# Thermal lensing-induced bifocusing of spatial solitons in Kerr-type optical media

###### Abstract

Thermo-optical effects cause a bifocusing of incoming beams in optical media, due to the birefringence created by a thermal lens that can resolve the incoming beams into two-component signals of different polarizations. We propose a non-perturbative theoretical description of the process of formation of double-pulse solitons in Kerr optical media with a thermally-induced birefringence, based on solving simultaneously the heat equation and the propagation equation for a beam in a one-dimensional medium with uniform heat flux load. By means of a non-isospectral Inverse Scattering Transform assuming an initial solution with a pulse shape, a one-soliton solution to the wave equation is obtained that represents a double-pulse beam which characteristic properties depend strongly on the profile of heat spatial distribution.

###### pacs:

42.65.Tg, 42.70.Gi, 44.10.+iIn dielectric media with thermo-optical
effects, the modulations of incoming beams can lead to a wavefront
distortion dist1 (); dist3 (); dist4 ()
reflecting their instability. Generally
this instability gives rise to a depolarization of a high-power
field hans (); menzel1 (); moshe1 (); moshe2 (); wang (); resolv2 ()
due to a thermally-induced birefringence which is
attributed biref1 (); biref2 (); biref3 () to a
change in the refractive index of the medium. It has been
established dist1 () that
this change in the refractive index originates from heat deposition in the
propagation medium, resulting in a space-dependent
temperature gradient (so-called thermal lensing) dist1 (); hans (); koechner (). For
media with linear indices,
the thermal lens leads to a drastic change in the
irradiance along
the beam axis so that the resulting depolarization can strongly degrade
the beam quality requiring thermal lensing compensation. However in nonlinear
media such as Kerr media, the nonlinearity can be a relevant
self-compensaton factor alex () stabilizing
multi-wave modes
generated by the thermal birefringence.

While several materials exhibiting thermo-optical effects are known in the
literature dist1 (), a most investigated one is the
solid-state laser Nd:YAG dist1 (); hans (); koechner (); eich (). This material is
represented as a rod crystal with a cylindrical geometry, where the change in
temperature induces thermal distortion of
incoming laser beams resolv2 ().
For this particular material, several theoretical attempts have been made to
formulate the spatial profile of the temperature gradient
along the rod exploiting available experimental data. In particular, in
refs. dist1 (); koechner (); eich () it was found that in
the cylindrical rod configuration where the heat is
generated at a constant rate koechner (); resolv2 (), a
quadratic spatial distribution gives a very good
description of the experimentally
observed birefringence and the resulting beam
bifocusing resolv2 (); maik ().

But thermo-optical processes are actually common to a broad class of
materials,
not just solid-state lasers. Indeed,
photonic crystals and optical fibers (including laser fibers) displaying
nonlocal
thermal and photothermal properties have been
considered in the recent past, from both
experimental and theoretical points of view chen (); martha (). These
materials share in
common the fact that heat resulting from the input pump source
causes physical variation of the material. Namely, the material expands
with the heat load due to a stress gradient formed which produces
space-dependent birefringence in the material. Quite remarkably when
the thermal gradient is strong enough the thermally-induced
birefringence can resolve a polarized high-power input beam into two-component
beams hans (); resolv2 (); resolv1 (), moreover upon recombination after
traversing the bulk the
two beams with two polarization directions are no longer in phase with one
another such that the polarization
state of the
input beam cannot be recovered dist4 (); resolv1 ().

The thermally-induced depolarization phenomenon together with the resulting
beam bifocusing have been investigated experimentally and theoretically
by several authors hans (); menzel1 (); moshe1 (); moshe2 (); wang (); resolv2 (). In photonic
crystals with a nonlinear Kerr-type
dielectric susceptibility
competing with diffraction, nonlocal thermal properties of the propagation
medium have very recently been shown to induce twin-mode spatial solitons
so-called dipole solitons yak (); cohen (); skupa (); bambi (). The robustness of these
double-pulse structures provides strong indication that
the competition between nonlinearity and thermal lensing might be a stabilizing
factor for the double-polarized modes, potentially
observable in several optical materials including solid-state and optical fibers
with thermally-induced birefringence.

