Download Localized superluminal solutions to the wave equation in

Optics Communications 226 (2003) 15–23
www.elsevier.com/locate/optcom
Localized superluminal solutions to the wave equation
in (vacuum or) dispersive media, for arbitrary frequencies
and with adjustable bandwidth q
M. Zamboni-Rached
a,*
brega a, H.E. Hern
, K.Z. No
andez-Figueroa a,
E. Recami b,c
a
b
DMO-FEEC, State University at Campinas, Campinas, S.P., Brazil
Facolt
a di Ingegneria, Universita statale di Bergamo, Dalmine (BG), Italy
c
INFN-Sezione di Milano, Milan, Italy
Received 7 February 2003; received in revised form 18 July 2003; accepted 20 July 2003
Abstract
In this paper we set forth new exact analytical superluminal localized solutions to the wave equation for arbitrary
frequencies and adjustable bandwidth. The formulation presented here is rather simple and its results can be expressed
in terms of the ordinary, so-called ‘‘X-shaped waves’’. Moreover, by the present formalism we obtain the first analytical
localized superluminal approximate solutions which represent pulses propagating in dispersive media. Our solutions
may find application in different fields, like optics, microwaves, radio waves, and so on.
Ó 2003 Elsevier B.V. All rights reserved.
PACS: 03.50.De; 41.20.Jb; 83.50.Vr; 91.30.Fn; 04.30.Nk; 42.25.Bs
Keywords: Wave equation; Wave propagation; Localized beams; Superluminal waves; Bessel beams; X-shaped waves; Optics;
Dispersion compensation
1. Introduction
For many years it has been known that localized
(non-dispersive) solutions exist to the (homogeq
Work supported by FAPESP (Brazil), and by MIUR,
INFN (Italy). This paper did first appear as e-print physics/
0209101.
*
Corresponding author. Tel./fax: +551932560338.
E-mail address: [email protected] (M. Zamboni-Rached).
neous) wave equation [1–7], endowed with subluminal or superluminal [8–13] velocities. These
solutions propagate without distortion for long
distances in vacuum.
Particular attention has been paid to the superluminal localized solutions (SLS) like the socalled X-waves [9,10,12] and their finite energy
generalizations [11,12]. It is well known that such
(SLS) have been experimentally produced in
acoustics [14], optics [15] and more recently microwave physics [16,17].
0030-4018/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved.
doi:10.1016/j.optcom.2003.08.022
16
M. Zamboni-Rached et al. / Optics Communications 226 (2003) 15–23
As is well known, the standard X-wave has a
broad band frequency spectrum, starting from
zero [12,13] (it being therefore appropriate for low
frequency applications). This fact can be viewed as
a problem, because it is difficult or even impossible
to define a carrier frequency for that solution, as
well as to use it in high frequency applications.
Therefore, it would be very interesting to obtain
exact SLSs to the wave equations with spectra
localized at higher (and arbitrary) frequencies and
with an adjustable bandwidth (in other words,
with a well-defined carrier frequency).
To the best of our knowledge, only three attempts were made in this direction: one by Zamboni-Rached et al. [12,18], another by Salo et al.
[19,20], and the last one by S~
onajalg et al. [21,22] .
The first two authors showed how to shift the
spectrum to higher frequencies without dealing with
its bandwidth, while the latter worked out computer
simulations [21] (not analytically) and experiments
[22] on optical X pulses in a dispersive medium,
obtaining very important results which confirm the
efficacy of these superluminal localized waves.
In this work, however, we are presenting analytical and exact superluminal localized solutions in
vacuum, whose spectra can be localized inside any
range of frequency with adjustable bandwidths,
and therefore with the possibility of choosing a
well-defined carrier frequency. In this way, we can
get (without any approximation) radio, microwave,
optical, etc., localized superluminal waves.
Taking advantage of our methodology, it
means, extending the solution found for the vacuum, we obtain the first analytical approximations
to the SLSs in dispersive media (i.e., in media with
a frequency dependent refractive index).
One of the interesting points of this work, let us
stress, is that all results are obtained from simple
mathematical operations on the standard ‘‘Xwave’’, using simple analytical expressions to do it.
