Ardavan et al. Vol. 24, No. 8 /August 2007 /J. Opt. Soc. Am. A 2443
Morphology of the nonspherically decayingradiation beam generated by a rotating
superluminal source
Houshang Ardavan,1 Arzhang Ardavan,2 John Singleton,3,* Joseph Fasel,4 and Andrea Schmidt4
1Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK2Clarendon Laboratory, Department of Physics, University of Oxford, Parks Road, Oxford OX1 3PU, UK
3National High Magnetic Field Laboratory, MS-E536, Los Alamos National Laboratory, Los Alamos,New Mexico 87545, USA
4Modeling and Analysis, MS-F609, Los Alamos National Laboratory, Los Alamos,New Mexico 87545, USA
Received August 11, 2006; revised January 28, 2007; accepted March 20, 2007;posted April 10, 2007 (Doc. ID 73982); published July 11, 2007
We consider the nonspherically decaying radiation field that is generated by a polarization current with a su-perluminally rotating distribution pattern in vacuum, a field that decays with the distance RP from its sourceas RP
−1/2, instead of RP−1. It is shown (i) that the nonspherical decay of this emission remains in force at all
distances from its source independently of the frequency of the radiation, (ii) that the part of the source thatmakes the main contribution toward the value of the nonspherically decaying field has a filamentary structurewhose radial and azimuthal widths become narrower (as RP
−2 and RP−3, respectively) the farther the observer is
from the source, (iii) that the loci on which the waves emanating from this filament interfere constructivelydelineate a radiation subbeam that is nondiffracting in the polar direction, (iv) that the cross-sectional area ofeach nondiffracting subbeam increases as RP, instead of RP
. INTRODUCTION. Preambleaxwell’s generalization of Ampère’s law [1] establishes
hat electromagnetic radiation can be equally well gener-ted by a time-dependent electric polarization current,ith a density �P /�t, as by a current of accelerated free
harges with the density j:
� � H =4�
cj +
1
c
�D
�t=
4�
c �j +�P
�t � +1
c
�E
�t; �1�
ere, E and H are the electric and magnetic fields, D ishe displacement, and c is the speed of light in vacuo. Aemarkable aspect of the emission from such polarizationurrents is that the motion of the radiation source is notimited by c. Although the speed of charged particles can-ot exceed c, nothing prevents the distribution pattern ofpolarization current, created by the coordinated motion
f subluminal particles, from moving faster than light2–4]. Indeed, radiation from such superluminal polariza-ion currents has been observed in the laboratory [5–8].
Since electric polarization arises from separation ofharges, a polarization current is by its nature volumeistributed. In fact, no superluminal source can be point-
ike, for, if a point source were to move faster than its ownaves, it would generate caustics on which the field
trength would diverge [2,9].There is growing experimental and theoretical interest
n radiation by polarization currents whose distributionatterns move at a superluminal speed with acceleration8]. One of the simplest implementations of such sourcesmploys distribution patterns that have the time depen-ence of a traveling wave with circular superluminal mo-ion; here, the acceleration is centripetal. We are investi-ating the use of polarization currents with suchuperluminally rotating distribution patterns in applica-ions relating to communications and radar [6,10]. Fur-hermore, one of the proposed models of the radio emis-ion from pulsars postulates the presence of sources ofhis type in the magnetospheres of rapidly rotating neu-ron stars [11,12]. The clarification of a diverse set of cur-ent questions, therefore, hinges on an understanding ofhe radiation from superluminal polarization currents un-ergoing circular motion [13–15].Our purposes in the present paper are (i) to examine
he geometry of those regions within such extendedources that make the dominant contribution toward theadiation field observed at a given point and time and (ii)o identify the salient features of the angular distribution
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2444 J. Opt. Soc. Am. A/Vol. 24, No. 8 /August 2007 Ardavan et al.
f this radiation. A detailed knowledge of the extent andeometry of the contributing part of the source is requiredot only for the efficient design of practical superluminalources of this type (e.g., for the design of the dielectric inhich the polarization current is generated) [6] but also
or understanding the narrow widths of the giant pulseshat are received from pulsars [16]. Likewise, a knowl-dge of the evolution of the angular distribution of the ra-iation with distance both facilitates the experimental de-ection of the tightly beamed large-amplitude componentf the emission from such sources and establishes a con-ection between two observed features (the nanostructurend the high brightness temperature) of the pulsar emis-ion [17–20].
In [13], the field of a superluminally rotating extendedource was evaluated by superposing the fields of its con-tituent volume elements, i.e., by convolving its densityith the familiar Liénard–Wiechert field of a rotatingoint source. This Liénard–Wiechert field is described byn expression essentially identical to that which is en-ountered in the analysis of synchrotron radiation, excepthat its value at any given observation time receives con-ributions from more than one retarded time. The multi-alued nature of the retarded time is an important fea-ure of all superluminal emission; we shall begin,herefore, by describing the relationship between observa-ion (reception) time and retarded (emission) time for thearticular case of a rotating source with the aid of Fig. 1.
