Bibliography |

[1]
J. Garnier,
Stochastic invariant imbedding.
Application to stochastic differential equations with boundary
conditions,
Prob. Th. Rel. Fields **103** (1995), pp. 249-271.

** Abstract:**
We study stochastic differential equations of the type:

[2]
J. Garnier,
Homogenization in a periodic and time-dependent potential,
SIAM, J. Appl. Math. **57** (1997), pp. 95-111.

** Abstract:**
This paper contains a study of the long time
behaviour of a diffusion process in a periodic potential.
The first goal is to determine a
suitable rescaling of time and space so that
the diffusion process converges to some homogeneous limit.
If the potential depends on time, then
the usual diffusive scaling may not be the right one.
Namely some drift may appear with the result that
the asymptotic behaviour of the process is superdiffusive.
This is applied to the homogenization of parabolic
differential equations.

[3]
J. Garnier,
A multi-scaled diffusion-approximation theorem.
Applications to wave propagation in random media,
ESAIM Probab. Statist. **1** (1997), pp. 183-206,
available on
http://www.emath.fr/ps/.

**Abstract:**
In this paper a multi-scaled diffusion-approximation theorem is proved
so as to unify various applications in wave propagation in random media:
transmission of optical modes through random planar waveguides;
time delay in scattering for the linear wave equation;
decay of the transmission coefficient for large lengths with fixed
output and phase difference in weakly nonlinear random media.

[4]
J. Garnier, L. Videau, C. Gouédard, and A. Migus,
Statistical analysis of beam smoothing and some applications,
J. Opt. Soc. Am. A **14** No 8 (1997), pp. 1928-1937.

**Abstract:**This paper aims at developing statistical tools
for beam smoothing analysis.
As applications we study the respective performances
of two-dimensional smoothing by spectral dispersion and
smoothing by optical fiber.
The calculations are valid in the asymptotic
framework of a large number of elements of the random phase plate
and of excited optical modes of the fiber.
Theoretical results and closed form expressions
for the contrast and spatial spectrum of the integrated intensity
of the speckle pattern are derived so as to put into evidence performance
differences between these methods, which are essentially based on
the longer time delay induced by the multimode
fiber with respect to the one induced by the gratings
and on the nature of the spectral broadening.

[5]
J. Garnier, J.-P. Fouque, L. Videau, C. Gouédard, and A. Migus,
Amplification of broadband incoherent light
in homogeneously broadened media in presence of Kerr nonlinearity,
J. Opt. Soc. Am. B **14** No 10 (1997), pp. 2563-2569.

**Abstract:**We have developed a statistical nonlinear
model in order to explain
an anomalous intensity saturation observed in the amplification
of intense broadband incoherent pulses on neodynium-doped glass power chains.
The physics behind this model is basically self-phase modulation creating
new wavelengths scattered in the tail of the gain profile.
The theory shows qualitative agreement with the experimental results.

[6]
J. Garnier,
Asymptotic transmission of solitons through random media,
SIAM, J. Appl. Math. **58** (1998), pp. 1969-1995.

**Abstract:**
This paper contains a study of
the transmission of a soliton through a slab of
nonlinear and random medium.
A random nonlinear Schrodinger equation is considered,
where the randomness holds in the potential and the nonlinear coefficient.
Using the inverse scattering transform,
we exhibit several asymptotic behaviors
corresponding to the limit
when the amplitudes of the random fluctuations go to zero and
the size of the slab goes to infinity.
The mass of the transmitted soliton may tend to zero
exponentially (as a function of the size of the slab)
or following a power law;
or else the soliton may keep its mass,
while its velocity decreases at a logarithmic rate
or even slower.
Numerical simulations are in good agreement
with the theoretical results.
Ref. [41] is a long version of this paper.

[7]
J. Garnier, L. Videau, C. Gouédard, and A. Migus,
Propagation and amplification
of incoherent pulses in dispersive and nonlinear media,
J. Opt. Soc. Am. B **15** (1998), pp. 2773-2781.

**Abstract:**
A statistical model is developed so as
to study all the relevant phenomena which can give rise to
an anomalous intensity saturation
in the propagation of incoherent pulses
in a laser amplifier.
The interplay between diffraction,
self-focusing, group velocity dispersion,
gain narrowing,
and gain saturation is investigated.
Changes in the temporal and spatial characteristics
of the pulses are shown.

[8]
J. Garnier,
Asymptotic behavior of the quantum
harmonic oscillator driven by a random time-dependent
electric field,
J. Statist. Phys. **93** (1998), pp. 211-241.

