Download Subluminal wave bullets: Exact localized subluminal

PHYSICAL REVIEW A 77, 033824 共2008兲
Subluminal wave bullets: Exact localized subluminal solutions to the wave equations
Michel Zamboni-Rached*
Centro de Ciências Naturais e Humanas, Universidade Federal do ABC, Santo André, SP, Brazil
and DMO-FEEC, State University at Campinas, Campinas, SP, Brazil
Erasmo Recami†
Facoltà di Ingegneria, Università Statale di Bergamo, Bergamo, Italy and INFN, Sezione di Milano, Milan, Italy
共Received 19 September 2007; published 11 March 2008兲
In this work it is shown how to obtain, in a simple way, localized 共nondiffractive兲 subluminal pulses as exact
analytic solutions to the wave equations. These ideal subluminal solutions, which propagate without distortion
in any homogeneous linear media, are herein obtained for arbitrarily chosen frequencies and bandwidths,
avoiding in particular any recourse to the noncausal 共backward moving兲 components that so frequently plague
the previously known localized waves. Such solutions are suitable superpositions of—zeroth order, in
general—Bessel beams, which can be performed either by integrating with respect to 共w.r.t.兲 the angular
frequency ␻, or by integrating w.r.t. the longitudinal wave number kz: Both methods are expounded in this
paper. The first one appears to be powerful enough; we study the second method as well, however, since it
allows us to deal even with the limiting case of zero-speed solutions 共and furnishes a way, in terms of
continuous spectra, for obtaining the so-called “frozen waves,” so promising also from the point of view of
applications兲. We briefly treat the case, moreover, of nonaxially symmetric solutions, in terms of higher-order
Bessel beams. Finally, some attention is paid to the known role of special relativity, and to the fact that the
localized waves are expected to be transformed one into the other by suitable Lorentz transformations. In this
work we fix our attention especially on acoustics and optics: However, results of the present kind are valid
whenever an essential role is played by a wave equation 共such as electromagnetism, seismology, geophysics,
gravitation, elementary particle physics, etc.兲.
DOI: 10.1103/PhysRevA.77.033824
PACS number共s兲: 42.25.Bs, 42.25.Fx, 41.20.Jb, 46.40.Cd
I. INTRODUCTION
For more than 10 years, the so-called 共nondiffracting兲 “localized waves” 共LW兲, which are solutions to the wave equations 共scalar, vectorial, spinorial, etc.兲, are in fashion, both in
theory and in experiment. In particular, rather well known
are the ones with luminal or superluminal peak velocity 关1兴,
such as the so-called X-shaped waves 共see 关2,3兴 and references therein; for a review, see, e.g., Ref. 关4兴兲, which are
supersonic in acoustics 关5兴, and superluminal in electromagnetism 共see 关6兴 and references therein兲.
Since Bateman 关7兴 and later on Courant and Hilbert 关8兴, it
was already known that luminal LWs exist, which are as well
solutions to the wave equations. More recently, some attention 关9–13兴 started to be paid to subluminal solutions too. Let
us recall that all the LWs propagate without distortion—and
in a self-reconstructive way 关14–16兴—in a homogeneous linear medium 共apart from local variations兲: In the sense that
their square magnitude keeps its shape during propagation,
while local variations are shown only by its real, or imaginary, parts.
As in the superluminal case, the 共more orthodox, in a
sense兲 subluminal LWs can be obtained by suitable superpositions of Bessel beams. They have been until now almost
*[email protected]; [email protected]
†
[email protected]
1050-2947/2008/77共3兲/033824共14兲
neglected, however, for the mathematical difficulties met in
getting analytic expressions for them, difficulties associated
with the fact that the superposition integral runs over a finite
interval. We want here to readdress the question of such subluminal LWs, showing, by contrast, that one can indeed arrive at exact 共analytic兲 solutions, both in the case of integration over the Bessel beams’ angular frequency ␻, and in the
case of integration over their longitudinal wave number kz.
The first approach, herein investigated in detail, is enough to
get the majority of the desired results; we study also the
second one, however, since it allows treating the limiting
case of zero-speed solutions, which have been called “frozen
waves” 共and also furnishes a second method—based on a
continuous spectrum—for obtaining such waves, so promising also from the point of view of applications兲. Moreover,
we shall briefly deal with nonaxially symmetric solutions, in
terms of higher-order Bessel beams. At last, some attention is
paid to the known role of special relativity, and to the fact
that the localized waves are expected to be transformed one
into the other by Lorentz transformations.
As already claimed, the present paper is devoted to the
exact, analytic solutions, i.e., to ideal solutions. In another
article, we shall go on to the corresponding pulses with finite
energy, or truncated, sometimes having recourse—in those
cases, only—to approximations. We shall fix our attention
especially on acoustics and optics: However, our results are
valid whenever an essential role is played by a wave equation 共such as electromagnetism, seismology, geophysics,
gravitation, elementary particle physics, etc.兲.
Let us recall that, in the past, too much attention was not
even paid to 1983 Brittingham’s paper 关17兴, wherein he
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©2008 The American Physical Society
PHYSICAL REVIEW A 77, 033824 共2008兲
MICHEL ZAMBONI-RACHED AND ERASMO RECAMI
showed the possibility of obtaining pulse-type solutions to
the Maxwell equations, which propagate in free space as a
kind of speed-c “soliton.” This lack of attention was partially
due to the fact that Brittingham had been able neither to get
correct finite-energy expressions for such “wavelets,” nor to
make suggestions about their practical production. Two years
later, however, Sezginer 关18兴 was able to show how to obtain
quasinondiffracting luminal pulses endowed with a finite energy. Finite-energy pulses no longer travel undistorted for an
infinite distance, but they can nevertheless propagate without
deformation for a long field depth, much larger than the one
achieved by ordinary pulses, such as the Gaussian ones: cf.,
e.g., Refs. 关19–24兴 and references therein.
Only after 1985 the general theory of all LWs started to be
extensively developed 关25–31,2,6,3兴 both in the case of
beams, and in the case of pulses. For reviews, see for instance, Refs. 关4,32,23,21,24兴 and references therein. For the
propagation of LWs in bounded regions 共such as guides兲, see
Refs. 关33–36兴 and references therein. For the focusing of
LWs, see Refs. 关37,38兴 and references therein. As to the construction of LWs propagating in dispersive media, cf. Refs.
关39–47兴; and, for lossy media, see Ref. 关16兴 and references
therein. At last, for finite energy, or truncated, solutions see
Refs. 关48–50,24,3,34兴.
By now, the LWs have been experimentally produced
关5,51,52兴, and are being applied in fields ranging from ultrasound scanning 关53,54,11兴 to optics 共for the production, e.g.,
of new type of tweezers 关55兴兲. All these works demonstrated
that nondiffracting pulses can travel with an arbitrary peak
speed v, that is, with 0 ⬍ v ⬍ ⬁; while Brittingham and
Sezginer had confined themselves to the luminal case
共v = c兲 only. As already commented, the superluminal and
luminal LWs have been, and are being, intensively studied,
while the subluminal ones have been neglected: Almost all
the papers dealing with them had until now recourse to the
paraxial 关56兴 approximation 关57兴, or to numerical simulations 关12兴, due to the above-mentioned mathematical difficulty in obtaining exact analytic expressions for subluminal
pulses. Actually, only one analytic solution is known
关9–11,28,57,58兴 biased by the physically inconvenient facts
that its frequency spectrum is very large, it does not even
possess a well-defined central frequency, and, even more,
that backward-traveling 关26,24兴 components 共ordinarily
called “noncausal,” since they should be entering the antenna
or generator兲 are a priori needed for constructing it. The aim
of the present paper is to show how subluminal localized
exact solutions can be constructed with any spectra, in any
frequency bands, and for any bandwidths; and, moreover,
without employing 关3,23兴 any backward-traveling components.
over the angular frequency ␻ and the longitudinal wave
number kz, i.e., in cylindrical coordinates,
⌿共␳,z,t兲 =
0
Axially symmetric solutions to the scalar wave equation
are known to be superpositions of zero-order Bessel beams
d␻
␻/c
−␻/c
冉冑
dkzS̄共␻,kz兲J0 ␳
冊
␻2
− kz2 eikzze−i␻t ,
c2
共1兲
where k␳2 ⬅ ␻2 / c2 − kz2 is the transverse wave number; quantity k␳2 must be positive since evanescent waves cannot come
into play.
