Stability of narrow beams in bulk Kerr-type nonlinear media

Abstract

We consider (2+1)-dimensional beams, whose transverse size may be comparable to or smaller than the carrier wavelength, on the basis of an extended version of the nonlinear Schr\"{o}dinger equation derived from the Maxwell`s equations. As this equation is very cumbersome, we also study, in parallel to it, its simplified version which keeps the most essential term: the term which accounts for the {\it nonlinear diffraction}. The full equation additionally includes terms generated by a deviation from the paraxial approximation and by a longitudinal electric-field component in the beam. Solitary-wave stationary solutions to both the full and simplified equations are found, treating the terms which modify the nonlinear Schr\"{o}dinger equation as perturbations. Within the framework of the perturbative approach, a conserved power of the beam is obtained in an explicit form. It is found that the nonlinear diffraction affects stationary beams much stronger than nonparaxiality and longitudinal field. Stability of the beams is directly tested by simulating the simplified equation, with initial configurations taken as predicted by the perturbation theory. The numerically generated solitary beams are always stable and never start to collapse, although they display periodic internal vibrations, whose amplitude decreases with the increase of the beam power. Comment: 7 pages, 6 figures Accepted for publication in PRA

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arXiv:nlin/0103053v1 [nlin.PS] 28 Mar 2001
Stability of narrow beams in bulk Kerr-type nonlinear media
B. A. Malomed
Department of Interdisciplinary studies
Faculty of Engineering
Tell Aviv University, Tel Aviv 69978, Israel
K. Marinov∗and D. I. Pushkarov
Institute of Solid State Physics
Bulgarian Academy of Sciences
BG-1784 Sofia, Bulgaria
A. Shivarova
Faculty of Physics, Sofia University
BG-1164 Sofia, Bulgaria
(February 3, 2008)
We consider (2+1)-dimensional beams, whose transverse size may be comparable to or smaller
than the carrier wavelength, on the basis of an extended version of the nonlinear Schr¨ odinger equa-
tion derived from the Maxwell‘s equations. As this equation is very cumbersome, we also study, in
parallel to it, its simplified version which keeps the most essential term: the term which accounts
for the nonlinear diffraction. The full equation additionally includes terms generated by a devia-
tion from the paraxial approximation and by a longitudinal electric-field component in the beam.
Solitary-wave stationary solutions to both the full and simplified equations are found, treating the
terms which modify the nonlinear Schr¨ odinger equation as perturbations. Within the framework
of the perturbative approach, a conserved power of the beam is obtained in an explicit form. It is
found that the nonlinear diffraction affects stationary beams much stronger than nonparaxiality and
longitudinal field. Stability of the beams is directly tested by simulating the simplified equation,
with initial configurations taken as predicted by the perturbation theory. The numerically generated
solitary beams are always stable and never start to collapse, although they display periodic internal
vibrations, whose amplitude decreases with the increase of the beam power.
PACS 42.65.Jx, 42.65.Tg, 42.25.Fx
I. INTRODUCTION
Self-focusing of optical beams in nonlinear media [1] is a subject of intensive study since the critical nature of this
phenomenon (a collapse of the beam) has been identified in the first studies [2–6]. Different theoretical approaches
have been developed for the analysis of self-focusing, including the moment theory [7], variational method (see, e.g.,
[8] and references therein), paraxial-ray approximation [9], ”adiabatic” description [10], etc. With a probability for a
collapse, no stable self-trapped beams can exist in bulk Kerr-media. Therefore, a considerable part of the studies in
the field has been directed towards finding mechanisms that limit the development of the collapse and thus provide
for the existence of stable self-trapped beams in two transverse dimensions. It has been shown [11] that the saturation
of the nonlinearity of the medium arrests the collapse and stable two-dimensional self-trapped beams can indeed exist
in such media [8]. The plasma generation by a self-focusing laser beam [12–14] through multiphoton and avalanche
ionization, with its negative (exponential) contribution to the refractive index of the media, can also stop the collapse
by leading to formation of filaments.