**Abstract:**
This paper investigates the evolution of the state vector
of a charged quantum particle in a harmonic oscillator driven
by a time-dependent electric field.
The external field randomly oscillates and its amplitude is small,
but it acts long enough so that we can solve
the problem in the asymptotic framework corresponding
to a field amplitude which tends to zero and a field duration
which tends to infinity.
We describe the effective evolution equation of the state vector
which reads as a stochastic partial differential equation.
We explicitly describe the transition probabilities which are
characterized by a polynomial decay of the probabilities
corresponding to the low-energy eigenstates
and give the exact statistical distribution of the energy
of the particle.

[9]
J. Garnier, L. Kallel, and M. Schoenauer,
Rigorous hitting times for binary mutations,
Evolutionary Computation **7** (1999), pp. 173-203 (electronic version unavailable
due to copyright transfert).

**Abstract:**
In the binary evolutionary optimization framework,
two mutation operators are theoretically investigated. For both
the standard mutation, in which all bits are flipped
independently with the same probability, and the 1-bit-flip mutation,
which flips exactly one bit per bitstring, the statistical
distribution of the first hitting times of
the target are thoroughly computed (expectation and variance) up to
terms of order *l* (the size of the bitstrings) in two distinct
situations: without any selection, or with the deterministic (1+1)-ES
selection on the OneMax problem. In both cases, the 1-bit-flip
mutation convergence time is smaller by a constant (in terms of
*l*) multiplicative factor.
These results extend to the case of multiple independent optimizers.

[10]
J. Garnier,
Statistics of the hot spots of smoothed beams produced
by random phase plates revisited,
Phys. Plasmas **6** (1999), pp. 1601-1610.

**Abstract:**
This paper revisits and corrects the statistical theory
of hot spots of speckle patterns such as those
produced by a random phase plate.
Analytical expressions are derived which are sensitively different
from the previous results of Rose and DuBois
(Phys. Fluids B **5**, 590 (1993)).
The departure essentially originates from a careful approach which takes
into account the fact that the fields are complex-valued,
while the standard mathematical theory deals with the maxima of
real-valued Gaussian fields.
This gives rise to an enhancement of the number of the most intense hot spots.
Excellent agreements between the theoretical
formulae and numerical simulations are shown.

[11]
J. Garnier, C. Gouédard, and A. Migus,
Statistics of the hottest spot of speckle patterns generated by smoothing techniques,
Journal of Modern Optics **46** (1999), pp. 1213-1232.

** Abstract:**
This paper is concerned with the statistical distribution
of the maximal hot spots
of speckle patterns such as those generated by optical smoothing methods
designed for inertial confinement fusion.
It is proved that the maximal intensity at the first order is proportional
to the logarithm of the ratio of the pulse volume over the mean hot
spot volume.
Nevertheless the complete description of the maximal intensity exhibits a
quite important variance.
Different ways for reducing either the maximal fluence or the maximal
intensity are investigated, which are based upon time incoherence or
polarization smoothing.

[12]
L. Videau, C. Rouyer, J. Garnier, and A. Migus,
The motion of hot spots in smoothed beams,
J. Opt. Soc. Am. A **16** (1999), pp. 1672-1681.

**Abstract:**
We develop a statistical model which describes the motion of a hot spot
created by smoothing techniques.
We define properly the transverse and longitudinal instantaneous velocities
of a hot spot and quantify its life time.
This relevant parameter is found to be longer than the laser coherence time
defined as the inverse of the spectrum bandwidth.
We apply this model to the most usual smoothing techniques,
using a sinusoidal phase modulation or a random spectrum.
In case of the one-dimensional Smoothing by Spectral Dispersion,
the Smoothing by Longitudinal Spectral Dispersion and the Smoothing
by Optical Fiber, we give asymptotic results for hot spot velocities
and life time.

[13]
F. Kh. Abdullaev and J. Garnier,
Modulational instability in birefringent fibers with periodic and random dispersion,
Phys. Rev. E. **60** (1999), pp. 1042-1050.

**Abstract:**
Modulational instability (MI) of electromagnetic waves in a birefringent fiber
with a periodic dispersion (two-step dispersion management scheme)
is investigated.
The properties of new sidebands are
studied. The strong variation of dispersion leads to the decreasing of the
main MI region and suppression of additional resonance.
In the random dispersion case the MI of all frequencies of modulation
in the normal dispersion region is predicted.
In the anomalous dispersion case the decreasing
of the main MI peak is calculated and changes in the spectral bandwidth
of MI gain are found.
The analytical predictions are confirmed by the numerical simulations
of the full coupled nonlinear Schrodinger equations
with periodic coefficients.

