200 13 Three-Dimensional Patterns

However, these unstable patterns correspond not to local potential minima in the parameter space of Aj but to saddle points.

Numerical integration of (13.8) confirms the stability of the tetrahedral structure. The numerical results are given in Fig. 13.4b, in the form of isolines at 85% of the maximum field intensity. This intensity structure actually consists of two nested structures, shown in Fig. 13.4c and Fig. 13.4d, where the isolines at 93% of the maximum and minimum amplitude are plotted.

13.3 The Nondegenerate OPO

In the case of a nondegenerate OPO, the interaction between the slowly varying envelope of the 3D pump, signal and idler pulses, A0(r , τ, z), A1(r , τ, z) and A2(r , τ, z), respectively, must be considered. This is described by the following set of equations:

∂A

∂2A

2

0

0

= ia ,0

+ ia ,0 A0 − χA1A2

,

(13.12a)

∂z

∂τ2

∂A1

= (v

v )

∂A1

+ ia

∂2A1

+ ia

2

A

+ χA A ,

(13.12b)

∂z

−

∂τ

,1

∂τ2

,1

0

1

1

0 2

∂A2

= (v

v )

∂A2

+ ia

∂2A2

+ ia

2

A

+ χA A .

(13.12c)

∂z

−

∂τ

,2

∂τ2

,2

0

2

2

0 1

Here the coe cients are analogous to those in (13.1), but now correspond to the pump (j = 0), signal (j = 1) and idler (j = 2) waves. The assumption that the changes in the fields during one resonator round trip are small may be made as in (13.1), which allows us to obtain a mapping describing the discrete changes of the subharmonic pulse in successive resonator round trips, and to derive equations of continuous evolution (the order parameter equation).

The analogue of (13.3) is

A0(r , τ, z) = A0(r , τ, 0) − χA1(r , τ)A2(r , τ)

∆l

,

(13.13)

2

and the analogue of (13.5) is

∂A

+ v1

∂A

= P B − A + i( 12 + ∆1)A − |B|2 A ,

(13.14a)

∂t

∂η

∂B

+ v2

∂B

= P A − B + i( 22 + ∆2)A − |A|2 B ,

(13.14b)

∂t

∂η

which is a system of two coupled Ginzburg–Landau equations for the variables

A(r , τ, z) = A1						∆l					(13.15a)
			(r , η, t)χ √						,
						2
B(r		, τ, z) = A (r			, η, t)χ	∆		l		.	(13.15b)
		2				√2