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Fig. 6.4. Mode-locked patterns: vortex quadrupole (left ) and a vortex quadrupole that has degenerated into two vortices of the same charge (middle). Both of these plots belong to the mode family 2p + l = 2. The distribution at the right belongs to transverse mode family 2p + l = 3

When the frequency detuning is larger than the threshold value given by (6.17), the modes continue to beat, despite their phase-sensitive coupling. Figure 6.5 shows the evolution of the phase di erence between two modes when the mode detuning is close to the locking threshold. The modes are unlocked but do not evolve freely: the phase di erence, during its cycle of 2π, sometimes varies faster, and sometimes varies more slowly (for favored values of the phase). The intensity of the field also oscillates correspondingly.

Fig. 6.5. Evolution of the phase di erence between the two modes, and of the intensity of one (arbitrary) mode close to the locking threshold

6.3 Degenerate Resonators

The use of resonators with curved mirrors allows one to observe the simplest transverse patterns containing several optical vortices (e.g. as shown in Figs. 4.2 and 4.3). The observed patterns are the result of the interference of transverse modes of the resonator. The patterns are either stationary, if the

98 6 Resonators with Curved Mirrors

modes are locked, or periodic, if the modes are beating. These patterns are weakly nonlinear patterns: role of the nonlinearity is only that of allowing mode competition. The spatial scale of the patterns (e.g. the size of the vortices) does not depend on the nonlinearity, but only on the boundary conditions. This means, equivalently, that the temporal spectrum is a collection of discrete frequencies. These frequencies may be shifted (for frequency-pulling modes) or locked. However, these patterns have a discrete set of frequencies and not a continuous spectrum, which indicates a dependence on the boundaries.

Patterns that are essentially nonlinear, such as those investigated in Chaps. 3 and 4, are of a di erent nature. They depend weakly on the boundaries and, equivalently, their temporal spectrum degenerate into a continuum. These patterns were predicted and calculated for a plane–plane mirror resonator.

Let us now estimate the width of the aperture of a plane–plane resonator

	necessary to observe at least one vortex. The evaluation of the vortex core
	2	√
			2a/p, which for a pump intensity of two
	radius in Chap. 4 gives r0	= 3
	times the threshold value (p = 1)		yields r0 ≈ 2√			. Remembering that the
					a

di raction coe cient is a = QLλ/(4π) (where L ≈ 1 m is a typical resonator length, λ ≈ 1 m is a typical wavelength of the radiation, and Q ≈ 10 is a typical resonator finesse), we obtain r0 ≈ 3 mm. A minimum width of the aperture of order of 1 cm is then necessary to observe one vortex. Correspondingly, for observation of ensembles of vortices, such as those calculated in Chaps. 4 and 5, unrealistically broad apertures are needed.

It is di cult experimentally to build a laser with such a broad aperture, and therefore one must think of some other configuration. In fact, the requirement of a broad-aperture cavity can be achieved not only with a plane–plane mirror resonator, but also with a curved cavity in a self-imaging or near-self- imaging configuration, as will be shown in this section.

The main reason why configurations other than a quasi-planar resonator are mistrusted for experimental investigation of transverse pattern formation concerns the theoretical assumptions. In deriving the order parameter equation (2.26) for lasers and other nonlinear optical systems, the mean-field approximation is used. It is assumed that the radiation changes very little during a resonator roundtrip. The question is whether order parameter equations derived for a mean-field case can be used to describe resonators where the fields vary strongly with propagation. Indeed, for self-imaging resonators (e.g. a confocal resonator) the field changes significantly over a roundtrip. On one roundtrip, the field changes from the near field in a reference plane to the far field in the Fourier-conjugated plane, and back to the near-field.

However, the variation of the field along the cavity can be neglected if the nonlinear material is short enough compared with the length of the cavity. In this case the di ractive propagation in the linear part of the resonator is completely irrelevant. What is significant is how much the fields change over

6.3 Degenerate Resonators

99

a complete roundtrip in the cavity. If the fields change just a little, then order parameter equations derived using a mean-field approximation are valid. In near-self-imaging resonators, the fields indeed change very little in a round trip. In a precisely self-imaging resonator, the fields do not change at all (they are imaged on themselves).

