# Femtosecond laser inscriptions in Kerr nonlinear transparent media: dynamics
in the presence of K-photon absorptions, radiative recombinations and electron
diffusions

Emmanuel O. Akewejea, G. Baderb, Alain M. Dikandéc and P. Kameni Nteutsed
CONTACT A. M. Dikandé. Email<EMAIL_ADDRESS>aAfrican Institute for
Mathematical Sciences (AIMS-Ghana), East Legon Hills, Accra, Ghana.;
bInstitute of Mathematics, Brandenburg Technical University, Cottbus-
Senftenberg, Germany;
cLaboratory of Research on Advanced Materials and Nonlinear Science
(LaRAMaNS), Department of Physics, Faculty of Science, University of Buea P.O.
Box 63 Buea, Cameroon;
cAfrican Institute for Mathematical Sciences (AIMS-Cameroon), Crystal Garden
P.O. Box 608 Limbe, Cameroon.

###### Abstract

Femtosecond lasers interacting with Kerr nonlinear optical materials,
propagate in form of filaments due to the balance of beam diffraction by self-
focusing induced by the Kerr nonlinearity. Femtosecond laser filamentation is
a universal phenomenon that belongs to a general class of processes proper to
ultrashort lasers processing systems, associated with the competition between
nonlinearity and dispersion also known to promote optical solitons. The
present work considers a model describing femtosecond laser inscriptions in a
transparent medium with Kerr nonlinearity. Upon inscription, the laser stores
energy in the optical material which induces an electron plasma. The model
consists of a cubic complex Ginzburg-Landau equation, in which an additional
$K$-order nonlinear term takes into account $K$-photon absorption processes.
The complex Ginzburg-Landau equation is coupled to a time first-order
nonlinear ordinary differential equation, accounting for time evolution of the
plasma density. The main objective of the study is to examine effects of the
competition between multi-photon absorptions, radiative recombination and
electron diffusion processes, on temporal profiles of the laser amplitude as
well as of the plasma density. From numerical simulations, it is found that
when the photon number (i.e. $K$) contributing to multiphoton ionization is
large enough, taking the electron diffusion processes into account favours
periodic structures in temporal profiles both of the laser and the plasma
density. The pulse repetition rate in the optical soliton train is increased
with increase of the electron diffusion coefficient, while the plasma density
is a train of multi-periodic anharmonic wave patterns.

###### keywords:

Femtosecond laser inscription; Kerr nonlinear materials, electron plasma;
Complex Ginzburg-Landau equation; Drude equation; optical solitons; electron-
hole radiative recombinations; multiphoton absorptions.

##  1 Introduction

Femtosecond lasers have become one of most efficient tools in a large variety
of modern industrial material processings [1, 2, 3, 4]. Their applications
extend from accurate manufacturing of electronic devices to fine cutting,
polishing and machining of hard metals, ceramics and soft plastics for
fabrications of various micro textures with fine and stylished designs [2, 3,
4, 5]. These applications are the fruit of outstanding progress in femtosecond
laser micromachining technology observed over the recent past [1, 2, 3, 4, 5,
6], that motivated a renewed interest in ultrashort laser interactions with
dielectric media [1, 2, 3, 4, 5, 6].

In particular, the interaction of femtosecond lasers with a transparent
material is a strongly nonlinear process that leads to a highly localized
energy storage in the optical material [7, 8, 9, 10, 11, 12]. In this
nonlinear process two physical phenomena are crucial for an optimum energy
storage, namely multiphoton ionizations and inverse Bremsstrahlung absorption
[10]. When the first free electrons, produced via multiphoton ionization
processes, accumulate sufficient energy by inverse Bremsstrahlung absorption,
they trigger avalanche ionization that causes the generation of an electron
plasma [10]. This electron plasma further absorbs energy from the incident
laser field to move toward the incoming laser field, thus forming new layers
of electron plasma in the material [10]. The generation of electron plasma can
be accompanied with an increase of the absorption coefficient of the optical
medium, favoring energy deposition from the radiation field at a relatively
short time scale [10].