However, as it is pointed out in Ref. [15], the occurrence of the self-focusing in different media (gases, liquids,
solids) makes it insufficient to identify a stabilizing effect for each separate case, and calls to search for universal, i.e.,
medium-independent, mechanisms which yield nonsingular behavior of the beams. In this connection, it is necessary
to stress on that the dynamics of narrow beams described by the full system of the Maxwell‘s equations may be
drastically different from the predictions of the usual parabolic nonlinear Schr¨ odinger (NLS) approximation valid for
broad beams (see, e.g., [16] and references therein). In fact, the Maxwell‘s equations implicitly contain few different
mechanisms that can check the collapse of the narrow beams.
One of such mechanisms is nonparaxiality of the beam propagation, which becomes important at the advanced
stages of the self-focusing. It was studied both numerically [15,17–19] and analytically [20], concluding that the
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nonparaxiality replaces the catastrophic focusing with a sequence of focusing-defocusing cycles. Beam filamentation
due to the nonparaxiality was demonstrated as well [15].
It was also shown that when the vector nature of the beam field is taken into regard along with the nonparaxiality
[21], the peak intensity occuring in the self-focusing is almost ten times lower than that produced by the nonparax-
iality alone for the same input beam, suggesting that other effects, which limit the self-focussing stronger than the
nonparaxiality, could exist. Recent analysis [22,23] has demonstrated that the efficiency of the vector model in limiting
the narrowing of the beam in the (1+1)-dimensional case is due not to the small longitudinal electric-field component
but rather to a scalar term which is present in the model. This scalar term, accounting for the rate of the spatial
variation of the nonlinear polarization, combines effects of nonlinearity and diffraction. A notation ”nonlinearly in-
duced diffraction” was introduced [22,23] to stress on this factor which restricts the beam narrowing predicted by
the usual NLS equation. In this connection it should be mentioned that a fundamental limit for the soliton width
has been found [24–26] to exist in the ”subwavelength” range (beams with a transverse size comparable with or
smaller than the carrier wavelength) due to the terms that come from Maxwell‘s equations but are absent in the usual
(1+1)-dimensional NLS equation. Given the results in Ref. [21] and conclusions drawn in Refs. [22–26], the existence
of a very narrow stable self-trapped beam in an ideal bulk Kerr-medium is still an open problem which needs to be
addressed, which is the subject of the present work.
The rest of the paper is organized as follows. In Section 2, we derive, from the Maxwell‘s equations, an equation
to govern the evolution of narrow (2+1)-dimensional beams in Kerr media. This is a very cumbersome generalization
of the NLS equation, with extra terms representing three different effects: nonlinear diffraction, nonparaxiality, and
the influence of the small longitudinal component of the electric field. In view of the very complex character of the
equation, we also consider a simplified version, that keeps only the nonlinear diffraction as a new effect. In section 3,
we develop a perturbation theory which treats all the new terms in both equations as a perturbation, which is true in
the case when the beam is still relatively broad. Further analysis in section 4 shows that the effect produced by the
nonlinear diffraction is much stronger than those generated by the nonparaxiality and longitudinal field, which justifies
the use of the simplified equation, at least for beams that are not extremely narrow. Another noteworthy result of
the perturbation theory is that, although the terms which modify the NLS equation destroy the circular symmetry
of the beam, the deviation from the symmetry is, in fact, fairly weak. In section 5, we show that the perturbative
approach also allows one to find an expression for the conserved power of the beam, which is not a trivial issue in
the present model. Lastly, in section 6 we display result of direct numerical simulations of the beam propagation,
starting from initial configurations predicted by the perturbation theory. The direct simulations are performed within
the framework of the simplified equation only. As a result, we conclude that the beams never show collapse and are
always stable, showing small internal vibrations, which are generated by a deviation of the initial configuration from
an exact solitary-beam shape.
II. THE BEAM PROPAGATION EQUATION
We consider the propagation - along the z-axis - of a light-beam, extended in the (x, y)-plane, in a bulk isotropic
Kerr-medium, characterized by the usual relation between the nonlinear polarization?PNLand electric filed?E,?PNL=
ε0χ?E2
x+ E2
z
??E, where χ is the χxxxx= χzzzz component of the third-order susceptibility tensor [27]. The wave
∂2?E
∂t2+ µ0∂2?PNL
equation,
∇2?E − ∇(∇.?E) =εL
c2
∂t2
,(1)
combined with the relation
∇.?E = −
1
ε0εL∇.?PNL,(2)
yield a scalar nonlinear wave equation in the following form:
2iβ∂E
∂z+ ∆⊥E +3ω2χ
4c2|E|2E +3χ
?2
4εL
∂2?|E|2E?
