9 Subcritical Solitons I: Saturable Absorber

In the previous chapter, the existence of solitons in subcritical systems was discussed from a general viewpoint. In this and the next chapter, we apply these concepts to concrete nonlinear optical systems. In the present chapter, a laser with a saturable absorber is analyzed. This system shows amplitude bistability, owing to the subcritical character of the bifurcation. Experimental results are also discussed, confirming the theoretical predictions.

9.1 Model and Order Parameter Equation

A simple theoretical model of a laser with an intracavity saturable absorber follows from the Maxwell–Bloch equation system after the adiabatic elimination of the fast atomic variables [1]:

where β represents the nonlinear saturable absorption (nonlinear losses), which relaxes at a rate γ to β0, its maximum value, in the absence of a field; p is the gain parameter; Ip and Is are the gain and absorption saturation intensities, respectively; α represents the linear losses; and aRe and aIm are the di usion and di raction coe cients, respectively.

A single order parameter equation for a laser with a saturable absorber can be found by neglecting the inertia of the absorber (i.e. when γ 1). In this case, the adiabatic elimination of β in (9.1b) leads to [2, 3]

In the derivation of (9.1) and (9.2), a cavity with plane mirrors was assumed. However, as discussed in Sect. 6.3, the validity of these equations can be extended to systems in a self-imaging resonator, just by using a di raction coe cient aIm = l/2k, where l is the displacement of the mirrors from the

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

Transverse Patterns in Nonlinear Optical Resonators, STMP 183, 125–138 (2003)c Springer-Verlag Berlin Heidelberg 2003

126 9 Subcritical Solitons I: Saturable Absorber

self-imaging case, instead of the total cavity length in a plane–plane mirror cavity. Then, in a quasi-self-imaging configuration (with l small), the di raction is correspondingly small.

The di usion term aRe is introduced phenomenologically in (9.2). Its physical sense is a spatial frequency filtering. In the case of a self-imaging resonator with an aperture located in the far field, the aperture imposes a localized profile of the gain in the Fourier domain. In the most convenient case of a parabolic profile, we can write

∂A(k , t) = −aRek2A(k , t) , (9.3) ∂t

where aRe is related to the size of the aperture. Converting (9.3) to the spatial domain by an inverse Fourier transform, we obtain

∂A(r, t) = aRe 2A(r, t) , (9.4)

∂t

which is a di usion term, phenomenologically included in (9.1a).

Owing to this equivalence, the di usion of the field can be controlled by varying the boundary conditions, either by using a finite pump area or by introducing an aperture into the resonator.

The model (9.2), although derived for a class A laser, is also valid, as shown in [4], for a photorefractive oscillator. This equivalence was discussed in Chap. 3.

An important restriction on the validity of (9.1) and (9.2) is that both nonlinear processes, gain and absorption saturation, must occur in the same location of the resonator along its optical axis. In some experimental situations, however, it is convenient to place the nonlinear elements in Fourierconjugate planes, and thus at di erent locations. This near-field–far-field separation can be taken into account if, in (9.1a), the nonlinear gain operator

ˆ ( ), defined as

N p, Ip

acts not on the field but on its Fourier image. This particular configuration is then modeled by the equation

together with (9.1b). Here the operators ˆ and ˆ−1 represent the direct and

F F

The models (9.2) and (9.6), which correspond to two di erent experimental configurations, are the basic models used for the investigation of amplitude domains and spatial solitons in this chapter.

9.2 Amplitude Domains and Spatial Solitons

In both configurations described above, when the system starts from a random initial condition, an ensemble of solitons can be excited in the near field, a situation that dominates in the initial stage of the evolution. For the first configuration, when both nonlinearities act in the near-field domain of a selfimaging resonator, solitons behave nearly independently, and the nonlinear evolution leads to an ensemble of weakly interacting solitons.

A di erent scenario is observed when the gain medium is located in the far-field domain. In this case, in the nonlinear evolution, a strong competition between solitons occurs. The solitons, although well separated in the spatial domain, overlap completely in the focal plane, where the gain medium is placed. The gain saturation depends on the total energy of the radiation. As a result, several solitons well separated from one another in the spatial domain share the same population inversion. The more solitons in the ensemble, the smaller the average energy (and the peak intensity) of a soliton. Owing to the nonlinear absorption, the weaker solitons are more strongly discriminated, initiating a competition, which proceeds until a single soliton survives.

We start the analysis by obtaining the homogeneous lasing solution of (9.2). Neglecting spatial and temporal derivatives, we find that the intensity is given by

The bifurcation from the trivial solution to (9.8) is subcritical (and the lasing solution is bistable) if

When (9.9) holds, bistability occurs in a pump parameter range given by

which corresponds to the parameter region where bright solitons can exist. An exact soliton solution does not exist for (9.2) or (9.6), and therefore

an analytical treatment is possible only using an approximate profile of the

128 9 Subcritical Solitons I: Saturable Absorber

soliton. One reasonable choice is a Gaussian envelope with unknown timedependent parameters,

Inserting the ansatz (9.11) into (9.6) leads to a set of equations for the complex parameters of the soliton. A handleable system of equations can be found by assuming the following conditions:

1.A fast saturable absorber (γ 1). In this case, the absorption variable can be adiabatically eliminated from (9.1b), leading to (9.2).

2.Di usion is strong compared with di raction, aRe aIm. This is the case when the resonator length is tuned to correspond to a self-imaging resonator, where di raction almost vanishes. In this limit the parameters of the soliton take real values.

3.The soliton can be approximated by a parabolic profile if we use, in-

Finally, gathering terms in r, we obtain the following system:

The system (9.12) is still too complicated to obtain analytically tractable steady-state solutions. However, an analysis of this system is useful for obtaining some preliminary conclusions concerning the role of the di usion parameter, aRe.

If di usion is absent (aRe = 0) , the system (9.12) leads to singular solutions, as (9.12b) results in a continuous increase of the curvature c(t) and a corresponding shrinking of the soliton. For p > α+β0/(1+I/Is), the intensity I(t) also grows, leading to an unphysical singularity. This can be understood by analyzing the di erent saturating e ects involved: the absorption leads to a narrowing of the soliton, since it acts more strongly in the parts with less amplitude (far from the peak). If the saturating gain were at the same location in the resonator, it would lead to a broadening of the soliton, since in this case the pumping would be stronger in the tails and weaker at the peak. The balance of these two e ects allows soliton formation in a quasi-planar laser cavity. However, in the present case the saturating gain acts in the Fourier plane, and a broadening in the Fourier domain corresponds to a narrowing of the soliton in the spatial domain. Hence, both nonlinear processes contribute to narrowing the soliton, and lead to a singularity if no other physical