[14]
F. Kh. Abdullaev and J. Garnier,
Solitons in media with random dispersive perturbations,
Phys. D **134** (1999), pp. 303-315.

**Abstract:**
A statistical approach of the propagation of solitons in media
with spatially random dispersive perturbations is developed.
Applying the inverse scattering transform several regimes are
put into evidence which are determined by the mass and the velocity
of the incoming soliton and also by the correlation length of
the perturbation.
Namely, the mass of the soliton is almost conserved if it is initially
large. If the initial mass is too small, then the mass decays
with the length of the system.
The decay rate is exponential in case of a white noise perturbation,
but the mass will decrease as the inverse of the square root
of the length if the central wavenumber of the soliton lies in the tail
of the spectrum of the perturbation.

[15]
J. Garnier,
Energy distribution of the quantum harmonic oscillator under random time-dependent perturbations,
Phys. Rev. E **60** (1999), pp. 3676-3687.

**Abstract:**
This paper investigates the evolution of
a quantum particle in a harmonic oscillator driven by time-dependent forces.
The perturbations are small, but they act long enough so that we can solve
the problem in the asymptotic framework corresponding
to a perturbation amplitude which tends to zero and a perturbation duration
which tends to infinity.
We describe the effective evolution equation of the state vector
which reads as a stochastic partial differential equation.
We exhibit a closed-form equation for the transition probabilities,
which can be interpreted in terms of a jump process.
Using standard probability tools, we are then able to
compute explicitly the probabilities for observing
the different energy eigenstates
and give the exact statistical distribution of the energy
of the particle.

[16]
J. Garnier,
Light propagation in square law media with random imperfections,
Wave Motion **31** (2000), pp 1-19.

**Abstract:**
This paper investigates the deformation of the wavefield
transmitted through a square law medium waveguide.
We consider the situation where the center of the waveguide
randomly oscillates around the optical axis or the radius
of the waveguide randomly pulsates.
The random perturbations are small, but the waveguide is long,
which gives rise to a macroscopic effect of the inhomogeneities.
This effect is characterized by coupling mechanisms between
optical modes, which tend to strengthen high order modes.
Precise expressions for the transmitted wave
are derived which exhibits some remarkable regimes,
where unexpected behaviors such as shift, spreading or even focusing
of the wavefield can be observed.
Numerical simulations are in good agreement with the
theoretical results.

[17]
J. Garnier, C. Gouédard, and L. Videau,
Propagation of a partially
coherent beam under the interaction of small and large scales,
Opt. Commun. **176** (2000), pp. 281-297.

**Abstract:**
This paper deals with the propagation
of Schell-model sources.
Two different and complementary approaches are developed.
The first one is standard and
based on the study of the Wigner distribution function.
The second one follows from a generic statistical representation
of the speckle pattern as the superposition of elementary
and independent modes.
Precise results are obtained for
the macroscopic and microscopic characteristics of the beam:
optical intensity profile, Rayleigh distance,
speckle radius and intensity profiles of the speckle spots.
These results are finally applied to the determination
of the main characteristics of the focal spot
generated by a Kinoform Phase Plate.
We also give the complete expressions of the above quantities
when the conditions of paraxial approximation are not fulfilled.

[18]
L. Videau, C. Rouyer, J. Garnier, and A. Migus,
Generation of a pure phase modulated pulse by cascading effect.
A theoretical approach,
J. Opt. Soc. Am. B **17** (2000), pp. 1008-1017.

**Abstract:**
New techniques to produce a spatio-temporal phase modulation
without using electro-optic devices are proposed and discussed.
By using nonlinear second order effect in crystal, it is possible
to transfer amplitude modulations of a pump wave to the phase
of a signal wave.
For that, we propose the use of a well-known cascading configuration
for which the phase mismatch is high.
Analytical results for spatial and/or temporal incoherent
phase modulation are developed with the correlation functions
formalism. Furthermore highly accurate expansions of the signal
phase and intensity are derived.
The effects of the group velocity difference, the group velocity dispersion
and the diffraction on the transfer of amplitude to phase
modulation are studied.
Finally an experimental demonstration into a KDP crystal
with a sinusoidal pump modulation that creates sinusoidal
phase modulation is proposed.

[19]
J. Garnier and L. Kallel,
Statistical distribution of the convergence
time of evolutionary algorithms for longpath problems,
IEEE Transactions on Evolutionary Computation **4** (2000), pp. 16-30.