2. Superluminal localized waves in dispersionless
media for arbitrary frequencies and adjustable
bandwidths
tion to the wave equation in vacuum (n ¼ n0 ), in
cylindrical coordinates, one can easily find that
wðq; z; tÞ ¼ J0 ðkq qÞeþikz z eixt with kq2 ¼ n20 ðx2 =c2 Þ
kz2 ; kq2 P 0, where J0 is the zeroth-order ordinary
Bessel function, kz and kq are the axial and the
transverse wavenumber, respectively, x is the angular frequency and c is the light velocity. Using
the following transformation:
kq ¼ xc n0 sin h;
ð1Þ
kz ¼ xc n0 cos h;
wðq; z; tÞ can be rewritten as the well-known Bessel
beam
x
x
ð2Þ
wðq; fÞ ¼ J0 n0 q sin h eþin0 c f cos h ;
c
where f z Vt while V ¼ c=ðn0 cos hÞ is the
phase velocity, quantity hð0 < h < p=2Þ being the
cone angle of the Bessel beam.
As it is well known [14,21], a SLS (with axial
symmetry) to the wave equation in a dispersionless
medium can be written as
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!
Z 1
x
V2
x
wðq; fÞ ¼
q n20 2 1 eþiV f dx;
SðxÞJ0
V
c
0
ð3Þ
where SðxÞ is the frequency spectrum.
In fact, such solutions are pulses propagating in
free space without distortion and with the superluminal velocity V ¼ c=ðn0 cos hÞ. The most popular spectrum SðxÞ is that one given by SðxÞ ¼ eax ,
which provides the ordinary X-shaped waves
V
X wðq; fÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :
2
2
ðaV ifÞ þ q2 ðn20 Vc2 1Þ
Because of its non-dispersive properties and its low
frequency spectrum, 1 the X-wave is being particularly applied in fields like acoustics [9].
To find analytical expression for Eq. (3) is a
difficult task, being possible or not. Its numerical
solutions usually brings some inconveniences for
further analysis, uncertainties concerning the fast
1
Let us start by dealing with SLSs in dispersionless media. From the axially symmetric solu-
ð4Þ
It is easy to see that this spectrum starts from zero, it being
suitable for low frequency applications, and has the bandwidth
Dx ¼ 1=a.
M. Zamboni-Rached et al. / Optics Communications 226 (2003) 15–23
17
oscillating field components, etc.; besides implying
a loss in the physical interpretation of the results.
Thus, it is always worth looking for analytical
expressions.
In this way, one of our main objectives is finding out a spectrum which can preserve the integrability of Eq. (3) for any frequency range. In
order to be able to shift our spectrum towards the
desired frequency, let us locate it around a central
frequency, xc , with an arbitrary bandwidth Dx
(denoting its full ‘‘standard deviation’’, i.e., the
spectral function full-width when its height equals
1=e of its maximum value).
2.1. The SðxÞ spectrum
Let us choose the spectrum
x m
SðxÞ ¼
eax ;
ð5Þ
V
where V is the wave velocity, while m and a are free
parameters. For m ¼ 0; SðxÞ ¼ expðaxÞ, and
one gets the (standard) X-wave spectrum.
After some mathematical manipulations, one
can easily find the following relations, valid for
m 6¼ 0:
1
m ¼ ðDx =xc Þlnð1þðDx
;
=xc ÞÞ
ð6Þ
m
xc ¼ a ;
where Dx Dxþ or Dx . Because of the nonsymmetric character of spectrum (5), let us call
Dxþ ð> 0) the bandwidth to the right, and Dx
(< 0) the bandwidth to the left of the spectrum
central frequency xc ; so that Dx ¼ Dxþ Dx . It
should be noted, however, that, already for small
values of m (typically, for m P 10), one has
Dxþ Dx . Once defined xc and Dx, one can
determine m from the first equation and then, using the second expression, to find a.
Fig. 1 illustrates the behavior of the first relation of Eq. (6). From this figure, one can observe
that the smaller Dx=xc is, the higher m must be.
Thus, one can notice that m plays the fundamental
role of controlling the spectrum bandwidth.
