6 Resonators with Curved Mirrors
All the previous chapters have dealt with pattern formation in plane–plane mirror resonators. The order parameter equations (Chaps. 2 and 3) were derived assuming a plane–plane mirror cavity. The vortex dynamics (Chaps. 4 and 5) were analyzed for a plane–plane mirror resonator too. In reality, however, most nonlinear resonators contain curved mirrors. This chapter is devoted to the transverse patterns in nonplanar resonators.
In the presence of curved mirrors, the Maxwell–Bloch equation system
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∂E |
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icr2E + ia 2E + P |
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(6.1a) |
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∂t |
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∂P |
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= −γ (P − ED) , |
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(6.1b) |
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∂t |
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∂D |
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(EP + E P ) . |
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(6.1c) |
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∂t |
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The only di erence between (6.1) and (2.1) is the additional term icr2E in (6.1a), which takes into account the presence of a parabolic mirror in the resonator. Here c = kC/2κ is the focusing parameter, proportional to the total curvature of the mirrors C (positive for a cavity with focusing mirrors, and negative for a cavity with defocusing mirrors).
The resonator equation, in the absence of nonlinearity (P = 0 in (6.1a)), has a simple solution in the form of a decaying Gaussian beam,
E (r, t) = E0 e−(1+iω)te−(r/r0)2 , |
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(6.2) |
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with a frequency ω = ω0 −2√ |
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and a half-width r02 = 2 |
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. The Gaussian |
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a/c |
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ac |
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beam is actually the lowest, fundamental transverse |
mode of the resonator, |
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named the TEM00 mode. In general, the higher-order transverse modes are also exponentially decaying solutions of the linearized version of (6.1a).
In the first two sections of this chapter the case of a quasi-planar resonator, such that the curvature of the resonator mirrors is small, is discussed. The e ects of the curved mirrors are weak in one resonator round trip, and can be calculated perturbatively. This fact allows one to derive the order parameter equation for a laser with curved mirrors (Sect. 6.1). It also allows the application of mode expansion techniques in the theoretical treatment of
K. Stali¯unas and V. J. S´anchez-Morcillo (Eds.):
Transverse Patterns in Nonlinear Optical Resonators, STMP 183, 91–102 (2003)c Springer-Verlag Berlin Heidelberg 2003
92 6 Resonators with Curved Mirrors
laser patterns (Sect. 6.2). In Sect. 6.3 the case of a resonator with strongly curved mirrors is investigated, which in some limits (confocal and self-imaging resonators) also leads to important simplifications of the problem.
6.1 Weakly Curved Mirrors
In this case the e ects of the curvature may be treated perturbatively. We can take into account of these e ects by assuming a weakly varying detuning in the transverse cross section of the resonator, ω(r) = ω0 +cr2. We then (see Fig. 6.1) consider a resonator with focusing parabolic mirrors as a resonator with a length thats varies over the transverse cross section: the resonator is of maximum length along the optical axis, and shorter at some distance from it. This corresponds to a negative detuning that is maximum on the optical axis and decreases away from it.
Fig. 6.1. A resonator with focusing curved mirrors can be considered as a resonator with laterally varying length: on the optical axis the length of the resonator is maximum
If the variation of the resonator length over the beam width is small, then the derivation of the order parameter equation is straightforward. We can repeat the adiabatic-elimination procedure in Chap. 2 by using a coordinate-
dependent resonator detuning ω(r) = ω0 + cr2. This leads to |
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∂A |
= pA + i(a 2 − ω0 − cr2)A |
(6.3) |
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∂τ |
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κ2 |
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− |
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(a 2 − ω0 − cr2)2A − |A|2 A . |
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(κ + γ )2 |
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The same result was obtained in [1] by using a multiscale expansion. In both derivations the smallness parameter is ε = √ac, which is actually the
frequency separation of two adjacent transverse modes. |
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ˆ |
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2 The transverse modes are the eigenfunctions of the operator P |
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cr |
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), in both (6.3) and (6.1a). P is the propagation operator in a linear |
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resonator without gain or decay. The eigenfunctions of the propagator ˆ are
P the transverse modes of the resonator, such that the field distributions do
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ˆ |
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not vary in time, and P An(r) = −iωnAn(r). For a cylindrically symmetric |
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resonator, the Gauss–Laguerre mode sets can be used: |
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Apl(ρ, ϕ) = √π (2ρ2)1/2 |
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Lpl (2ρ2) exp(−ρ2) exp(ilϕ) , (6.4) |
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2 |
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p! |
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