**Abstract:**
The asymptotical behavior of a (1+1)-ES process on Rudolph's long
*k*-paths is investigated extensively in this paper. First, in the case of
,
we prove that the long *k*-path is a longpath for the
(1+1)-ES, in the sense that the entire path has to be followed before
convergence occurs. For ,
expected convergence time is
still exponential but some shortcuts will occur meanwhile which speeds up
the process.

Second, in the case of constant *k*, the statistical distribution of
convergence time is calculated, and the influence of population size
is investigated for different
.
Besides, the
histogram of the first hitting time of the solution shows an anomalous
peak close to zero, which corresponds to an exceptional set of events that
speed up the expected convergence time with a factor of *l*^{2}. A
direct consequence of this exceptional set is that performing
independent (1+1)-ES processes proves to be more advantageous than any
population based
.

[20]
J. Garnier and F. Kh. Abdullaev,
Modulational instability
induced by randomly varying coefficients
for the nonlinear Schrodinger equation,
Phys. D **145** (2000), pp. 65-83.

**Abstract:**
We introduce the theory of modulational
instability (MI) of electromagnetic waves in optical fibers.
The model at hand is the one-dimensional nonlinear Schrodinger equation
with random group velocity dispersion and random nonlinear coefficient.
We compute the MI gain which reads as the Lyapunov exponent
of a random linear system.
The sample and moment MI gains appear to be very different.
In the anomalous dispersion regime,
random fluctuations of the nonlinear coefficient
reduces the sample MI gain peak,
although the moment MI peak is enhanced, and
the unstable bandwidth is widened.
Still in the anomalous dispersion regime,
random fluctuations of the group velocity dispersion reduces
both the sample MI gain peakand the moment MI peak.
Finally, in the normal dispersion regime, randomness extends the MI
domain to the whole spectrum of modulations, and increases
the MI gain peak.
The linear stability analysis
is confirmed by numerical simulations of the
full stochastic nonlinear Schrodinger equation.

[21]
J. Garnier,
Propagation of solitons in a randomly perturbed Ablowitz-Ladik chain,
Phys. Rev. E **63** (2001), 026608.

**Abstract:**
This paper deals with
the transmission of a soliton in a
discrete, nonlinear and random medium.
A random lattice nonlinear Schrödinger equation is considered,
where the randomness holds in the on-site potential
or in the coupling coefficients.
We study the interplay of nonlinearity, randomness and
discreteness.
We derive effective evolution equations for the soliton parameters
by applying a perturbation theory of
the inverse scattering transform and
limit theorems of stochastic calculus.

[22]
J. Garnier,
High-frequency asymptotics
for Maxwell's equations
in anisotropic media. Part I: Linear geometric and diffractive
optics,
J. Math. Phys. **42** (2001), pp. 1612-1635.

**Abstract:**
This paper is devoted to the derivations
of the equations that govern the propagation
of pulses in noncentrosymmetric crystals.
The method is based upon high-frequency expansions techniques
for Maxwell equations.
By suitable choices of the scalings we are able to derive
two classical models: geometric optics and
diffractive optics (Schrodinger-like equations).
In the so-called geometric regime we recover the standard results
on the propagation of pulses in crystals (dispersion equation,
polarization states, group velocity).
In the diffractive regime we exhibit original results and
give a closed-form expression for the diffraction
operator which reads as an anisotropic operator.
Given this expression we identify a critical configuration
where the diffraction reduces to a one-dimensional second-order
operator instead of the standard transverse Laplacian.

[23]
J. Garnier,
High-frequency asymptotics for Maxwell's equations
in anisotropic media. Part II: Nonlinear propagation and frequency
conversion,
J. Math. Phys. **42** (2001), pp. 1636-1654.

**Abstract:**
This paper is devoted to the
derivations of the equations that govern
the propagation and frequency conversion
of pulses in noncentrosymmetric crystals.
The method is based upon high frequency expansions techniques
for hyperbolic quasi-linear and semi-linear equations.
In the so-called geometric regime we recover the standard results
on the frequency conversion of pulses in nonlinear crystals.
In the diffractive regime we show that the anisotropy of
the diffraction operator involves remarkable phenomena.
In particular the phase matching angle of a divergent pulse
depends on the distance between the waist and the crystal plate.
Finally we detect a configuration where the beam propagation
in a biaxial crystal involves the generation of spatial solitons
thanks to an anomalous one-dimensional diffraction.

[24]
J. Garnier, F. Kh. Abdullaev, E. Seve, and S. Wabnitz,
Role of polarization mode dispersion
on modulational instability in optical fibers,
Phys. Rev. E **63** (2001), 066616.