. Multivalued Retarded Times, the Cusp, andemporal Focusingigures 1(a) and 1(b) show the wavefronts that emanate
rom a small, circularly moving superluminal source S. Ase have already pointed out, no superluminal source cane truly pointlike. Here we are considering a volume ele-
Pc
ent of an extended source whose linear dimensions areuch smaller than the other length scales of the problem.The emission of waves by any moving point source
hose speed exceeds the wave speed is described by aiénard–Wiechert field that has extended singularities.hese singularities occur on the envelope of wavefrontshere the Huygens wavelets emitted at differing retarded
imes interfere constructively and so form caustics. Aell-understood example is the emission of acousticaves by a point source that moves along a straight lineith a constant supersonic speed. In this case, a simple
austic forms along a cone issuing from the source, the so-alled Mach cone, and most of the emitted energy is con-ned to the vicinity of this propagating shock front. An-ther, similar, example is the formation of the Cerenkovone in the electromagnetic field of a uniformly movingoint charge whose speed exceeds the speed of light insidedielectric medium.When the supersonic or superluminal motion of such
ources is in addition accelerated, the simple conical caus-ic that occurs in the Mach or Cerenkov radiation is re-laced by a two-sheeted envelope with a cusp [9,21,22].he effect of acceleration is to give rise to a one-imensional locus of observation points at which morehan two simultaneously received wavefronts meet tan-entially. The spherical wavefronts that are centered athe retarded positions of the source neighboring a pointrom which such coalescing wavefronts emanate cannotut be mutually tangential (in pairs) to two distinct sur-aces, surfaces that constitute the separate sheets of ausped envelope.
More specifically, the Cerenkov-like envelope that isenerated by a uniformly rotating superluminal sourceonsists of a tubelike surface whose two sheets meet, andre tangent to each other, along a spiraling cusp curve;
ig. 1. (a) Cross section of the Cerenkov-like envelope (bold curves) of the spherical Huygens wavefronts (fine circles) emitted by a smalllement S within an extended, rotating superluminal source of angular velocity �. S is on a circle of radius r=2.5c /�, or, in our dimen-ionless units, r�r� /c=2.5; i.e., its instantaneous linear velocity is r�=2.5c. The cross section is in the plane of S’s rotation; dashedircles designate the light cylinder rP=c /� �rP=1� and the orbit of S. (b) Three-dimensional view of the light cylinder, the envelope ofavefronts emanating from S, and the cusp along which the two sheets �± of this envelope meet tangentially. (c) The relationship be-
ween reception time tP and source (retarded) time t [Eq. (4)] plotted for r=2.5 and three different observation points. The maxima andinima of curve (i) occur on the sheets �± of the envelope, respectively. Curve (ii) corresponds to an observation point that is located on
he cusp. Note that the waves emitted during an interval of retarded time centered at tc are received over a much shorter interval ofbservation time at t . Curve (iii) is for an observation point that is never crossed by the rotating sheets of the envelope (after [13].)
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Ardavan et al. Vol. 24, No. 8 /August 2007 /J. Opt. Soc. Am. A 2445
his envelope is depicted in Fig. 1 and mathematically de-cribed in Eqs. (9)–(13) below. At any given observationime, three wavefronts pass through an observation pointnside the envelope, while only one wavefront passeshrough a point outside this surface. The envelope and itsusp are the loci of observation points at which two orhree of the simultaneously received wavefronts are tan-ential to one another. To specify the retarded times t athich various wavefronts are emitted, let us adopt a cy-
indrical coordinate system based on the axis of rotationnd denote the trajectory of the volume element S, shownn Fig. 1, by
r = const., ��t� = � + �t, z = const., �2�
here � denotes the initial value of � and � is the angularelocity of S. Let a stationary observer be positioned at aoint P, with cylindrical polar coordinates �rP,�P,zP�. Theetarded-time separation R�t� between the source volumelement and the observer (i.e., their instantaneous sepa-ation at the time t of emission) will therefore be
he relationship between the retarded time t and the ob-ervation time tP, i.e.,
tP = t + R�t�/c, �4�
s plotted in Fig. 1(c) for the source speed r�=2.5c and forhree classes of stationary observation points: those, lo-ated sufficiently close to the plane of rotation, that areeriodically crossed by the two sheets of the rigidly rotat-ng envelope [curve (i)] or by just the cusp curve of the en-elope [curve (ii)] and those at higher latitudes that areever crossed by the envelope [curve (iii)].The ordinates of the neighboring extrema of curve (i) in
ig. 1(c) designate those observation times, during eachotation period, at which the two sheets of the envelope goast the stationary observer [see Eqs. (6)–(9) below].hus, the field inside the envelope receives contributions
rom three distinct values of the retarded time [curve (i)],hile the field outside the envelope is influenced by only a
ingle instant of emission time [curves (i) and (iii)]. Theonstructive interference of the emitted waves on the en-elope (where two of the contributing retarded times coa-esce) and on its cusp {where all three of the contributingetarded times coalesce [curve (ii)]} gives rise to the diver-ence of the Liénard–Wiechert field on these loci. There ishigher-order focusing of the waves, and so a higher-
rder mathematical singularity, on the cusp than on thenvelope itself. While the singularity that occurs on thenvelope is integrable, that which occurs on the cusp isot. In that it occurs in the temporal as well as the spatialomain, this focusing is distinct from that produced by aonventional horn, mirror, or lens. The enhanced ampli-ude on the cusp is due to the contributions from emissionver an extended period of source time, reaching the ob-erver over a significantly shorter period of observationime.
The Liénard–Wiechert field derived in [13] was used ashe Green’s function for calculating the emission from auperluminal polarization current, comprising both poloi-al and toroidal components, whose distribution pattern
otates (with an angular frequency �) and oscillates (withfrequency �) at the same time [13]. It was found that
he convolution of the density of this current with thereen’s function described above results in a field that de-
ays nonspherically: a field whose strength diminishesith the distance RP from the source as RP
−1/2, rather than
P−1, within the bundle of cusps that emanate from the
onstituent volume elements of the source and extendnto the far zone. This result, which has now been dem-nstrated experimentally [6,7], was derived in [13] by set-ing the observation point within the bundle of generatedusps and evaluating the convolution integrals over vari-us dimensions of the source [13]. The steps in this pro-edure are listed below.
