56 3 Order Parameter Equations for Other Nonlinear Resonators

A1 = 1 + ω02Y eiΩt, A2 =				1 + ω02Ze−iΩt .		(3.25)

The changes now include a change in the reference frequency Ω, given by

	Ω =	a1ω2 − a2ω1	,	(3.26)
		a

where a = (a1γ1 + a2γ2) / (γ1 + γ2). This corresponds physically to the elimination of the frequency shift of the signal and idler waves at the generation threshold, which appears at a negative value of the e ective detuning parameter, defined by

ω =

ω1γ1 + ω2γ2

.

(3.27)

γ1 + γ2

With these changes, the equations (3.1) read

∂X

= γ0

−

(1 + iω0) (X + Y Z) + i˜a0

2X ,

(3.28a)

∂Y

∂t

−

−

−

∂Z

= γ1

Y

i˜a1

ω

2

Y + EZ + i (1 + i∆0) XZ

,

(3.28b)

∂t

= γ2

−Z − i˜a2

ω − 2 Z + EY + i (1 + i∆0) XY ,

(3.28c)

∂t

where X, Y and Z are the new pump, signal and idler fields, respectively, and a˜i = ai/a. For the new model (3.28), the trivial nonlasing solution again takes the simple form

X = Y = Z = 0 .

(3.29)

3.3.1 Linear Stability Analysis

The linearization of (3.28) around (3.29), with spatially dependent perturbations, leads to the following growth rates for the perturbations:

λ0 = γ0				−1 + i ∆0 + ak2								,								(3.30)
and
λ	=		1	(γ		+ γ	) i	1	(˜a	γ			a˜		γ	) ω + k2
					1						1			2
1		−	2			2	−	2	1			−			2
			1
																		2
		±		[(γ1 − γ2) +i (˜a1γ1 + a˜2γ2) (ω + k2)]														+ 4γ1	γ2E2.	(3.31)
			2

An instability of (3.29) can be caused by the upper (plus sign) branch of (3.31). The threshold for this instability of (3.29) is

EB(k) = 1 + (ω + k2)2 . (3.32)

3.3 The Complex Swift–Hohenberg Equation for OPOs

57

Unlike the degenerate case, where the bifurcation is static (the eigenvalue is real), the bifurcation here is oscillatory (Hopf), since the perturbations grow with a frequency given by

ν = −	2γ1γ2	(˜a1 − a˜2) ω + k2 .	(3.33)
	γ1 + γ2

Note that in the degenerate case ω1 = ω2, γ1 = γ2 and a1 = a2 (ν = 0), the eigenvalue (3.31) converts into (3.7), obtained in the analysis of the degenerate case, which is real. In fact, the expression (3.31) becomes identical to that obtained for the DOPO, (3.8), with ω1 replaced by ω.

3.3.2 Scales

We assume again the near-to-threshold condition (3.9) and the close-to- resonance condition. The latter now takes the form

2 − ω = εΘ ,

(3.34)

with the additional condition a˜0 2 O(ε), as discused in Sect. 3.2.2. Under these smallness assumptions, the eigenvalue (3.31) can be approximated by

−

2

−

−2

γ−1

λ =

i

a˜1 − a˜2

ω + k2

+

(E

1)

1

ω + k2

2

,

(3.35)

which is similar to (2.9) for the laser case.

The eigenvalue (3.35) is now complex. The imaginary part is O(ε), while the real part is O(ε2). This suggests the introduction of two di erent temporal scales, T1 = εt and T2 = ε2t, and consequently the following expansion for the temporal derivative

∂		= ε	∂	+ ε2	∂	.	(3.36)
			∂T 1
∂t					∂T 2

Again, the order of magnitude of the pump detuning can be chosen freely. For simplicity, in the following we restrict the analysis to the case ω0 O (1).

3.3.3 Derivation of the OPE

Consider the system (3.28), with the smallness conditions described above, together with a power expansion of the fields in the form

∞	∞	∞
X = εnxn , Y =	εn yn , Z =	εnzn .	(3.37)
n=1	n=1	n=1

At the first order

58 3 Order Parameter Equations for Other Nonlinear Resonators

x1 = 0 ,										(3.38a)
y1 = z1 .										(3.38b)
At the second order
	x2 = −y1z1 = − |y1|2 .									(3.39)
The other fields evolve with respect to the slow time T1:
	∂y1	= γ1		(z2 − y2 − i˜a1Θy1) ,						(3.40a)
	∂T 1
	∂z	= γ		(	z + y		+ i˜a		Θz ) .	(3.40b)
	1
	∂T 1
			2	−	2	2		2	1

Taking into account (3.38b), and adding (3.40a) to (3.40b), we obtain a closed equation for the evolution of the signal with respect to the slow time T1,

	∂y1	= −i	γ1γ2	(˜a1 − a˜2) Θy1 .	(3.41)
	∂T 1		γ1 + γ2
Subtracting (3.40a) from (3.40b) gives
z2 = y2 + iΘy1 ,					(3.42)

where we have used the relation (γ1a˜1 + γ2a˜2) / (γ1 + γ2) = 1. At the third order, only the equations for the signal and idler fields are relevant:

1

∂y2

+

∂y1

=

(3.43a)

γ1

∂T1

∂T 2

z

−

y

3 −

i˜a

Θy

2

+ E z + i (1

−

iω

) x z ,

3

1

2

1

0

2

1

1

∂z2

+

∂z1

=

(3.43b)

γ2

∂T 1

∂T 2

−z3 + y3 + i˜a2Θz2 + E2y1 − i (1 + iω0) x2y1 .

The solvability condition is obtained by adding (3.43a) to (3.43b), resulting

in

γ1

∂T1

+ ∂T 2

+ γ2

∂T

+ ∂T 2

(3.44)

1

1

∂y2

∂y1

1

∂z2

∂y1

=

−

i˜a

Θy

2

+ i˜a

Θz

+ 2E y

1 −

2y

y

2 .

1

2

2

2

1 |

1|

The dependence of (3.44) on z2 can be eliminated by substitution of (3.42),

leaving	+ γ2	∂T1		+ ∂T2
γ1
1	1		∂y2		∂y1
= 2E2y1 − 2y1 |y1|2 − i (˜a1 − a˜2) Θy2 − Θ2y1.						(3.45)

We now define the order parameter A as A = εy1 + ε2y2. We can express its evolution on the original timescale as