48 2 Order Parameter Equations for Lasers

Both of these relations will be useful later, at the next order. From (2.35c), we obtain

d2 = − |e1|2 .

(2.38)

At the order O(ε3), the equations read

1 ∂e1

+

1 ∂e2

= −e3 + p3 + iΘe2 ,

(2.39a)

κ ∂T2

κ

∂T1

1 ∂p1

1 ∂p2

+

= e3 − p3 + µe1 + d2e1 .

(2.39b)

γ

∂T2

γ

∂T1

This system reduces, by applying a solvability condition that actually consists in eliminating the dependence on third-order contributions, to

1

1

∂e1

∂e2

κ2

+

+

= µe1 −e1 |e1|2 +iΘe2 −

Θ2e1 , (2.40)

κ

γ

∂T2

∂T1

(κ + γ )2

where, in obtaining the last term, we have used

1 ∂p2

1 ∂e2

1

κ

∂e1

1 ∂e2

κ2

=

−

iΘ

=

+

Θ2e1 . (2.41)

γ

∂T

1

γ

∂T

1

γ

κ + γ

∂T

1

γ

∂T

1

2

(κ + γ )

Equation (2.40) depends now only on the field amplitude at di erent orders. Let us now define an order parameter A = εe1 + ε2e2. The evolution of the order parameter with respect to the original time t is given by

∂t

= ε

∂T1 + ε

∂T1

+ ∂T2

= ε

κ + γ iΘe1

+ ε

∂T1

+ ∂T2 .

∂A

2

∂e1

3

∂e2

∂e1

2 κγ

3

∂e2

∂e1

(2.42)

Finally, (2.42), expressed in terms of the original variables, gives the evolution equation of the order parameter,

1		1		∂A			2
	+					= (D0 − 1) A − A |A|
κ		γ		∂t
						+i a 2 − ω A −	κ2	a 2 − ω	2	A , (2.43)
							(κ + γ )2

which coincides with (2.26), obtained by using the adiabatic-elimination procedure.

References

1.M.C. Cross and P.C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys. 65, 851 (1993). 33

References 49

2.R. Graham and H. Haken, Laserlight – first example of a second order phase transition far from thermal equilibrium, Z. Phys. 237, 31 (1970). 34, 41

3.K. Staliunas, Laser Ginzburg–Landau equation and laser hydrodynamics, Phys. Rev. A 48, 1573 (1993). 41

4.J. Lega, J.V. Moloney and A.C. Newell, Swift–Hohenberg equation for lasers, Phys. Rev. Lett. 73, 2978 (1994). 41, 46

5.J.B. Swift and P.C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A 15, 319 (1977). 45

6.L.A. Lugiato and R. Lefever, Spatial dissipative structures in passive optical systems, Phys. Rev. Lett. 58, 2209 (1987). 46