46 2 Order Parameter Equations for Lasers

negative detuning emits waves at an angle to the optical axis of the resonator (conical emission). Such a detuning-caused pattern-forming instability of lasers was first predicted in [6].

The first and last terms on the right-hand side of (2.26) give the normal form of a supercritical Hopf bifurcation. When the control parameter p = D0 − 1 goes through zero a bifurcation occurs, bringing the system from a stable point corresponding to the nonlasing solution A = 0 to a ring corresponding to the lasing solution |A|2 = p, characterized by a fixed amplitude but arbitrary phase.

2.3.2 Multiple-Scale Expansion

Another method that allows one to derive the OPE is the multiple-scale expansion technique, widely used in nonlinear analysis. The starting point is again a linear stability analysis, but the evolution equation of the order parameter is found as a solvability condition.

This technique consists of the following steps:

1.The relevant variables and parameters of the system are expressed in terms of a smallness parameter ε. This allows one to write the fields as an asymptotic expansion,

∞
	(2.28)
v = εnvn .

n=1

2.The original equations are expanded, and the coe cients of powers of ε are gathered. At order n, the equation has the form Lvn = gn, which is linear in vn, where gn contains the nonlinear interactions and variations of the fields at lower orders, and L is a linear operator.

3.A solvability condition is applied at some order n, to require the exis-

tence of solutions. This is done by requiring that gn be orthogonal to the solutions of the adjoint homogeneous problem, L vn = 0. This process is also known as the Fredholm alternative theorem.

4.Finally, at a given order, a closed equation is obtained for the evolution of one single variable, namely the order parameter.

In the following, we apply this method to the Maxwell–Bloch equations (2.1) [4].

First, we assume that the near-to-resonance condition holds, requiring that ∆ω = a 2 − ω be a small quantity:

∆ω = εΘ .

(2.29)

In the original paper [4], ω and 2 were both required to be small. This restricts seriously the validity of the order parameter equation. Here we note that requiring only ∆ω to be small, leads to the same result, and this allows

2.3 Derivation of the Laser Order Parameter Equation

47

us to consider cases where ω and 2 are moderate or large, as long as their di erence is small.

We make also the near-to-threshold assumption,

D0 = 1 + p , p = µε2 .

(2.30)

With these assumptions, the laser variables depend on slow temporal and spatial scales, which can be determined by using the results of the linear stability analysis. The eigenvalue (2.8), which determines the temporal evolution in the linear stage, has terms of first and second order in ε. We can then define two temporal scales,

T1 = εt , T2 = ε2t ,							(2.31)
which allows us to expand the temporal derivative as
	∂	= ε	∂	+ ε2	∂	.	(2.32)
			∂T1
	∂t				∂T2

Finally, we expand the fields around the trivial solution:

∞	∞	∞
E =	εnen , P =	εnpn , D = D0 + εndn .	(2.33)
n=1	n=1	n=1

All the definitions (2.29)–(2.33) are now introduced into the Maxwell– Bloch equations (2.1). Powers of ε are gathered, and the equations are solved recursively at each order.

At the first order,

e1 = p , d1 = 0 .

At the second order,

1 ∂e1 = −e2 + p2 + iΘe1 , κ ∂T1

1 ∂p1 = e2 − p2 , γ ∂T1

0 = d2 − 12 (e1p1 + p1e1) .

The compatibility of (2.35a) and (2.35b) requires that

∂e1	= i	κγ	Θe1 ,
∂T1		κ + γ

while the polarization is related to the field through

κ

p2 = e2 − iκ + γ Θe1 .

(2.34)

(2.35a)

(2.35b)

(2.35c)

(2.36)

(2.37)