The present work aims at proposing a non-perturbative description
of the generation of a double-polarized pulse beam in optical materials with
thermally-induced linear
birefringence, paying particular attention to nonlinear optical media
with a quadratic spatial profile of temperature distribution. Our objective is
to provide a consistent theoretical framework for understanding better the
effects of
thermal lensing on beam propagation in nonlinear optical materials, of which the
YAG fiber
laser which temperature profile was exprimentally
established dist1 (); koechner (); eich ().
We will first demonstrate, by solving the heat equation, that the topography
of temperature distribution is quadratic in the case of a one-dimensional ()
anisotropic material uniformly
loaded
at a constant heat flux rate. Next, using the relationship between the temperature gradient
and the induced inhomogeneous optical index,
we shall derive an effective refractive index for the
material by combining the thermally-induced optical index and
a Kerr-type
refractive index reflecting the intrinsic nonlinearity of optical properties of
the
anisotropic material. With the help of this effective refractive index we shall formulate
the
propagation equation for beams in the thermal nonlinear medium. Note that this equation is a
Nonlinear Schrödinger equation (NLSE) with a repulsive
external
quadratic potential, that has already been derived in ref. cari () but solved
perturbatively. Here instead, a
non-perturbative treatment will be proposed based on a non-isospectral Inverse
Scattering Transform (IST) method rada (); dika1 () with emphasis on
IST’s initial solution being single-pulse shaped.

Thermal lensing can be described in simple words as a thermo-optical process
associated with a weak absorption of an
input beam that
induces a nonzero temperature gradient across the material, leading to a spatial
variation of its refractive
index dist1 (); koechner (); cohen (). Recent experimental as well as theoretical
developments on this process suggest that the
underlying mechanism involves a local refractive index change
which increases linearly cohen (); eich () with the temperature change i.e. , where refers to the
thermo-optic coefficient cohen (); sowade ().
Thus
when the optical
beam of a uniform thermal load gets slightly absorbed and heats the material,
this produces heat that is conveyed by the electromagnetic wave. If
denotes the uniform heat flux density and the
heat conductivity coefficient, the heat diffusion in the material along a
preferred direction
(for anisotropic materials of current interest), driven by the
uniform heat flux load, is determined
by the heat equation:

(1) |

Since eq. (1) is key to the current analysis we consider its most general solution given by:

(2) |

where , and being two arbitrary real constants.
Formula (2) is
consistent with the quadratic law of temperature variation found for most laser
fibers in the presence of a uniform
head load. More specifically in YAG fiber lasers this law
is common dist1 (); koechner (); eich () and is consistent with the optical
bifocalility associated with a thermally-induced
birefringence, that promotes double-polarized laser beams from an input laser
field.

For the sake of simplicity we require
the temperature gradient to be zero
and temperature to take a bare value at (i.e. the ambient
temperature). The change of temperature
in the material along the axis, hereafter assumed to be the axis of beam
propagation, then reads:

(3) | |||||

With the last formula we derive the following expression for the local refractive index change:

(4) |

Now if the intrinsic optical properties of the material are dominated by Kerr-type phenomena, the homogeneous part of the refractive index can be expressed as where is the beam intensity. With the help of (4) we can readily define an effective refractive index for the thermal nonlinear material viz:

(5) |

Assuming that the wave motion is fast along the axis of anisotropy (i.e. ) but very slow along cohen (), the paraxial approximation on the wave equation for an electromagnetic field leads to:

(6) |

As already stressed eq. (6) is actually not new, indeed the same equation
was obtained cari () for the same problem but solved
following the collective-coordinate method. To this last point, eq. (6) is an inhomogeneous NLSE and so can in principle be solved using the
collective-coordinate method. However this is a perturbative
method and consequently requires that the thermal lensing
is sufficiently weak, so that the Kerr nonlinearity remains the main
governing factor in the modulation and stability of signals in the
thermal nonlinear medium. So to say any
input beam sent in the medium must be modulated into a signal of
permanent single-pulse shape, with eventually an acceleration or slow down of
the
pulse due to the
thermal lensing. In fact this
consideration is very far from any acceptable consistency with the physics of
the
process under study, which the double-polarization of the incident beam is a
most salient aspect.