**Abstract:**
We introduce the theory of modulational
instability (MI) of electromagnetic waves in fibers with
random polarization mode dispersion.
Applying a linear stability analysis and stochastic calculus
we show that the MI gain spectrum reads as the maximal eigenvalue of
a constant effective matrix.
In the limit of small or large fluctuations, we give explicit
expressions for the MI gain spectra.
In the general configurations we give the explicit form of the
effective matrix and compute numerically the maximal eigenvalue.
In the anomalous dispersion regime, polarization
dispersion widens the unstable bandwidth.
Depending on the type of variations of the birefringence parameters,
polarization dispersion reduces or enhances the MI gain peak.
In the normal dispersion regime, random effects may extend the instability
domain to the whole spectrum of modulations.
The linear stability analysis
is confirmed by numerical simulation of the
full stochastic coupled nonlinear Schrödinger equations.

[25]
J. Garnier,
Solitons in random media with long-range correlation,
Waves Random Media **11** (2001), pp. 149-162.

**Abstract:**
A statistical approach of the propagation of solitons in media
with spatially random potential is developed.
Applying the inverse scattering transform several regimes are
put into evidence which are determined by the mass and the velocity
of the incoming soliton as well as by the correlation length of
the random potential.
Namely, the mass of the soliton is conserved if its initial amplitude
is large enough.
If the initial mass is small, then the mass decays
with the length of the system.
The decay rate is exponential in case of a white noise perturbation,
but it obeys a power law
if the carrier wavenumber of the soliton lies in the tail
of the spectrum of the potential.
Furthermore, the scattered radiation propagates in backward direction
in case of a white noise perturbation,
while it propagates in forward direction (with the same carrier wavenumber
as the soliton) in case of a colored noise with long range
correlation.

[26]
M.-O. Bernard, J. Garnier, and J.-F. Gouyet,
Laplacian growth of parallel needles.
A Fokker-Planck equation approach,
Phys. Rev. E **64** (2001) 041401.

** Abstract:**
Using a conformal transformation to set up the iterative nonlinear equations,
we study analytically the kinetics of growth of parallel needles.
We establish a discrete Fokker-Planck equation for the probability
of finding at time t a given distribution of needle lengths.
In the linear regime, it shows
a short-wavelength Laplacian instability which we investigate in detail.
From the crossover of the solutions to the nonlinear regime, we deduce
analytically the general scale invariance of the two-dimensional
models.

[27]
J. Garnier and L. Videau,
Statistical analysis of the sizes and
velocities of laser hot spots of smoothed beams,
Phys. Plasmas **8** (2001), pp. 4914-4924.

**Abstract:**
This paper presents a precise description of the characteristics
of the hot spots of a partially coherent pulse. The average values
of the sizes and velocities of the hot spots are computed,
as well as the corresponding probability density functions.
Applications to the speckle patterns generated by optical smoothing techniques
for uniform irradiation in plasma physics are discussed.

[28]
J. Garnier,
Long-time dynamics of Korteweg-de Vries
solitons driven by random perturbations,
J. Statist. Phys **105** (2001), pp. 789-833.

**Abstract:**
This paper deals with
the transmission of a soliton in a random medium described
by a randomly perturbed Korteweg-de Vries equation.
Different kinds of perturbations are addressed,
depending on their specific time or position dependences,
with or without damping.
We derive effective evolution equations for the soliton parameter
by applying a perturbation theory of
the inverse scattering transform and
limit theorems of stochastic calculus.
Original results are derived that are very different
compared to a randomly perturbed Nonlinear Schrödinger equation.
First the emission of a soliton gas is proved to be a very general
feature.
Second some perturbations are shown to involve
a speeding-up of the soliton, instead
of the decay that is usually observed in random media.

[29]
J. Garnier and J.-P. Fouque,
Amplification of incoherent light with wide spectrum,
Proceedings of the *Third International Conference
on Mathematical and Numeriacl Aspects of Wave Propagation
Phenomena*,
G. Cohen, ed., SIAM-INRIA, 1995, pp. 584-593.

**Abstract:**
We consider a large-band incoherent pulse
and its propagation in an amplifier.
We show how the intensity grows and how the correlation function behaves
in the medium.
We see how a small nonlinearity may greatly affect the amplification.

[30]
J.-P. Fouque and J. Garnier,
On waves in random media in the diffusion-approximation regime,
Proceedings of the meeting
*Waves in Random and other Complex Media*,
R. Burridge, G. Papanicolaou, and L. Pastur, eds.,
IMA Vol. 96, Springer Verlag, New York, 1997, pp. 31-48.