1. The integration with respect to the azimuthal extentf the source was performed by means of Hadamard’sethod [23,24]. It was shown that the Hadamard finite
art of the divergent integral that describes the field of auperluminally rotating ring with a sinusoidal densityistribution consists of two parts: one part is exclusivelyontributed by the two elements on the ring that ap-roach the observer along the radiation direction with thepeed of light at the retarded time (i.e., the elements forhich dR /dt=−c), and the other part is contributed by thentire extent of the ring.
2. The integration with respect to the radial dimensionf the source was subsequently performed by the methodf stationary phase [25].
It was found that, when the radiation frequency isuch higher than the rotation frequency �, the main con-
ribution toward the field of a superluminally rotating an-ular ring comes from the vicinity of the point on the ringhat approaches the observer not only with the wavepeed but also with zero acceleration (i.e., the point athich dR /dt=−c and d2R /dt2=0 simultaneously).These contributing source elements are the ones for
hich the time-domain phase tP= t+R�t� /c is doubly sta-ionary. Differentiating Eq. (4) with respect to t, we canee that
dR
dt= − c,
d2R
dt2 = 0 �5�
re equivalent to
dtP
dt= 0,
d2tP
dt2 = 0. �6�
hese conditions jointly define the point of inflection inurve (ii) of Fig. 1(c), corresponding to the cusp passinghrough the point of observation P.
The collection of volume elements satisfying Eqs. (5)ithin an extended source has a filamentary locus that ispproximately parallel to the axis of rotation for an obser-ation point located in the far zone (Fig. 2). The non-pherically decaying field that is generated by a volume-istributed source arises almost exclusively from thelements in the vicinity of this narrow filament, a fila-ent whose position within the source depends on the lo-
ation of the observer.
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2446 J. Opt. Soc. Am. A/Vol. 24, No. 8 /August 2007 Ardavan et al.
. Zeroth-Order Evaluation of the Angular Position ofhe Nonspherically Decaying Beamhe angle of observation corresponding to the cusp andhe reason for the filamentary structure of the contribut-ng parts of the extended source may be inferred from thebove equations. Applying the first condition in Eqs. (5) toq. (3) and solving the resulting equation for the retarded
ime t, or, equivalently, the retarded position �=�t+ �, webtain
� = �± � �P + 2� − arccos�1 � �1/2
rrP� , �7�
here
� � �rP2 − 1��r2 − 1� − �z − zP�2. �8�
n these expressions, �r , z ; rP, zP� stand forr� /c ,z� /c ;rP� /c ,zP� /c�, i.e., for the coordinatesr ,z ;rP,zP� of the source point and the observation pointn units of the light-cylinder radius c /�. (This radius,hich automatically appears in the present calculations,
urns out to be the main length scale of the problem.)The retarded times t±���±− �+2n�� /�, respectively,
epresent the maxima and minima of curve (i) in Fig. 1(c)here n is an integer. Applying both conditions of Eqs. (5)
o Eq. (3), we obtain Eq. (7) and �=0. The retarded timec��t±��=0 represents the inflection point of curve (ii) inig. 1(c). Curve (iii) in Fig. 1(c) corresponds to an obser-ation point for which �0, and so �± are not real.
The envelope of wavefronts comprises those observa-ion points at which two retarded times coalesce, i.e., at
ig. 2. Schematic illustration of the light cylinder r=c /�, thelamentary part of the source that approaches the observerationoint with the speed of light and zero acceleration at the retardedime, the orbit of this filamentary source, and the subbeamormed by the bundle of cusps that emanate from the constituentolume elements of this filament. The subbeam is diffractionlessn the direction of P. The figure represents a snapshot corre-ponding to a fixed value of the observation time tP. The polaridth �P of this subbeam decreases with the distance RP in suchway that the thickness RP�P of the subbeam in the polar di-
ection remains constant: it equals the projection, �z sin P, of theˆ extent, �z, of the contributing filamentary source onto a direc-ion normal to the line of sight. The azimuthal width of the sub-eam, on the other hand, is subject to diffraction as in any otheradiation beam: ��P is independent of RP.
hich t= t±. Inserting these values of the retarded time inq. (4) and solving the resulting equation for �P as a func-
ion of �rP,zP� at a fixed observation time tP, we find that
hese equations describe a rigidly rotating surface in thepace �rP,�P,zP� of observation points that extends fromhe light cylinder rP=1 to infinity (see Fig. 1).
The two sheets �± of this envelope meet at a cusp. Theusp occurs along the curve
� = 0, �P = �tP + � − ��±�rP,zP���=0, �12�
hown in Fig. 4(a) below. It can be easily seen that, for aar-field observation point with the spherical polar coordi-ates RP��rP
2 +zP2�1/2, P�arccos�zP/RP�, �P, Eqs. (12) re-
uce to
P = arcsin�r−1� + ¯ , �P = � − 32� + ¯ , �13�
o within the zeroth order in the small parameter RP−1,
here RP�RP� /c. [The higher-order terms of this expan-ion are given in Eqs. (67) and (68).] In other words, theusp that is detected at an observation point �RP,P,�P�n the far zone arises from the constructive interference ofhe waves that were emitted by the volume elements at
ˆ =csc P, �=�P+ 32�, regardless of what their z coordi-
ates may be. These volume elements therefore have alamentary locus parallel to the axis of rotation whose
ength is of the order of the z extent of the source distri-ution along the line r=csc P, �=�P+ 3
2� (see Fig. 2).
. Filamentary Locus of the Contributing Sourcelementshe locus of source elements that approach the observerith the wave speed and zero acceleration at the retarded
ime has a filamentary shape not only within the zeroth-rder approximation in the small parameter RP
−1 but ineneral. To demonstrate this, we need to introduce the no-ion of bifurcation surface [9].
