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.
1. Introduction
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 [1–7]. Despite their
amazing simplicity, these equations
deliver a remarkably broad range of solutions in their unstable regime of
operation, ranging from stable period one to typical hierarchical cascades
ending in erratic time traces.
The pulsing solutions of these
equations must be found by numerical integration [5]. That was the generally accepted line of thoughts which
prevailed among the scientific community involved in the laser dynamic issue
ever since Haken proved the analogy between the single-mode laser equations and
the Lorenz model of fluid turbulence [3]. Immediately after the recognition of
such an analogy, an unprecedented rush into computers ensued (see [2, 5], and
references therein for more details). Despite the huge literature that was
devoted to the system for the past three decades, the interest towards the
subject does not seem to dwindle away over the years [8–11].
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 [10], which revealed
to be quite genuine to handle some features of the system unstable behavior.
The procedure, carried out to third-order in field amplitude, has led us to
extract, for the first time in the dynamic literature, a second eigenfrequency,
which has been shown to be an intrinsic property of the unstable permanent
solution of the system. This new natural frequency, added to the well-known
transient frequency, which is rooted in standard first-order small signal
analyses, gives the system all the properties of a two-frequency scheme, from
which most of the physical interpretation of its dynamics have been pulled out
[11].
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.
2. Reminder of the Laser-Lorenz Equations
In their
simplest normalized form, the well-known single-mode homogeneously broadened
laser equations write [2, 3, 10, 11]
()
where E(t), P(t), D(t) represent,
respectively, the laser field-amplitude, the polarization, and the population inversion of the amplifying medium, κ and ℘ are, respectively, the cavity decay-rate and
the population relaxation rate, both scaled to the polarization relaxation
rate, while 2C is an
excitation parameter associated with the external pumping mechanism that is
responsible for population inversion. No attempt is made, here, to recall the
full properties of these equations, since these equations were described in many books and volume
chapters, in addition to the so many journal contributions.
3. Typical Hierarchies of Numerical Solutions
Regular and
irregular pulsing solutions are known to characterize (1) for low ℘ values. Let us center our attention on a period-doubling
sequence that structures itself following an increase of the excitation
parameter 2C,
obtained with cavity and material parameters κ = 3 and ℘ = 0.1.
Figure 1 represents the evolution of the laser-field variable along with the
corresponding frequency spectra, obtained with a standard fast Fourier transform
algorithm. Up to 2C = 18.4, the solution is a symmetric period one solution (see Figure
1(a)) and the frequency spectrum consists of components at Δ, 3Δ, 5Δ, and so forth (see Figure 1(d)). Δ is the
fundamental angular frequency of the signal-time trace. Increasing the
excitation parameter transforms the symmetric solution into an asymmetric
signal, an example obtained with 2C = 27.7 is represented in Figure 1(b). Note
that the corresponding frequency spectrum (see Figure 1(e)) exhibits additional
components at 2Δ, 4Δ, and 6Δ. This constitutes the first signature of a
period-doubling sequence which is observed in the output intensity. For 2C = 29.5, the asymmetric signal
shows period-two oscillations (see Figure 1(c)), while the frequency spectrum
shows additional components at 1.5Δ, 2.5Δ, 3.5Δ, and so forth (see Figure 1(f)). The
corresponding intensity is a period-four signal (as may be checked from Figure 2(c)).
Typical hierarchy
of pulsing solutions obtained with increasing excitation parameter and
corresponding frequency spectra. (a) Symmetric solution with period one,
obtained with 2C = 18.4, (b) asymmetric period one solution, obtained with 2C = 27.9, (c) asymmetric solution with period two, obtained with 2C = 29.5, (d) frequency spectrum of the period one solution, (e) frequency spectrum of the asymmetric solution, showing the emergence of small even
harmonics, and (f) frequency spectrum of the asymmetric period two solution,
where, in addition to the even harmonics of (e), additional peaks appear at
intermediate values between the odd and even components (subharmonic
components).
Typical hierarchy
of pulsing solutions obtained with increasing excitation parameter and
corresponding frequency spectra. (a) Symmetric solution with period one,
obtained with 2C = 18.4, (b) asymmetric period one solution, obtained with 2C = 27.9, (c) asymmetric solution with period two, obtained with 2C = 29.5, (d) frequency spectrum of the period one solution, (e) frequency spectrum of the asymmetric solution, showing the emergence of small even
harmonics, and (f) frequency spectrum of the asymmetric period two solution,
where, in addition to the even harmonics of (e), additional peaks appear at
intermediate values between the odd and even components (subharmonic
components).
Typical hierarchy
of pulsing solutions obtained with increasing excitation parameter and
corresponding frequency spectra. (a) Symmetric solution with period one,
obtained with 2C = 18.4, (b) asymmetric period one solution, obtained with 2C = 27.9, (c) asymmetric solution with period two, obtained with 2C = 29.5, (d) frequency spectrum of the period one solution, (e) frequency spectrum of the asymmetric solution, showing the emergence of small even
harmonics, and (f) frequency spectrum of the asymmetric period two solution,
where, in addition to the even harmonics of (e), additional peaks appear at
intermediate values between the odd and even components (subharmonic
components).
