seeded second-harmonic generation, frequency division, and spatial chaos in a medium with periodic mismatch. The experimental observation by Franken et al.
May 1, 1992 / Vol. 17, No. 9 / OPTICS LETTERS
637
Two-wave mixing in a quadratic nonlinear medium:
bifurcations, spatial instabilities, and chaos S. Trillo, S. Wabnitz, and R. Chisari Fondazione Ugo Bordoni, Via B. Castiglione 59, 00142 Rome, Italy
G. Cappellini Dipartimento di Fisica, Universitadegli Studi di Roma, Piazza A. Moro 3, 00185 Rome, Italy Received November 21, 1991
We present the qualitative dynamics of parametric mixing of two waves in a quadratic nonlinear medium by introducing a reduced phase-plane description. We find the nonlinear frequency-conversion eigenmodes and their bifurcations and instabilities. We discuss applications of this description to the phase dependence of seeded second-harmonic generation, frequency division, and spatial chaos in a medium with periodic mismatch.
The experimental observation by Franken et al.' of
second-harmonic generation (SHG) in a crystal
started the field of nonlinear optics. The theory of optical parametric frequency conversion in a
the dimensionless fields ao = (2)"12Ao/(Po)"12 and a 2 = A 2 /(Po)"/2 exp(iAkz). Then we may write daj/d4 = iaH/Oaj* (j = 0, 2), where H = H(aoa
quadratic nonlinear medium was developed by
Armstrong et al.,2 who obtained exact solutions for the evolution of the wave amplitudes in terms of Jacobian elliptic functions. These solutions involve rather cumbersome expressions and the inversion of an integral' and do not lead to an immediate insight into the physics of the conversion process. We introduce here a reduced geometrical description of the parametric mixing between a wave and its second-harmonic (i.e., two-wave mixing) in a quadratic nonlinear medium. The spatial evolution of the two electric fields is represented by the motion of a point on a given trajectory in the phase plane.' This reduced representation permits us to discover the nonlinear eigenmodes of the mixing and their bifurcations and instabilities. The qualitative description of the nonlinear dynamics of parametric mixing may give insight into the phase dependence of seeded SHG and spontaneous frequency division and leads to the prediction of parametric spatial chaos. The electric-field amplitudes of two monochromatic plane waves at frequencies (oo and C02 (w2 = 2coo) in a medium with second-order nonlinearity obey the familiar equations5
-i
d °=
.dA 2
WORA 2 Ao* exp(iAkz), 2
=d
2 exp(-iAkz), cooRAo
(la)
2,ao*,a 2 *)
= 2[aO2a 2 * + 2
(aO*) 2a 2 ] + Kla 2 12.
(2)
Here 4 = pz, p = coR(PO)1 2, and K = Ak/p. By exploiting the conservation of both the Hamiltonian H and the optical power PO,one may reduce Eqs. (1) to a quadrature. Here we want to gain a global understanding into the qualitative properties of the wavemixing process. In order to do this, we follow a geometrical approach: we represent the solution of Eqs. (1) as a function of 4 as a trajectory in phase
space. This is particularly effective in the present case because the conservation laws permit us to reduce Eqs. (1) to a system of just two differential 2 equations in the real variables 77(4)a 1a 2()1 and
4)(4) =K
+ 42(4) -
20o(). In fact,dq/d=
-aO a/o4 and d)/d6 = ad/a-q, where the new Hamiltonian We,obtained from Eq. (2)by substitution, reads We= Xq1,o) = 2Vfi;(1 - ,q)cos4)+
KJo.
(3)
Quantitative information on the mixing process, e.g., the fraction of second-harmonic power q,versus
the propagation distance 4, may be obtained in terms of Jacobian elliptic functions by inverting the integral (see also Ref. 2),
g= 0raL d ) 1 -R(E) 1
(lb)
do
=
where Ak = k2-
2ko is a mismatch of the wave vec-
tors and R is a constant that is proportional to the
second-order susceptibility.5 Note that the total 2 + IA 12 is a conserved power Po = JAol quantity of 2 Eqs. (1), unlike the total number of photons. Let us first recast Eqs. (1) in Hamiltonian form. Consider
0146-9592/92/090637-03$5.00/0
,o0 N/E -V(V7)
where E - V(_q)=
(4)
q -_q2 (2 + K2 /4) + q(1 + has the meaning of an equivalentpotential, and We= XC('qo, 4o),where 'io _q(e = 0) and KXC/2)
-
0o = 0(6
X2/4
= 0).