Theoretical investigations of femtosecond laser interaction with transparent
media have been carried out in some past studies [7, 8, 9, 10, 11, 12]. In
these studies the optical field propagation in the transparent medium is
described by the complex Ginzburg-Landau equation, while time evolution of the
plasma density is represented by a Drude-type equation, with specific terms
accounting for physical ingredients assumed to contribute to photoionization
processes. In principle, these models are intended to give account of the
laser self-focusing due to Kerr nonlinearity of the transparent medium,
enhanced by multiphoton absorption processses [13, 14]. However most of these
past studies [7, 8, 9, 10, 11, 12] were focused mainly on assessing thermal
effects related to material processings using femtosecond lasers, and
particularly their impact on the quality (e.g. the degree of fineness and
refinement) of processed materials. Nevertheless in relatively more recent
works [15, 16], attention was paid on dynamical properties of some among the
models proposed e.g. in refs. [7, 8, 9, 10, 11, 12]. Thus in ref. [15] the
laser dynamics, together with time evolution of the plasma density, were
investigated considering the model proposed by Petrovic et al. [12], according
to which the plasma generation was governed essentially by avalanche
ionization and K-photon absorption processes. From the assumption that
depending on conditions in the optical medium, the femtosecond laser will
operate in one of two distinct regimes i.e. either continuous-wave or soliton
regimes, it was established [15] that for this specific model a decrease in
$K$, the number of simultaneously absorbed photons, would stabilize continuous
waves in the anomalous dispersion regime. However any continuous wave
propagating in the transparent medium in the normal dispersion regime, is
expected to be always unstable irrespective of the value of $K$. The same
study [15] also established that large values of the phase shift per laser
roundtrip, will favor a stronger stability of continuous-wave operation in the
normal dispersion regime for any value of $K$, whereas in the anomalous
dispersion regime only for small values of $K$ continuous waves would be
stable. Extending this first study, in ref. [16] the same authors considered
the problem but including electron-hole radiative recombination processes
[10]. In this second work they found that in steady state, the electron plasma
density will strongly depend on the electron-hole radiative recombination
coefficient. They obtained that an increase of the electron-hole radiative
recombination coefficient causes the laser amplitude to increase in the
soliton regime, whereas the electron plasma density is more and more strongly
decreased in time as the radiative recombination coefficient is increased.

In the studies of refs. [15, 16] however, dynamical properties of femtosecond
lasers in optical Kerr media were investigated ignoring elctron diffusion
processes. Yet in several studies and particularly the work of ref. [10],
reviewing recent experimental and theoretical developments on the topic, the
role of electron diffusions in the buildup of the electron plasma was
unambiguously emphasized. In fact, taking into consideration the contribution
of electron diffusion processes in the generation of the electron plasma,
could be relevant for a better description of dynamical properties of the
optical field and the plasma density.

In this work we investigate the dynamics of femtosecond lasers interacting
with transparent material with optical Kerr nonlinearity, in the presence of
avalanche ionizations, electron-hole radiative recombinations and electron
diffusion processes [10]. Since in two previous studies, we considered the
same problem restricting analysis to avalanche ionizations [15], and to a
combination of avalanche ionizations and radiative recombination processes
[16] as being responsible for the electron plasma generation, in the present
study our attention will be focused on the influence of electron diffusion
processes on dynamical properties of the laser and the plasma density.
Starting we look at characteristic properties of the model in the steady-state
regime. This first analysis will provide relevant insight onto the influence
of electron diffusions on fixed-point solutions to the model equations. Then
temporal profiles of the laser amplitude and the plasma density will be
generated by solving numerically the equations of motion. In this second
analysis, emphasis will be on the effects of varying the electron diffusion
coefficient on temporal profiles of the laser amplitude and the plasma
density. For this dynamical analysis numerical solutions will be sought in two
distinct regimes, namely the regime of very small diffusion coefficient and
the regime of relatively large diffusion coefficient.

##  2 The model

Femtosecond laser interaction with transparent media is a nonlinear process
that involves an ultrashort laser field (of the order of femtosecond), focused
on an optical material with a nonlinear refractive index. Upon absorption, the
laser field propagates in the optical material storing heat along its path,
thus ionizing the material and releasing electrons via inverse Bremsstrahlung
absorption. Once released free electrons contribute to the avalanche
ionization by favoring multiphoton recombination processes, and their
diffusions in the nonlinear optical medium.

