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A newly proposed strong harmonic-expansion method is applied to the laser-Lorenz equations to analytically construct a few typical solutions, including the first few expansions of the well-known period-doubling cascade that characterizes the system in its self-pulsing regime of operation. These solutions are shown to evolve in accordance with the driving frequency of the permanent solution that we recently reported to illustrate the system. The procedure amounts to analytically construct the signal Fourier transform by applying an iterative algorithm that reconstitutes the first few terms of its development.

The standard laser equations describe semiclassical
atom-field interactions inside a unidirectional ring cavity. In the case of a
homogeneously broadened system, these equations are transformed,
after adequate approximations, into a simple set of three nonlinearly coupled
differential equations, the so-called laser-Lorenz equations [

The pulsing solutions of these
equations must be found by numerical integration [

In connection with analytical
aspects of the laser-Lorenz dynamics, we have recently revisited the system
with the application of a strong harmonic expansion method [

This paper aims at demonstrating that the analytical information extracted from such third-order expansion analysis naturally yields a simple method to construct periodic solutions. Typical as well as peculiar examples, including the well-known period-doubling that structures following an increase of the excitation parameter are directly built.

In their
simplest normalized form, the well-known single-mode homogeneously broadened
laser equations write [

Regular and
irregular pulsing solutions are known to characterize (

Figure

Typical hierarchy
of pulsing solutions obtained with increasing excitation parameter and
corresponding frequency spectra. (a) Symmetric solution with period one,
obtained with

Comparison between the field-intensities, obtained (a), (b), (c) numerically and (d), (e), (f) analytically. The period-doubling sequence appears clearly in both series of traces.

Let
us point out that, for all the solutions represented in Figure

Representations
of the population-inversion variable show an evolution according to the
following scheme. Up to

Before constructing the corresponding analytical structures, let us turn into basic analytical considerations. In order to characterize the essential features of the pulsing solutions, we will first give a brief outline of the essential steps of an adapted strong harmonic-expansion method that yields analytical expressions for the angular frequency of the pulsing state as well as for the first few harmonic components of the corresponding analytical solutions.

Adapted to the long-term
solutions of Figure

When the above expansions are inserted into (

Let us focus on the period-one
example of Figure

In order to obtain a
better fit, the calculations are extended towards fifth order in field
amplitude. For the same parameter values, (

The final analytical
expansion of the period-one oscillations thus writes

The odd-order expansion (

Now,
let us focus on the asymmetric and the period-two signals of Figures

The asymmetric solution of Figure

The
amplitudes are evaluated from the fast Fourier transform of Figure

Finally,
the period-two solution of Figure

The amplitudes are obtained from the fast Fourier transform of Figure

As
a final illustration of the period-doubling cascade, we represent the
field-intensity signals corresponding to the hierarchy of Figure

We have extended a recently introduced strong harmonic-expansion
technique to the laser-Lorenz equations to find an analytical description of a
few typical solutions that characterize the self-pulsing regime of the single mode
homogeneously broadened laser operating in a bad-cavity configuration. The
method presented here is fairly general to be applicable to other differential
sets of equations that qualitatively possess the same self-pulsing properties.
In particular, the well-known integro-differential “Maxwell-Bloch” equations
adapted to describe the unstable state of a single-mode inhomogeneously
broadened laser have been, as well, handled with the same approach [

Even in the much simpler cases, presented in
this paper, the very lengthy and awkward algebra involved in the determination
of high-order terms has not allowed us to evaluate higher than the fifth-order
components. Despite these limitations, we have proposed a fairly simple method
to determine the complete solutions by inferring the field-amplitudes directly
from the field-amplitude spectra obtained with a fast Fourier transform of the
temporal signals. The method has been applied to describe, analytically, the
first few solutions of a period-doubling sequence, peculiar to the Lorenz
equations, which takes place following an increase of the excitation parameter.
Even though most of the analysis focused on fixed control parameters
(cavity-decay rate and population inversion relaxation rate), the method
contains enough generality to be extended to the entire parameter space that
exhibits regular pulsing solutions. The examples chosen in Section