The condition characterizing a nondiffracting wave is the
existence 关24,59兴 of a linear relation between longitudinal
wave number kz and frequency ␻ for all the Bessel beams
entering superposition 共1兲; that is to say, the chosen spectrum
must entail 关3,21兴 for each Bessel beam a linear relationship
of the type1
␻ = vkz + b,
共2兲
where, as we will see, the constant v is the wave velocity
共i.e., the pulse peak velocity兲 and b ⱖ 0. Requirement 共2兲 can
be regarded also as a specific space-time coupling, implied
by the chosen spectrum S̄. Equation 共2兲 must be obeyed by
the spectra of any one of the three possible types 共subluminal, luminal or superluminal兲 of nondiffracting pulses. Let us
mention that with the choice in Eq. 共2兲 the pulse regains its
initial shape after the space interval ⌬z1 = 2␲v / b; the more
general case can be, however, considered 关3,47兴 when b assumes any values bm = mb 共with m an integer兲, and the periodicity space interval becomes ⌬zm = ⌬z1 / m. We are referring ourselves, of course, to the real 共or imaginary兲 part of
the pulse, since its modulus is known to be endowed with
rigid motion.
In the subluminal case, of interest here, the only exact
solution known until now, represented by Eq. 共10兲 below, is
the one found by Mackinnon 关9兴. Indeed, by taking into explicit account that the transverse wave number k␳ of each
Bessel beam entering Eq. 共1兲 must be real, it can be easily
shown 共as first noticed by Salo et al. for the analogous
acoustic solutions 关12兴兲 that in the subluminal case b cannot
vanish, but must be larger than zero: b ⬎ 0. Then, by using
conditions 共2兲 and b ⬎ 0, the subluminal localized pulses can
be expressed as integrals over the frequency only,
冉 冊冕
⌿共␳,z,t兲 = exp − ib
z
v
␻+
␻−
冉 冊
d␻S共␻兲J0共␳k␳兲exp i␻
␨
,
v
共3兲
1
II. FIRST METHOD FOR CONSTRUCTING PHYSICALLY
ACCEPTABLE, SUBLUMINAL LOCALIZED PULSES
冕 冕
⬁
More generally, as shown in Ref. 关3兴, the chosen spectrum must
call into play, in the plane ␻ , kz, if not exactly the line 共2兲, at least
a region in the proximity of a straight line of such a type. It is
interesting that in the latter case one obtains solutions endowed with
finite energy, but possessing a finite “depth of field,” that is, nondiffracting only until a certain finite distance.
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PHYSICAL REVIEW A 77, 033824 共2008兲
which means nothing but S共␻兲 = const: In this case, one obtains indeed the Mackinnon solution 关9,28,11,50兴
where now
k␳ =
1
v
冑2b␻ − b2 − 共1 − v2/c2兲␻2
共4兲
with
册
b2 2 2
␥ 共 ␳ + ␥ 2␨ 2兲 ,
c2
共10兲
␨ ⬅ z − vt
共5兲
and with
␻− =
␻+ =
b
.
1 − v/c
␻⬅
冉 冊
b
v
1+ s ,
1 − v2/c2
c
and
⫻
冕
−1
dsS共s兲J0
冉
⫻exp i
v
␥⬅
1
冑1 − ␤2 .
共11兲
冉
S共␻兲 = exp
冊
i2n␲␻
,
⍀
共12兲
⍀ ⬅ ␻+ − ␻− .
The solution is written in this more general case as
冉
冋冑
共9兲
As already mentioned, Eq. 共8兲 has until now yielded one
analytic solution for S共s兲 = const, only 关for instance, S共s兲 = 1兴;
冊 冉 冊
冉 冊册
␲
b
⌿共␳, ␨, ␩兲 = 2b␤␥2 exp i ␤␥2␩ exp in
c
␤
共8兲
In the following we shall adhere to some symbols standard in
special relativity 关since the whole topic of subluminal, luminal, and superluminal LWs is strictly connected 关4,6,60兴 with
the principles and structure of special relativity 共cf. 关61,62兴
and references therein兲, as we shall mention in the conclusions兴; namely,
␤⬅ ,
c
c2
,
v
with n any integer number, which means for any spectra of
the type S共s兲 = exp共in␲ / ␤兲exp共in␲s兲, as can be easily seen
by checking the product of the various exponentials entering
the integrand. In Eq. 共12兲 we have set
␳
b
1 − s2
c 1 − v2/c2
1
b
␨s .
c 1 − v2/c2
with V ⬅
where V and v are related by the de Broglie relation. 关Notice
that ⌿ in Eq. 共10兲, and in the following ones, is eventually a
function 共besides of ␳兲 of z , t via ␨ and ␩, both functions of
z and t.兴
We can, however, construct by a very simple method subluminal pulses corresponding to whatever spectrum, and devoid of backward-moving 共i.e., “entering”兲 components, by
also taking advantage of the fact that in our Eq. 共8兲 the integration interval is finite: That is, by transforming it in a good,
instead of an evil. Let us first observe that Eq. 共8兲 does not
admit only the already known analytic solution corresponding to S共s兲 = const, and more in general to S共␻兲 = const, but it
will also yield an exact, analytic solution for any exponential
spectra of the type
共7兲
1
b
b
b
v
␨
2 2 exp − i z exp i
c 1 − v /c
v
v 1 − v2/c2
1
␩ ⬅ z − Vt,
共6兲
Eq. 共3兲 becomes 关12兴
冉 冊 冉
冊
冉冑 冑 冊
冊
which however, for its above-mentioned drawbacks, is endowed with little physical and practical interest. In Eq. 共9兲
the sinc function has the ordinary definition
sinc x ⬅ 共sin x兲/x
b
,
1 + v/c
As anticipated, the Bessel beam superposition in the subluminal case results to be an integration over a finite interval of
␻, which does clearly show that the backward-traveling
共noncausal兲 components correspond to the interval ␻− ⬍ ␻
⬍ b. It could be noticed that Eq. 共3兲 does not represent the
most general exact solution, which on the contrary is a sum
关47兴 of such solutions for the various possible values of b
just mentioned above: That is, for the values bm = mb and
spatial periodicity ⌬zm = ⌬z1 / m; but we can confine ourselves
to solution 共3兲 without any real loss of generality, since the
actual problem is evaluating in analytic form the integral
entering Eq. 共3兲. For any mathematical and physical details,
see Ref. 关47兴.
Now, if one adopts the change of variables
⌿共␳,z,t兲 =
冊 冋冑
冉
b
b
⌿共␳, ␨, ␩兲 = 2 v␥2 exp i ␤␥2␩ sinc
c
c
⫻sinc
b2 2 2
b 2
␥ ␨ + n␲
2␥ ␳ +
c
c
2
. 共13兲
Let us explicitly notice that also in Eq. 共13兲 quantity ␩ is
defined as in Eqs. 共11兲, where V and v obey the de Broglie
relation vV = c2, the subluminal quantity v being the velocity
of the pulse envelope, and V playing the role 共in the envelope’s interior兲 of a superluminal phase velocity.
The next step, as anticipated, consists just in taking advantage of the finiteness of the integration limits for expanding any arbitrary spectra S共␻兲 in a Fourier series in the interval ␻− ⱕ ␻ ⱕ ␻+,
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PHYSICAL REVIEW A 77, 033824 共2008兲
MICHEL ZAMBONI-RACHED AND ERASMO RECAMI
冉
⬁
S共␻兲 =
兺 An exp
n=−⬁
+ in
冊
2␲
␻ ,
⍀
共14兲
where 共we can go back, now, from the s to the ␻ variable兲,
An =
1
⍀
冕
␻+
␻−
冉
d␻S共␻兲exp − in
2␲
␻
⍀
冊
共15兲
quantity ⍀ being defined as above.