From the X-wave spectrum, it is known that a is
related to the (negative) slope of the spectrum.
Contrarily to a, quantity m has the effect of rising
the spectrum. In this way, one parameter com-
Fig. 1. Behavior of the derivative number, m, as a function of
the normalized bandwidth frequency, Dx =xc . Given a central
frequency, xc , and a bandwidth, Dx , one finds the exact value
of m by substituting these values into the first expression of Eq.
(6).
pensates the other, producing the localization of
the spectrum inside a certain frequency range.
At the same time, this fact also explains (because
of the second expression of Eq. (6)) why an increase of both m and a is necessary to keep the
same xc . This can be inferred from Fig. 2.
In Fig. 2, both spectra have the same central
value xc . Taking the narrow spectrum as a reference, one can observe that, to get such a result,
both quantities m and a have to increase. More-
Fig. 2. Normalized spectra for xc ¼ 23:56 1014 Hz and different bandwidths. The first with m ¼ 27 (solid line) and the
second spectrum with m ¼ 45 (dotted line). See the text.
18
M. Zamboni-Rached et al. / Optics Communications 226 (2003) 15–23
over, this figure shows the important role of m for
generating a wider, or narrower, spectrum.
Let us notice that the spectrum (5) was first
considered in three relevant works by Friberg and
co-workers [19,20,23], as well as in another paper
by Zamboni-Rached et al. [12]. However, in those
previous works only the spectrum shifting was
discussed (second expression of Eq. (6)), meanwhile nothing was mentioned about its bandwidth
(first expression of Eq. (6)). To our knowledge, the
present work is the first effort to completely describe the relationship between the central frequency and its bandwidth of the spectrum given by
Eq. (5).
Fig. 3. The real part of an X-shaped beam for microwave frequencies in a dispersionless medium.
2.2. X-type waves in a dispersionless medium
Applying our spectrum expressed by Eq. (5),
Eq. (3) can be written as
Z 1 m
x
wðq; fÞ ¼ V
V
0
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!
x
x
V2
q n20 2 1 eðaV ifÞx=V d
J0
:
V
V
c
ð7Þ
Therefore, we have seen that the use of a spectrum
like (5) allows shifting it towards any frequency
and confining it within the desired frequency
range. In fact, this is one of its most important
characteristics.
It can be noticed that, using SðxÞ ¼ expðaxÞ,
Eqs. (3) and (4) are equivalent. Thus, differentiating Eq. (3) with respect to ðaV ifÞ, a multiplicative factor ðx=V Þ is each time produced (an
obvious property of Laplace transforms). In this
way, it is possible to write Eq. (7) as
wðq; fÞ ¼ ð1Þ
m
om X
:
oðaV ifÞm
ð8Þ
A different expression for Eq. (7), without any
need of calculating the mth differentiation of the Xwave, can be found by using identity (6.621) of [24]
Cðm þ 1ÞX mþ1
mþ1
m
; ; 1;
wðq; fÞ ¼
F
2
2
Vm
2
2
V
X
0
n20 2 1 q2 2 ;
ð8 Þ
c
V
where X is the ordinary X-wave given in Eq. (4),
and F is a GaussÕ hypergeometric function. Eq. (80 )
can be useful in the cases of large values of m.
Fig. 3 shows an example of an X-shaped wave
for microwave frequencies. To that aim, it was
chosen xc ¼ 6 GHz, Dx ¼ 0:9xc , V ¼ 5c and the
values of m and a were calculated by using Eq. (6):
thus obtaining m ¼ 10 and a ¼ 1:6667 109 s.
As one can see, the resulting wave has really the
same shape and the same properties as the classical
X-waves: namely, both a longitudinal and a
transverse localization.
3. Superluminal localized waves in dispersive media
In Section 2, Eq. (1) was written for a dispersionless medium (n0 ¼ constant, independent of the
frequency). However, for a typical medium, when
the refractive index depends on the wave frequency, nðxÞ, those equations become [21]
kq ðxÞ ¼ xc nðxÞ sin h;
ð9Þ
kz ðxÞ ¼ xc nðxÞ cos h:
The above equations describe one of the basic
points of this work. In Section 2 it was mentioned
that h determines the wave velocity: a fact that can
be exploited when one looks for a localized wave
that does not suffer dispersion. In other words, one
can choose a particular frequency dependence of h
to compensate the (‘‘material’’) dispersion due to
M. Zamboni-Rached et al. / Optics Communications 226 (2003) 15–23
the variation with the frequency of the refractive
index [21].