**Abstract:**
The aim of this contribution is to present recent
results obtained at the "Centre de Mathématiques Appliquées
de l'Ecole Polytechnique" by the group working on waves in random media
(F. Bailly, J. Chillan, J.F. Clouet, J.P. Fouque and J. Garnier).
These results are based
on various generalizations of classical diffusion-approximation results.
In the first section
we study the spreading of an acoustic pulse travelling through a
randomly layered medium
In the second section we present
a justification of the parabolic and white noise approximation
for waves in random media in the high frequency
regime leading to a stochastic Schrodinger equation
The third section is devoted to the effect of a weak
nonlinearity on a wave equation with a random potential.
In the last section we study the amplification
of an incoherent optical pulse propagating in a nonlinear Kerr medium.

[31]
J. Garnier, L. Videau, C. Gouédard, and A. Migus,
Which optical smoothing for LMJ and NIF ?,
Proceedings of the meeting
*Solid state lasers for applications to ICF 1996*,
M. André and H.T. Powell, eds., SPIE, Vol. 3047, 1997,
pp. 260-271.

**Abstract:**
This paper uses statistical theory to investigate the respective performances
of two-dimensional smoothing by spectral dispersion and
smoothing by optical fiber,
both techniques being proposed and implemented for uniform
irradiation in plasma physics.
The calculations are valid in the asymptotic
framework of a large number of elements of the random phase plate
or the excited optical modes of the fiber.
Theoretical results and closed-form expressions
for the contrast and spatial spectrum of the integrated intensity
of the speckle pattern are derived so as to put into evidence performance
differences between these methods.
These differences essentially originate from
the much longer time delay induced by the multimode
fiber with respect to the one induced by the gratings
and from the interplay between the nature of the delay line
vs. the nature of the spectral broadening.

[32]
L. Videau, A. Boscheron, J. Garnier, C. Gouédard, C. Feral,
M. Laurent, J. Paye, C. Sauteret, and A. Migus,
Recent results of optical smoothing on the Phebus Laser,
Proceedings of the meeting
*Solid state lasers for applications to ICF 1996*,
M. André and H.T. Powell, eds., SPIE, Vol. 3047, 1997,
pp. 757-762.

**Abstract:**

[33]
L. Videau, J. Garnier, C. Feral, C. Gouédard,
C. Sauteret, and A. Migus,
Spectral broadening and nonlinear limitation of partially
incoherent pulses in high power amplifiers,
Proceedings of the meeting *CLEO'97*,
OSA Technical Digest Series, Vol. 11, 1997, pp. 353-354.

**Abstract:**
Spatial and temporal incoherent pulses have been amplified in a high
power glass laser up to 1.5 kJ. Performance
has been studied as a function of initial bandwidth and energy input,
and are compared to a statistical model of amplification.

[34]
L. Videau, E. Bar, C. Rouyer, C. Gouédard, J. Garnier, and A. Migus,
Control of the amplification of large band amplitude modulated
pulses in Nd-glass amplifier chain,
Proceedings of the meeting
*Solid state lasers for applications to ICF 1998*,
W. Howard Lowdermilk, ed., SPIE, Vol. 3492, 1999, pp. ???-???.

**Abstract:**
The development of the coming generation
of the Megajoule-class laser requires optical smoothing to obtain
a focal spot large enough with a good uniformity.
Different optical smoothing techniques have been proposed and experimented,
such as Smoothing by Optical Fiber (SOF) and Smoothing by Spectral Dispersion
(SSD).
SOF seems to be an efficient method but experiments have shown
a limitation in term of the amplification performance.
In this paper we present recent results obtained on the Phebus
facility and compare them with a theoretical
model which takes into account the interaction between
time-space incoherence and nonlinear effects.
We also discuss different techniques that can be applied to circumvent
the anomalous intensity saturation.

[35]
L. Videau, J. Garnier, C. Rouyer, and A. Migus,
Speckle movement description in case
of 1D-SSD and longitudinal-SSD for a temporal sinusoidal phase modulation,
Proceedings of the meeting
*Solid state lasers for applications to ICF 1998*,
W. Howard Lowdermilk, ed., SPIE, Vol. 3492, 1999, pp. 277-284.