Then, on remembering the special solution 共13兲, we can
infer from expansion 共14兲 that, for any arbitrary spectral
function S共␻兲, a rather general, axially symmetric, analytic
solution for the subluminal case can be written as
Eq. 共3兲 is made, it being ␻− = b / 共1 + ␤兲 and ␻+ = b / 共1 − ␤兲. Of
course, relation 共2兲 must still be satisfied, and with b ⬎ 0, for
getting an ideal subluminal localized solution. Notice that,
with spectrum 共17兲, the bandwidth 关actually, the full width at
half-maximum 共FWHM兲兴 results to be ⌬␻ = 2 / a. Let us emphasize that, once v and b have been fixed, the values of a
and ␻0 can then be selected in order to kill the backwardtraveling components, that exist, as we know, for ␻ ⬍ b.
The Fourier expansion in Eq. 共14兲, which yields, with the
above spectral function 共17兲, the coefficients
An ⯝
冉
冊 冉
冊
2␲
1
− n 2␲ 2
exp − in ␻0 exp 2 2 ,
⍀
⍀
a⍀
共18兲
共17兲
constitutes an excellent representation of the Gaussian spectrum 共17兲 in the interval ␻− ⬍ ␻ ⬍ ␻+ 共provided that, as we
requested, our Gaussian spectrum does get negligible values
outside the frequency interval ␻− ⬍ ␻ ⬍ ␻+兲.
In other words, on choosing a pulse velocity v ⬍ c and a
value for the parameter b, a subluminal pulse with the above
frequency spectrum 共17兲 can be written as Eq. 共16兲, with the
coefficients An given by Eq. 共18兲: The evaluation of such
coefficients An being rather simple. Let us repeat that, even if
the values of the An are obtained via a 共rather good兲 approximation, we based ourselves on the exact solution equation
共16兲.
One can, for instance, obtain exact solutions representing
subluminal pulses for optical frequencies. Let us get the subluminal pulse with velocity v = 0.99c, angular carrier frequency ␻0 = 2.4⫻ 1015 Hz 共that is, ␭0 = 0.785 ␮m兲, and
bandwidth 共FWHM兲 ⌬␻ = ␻0 / 20= 1.2⫻ 1014 Hz, which is
an optical pulse of 24 fs 共that is the FWHM of the pulse
intensity兲. For a complete pulse characterization, one must
choose the value of the frequency b: Let it be b = 3
⫻ 1013 Hz; as a consequence one has ␻− = 1.507⫻ 1013 Hz
and ␻+ = 3 ⫻ 1015 Hz. 共This is exactly a case in which the
considered pulse is not plagued by the presence of backwardtraveling components, since the chosen spectrum forwards
totally negligible values for ␻ ⬍ b.兲 The construction of the
pulse does already result satisfactory when considering approximately 51 terms 共−25ⱕ n ⱕ 25兲 in the series entering
Eq. 共16兲.
Figure 1 shows our pulse, plotted by considering the mentioned 51 terms. Namely, Fig. 1共a兲 depicts the orthogonal
projection of the pulse intensity; Fig. 1共b兲 shows the threedimensional intensity pattern of the real part of the pulse,
which reveals the carrier wave oscillations.
Let us stress that the ball-like shape2 for the field intensity
should be typically associated with all the subluminal LWs,
while the typical superluminal ones are known to be X
shaped 关2,6,60兴, as predicted, since long, by special relativity
in its “nonrestricted” version: See Refs. 关61,62,6,4兴 and references therein.
A second spectrum S共␻兲 would be, for instance, the “inverted parabola” one, centered at the frequency ␻0, that is,
whose values are negligible outside the frequency interval
␻− ⬍ ␻ ⬍ ␻+ over which the Bessel beams superposition in
2
It can be noted that each term of the series in Eq. 共16兲 corresponds to an ellipsoid or, more specifically, to a spheroid, for each
velocity v.
冉
冋冑
b
⌿共␳, ␨, ␩兲 = 2b␤␥2 exp i ␤␥2␩
c
冊兺
冉
⬁
n=−⬁
b2 2 2
b
␥ ␳ + ␥ 2␨ + n ␲
c2
c
⫻sinc
冉 冊
冊册
An exp in
␲
␤
2
,
共16兲
in which the coefficients An are still given by Eq. 共15兲. Let us
repeat that our solution is expressed in terms of the particular
Eq. 共13兲, which is a Mackinnon-type solution.
The present approach presents many advantages. We can
easily choose spectra localized within the prefixed frequency
interval 共optical waves, microwaves, etc.兲 and endowed with
the desired bandwidth. Moreover, as already said, spectra can
now be chosen such that they have zero value in the region
␻− ⱕ ␻ ⱕ b, which is responsible for the backward-traveling
components of the subluminal pulse.
Let us stress that, even when the adopted spectrum S共␻兲
does not possess a known Fourier series 关so that the coefficients An cannot be exactly evaluated via Eq. 共15兲兴, one can
calculate approximately such coefficients without meeting
any problem, since our general solutions 共16兲 will still be
exact solutions.
Let us set forth in the following some examples.
Examples. In general, optical pulses generated in the laboratory possess a spectrum centered on some frequency value,
␻0, called the carrier frequency. The pulses can be, for instance, ultrashort, when ⌬␻ / ␻0 ⱖ 1; or quasimonochromatic,
when ⌬␻ / ␻0 1, where ⌬␻ is the spectrum bandwidth.
These kinds of spectra can be mathematically represented
by a Gaussian function, or functions with similar behavior.
First two examples. Let us first consider a Gaussian spectrum
S共␻兲 =
a
冑␲ exp关− a 共␻ − ␻0兲 兴
2
2
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FIG. 1. 共Color online兲 共a兲 The intensity orthogonal projection for the pulse corresponding to Eqs. 共17兲 and 共18兲 in the case of an optical
frequency 共see the text兲; 共b兲 the three-dimensional intensity pattern of the real part of the same pulse, which reveals the carrier wave
oscillations.
S共␻兲 =
冦
− 4关␻ − 共␻0 − ⌬␻/2兲兴关␻ − 共␻0 + ⌬␻/2兲兴
for ␻0 − ⌬␻/2 ⱕ ␻ ⱕ ␻0 + ⌬␻/2,
⌬␻2
0
otherwise,
where ⌬␻, the distance between the two zeros of the parabola, can be regarded as the spectrum bandwidth. One can
expand S共␻兲, given in Eq. 共19兲, in the Fourier series 共14兲, for
␻− ⱕ ␻ ⱕ ␻+, with coefficients An that—even if straightforwardly calculable—results to be complicated, so that we skip
reporting them here explicitly. Let us here only mention that
spectrum 共19兲 may be easily used to get, for instance, an
ultrashort 共femtasecond兲 optical nondiffracting pulse, with
satisfactory results even when considering very few terms in
expansion 共14兲.
Third example. As a third, interesting example, let us consider the very simple case when—within the integration limits ␻−, ␻+—the complex exponential spectrum 共12兲 is replaced by the real function 共still linear in ␻兲
S共␻兲 =
a
exp关a共␻ − ␻+兲兴,
1 − exp关− a共␻+ − ␻−兲兴
共20兲
with a a positive number 共for a = 0 one goes back to the
Mackinnon case兲. Spectrum 共20兲 is exponentially concentrated in the proximity of ␻+, where it reaches its maximum
value; and becomes more and more concentrated 共on the lefthand side of ␻+, of course兲 as the arbitrarily chosen value of
a increases, the frequency bandwidth being ⌬␻ = 1 / a. Let us
recall that, on their turn, quantities ␻+ and ␻− depend on the
pulse velocity v and on the arbitrary parameter b.
