaries. Thethe-
The spectrum obtained for a sample with 0.55 µm di- oreticalcurveinFigure3aisthesumofthesetwocurves
ameter is shown in Figure 3a, along with the theoretical from Figure 3b. The theoretical spectrum explains the
spectrumcalculatedforthesamenanowirediameter(the oneobservedexperimentallywithremarkableagreement,
theoretical linewidth were obtained directly from the ex- and no fitting parameter was required (except for a 1%
perimentalspectrum). Thefirstpointtohighlightisthat adjustment on the bulk acoustic velocities to match the
several peaks in the experimental spectrum do not cor- calculated and measured frequencies).
respond to the frequencies of acoustic modes calculated TodemonstrateBrillouinSelf-Cancellationinoursilica
for a wire with the same diameter (which were shown in nanowires, we show in Figure 4a the calculated Brillouin
Figure 2c). These measured additional peaks can be ex- spectrum evolution as a function of the nanowire diam-
plained by considering the single-mode fiber pigtail and eter for the two fundamental acoustic modes (R and
01
thetransitionregions. Thesingle-modefiberattheinput TR ). Slightly above their accidental frequency cross-
21
givesorigintothepeakat10.8-10.9GHz(shadedingray ing (at d = 0.9 µm), the BSC effect of the R mode
01
as this is not the focus of this paper). In Figure 3b, we is predicted for a roughly 50 nm diameter range around
5
1.6 (a) s/PpL (m1-10H-1z6-1) 5 dB (b) 1.34 µm 1.6 (c)
0.
R 10-17 )1.4
01 m
10-18 B µ
1.4 TR21 10-19 0.5 dB0.5 d 11..2279 µ µmm Diameter (011...802 RTR0121
●
1.2 B
1 d 1.16 µm 0.6
)
m
µr ( 2 dB 1.06 µm 0.45.0 5.5 6.0 6.5 7.0
ete1.0 5 Frequency (GHz)
am 3 dB 0.91 µm
Di 4 (d)
B
d
0.8 5 0.86 µm
13
2
B P
5 d 0.71 µm P/01
2
0.6 B
1 d 0.55 µm 1
B
d 0.51 µm
0.4 1 0
5.0 5.5 6.0 6.5 7.0 5.0 5.5 6.0 6.5 7.0 0.4 0.6 0.8 1.0 1.2 1.4 1.6
Frequency (GHz) Frequency (GHz) Diameter (µm)
FIG. 4. Observation of the Brillouin Self-Cancellation effect: (a) theoretical evolution of the Brillouin backscattering
spectrum as a function of the nanowire diameter for both Rayleigh modes R and TR . The color-scale represents the
01 21
scattered power spectral density (normalized by the pump power and wire length); (b) series of experimental spectra for
nanowires with different diameters(as indicated ineachspectrum). The peaks due tothe R or theTR are coloredin blue
01 21
and red, respectively; (c) measured central frequencies for both R and TR (dotted blue and red points) compared with
01 21
the expected theoretical frequencies (solid curves) as a function of diameter; (d) ratio between the R and the TR total
01 21
scattered power for all samples, and the theoretical result (solid curve) given by κ2 /κ2 .
01 21
1.1µm. Thisbehaviorispreciselyconfirmedintheseries normalization eliminates the uncertainty with respect to
of measured spectra, shown in Figure 4b. Each spec- parameters that directly influence the absolute scattered
trum was obtained for a nanowire with a distinct di- power such as slightly differences in wire length, differ-
ameter, as indicated. Each spectrum generally shows ences in the transition region optical attenuation, exact
two well-defined peaks (except for the smallest diame- power at the wire input and the wire attenuation itself.
ters that more clearly shows a spectral band due to the The solid curve is the theoretical result, which is simply
transition regions). The two peaks correspond to the given by κ2 /κ2 . A remarkable agreement is observed
01 21
Rayleigh modes R and TR (colored in blue and red, and clearly the R mode intensity approaches the noise
01 21 01
respectively). Their central frequencies were measured limit at 1.1 µm, confirming the BSC effect.