Couairon and Mysyrowicz [10] suggested that in the case of uni-directional
propagation, the dynamics of a femtosecond laser and of the density of the
electron plasma, could be represented by the following set of coupled
nonlinear equations:

$\displaystyle i\frac{\partial u}{\partial z}$ $\displaystyle=$
$\displaystyle\delta\frac{\partial^{2}u}{\partial
t^{2}}-\sigma|u|^{2}u-i\gamma(1-i\omega_{0}\tau_{0})\rho u-i\mu|u|^{2K-2}u,$
(1) $\displaystyle\frac{\partial\rho}{\partial t}$ $\displaystyle=$
$\displaystyle\nu|u|^{2}\rho+\alpha|u|^{2K}-\eta\rho-a\rho^{2}.$ (2)

More explicitely eq. (1) describes a complex field $u(z,t)$ propagating along
$z$, where the first term in the right-hand side accounts for the group-
velocity dispersion, the second term describes the Kerr nonlinearity of the
transparent medium, the third term describes both the plasma absorption and
laser defocusing, and the last term accounts for multiphoton absorption
processes. The second equation, i.e. eq. (2), governs time evolution of the
electron plasma density $\rho$. In this equation, the first term in the right-
hand side represents the avalanche impact ionization, the second term
describes multiphoton ionization processes, the third term accounts for
electron diffusion processes and the fourth term represents electron-hole
radiative recombination processes. Characteristic parameters in eqs. (1) and
(2) are defined as follow (their units are also found in refs. [10, 16]):

  * •

$K$ is the photon number contributing to multiphoton absorption processes,

  * •

$\delta$ is the coefficient of group-velocity dispersion,

  * •

$\sigma$ is the Kerr nonlinearity coefficient,

  * •

$\omega_{0}$ (in MHz) and $\tau_{0}$ (in fs) are the frequency and
characteristic relaxation time of the electron plasma respectively,

  * •

$\mu$ is the $K$-photon ionization coefficient,

  * •

$\nu$ is the coefficient of avalanche impact ionization,

  * •

$\alpha$ is the coefficient of plasma balance due to multiphoton absorption,

  * •

$\eta$ is the electron diffusion coefficient,

  * •

$a$ is the electron-hole radiative recombination coefficient.

Experimental values of these characteristic parameters are given in most works
where the models are proposed from experiments, and also in the review paper
ref. [10]. Here we are interested in a qualitative analysis of the system
dynamics, reason why we will not bother about exact values of these
characteristic parameters. However our analysis will take care of their
appropriate respective signs.

We are interested in solutions to eq. (1) that describe waves of real
amplitude, modulated in space and time according to the following ansatz [16,
17, 18, 19, 20]:

$u(z,t)=A(z,t)\exp i\left[\phi(z,t)-\omega z\right],$ (3)

where $A(z,t)$ is the real amplitude, $\phi(z,t)$ is the instantaneous phase
also assumed real, and $\omega$ is a nonlinear shift in the propagation
constant. Let us introduce a reduced time as [16, 18, 19, 20]
$\tau=t-\vartheta z$, where $\vartheta$ is the inverse pulse velocity. Using
this new time variable we can write $A(z,t)\equiv A(\tau)$ and
$\phi(z,t)\equiv\phi(\tau)$. Substituting eq. (3) in eqs. (1) and (2) and
separating real from imaginary parts, we obtain the following set of coupled
first-order nonlinear ordinary differential equations:

$\displaystyle M^{\prime}$ $\displaystyle=$ $\displaystyle-\frac{\vartheta
y}{\delta A}+\frac{\gamma\rho}{\delta}+\frac{\mu
A^{2(K-1)}}{\delta}-\frac{2My}{A},$ (4) $\displaystyle y^{\prime}$
$\displaystyle=$ $\displaystyle\frac{MA\vartheta}{\delta}+\frac{\omega
A}{\delta}+\frac{\gamma\omega_{o}\tau_{o}\rho A}{\delta}+\frac{\delta
AM^{2}}{\delta}+\frac{\sigma A^{3}}{\delta},$ (5) $\displaystyle\rho^{\prime}$
$\displaystyle=$ $\displaystyle\nu A^{2}\rho+\alpha
A^{2K}-\eta\rho-a\rho^{2},$ (6) $\displaystyle A^{\prime}$ $\displaystyle=$
$\displaystyle y,\hskip 8.5359pt\phi^{\prime}=M,$ (7)

where prime symbols denote first-order derivatives with respect to $\tau$. In
our study we shall be intersted mainly in the stationary solutions for which
$\vartheta=0$. [16, 17]