冧
共19兲
By performing the integration as in the case of spectrum
共12兲, instead of solution 共13兲 in the present case one eventually gets the solution
⌿共␳, ␨, ␩兲 =
冉
2ab␤␥2 exp共ab␥2兲exp共− a␻+兲
b
exp i ␤␥2␩
c
1 − exp关− a共␻+ − ␻−兲兴
冉
⫻sinc
冊
b 2 −2 2
␥ 冑␥ ␳ − 共av + i␨兲2 .
c
冊
共21兲
After Mckinnon’s, Eq. 共21兲 appears to be the simplest
closed-form solution, since both of them do not need any
recourse to series expansions. In a sense, our solution 共21兲
might be regarded as the subluminal analogous of the 共superluminal兲 X-wave solution; a difference being that the standard X-shaped solution has a spectrum starting with 0, where
it assumes its maximum value, while in the present case the
spectrum starts at ␻− and gets increasing afterwards until ␻+.
More important is to observe that the Gaussian spectrum has
a priori two advantages with respect to 共w.r.t.兲 Eq. 共20兲: It
may be more easily centered around any value ␻0 of ␻, and,
when increasing its concentration in the surrounding of ␻0,
the spot transverse width does not increase indefinitely, but
tends to the spot width of a Bessel beam with ␻ = ␻0 and
kz = 共␻0 − b兲 / v, at variance with what happens for spectrum
共20兲; however, solution 共21兲 is noticeable, since it is really
the simplest one.
033824-5
PHYSICAL REVIEW A 77, 033824 共2008兲
MICHEL ZAMBONI-RACHED AND ERASMO RECAMI
⌿共␳,z,t兲 = exp共− ibt兲
冕
kz max
dkzS共kz兲J0共␳k␳兲exp共i␨kz兲,
kz min
共22兲
with
kz min =
−b 1
,
c 1+␤
b 1
,
c1−␤
kz max =
共23兲
and with
k␳2 = −
FIG. 2. The intensity of the real part of the subluminal pulse
corresponding to spectrum 共20兲, with v = 0.99c, b = 3 ⫻ 1013 Hz
共which result in ␻− = 1.5⫻ 1013 Hz and ␻− = 3 ⫻ 1015 Hz兲, ⌬␻ / ␻+
= 1 / 100 共i.e., a = 100兲.
Figure 2 shows the intensity of the real part of the subluminal pulse corresponding to this spectrum, with v = 0.99c,
b = 3 ⫻ 1013 Hz 共which result in ␻− = 1.5⫻ 1013 Hz and ␻+
= 3 ⫻ 1015 Hz兲, ⌬␻ / ␻+ = 1 / 100 共i.e., a = 100兲. This is an optical pulse of 0.2 ps.
where quantity ␨ is still defined according to Eq. 共5兲, always
with v ⬍ c.
One can show that the unique exact solution previously
known 关9兴 may be rewritten in the form of Eq. 共22兲 with
S共kz兲 = const. Then, on following the same procedure exploited in our first method 共preceding section兲, one can find
out exact solutions corresponding to
冉
S共kz兲 = exp
冊
i2n␲kz
,
K
共25兲
K ⬅ kz max − kz min,
The previous method appears to be very efficient for finding out analytic subluminal LWs, but it loses its validity in
the limiting case v → 0, since for v = 0 it is ␻− ⬅ ␻+ and the
integral in Eq. 共3兲 degenerates, furnishing a null value. By
contrast, we are interested also in the v = 0 case, since it
corresponds to some of the most interesting, and potentially
useful, LWs: Namely, to the so-called “frozen waves,” which
are stationary solutions to the wave equations, possessing a
static envelope.
Before going on, let us recall that the theory of frozen
waves was developed in Refs. 关49,63,16兴, by having recourse
to discrete superpositions in order to bypass the need of numerical simulations. In the case of continuous superpositions, numerical simulations were performed in Ref. 关64兴.
However, the method presented in this section does allow us
to find analytical exact solutions 共without any further need,
now, of numerical simulations兲 even for frozen waves consisting in continuous superpositions. Actually, we are going
to see that the present method works whatever is the chosen
field-intensity shape, also in regions with size of the order of
the wavelength.
It is possible to obtain such results by starting again from
Eq. 共1兲, with constraint 共2兲, but going on—this time—to integrals over kz, instead of over ␻. It is enough to write relation 共2兲 in the form
for expressing the exact solutions 共1兲 as
共24兲
where
III. SECOND METHOD FOR CONSTRUCTING
SUBLUMINAL LOCALIZED PULSES
1
kz = 共␻ − b兲
v
kz2
b
b2
2 + 2 ␤kz + 2 ,
␥
c
c
共2⬘兲
by performing the change of variable 关analogous, in its finality, to the one in Eq. 共7兲兴
b
kz ⬅ ␥2共s + ␤兲.
c
共26兲
At the end, the exact subluminal solution corresponding
to the spectrum 共25兲 results to be
冉
冋冑
冊
b
b
⌿共␳, ␨, ␩兲 = 2 ␥2 exp i ␤␥2␩ exp共in␲␤兲
c
c
⫻sinc
冉
b2 2 2
b
␥ ␳ + ␥ 2␨ + n ␲
c2
c
冊册
2
. 共27兲
We can again observe that any spectra S共kz兲 can be expanded, in the interval kz min⬍ kz ⬍ kz max, in a Fourier series
冉
⬁
S共kz兲 =
兺 An exp
n=−⬁
with coefficients now given by
An =
1
K
冕
kz max
kz min
+ in
冉
冊
2␲
kz ,
K
dkzS共kz兲exp − in
共28兲
冊
2␲
kz ,
K
共29兲
quantity K having been defined above.
At the very end of the whole procedure, the general exact
solution representing a subluminal LW, for any spectra S共kz兲,
can be eventually written as
033824-6
SUBLUMINAL WAVE BULLETS: EXACT LOCALIZED …
冉
冋冑
b
b
⌿共␳, ␨, ␩兲 = 2 ␥2 exp i ␤␥2␩
c
c
⫻sinc
冊兺
冉
PHYSICAL REVIEW A 77, 033824 共2008兲
⬁
An exp共in␲␤兲
n=−⬁
b2 2 2
b 2
␥ ␨ + n␲
2␥ ␳ +
c
c
冊册
2
共30兲
,
whose coefficients are expressed in Eq. 共29兲, and where
quantity ␩ is defined as above, in Eq. 共11兲.
Interesting examples could be easily worked out, as we
did at the end of the preceding section.
IV. STATIONARY SOLUTIONS WITH ZERO-SPEED
ENVELOPES (“FROZEN WAVES”)
Here, we shall refer to the 共second兲 method, expounded in
the preceding section. Our solution 共30兲, for the case of envelopes at rest, that is, in the case v = 0 共which implies ␨ = z兲,
becomes
b
⌿共␳,z,t兲 = 2 exp共− ibt兲
c
⬁
⫻
冋冑 冉
兺 An sinc
n=−⬁
b2 2
b
␳ + z + n␲
c2
c
冊册
2
,
共31兲
imaginary, parts carry energy and momentum, too. We had
just recourse to the real part only, since the particular fields
considered by us, for the scalar optical or acoustic cases, are
obviously real 共even if one always adopts the customary
complex formalism, mainly for elegance reasons兲, and the
power flux associated with ⌿ must be obtained from the real
part.兴
It may be stressed that the present 共second兲 method does
yield exact solutions, without any need of the paraxial approximation, which, on the contrary, is so often used when
looking for expressions representing beams, such as the
Gaussian ones. Let us recall that, when having recourse to
the paraxial approximation, the obtained beam expressions
are valid only when the envelope sizes 共e.g., the beam spot兲
vary in space much more slowly than the beam wavelength.
