Stability of the self-phase-locked pump-enhanced singly resonant parametric oscillator

4 Pages·0.28 MB·English

by  Jean-Jacques Zondy

Stability of the self-phase-locked pump-enhanced singly resonant parametric oscillator
Jean-Jacques Zondy∗
BNM-SYRTE (UMR-CNRS 8630), 61 avenue de l’Observatoire, F-75014 Paris, France
(Dated: February 2, 2008)
Steady-stateand dynamics of theself-phase-locked (3ω −→2ω,ω) subharmonic optical parametric
oscillator are analyzed in the pump-and-signal resonant configurations, using an approximate ana-
lytical model and a full propagation model. The upper branch solutions are found always stable,
regardlessofthedegreeofpumpenhancement. Thedomainofexistenceofstationarystatesisfound
3
tocritically depend on thephase-mismatch of thecompeting second-harmonic process.
0
0
PACSnumbers: 42.65.Yj,42.65.Sf,42.65.Ky
2

n
a A new class of subharmonic (frequency divide-by-three, DRO/TROs(subjecttomodepairinstabilities)orSROs
J
or 3 1) optical parametric oscillators (OPOs), namely (requirement of high pump thresholds), it is interesting
6 self-p÷hase-locked (SPL-OPOs), has aroused lately much to check how the dynamics of the doubly/triply reso-
1
interest from both the experimental [1, 2] and theo- nant SPL-OPOs is affected when only one of the sub-

retical [3, 4, 5] standpoints. Consider a 3 1 OPO harmonicwavesoscillateswith a varyingpump enhance-
] ÷
s pumped at an angular frequency ωp = 3ω (generating ment, and to compare with the behaviour of the purely
c
signal and idler waves at ω = 2ω and ω = ω) con- idler-resonant (SPL-IRO) case treated by Longhi in the
i s i
t taining a secondnonlinear crystalthat is phase-matched mean-field approximation and neglecting pump deple-
p
for the competing second-harmonic (degenerate down- tion [5].
o
. conversion) of the idler (signal) waves. The loss-free The analysis starts with the solutions of the reduced
s χ(2) medium of total length L = L +L consists of a propagation equations for the normalized slowly vary-
c 1 2
′ ′
i dual-grating periodically-poled (PP) crystal comprising ing field envelopes Aj(t,Z) = g1L1Nj(t,Z) (j = p,s,i)
s
a first section of length L perfectly phase-matched for throughout the dual-section medium. The complex am-
y 1
h the χ(2) χ(2)( 3ω;2ω,ω) down-conversion, followed plitudes are scaled such that their intensities Ij = Aj 2
p by a sOePcOon≡d sectio−n of length L2 nearly phase-matched are proportional to the number of photons |Nj|2 |in j|-
[ for the competing χ(2) χ(2)( 2ω;ω,ω) process with thmode, times the squareofthe small-signalparametric
v1 tahweamvuevtuecatlosrelmf-iisnmjeacSttHciohGn∆≡okf t=heks−2uωbh−ar2mkωon6=ic 0w.avDesu,ethtoe gthaeinpgh1aLse1-r(≪eta1rd)eodf tthimee3f÷ra1mper(otce=sst′(g1n¯∝Zχ/c(O2;P)zO)=[3Z]/.LIn1
5 dynamics of the signal-and-idler resonant devices was orz′ =Z/L2),wheren¯ istheaveragei−ndexofrefraction,
3 recently shown to depart from that of a conventional the 3 plane-wave equations for 0 z L1/L1 =1 are
0 ≤ ≤
1 (L2 = 0) non-degenerate OPO [3]. The main differ- ∂ A =iA A ; ∂ A =iA A∗ , (1)
0 ence is thatSPL-OPOsarecharacterizedby anintensity z p i s z s,i p i,s
3 bistability (sub-criticalbifurcation) whatever the config- with the initial condition Ai(z = 0) = 0 (non-resonant
0 uration [3, 4, 5]. Secondly, each intensity state of the idler). In the SHG section (0 z′ L /L = 1), the
2 2
≤ ≤