When deriving the equation describing the envelope ofavefronts, we kept the coordinates �r , � ,z�, which label a
otating source element, fixed and found the surface inhe space �rP,�P,zP� of observation points on whichR /dt=−c at a given time tP. If we keep �rP,�P,zP� and tPxed, then dR /dt=−c would describe a surface that re-ides in the space �r ,� ,z� of source points: the so-calledifurcation surface of the observation point P. Like the en-elope, the bifurcation surface consists of two sheets thateet tangentially along a cusp (a spiraling curve onhich d2R /dt2=0), but the bifurcation surface issues from
he observation point P (rather than the source point S)nd spirals about the rotation axis in the opposite direc-ion to the envelope (see Fig. 3). The similarity between
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Ardavan et al. Vol. 24, No. 8 /August 2007 /J. Opt. Soc. Am. A 2447
he two surfaces stems from the following reciprocityroperties of P and S: the equation describing the enve-ope, Eq. (9), remains invariant under the interchanges↔rP, z↔zP, �↔−�P, t↔−tP.
The locus of source elements that approach an observerwith the wave speed and zero acceleration at the re-
arded time is given by the intersection of the cusp curvef the bifurcation surface of P with the volume of theource. This filamentary locus has exactly the same shapes the cusp curve of the envelope [shown in Fig. 4(a)], ex-ept that it resides in the space of source points, insteadf the space of observation points, and points in the direc-ion of the source velocity. The projection of this curvento the �r ,z� plane consists of a branch of a hyperbolaith asymptotes that lie along the angles arcsin�rP
−1� and−arcsin�rP
−1� with respect to the z axis [see Fig. 4(b)]. Forn observation point that is located in the far zone, there-ore, the projection of the cusp curve of the bifurcationurface onto the �r ,z� plane is virtually parallel to the ro-ation axis.
The reciprocity relations referred to above ensure thatf a source element S is located on the cusp curve of theifurcation surface of an observer P, then the envelope ofhe wavefronts emitted by S would have a cusp passinghrough P (or, conversely, if an observer P is located on theusp curve of the envelope of wavefronts emitted by aource element S, then the cusp curve of the bifurcationurface of P would pass through S). In the case of a singleoint source, the retarded position � of the source linearlyhanges with time ��= �+�t�, and so the cusp that it gen-rates both is spiral shaped and rigidly rotates about theaxis. In the case of an extended source, on the other
and, the position � of each contributing source elementan element that lies on the cusp curve of the bifurcationurface of a far-field observer P) is fixed (�=�P+3� /2, rcsc ), and the elements that occupy that position are
ig. 3. Bifurcation surface of the observation point P for aource whose rotational motion is counterclockwise. The sourceoints that lie inside this surface influence the field at P at threeistinct values of the retarded time, while those that lie outsidehis surface influence the field at only a single value of the re-arded time. The source elements on the filamentary locus athich the cusp curve of this surface intersects the source distri-ution approach P with the speed of light and zero accelerationt the retarded time and so generate a nonspherically decayingeld at P.
P
onstantly changing. The cusps generated by the movingource elements that pass through this fixed position atarious retarded times have a locus, at any given obser-ation time, that is straight and stationary as shown inig. 2. In other words, of the source elements constitutinghe filament at �=�P+3� /2, r=csc P, each contributes auasi-instantaneous pulse of nonspherically decayinglectromagnetic radiation that in the far field appears toave propagated out along a virtually straight-line locusefined by the angle P=arcsin�r−1�.
. Objectives and Organization of the Paperhe objectives of the present paper are as follows (the lo-ation of the resolution of each objective is given in brack-ts):
1. to show that the nonspherical decay of the radiationeld that arises from a rotating superluminal source re-ains in force at all distances from this source indepen-
ently of the frequency of the radiation (Subsection 3.D);2. to specify the dimensions of the filamentary part of
he source that makes the main contribution toward thealue of the nonspherically decaying field [Eqs. (58) and59)];
3. to show that the bundle of cusps emanating from thislament delineates a radiation subbeam that is nondif-racting in one dimension; that is to say, the width of this
ig. 4. (a) Segment of the cusp of the envelope of wavefrontsmitted by a rotating point source with the speed r�=3c. Thisurve is tangent to the light cylinder at the point (rP=1, �P=�3� /2, zP= z) on the plane of the orbit and spirals outward into
he far zone. Note that this figure represents a snapshot at axed value of the observation time tP. The cusp curve of the bi-urcation surface of an observer P shown in Fig. 3 has preciselyhe same shape, except that it resides in the space of sourceoints, instead of the space of observation points, and spirals inhe counterclockwise direction: it is tangent to the light cylindert the point (r=1, �=�P+3� /2, z= zP). (b) The projections of theusp curve of the bifurcation surface and a localized source dis-ribution onto the �r , z� plane. Only the part of the source thaties close to the cusp in ��0 contributes to the nonsphericallyecaying radiation. The source elements whose �r , z� coordinatesall in �0 approach the observer with a speed dR /dtc at theetarded time and so make contributions toward the field thatre no different from those made in the subluminal regime. Thesymptotes of the hyperbola �=0 make the angles arcsin�1/ rP�nd �−arcsin�1/ rP� with the z axis, so that for an observationoint in the far zone �rP 1� the projection of the cusp onto ther , z� plane is (as depicted in Fig. 2) effectively parallel to the ro-ation axis.
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2448 J. Opt. Soc. Am. A/Vol. 24, No. 8 /August 2007 Ardavan et al.
eam in the polar direction remains the same at all dis-ances from the source [Fig. 2, Eqs. (70)];
4. to clarify how the requirements of conservation of en-rgy are met by the nonspherically decaying radiation:he cross-sectional area of each nondiffracting subbeamncreases as RP, rather than RP
2, with the distance RProm the source (Section 4); and
5. to show that the overall radiation beam within whichhe field decays nonspherically consists, in general, of anncoherent superposition of the coherent nondiffractingubbeams described above (Section 4).