**Abstract:**
Smoothing techniques are important for Ignition Confinement Fusion
in order to reduce instabilities in the plasma interaction.
The future ICF configurations (French LMJ and US NIF)
are designed for the indirect drive scheme
so that high laser intensitites are likely to induce parametric
instabilities in the extended window and in the hohlraum gas.
A lot of work have been concerned with the effects of smoothing techniques
for reducing parametric instabilities.
Very often theoretical papers consider speckle patterns as
a collection of hot spots
moving in the forward direction.
We have developed a statistical formalism which
is based on the study of the time-space autocorrelation function of
the field.
We are then able to compute the motions of the hot spots.
We apply this method in different 1D-types of SSD techniques
with sinusoidal phase modulation.
Results show that the motions may be backward and/or with speed larger
than light velocity and that the hot spot lifetime may be
longer than the coherence time of the laser.
The relevant parameters are the modulation frequency *f* and depth *b*.
For a given spectrum (equal to the product *f b *)
different speeds and lifetimes are possible,
so the choice of the couple (*f*,*b*)
is crucial for reducing the interaction length between the laser
and parametric instabilitites.

[36]
F. Abdullaev and J. Garnier,
Modulational instability in
birefringent fibers with strong dispersion management,
to appear in the proceedings of the conference
SCT'99 (Solitons, Collapses, and Turbulence, Chernogolovka, Russia, 1999).

**Abstract:**
Strong dispersion management in birefringent fibers
with periodic dispersion is shown to reduce modulational instability domain
and suppress additional resonances.

[37]
F. Abdullaev, J. Garnier, E. Seve, and S. Wabnitz,
Modulational instability in
optical fibers with polarization mode dispersion,
proceedings of the conference NLGW'99
(Nonlinear Guided Waves, Dijon, 1999),
OSA Technical Digest Series, paper WB3.

**Abstract:**
Random polarization
mode dispersion leads to a substantial extension
of the modulational instability
domain in both the normal and anomalous dispersion regime of fibers.

[38]
J. Garnier and L. Kallel,
How to detect all maxima of a function ?,
proceedings of the Second EVONET Summer School on
Theoretical Aspects of Evolutionary Computing (Anvers, 1999),
Springer, Berlin, 2001, pp. 343-370.

**Abstract:**
This paper provides a new methodology allowing one to estimate the number
and the sizes of the attraction basins of a landscape specified
in relation to some modification operator.

[39]
J. Garnier and F. Kh. Abdullaev,
Long-range transmission of solitons in random media,
proceedings of the conference
Photonics West (San José, 2001),
SPIE Proceedings Series, Vol. 4271 (2001), pp. 32-42.

**Abstract:**
A statistical approach of the propagation of solitons in media
with spatially random perturbations is developed.
Applying the inverse scattering transform several regimes are
put into evidence which are determined by the mass and the velocity
of the incoming soliton and also by the correlation length of
the perturbation.
The mass of the transmitted soliton may tend to zero
exponentially (as a function of the size of the slab)
or following a power law;
or else the soliton may keep its mass if it is initially
large enough, while its velocity decreases at a logarithmic rate
or even slower.
Numerical simulations are in good agreement
with the theoretical results.

[40]
J. Garnier,
Some applications of the anisotropic
diffraction in biaxial crystals,
proceedings of the conference Photonics West (San José, 2001),
SPIE Proceedings Series, Vol. 4271 (2001), pp. 138-149.

**Abstract:**
We analyze the propagation
of pulses in noncentrosymmetric crystals
by applying high-frequency expansions techniques
for Maxwell equations.
As a first application we give a closed-form expression
for the anisotropic diffraction operator.
Given this expression we identify a critical configuration
in biaxial crystals where the diffraction reduces
to a one-dimensional second-order
operator for the ordinary wave instead of the standard
transverse Laplacian.
The beam propagation in such a configuration
involves the generation of spatial solitons because of this anomalous
one-dimensional diffraction.
As a second application we present closed-form formulas
for the interference patterns from biaxial crystal plates
between two polarizers. These formulas
agree with experimental patterns.

[41]
J. Garnier,
Theoretical and numerical study of solitons in random media,
technical report, long version of [6].

**Abstract:**
This paper contains a study of
the transmission of a soliton through a slab of
nonlinear and random medium.
A random nonlinear Schrodinger equation is considered,
where the randomness holds in the potential and the nonlinear
coefficient.
Using the inverse scattering transform,
we exhibit several asymptotic behaviors
corresponding to the limit
when the amplitudes of the random fluctuations go to zero and
the size of the slab goes to infinity.
The mass of the transmitted soliton may tend to zero
exponentially (as a function of the size of the slab)
or following a power law;
or else the soliton may keep its mass,
while its velocity decreases at a logarithmic rate
or even slower.
Numerical simulations are in good agreement
with the theoretical results.
A short version of this paper has been published in SIAM,
Journal on Applied Mathematics [6].
We present here more numerical simulations and give the proofs of all
the technical estimates.