/ stablebranchcantake3possibledeterministicphaseval- subharmonic amplitudes evolve as
s
ic suuebsjeeqcutatlolyassptaoccehdasbtyic2pπh/a3s,ewdhiffiluescioonnvpernotcioenssal[6O].PEOxspaerre- ∂z′As = iSA2i exp(+i2ξz′), (2)
s ∗ ′
hy mimeetnrotalollgyy, 3[1÷, 71,S8P],LF-oOuPriOerssayrnetihnevseisstoigfaattetdosinecfornedqupeunlcsye with the initi∂azl′Acoin=ditiioSnAssAAi e(xzp′ (=−i02)ξz=)A (z = (13)),
p s,i s,i
[2],transversepatternformation[4,5]andpotentiallyfor
: while the pump amplitude keeps its value at z = 1
v newfeaturesinsqueezedstatesoflight. Finally,owingto
throughout the second section (∂z′Ap = 0). The pa-
i the much lower pump intensity requirement, SPL-OPOs
X rameter ξ = ∆kL /2 is the phase mismatch of the com-
2
should be also suited for the first experimental evidence
peting SHG process. The nonlinear coupling parame-
r
ofaHopfbifurcationincwOPOs,predictedtoonlyoccur
a ter S = g L /g L is the ratio of the SHG to OPO
2 2 1 1
in triply-resonant OPOs under extreme detuning condi-
(2)
small signal gains (g χ ). Its expression reduces
tions [9]. 2 ∝ SHG
to S (L /L )/√3 for a 3 1 OPO employing a PP
2 1
≃ ÷
material [3]. The time-dependent cavity dynamics is ob-
Inthis briefreport,I provideananalysisofthe pump-
enhanced singly resonant device (SPL-PRSRO) which tained from an iterative mapping of the resonating field
amplitudes at z′ = 1 and at a time t to their values at
was only over-viewedin the conclusion of Ref. [3]. Bear-
z =0 after one roundtrip time τ of the ring-type cavity,
ing in mind that PRSROs are easier to implement than
′
A (t+τ,z =0) = r exp(i∆ )A (t,z =1)+A ,(4)
j j j j in
with j = p,s and with r being the (real) amplitude re-
j
∗Electronicaddress: [email protected] flectivity from z′ =1 back to z =0. The constant input
2
field A stands for the driving pump field and is null The above Mac-Laurin solutions (as compared with the
in
for the signal wave. The r ’s are related to the cavity mean-fieldapproachbasedonamplitudeexpansioninthe
j
loss parameters κ ’s by κ = 1 r . For the resonat- power of the κ’s [5]) converge satisfactorily to the full
j j j
−
ing subharmonic,it will be alwaysassumedthatκ 1, propagationmodelaslongasr 0.8. Excellentconver-
s p
≪ ≥
r 1. The amplitude loss parameter κ is then related gence (to 3% for the parameters of Fig.3, for instance)
s s
≃ ±
to the cavity finesse by F =π/κ and to the cavity half is found when both the pump and signal experience low
s s
linewithγ by2πγ =κ /τ. Thephase factors∆ corre- roundtriploss,notexceedingafewpercentevenforlarge
s s s j
spondtothelinearpropagation(andmirror)phaseshifts, pumping (I = A 2 upto 50times the thresholdfor
in in
| | ∼
modulo 2π. The ∆’s, also called cavity detuning param- oscillation). For decreasing pump resonance and mod-
eters, are equal to (ν ν )τ, e.g. to the wave frequency erate pumping they still provide a qualitative accountof
c
−
mismatchesfromthenearestcoldcavityfrequency,scaled thedynamicalbehaviorofthesystem,althoughresulting
to the free spectral range 1/τ. inhighervaluesoftheintensitiesI = A 2. Thedomain
j j
| |
The dynamicsofthe systems canbe numericallystud- of validity of the Mc Laurin model will be shown to de-
iedwithoutanyapproximationbysolvingEqs.(1-3)using pend on the value of r . In the true SRO limit (r 0),
p p
→
a fourth-order Runge-Kutta solver with the appropriate the full propagation model remains always valid.