We begin with the mathematical formulation of theroblem in Section 2. In Section 3, we show that objec-ives 1 and 2 can be achieved by replacing the method oftationary phase used in [13] with the method of steepestescents [26]. By converting the Fourier-type integralver the radial extent of the source into a Laplace-type in-egral and making use of contour integration, we presentn asymptotic analysis for which the large parameter ishe distance from the source (in units of the light-cylinderadius c /�) rather than the radiation frequency. Not onlys there no restriction on the range of frequencies forhich the emission from a rotating superluminal sourceecays nonspherically, but the more distant the observa-ion point, the more accurate the asymptotic analysis thatredicts this decay rate.The more poweful asymptotic technique we employ
ere establishes, moreover, that the transverse dimen-ions of the filamentary part of the source responsible forhe nonspherically decaying field are of the order of �rRP
−2 in the radial direction and ���RP−3 in the azimuthal
irection (see Section 4). The dimension of this filament inhe direction parallel to the rotation axis is of the order ofhe length scale of the source distribution in that direc-ion.
The corresponding dimensions of the bundle of cuspshat emanate from the contributing source elements cane easily inferred from the above dimensions of the fila-entary region containing these elements. The cusps oc-
upy a solid angle in the space of observation pointshose azimuthal width ��P has a constant value (as doesconventional radiation beam) but whose polar width �Pecreases with the distance RP as RP
−1. This may be seeny considering a cohort of propagating polarization-urrent volume elements that are at the same azimuthalngle � and radius r (possessing the same speed r�) butt differing heights z. Each will give rise to a cusp in thear zone that forms the angle P=arcsin�r−1� with the zxis but starts from a different height at the light cylindersee Fig. 2). The spatial extent in the direction of increas-ng P of the composite set of cusps from this cohort of vol-me elements (the subbeam) will therefore be determinedolely by the height �z of the region confining the polar-zation current. Projected onto a direction normal to theine of sight, this will result in a width w= ��z�sin P occu-ied by the cusps that is independent of the distance RProm the source. (Note that w is a fixed linear widthather than an angular width.)
Thus, the area RP2 sin P�P��P subtended by the
undle of cusps defining this subbeam increases as RP,ather than R2, with the distance R from the source. In
P P
rder that the flux of energy remain the same across aross section of the subbeam, therefore, it is essential thathe Poynting vector associated with this radiation corre-pondingly decay as RP
−1 rather than RP−2. This require-
ent is, of course, met automatically by the radiationhat propagates along the nondiffracting subbeam.
For a rotating superluminal source with the radialoundaries r�1 and r�� r, the nonspherically decay-ng radiation is detectable in the far zone only within theonical shell
arcsin�1/r�� � P � arcsin�1/r�. �14�
hese limits on P merely reflect the fact that a rigidly ro-ating extended source with finite radial spread entails aimited range of linear speeds r�; Eqs. (13) show that aimited range of speeds results in a limited spread in thengular positions of the generated subbeams. The overalleam described by Eq. (14) consists, in general, of a su-erposition of nondiffracting subbeams with widely differ-ng amplitudes and phases. The individual subbeamswhich would be narrower and more distinguishable, theurther away the observer is from the source) decay non-pherically, but the incoherence of their phase relation-hips ensures that the integrated flux of energy associ-ted with their superposition across this finite solid angleemains independent of RP.
Having made a preliminary description of the salienteatures of the analysis, we now embark on the detailedreatment of the problem in Sections 2–4. We conclude inection 5 with some remarks on the applicability of ournalysis to numerical calculations of the emission fromuperluminal sources and to the observational data onhe giant pulses received from pulsars.
. NONSPHERICALLY DECAYINGOMPONENT OF THE RADIATION FIELDROM A ROTATING SUPERLUMINALOURCEs in [13], we base our analysis on a polarization-currentensity j=�P /�t for which
here Pr,�,z are the components of the polarization P in aylindrical coordinate system based on the axis of rota-ion, s�r ,z� is an arbitrary vector that vanishes outside anite region of the �r ,z� space, and m is a positive integer.or a fixed value of t, the azimuthal dependence of theensity [Eq. (15)] along each circle of radius r within theource is the same as that of a sinusoidal wave train withhe wavelength 2�r /m whose m cycles fit around the cir-umference of the circle smoothly. As time elapses, thisave train both propagates around each circle with theelocity r� and oscillates in its amplitude with the fre-uency �. This is a generic source: one can construct anyistribution with a uniformly rotating pattern,
Pf
tbr
HssRnt
tsstgteeitsstw
bgistnffimsr
sc
w
wpaTo(
i
Ws
ositta
w
Ns�
v
[mp
tldpn
Ardavan et al. Vol. 24, No. 8 /August 2007 /J. Opt. Soc. Am. A 2449
r,�,z�r , � ,z�, by the superposition over m of terms of theorm sr,�,z�r ,z ,m�cos�m��.
The electromagnetic fields
E = − �PA0 −�A
��ctP�, B = �P � A �17�
hat arise from such a source are given, in the absence ofoundaries, by the following classical expression for theetarded four-potential:
A��xP,tP� = c−1 d3xdtj��x,t���tP − t − R/c�/R,
� = 0, . . . ,3. �18�
ere, �xP, tP�= �rP,�P,zP, tP� and �x , t�= �r ,� ,z , t� are thepace–time coordinates of the observation point and theource points, respectively, R stands for the magnitude of�xP−x, and �=1,2,3 designates the spatial compo-
ents, A and j, of A� and j� in a Cartesian coordinate sys-em [1].