##  3 Fixed-point solutions

In the steady-state regime, solutions to the system of first-order ordinary
differential equations (4)-(6) are fixed points of the system. These solutions
therefore correspond to the roots of the following set of three coupled
multivariate polynomials in $A$, $M$ and $\rho$:

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{\gamma\rho}{\delta}+\frac{\mu
A^{2(K-1)}}{\delta}-\frac{2My}{A},$ (8) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{\omega A}{\delta}+\frac{\gamma\omega_{o}\tau_{o}\rho
A}{\delta}+\frac{\delta AM^{2}}{\delta}+\frac{\sigma A^{3}}{\delta},$ (9)
$\displaystyle 0$ $\displaystyle=$ $\displaystyle\nu A^{2}\rho+\alpha
A^{2K}-\eta\rho-a\rho^{2},$ (10)

with in addition $y=0$ suggesting that at steady state, any arbitrary real
constant will be a suitable solution for the laser amplitude. From eqs.
(8)-(10), we find the fixed-point solutions in $M$ and $\rho$ as being:

$\displaystyle M^{2}$ $\displaystyle=$
$\displaystyle\frac{\omega_{o}\tau_{o}\mu A^{2(K-1)}-\sigma
A^{2}-\omega}{\delta}.$ (11) $\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{-\eta\pm\sqrt{\eta^{2}+4a\bigg{[}\alpha-\frac{\nu\mu}{\gamma}\bigg{]}A^{2K}}}{2a}.$
(12)

Instructively formula (12) suggests two possible fixed-point solutions for the
plasma density, i.e. a negative-valued solution and a positive solution. Since
$\rho$ must be positive or zero to be physically significant, the negative
solution is physically irrelevant. We must therefore keep the solution with
positive sign, such that the fixed-point solution for the plasma density is:

$\rho=\frac{-\eta+\sqrt{\eta^{2}+4a\bigg{[}\alpha-\frac{\nu\mu}{\gamma}\bigg{]}A^{2K}}}{2a}.$
(13)

The fixed point of $\rho$ will be zero when $\nu$ (the coefficient of
avalanche impact ionization), $\alpha$ (the coefficient of plasma balance due
to multiphoton absorption), $\mu$ (the mutiphoton ionization coefficient) and
$\gamma$ (the strength of coupling of the electron plasma to the optical
field), are related through:

$\alpha=\frac{\nu\mu}{\gamma},$ (14)

for nonzero values of the electron-hole radiative recombination coefficient
$a$ and the electron diffusion coefficient $\eta$. Note that the steady-state
value of $\rho$ given by eq. (13), implies that $\alpha$ should be greater
than $\nu\mu/\gamma$ and the electron-hole radiative recombination coefficient
$a$ should be positive. Since the multiphoton ionization coefficient $\mu$ can
be negtive, choosing $\alpha$, $\nu$ and $\gamma$ all positive will be enough
to ensure a physically consistent value of $\rho$ at steady state. Fig. 1
shows the variation of the fixed point eq. (13), as a function of the input
power $I=A^{2}$ of femtosecond laser for four different values of $K$, and
some values of the electron diffusion coefficient $\eta$. Values of other
characteristic parameters of the model are indicated in the caption.

Figure 1: Variation of the fixed point of the plasma density $\rho$, given by
eq. (13), as a function of the input power $I=A^{2}$ for $\eta=0$, $0.25$,
$0.5$, $0.8$, and for four different values of the photon number $K$. We have
used $\delta=-0.5$, $\sigma=1.5$, $\omega_{0}\tau_{0}=0.2$, $\mu=-0.25$.

According to fig. 1 the fixed point of the plasma density increases when the
input power $A^{2}$ is increased, irrespective of the value of $K$. However,
an increase of $K$ affects the law of dependence of $\rho$ with the input
power. Indeed, fig. 1 shows that $\rho$ is more and more strongly nonlinearly
dependent on $A^{2}$.