Typical hierarchy
of pulsing solutions obtained with increasing excitation parameter and
corresponding frequency spectra. (a) Symmetric solution with period one,
obtained with 2C = 18.4, (b) asymmetric period one solution, obtained with 2C = 27.9, (c) asymmetric solution with period two, obtained with 2C = 29.5, (d) frequency spectrum of the period one solution, (e) frequency spectrum of the asymmetric solution, showing the emergence of small even
harmonics, and (f) frequency spectrum of the asymmetric period two solution,
where, in addition to the even harmonics of (e), additional peaks appear at
intermediate values between the odd and even components (subharmonic
components).
Typical hierarchy
of pulsing solutions obtained with increasing excitation parameter and
corresponding frequency spectra. (a) Symmetric solution with period one,
obtained with 2C = 18.4, (b) asymmetric period one solution, obtained with 2C = 27.9, (c) asymmetric solution with period two, obtained with 2C = 29.5, (d) frequency spectrum of the period one solution, (e) frequency spectrum of the asymmetric solution, showing the emergence of small even
harmonics, and (f) frequency spectrum of the asymmetric period two solution,
where, in addition to the even harmonics of (e), additional peaks appear at
intermediate values between the odd and even components (subharmonic
components).
Typical hierarchy
of pulsing solutions obtained with increasing excitation parameter and
corresponding frequency spectra. (a) Symmetric solution with period one,
obtained with 2C = 18.4, (b) asymmetric period one solution, obtained with 2C = 27.9, (c) asymmetric solution with period two, obtained with 2C = 29.5, (d) frequency spectrum of the period one solution, (e) frequency spectrum of the asymmetric solution, showing the emergence of small even
harmonics, and (f) frequency spectrum of the asymmetric period two solution,
where, in addition to the even harmonics of (e), additional peaks appear at
intermediate values between the odd and even components (subharmonic
components).
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.
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.
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.
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.
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.
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 1, the field amplitude
undergoes perpetual switching around a zero-mean value. This behavior
constitutes the basis of the strong harmonic expansion method, which is outlined
in the following section.
Representations
of the population-inversion variable show an evolution according to the
following scheme. Up to 2C = 18.4, the time trace is a regular period one
oscillation, with sharp components in the frequency spectrum at 2Δ, 4Δ, 6Δ, and so forth. An increase of 2C transforms the period-one
solution into a period-two signal, the corresponding frequency spectrum shows
additional components at Δ, 3Δ, 5Δ, and so forth. For 2C = 29.5, a period-four
oscillation characterizes the solution while the frequency spectrum consists of
the previous components and emerging small peaks at 0.5Δ, 1.5Δ, 2.5Δ, and so forth.
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.
4. Strong Harmonic-Expansion Analysis of the Long-Term Solutions
Adapted to the long-term
solutions of Figure 1 are the following Fourier expansions:
()
()
()
where the
indexes ip and op, respectively, stand for in phase and out of phase.
When the above expansions are inserted into (1), one finds a hierarchical
set of algebraic relations. These show that the field and polarization expansions
(2a) and (2b) both contain odd terms while the population inversion development
(2c) contains even terms, in conformity with the nonlinear coupling of (1) and with
the spectra shown in Figure 1. The first-, third-, and fifth-order terms for
the laser-field expansion (2a) are derived after lengthy but straightforward
calculations and take the form
()
()
()
with
()
()
The above
procedure, limited to third-order, also gives a closed-form expression for the
long-term pulsation in terms of the fixed decay rates κ and ℘,
and the adjustable excitation parameter 2C, which writes (for details, see [10]):
()
Let us point out,
at this stage, that this last expression carries a fundamental significance in
pulse structuring of the permanentpulsing state, whereas the usual small-signal analyses only give the
transient oscillations of the relaxing solutions.
