We show that, in their unstable regime of operation, the "Maxwell-Bloch" equations that describe light-matter
interactions and dynamics inside a bad-cavity-configured laser carry the same resonance properties as any externally
driven mechanic or electric oscillator. This finding demonstrates that the non-linearly coupled Laser equations belong to
the same universal family of forced oscillatory systems. The primary difference is that while mechanical or electrical
systems are put into resonance with an external sinusoidal force with constant amplitude, the resonance-curve of the laser
equations is described exclusively in terms of linear pump scans, for fixed cavity and material decay rates. In both cases
however, the damping factors play the same fundamental role. In addition, the basic phase factor between the external
excitation mechanism and the mechanical or electric oscillator response is shown to play the same essential role in the
dynamic response of the "Maxwell-Bloch" equations with respect to the external driving pump level. Dephasing
mechanisms occur between successive-order components of an adapted strong-harmonic expansion that describes the
regular self-pulsing solutions of light-matter interactions inside a bad-cavity configured laser cavity.
This paper aims at revisiting the basic Lorenz-Haken equations with two-fold harmonic-expansion approaches, yielding
new analytical information on both the transient and the long term characteristics of the system pulse-structuring. First,
we extend the well-known Casperson Hendow-Sargent weak-sideband analysis to derive a general formula that gives
the value of the transient frequencies, characteristic of the laser relaxing towards its long-term state, either stable or
unstable. Its validity is shown to apply with a remarkable precision at any level of excitation, both beyond and below the
instability threshold. Second, we put forward a strong-harmonic expansion scheme to analyse the system long-term
solutions. Carried up to third order in field amplitude, the method allows for the derivation of a closed form expression
of the system eigen-frequency (derived here for the first time in three decades of laser dynamics) that naturally yields an
iterative algorithm to build, analytically, the regular pulsing solutions of the Lorenz-Haken equations. These solutions
are constructed for typical examples, extending well beyond the boundary region of the instability domain, inside which
the laser field amplitude undergoes regular pulsations around zero-mean values.
This paper aims at revisiting some of the self-pulsing properties of the integro-differential "Maxwell-Bloch" equations
that describe single-mode inhomogeneously broadened, including semi-conductor and fibre lasers, focusing mainly on
the dynamic gain contour, which is shown to undergo deep modifications during pulse build-up. First, we extend the
well-known Casperson Hendow-Sargent weak side-band approach and delimit its applicability with respect to dynamic
gain profiling. In particular, we demonstrate, both numerically and analytically that the small-sideband method only
describes centre-line saturation which occurs in the small periodic-oscillations regime. In the strong self-pulsing regime,
however, lateral saturation along the gain curve occurs as demonstrated with numerical simulations. The small side-band
approach fails to describe such gain structuring during pulse build-up. Second, we put forward a strong-harmonic
expansion method which reveals quite adapted for the description of the dynamic gain profiles. Extended to third-order
in field amplitude and to second order in population inversion, the method allows for the extraction of analytical
expressions that retrieve the lateral saturation effects with a remarkable precision.
We apply a simple harmonic expansion method to the single-mode laser equations to obtain a few analytical solutions in the self-pulsing regime of operation. The long-term solutions are derived for typical examples, characteristic of the boundary region of the instability domain where the laser field-amplitude undergoes regular oscillations around zero-mean values.
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