In [13], we first calculated the Liénard–Wiechert fieldhat arises from a circularly moving point source (repre-enting a volume element of an extended source) with auperluminal speed r��c, i.e., considered a generaliza-ion of the synchrotron radiation to the superluminal re-ime. We then evaluated the integral representing the re-arded field (rather than the retarded potential) of thextended source [Eq. (15)] by superposing the fields gen-rated by the constituent volume elements of this source,.e., by using the generalization of the synchrotron field ashe Green’s function for the problem (see also [15]). In theuperluminal regime, this Green’s function has extendedingularities, singularities that arise from the construc-ive intereference of the emitted waves on the envelope ofavefronts and its cusp.Labeling each element of the extended source [Eq. (15)]
y its Lagrangian coordinate � and performing the inte-ration with respect to t and � (or, equivalently, � and �)n the multiple integral implied by Eqs. (15)–(18), wehowed in [13] that the resulting expression for the radia-ion field B (or E) consists of two parts: a part whose mag-itude decays spherically, as RP
−1, with the distance RProm the source (as in any other conventional radiationeld) and another part Bns, with Ens= n�Bns, whoseagnitude decays as RP
−1/2 within the conical shell de-cribed by Eq. (14). (Here, n�R /R is a unit vector in theadiation direction.)
The expression found in [13] [Eq. (47)] for the non-pherically decaying component of the field within thisonical shell, in the far zone, is
Bns − 43i exp�i��/����P + 3�/2�� �
�=�±
� exp�− i��P�
� �j=1
3
qj��0
rdrdz�−1/2uj exp�− i��−�, �19�
here
�± � ��/�� ± m, �20�
� � � − �t , �21�
P P P
qj � �1 − i�/� i�/��, �22�
uj � �sr cos Pe + s�e�
− s� cos Pe + sre�
− sz sin Pe
� , �23�
ith j=1,2,3. In the above expression, e � e�P(which is
arallel to the plane of rotation) and e�� n� e comprisepair of unit vectors normal to the radiation direction n.he domain of integration in Eq. (19) consists of the partf the source distribution s�r ,z� that falls within ��0see Fig. 4).
The function �−�r , z� that appears in the phase of thentegrand in Eq. (19) is stationary as a function of r at
r = rC�z� � � 12 �rP
2 + 1� − � 14 �rP
2 − 1�2 − �z − zP�2�1/2�1/2.
�24�
hen the observer is located in the far zone, this isolatedtationary point coincides with the locus,
r = rS � �1 + �z − zP�2/�rP2 − 1��1/2, �25�
f source points that approach the observer with thepeed of light and zero acceleration at the retarded time,.e., with the projection of the cusp curve of the bifurca-ion surface onto the �r ,z� plane (see Fig. 4). For RP 1,he separation rC− rS vanishes as RP
−2 [see Eq. (39) below],nd both rC and rS assume the value csc P.It follows from Eq. (10) that
��−�r=rC� �C = RC + �C − �P, �26�
���−/�r�r=rC=0, and
� �2�−
�r2 �r=rC
� a = − RC−1��rP
2 − 1��rC2 − 1�−1 − 2�, �27�
here
�C = �P + 2� − arccos�rC/rP�, �28�
RC = rC�rP2 − rC
2 �1/2. �29�
ote that for observation points of interest to us (the ob-ervation points located outside the plane of rotation, P
� /2, in the far zone, RP 1), the parameter a has aalue whose magnitude increases with increasing RP:
a − RP sin4 P sec2 P �30�
see Eq. (27)]. In other words, the phase function �− isore peaked at its maximum, the farther the observation
oint is from the source.This property of the phase function �− distinguishes
he asymptotic analysis that will be presented in the fol-owing section from those commonly encountered in ra-iation theory. What turns out to play the role of a largearameter in this asymptotic expansion is distance �RP�,ot frequency �� ��.
±
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2450 J. Opt. Soc. Am. A/Vol. 24, No. 8 /August 2007 Ardavan et al.
. ASYMPTOTIC ANALYSIS OF THENTEGRAL REPRESENTING THEIELD FOR LARGE DISTANCE. Transformation of the Phase of the Integrand into aanonical Formhe first step in the asymptotic analysis of the integralhat appears in Eq. (19) is to introduce a change of vari-ble �=��r , z� that replaces the original phase �− of the in-egrand by as simple a polynomial as possible. This trans-ormation should be one to one and should preserve theumber and nature of the stationary points of thehase[25,26]. Since �− has a single isolated stationaryoint at r= rC�z�, it can be cast into a canonical form byeans of the following transformation:
�−�r, z� = �C�z� + 12a�z��2, �31�
n which a is the coefficient given in Eqs. (27) and (30).The integral in Eq. (19) can thus be written as
��0
rdrdz�−1/2uj exp�− i��−� =���S
dzd�A��, z�exp�i��2�,
�32�
n which
A��, z� � r�−1/2uj
�r
��exp�− i��C�, �33�
ith
�r
��= a�rR−�r2 − 1 − �1/2�−1, �34�
nd ��−�a /2. The stationary point r= rC and the bound-ry point r= rS, respectively, map onto �=0 and
� = �S � − �2a−1��S − �C��1/2, �35�
here
�S � ��−�r=rS= 2� − arccos�1/�rSrP�� + �rS
2rP2 − 1�1/2.