© 1992 Optical Society of America
638
OPTICS LETTERS
/ Vol. 17, No. 9 / May 1, 1992
matched case with K = 0. As can be seen, the instability of the circle with i-e l = 1 dramatically changes the nature of the motion in the plane. A
1 U)
z0
separatrix trajectory emanates from the point (1, - r/2) and returns to the circle in the point (1,7r/2). This separatrix divides two regions in
0U)
zU) C,
0 -8
-4
0 4 MISMATCH K
Fig. 1. Bifurcation diagram:
8
fraction of second-
harmonic power TYe of the stable (solid curves) and unstable (dashed line) nonlinear eigenmodes versus the normalized mismatch K. 1
I:: .
o~
0
;ZO
-1
-1
0
1-1
0
1
n cos $ n cos $ Fig. 2. Phase-plane portraits for (a) K = -5 and (b) K = 0.
However, as we see below, the real advar Ltage of the reduced description of the wave mixing iia terms of the variables (, 4))is that we may obtain an immediate insight into the properties of the frei quencyconversion process, given an arbitrary inpulI power ratio and relative phase between the two fi 31ds,by a simple visual inspection of the constant Welevel curves in the circle 17[1 1. In turn, the s]lhape of these curves may be easily guessed once tl ie location and the stability of the extrema of We,say, {7)e, Oe}A are determined. These are the nonlinear '
eigenmodes of the mixing, since axe/a71
(wee
=
aWe/alIl ('le,'e)
=
0.
Figure 1 shows the coordinate 'he of the mixing eigenmodes as a function of the linear mismlatch K, which is the only free parameter in Eq. (3). Solid curves in the bifurcation diagram indicate t he spatially stable eigenmodes, whereas the dash ed line corresponds to an instability. The eigenmo le with Bqe = 1 is a stable center of motion for IKI> 2 . This eigenmode represents the second-harmonic c wave. Figure 2 shows the motion of the point (7r 4)) on the curves with constant We. As can be sseen in Fig. 2(a), where IKI> 2, only the phase 4):rotates with distance, whereas the energy conversiion between the two waves is always negligible. Whenever IKI = 2, the second harmonic bifiircates and loses its stability. As can be seen from Figs. 1 and 2(b), the bifurcation generates two stablee eigenmodes with ,e# 1. The coordinates of thei se centers may be easily obtained from the con litions
which periodic energy conversion takes place. The centers of these two regions are elliptic points that represent the stable eigenmodes with 4be = 07T of Fig. 1. We point out also that the location of the separatrix [i.e., the curve W = XC(10= 1, 4) = K] in the phase plane depends on K (IKI< 2).
Physically the spatial instability of the highfrequency wave is nothing but the spontaneous (i.e., initiated by a quantum noise seed) frequencydivision effect.4 The present description enlightens the dynamics of this process in the nonlinear regime, i.e., past the initial stage of amplification of the fluctuations at frequency wo by an undepleted pump at 2 Coo. Note that the wave at frequency Co0 (i.e., the point with bqe = 0) is never an exact eigenmode of the mixing; in fact, as is well known, SHG does not require a seed at frequency 2 coo. It is instructive to demonstrate by two examples the physical insight that may be gained in the behavior of the frequency-conversion process by using phase-plane portraits as in Fig. 2. Consider first
the influence of the phase of an initial secondharmonic seed on the upconversion of a strong pump. For simplicity, we only discuss here the phase-matched case with K = 0. In Fig. 3(a) we show the variation along the nonlinear medium of the power fraction in the second harmonic 77with 10% of the energy in the initial seed (i.e., 7r0= 0.1) and with three different values of the phase )o. The pump excitation factor is y = CORPo/ 2L = 8, where L is the length of the crystal. For 4o = 0, approximately 50% of the energy that is initially in the fundamental is periodically converted to the second harmonic. On the other hand, a proper choice of the phase 4)o, i.e., 4o = ±v/2, leads to complete and aperiodic conversion to 2 C)o. This behavior
!,
d4)/d6 = 0 and sin
O)e = 0,
which lead t() ?le=
+ 12)/2 + K]2 /36 for )e = 0 and 'ie = [(K2 + 12) /2 - K] 2 /36 for 4)e = r. Figure 2(b) sho ws the trajectories traced by the point (-q,4) f or the [(K 2
I
0.5
z0 U)
0
z0 U)
0.5
0 0
0.5
1
DISTANCEz/L
Fig. 3. Fraction of second-harmonic power versus the propagation distance zIL for different values of the initial relative phase for K = 0 and (a) qio= 0.1, y = 8, (b) ,qo= 0.999, y = 15.