If the frequency dependence of the refractive
index in a medium is known, within a certain
frequency range, let us see how the consequent
dispersion can be compensated for. When a dispersionless pulse is desired, the constraint kz ¼
d þ xb must be satisfied (in other words the group
velocity, ox=okz , does not depend on the frequency). And, by using the last term in Eq. (9), one
infers that such a constraint is forwarded by the
following relationship between h and x:
cos hðxÞ ¼
cðd þ bxÞ
;
xnðxÞ
ð10Þ
where d and b are arbitrary constants (and b is
related to the wave velocity: b ¼ 1=V ). For convenience, we shall consider d ¼ 0. Then, Eq. (3)
can be rewritten as
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!
Z 1
x
n2 ðxÞV 2
wðq; fÞ ¼
q
SðxÞJ0
1 eþixbf dx:
V
c2
0
ð11Þ
Let us stress that this equation is a priori suited
for many kinds of applications. In fact, whatever
its frequency be (in the optical, acoustic, microwave, etc., range), it constitutes the integral
formula representing a wave which propagates
without dispersion in a dispersive medium.
Let us recall, now, that there are at least three
different ways (see, e.g. [21,22]) for generating
these dispersionless localized superluminal pulses,
whose wave vectors and frequencies obey Eq. (9)
and whose cone angle is given by Eq. (10). Such
different approaches may be found discussed with
some details in [21,22,25–29], and here we shall
describe them briefly.
The first one uses the classical [10] set-up of a
multiannular aperture plus a lens, where each
frequency component of the pulse illuminates –
however – a different (but specific) annular slit [25],
in such a way that relations (9) and (10) get satisfied. In the known experiments by Durnin et al.
[2,3] a single annular slit and a lens were adopted
to generate a Bessel beam.
A second, simple approach, which works for a
medium with low dispersion, consists in using an
19
Axicon (see [21,22], and references therein). When
illuminated by a pulse, this conical lens gets relation Eq. (9) satisfied. Moreover, for small values of
the angle c between the conical and the flat surface,
the cone angle h can be expressed by the simple
expression hðx; cÞ ¼ ðnA ðxÞ 1Þc, where nA ðxÞ is
the refractive index of the axicon material
[21,22,26]. In the case of a low dispersion medium
and of group-velocities V > c, one can therefore
look for the best value of c to fit also Eq. (10) by
means of the previous relation h ¼ hðx; cÞ, which
is linear in c.
The last approach consists in using holographic
elements, which make possible to deal satisfactorily also with stronger dispersion media. Elements
of this kind have been considered in papers as [27–
29], whose transmission function can correspond
to two successive optical elements (an axicon and a
thin lens).
In general, one can follow a procedure similar
to the one illustrated in Fig. 4. In such a case, there
is a different deviation of the wave vector for each
spectral component in passing through the setup
chosen (one of the three listed above). Such deviation, associated with the dispersion due to the
medium, makes the phase velocity equal for each
frequency. This corresponds to no dispersion for
the group velocity. More details about the physics
under consideration can be found in [21,22].
Now, let us consider a nearly gaussian spectrum
as that one given by Eq. (5), and assume the
presence of a dispersive medium whose refractive
index (for the frequency range of interest) can be
written in the form
nðxÞ ¼ n0 þ xd;
ð12Þ
where n0 is a constant, while d is a free parameter
that makes it possible a linear behavior of nðxÞ:
something that is actually realizable for frequencies far from the resonances associated with the
used material. Notice that the linear relationship
between the refractive index and the wave frequency assumed in Eq. (12) is not necessary: but its
existence gets our calculations simplified.
In this way, on substituting Eq. (12) into Eq.