�36�
he upper limit of integration in Eq. (32) is determined byhe image of the support of the source density (s in uj) un-er the transformation (31).The Jacobian �r /�� of the above transformation is inde-
erminate at �=0. Its value at this critical point has to beound by repeated differentiation of Eq. (31) with respecto �,
��−
�r
�r
��= a�, �37�
�2�−
�r2 � �r
���2
+��−
�r
�2r
��2 = a, �38�
nd the evaluation of the resulting relation (38) at r= rCith the aid of Eq. (27). This procedure, which amounts topplying l’Hôpital’s rule, yields ��r /����=0=1: a result thatould have been anticipated in light of the coincidence ofransformation (31) with the Taylor expansion of � about
−
ˆ = rC to within the leading order. Correspondingly, themplitude A��� that appears in Eq. (33) has the value
ˆC�rC2 −1�−1�uj�r=rC
exp�−i��C� at the critical point C.When the observer is located in the far field �RP 1�,
he phase of the integrand on the right-hand side of Eq.32) is rapidly oscillating irrespective of how low the har-onic numbers �± (i.e., the radiation frequencies �±�)ay be. The leading contribution to the asymptotic value
f integral (32) from the stationary point �=0 can there-ore be determined by the method of stationary phase25]. However, in the limit RP→�, �S reduces to
�S − 3−1/2 cos4 P csc5 PRP−2, �39�
o that the stationary point �=0 is separated from theoundary point �=�S by an interval of the order of RP
−2
nly. To determine the extent of the interval in r fromhich the dominant contribution toward the value of the
adiation field arises, we therefore need to employ a moreowerful technique for the asymptotic analysis of integral32), a technique that is capable of handling the contribu-ions from both rC and rS simultaneously.
. Contours of Steepest Descenthe technique we shall employ for this purpose is theethod of steepest descents [26]. We regard the variable
f integration in
I�z� � �S
��
d�A��, z�exp�i��2� �40�
s complex, i.e., write �=u+iv, and invoke Cauchy’s inte-ral theorem to deform the original path of integrationnto the contours of steepest descent that pass throughach of the critical points �=�S, �=0, and �=��. Here, weave introduced the real variable ���z� to designate the
mage of r� under transformation (31), i.e., the boundaryf the support of the source term uj that appears in themplitude A�� , z�. We shall treat only the case in which �and hence �) is positive; I�z� for negative � can then bebtained by taking the complex conjugate of the derivedxpression and replacing �C by −�C [see Eq. (33)].
The path of steepest descent through the stationaryoint C at which �=0 is given, according to
i�2 = − 2uv + i�u2 − v2�, �41�
y u=v when � is positive. If we designate this path by C1see Fig. 5), then
C1
d�A��, z�exp�i��2� = �1 + i�−�
�
�dvA��=�1+i�v exp�− 2�v2�
�2�/��1/2 exp�− i���C − �/4��
�uj�C csc P�sec P�RP−1/2, �42�
or RP 1. Here, we have obtained the leading term in thesymptotic expansion of the above integral for large RP bypproximating �A��=�1+i�v by its value at C, where v=0,nd using Eq. (30) to replace � with its value in the farone. Note that the next term in this asymptotic expan-
st
p(i
ip+
t
Td(
afvt�
A
Iy
fc
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i
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ip+
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FcurTta
Ardavan et al. Vol. 24, No. 8 /August 2007 /J. Opt. Soc. Am. A 2451
ion is by a factor of order RP−1/2 smaller than this leading
erm.The path of steepest descent through the boundary
oint S, at which u�uS=�S and v=0 [see Eqs. (35) and39)], is given by u=−�v2+uS
2�1/2, i.e., by the contour des-gnated as C2 in Fig. 5. The real part of
�i�2�C2= 2v�v2 + uS
2�1/2 + iuS2 �43�
s a monotonic function of v and so can be used as a curvearameter for contour C2 in place of v. If we let 2v�v2
uS2�1/2�−�, then it follows from
���C2= − �uS
2 + i��1/2 �44�
hat
C2
d�A��, z�exp�i��2� = 12 exp�i��uS
2 − �/2��0
�
d��uS2
+ i��−1/2�A��=−�uS2 + i��1/2 exp�− ���.
�45�
he function �A�C2that here enters the integrand can be
etermined only by inverting the original transformation31).
However, since the dominant contribution toward thesymptotic value of the above integral for RP 1 comesrom the vicinity of the boundary point S, the required in-ersion of transformation (31) needs to be carried out onlyo within the leading order in �. The Taylor expansions of±�r , z� about r= rS�z� are of the forms
�± = �S + rS−1�rS
2 − 1��rS2rP
2 − 1�−1/2�r − rS�
± 13 �2rS�3/2�rP
2 − 1�3/2�rS2rP
2 − 1�−3/2�r − rS�3/2 + ¯ .
�46�
ccording to Eqs. (31) and (35), on the other hand,
�− − �S = 12a��2 − �S
2�. �47�
n the vicinity of �=�S, therefore, Eqs. (46) and (47) jointlyield
ig. 5. Integration contours in the complex plane �=u+iv. Theritical point C lies at the origin, and uS and u� are the imagesnder transformation (31) of the radial boundaries r= rS�z� and= r��z� of the part of the source that lies within ��0 (see Fig. 4).he contours C1, C2, and C3 are the paths of steepest descent
hrough the stationary point C and the lower and upper bound-ries of the integration domain, respectively.
r rS + 12 sin5 P sec4 PRP
2��S2 − �2�, �48�
or RP 1. Note that �S2−�2=−i� and that close to the
usp in the far zone
�1/2 �2 sin P�1/2RP�r − rS�1/2, �r − rS� � 1, RP 1.
