108 7 The Restless Vortex

Fig. 7.4. A restless vortex taken from the first plot in Fig. 7.2, and magnified four times. The vortex moves to the left, and is squeezed along the direction of motion

The radius of the circular orbit of the vortex depends in general on the di usion coe cient in the equation for the field (the g-factor), and also on the decay rate of the population inversion, γ. The exact dependence is unclear. The numerical calculations indicate that the radius of the vortex orbit increases with a decrease of the di usion factor g and the population inversion decay rate γ. However, even in the purely di ractive case g = 0, the vortex trajectory is a curved, not a straight line. This is related to the fact that a spatially dependent gain occurs because of the spatial mismatch between the field zero and the maximum of the population inversion. This creates a spatial profile of both the background field intensity and the background field phase. The gradients of both the intensity and the phase push the optical vortex in accordance with the results of Chap 2: the vortex motion induced by the phase inhomogeneity is parallel to the phase gradient, and should result in a parallel motion of the restless vortex in the purely di ractive case. However, the inhomogeneity of the intensity causes a vortex motion perpendicular to the intensity gradient. The amplitude and phase gradients together cause a circular vortex motion, even in the purely di ractive case.

7.3 Vortex Lattices

The nonstationarity of a single vortex in a class B laser suggests that vortices arranged in a lattice will also be, in general, nonstationary. Numerical integration of the class B laser equations confirms this suggestion.

As shown in [2], vortex oscillations in a vortex lattice can be synchronized in di erent ways. “Optical”, “acoustic” and several “mixed” oscillation modes have been identified here, on the basis of numerical integration of the laser equations with reflecting (zero flux) boundary conditions. When periodic boundaries are used, two pure cases of self-induced dynamics of vortex lattices are observed: (1) an “optical“ oscillation mode, where neighboring vortices along a diagonal (with the same topological charge) oscillate in antiphase, and (2) a parallel translation of the vortex lattice. The analogue of

7.3 Vortex Lattices

109

this translational motion in the case of reflecting boundaries is an “acoustic” oscillation mode.

7.3.1 “Optical” Oscillation Mode

Figure 7.5 shows four snapshots of a vortex lattice, where neighboring vortices along the diagonal oscillate in antiphase. The motion resembles the oscillation of atoms in a crystal when an “optical” oscillation mode is excited.

An interesting fact is that the vortices in the oscillating square vortex lattice are arranged hexagonally most of the time. The average intensity distribution is, however, of square symmetry.

Fig. 7.5. Oscillations of a vortex lattice as obtained by numerical integration of (7.8). The parameters are g = 0.2, D0 = 2, ω = 0.335, γ = 0.1. The time between snapshots is t = 5

The oscillation frequency of the “optical” oscillation mode can be found by noting that the far field of an oscillating vortex lattice, as shown in Fig. 7.6, is a superposition of two cross-roll patterns rotated by 45◦. One cross-roll

has a resonant transverse wavenumber, with eigenfrequency ω0 =√ak2; the wavenumber of the other pair of cross-rolls is larger by a factor of 2, which corresponds to the eigenfrequency ω1 = 2ak2. This leads to a beat frequency ∆ω = ak2.

Fig. 7.6. Spatial Fourier spectrum of the optical field in the oscillating vortex lattice shown in Fig. 7.5 (left). Right, the composition of the wavevectors in the oscillating vortex lattice

110 7 The Restless Vortex

7.3.2 Parallel translation of a vortex lattice

If the gain line is extremely narrow (∆ω ak2), then both cross-rolls cannot be simultaneously excited, and the “optical” oscillation mode does not appear. Instead of oscillating, the vortices start to drift. Drift occurs if the tilted-wave components of the vortex lattice have di erent frequencies.

To investigate the drift of a vortex lattice, we use the ansatz

	j	Ai exp(i kj · r − iωj t) ,	(7.10a)
	A(r, t) =
	=1,4	l
	D(r, t) = d0 +		(7.10b)
		dj,k exp [i (kj − kl ) · r − i (ωj − ωl) t] ,
		j=1,4 =1,4

where the frequencies obey ωj = 0. For a parallel translation of the vortex lattice, the frequencies of antiparallel TW components are of opposite sign: ω1 = −ω3 and ω2 = −ω4. The drift direction depends on the frequency ratio:

v · kj = ωj /ω, where ω = ω21 + ω22 is the total oscillation frequency, and

ω2 =	γ (D0 − 1)	.	(7.11)
	4 + γ

As (7.10) and (7.11) indicate, oscillations of the intensity of the optical field appear at every location, and are related to the translational motion of the pattern. The frequency of the intensity oscillations in the limit of slow

population inversion is ωosc = 2ω =						γ(D0	− 1), and thus this frequency is
smaller than the frequency of the					relaxation oscillation by a factor of				√
									2.
The corresponding velocity of the pattern is |v| = ω/k, given by
	=			.
v			γ (D0 − 1)						(7.12)
| |			2k

A parallel translation of a square vortex lattice is possible for periodic boundary conditions only. For reflecting boundaries, a square vortex lattice either is fixed if the area is too small, or oscillates periodically as a whole. The latter oscillation mode of the vortex lattice has been called an “acoustic” oscillation mode in [2].