(11), and considering the spectrum, shifted towards optical frequencies, given by Eq. (5), a relation similar to Eq. (13) is found
20
M. Zamboni-Rached et al. / Optics Communications 226 (2003) 15–23
Fig. 4. Sketch of a generic setup (axicon, hologram, etc.) suited to properly deviating the wave vector of each spectral component.
Z
1
x m
wðq;f;dÞ ¼ V
J0
V
0
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!
x
x
V2
q
ðn0 þ dxÞ2 1 eðaV ifÞx=V d
:
2
V
V
c
ð13Þ
To the purpose of evaluating Eq. (13), let us make
a Taylor expansion and rewrite it as
ow d2 o2 w þ
wðq; f; dÞ ¼ wðq; f; 0Þ þ d
od d¼0 2! od2 d¼0
d3 o3 w þ
þ :
ð14Þ
3! od3 d¼0
For the above equation it is known that, if d is
small enough, it is possible to truncate the series at
its first derivative. For the time being, let us assume this is the case and that there is no problem
on truncating Eq. (14). One can check Fig. 5,
which shows typical values of d for SiO2 , a typical
raw-material in fiber optics.
Looking at Eq. (14), one can notice that its first
term wðq; f; 0Þ is already known to us, because it
coincides with the solution given by our Eq. (7).
To complete the expansion (14), one must find
ow
j . After some simple mathematical manipulaod d¼0
tions, one gets that
Z 1 mþ2
ow V 4 qn0
x
¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
eðaV ifÞx=V
2
od d¼0
V
2V
2
0
c n0 c2 1
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!
x
x
V2
n20 2 1 d
J1 q
:
ð15Þ
V
V
c
This integral can be easily evaluated by using
identity 6.621-4 of [24], so to obtain
ow V 3 n0
¼ ð1Þmþ4 2 V 2
od d¼0
c2 n0 c2 1
omþ2
oðaV ifÞ
mþ2
½ðaV ifÞX :
ð16Þ
As in the case of Eqs. (8), (80 ), another form for
expressing Eq. (15) can be found by having recourse once more to the identity (6.621) of [24]
ow n0 q2 Cðm þ 4ÞX mþ4
¼
od d¼0
2c2 V m
m þ 4 m 1
;
; 2;
F
2
2
V2
X2
0
n20 2 1 q2 2 ;
ð16 Þ
c
V
where, as before, X is the ordinary X-wave given
by Eq. (4) and F is again a GaussÕ hypergeometric
M. Zamboni-Rached et al. / Optics Communications 226 (2003) 15–23
21
Fig. 5. Variation of the refractive index nðxÞ with frequency for fused silica. The solid line is its behaviour, according to SellmeierÕs
formulae. The open circles and squares are the linear approximations for m ¼ 45 and m ¼ 27, respectively.
function. Once more, Eq. (160 ) can be useful in the
cases of large values of m.
Finally, from our basic solution (8) and its first
derivative (16), one can write the desired solution
of Eq. (11) as
sive medium has been obtained from simple
mathematical operations (derivatives) applied to
the standard ‘‘X-wave’’.
4. Optical applications
om X
wðq; fÞ ¼ ð1Þ
oðaV ifÞm
V 3 n0
omþ2
mþ4
2 V2
þ ð1Þ
c2 n0 c2 1 oðaV ifÞmþ2
m
½ðaV ifÞX d:
ð17Þ
However, if one wants to use Eqs. (80 ) and (160 ),
instead of Eqs. (8) and (16), the solution (17) can
be written in the form
wðq;fÞ ¼
Cðm þ 1ÞX mþ1
mþ1 m
; ;1;
F
2
2
Vm
2
2
2
V
X
n0 q Cðm þ 4ÞX mþ4
n20 2 1 q2 2 d
c
V
2c2 V m
m þ 4 m 1
V2
X2
;
;2; n20 2 1 q2 2 ;
F
2
2
c
V
0
ð17 Þ
It is also interesting to notice that the approximate
superluminal localized solution (17) for a disper-
To illustrate what was said before, two practical
examples will be considered, both in optical frequencies. When mentioning optics, it is natural to
refer ourselves to optical fibers. Then, let us suppose the bulk of the dispersive medium under
consideration to be fused silica (SiO2 ).