For instance, the usual expression for a Gaussian beam 关56兴
holds only when the beam spot ⌬␳ is much larger than ␭0
⬅ ␻0 / 共2␲c兲 = b / 共2␲c兲, so that those beams cannot be very
localized. By contrast, our method overcomes such problems, since it yields, as we have seen above, exact expressions for 共well localized兲 beams with sizes of the order of
their wavelength. Notice, moreover, that the already known
exact solutions—for instance, the Bessel beams—are nothing
but particular cases of our solution 共31兲.
An example. On choosing 共with 0 ⱕ q− ⬍ q+ ⱕ 1兲 the spectral double-step function
with coefficients An given by Eq. 共29兲 with v = 0, so that its
integration limits simplify into −b / c and b / c, respectively;
thus, one obtains
An =
c
2b
冕
冉
b/c
dkzS共kz兲exp − in
−b/c
冊
c␲
kz .
b
冦
for q−␻0/c ⱕ kz ⱕ q+␻0/c,
S共kz兲 = ␻0共q+ − q−兲
0
elsewhere,
共29⬘兲
Equation 共31兲 is an exact solution, corresponding to stationary beams with a static intensity envelope. Let us observe,
however, that even in this case one has energy propagation,
as it can be easily verified from the power flux Ss
= −⵱⌿R ⳵ ⌿R / ⳵t 共scalar case兲 or from the Poynting vector
Sv = 共E ⫻ H兲 共vectorial case, the condition being that ⌿R be a
single component, Az, of the vector potential A兲 关6兴. We have
here indicated by ⌿R the real part of ⌿. For v = 0, Eq. 共2兲
becomes
␻ = b ⬅ ␻0 ,
so that the particular subluminal waves endowed with null
velocity are actually monochromatic beams. 关Incidentally, let
us seize the present opportunity for recalling that only for
superluminal LWs one can meet a rigid motion not only of
the field amplitude, but also of its real and imaginary parts:
In the most general case, the field magnitude does keep its
shape while propagating 共according to the definition of LW兲,
but its real and imaginary parts suffer local variations; in the
sense that the latter are still LWs, but their envelope 共which
determines the field shape兲 appears multiplied by a plane
wave, which entails the already mentioned local variations.
For instance, the real part of the simple Mackinnon solution,
Eq. 共10兲, appears as the product of a nondiffracting envelope
and a plane wave. This can be verified, more in general, by
inspection of Eq. 共8兲. Of course, both the single real, or
c
冧
共32兲
the coefficients of Eq. 共31兲 become
An =
ic
共e−iq+␲n − e−iq−␲n兲.
2␲n␻0共q+ − q−兲
共33兲
The double-step spectrum 共32兲, with regard to the
longitudinal wave number, corresponds to the mean value
k̄z = ␻0共q+ + q−兲 / 2c and to the width ⌬kz = ␻0共q+ − q−兲 / c. From
these relations, it follows that ⌬kz / k̄z = 2共q+ − q−兲 / 共q+ + q−兲.
For values of q− and q+ that do not satisfy the inequality
⌬kz / k̄z 1, the resulting solution will be a nonparaxial
beam.
Figure 3 shows the exact solution corresponding to ␻0
= 1.88⫻ 1015 Hz 共i.e., ␭0 = 1 ␮m兲 and to q− = 0.3, q+ = 0.9,
which results to be a beam with a spot-diameter of 0.6 ␮m,
and, moreover, with a rather good longitudinal localization.
In the case of Eqs. 共32兲 and 共33兲, about 21 terms 共−10ⱕ n
ⱕ 10兲 in the sum entering Eq. 共31兲 are quite enough for a
good evaluation of the series. The beam considered in this
example is highly nonparaxial 共with ⌬kz / k̄z = 1兲, and therefore could not have been obtained by ordinary Gaussian
033824-7
PHYSICAL REVIEW A 77, 033824 共2008兲
MICHEL ZAMBONI-RACHED AND ERASMO RECAMI
FIG. 3. 共Color online兲 共a兲 Orthogonal projection of the three-dimensional intensity pattern of the beam 共a null-speed subluminal wave兲
corresponding to spectrum 共32兲; 共b兲 3D plot of the field intensity. The beam considered in this example is highly nonparaxial.
beam solutions 共which are valid in the paraxial regime
only兲.3
Let us now emphasize that a noticeable property of our
present method is that it allows a spatial modeling even of
monochromatic fields 共that correspond to envelopes at rest;
so that, in the electromagnetic cases, one can speak, e.g., of
the modeling of “light-fields at rest”兲. Such a property—
rather interesting, especially for applications 关55兴—was already investigated, under different assumptions, in Refs.
关49,63,16兴, where the stationary fields with static envelopes
were called “frozen waves” 共FW兲. Namely, in the quoted
references, discrete superpositions of Bessel beams were
adopted in order to get a predetermined longitudinal 共onaxis兲 intensity pattern, inside the desired space interval 0
⬍ z ⬍ L. In other words, in Refs. 关49,63,16兴, the frozen waves
have been written in the form
N
⌿共␳,z,t兲 = e−i␻0teiQz
BnJ0共␳k␳n兲ei2n␲z/L
兺
n=−N
共34兲
with
Bn =
1
L
冕
L
dzF共z兲e−i2n␲z/L ,
共35兲
0
quantity 兩F共z兲兩2 being the desired longitudinal intensity
2
shape, chosen a priori. In Eq. 共34兲, it is k␳2n = ␻20 / c2 − kzn
, and
0 ⱕ Q + 2N␲ / L ⱕ ␻0 / c, where we set kzn ⬅ Q + 2n␲ / L. As we
see from Eq. 共34兲, the FWs have been represented in the past
in terms of discrete superpositions of Bessel beams. But,
now, the method exploited in this paper allows us to go on to
dealing with continuous superpositions. In fact, the continu3
We are considering here only scalar wave fields. In the case of
nonparaxial optical beams, the vector character of the field must be
considered.
ous superposition analogous to Eq. 共34兲 are written as
⌿共␳,z,t兲 = e−i␻0t
冕
␻0/c
−␻0/c
dkzS共kz兲J0共␳k␳兲eizkz ,
共36兲
which, actually, is nothing but our previous equation 共22兲
with v = 0 共and therefore ␨ = z兲: That is, Eq. 共36兲 does just
represent a null-speed subluminal wave. To be clearer, let us
recall that the FWs were expressed in the past as discrete
superposition, mainly because it was not known at that time
how to analytically treat a continuous superposition as Eq.
共36兲. Only by following the method presented in this work
one can eventually extend the FW approach 关49,63,16兴 to the
case of integrals: without numerical simulations, but in terms
once more of analytic solutions.
Indeed, the exact solution of Eq. 共36兲 is given by Eq. 共31兲,
with coefficients 共29⬘兲. One can choose the spectral function
S共kz兲 in such a way that ⌿ assumes the on-axis prechosen
static intensity pattern 兩F共z兲兩2. Namely, the equation to be
satisfied by S共kz兲, to such an aim, comes by associating Eq.
共36兲 with the requirement 兩⌿共␳ = 0 , z , t兲兩2 = 兩F共z兲兩2, which entails the integral relation
冕
␻0/c
−␻0/c
dkzS共kz兲eizkz = F共z兲.
共37兲
Equation 共37兲 would be trivially soluble in the case of an
integration between −⬁ and +⬁, since it would merely be a
Fourier transformation; but obviously this is not the case,
because its integration limits are finite. Actually, there are
functions F共z兲 for which Eq. 共37兲 is not soluble, in the sense
that no spectra S共kz兲 exist obeying the last equation. Namely,
if we consider the Fourier expansion
033824-8
SUBLUMINAL WAVE BULLETS: EXACT LOCALIZED …
PHYSICAL REVIEW A 77, 033824 共2008兲
bility of our method to model the field intensity shape also
under such strict requirements.
In the four examples below, we considered a wavelength
␭ = 0.6 ␮m, which corresponds to ␻0 = b = 3.14⫻ 1015 Hz.