Figure 7.7 illustrates the “acoustic” oscillation mode in the case of one spatial dimension. In this case we can excite a standing-wave pattern (owing to the reflecting boundaries). The standing wave oscillates back and forth, as shown in the figure. A vortex lattice in 2D oscillates in a similar way.

Summarizing, the vortex lattice either oscillates in an “optical” mode or undergoes a parallel translation, for periodic boundary conditions and a class B laser. The vortex lattice is never at rest. For reflecting (zero flux) boundary conditions, the lattice can also oscillate in an “optical” mode. A parallel translation is impossible in this case, and therefore the lattice either displays an “acoustic” oscillation mode or is at rest. For boundaries di erent

7.4 Experimental Demonstration of the “Restless” Vortex

111

Fig. 7.7. Temporal oscillations of a 1D optical pattern (a standing wave) as obtained by numerical integration of the 1D version of (7.8) with reflecting lateral boundaries. The spatial coordinate is horizontal; the time varies from top to bottom by t = 100. The parameters are D0 = 1.5, γ = 0.16 and g = 0.1

from those discussed above, the vortex oscillations can synchronize in di erent ways. For example, in the case of a rotationally symmetric boundary (a circular aperture, or a spherical mirror), small vortex ensembles can rotate.

All self-induced oscillations of vortices and vortex lattices in the case of a class B laser occur at a frequency of the order of the relaxation oscillation frequency.

7.4 Experimental Demonstration

of the “Restless” Vortex

The “restless vortex” phenomenon suggests that simple transverse patterns consisting of one or several optical vortices, which are stationary in class A lasers, may be nonstationary (periodic or chaotic) in class B lasers. In a class A laser, the frequencies of the transverse modes are pulled towards one another and may lock to a common frequency, owing to the nonlinear coupling via the population inversion common to both modes, as described in Chap. 6. In contrast, in a class B laser, a vortex is nonstationary, and transverse modes may not lock in this case.

In this section, a cavity with curved mirrors is considered. In this case, vortex solutions correspond to the excitation of high-order modes in the cavity. Many of the expressions used here have already been derived in Chap. 6.

7.4.1 Mode Expansion

We look for solutions of (7.8) containing a small number of transverse modes, using a mode expansion technique for the optical field, represented by

A(r, t) = fi(t)Ai (r) , (7.13)

i

112 7 The Restless Vortex

where i is an arbitrary combination of the mode indices p and l.1 The expansion (7.13) was used in Chap. 6 in the limit of a class A laser, where the population inversion can be adiabatically eliminated. In this case D(r, t) was expressed as a function of the field A(r, t),

		D0	2
D(r, t) =				≈ D0 − |A(r, t)|
		1 + |A(r, t)|2
	= D0 − i		fi(t)fj (t)Ai (r)Aj (r) .		(7.14)

Since the population inversion is not enslaved by the optical field in a class B laser, we expand the population inversion too:

D(r, t) = D0 − dij (t)Ai (r)Aj (r) . (7.15)

i

The motivation for the expansion (7.15) is the expression (7.14), since in the limiting case of a class A laser dij = fifj . We insert the expressions (7.13) and (7.15) into (7.8), multiply (7.8a) by Ak and integrate over the two-dimensional space. This leads to the following equations for the mode coe cients fi (t) and dij (t) :

		∂fi		= pi fi − i (ωi − ω0) fi −
					Γjkil fj dkl ,	(7.16a)
		∂τ
					jkl
	∂d
	ij			= −γ dij − fifj ,		(7.16b)
	∂τ				il
where pi and the nonlinear coupling coe cients Γjk						have been defined in
(6.8).

The model (7.16) is, to our knowledge, the simplest one capable of describing a multi-transverse-mode class B laser. In principle, it describes an arbitrary number of transverse modes, but in practice it is useful if a small number of modes is excited. We analyze two-mode states in this section, for

which the equations (7.16) read
		∂f1		= p1f1		− i(ω1 + ω0)f1					(7.17a)
		∂τ
				f	(G d + G d )		−	f	(G d + G		d ) ,
		∂f2		− 1		11 11 12 22		2	12 12	12	21
				= p2f2		− i(ω2 + ω0)f2					(7.17b)
		∂τ
	∂d			−f2(G22d22 + G12d11) − f1(G12d21 + G12d12) ,
		ij		= −γ		dij − fifj ,					(7.17c)
		∂τ

where the coupling coe cients are as defined in (6.9).

1Actually we expand (7.8a) here, but take into account of a spatially dependent detuning corresponding to the curved mirrors of the resonator.