Being interested in a solution to eq. (6) which is more consistent with
experiments we try for a non-perturbative approach. Remark
to start that this equation can be
rewritten in the following form:

(7) |

corresponding to a NLSE with an external potential

(8) |

which is quadratic in with a maximum at . The physics behind this
quadratic potential is contained both in its profile and the parameter
defined in (4), which determines the strength of the
thermal birefringence on the beam shape. One remarkable side of this
physics emerges from the assumption of the heat flux density and the heat
conductivity as being fixed, such that appears to be
increasing with an increase of the thermo-optic coefficient . Thus when
is
increased the curvature of the quadratic profile of heat distribution in the
material is more and more pronounced so that
the effect of thermal lensing on beam modulation is more and more strong. In
fact the external potential is expected to be much effective
on the beam position and according to the profile of this potential given
by (8),
the centre of mass of the input beam should experience a trapping
force from
the manifestly expulsive quadratic potential.

When is large enough such that the potential field erected by the
thermal lensing process on the beam path is strongly localized, its contribution
must be fully taken into consideration. In this last respect we follow
the non-isospectral IST technique proper rada (); dika1 () to equations of this
specific kind,
considering an initial signal of a permanent single-pulse shape
along at .
For eq. (6) this technique leads to the
following one-soliton solution:

(9) |

with

(10) |

(11) | |||||

(12) |

(13) |

(14) |

(15) |

If the ”sech” function in (9) reminds a pulse signal, the complicated dependence of its prefactor in and clearly suggests an actually complex pulse structure for the beam on the plane. To gain insight about what this dependence implies for the signal profile, in fig. 1 we plotted the beam amplitude as a function of at two different positions in the direction transverse to the beam propagation, and for four distinct values of .

More explicitely the four left graphs in fig. 1 represent versus at
for , , and , while the left graphs
represent versus
for same values of but at . As one sees, the signal intensity is
a two-component pulse which intensities are strongly
dependent on the strength of the thermal lens potential.
It is quite noticeable on exploring the six figures
that when increases, the intensities of the two-component pulse
increase while
their width at half tails diminish. The last behaviour is
consistent
with the dependence of parameter defined in (15), which
indeed represents the average spatial extension of the pulse along the
axis.

Another relevant feature emerging from graphs of fig. 1 is the fact
that when is decreased for a fixed value of , the two constituents
pulses in the double-polarized beam
preserve their shapes but their peak positions (i.e. centres) are gradually
shifted. The last feature is more transparent in the left graphs
corresponding to a relatively large value of . In fact, the last behaviour
can be interpreted in terms of a ring profile for the signal intensity
in the plane as reflected by the contour plots of fig. 2, where
shadows of the double-pulse
soliton in the plane are represented for different values of .

The figure clearly indicates an increase of the separation between pulses in the
double-pulse signal,
implying that the radius and curvature of the ring signal in the
plane are fixed by the magnitude of .

The double-pulse structure obtained as well as its ring profile emerging in are reminiscent of dipole and ring-vortex
solitons, predicted recently in some nonlocal nonlinear
media cohen (); bambi (); fei (). However in these previous studies the physical origins of nonlocalities
were generally not well specified, whereas in
our context the quadratic form of the optical inhomogeneity has an
experimental foundation dist1 (); koechner (); eich (). Also, a previous
attempt cari () to model the same problem led to
eq.(6). In this previous work the equation was treated perturbatively and so
results could not reflect the remarkable aspects underlying the physics of thermal lensing in Kerr media. As end remark, the
universality of the IST one-soliton
solution (9) can be checked by applying
any other exact spectral method to the generating equation, such as the Darboux method nakka () combined with a Lax-pair
formalism with non-conserved spectral parameters porse (); khawa1 ().

Part of this work was done at the Abdus Salam International Centre for Theoretical Physics (ICTP) Trieste, Italy. I thank Markus Muller, Matteo Marsili, M. Kiselev and V. Kravtsov for their kind hospitalities.

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