5. Analytical Structuring of the Pulsing Solutions
5.1. First Example: Period-One Solution
Let us focus on the period-one
example of Figure 1(a) and see how the analytical solution structures itself
with increasing order in laser-field expansion. The long-term operating frequency
is estimated from (4). Indeed if one limits expansion (2a) to first order only,
the solution is a mere sinusoidal trace, while the inclusion of the third-order
field component as evaluated from (3a) and (3b), respectively, gives plain
indications on how the solution structures itself. For example, for the regular
solution obtained just above the instability threshold at 2C = 10, κ = 3, and ℘ = 0.1,
the values of these components evaluate as E1 = 5.2 and E3 = 1.73,
while the corresponding long- term frequency is Δ ≈ 0.42. Thus, to third order, the analytical
field expansion writes
()
In order to obtain a
better fit, the calculations are extended towards fifth order in field
amplitude. For the same parameter values, (3c) yields E5 = 0.8. To fifth order, the analytical field expansion
thus writes
()
The very lengthy and time-consuming
calculations (with an increased rate of error occurrence with increasing order-term
evaluation) required to obtain the fifth-order terms are of no encouragement to
attempt higher-order-terms determination any further. Instead, a much more
convenient way to find the amplitudes of higher-order terms consists in a
direct evaluation of the components peak height from the frequency spectra of
the corresponding numerical solutions. Such a procedure straightforwardly
yields the analytical solution up to the desired order. The highest order term
is dictated by the importance of the corresponding frequency component emerging
from the frequency spectrum. The examples of Figure 1 suggest an expansion up
to the 11th term. Extending the expansion to ninth order still shows some
differences with the exact numerical solution. The adapted expansion requires
taking into account all the terms up to the eleventh order. This demonstrates
the importance of each order in pulse structuring.
The final analytical
expansion of the period-one oscillations thus writes
()
The odd-order expansion (7)
remains valid as long as the solution consists of symmetric period one
oscillations, that is, from 2C2th ≈ 10 (instability threshold) up to 2C = 18.4. Indeed, the field-amplitude components depend on the excitation
parameter and must be evaluated separately for each excitation level. The
values of these components obtained with 2C = 18.4 are directly extracted from the
corresponding Fourier components of Figure 1(d).
Now,
let us focus on the asymmetric and the period-two signals of Figures 1(b) and 1(c), respectively. The following subsection demonstrates that even for these seemingly more complicated examples; the above procedure appears to be fairly-well
adaptable to find the corresponding analytical Fourier expansions.
5.2. Construction of the Period-Doubling Cascade
The asymmetric solution of Figure 1(b)
stems from the appearance of additional and intermediate even components in the
corresponding frequency spectrum (see Figure 1(e)). Thus, obviously an
asymmetric regular solution must contain odd as well as even frequency
components that stem from stronger nonlinear interactions between the
laser-field variable, which evolves with an odd expansion structure, with the
population inversion whose Fourier development carries an even form. The field
expansion thus writes
()
The
amplitudes are evaluated from the fast Fourier transform of Figure 1(e). When
these amplitudes are
inserted into (8), one finds, with the use of (4), the complete analytical representation
of Figure 1(b).
Finally,
the period-two solution of Figure 1(c) is constructed by inferring terms that
contain frequency components of the form 1.5Δ, 2.5Δ, 3.5Δ, and so forth. These terms emerge from the corresponding
frequency spectrum of Figure 1(f). The period-two solution of Figure 1(c) is
described by the following field-expansion:
()
The amplitudes are obtained from the fast Fourier transform of Figure 1(f), and the obtained solution carries the same structure as Figure 1(c).
As
a final illustration of the period-doubling cascade, we represent the
field-intensity signals corresponding to the hierarchy of Figure 1 along with
their analytical counterparts for comparison. These are shown in Figure 2. The
first few period-doubling sequences show a perfect match between the numerical
and the analytical solutions. These representations demonstrate that the numerical
period-doubling sequence peculiar to (1) can be constructed analytically with
an adapted strong-harmonic expansion method. Indeed, the method is applicable
inside the whole control-parameter space that exhibits periodic solutions. Some
other interesting examples, obtained with the more complex infinite-dimensional
system, are given in [12].
6. Conclusion
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 [12].
However, the involved algebra is much more complicated, and the analytical
calculations were limited to third order in field amplitude.
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 5 show
strong enough evidence of the efficacy of the strong harmonic expansion
technique. It is worth to mention
again and stress on the fact that the main clue behind the construction
of analytical self-pulsing solutions is rooted on the analytical expression of
the driving angular frequency, which in turn is rooted in the first- and third-order
terms of the harmonic expansions. Once, the driving frequency is evaluated,
finding the complete solutions merely amounts to evaluate the laser-field
components of the corresponding Fourier expansions.
2Meziane B., Construction et etude de modèles simples en dynamique des lasers: réduction des systèmes de dimensions infinies, Ph.D. dissertation, 1992, Université de Rennes I, Rennes, France.
5Abraham N. B., Lugiato L. A., and Narducci L. M., Overview of instabilities in laser systems, Journal of the Optical Society of America B. (1985) 2, no. 1, 7–14.
8Tsang Y. H., [email protected], King T. A., Ko D.-K., and Lee J., Output dynamics and stabilisation of a multi-mode double-clad Yb-doped silica fibre laser, Optics Communications. (2006) 259, no. 1, 236–241, https://doi.org/10.1016/j.optcom.2005.08.040.
9El-Sherif A. F. and King T. A., Dynamics and self-pulsing effects in Tm3+-doped silica fibre lasers, Optics Communications. (2002) 208, no. 4–6, 381–389, https://doi.org/10.1016/S0030-4018(02)01587-0.
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