The increase of the fixed-point value of the plasma density with increase of
the input power, can be understood by invoking the fact that the higher the
intensity of femtosecond laser, the higher the amount of energy stored by the
femtosecond laser in the material. Considering such higher energy stored in
the material, photoionization processes should be stronger and consequently
the density of the electron plasma too should be higher. When both $K$ and
$\eta$ are very small, the fixed-point solution $\rho$ is seen to increase
linearly with the laser input power $A^{2}$. However, as $K$ is increased the
fixed-point value of $\rho$ turns gradually from linear to parabolic
(nonlinear) in $A^{2}$. The flexion marking the deviation to a parabolic curve
is, as evidenced in figure 1, more and more sharp as the electron diffusion
coefficient $\eta$ is increased. It is to be remarked that in the nonlinear
regime, the fixed point of the plasma density at the same input laser
intensity will be lowered down as $\eta$ increases. These behaviours clearly
suggest that the electron diffusion processes play a key role in the process,
namely the fixed-point analysis carried out in this section suggests that the
relationship between the threshold value of the plasma density and the input
laser intensity can be controlled by electron diffusion processes.

##  4 Soliton-train solutions

In section 3, fixed-point solutions to the coupled first-order ordinary
differential equations (4)-(7), describing the laser dynamics together with
temporal evolution of the plasma density, were examined. We obtained that any
arbitrary real constant could be a fixed point of the laser amplitude, whereas
the fixed-point solution for the plasma density $\rho$ was a function of the
input laser intensity $A^{2}$. The dependence of $\rho$ on $A^{2}$ emerged to
be linear for small $K$ and $\eta$, however increasing $K$ and $\eta$
simultaneously, turns the linear dependence into a parabolic function
reflecting a nonlinear variation of the steady-state value of the plasma
density with the input power.

In this section, we wish to look at temporal profiles of the laser amplitude
$A(\tau)$ and of the plasma density $\rho(\tau)$. For this purpose, we solved
numerically the set of coupled first-order ordinary differential equations
(4)-(7), using an explicit sixth-order Runge-Kutta scheme [21]. To highlight
the influence of electron diffusion processes on shape profiles of the laser
amplitude and the plasma density, the numerical solutions $A(\tau)$ and
$\rho(\tau)$ were generated considering a very small value of $\eta$ and a
relative large value of this parameter. In figs. (2), (3) and (4), we display
time series of $A(\tau)$, $M(\tau)$ and $\rho(\tau)$ corresponding to temporal
profiles of the laser amplitude, the laser instantaneous frequency
$M(\tau)=\partial\phi/\partial\tau$ and plasma density $\rho(\tau)$, for four
different values of $K$ (i.e. $K=2$, 3, 4, 5) and for $\eta=0.005$.

Figure 2:  Temporal profile of the laser amplitude $A(\tau)$, generated
numerically for different values of $K$ and for $\eta=0.005$.

Figure 3: Temporal profile of the instantaneous frequency $M(\tau)$ of the
laser, generated numerically for different values of $K$ and for $\eta=0.005$.

Figure 4:  Temporal profile of the plasma density $\rho(\tau)$, generated
numerically for different values of $K$ and for $\eta=0.005$.

Fig. 2 features a laser amplitude $A(\tau)$ forming a periodic train of
anharmonic oscillations, with an exponential amplification as time goes when
$K=2$. An increase of $K$ gradually softens the exponental amplification and
for large values of $K$, the laser amplitude deploys as a regular train of
quasi-periodic temporal pulses. Likewise, the instantaneous frequency $M$ and
the plasma density $\rho$, shown respectively in figs. 3 and 4, evolve from
irregular to regular quasi-periodic nonlinear wave patterns as $K$ is
increased from 2 to 5. Note the very small damping of the plasma density with
time for $K=5$, contrasting with the very strong exponential amplification
observed in the first graph of the same figure when $K=2$.

To highlight the influence of $\eta$ on numerical solutions, in figs (5), (6)
and (7), time series of $A(\tau)$, $M(\tau)$ and $\rho(\tau)$ are represented
now for $\eta=0.25$ and keeping all other characteristic parameters fixed.