∂x2
∂E∗
∂x
+∂2E
∂z2+
ω2χ
2c2β2
????
∂E
∂x
????
2
E
−ω2χ
4c2β2E∗
?∂E
∂x
+χ
εL
∂
∂x
?E2
4
?
−χ
εL
∂
∂x
?|E|2
2
∂E
∂x
?
= 0.(3)
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In Eq.(3), ω is the frequency of the carrier wave, E ≡ Ex, ∆⊥ ≡
operator, εLand β = (ω/c)εLare the linear permittivity and the linear propagation constant (wavenumber), respec-
tively. After rescaling 2βx → x, 2βy → y, 2βz → z, γE → E, with γ2=?3ω2χ?/?16β2c2?, and some manipulation,
∂z+ ∆⊥E + |E|2E +∂2E
∂x
?∂2/∂x2?+?∂2/∂y2?
is the transversal Laplace
Eq. (3) transforms into
i∂E
∂z2+ 4
????
∂E
????
2
E −4
3
∂|E|2
∂x
∂E
∂x+16
3
∂
∂x
?
E∂|E|2
∂x
?
= 0.(4)
The modifications of the (2+1)-dimensional NLS equation involved in Eq. (4) stem from nonparaxiality (the fourth
term) and nonlinear diffraction and longitudinal field component included in a combined manner in the last three
terms. It makes sense to consider Eq. (4) parallel to its ”model,” simplified, version
i∂E
∂z+ ∆⊥E + |E|2E + 4∂2?|E|2E?
∂x2
= 0,(5)
where only the nonlinear diffraction [the last term in Eq. (5)] is taken into account to modify the NLS equation [22,23].
The relevance of approximating Eq. (4) by Eq. (5) will be verified a posteriori.
III. SOLITARY-WAVE SOLUTIONS AND PERTURBATION THEORY
Seeking for solitary-wave solutions to Eqs. (4) and (5) in the form E(x,y,z) = F(x,y)exp(iµz), where µ is a
nonlinear wavenumber shift of the solitary-wave solutions considered, one obtains
?1 + aF2?∂2F
∂x2+∂2F
∂y2− kF + F3+ bF
?∂F
∂x
?2
= 0.(6)
In the case of Eq. (5), a = 12, b = 24, and k = µ, while Eq. (4) corresponds to a = 32/3, b = 68/3 and k = µ + µ2.
Besides the linear diffraction in the second (y-) transverse direction, Eq. (6) exactly coincides with the one-dimensional
equation derived in Ref. [22] for both the vector nonparaxial model and for the scalar ”nonlinear diffraction” model.
Generally speaking, Eq. (6) must be solved numerically as it stands. However, for the case of small gradients a
perturbation scheme can be developed, which simplifies the problem considerably. In fact, this implies seeking for
first corrections to the usual broad solitons, generated by the new terms obtained from the Maxwell‘s equations. To
this end, F(x,y) and k are presented in the form F(x,y) = F0(x,y) + f(x,y), k = k0+ ∆k, where f and ∆k are the
first order corrections, so that the substitution of this into Eq. (6) yields
∂2F0
∂x2+∂2F0
∂y2− k0F0+ F3
0= 0,(7)
and
∂2f
∂x2+∂2f
∂y2+?3F2
0− k0
?f = ∆kF0− aF2
0
∂2F0
∂x2− bF0
?∂F0
∂x
?2
.(8)
In Eq. (7), k0= µ0is the zeroth-order approximation for the nonlinear wavenumber shift. The transformation to the
polar coordinates x = rcosϕ, y = rsinϕ transforms Eqs. (7) and (8) into
d2F0
dr2+1
r
dF0
dr
− k0F0+ F3
0= 0 (9)
and
∂2f
∂r2+1
r
∂f
∂r+1
?2
r2
∂2f
∂ϕ2+?3F2
?
2
0− k0
?f = ∆kF0−aF2
dr2−1
r dr
0
2
?d2F0
?2?
dr2+1
r
dF0
dr
?
−bF0
2
?dF0
dx
−
aF2
0
?d2F0
dF0
?