May 1, 1992 / Vol. 17, No. 9 / OPTICS LETTERS I
I
1
0 U)
I
,
I
I I
Ti
0
Figure 4 shows the degree of conversion from the fundamental to the second harmonic versus the distance 4,with e = 0.1, for two different values (Q = 0.5 and 1) of the period of the mismatch. As can be seen, the conversion process is not periodic;
I
-
III
0.5 I I
C)
U)
rather, it exhibits an irregular behavior. Moreover
I
II
I , I II
0
0.5
0
1
DISTANCE z/L
Fig. 4. Fraction of second-harmonic power versus the propagation distance zIL with e = 0.1 and y = 32 different values of the spatial frequency of the periodic mismatch: El = 0.5 (solid curve) and fl = 1 (dashed curve).
may be immediately predicted by looking at Fig. 2(b). In fact, in the last two cases the initial point sits right on the separatrix.
Figure 3(b) shows the influence of the initial
for large distances the degree of conversion will depend sensitively on the initial conditions. Clearly in practice fluctuations of the fundamental pump power may also lead to randomly varying conversion for large distances, which will mask the inherent stochasticity of the SHG process. We have verified that the irregular properties of the dynamical system (5) originate from homoclinic crossings that lead to horseshoe chaos.8 This may be analytically checked by calculating the Melnikov integral,'
7T/2,virtually no conversion takes place.
Again this curious behavior could have been easily anticipated by inspecting the fate of the different points on the 1X1 = 1 circle that correspond to the initial conditions in Fig. 3(b). Parametric instabilities lead to a strong sensitivity in the frequency conversion of a wave at frequency Coi to phase changes of a small seed at frequency c02. This effect may be important in improving the design of all-optical switching devices based on second-order nonlinearities. For example, passive AM and FM mode locking have been recently demonstrated by exploiting the phase depen-
dence of the reflectivity by a parametric mirror
with a KTP crystal.6 Moreover these instabilities may even be responsible for spatially chaotic behavior in the wavemixing process, in the presence of a small periodic longitudinal variation of the linear mismatch (or, equivalently, of the nonlinearity). Although spatial chaos is perhaps difficult to observe in practice (the exponential separation of nearby trajectories only occurs after several conversion periods), we believe that pointing out the inherent chaoticity of SHG is relevant from a general physical point of view. Periodic variations of the linear refractive index are introduced in practice by the quasi-phase-matching technique.7 We consider here the case of a small periodic variation about the zero value of the lin-
ear mismatch, which may be described by the Hamiltonian 2N/;(1
-
-q)cos
4+
e cos(f16)'r.
sinh(rfl/2)
(6)
on the separatrix trajectory -q(4)= tanh(6)2 and 4)= 4(P4) = +±-/2. The MelnikovfunctionM has an infinite number of zeros. Therefore the stable and unstable manifolds that enter and exit, respec-
tively, from the saddle point that represents the second harmonic do intersect for a countable number of times. In conclusion, by means of a geometrical description of two-wave mixing, we have studied spatial in-
stabilities and chaos in a quadratic nonlinear medium. The qualitative information that is provided by the phase-plane representation may facilitate the design of devices for parametric conversion.
This research was carried out under the agreement between the Fondazione Ugo Bordoni and the Istituto Superiore Poste e Telecomunicazioni.
References 1. P. A. Franken, A. E. Hill, C. W Peters, and G. Weinreich, Phys. Lett. 7, 118 (1961).
2. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, Phys. Rev. 6, 1918 (1962).
3. P. F. Byrd and M. D. Friedman, Handbook of Elliptic
Integrals for Engineers and Scientists (Springer-
Verlag, Berlin, 1971). 4. S. A. Akhmanov and R. V Khokhlov, Problems of Non-
linear Optics (Gordon and Breach, New York, 1972). 5. A. Yariv, Quantum Electronics (Wiley, New York, 1989). 6. X. M. Zhao and D. J. McGraw, "Parametric mode lock-
ing," IEEE J. Quantum Electron. (to be published). 7. S. Somekh and A. Yariv, Appl. Phys. Lett. 21, 140 (1972).
8. J. Guckenheimer and P. Holmes, Nonlinear Oscilla-
e= xo(q, 4) + 1(m 4, 4) =
aq1 as
2 e'7rfQ sin(fQ4)
with a small seed at oo(here io= 0.999 and y = 15). Whenever 4o= 0, the frequency conversion is periodic and almost complete. Instead, for 4o = -iT/2, the field returns to the second harmonic for all distances after just one cycle. Finally, for
( 0anaw4
M(0) =
phase 4o on the frequency division of a wave at 2wo
(Po =
639
tions, Dynamical Systems, and Bifurcations of Vector (5)
Fields (Springer-Verlag, New York, 1983), p. 184.