The first longitudinal 共on-axis兲 pattern considered by us is
that given by
F共z兲 =
再
ea共z−Z兲 for 0 ⱕ z ⱕ Z,
0
elsewhere,
冎
i.e., a pattern with an exponential increase, starting from z
= 0 until z = Z. The chosen values of a and Z are Z = 10 ␮m
and a = 3 / Z. The intensity of the corresponding frozen wave
is shown in Fig. 4共a兲.
The second longitudinal pattern 共on axis兲 taken into consideration is the Gaussian one, given by
F共z兲 =
FIG. 4. Frozen waves with the on-axis longitudinal field pattern
chosen as 共a兲 exponential; 共b兲 Gaussian; 共c兲 super-Gaussian; 共d兲
zero-order Bessel function.
冕
F共z兲 =
⬁
dkzS̃共kz兲e
izkz
,
when S̃共kz兲 does assume non-negligible values outside the
interval −␻0 / c ⬍ kz ⬍ ␻0 / c, then in Eq. 共37兲 no S共kz兲 can forward that particular F共z兲 as a result.
However, a way out can be devised, such that one can
nevertheless find out a function S共kz兲 that approximately 共but
satisfactorily兲 complies with Eq. 共37兲.
The first way out consists in writing S共kz兲 in the form
⬁
冉 冊
1
2n␲ −i2n␲k /K
z .
e
S共kz兲 = 兺 F
K n=−⬁
K
共38兲
where, as before, K = 2␻0 / c. Then, one can easily verify Eq.
共38兲 to guarantee that the integral in Eq. 共37兲 yields the values of the desired F共z兲 at the discrete points z = 2n␲ / K. Indeed, the Fourier expansion 共38兲 is already of the same type
as Eq. 共28兲, so that in this case the coefficients An of our
solution 共31兲, appearing in Eq. 共29兲, do simply become
An =
冉 冊
1
2n␲
F −
.
K
K
共39兲
This is a powerful way for obtaining a desired longitudinal 共on-axis兲 intensity pattern, especially for very small spatial regions, because it is not necessary to solve any integral
to find out the coefficients An, which by contrast are given
directly by Eq. 共39兲.
Figure 4 depicts some interesting applications of this
method. A few desired longitudinal intensity patterns 兩F共z兲兩2
have been chosen, and the corresponding frozen waves calculated by using Eq. 共31兲 with the coefficients An given in
Eq. 共39兲. The desired patterns are enforced to exist within
very small spatial intervals only, in order to show the capa-
2
for − Z ⱕ z ⱕ Z,
elsewhere,
冎
with q = 2 and Z = 1.6 ␮m. The intensity of the corresponding frozen wave is shown in Fig. 4共b兲.
In the third example, the desired longitudinal pattern is
supposed to be a super-Gaussian,
F共z兲 =
−⬁
再
e−q共z/Z兲
0
再
e−q共z/Z兲
0
2m
for − Z ⱕ z ⱕ Z,
elsewhere,
冎
where m controls the edge sharpness. Here we have chosen
q = 2, m = 4, and Z = 2 ␮m. The intensity of the frozen wave
obtained in this case is shown in Fig. 4共c兲.
Finally, in the fourth example, let us choose the longitudinal pattern as being the zero-order Bessel function
F共z兲 =
再
J0共qz兲 for − Z ⱕ z ⱕ Z,
0
elsewhere,
冎
with q = 1.6⫻ 106 m−1 and Z = 15 ␮m. The intensity of the
corresponding frozen wave is shown in Fig. 4共d兲.
Let us observe that, of course, any static envelopes of this
type can be easily transformed into propagating pulses by the
mere application of Lorentz transformations.
Another way out exists for evaluating S共kz兲, based on the
assumption that
S共kz兲 ⯝ S̃共kz兲,
共40兲
which constitutes a good approximation whenever S̃共kz兲 assumes negligible values outside the interval 关−␻0 / c , ␻0 / c兴.
In such a case, one can have recourse to the method associated with Eq. 共28兲 and expand S̃共kz兲 itself in a Fourier series,
obtaining eventually the relevant coefficients An by Eq. 共29⬘兲.
Let us recall that it is still K ⬅ kz max− kz min= 2␻0 / c.
It may be interesting to call attention to the circumstance
that, when constructing FWs in terms of a sum of discrete
superpositions of Bessel beams 共as it was done by us in Refs.
关49,63,16,55兴兲, it was easy to obtain extended envelopes
such as, e.g., “cigars,” where easy means having recourse to
a few terms of the sum. By contrast, when we construct
FWs—following this section—as continuous superpositions,
then it is easy to get highly localized 共concentrated兲 enve-
033824-9
PHYSICAL REVIEW A 77, 033824 共2008兲
MICHEL ZAMBONI-RACHED AND ERASMO RECAMI
lopes. Let us explicitly mention, moreover, that the method
presented in this section furnishes FWs that are no longer
periodic along the z axis 共a situation that, with our old
method 关49,63兴, was obtainable only when the periodicity
interval tended to infinity兲.
V. MENTIONING THE ROLE OF SPECIAL RELATIVITY
AND OF LORENTZ TRANSFORMATIONS
Strict connections exist between, on one hand, the principles and structure of special relativity and, on the other
hand, the whole subject of subluminal, luminal, superluminal
localized waves, in the sense that it is expected for a long
time that a priori they are transformable one into the other
via suitable Lorentz transformations 共cf. Refs. 关61,62,65兴,
besides work of our own in progress兲.
Let us first confine ourselves to the cases faced in this
paper. Our subluminal localized pulses, that may be called
“wave bullets,” behave as particles: Indeed, our subluminal
pulses 关as well as the luminal and superluminal 共X-shaped兲
ones, that have been amply investigated in the past literature兴
do exist as solutions of any wave equations, ranging from
electromagnetism and acoustics or geophysics, to elementary
particle physics 共and even, as we discovered recently, to
gravitation physics兲. From the kinematical point of view, the
velocity composition relativistic law holds also for them. The
same is true, more in general, for any localized waves 共pulses
or beams兲.
Let us start for simplicity by considering, in an initial
reference frame O, just a 共␯-order兲 Bessel beam
⌿共␳, ␾,z,t兲 = J␯共␳k␳兲ei␯␾eizkze−i␻t;
共41兲
in Ref. 关66兴—whose philosophy, which in part goes back to
Refs. 关61,62兴, has been constantly shared by us—it was first
shown, by applying the appropriate Lorentz boost, that a
second reference frame O⬘, moving with respect to O with
speed u—along the positive z axis and in the positive direction, for simplicity’s sake—will observe the Bessel beam
⌿共␳⬘, ␾⬘,z⬘,t⬘兲 = J␯共␳⬘k␳⬘⬘兲ei␯␾⬘eiz⬘kz⬘⬘e−i␻⬘t⬘ .
共42兲
Let us now pass to subluminal pulses. One can investigate
the action of a Lorentz transformation 共LT兲, by expressing
them either via the first method 共Sec. II兲 or via the second
one 共Sec. III兲. Let us consider for instance, in the frame O, a
v-speed 共subluminal兲 pulse, given by Eq. 共3兲. When one goes
on to a second observer O⬘ moving with the same speed v
w.r.t. frame O, and, still for the sake of simplicity, passing
through the origin O of the initial frame at time t = 0, the
observer O⬘ will see, as explicitly noticed 关66兴 by applying
again the suitable LT, the pulse
⌿共␳⬘,z⬘,t⬘兲 = e−it⬘␻0⬘
with
冕
kz⬘⬘ = ␥−1␻/v − ␥b/v,
␻−
d␻S共␻兲J0共␳⬘k␳⬘⬘兲eiz⬘kz⬘⬘ ,
共43兲
k␳⬘⬘ = ␻0⬘/c2 − kz⬘⬘ .