initialconditionsandmakinguseofthe boundarycondi- Besides the trivial (non-lasing) solutions A = 0,
s,i
tions (4) (propagation model). Unstable fixed point can- A = A /(κ ir ∆ ), Eqs.(8) admit non-zero inten-
p in p p p
not be found numerically so that approximate analytic sity states I −= A 2 with well-defined phases. It is
p,s p,s
solutions of the steady state equations must be worked convenienttointro|duce|thescaledintensitiesI¯ =I /Ith,
p p p
out by expanding the amplitudes in Mac Laurin series I¯ =I /I ,I¯ =I /(IthI ),withIth =2κ ,I =2κ /r ;
of z, e.g. Aj(z) = Aj(0)+P∞n=1[∂z(n)Aj]0zn/n!, which insputspumSpiintenisitypI¯inS = Iin/(p2κ2pκs)s; aSnd norpmapl-
allowsto integrateEqs.(1-3). Suchanexpansionis justi- ized cavity detuning ∆¯ = ∆ /κ . Introducing
p,s p,s p,s
fied by the smallness of the scaled′ amplitudes (|Aj|≤1) Cs = 4 χ2(κp/rp), and taking the modulii of (8), the
since g1L1 ≪ 1 and because z,z < 1. The n-th order scaled si|gn|al intensity I¯s is then the solution of
derivatives can be evaluated in terms of the field prod-
uthcetsleaatdzin=g(0fouusritnhg)-tohredegrenceoruipcleinqguatteiromnss(f1ro)-m(3)t.hiTsopgeert- (cid:2)F(1−rsκs∆¯2s)−rs(1+κ2s∆¯2s)I¯in(cid:3)2+(∆¯sF)2 (9)
turbative approach, only the n = 1 terms in the field =2Csrs2(1+κ2s∆¯2s)2I¯sI¯i2n,
expansions need to be kept. After some algebra, the ap-
where the symbol F stands for
proximate solutions of (1)-(3) are
Ap(t,L1+L2) = Ap(t,0) (1/2)Ap(t,0) As(t,0)2,(5) F =(1+I¯s)2+∆¯2p(κpI¯s−rp)2. (10)
− | |
As(t,L1+L2) = As(t,0)+(1/2)As(t,0)|Ap(t,0)|2 TheintracavitypumpisgivenbyFI¯p =I¯in andtheidler
iχ∗A2(t,0)[A∗(t,0)]2, (6) intensity by I¯i =I¯pI¯s[1+I¯S χI¯s+(I¯p 1)/I¯p].
− p s Phase relationships, demo|n|strating −phase-locking of
A (t,L +L ) = iA (t,0)A∗(t,0)+χA∗(t,0)A2(t,0()7.)
i 1 2 p s p s the subharmonic waves to the pump laser, can be de-
where the nonlinear coupling parameter is χ = rived from Eqs.(8) by writing Aj = αjexp(iϕj) where
Sexp( iξ)(sinξ/ξ). Thesesolutionswhichassumealin- αj are the amplitude modulii. Defining ϕin as the arbi-
ear z-v−ariation of the fields depart from the numerical trary phase of the pump laser and ϕD =ξ+2ϕp 3ϕs,
−
onesfordecreasingpumpresonance(SROlimit),butstill one obtains
accountforpumpdepletiontofirstorder. Thecubicterm
tan(ϕ ϕ )= [∆¯ (r κ I¯)]/[1+I¯], (11)
in Eq.(5), present in conventional PRSRO model [10], is in− p − p p− p s s
due to the usual cascading (3ω −2ω = ω) followed by cotϕD =∆¯s/[1−rsκs∆¯2s−rs(1+κ2s∆¯2s)I¯p]. (12)
the re-combinationprocess(ω+2ω=3ω),while the last
Whensolvedforϕ ,theserelationsyield3possiblevalues
quartictermin Eq.(6), describingthe two-stepprocesses s
ϕ = ϕ +2kπ/3 (ϕ being a constant and k = 0, 1),
(3ω 2ω =ω)followedby(ω+ω =2ω),leadstoinjection- s 0 0
±
− while for χ =0 only the sum phase ϕ +ϕ =ϕ +π/2
locking. Note that this term is quadratic in the signal- s i p
| |
is deterministic as predicted for conventional oscillators,
and-idler resonant cases (see Eqs.(9) in Ref. [3]) or in
duetothephasediffusionnoisestemmingfromthespon-
the pure IRO case without pump depletion (see Eq.(11)
taneous parametricfluorescence [6]. Furthermore,when
of Ref. [5]). Stationary solutions to Eqs.(4)-(6) are ob-
χ =0(C =0),Eq.(9)impliesthatthesignalresonates
tainedbyrequiringthatA (t+τ,0)=A (t,0) A . s
p,s p,s ≡ p,s n|e|cessarily with zero detuning (∆¯ = 0), and one re-
Considering smallenoughdetuning (∆ 2π), the ex- s
p,s
≪ trievestheresultthattheintracavitypumpisclampedto
ponentialphasefactorinEq.(4)isexpandedas 1+i∆ .