Figure 5:  Temporal profile of the laser amplitude $A(\tau)$, generated
numerically for different values of $K$ and for $\eta=0.25$.

Figure 6:  Temporal profile of the instantaneous frequency $M(\tau)$ of the
laser, generated numerically for different values of $K$ and for $\eta=0.25$.

Figure 7:  Temporal profile of the plasma density $\rho(\tau)$, generated
numerically for different values of $K$ and for $\eta=0.25$.

Temporal profiles of the laser amplitude $A(\tau)$, the laser instantaneous
frequency $M(\tau)$ and the plasma density $\rho(\tau)$ for $\eta=0.25$,
represented in figs. 5, 6 and 7 respectively, are very clearly different from
results obtained when $\eta=0.005$: the exponential growth of $A(\tau)$ that
characterized this quantity for small values of the electron diffusion
coefficient $\eta$ and for $K=2$, is completely wiped out at higher values of
$\eta$. In this later case, the laser amplitude $A(\tau)$ instead reproduces a
regular train of pulse structures, whose temporal period (i.e. the pulse
repetition rate in the periodic train) is increased as $K$ increases. So to
say, temporal profiles of the laser amplitude obtained from numerical
simulations suggest that stronger electron diffusion processes will favor
trains of well separated pulses with large repetition rates. The plasma
density in this case too displays a relatively more regular structure, as seen
in fig. 7, compared with numerical results of the same quantity for smaller
value of $\eta$ and represented in fig. 4. The laser instantaneous frequency
$M(\tau)$ also becomes more regular as $\eta$ is increased, as seen in fig. 6
compared with the behaviour observed in fig. 3 when $\eta$ is relatively
smaller.

##  5 Conclusion

Nonlinear phenomena in artifical optical systems have attracted a great deal
of interest over the past years, due to their contributions to innovative
technologies in optics and photonics in general for all‐optical signal
processings. These technologies rely on exotic optical circuits fabricated via
different methods, including femtosecond laser inscriptions in optical and
photonic materials [22]. Femtosecond laser inscription technique rests on a
nonlinear absorption of an ultrashort laser pulse, focused in the bulk of a
transparent material with eventually an optical Kerr nonlinearity [10]. Since
the laser interaction with the transparent material is a nonlinear process [7,
8, 9, 10, 11, 12], a localized modification of the bulk material is favored
which generates a positive refractive index change [10, 23, 24, 25, 26].

To gain a better understanding of the laser dynamics interacting with
transparent materials, recently some theoretical models were proposed [7, 8,
9, 10, 11, 12]. In these models it is assumed that upon absorption, the
femtosecond laser propagates in the transparent material storing energy along
its path. The energy stored causes an ionization of the bulk material and
subsequently induces an electron plasma in the material. Theoretically, the
femtosecond laser propagation in the optical Kerr medium can be represented by
a cubic complex Ginzburg-Landau equation, with an additional term accounting
for multiphoton absorption processes. The plasma density is assumed
inhomogeneous, and its time evolution is governed by a first-order ordinary
differential equation, where distinct terms can be present accounting for
distinct physical processes assumed to contribute to the generation of the
electron plasma. In this respect, in ref. [10] it was stressed that a
physically acceptable model for the dynamics of plasma density should include
simultaneously the avalanche ionization, electron-hole radiative recombination
processes, the multiphoton ionization and electron diffusion processes. In the
present work we examined the dynamics of this last model, laying emphasis on
the influence of electron diffusion processes on temporal profiles of the
amplitude and instantaneous frequency of the femtosecond laser, as well as of
the plasma density. We found that electron diffusion processes favor pulse
trains with relatively large pulse repetition rates, and weakly damped
periodic anharmonic oscillations of the plasma density. The present study
therefore clearly shows that electron diffusion processes play a relevant role
in femtosecond laser inscriptions in transparent media with Kerr nonlinearity.

## Acknowledgments

E. O. Akeweje wishes to thank AIMS-Ghana for the opportunity of visit, and E.
Opoku Gyasi for enriching discussions during his stay at AIMS Ghana.

## Disclosure statement

The authors declare no potential conflict of interest.

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