2
共44兲
Notice that kz⬘⬘ is a function of ␻, as expressed by the first
one of the three relations in the Eqs. 共44兲; and that here ␻⬘ is
a constant. It is interesting that Eq. 共43兲 can be written as
⌿共␳⬘,z⬘,t⬘兲 = ␥ve−it⬘␻0⬘
冕
␻0⬘/c
−␻0⬘/c
dkz⬘⬘S̄共kz⬘⬘兲J0共␳⬘k␳⬘⬘兲eiz⬘kz⬘⬘ ,
共45兲
with S̄共kz⬘⬘兲 = S共␥vkz⬘⬘ + ␥2b兲. Equation 共45兲 describes monochromatic beams with axial symmetry 共and does coincide
also with what was derived within our second method, in
Sec. III, when posing v = 0兲.
One can therefore conclude, in agreement with Ref. 关66兴,
that a subluminal pulse, given by Eq. 共3兲, which appears as a
v-speed pulse in a frame O, will appear in another frame O⬘
共traveling w.r.t. observer O with the same speed v in the
same direction z兲 just as the monochromatic beam in Eq. 共45兲
endowed with angular frequency ␻0⬘ = ␥b, whatever be the
pulse spectral function in the initial frame O: Even if the
kind of monochromatic beam one arrives to does of course
depend on the chosen S共␻兲. 关One gets in particular a Besseltype beam when S is a Dirac’s ␦ function, S共␻兲 = ␦共␻ − ␻0兲;
let us moreover notice that, on applying a LT to a Bessel
beam, one obtains another Bessel beam, with a different axicon angle.兴 The vice versa is also true, in general.
Let us set forth explicitly an observation that has not been
noticed in the existing literature yet. Namely, let us mention
that, when starting not from Eq. 共3兲 but from the most general solutions which—as we have already seen—are sums of
solutions 共3兲 over the various values bm of b, then a Lorentz
transformation will lead us to a sum of monochromatic
beams; actually, of harmonics 共rather than to a single monochromatic beam兲. In particular, if one wants to obtain a sum
of harmonic beams, one must apply a LT to more general
subluminal pulses.
Let us add that also the various superluminal localized
pulses get transformed one into the other by the mere application 关66兴 of ordinary LTs; while it may be expected that the
subluminal and the superluminal LWs are to be linked 共apart
from some known technical difficulties, that require a particular caution兲 by the superluminal Lorentz “transformations” expounded long ago, e.g., in Refs. 关62,67,65,61兴 and
references therein.4 Let us recall once more that, in the years
1980–1982, special relativity, in its nonrestricted version,
predicted that, while the simplest subluminal object is obviously a sphere 共or, in the limit, a space point兲, the simplest
4
␻+
␻ ⬘ = ␥ b ⬅ ␻ 0⬘,
One should pay due attention to the circumstance that, as we
mention, the topic of superluminal LTs is a delicate 关62,67,65,61兴
one, at the extent that the majority of the recent attempts to readdress this question and its applications seem to be defective 共sometimes they do not even keep the necessary covariance of the wave
equation itself兲.
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PHYSICAL REVIEW A 77, 033824 共2008兲
FIG. 5. Let us consider an object that is intrinsically spherical,
i.e., that is a sphere in its rest frame 关panel 共a兲兴. After a generic
subluminal LT along z, i.e., under a subluminal z boost, it is predicted by special relativity 共SR兲 to appear as ellipsoidal due to
Lorentz contraction 关panel 共b兲兴. After a superluminal z boost
关62,67,65兴 共namely, when this object moves 关60兴 with superluminal
speed V兲, it is predicted by SR, in its nonrestricted version 共ER兲, to
appear 关61兴 as in panel 共d兲, i.e., as occupying the cylindrically symmetric region bounded by a two-sheeted rotation hyperboloid and
an indefinite double cone. The whole structure, according to ER, is
expected to move rigidly and, of course, with the speed V, the
cotangent square of the cone semiangle being 共V / c兲2 − 1. Panel 共c兲
refers to the limiting case when the boost speed tends to c, either
from the left or from the right 共for simplicity, a space axis is
skipped兲. It is remarkable that the shape of the localized 共subluminal and superluminal兲 pulses, solutions to the wave equations, appears to follow the same behavior; this can have a role for a better
comprehension even of de Broglie and Schrödinger wave mechanics. The present figure is taken from Refs. 关61,62兴. See also Fig. 6.
superluminal object is on the contrary an X-shaped pulse 共or,
in the limit, a double cone兲: cf. Fig. 5, taken from Refs.
关61,62兴.
The circumstance that also the pattern of the localized
solutions to the wave equations does meet this prediction is
rather interesting, and is expected to be useful—in the case,
e.g., of elementary particles and quantum physics—for a
deeper comprehension of de Broglie’s and Schrödinger’s
wave mechanics. With regard to the fact that the simplest
subluminal LWs, solutions to the wave equation, are “balllike,” let us depict by Fig. 6, in the ordinary threedimensional 共3D兲 space, the general shape of the Mackinnon’s solutions as expressed by Eq. 共10兲 for v c: In such
figures we graphically represent the field isointensity surfaces, which in the considered case result to be 共as expected兲
just spherical.
We have also seen that, even if our first method 共Sec. II兲
cannot yield directly zero-speed envelopes, such envelopes
“at rest,” in Eq. 共31兲, can be however obtained by applying a
v-speed LT to Eq. 共16兲. In this way, one starts from many
frequencies 关Eq. 共16兲兴 and ends up with one frequency only,
since ␥b gets transformed into the frequency of the monochromatic beam.
VI. NONAXIALLY SYMMETRIC SOLUTIONS: THE CASE
OF HIGHER-ORDER BESSEL BEAMS
Let us stress that until now we have paid attention to
exact solutions representing axially symmetric 共subluminal兲
pulses only: That is to say, to pulses obtained by suitable
superpositions of zero-order Bessel beams.
It is however interesting to look also for analytic solutions
representing nonaxially symmetric subluminal pulses, which
can be constructed in terms of superpositions of ␯-order
Bessel beams, with ␯ a positive integer. This can be attempted both in the case of Sec. II 共first method兲, and in the
case of Sec. III 共second method兲.
FIG. 6. 共Color online兲 In Fig. 5 we have seen how SR, in its nonrestricted version 共ER兲, predicted 关61,62兴 that, while the simplest
subluminal object is obviously a sphere 共or, in the limit, a space point兲, the simplest superluminal object is on the contrary an X-shaped pulse
共or, in the limit, a double cone兲. The circumstance that the localized solutions to the wave equations do follow the same pattern is rather
interesting, and is expected to be useful—in the case, e.g., of elementary particles and quantum physics—for a deeper comprehension of de
Broglie’s and Schrödinger’s wave mechanics. With regard to the fact that the simplest subluminal LWs, solutions to the wave equations, are
“ball-like,” let us depict by these figures, in the ordinary 3D space, the general shape of the Mackinnon’s solutions as expressed by Eq. 共10兲,
numerically evaluated for v c. In 共a兲 and 共b兲 we graphically represent the field isointensity surfaces, which in the considered case result to
be 共as expected兲 just spherical.
033824-11
PHYSICAL REVIEW A 77, 033824 共2008兲
MICHEL ZAMBONI-RACHED AND ERASMO RECAMI
For brevity’s sake, let us take only the first method 共Sec.
II兲 into consideration. One is immediately confronted with
the difficulty that no exact solution is known for the integral
in Eq. 共8兲 when J0共¯兲 is replaced with J␯共¯兲.
One can overcome this difficulty by following a simple
method, which will allow us to obtain “higher-order” subluminal waves in terms of the axially symmetric ones. Indeed,
it is well known that, if ⌿共x , y , z , t兲 is an exact solution to the
ordinary wave equation, then ⳵n⌿ / ⳵xn and ⳵n⌿ / ⳵y n are also
exact solutions.5 One should notice that, on the contrary,
when working in cylindrical coordinates, if ⌿共␳ , ␾ , z , t兲 is a
solution to the wave equation, ⳵⌿ / ⳵␳ and ⳵⌿ / ⳵␾ are not
solutions, in general. Nevertheless, it is not difficult at all to
reach the noticeable conclusion that, once ⌿共␳ , ␾ , z , t兲 is a
solution, then also
⌿̄共␳, ␾,z,t兲 = ei␾
冉
⳵⌿ i ⳵⌿
+
⳵␳ ␳ ⳵ ␾
冊
共46兲
is an exact solution. For instance, for an axially symmetric
solution of the type ⌿ = J0共k␳␳兲exp共ikz兲exp共−i␻t兲, Eq. 共46兲
yields ⌿̄ = −k␳J1共k␳␳兲exp共i␾兲exp共ikz兲exp共−i␻t兲, which is actually another analytic solution.