[42]
J. Garnier,
Approche probabiliste du séquencage à grande échelle,
technical report (in French).

**Abstract:**
Cette première approche a pour but de présenter
le type de résultats qu'on peut obtenir par une approche probabiliste
du problème, qui repose essentiellement sur une analyse statistique
dans le cadre asymptotique où la longueur
de génome est nettement plus grande que toutes
les autres échelles caractéristiques.
Afin de simplifier cette étude préliminaire,
on a négligé toutes les sources d'erreurs.
On propose donc dans ce document des résultats théoriques
qui décrivent les distributions statistiques des quantités
intéressantes :
les nombres et tailles des trous, des contigs de séquences et de clones,
la longueur du plus grand contig de clones et plus particulièrement
l'"instant" où le contig maximal recouvre
entièrement le génome.
On va commencer par examiner dans la Section 1
la stratégie de séquençage
la plus simple, à savoir le shot gun simple.
On présentera ensuite dans la Section 2
des résultats encore partiels sur le séquençage par
paires d'extrémités.

Avertissement :
Ce rapport est plus un document de travail utile pour la suite
qu'un objet fini.
C'est pourquoi la présentation est encore un peu sèche.

[43]
J. Garnier and L. Kallel,
Efficiency of local search with multiple local optima,
submitted to SIAM Journal on Discrete Mathematics.

**Abstract:**
The first contribution of this paper is a theoretical investigation
of combinatorial optimization problems. Their landscapes are specified by
the set of neighborhoods of all points of the search space.
The aim of the paper consists in the estimation of the number *N* of local
optima and the distributions of the sizes
of their attraction basins.
For different types of landscapes we give precise
estimates of the size of the random sample that ensures that at
least one point lies in each attraction basin.

A practical methodology is then proposed for identifying these
quantities (*N* and
distributions) for an unknown landscape,
given a random sample of starting points and a local steepest ascent search.
This methodology can be applied to any
landscape specified with a modification operator and provides bounds
on search complexity to detect all local optima.
Experiments demonstrate the efficiency of this methodology for
guiding the choice of modification operators, eventually leading to
the design of problem-dependent optimization heuristics.

[44]
J. Garnier,
Instability
of a quantum particle induced by a randomly varying spring coefficient,
to appear in the proceedings of the conference Ascona'99.

**Abstract:**
This paper investigates the evolution of
a quantum particle in a harmonic oscillator whose
spring coefficient randomly fluctuates around its mean value.
The perturbations are small, but they act long enough so that we can solve
the problem in the asymptotic framework corresponding
to a perturbation amplitude which tends to zero and a perturbation duration
which tends to infinity.
We describe the effective evolution equation of the state vector
which reads as a stochastic partial differential equation.
We exhibit a closed-form equation for the transition probabilities,
which can be interpreted in terms of a jump process.
Using standard probability tools, we are then able to
compute explicitly the probabilities for observing
the different energy eigenstates
and give the exact statistical distribution of the energy
of the particle.

[45]
J. Garnier and L. Kallel,
Optimization of binary mutations for evolutionary algorithms,
submitted to Evolutionary Computation.

**Abstract:**
This paper deals with theoretical parameter design for Evolutionary
Algorithms, in the case of the mutation parameter.
First, we prove the explicit relationship between the convergence time
of an Evolutionary Algorithm and its mutation operator, in the case of
the Onemax problem and for all possible memoryless binary mutations.
Second, this relationship is used for deriving the optimal mutation
strategy that minimizes the convergence time of the algorithm
towards the optimum.
We address both the static and time dependent mutation cases, and we
show that the optimal mutation strategy does not depend on the string
size.

[46]
J. Garnier,
Exponential localization versus soliton propagation,
to appear in the NATO ARW proceedings "Nonlinearity and disorder".

**Abstract:**
The scattering of a wavepacket by a random nonlinear medium is analyzed.
In the linear limit strong localization occurs, which means
that the transmission coefficient decays exponentially
with a characteristic localization length.
In some nonlinear homogeneous media
solitons propagate without changes in their shape or velocity.
Solitons are therefore candidates to test the robustness of the
exponential localization in random nonlinear media.
Using the inverse scattering transform for the nonlinear Schrödinger
equation different typical behaviors can be exhibited
depending on the amplitude of the incoming soliton.