In order to derive an order parameter equation for a near-self-imaging resonator, we must rewrite the field propagation equation (6.1a) for a more general case:

∂E	= κ − (1 + iω) E −	ikCr2	E + i	B	2E + P	,	(6.18)
∂t		2κ		2kκ

where the coe cients B and C are the o -diagonal elements of the propagation (ABCD) matrix. For a near-planar resonator, the ABCD matrix is

ABCD =	C	D	=	c	1		,	(6.19)
	A	B		1	L

−

where the element B is equal to the total resonator length, and the element C is the total curvature of the resonator c. From this, one can retrieve the propagation equation from (6.18), and for small L and c one can derive the CSH equation as the order parameter equation (6.3). In experiments, the so-called 8f resonator, shown in Fig. 6.6, is very convenient.

Fig. 6.6. The 8f resonator. The cavity is self-imaging for a linear length equal to 4f (the full length is equal to 8f); dL is a small deviation from the self-imaging length

The ABCD matrix for a perfectly self-imaging case is the diagonal unitary matrix. If a small deviation from the self-imaging length is present, then the ABCD matrix is

1 dL

ABCD =

,

(6.20)

01

100 6 Resonators with Curved Mirrors

where the element B, which is proportional to the di raction, is equal to the deviation from the focal length. In the precise self-imaging case, the di raction vanishes. A slightly longer resonator is equivalent to a planar resonator with a total length equal to the deviation dL. A resonator slightly shorter than the length of a self-imaging resonator is equivalent to a planar resonator with a negative length. A negative length of the equivalent resonator means a negative di raction coe cient.1

The sign of the di raction has no influence on the linear propagation (spreading) properties. However, combined with a focusing nonlinearity, a change between di raction and antidi raction allows one to change between focusing and defocusing.

Some experiments have been performed with a nearly confocal resonator, as shown in Fig. 6.7. The ABCD matrix in this case is

	−1	dL	(6.21)
ABCD =	−dL/f2	,
		−1

which, for the precisely confocal configuration, corresponds to an antiunitary matrix. Confocal resonators are not completely degenerate. They self-image only after two resonator roundtrips. In one roundtrip, they invert the image. Therefore a confocal resonator can support only patterns that have central symmetry or antisymmetry.

Fig. 6.7. Confocal resonator

Summarizing, one can expect that the nonlinear light dynamics in a selfimaging (4f) resonator are described by the CSH equation (2.26), with a

1A negative di raction coe cient has the following physical sense: the usual di raction of a resonator, in a geometrical approach, relates the angle of the incident ray (with respect to the optical axis) to the lateral shift of the ray in the resonator after one round trip. An incident ray inclined to the right normally returns shifted to the right after a round trip. The larger the resonator di raction is, the larger is the shift. For a resonator slightly shorter than the corresponding self-imaging resonator, an incident ray inclined to the right returns shifted to the left. This can be interpreted as propagation in a resonator with negative di raction.

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101

di raction coe cient calculated using the deviation from the self-imaging length instead of the total length. The dynamics in a confocal resonator can also be described by (2.26), but with additional symmetry restrictions, such that central symmetry, A(r, t) = A(−r, t), or antisymmetry A(r, t) = −A(−r, t), is imposed.

This insight inspired the use of self-imaging resonators for observing essentially nonlinear vortex patterns (Chaps. 4 and 5), bright spatial solitons (Chap. 9), and phase domains and phase solitons (Chap. 11). For example, in some experiments to observe essentially nonlinear patterns we have used a photorefractive oscillator with BaTiO3 as the active medium. In the confocal case, the resonator consisted of two highly reflecting mirrors with a radius of curvature of 350 mm. In the self-imaging case, a resonator with four highly reflecting plane mirrors and four identical intracavity lenses (with focal length f = 100 mm), arranged in a near-self-imaging geometry, was used (Fig. 6.8). The total length of the resonator is L = 8f + l, where l is a small shift from the self-imaging configuration (l f).

Fig. 6.8. Schematic illustration of the experimental arrangement used in the 8f resonator geometry

For a totally open aperture, we observe in the near field (at the plane where the crystal is located) a random small-scale structure, as is typically observed in large-Fresnel-number photorefractive oscillators. In the far field (Fourier-conjugated plane), we observe a set of concentric rings (Fig. 6.9). The rings in the far field indicate the slight deviation from the self-imaging length and are comparable to the rings observed in plane Fabry–P´erot resonators. Di erent rings correspond to di erent longitudinal orders of the resonant spatial wavevectors (or di erent longitudinal modes). A variation of the resonator length leads to a change of the tilt angles of the resonant wavevectors and a change of the ring diameters in the far field.