In other words, it is enough to start for simplicity from a
zero-order Bessel beam, and to apply Eq. 共46兲, successively,
␯ times, in order to obtain as a solution ⌿̄
= 共−k␳兲␯J␯共k␳␳兲exp共i␯␾兲exp共ikz兲exp共−i␻t兲, which is a
␯-order Bessel beam.
In such a way, when applying ␯ times Eq. 共46兲 to the
共axially symmetric兲 subluminal solution ⌿共␳ , z , t兲 in Eqs.
共16兲, 共15兲, and 共14兲 关obtained from Eq. 共3兲 with spectral
function S共␻兲兴, we are able to obtain the subluminal nonaxially symmetric pulses ⌿␯共␳ , ␾ , z , t兲 as analytic solutions,
consisting as expected in superpositions of ␯-order Bessel
beams,
⌿␯共␳, ␾,z,t兲 =
冕
␻+
␻−
d␻S⬘共␻兲J␯共k␳␳兲e
i␯␾ ikzz −i␻t
e
e
⌿1共␳, ␾,z,t兲 =
Let us mention that even ⳵n⌿ / ⳵zn and ⳵n⌿ / ⳵tn will be exact
solutions.
⳵ ⌿ i␾
e ,
⳵␳
共48兲
which actually yields the “first-order” pulse ⌿1共␳ , ␾ , z , t兲,
which can be more compactly written in the form
冉
b
b
⌿1共␳, ␾, ␩, ␨兲 = 2 v␥2 exp i ␤␥2␩
c
c
冊兺
冉 冊
⬁
An exp in
n=−⬁
␲
␺
␤ 1n
共49兲
with
␺1n共␳, ␾, ␩, ␨兲 ⬅
, 共47兲
with k␳共␻兲 given by Eq. 共4兲, and quantities S⬘共␻兲
= 关−k␳共␻兲兴␯S共␻兲 being the spectra of the pulses. If S共␻兲 is
centered at a certain carrier frequency 共it is a Gaussian spectrum, for instance兲, then S⬘共␻兲 too will approximately be of
the same type.
Now, if we wish the solution ⌿␯共␳ , ␾ , z , t兲 to possess a
predefined spectrum S⬘共␻兲 = F共␻兲, we can first take Eq. 共3兲
and set S共␻兲 = F共␻兲 / 关−k␳共␻兲兴␯ in its solution 共16兲, and afterwards apply to it, ␯ times, the operator U ⬅ exp共i␾兲关⳵ / ⳵␳
+ 共i / ␳兲 ⳵ / ⳵␾兴: As a result, we will obtain the desired pulse,
⌿␯共␳ , ␾ , z , t兲, endowed with S⬘共␻兲 = F共␻兲.
An example. On starting from the subluminal axially symmetric pulse ⌿共␳ , z , t兲, given by Eq. 共16兲 with the Gaussian
spectrum 共17兲, we can obtain the subluminal, nonaxially
symmetric, exact solution ⌿1共␳ , ␾ , z , t兲 by simply calculating
5
FIG. 7. 共Color online兲 Orthogonal projection of the field intensity corresponding to the higher-order subluminal pulse,
⌿1共␳ , ␾ , ␩ , ␨兲, represented by the exact solution equation 共49兲,
quantity ⌿ being given by Eq. 共16兲 with the Gaussian spectrum
共17兲. The pulse intensity happens to have a “donut”-like shape.
where
Z⬅
冑
b2 2 −3
␥ ␳Z 共Z cos Z − sin Z兲ei␾ ,
c2
冉
冊
2
b 2
b2 2 2
␥
␳
+
␥
␨
+
n
␲
.
c2
c
共50兲
共51兲
This exact solution, let us repeat, corresponds to superposition 共47兲, with S⬘共␻兲 = k␳共␻兲S共␻兲, quantity S共␻兲 being given
by Eq. 共17兲. It is represented in Fig. 7. The pulse intensity
has a “donutlike” shape.
VII. CONCLUSIONS
As in the well-known superluminal case 关1兴, the subluminal localized waves can be obtained by superposing Bessel
beams. However, they have been scarcely considered in the
past, for the reason that the superposition integral must run in
this case over a finite interval 共which makes it mathematically difficult to work out analytic expressions for them兲. In
this paper, however, we have obtained nondiffracting subluminal pulses as exact analytic solutions to the wave equa-
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PHYSICAL REVIEW A 77, 033824 共2008兲
tions: For arbitrarily chosen frequencies and bandwidths,
avoiding any recourse to the backward-traveling components, and in a simple way.
Indeed, only one closed-form subluminal LW solution,
␺cf, to the wave equations was known 关9兴: It was obtained by
choosing, in the relevant integration, a constant weight function S共␻兲; while all other solutions had been previously obtained only by numerical simulations. By contrast, we have
shown that, for instance, a subluminal LW can be obtained in
closed form by adopting any spectra S共␻兲 that are expansions
in terms of ␺cf. In fact, the initial disadvantage, of having to
deal with a limited bandwidth, may be turned into an advantage, since in the case of “truncated” integrals the spectrum
S共␻兲 can be expanded in a Fourier series.
More in general, it has been shown in this paper how one
can arrive at exact solutions both by integration over the
Bessel beams’ angular frequency ␻, and by integration over
their longitudinal wave number kz. Both methods are expounded above. The first one appears to be comprehensive
enough; we have studied the second method as well, however, since it allows tackling also the limiting case of zerospeed solutions 共thus furnishing a second way, in terms of
continuous spectra, for obtaining the so-called “frozen
waves,” quite promising also from the applicative point of
view兲. We have briefly treated the case, moreover, of nonaxially symmetric solutions, that is, of higher-order Bessel
beams.
At last, some attention has been paid to the role of special
relativity, and to the fact that the localized waves are to be
transformed one into the other by suitable Lorentz transfor-
mations. Moreover, our results seem to show that in the subluminal case the simplest LW solutions are 共for v c兲
“ball”-like, as expected since long ago 关61兴 on the mere basis
of special relativity 关62兴: More precisely, already in the years
1980–1982 it had been predicted that, if the simplest subluminal object is a sphere 共or, in the limit, a space point兲, then
the simplest superluminal object is an X-shaped pulse 共or, in
the limit, a double cone兲; and vice versa: cf. Fig. 5. It is
rather interesting that the same pattern appears to be followed by the localized solutions of the wave equations. For
the subluminal case, see, e.g., Fig. 6. The localized pulses,
endowed with a finite energy, or merely truncated, will be
constructed in another presentation.
In the present work we have fixed our attention on acoustics and optics. However, analogous results are valid whenever an essential role is played by a wave equation 共such as
electromagnetism, seismology, geophysics, gravitation, elementary particle physics, etc.兲.
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ACKNOWLEDGMENTS
This work was partially supported by FAPESP and CNPq
共Brazil兲, and by INFN, MIUR 共Italy兲. The authors are indebted to H. E. Hernández-Figueroa for his continuous collaboration, and thank L. R. Baldini, I. M. Besieris, A. Castoldi, C. Conti, J-y. Lu, G. C. Maccarini, R. Riva, P. Saari, A.
M. Shaarawi, and M. Tygel for useful discussions or kind
interest. The authors are moreover deeply grateful to an
anonymous referee for many, very useful remarks.
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MICHEL ZAMBONI-RACHED AND ERASMO RECAMI
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