Far from the medium resonances (which is our
case), the refractive index can be approximated by
the well-known Sellmeier equation [30]
n2 ðxÞ ¼ 1 þ
N
X
j¼1
Bj x2j
;
x2j x2
ð18Þ
where xj is the resonance frequency, Bj is the
strength of the jth resonance, and N is the total
number of the material resonances that appear in
the frequency range of interest. For typical frequencies of ‘‘long-haul transmission’’ in optics, it
is necessary to choose N ¼ 3, which leads us [29] to
the values B1 ¼ 0:6961663, B2 ¼ 0:4079426, B3 ¼
22
M. Zamboni-Rached et al. / Optics Communications 226 (2003) 15–23
0:8974794, k1 ¼ 0:0684043 lm, k2 ¼ 0:1162414
lm and k3 ¼ 9:896161 lm.
Fig. 5 illustrates the variation of the refractive
index nðxÞ with frequency for fused silica. The
solid line is its behaviour, according to SellmeierÕs
formulae. The open circles and squares are the
linear approximations for m ¼ 45 and m ¼ 27, respectively, and specifies the ranges that will be
adopted here.
In the two examples, the spectra are localized
around the angular frequency xc ¼ 23:56 1014
Hz (which corresponds to the wavelength kc ¼
0:8 lm), with two different bandwidth Dx1 ¼
0:60xc and Dx2 ¼ 0:45xc . The values of a and m
corresponding to these two situations are a ¼
1:14592 1014 , m ¼ 27, and a ¼ 1:90986 1014 ,
m ¼ 45, respectively. We also have kept 2 the same
velocity, b ¼ 1=V ¼ 1=ð5cÞ.
Looking at these ‘‘windows’’, one can notice that
silica does not suffer strong variations of its refractive index. As a matter of fact, a linear approximation to n ¼ nðxÞ is quite satisfactory in these cases.
Moreover, for both situations, and for their respective n0 values, the value of parameter d results to
be very small, verifying condition (11): which means
that it is quite acceptable our truncation of the
Taylor expansion. The beam intensity profiles for
both bandwidths are shown in Figs. 6 and 7.
In the first figure, one can see a pattern similar
to the fundamental X-wave, but here, of course,
the pulse is much more localized spatially and
temporally (typically, it is a femtosecond pulse).
In the second figure, one can observe some little
differences with respect to the first one, mainly in
the spatial oscillations inside the wave envelope
[18]. This may be explained by taking into account
that, for certain values of the bandwidth, the carrier wavelength become shorter than the width of
the spatial envelope; so that one meets a well-defined carrier frequency.
Let us point out that both these waves are
transversally and longitudinally localized, and
2
Suitable ways for generating such pulses can be the first and
especially the third one listed in Section 3, after Eq. (11). By
contrast, the second approach (with an Axicon) is not appropriate, whenever the considered medium has non-negligible
dispersive effects.
Fig. 6. The real part of an X-shaped beam for optical frequencies in a dispersive medium, with m ¼ 27. It refers to the
outside window in Fig. 5.
Fig. 7. The real part of an X-shaped beam for optical frequencies in a dispersive medium, with m ¼ 45. It refers to the
inner window in Fig. 5.
that, since the dependence of w on z and t is given
by f ¼ z Vt, they are free from dispersion, just
like a classical X-shaped wave.
5. Conclusions
In this paper we have first worked out analytical
superluminal localized solutions to the wave
equation for arbitrary frequencies and with adjustable bandwidth in vacuum. The same methodology has been then used to obtain new, analytical
expressions representing X-shaped waves (with
arbitrary frequencies and adjustable bandwidth)
M. Zamboni-Rached et al. / Optics Communications 226 (2003) 15–23
which propagate in dispersive media. Such expressions have been obtained, on one hand, by adopting the appropriate spectrum (which made possible
to us both choosing the carrier frequency rather
freely, and controlling the spectral bandwidth),
and, on the other hand, by having recourse to
simple mathematics. Finally, we have illustrated
some examples of our approach with applications
in optics, considering fused silica as the dispersive
medium.
