Accurate switching intensities and length scales in quasi-phase-matched materials

3 Pages·0.16 MB·English

by  Ole Bang

Accurate switching intensities and length scales in quasi-phase-matched materials
Ole Bang, Torben Winther Graversen,and Joel F. Corney
Department of Informatics and Mathematical Modelling, Technical University of Denmark, DK-2800 Lyngby, Denmark
We consider unseeded Type I second-harmonic generation in quasi-phase-matched (QPM) quadratic nonlinear ma-
1 terials and derive an accurate analytical expression for the evolution of the average intensity. The intensity-dependent
0 nonlinearphasemismatchduetotheQPMinducedcubicnonlinearityisfound. Theequivalentformulafortheintensity
0 for maximum conversion, the crossing of which changes the nonlinear phase-shift of the fundamental over a period
2 abruptlybyπ,corrects earlier estimates bymore thana factor of 5. Wefindthecrystal lengthsnecessary toobtain an
n optimal flat phase versusintensity response on either side of this separatrix intensity.
a
J Since the observation of nonlinear phase-shifts in ex- second-harmonicgeneration(SHG).Weaddressthisdis-

0 cess of π through cascading close to phase-matching1,2, crepancy and find the exact separatrix intensity for un-
3 quadratic nonlinear or χ(2) materials have been of sig- seeded Type I SHG in QPM samples. We further show
nificant interest in photonics3. With the maturing of that the averaged model gives also quantitatively inac-
]
S the quasi-phase-matching (QPM) technique4, in partic- curate results for no seeding of the average SH, since a
P ular by electric-field poling of ferro-electric materials, basic assumptionin the averagingprocedureis violated,
n. such as LiNbO35, and by quantum-well disordering in and we find the optimum crystal lengths for using the
semiconductors6, the number of possible applications of induced separatrix intensity for all-optical switching.
i
l cascading in χ(2) materials has increased even more.
n We consider a linearly polarized electric field E~ =
[ An effect of QPM gratings is to generate cubic non- eˆ[E (z)exp(ik z iωt)+E (z)exp(ik z i2ωt)+c.c.]/2,
1 1 2 2
1 linearities in the equations for the average field due to propagating in a−lossless QPM χ(2) med−ium under con-
v non-phase-matched coupling between modes7. This cu- ditions fortypeI SHG.The dynamicalequationsforthe
2 bicnonlinearityappearsinQPMwithlinearand/ornon- slowly varying envelopes take the form14
5 linear gratings8, it can be focusing or defocusing, de-
0 pending on the sign of the phase mismatch8, and its idE1/dz+G(z)χ1E1∗E2ei∆kz =0, (1)
01 strengthcanbesignificantlyincreasedbymodulatingthe idE2/dz+G(z)χ2E12e−i∆kz =0, (2)
grating9. Simulations of QPM systems have confirmed
1
0 the presence of an intensity dependent nonlinear phase where E1(z) is the fundamental wave (FW) with fre-
/ mismatch10andofsolitonproperties7,8,thatcanonlybe quency ω and wavevector k1, E2(z) is the SH with
n
describedbyincludingthecubicterms. TheKerrnonlin- wavevectork2,∆k=k2 2k1 isthewavevectormismatch,
i −
nl earityisalsoknowntodistortthesecond-harmonic(SH) andχj=ωdeff/(njc),withnj=n(jω)beingthe refractive
spectrum and introduce an intensity dependent nonlin- index and d =χ(2)/2 being in MKS units. The to-
: eff
v ear phase mismatch, as found both theoretically11 and tal intensity I = 1η (n E 2 +n E 2) is conserved,
Xi experimentally12. Thus, although the physical origin where η = ǫ /µ 2is0 th1e| s1p|ecific2|ad2m|ittance of vac-
0 0 0
r of the Kerr and the induced cubic nonlinearities is dis- uum. The pχ(2) susceptibility is modulated by the grat-
a tinctlydifferent,theyhavequalitativelysimilareffectson
ing function G(z) with unit amplitude and Fourier se-
CW waves. ries G(z)=σ g einκz, where g =0 for n even and
n n n
Inthisletterwefocusontheintensitydependentnon- gn=2/(iπn) fPor n odd. This zero-average square-wave
linear phase-mismatch, which implies a finite separatrix modulation is typical for QPM by domain inversion
intensity that can be used in efficient all-optical switch- in ferro-electric materials, such as LiNbO3. We define
ing, as was shown for the Kerr nonlinearity in poled σ=sign(κ) so that G(z) is positive in the first domain.