+bF0
2
?dF0
dr
cos2ϕ,(10)
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respectively. Equation (9), which is azimuthally symmetric, gives rise to the classical Townes soliton [5], while Eq.
(10) generates a first-order correction to it. In Eq. (10) the variables can be separated by means of the substitution
f(r,ϕ) = f0(r) + f1(r)cos2ϕ, the resulting equations for f0and f1being
d2f0
dr2+1
r
df0
dr+?3F2
0− k0
?f0= ∆kF0−aF2
0
2
?d2F0
dr2+1
r
dF0
dr
?
−bF0
2
?dF0
dr
?2
(11)
d2f1
dr2+1
r
df1
dr+
?
3F2
0− k0−4
r2
?
f1= −aF2
0
2
?d2F0
dr2−1
r
dF0
dr
?
−bF0
2
?dF0
dr
?2
. (12)
Thus, the first-order perturbative solution to Eq. (6) amounts to solving three ordinary differential equations, viz.,
Eqs. (9), (11) and (12). This can be easily done by means of the shooting method [28]. We supplement the equations
with obvious boundary conditions that single out soliton solutions:
dF0(r = 0)
dr
F0(r = 0) = A > 0,f0(r = 0) = f1(r = 0) = 0,
F0(r → ∞) → 0, f0(r → ∞) → 0, f1(r → ∞) → 0.
=df0(r = 0)
dr
=df1(r = 0)
dr
= 0,
IV. RESULTS OF THE PERTURBATION THEORY
The solution of Eq. (9) is the same as that obtained in Ref. [5]. For this solution, k0≈ A2/4.86, and the power
∞ ?
two-dimensional NLS equation [5,7,8].
Solutions to Eqs. (11) and (12) are displayed in Figs. 1(a) and 1(b), respectively. The solutions marked by ”NLD”
pertain to the case of a = 12, b = 24, which corresponds to the simplified equation (5), and those marked by ”vector”
pertain to a = 32/3 ≈ 10.67, b = 68/3 ≈ 22.67, which correspond to the full equation (4). For comparison, the
zeroth-order solution (the Townes soliton) is shown in Fig. 1(c). It is obvious that the nonlinear diffraction is the
main factor which causes deviations from the solution of the NLS equation and the effect of the longitudinal field
component is quite small.
In Fig. 2, results for the half-widths Rx, Ry of the beam (defined at half-maximum of its amplitude) are shown.
It can again be concluded that the nonlinear diffraction term is the main reason which restricts the narrowing of
the Townes soliton with the increase of its on-axis intensity (denoted as ”standard” in Fig. 2). The influence of
the longitudinal field component shows up into a small reduction of the effect of the nonlinear diffraction. It is also
worth noting that the half-width Rxin the direction of the main electric-filed component (x) is much more affected
by the nonlinear-diffraction term than that (Ry) in the other transverse direction (y). The latter feature characterizes
an effective anisotropy induced by the new terms. Note that deviations from the circular symmetry of a beam
propagating according to an equation similar to Eq. (4) was reported in Refs. [1,21].
The comparison of the contributions to the nonlinear wavenumber shift µ stemming from nonparaxiality, nonlinear
diffraction and longitudinal electric field offers an additional way to estimate a relative strength of effects generated
by these factors. In the case when the simplified model
neglects the nonparaxiality and the influence of the longitudinal field, i.e., a = 12 and b = 24 in Eq. (6), the
result is µNLID ≈
4.86+
i.e., with a = 32/3 and b = 68/3 in Eq. (6), yields µV N ≈
the parenthesis is due to the vectorial structure of the field, and the second one is an effect of the nonparaxiality.
This result demonstrates, that within the framework of the first-order perturbation theory, the contribution to the
nonlinear wavenumber shift stemming from the nonlinear diffraction exceeds those originating from nonparaxiality
and longitudinal field component by approximately 13 and 20 times, respectively, thus validating the use of Eq. (5)
instead of the full equation (4) for the study of the solitary waves.