∼ j the constantvalue I¯ =1 foranyI¯ [10]. FromEq.(10),
The resulting steady state amplitude equations are p in
I¯ is then the solution of a quadratic equation which ad-
s
(κp irp∆p)Ap = (1/2)rp(1+i∆p)Ap As 2+Ain, mits a single positive solution (supercritical bifurcation)
− − | |
rsκ(s1−+ii∆∆ss)As = [+(1/2)AsAp|2−iχ∗A2p[A∗s]2. (8) iwfaaynsdsoantilsyfiiefdI¯ifno(r1I¯+inκ≥2p∆¯I¯2pth)−,w∆¯h2per≥et0h.eTihnipsucto(nidnittriaocnavisitayl)-
3

I2S Ip 12I0 = 0.927 s = 0 1.I0S (z’=1)I 1p s = 0 rp = 0.3
1 I00 = 11.472 Iin3 43 rp = 0.3 0.5 I0 = 00.9613 2 Iin3 24 3
0 1 2 0 1 I0 = 1.993
0 1 2 3 4 5 0 1 2 3 4 5
pump input Iin pump input Iin
FIG. 1: Bifurcation diagram of signal intensity versus pump FIG. 2: Stable (upper) branches of signal intensity versus
parameter, computed from theMc Laurin solutions, for rp = pumpparameter,computedfromthepropagationmodelwith
0.3, ∆¯p =0, κs =0.005, χ=S =0.2. Curves (1)-(3) are for thesameparametersasforFig.1. Thehatcheddomainatthe
∆¯s = 0;0.4;0.5. The inset plots show the intracavity pump left ofcurve(2) givestheamplitudeof thelimit cyclesbelow
stable(solidline)andunstable(dashed)fixedpointsI¯p±. Note thesaddle-nodeintensityI¯0(non-stationarySPLstates). The
that I¯s+ diverges at I¯in ≥ 3.8;4.4;4.7 for curves 1;2;3 from inset plots show theoutput (z′ =1) intracavity pumpI¯p.
either the LSA analysis or the time mapping of Eqs.(5)-(7),
setting thevalidity range of the Mc Laurin approximation.
full propagation model converge to a fixed point for any
input intensity value (Fig. 2). Hence the whole upper
pumpthresholdexpressesasI¯ =1+r2∆¯2. Considering branch of the SPL-PRSRO is actually stable, the insta-
th p p
nowthecase χ =0,thesignalwaveisapriori nolonger bilitypredictedbytheMcLaurinmodelbeingmerelyan
constrainedt|o o|s6cillate withzerodetuning andits inten- artefactoftheapproximation. Actually,asthepumpen-
sityisthesolutionofaquarticequation,P4n=0anI¯sn =0, hancementisdecreasedthevalidityrangeoftheMcLau-
obtainedbyexpandingEq.(9)usingEq.(10). Thenumer- rin model is restricted to pump parameters lying closer
ical resolution of this equation for a wide range of signal and closer to the threshold. In Fig. 2, obtained by back-
detuning ordriving pumpintensity alwaysyieldtworeal ward adiabatic following of the stationary solutions, all
positive roots I¯s±, defining two branches of solutions, for curves end at their saddle-node point I¯0 since critical
pump intensities I¯ I¯ (see below for definition of slowingdownisobservedasthesepointsareapproached.