fibres13 and the QPM induced cubic nonlinearity10. A
We further consider first order forward QPM with a
distinct feature of the averaged QPM model with induced
shortcoherencelengthL L L,whereListhecrystal
d c
cubic nonlinearities is that there is not aone-to-one cor- ∼ ≪
length, L =π/κ is the domain length, and L =π/∆k
d c
respondance between the physical field and the averaged | | | |
is the coherence length. Expanding the fields in Fourier
field. TheseparatrixintensityobtainedwithnoSHseed-
series in the grating wavenumber κ,
ing in the averagedmodel does therefore not accurately
predicttherealphysicalseparatrixintensityofunseeded E = w (z)einκz, E =σ v (z)ei(nκ−β)z, (3)
1 n 2 n
Xn Xn
1
where the harmonics are small compared to the dc- and found by numerical integration of the correspond-
component, a simple first-order perturbation theory7,8 ing physical system (1-2) with initial condition de-
givesthe dynamicalequationsforthe slowlyvarying(on termined from Eqs. (3) and (6). With the domain
the scale of the domain length) average field length L =8.8µm the predicted separatrix intensity
d
I0=231GW/cm2 is 26GW/cm2 too high compared to
idw0/dz+iρ1w0∗v0+(γ1|v0|2−γ2|w0|2)w0 =0, (4) wshatisactuallyfoundinthephysicalsystem(1-2). This
idv /dz+βv iρ w2+2γ w 2v =0, (5) inaccuracy has not been observed before and is mainly
0 0− 2 0 2| 0| 0
duetotheaverageSHbeinginitiallyzero,whichviolates
the requirement that the harmonics are small compared
whereβ=∆k κ κistheresidualmismatch,ρ =χ 2/π,
j j
andγ =χ χ −(1 ≪8/π2)/κ. SinceL L Lthesignofκ to the average field.
j j 1 d c
− ∼ ≪
isthesignofthemismatch,sign(κ)=sign(∆k). Thusthe
QPM induced cubic nonlinearity can be both focusing
and defocusing, depending on the sign of the mismatch
∆k, just as the effective cubic nonlinearity obtained in
the cascading limit3. The harmonics are given by
nκw =χ g w∗v , nκv =χ g w2, (6)
n6=0 1 n−1 0 0 n6=0 2 n+1 0
The average model (4-5) conserves the total average
intensity I = 1η (n w 2 +n v 2) and the quantity
0 2 0 1| 0| 2| 0|
ǫ = (1 u)√usin(φ 2φ )+Cu+Du2. Here φ (z)
2 1 1
− −
and φ (z) are the phases of w (z) and v (z), respec-
2 0 0
tively, and u(z) = 1η n v (z)2/I is the fraction of
2 0 2| 0 | 0 Fig.1. Nonlinearphase-shiftoftheFWversusintensity
average intensity I0 in the average SH. The parameters I forbulkLiNbO3,aspredictedbytheaveragedmodel(4-5)
are D = 32γ1√J/ρ1 and C =−12β/(ρ1√J)− 43D, where (dashed)andfoundnumerically from thephysicalEqs.(1-2)
I0 = 12η0n2J. In the physical regime, where the polyno- (solid). TheQPMdomainlengthisLd=8.8µmandthecrys-
mialf(u)= D2u4+(1 2CD)u3 (2+C2 2ǫD)u2+ tallength isL=351µm (174µm) for unseeded(seeded)SHG.
(1+2ǫC)u ǫ−2hasfourre−alrootsu − u 1− u u , The vertical dotted lines mark the separatrices Is and Is0.
0 1 2 3
− ≤ ≤ ≤ ≤
we solve Eqs. (4-5) by quadrature and find the average
SH intensity
At this point we stress that no seeding in the average
model(4-5)implies aseedingofthephysicalSH,asgiven
u(z)=[u sn2(rz m)+u p]/[sn2(rz m)+p], (7)
3 | 0 | by the transformations (3) and (6), i.e. v0(0)=0 gives
E (0)=w (0) and E (0)= i(σρ /κ)w2(0). The experi-
wherep=(u3 u1)/(u1 u0),mp=(u3 u2)/(u2 u0), 1 0 2 − 2 0
andr = 3 γ J− (u u−)(u u )arer−ealpositiv−epa- mentally relevantsetup is unseededSHG with E2(0)=0,
2| 1| 2− 0 3− 1 forwhichFig.1showsthatthephysicalphaseversusin-
rameters. TheJpacobianEllipticsn(z m)functionisperi-
| tensity curve is indistinguishable from the prediction of
odicwithperiod4K(m),whereK(m)isthe completeel-
lipticintegralofthefirstkind15. Thesolution(7)isthus theaveragemodel(v0isnowseededandthusnoassump-
tions areviolated). The physicalseparatrixintensity for
periodicwiththeperiod2K(m)/randbecomesaperiodic
unseeded SHG is 44GW/cm2, i.e., 5.3 times lower than
when u =u =1 (m=1), in which case 100% transfer of
1 2 the prediction I0 of the unseeded averagedmodel. Note
powertotheSHispredictedbytheaveragemodel. From s
that the averagedmodel without the cubic terms would
f(1)=0 we obtain the separatrix C+D=ǫ.
not predict any separatrix.