Acknowledgements
The authors are grateful to the anonymous referees of this article, as well as to C.A. Dartora and
Amr Shaarawi for continuous scientific collaboration; and to Jane M. Madureira for stimulating
discussions. For useful discussions they also thank
V. Abate, C. Becchi, M. Brambilla, C. Cocca, R.
Collina, G.C. Costa, G. Degli Antoni, F. Fontana,
M. Pernici, M. Villa and M.T. Vasconselos.
References
[1] J.N. Brittingham, J. Appl. Phys. 54 (1983) 1179.
[2] J. Durnin, J.J. Miceli, J.H. Eberly, Phys. Rev. Lett. 58
(1987) 1499.
[3] J. Durnin, J.J. Miceli Jr., J.H. Eberly, Opt. Lett. 13 (1988)
79–80.
[4] R.W. Ziolkowski, J. Math. Phys. 26 (1985) 861.
[5] R.W. Ziolkowski, Phys. Rev. A 39 (1989) 2005.
[6] A. Shaarawi, I.M. Besieris, R.W. Ziolkowski, J. Math.
Phys. 30 (1989) 1254.
[7] A. Shaarawi, R.W. Ziolkowski, I.M. Besieris, J. Math.
Phys. 36 (1985) 5565.
23
[8] R. Donnelly, R.W. Ziolkowski, Proc. R. Soc. Lond. A 440
(1993) 541.
[9] J.-y. Lu, J.F. Greenleaf, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39 (1992) 441.
[10] E. Recami, Physica A 252 (1998) 586, and references therein.
[11] I.M. Besieris, M. Abdel-Rahman, A. Shaarawi, A. Chatzipetros, Prog. Electromagn. Res. 19 (1998) 1.
[12] M. Zamboni-Rached, E. Recami, H.E. Hernandez-Figueroa, Eur. Phys. J. D 21 (2002) 217.
[13] M. Zamboni-Rached, Localized solutions: structure and
applications, M.Sc. Thesis, Physics Department, State
University of Campinas, 1999.
[14] J.-y. Lu, J.F. Greenleaf, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39 (1992) 19.
[15] P. Saari, K. Reivelt, Phys. Rev. Lett. 79 (1997) 4135.
[16] D. Mugnai, A. Ranfagni, R. Ruggeri, Phys. Rev. Lett. 84
(2000) 4830.
[17] E. Recami, Found. Phys. 31 (2001) 1119.
[18] M. Zamboni-Rached, K.Z. N
obrega, E. Recami, H.E.
Hernandez, Phys. Rev. E 66 (2002), article 046617.
[19] J. Salo, J. Fagerholm, A.T. Friberg, M.M. Salomaa, Phys.
Rev. E 62 (2000) 4261.
[20] J. Salo, A.T. Friberg, M.M. Salomaa, J. Phys. A 34 (2001)
9319.
[21] H. S~
onajalg, P. Saari, Opt. Lett. 21 (1996) 1162.
[22] H. S~
onajalg, M. Ratsep, P. Saari, Opt. Lett. 22 (1997) 310.
[23] J. Fagerholm, A.T. Friberg, J. Huttunen, D.P. Morgan,
M.M. Salomaa, Phys. Rev. E 54 (1996) 4347.
[24] I.S. Gradshteyn, I.M. Ryzhik, Integrals, Series and Products, fourth ed., Academic Press, New York, 1965.
[25] H. S~
onajalg, A. Gorokhovskii, R. Kaarli, et al., Opt.
Commun. 71 (1989) 377.
[26] R.M. Herman, T.A. Wiggins, J. Opt. Soc. Am. A 8 (1991)
932.
[27] A. Vasara, J. Turunen, A.T. Friberg, J. Opt. Soc. Am. A 6
(1989) 1748.
[28] R.P. MacDonald, J. Chrostowski, S.A. Boothroyd, B.A.
Syrett, Appl. Opt. 32 (1983) 6470.
[29] See, e.g., V.P. Koronkevich, I.A. Mikhaltsova, E.G.
Churin, Y.I. Yurlov, Appl. Opt. 34 (1995) 5761, and
references therein.
[30] G.P. Agrawal, Nonlinear Fiber Optics, second ed., Academic Press, New York, 1995.