in 0
I¯0). The stability of t≥hese two fixed points, each asso- The limit cycles occurring for I¯in < I¯0 (hatched area)
ciated with the 3 possible phase states, was investigated are merely due to the non-existence of stationary states.
using a linear stability analysis (LSA) [3] that leads to The upper branch is found stable even in the SRO limit
a quartic characteristic equation, P4n=0ΦnΛn = 0. The (rp = 0), whatever the detunings, in contrast with the
SPL-IRO mean-field analysis results [5]. Let us remind
LSA results were double-checked by a direct time map-
thatintheSPL-DRO/TROstheupperbranchwasfound
ping of Eqs.(5)-(6) using the boundary conditions (4).
todestabilizeviaaHopfbifurcation[3]. Thedifferencein
From the LSA of the trivial state, the threshold for os-
dynamical behavior is due to the stronger self-injection
cillation(notnecessarilyonastablefixedpoint)remains
regime in these latter devices. The inset frame in Fig. 2
the same as for conventional devices. Fig.1 shows the
gives the output pump intensity (at z′ = 1) correspond-
analytical bifurcation plot versus the pump parameter
for r = 0.3 and 3 different values for ∆¯ , as compared ing to ∆¯p,s = 0 (curve (1)), it can be seen that the
p s
pump is no longer clamped to unity as in conventional
withthesolutionscomputedwiththepropagationmodel
(Fig.2). The bifurcation diagram versus I¯ displays a PRSROs [10], meaning that the competing nonlinearity
in
saddle-node region, with I¯ being the input intensity at enhances the down-conversion efficiency. In Fig. 3, as
0
the saddle-node point where I¯+ = I¯−. For zero or van-
s s
ishingly small ∆¯ , one has I¯ < I¯ but as the detun-
ing is increased,sI¯0 > I¯th. Th0is behthavior contrasts with 1I.s0 00..6601 I0 = 2.551 (A) rp = 0.95 (b)
thesignal-and-idlerresonantcases,forwhichatransition 0.59
fromsub-criticalityto super-criticalityis predictedwhen 0.5 0.58 2.55Iin2 .56 2.57 0.6
n∆¯osd,ie→poi0n,tcoofrrceosoprodnindainteg(tI¯o0,tI¯hse=m0e)rgwinitghotfhtehtehrseasdhdolled- 0 0 (a1) 2 2.5 2 .36(B)0.5 4
point (I¯ ,I¯ = 0) [3]. Note also6 the tiny range of sub- pump inp ut Iin
th s
threshold states in curve (1), resulting from the weak
FIG. 3: Bifurcation diagram of signal intensity versus pump
self-injection regime. From the LSA of the Mc Laurin
solutions, the I¯− branches (dashed curves) are always parameter,computedfromtheanalyticalmodelforrp =0.95
s and same other parameters as in Fig.1, excepted that curves
unstable, due to a positive real eigenvalue, but only a (a)-(b) are for ∆¯s = 0;0.1. The thin solid lines under-riding
portion of the upper branch (solid lines) extending from the(a)-(b)solid lines show theupperbranchcomputed from
the saddle-node intensity I¯0 to some criticalintensity I¯C the propagation model. The insets (A: Mc Laurin model, B:
is found stable. The instability (dotted lines) beyond I¯ propagation model) are blow-ups of the saddle-node region
C
(characterized by 3 real positive eigenvalues of the LSA of (b) case. In (B), critical slowing-down characteristic of
equation) was confirmed by the time mapping of the Mc saddle-node pointsoccurs.