The solution (7) was found earlier for no seeding of
the average SH, i.e. with u(0)=u =ǫ=010. In this case From Eqs. (3) and (6) with E2(0)=0 we find that
0 v (0)=i(ρ /κ)x2 where x=w (0) is real and E (0) =
I=I to lowest order and the separatrix C+D=0 corre- 0 2 0 1
0
sponds to the physical intensity I=I0=1η n J0, where x+(ρ1ρ2/κ2)x3. Thenǫ=n2ρ2√J/(n1κ)tolowestorder
Js0=−β/γ1. Itwasfurtherproven10 tshat2w0he2ntshe sepa- asenpdartahtursixJisn=te−n(s1it−y8f/oπr2u)nβs/eγe1d.edTShHisGgives the physical
ratrixis crossedthe phase-shiftofthe FW overa period
changed by exactly π, a result that is also valid for our η n 8 β
0 2
general solution (7). However, it was never investigated Is =− 2 (cid:18)1− π2(cid:19)γ (8)
whether I0= 1η n β/γ was an accurate prediction of 1
s −2 0 2 1
the separatrix intensity in the physical system (1-2). since again I=I to lowest order. For L =8.8µm this
0 d
gives I =44GW/cm2, which is exactly the numerically
As a typical sample we consider bulk LiNbO , a s
3 foundvalue(seeFig.1). Ournumericalsimulationscon-
FW wavelengthof 1.064µm,d =30pm/V, n =2.2,and
eff 1 firm this accuracy for all values of the domain length
n =2.23, which gives the coherence length L =8.9µm.
2 c satisfying L L L and also for negative ∆k.
In Fig. 1 we show the nonlinear phase-shift of the d ∼ c ≪
FW versus the physical intensity I as predicted by AfirstrequirementfortheQPM-inducedseparatrixto
the unseeded average model (4-5) with u(0)=v (0)=0 berelevantforswitchingpurposesisthattheintensityI
0 s
2
is low,whichrequiresthe ratioβ/γ to be small, i.e.the tion (7) versus intensity. The minimum period between
1
effective mismatch must be small (but nonzero) and the I=0andtheseparatrixI=I providesagoodmeasureof
s
induced cubic nonlinearity strong. We will not discuss the optimal crystal length observed in Fig. 2, for which
more appropriate materials than LiNbO , but we note the π-shift due to the separatrix is the clearest and the
3
that the averagemodel is verygeneral9,8, allowing γ to flatphaseversusintensityplateausoneithersidearethe
1
be significantly increased by, e.g., modulating the QPM broadest. FromFig.3weseehowthisoptimallengthin-
grating9 or by a strong dc-value of the grating8. creases as the domain length approaches the coherence
length 8.9µm, i.e. as exact effective phase-matching is
approached. At the same time the separatrix intensity
decreases.
Thus it is desirable to work close to, but not exactly
at,exacteffectivephase-matchinginordertoreducethe
holding intensity in the switching process and keep a
reasonable crystal length. Clearly the resolution in the
photolithographic process could be an issue if the sepa-
ratrix is to be measured and used.
TheresearchissupportedbytheDanishTechnicalRe-
search Council through Talent Grant No. 5600-00-0355.
Fig. 2. Nonlinear phase-shift of the FW versus inten-
sityI forunseededSHGinbulkLiNbO3. TheQPMdomain
lengthisLd=8.8µm. Thecrystallengthisgiveninthefigure.
TheverticaldottedlinemarkstheseparatrixIs=44GW/cm2.
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Sofartheinducedseparatrixhasnotbeenobservedin
Spektrosk. 66 (1989) 1383 [Opt. Spectrosc. 66 (1989)
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806].
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χ(2) materials are self-focusing and have normal disper-
(1992).
sion and thus the inherent SPM coefficient is positive, 3. For a review see G. Stegeman, D.J. Hagan, L. Torner,
whiletheinducedSPMcoefficient( γ2)isnegative;(iii) Opt.Quantum Electron. 28, 1691 (1996).
−
the crystal length was inappropriate. In Fig. 2 we illus- 4. J.A.Armstrong,N.Bloembergen, J.Ducuing,P.S.Per-
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variation due to the quadratic nonlinearity, and the flat linearOptics,F.Kajzar andR.Reinisch,eds.(Plenum,
plateausoneachsidebecomenarrower,asLisincreased New York,1998) pp.375-406.
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short the separatrix is lost. J.H. Marsh, A.C. Bryce, J.M. Arnold, C.R. Stan-
ley, J.S. Aitchison, C.T.A. Brown, K. Moutzouris, M.
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Fig.3. Left: PeriodversusintensityI forunseededSHG 13. Y. Zhao, G. Town, M. Sceats, Opt. Commun. 115, 129
in bulk QPM LiNbO3 with the QPM domain length given (1995).
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In Fig. 3 we show the period 2K(m)/r of the solu-
3