4 Zero Detuning: Laser Hydrodynamics

and Optical Vortices

In this chapter, the properties of vortices in class A and class C lasers at zero detuning are investigated. As shown in the previous chapters, these classes of laser can be described by the complex Swift–Hohenberg equation (2.26), which in the zero-detuning case reads

∂A			κ2
		= pA + ia 2A −		a2 4A − |A|2 A .	(4.1)
∂τ			(κ + γ )2

This equation is similar to the complex Ginzburg–Landau equation, except for the di usion term. Instead of the Laplace operator describing the usual di usion, here one has a second-order Laplace operator, corresponding to super-di usion. Therefore, adopting the terminology of [1], we call (4.1) the laser Ginzburg–Landau (LGL) equation.

Equation (4.1) can be simplified by using the following normalizations for
time, space and the order parameter: τ → t/p, x → x						and A → A√		.
					a/p
							p
Now, instead of (4.1), we can deal with an LGL			equation with only one free
parameter,
	∂A
		= A + i 2A − g 4A − |A|2 A .		(4.2)
	∂t

The remaining parameter in (4.2) is g = pκ2/(κ + γ )2, and thus all the properties of the solutions of the LGL equation depend on this g-factor, which has the meaning of a super-di usion coe cient.

4.1 Hydrodynamic Form

The LGL equation (4.2) can be converted into a hydrodynamic form by using the Madelung transformation. Originally, Madelung demonstrated [2] that

the transformation of the order parameter A(r, t) = ρ(r, t) exp [iΦ(r, t)] brings the nonlinear Schr¨odinger equation into a hydrodynamic form, where the intensity plays the role of a (super)fluid density, and the phase gradientΦ(r, t) the role of a velocity. Performing the same transformation in (4.2), we obtain

K. Stali¯unas and V. J. S´anchez-Morcillo (Eds.):

Transverse Patterns in Nonlinear Optical Resonators, STMP 183, 65–79 (2003)c Springer-Verlag Berlin Heidelberg 2003

66

4 Zero Detuning: Laser Hydrodynamics and Optical Vortices

∂ρ

= 2ρ − 2ρ2 − 2 (ρ Φ) ,

(4.3a)

∂t

∂Φ

(2√

)

4

ρ

2

= −g Φ +

− ( Φ)

.

(4.3b)

∂t

√

ρ

In (4.3) the super-di usion has been assumed to be small: all the terms containing the coe cient g have been neglected, except for the most sig-

nificant one (g 4Φ in (4.3b)). Normalizing again the spatial variables with

√

the change r/ 2 → r, and rewriting (4.3) in terms of the velocity in the transverse space v = v = Φ, we obtain

∂ρ

+ (ρv) = 2ρ − 2ρ2 ,

(4.4a)

∂t

(√

)

∂v

g

ρ

+ (v )v = −

4v +

.

(4.4b)

∂t

2

2√

ρ

Equations (4.4) describe, in the hydrodynamic analogy, the evolution of a “photon fluid”. The left parts of the equations can be interpreted as the conservation of mass and momentum (the analogues of the continuity and Euler equations, respectively). The right part of (4.4a) describes the presence of sources and sinks: “mass” is created owing to the linear gain and dissipates owing to the saturation of the inversion in the laser. The right part of (4.4b) can be interpreted as the dissipation of momentum due to super-viscosity, which is proportional to the g-factor. The last term in (4.4b), called the “quantum pressure” term, has no analogue in standard fluid mechanics.

The “photon fluid” in a laser as described by the LGL equation does not possess the usual compressibility, where the internal pressure is proportional to some local function of the density (note that the quantum pressure in (4.4b) is nonlocal). However, a classical pressure can occur if we consider an additional self-focusing or self-defocusing mechanism, e.g., if a focusing– defocusing Kerr material is present in the laser resonator. In this case the LGL equation becomes

∂A	= A + i 2A − g 4A − (1 + iα) |A|2 A .	(4.5)
∂t

The Euler equation (4.4b) is modified because of this self-focusing/defo- cusing to

∂t

− ρ

− 2

2√ρ

∂v

+ (v

)v =

p

g

4v +

(√

ρ

)

,

(4.6)

where the classical pressure is proportional to the fluid density, since p = αρ2/2. For a defocusing medium (α > 0) the compressibility relation is “normal”, while for a focusing medium (α < 0) it is “anomalous”.

Looking ahead (this topic will be treated in detail in Chap. 6), in the case of a curved (parabolic) resonator, an additional external potential appears in