Laurin solutions which diverges at I¯ , wherewhile the
C
4
defined as the maximum allowed ∆¯ for a given pump
De C 45 k s = 0.005 inputintensity,wouldbesmallerthans insignal-and-idler
ang 3 D p = 0 1 resonant set-ups. Indeed large pump enhancement is
PL r 2 Iin = 4 2 paid back with a shrinking self-locking range. Fig.4 dis-
S 1 3 plays the critical ∆¯ (S) detuning for both r =0.3 and
C p
00 0.2 0.4 0.6 0.8 1 rp =0.95, when ∆¯p =0 and when the device is pumped
coupling strength S 4 times above threshold. Surprisingly, for the same non-
linearcouplingstrengthS,thehighpumpresonancecase
FIG. 4: Critical values of the signal detuning parameter leadstolessthanacavitylinewidthSPLrange,whilefor
delimiting the boundary of stable (underneath the curves) the SRO limiting case the SPL range is 10-fold wider,
and oscillatory SPL states (above) in the (∆¯s,S) plane at as with SPL-DRO/TROs. Hence a large SPL range
fixed pump input/detuning, computed from the propaga- is not necessarily associated with the double-resonance
tion model. The thick solid lines are for rp = 0.3 and the conditionthatinvolvesastrongmutual-injectionprocess
thin lines for rp = 0.95. The curve labels (1)-(3) stand for between the subharmonics. The small (< 1-MHz, e.g.
ξ=0; π/2, 3π/4.
less than a cavity linewidth) self-locking range reported
by Boller et al [1] for a SPL-PRSRO is in agreement
2 with this expectation. As a noticeable difference with
)ad 011...505(a)S = 0.2 (b)S = 0.75 tLhSeAsiegqnuaal-tainond-iddelperenrdesoonnanthtecapsheas,sethoefctoheefficcoieunptlsinogftphae-
x ( p r011...50520(c)S = 0.2 Ds = 0.05 (d)S = 0.75 Ds = 0.05 rDOaRnmeOetheesrtnaχcbe,iwleitxhypileeacχntsaelnythtseiasrts(toshneeelydtayhsneiatmAsmpicpsoednoufdliuSxsPiiLnn-PtRhReeSfS.R[P3OL])s-.
00 1 2 3 40 1 2 3 4 will be sensitive to the SHG phase mismatch ξ. Fig.5
pump input Iin shows the steady-state signal intensity contour plots in
the parameter space (I¯ ,ξ), computed with the propa-
in
FIG. 5: Domain of stability (dark region) of phase-locked gation model for I¯ = 4, r = 0.95, ∆¯ = 0 and two
in p p
states in the (ξ,I¯in) plane (propagation model), with rp = values S = 0.2 (a,c) and S = 0.75 (b,d) of the coupling
0.95, κs =0.005, ∆¯p =0: (a),(c) for S =0.2 and (b),(d) for parameter. For the stronger coupling S = 0.75 (panels
S =0.75. Upper (lower) frames are for ∆¯s =0 (∆¯s =0.05). b-d), small amplitude limit cycles arise around ξ = π,
The solid lines give the boundary of existence of stationary even for δ¯ = 0 (panel b). To avoid such non-station±ary
s
states from the Mc Laurin model. The plots are symmetric
states,it is importantto tailoraccuratelythe simultane-
for negative phase mismatch ξ.
ous phase-matching of both competing processes.
In conclusion, it is found that the whole upper branch
the pump finesse increases (κ = 0.05), the Mc-Laurin of the SPL-(PR)SRO is stable and that the self-locking
p
(thick solid lines) and the propagation model (thin solid rangeshrinkswithincreasingpumpenhancement. Inthe
lines) converge excellently for any input intensity. Both SRO limit, the approximatedmodel may fail to describe
models predict then a whole upper branch stability, for correctlythedynamicalbehavioroveralargepumpinput
a wide range of input intensity (up to tested I¯ = 50 range. Ofmoreconcern,incontrastwithsignal-and-idler
in
at least). The Mc Laurin model converges to the prop- resonant devices, the domain of existence of stationary
agation model because when κ 0 the assumption of phase-locked states is found sensitive to the value of the
p
→
a linear z-dependence of the resonating fields inside the residual phase mismatch of the competing SHG nonlin-
medium is fully justified (uniform field limit). Notice earity. The author is indebted to one of the referees for
thatthethresholdvalueisthenmoresensitivetothesig- his personal involvement in the improvement of this re-
nal detuning than in Fig. 2. As a consequence of the port. This work has benefited from a partial support
reduced sub-threshold state range due to the weak SPL from an European Union INCO-Copernicus grant (Con-
regime one expect that the self-locking detuning range, tract No. ERBIC15CT980814).
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