P that it carries is P = 2π
0
rF2
0dr ≈ 11.70, which is the known ”critical power” for the weak collapse in the
(5) is used, which keeps the nonlinear diffraction but
A2
A4
1.74up to the A4-order. On the other hand, the full model corresponding to Eq. (4),
A2
4.86+
1.74
A4
?1 −
1
20.52−
1
13.57
?. The first correction in
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V. AN APPROXIMATE POWER-CONSERVATION LAW
According to the results of the perturbation theory for the case of small gradients (Figs. 1 and 2), the deviation
of the shape of the beam from the circular symmetry is small (although existing). Introducing the polar coordinates
into Eq. (5) and assuming weak dependence of the field envelop on the angular variable ϕ, one obtains
i∂E
∂z+
?∂2
∂r2+1
r
∂
∂r
??E + 2|E|2E?+1
r2
∂2E
∂ϕ2+ |E|2E + 2cos2ϕ
?∂2
∂r2−1
r
∂
∂r
??|E|2E?= 0. (13)
Multiplying Eq. (13) by?E∗+ |E|2E∗?, subtracting the complex conjugate and integrating over the transversal plane
2π
?
0
0
while keeping the first order corrections only, a power conservation law in the form
dP
dz
≡
d
dz
∞
?
r?|E|2+ |E|4?drdϕ = 0(14)
is obtained. Note that the second term in the integrand is due to the nonlinear diffraction which comes to involve the
power density associated with the nonlinear polarization into the power conservation law.
In Fig. 3 the dependence of the power invariant, given by Eq. (14), on the on-axis intensity of the beam is shown.
For comparison, the same dependence for the case of the standard NLS equation (
2π ?
0
∞ ?
0
r|E|2drdϕ) is also presented.
It is evident that the solitary waves studied here exist for power levels above the critical. The fact that the power is
a growing function of the on-axis intensity suggests stability for the solitary wave solutions (see also the discussions
in Ref. [19] in the case of the scalar nonparaxial model).
VI. NUMERICAL SIMULATIONS OF THE BEAM PROPAGATION
As Fig. 3 shows, the presence of nonlinear diffraction increases the power necessary for self-trapping (i.e., the power
at which the diffraction and nonlinearity balance each other). This is demonstrated by the numerical simulations
displayed in Fig. 4(a), where an initial configuration in the form of the Townes soliton, see Eq. (9), propagates
according to the Eq. (5). We do not simulate the full equation (4), which has an extremely complex form, since the
results presented above strongly suggest that the simplified equation (5) may describe essential features of the beam
dynamics quite adequately. Figure 4 shows that in this case the beam diffracts away, while, as it is well known, the
usual NLS equation predicts collapse for the same initial conditions [see Fig. 4(c)]. In Fig. 4(b), the approximate
invariant given by Eq. (14) and the invariant of the NLS equation in its standard form,
∞ ?
−∞
∞ ?
−∞
|E|2dxdy, are shown
vrs. the propagation distance for the case when Eq. (5) is solved. The fact that the NLS invariant increases when
the beam diffracts can be explained in the following way. As it has been mentioned above, the full invariant, given
by Eq. (14), takes into account the power density associated with both the linear and nonlinear polarization of the
medium and is conserved, provided the gradients remain small enough. When the beam diffracts, the contribution of
the nonlinear polarization decreases, transferring the power it carries to the linear-polarization part, which therefore
increases in accord with what is seen in Fig. 4(b). When the beam is self-focusing, the computations show that the
NLS invariant decreases, as one should expect.
In order to test directly the dynamical properties of the solitary waves in the present model (first of all, their stability
against the collapse), equation (5) is solved numerically with initial conditions produced by the perturbation scheme
developed above [i.e., the solitary-wave solutions generated by Eqs. (9) - (12) have been used as initial configurations].
Figure 5 shows results for the evolution of the beam intensity and of the approximate power invariant given by Eq.
(14). For comparison, the usual NLS invariant is also displayed. As is seen in Fig. 5, the beams experience periodic
focusing and defocusing with a relatively small amplitude of the corresponding internal vibrations, rather than the
collapse that would occur in the case of the NLS equation. The period of the oscillations rapidly decreases with the
increase of the input power.
The results presented in Figs. 5(a) and 5(b) are in a qualitative agreement with those in Ref. [18] where the
non-catastrophycal propagation of Gaussian beams within the scalar nonparaxial model has been obtained. The
soliton-like propagation studied later [19] within the same model shows the same type of oscillating behaviour. The
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