and compared with the theoretically prediction in Fig-
ure 4c, with quite good agreement. The particular fre-
quencyversusdiameterevolutionisdiscussedindetailin III. DISCUSSION
theSupplementarymaterial. TheTR peakisobserved
21
in all experimental spectra, in agreement with the the-
Tounderstandthecancellationeffectindetail,itises-
ory, since there is no BSC expected for this mode. In
sential to understand the acoustic profile and associated
contrast, the R is observed for small and large diame-
01 index perturbation profile. The fundamental R mode
01
ters however it is not observed (within the experimental
hasnodependenceontheazimuthalangleφ,anditsdis-
noise limit) in the region around 1.1 µm – a clear ev-
placement components are the axial u sin(β z−Ωt)
z ac
idence of the Brillouin Self-Cancellation effect. This is
and the radial u cos(β z−Ωt) . Figure 5a shows the
r ac
morereadilyobservedinFigure4d,wherethetotalscat-
transverse profile for each component and Figure 5c il-
tered power in the R mode (P ) is normalized to the
01 01 lustratesthewiredeformationunderthisparticularmode
total scattered power in the TR mode (P ). Such
21 21 (with amplitude arbitrarily large for better visualiza-
6
nt 6 (a) ur agonal term is ∆(cid:15)rr = −n4(p11Srr+p12Sφφ+p12Szz).
e Similar expressions hold for the ∆(cid:15) , ∆(cid:15) and ∆(cid:15)
emm) 4 uz components. Note that the sign of thφφe photzoz-elastic croz-
ac-170 2 efficients determines whether a positive (tensile) strain
spl(1 0 increases or reduces the dielectric constant. In silica
Di 0.2 0.4 0.6 0.8 1.0 (which is an isotropic medium), there are only two in-
-2 (2r/d) dependent photo-elastic coefficients, p and p , and
11 12
4 they are both positives (except for the small crossed
(b) S S S
rr φφ zz term ∆(cid:15) that depends on p = 1(p −p ), which
3 (p S + p S + p S ) rz 44 2 11 12
11 rr 12 zz 12 φφ isnegative). Thereforeanexpansionleadstoareduction
2
n) in the dielectric constant. More precisely, this means
Strai-10(10 01 bthyatth∆e(cid:15)prrhoiston-eelgaasttiivceciofetffiheciseunmts)ofisthpeossittriavien.sB(wyeiegxhatmed-
0.2 0.4 0.6 0.8 1.0
-1 ining the wire deformation profile in Figure 5c, it be-
(2r/d) comes very clear which regions have positive or nega-
-2
(c) λa tive strains fields. In the center of the wire, while it
is compressed in the axial direction, it expands out ra-
r
e dially (and thus azimuthally). This means that these
et
m strains fields counteract each other in the central region,
a leading to relatively smaller net contribution to the di-
Di
electric perturbation in the center of the wire, as shown
in the black solid curve in Figure 5b. Note that this
is just completely counter-intuitive for those accustomed
(d) 1.0
to photo-elastic effect in conventional fibers, where ax-
0.5
+ + = ial strain completely dominates the effect. Near the wire
0.0
surface, the strain competition picture changes radically.
-0.5
longitudinal crossed transverse total There is mostly simultaneous expansion in all (axial, az-
-1.0
imuthal and radial) directions, and thus the strain fields
FIG. 5. Physical understanding of the photo-elastic contribute to a reduction on the dielectric constant (as
contribution: (a) radial and axial displacement field pro- required to achieve the BSC effect). One particular as-
file for the fundamental R01 acoustic mode for a wire with pectforbackwardscattering(asopposedtoforwardscat-
d = 1.1 µm diameter (absolute amplitude given by thermal
tering) is that E – the longitudinal component of the
z
energynormalization);(b)resultingstrainprofilesalongwith
electric field – changes sign for the backward wave. In
the total strain profile weighted by the photo-elastic coeffi-
cylindricalcoordinates,theintegrandinEquation1then
cients; (c) illustrative wire deformation (λ is the acoustic
a becomes∆(cid:15) |E |2+∆(cid:15) |E |2−∆(cid:15) |E |2−2∆(cid:15) E E .
wavelength); (d) each integrand term in the coupling co- rr r φφ φ zz z rz r z
These terms are shown in the two-dimensional profile in
efficient overlap integral (Equation 1) is shown separately.
The longitudinal term (−∆(cid:15) |E |2) mostly cancels out the Figure5d(normalizedtounit). Forthisparticulardiam-
zz z
crossed term (−2∆(cid:15) E E .). As a result, the total pertur- eter, the last two terms (longitudinal and crossed-terms)
rz r z
bation profile is mostly dominated by the transverse term mostly cancel out each other, and the net result is dom-
(∆(cid:15)rr|Er|2+∆(cid:15)φφ|Eφ|2). inated by the transverse terms ∆(cid:15)rr|Er|2 +∆(cid:15)φφ|Eφ|2.
