7.4 Experimental Demonstration of the “Restless” Vortex

113

We recall that the coe cient G12 vanishes for phase-insensitive (helical) modes (6.3), and is nonzero for phase-sensitive (flower) modes (6.12).

7.4.2 Phase-Insensitive Modes

In a class A laser, the frequencies of phase-insensitive modes are not a ected by the nonlinear coupling. The vortices rotate around the optical axis with a frequency exactly equal to the di erence of the eigenfrequencies of the two corresponding modes. In the degenerate case the vortices stop rotating. We show below that for a class B laser, the vortices behave di erently.

We look for solutions of (7.17) in the form fi(t) = ni exp (iΩit). The coe cients of the population inversion become

dii = fifi ,				γ						(7.18a)
dij = −fifj							,			(7.18b)
		γ + i (Ωi − Ωj )
and the mode beat frequency ∆Ω = Ωi − Ωj										obeys the equation
1 − ∆Ω γ2			G12		+ 2		+ ∆Ω2	G12		+ 1 = 2γ (D0 − 1) , (7.19)
	∆ω			G11					G11

where ∆ω = ωi − ωj is the mode degeneracy here 2, or, in other words, the di erence of the eigenfrequencies of the modes.

Fig. 7.8. The mode beat frequency ∆Ω versus the mode frequency detuning ∆ω for di erent values of γ. The mode-coupling coe cients correspond to the beating of the TEM01 and TEM01 modes (a circling vortex)

A family of curves of ∆Ω versus ∆ω for di erent values of γ is plotted in Fig. 7.8. The curves clearly show the mode-pushing phenomenon in the case of a class B laser. The pushing is weaker in the intermediate case between a class A and a class B laser (γ ≈ 1) and grows with decreasing γ. When

2 Do not confuse this ∆ω with ∆ω = ω − a 2 used previously in this chapter.

114 7 The Restless Vortex

	γ <	2 (D0 − 1)	,	(7.20)
		G11/G12 + 2

bistability appears. In this case the beat frequency is never zero, which means that a stationary pattern consisting of helical modes is never generated.

The frequency of the self-induced mode beat in the case ∆ω = 0 is given

by

	∆Ω2	=	2γ (D0 − 1) − γ2 (G11/G12 + 2)	,	(7.21)
			G11/G12 + 1
	0

which is proportional to the frequency of the relaxation oscillations in the limit γ 1.

7.4.3 Phase-Sensitive Modes

Mode pushing also occurs for the modes which frequency-pull for class A lasers. For a class B laser, the relation between the mode beat frequency and the detuning is modified with respect to the class A laser, as shown in Fig. 7.9. The mode-locking region shrinks with decreasing γ and bistability appears, similarly to the case of no mode-locking.

Fig. 7.9. The mode beat frequency ∆Ω versus the mode frequency detuning ∆ω for di erent values of γ in the mode-pulling case. The modecoupling coe cients correspond to the TEM10 and TEM01 Gauss–Hermite modes, and their locking to the helical TEM01 mode

A numerical analysis leads to the conclusion that the transition between mode locking and unlocking occurs at a mode beat frequency proportional to the relaxation oscillation frequency: the pattern either oscillates at a frequency larger than the relaxation frequency or locks to a stationary pattern. A pattern consisting of nonlocking modes is never at rest for a class B laser.

The mode locking was investigated experimentally by studying the vortex behavior in a CO2 laser [3]. Figure 7.10 shows the frequency of the spatial