Moreover,thetransverseopticalfieldprofileprovidesthe
rightbalancebetweenthepositive(central)andnegative
(edges) perturbation regions. As a result, for a nanowire
tion). The maximum radial displacement, shown in
with 1.1 µm diameter, the negative outer region domi-
Figure 5a, is near the wire surface and therefore in-
nates over the positive center region by just the exact
duces significant moving-boundary effect. As mentioned
amountnecessarytocancelthepositivedielectricpertur-
before, the BSC effect shall occur only if the photo-
bationduetothemb-effect. Thissamedynamicsexplains
elastic effect causes a negative net dielectric perturba-
why the pe-effect has a zero-point near 0.51 µm diame-
tion in order to compensate the positive mb-effect per-
ter(aspreviouslyshowninFigure2f), whenthepositive
turbation. For the R mode, the only nonzero strain
01 centralregionjustbalancesoutthenegativeouterregion.
components are the diagonal terms S = ∂ u (strain
rr r r Note that Figure 4b shows the measured spectrum of a
in the radial direction due to a radial displacement),
0.51µmdiameternanowire,thepointwheretheBrillouin
S = u /r (strain in the azimuthal direction due to a
φφ r scattering is totally due to the moving-boundary effect.
radialdisplacement)andS =∂ u (strainintheaxial-
zz z z
direction due to axial displacement), and the cross-term In conclusion, we have demonstrated experimentally
S = 1(∂ u +∂ u ). The strain profiles are shown in the Brillouin Scattering Self-Cancellation effect in silica
rz 2 z r r z
Figure 5b. The only nonzero terms in the dielectric per- nanowires. Precise control of both optical and acoustic
turbation tensor are also the diagonal terms ∆(cid:15) , ∆(cid:15) fieldprofileturnsouttocreateexactlyopposingcontribu-
rr φφ
and ∆(cid:15) , and the cross-term ∆(cid:15) . Explicitly, the di- tions due to photo-elastic and moving-boundary effects.
zz rz
7
Incylindricalgeometry,wehavedemonstratedthiseffect to selectively control the photon-phonon interaction in
forthefundamentalaxial-radialRayleighacousticmode. photonic waveguides and cavities.
A detailed understanding of the observed Brillouin spec-
trumandofthephysicalmechanismbehindthecancella- Acknowledgements: this work was supported by
tion effect is presented. We have also demonstrated that the S˜ao Paulo Research Foundation (FAPESP) un-
for 0.51 µm diameter, the photo-elastic effect is zero (to der grants 2013/20180-3, 2012/17765-7, 2012/17610-3,
first order) and the observed Brillouin scattering is pro- and 08/57857-2, and by the National Council for Sci-
duced completely by moving-boundary effect only. Such entific and Technological Development (CNPq), grant
rich interplay between photo-elastic and moving bound- 574017/2008-9. O. Florez and Y.A.V. Espinel wish to
ary effects can be further explored by modal engineer- thank the Coordination for the Improvement of Higher
ing and the use of materials with different photo-elastic Education Personnel (CAPES) for financial support.
coefficients and refractive indices to observe the effect ScanningElectronMicroscopywasperformedattheCen-
of Brillouin scattering Self-Cancellation in a variety of terforSemiconductorComponents(UniversityofCamp-
structure. Moreover, it can be used as a powerful tool inas).
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9
SUPPLEMENTARY INFORMATION
S1. NONLINEAR POLARIZATION
The macroscopic Maxwell equations in the time domain for a nonmagnetic medium with no free-charges are:
∇·D =0,
∇·H =0,
(S1)
∇×E =−µ ∂ H,
0 t
∇×H =∂ D.
t
Here D = ε (cid:15)E+P is the electric displacement field, ε is the vaccum permittivity, (cid:15) is the relative dielectric
0 NL 0
permittivity and µ is the vacuum permeability. The nonlinear polarization P induced by an acoustic wave is
0 NL
written in the time domain as:
P (t)=ε ∆(cid:15)(t)·E(t),
NL 0
where ∆(cid:15) is the relative permittivity perturbation caused by the acoustic wave. For a harmonic perturbation, the
relative permittivity perturbation can be written as:
1
∆(cid:15)(t)= ∆(cid:15)(x,y)e−i(Ωt−βaz)+c.c.,
2
where ∆(cid:15)(x,y), Ω and β are the perturbation transverse profile, angular frequency and propagation constant
a
respectively. Since our interest is to investigate backward Brillouin scattering, we write the total field as the sum of a
pump signal at ω and a backward propagating signal at ω . We restrict our development to the Stokes line so that
p s
ω = ω −Ω (generalization to the anti-Stokes line at ω = ω +Ω is straightforward). We use the sub-index p to
s p as p
denote the pump field (propagating in the forward direction) and s to denote the scattered field propagating in the
backward direction. We assume both fields as pure harmonics:
1
E (r,t)= a (z)E (x,y)e−i(ωpt−βpz)+c.c.
p 2 p p
1
E (r,t)= a (z)E (x,y)e−i(ωst+βsz)+c.c..
s 2 s s
Note that the power carried by each signal is simply P = s 1|a |2Re (cid:2)(cid:82) (E ×H∗)·zdA(cid:3). From now on, E
n n2 n n n p,s
denotes the transverse field profile E (x,y). The factor s = +1 for the pump (co-propagating) and s = −1 for
p,s p s
the Stokes signal (back-propagating) are used so that the optical power is always positive. We can always normalize
the eigenmodes so that s 1Re (cid:2)(cid:82) (E ×H∗)·zdA(cid:3)=1 W, and then at any position along the waveguide the power
n2 n n
carried by the eigenmode n is simply P =|a |2·1 W. We calculate the polarization generated by the pump field (i.e.
n n
neglecting the nonlinear polarization caused by the Stokes and anti-Stokes fields since in our experiments they are
much weaker than the pump), obtaining:
ε ∆(cid:15) ε ∆(cid:15)∗
P (t)= 0 a E (x,y)e−i[ωast−(βp+βa)z]+ 0 a E (x,y)e−i[ωst−(βp−βa)z]+c.c..
NL 4 p p 4 p p
The first and the second terms are respectively the source terms for the anti-Stokes and Stokes scattered signals.
Since we neglected the nonlinear polarization caused by the Stokes and anti-Stokes, there is no source term at the
pump frequency ω (neither at high-order scattering such as ω ±2Ω). This approximation implies a constant pump
p p
field amplitude along the waveguide, a (z)=a (in other words, pump depletion is neglected).
p p
A. Perturbation theory
ThenonlinearpolarizationistreatedasaperturbationterminMaxwellequations. Weapplystandardperturbation
theory to calculate the evolution of the Brillouin backscattered signal amplitude a (z) along the waveguide. In the
s
10
linearregime(P =0),themacroscopicMaxwellequationsforaharmonicfieldEeiωt (specificallythecurlequations
NL
in S1) can be written in terms of operators as:
A|ψ(cid:105)=−i∂ B |ψ(cid:105), (S2)
z
(cid:20) (cid:21)
E
where |ψ(cid:105)= , and the operators are defined as:
H
(cid:20) (cid:21)
ωε (cid:15) −i∇ ×
A= 0 t ,
i∇ × ωµ
t 0
(cid:20) (cid:21)
0 −z×
B = .
z× 0
For a waveguide, the solution to Equation S2 is of the form:
(cid:20) (cid:21)
1 E (x,y)
|ψ (cid:105)= a eiβnz |ϕ (cid:105), with |ϕ (cid:105)= n , (S3)
n 2 n n n Hn(x,y)
where E and H are the field profiles for a specific eigenmode, and β is the propagation constant (eigenvalue)
n n n
at frequency ω. In the linear regime a is a constant. Substituting Equation S3 into S2, we obtain a generalized
n
eigenvalue problem:
A|ϕ (cid:105)=β B |ϕ (cid:105), (S4)
n n n
which for the back-propagating Stokes line becomes A|ϕ (cid:105) = −β B |ϕ (cid:105) (where the negative sign arises naturally
s s s
from the e−iβsz dependence). The nonlinear polarization can be formally introduced as a perturbation term ∆A
added to the operator A. For instance, from all the terms in the expression for the nonlinear polarization, the term
oscillating at the Stokes field (e−iωst) is:
ε ∆(cid:15)∗
P (t)= 0 a E (x,y)e−i[ωst−(βp−βa)z],
NL 4 p p
Inthepresenceofthisterm, Maxwellequationsforthefieldoscillatingase−iωst canagainbecastinoperatorform
as:
a e−iβszA|ϕ (cid:105)+a ei(βp−βa)z∆A|ϕ (cid:105)=−i(∂ a −iβ a )e−iβszB |ϕ (cid:105), (S5)
s s p p z s s s s
where the perturbation operator is:
(cid:20) ωsε0∆(cid:15)∗ 0(cid:21)
∆A= 2 .
0 0
UsingthegeneralizedeigenvalueEquationS4fortheStokessignal,thefirsttermontheleft-handsideofEquationS5
cancels with the second term on the right-hand side, which leads to:
(cid:104)ϕ |∆A|ϕ (cid:105)
∂ a =−iκa ei∆βz, with κ= s p . (S6)
z s p 4
The phase mismatch is ∆β = β +β −β and κ is the coupling coefficient. In the definition of the coupling
p s a
coefficient κ, we used the mode normalization
(cid:20)(cid:90) (cid:21)
(cid:104)ϕ |B|ϕ (cid:105)=2(cid:60) (E ×H∗)·zˆdA =4s [W]=−4[W],
s s n n s