�49�
ence, inserting Eqs. (48) and (49) in Eq. (33), taking theimit RP→�, and expressing � in terms of �, we find that
�A�C2 exp�− i���C − �/4���uj�S sin P sec2 P
��uS2 + i��1/2�−1/2 �50�
n the immediate vicinity of the point S at which �=0.Strictly speaking, we should excise the singularity of A
t �=0 by means of an arc-shaped contour. However,ince this singularity is integrable and so has no associ-ted residue, the contribution from such a contour van-shes in the limit that its arc length tends to zero. An al-ernative way of handling the removable singularity at=0, followed below, is to introduce a change of integra-
ion variable. If we let �=�2, then the integral in Eq. (45)ssumes the form
here use has been made of Eq. (30) and the definition�−�a /2. Note that this differs from the correspondingxpression in Eq. (42) for the integral over C1 by the fac-or 1
2 exp�−i� /2�.The path of steepest descent through the boundary
oint �=��, at which u=u�, v=0, is given by u= �v2
u�2 �1/2, i.e., by the contour designated as C3 in Fig. 5. The
eal part of the exponent
�i�2�C3= − 2v�v2 + u�
2�1/2 + iu�2 �52�
s again a monotonic function of v and so can be used toarametrize contour C3 in place of v. If we let 2v�v2
u�2 �1/2��, then it follows from
���C3= �u�
2 + i��1/2 �53�
hat
C3
d�A��, z�exp�i��2�
= 12 exp�i��u�
2 − �/2��0
�
d�
��u�2 + i��−1/2�A��=�u
�2 + i��1/2 exp�− ���. �54�
Ti
ba
[l
t
CTtcCcotftalfi(
io
eis�eoat
ito
agtatti
a(
wgftsRi�
tu
Tepioaepoaat
DAgtp(do�if
tRpttdo
2452 J. Opt. Soc. Am. A/Vol. 24, No. 8 /August 2007 Ardavan et al.
he asymptotic value of this integral for RP 1 receivests dominant contribution from �=0. Because the function�A�C3
is regular, on the other hand, its value at �=0 cane found by simply evaluating the expression in Eq. (33)t r= r�. The result, for RP→�, is
�A�C3,�=0 r�2 sin4 P sec2 P�r�
2 sin2 P − 1�−1�uj�r=r�
�exp�− i��C�u� �55�
see Eqs. (8) and (30)]. This, in conjunction with Watson’semma, therefore implies that
C3
d�A��, z�exp�i��2� r�2 �r�
2 sin2 P − 1�−1�uj�r�
�exp�− i����−�r�+ �/2���−1RP
−1,
�56�
o within the leading order in RP−1.
. Asymptotic Value of the Radiation Fieldhe integral in Eq. (32) equals the sum of the three con-
our integrals that appear in Eqs. (42), (51), and (56); theontributions of the contours that connect C1 and C2 and1 and C3, at infinity (see Fig. 5) are exponentially small
ompared with those of C1, C2, and C3 themselves. On thether hand, the leading term in the asymptotic value ofhe integral over C3 decreases (with increasing RP) muchaster than those of the integrals over C1 and C2: the in-egral over C3 decays as RP
−1, while the integrals over C1
nd C2 decay as RP−1/2 [see Eqs. (42), (51), and (56)]. The
eading term in the asymptotic expansion of the radiationeld Bns for large RP is therefore given, according to Eqs.
18), (32), and (42), by
Bns −2
3�1 + 2i��2��1/2RP
−1/2�sec P�csc P
�exp�i��/����P + 3�/2�� ��=�±
���1/2sgn���
�exp�i�
4sgn ���
j=1
3
qj−�
�
dz�uj�C exp�− i���C + �P��,
�57�
n which �± can also be negative (see the first paragraphf Subsection 3.B).
This result agrees with that in Eq. (55) of [13]. The twoxpressions differ by a factor of 2−i because we have herencluded the additional contribution that arises from theource elements in the (vanishingly small) interval rSr� rC. The integration with respect to r in Eq. 52 of [13]
xtends over rC� r� r�, while that in Eq. (40) extendsver rS� r� r�. The contribution from rS� r� rC is given,ccording to Cauchy’s theorem, by the contribution fromhe lower half of C1 plus the contribution from C2.
Even though the length of the interval rS� r� rC van-shes as RP
−2 as RP tends to infinity [see Eq. (39)], the con-ribution that arises from this interval toward the valuef the field has the same order of magnitude as that which
rises from rC� r� r� and is by a factor of order RP1/2
reater than that which arises from the open interval rCr� r�. Thus, the nonspherically decaying component of
he radiation field that is observed at any given �xP, tP�rises from those elements of the source, located at the in-ersection of the cusp curve of the bifurcation surface withhe volume of the source (Fig. 4), that occupy the vanish-ngly small radial interval
�r � rC − rS 12 cos4 P csc5 PRP
−2 �58�
djacent to the cusp at r= rScsc P [see Eqs. (24) and25)].
The corresponding azimuthal extent of the source fromhich the contribution described by Eq. (19) arises isiven by the separation �+−�− of the two sheets of the bi-urcation surface shown in Fig. 3 close to the cusp curve ofhis surface: the contribution of the source elements out-ide the bifurcation surface is by a factor of the order ofˆ
P−1/2 smaller than those of the elements close to the cusp
nside this surface [see Eqs. 41 and 42 of Ref. [13]]. Since
+−�−�25/2 /3��csc P�−3/2�r− rS�3/2 for �r− rS��1 and RP1 [see Eq. (46)] and the contributing interval in r is of
he order of RP−2 [see Eq. (58)], it follows that the contrib-
ting interval in � is
�� � ��+ − �−�r=rC 2
3 cot6 PRP−3. �59�
he contribution from this vanishingly small azimuthalxtent of the rotating source is made when the retardedosition of this part of the source is �=�C [see Eq. (28)],.e., when the contributing source elements approach thebserver with the speed of light and zero accelerationlong the radiation direction. Thus, the source that gen-rates the nonspherically decaying field observed at aoint �RP,P,�P� in the far zone �RP 1� consists entirelyf the narrow filament parallel to the z axis that occupiesradial interval �r�RP
−2 encompassing rcsc P and anzimuthal interval ���RP
−3 encompassing ��P+3� /2 athe retarded time.
. Frequency Independence of the Nonspherical Decayfurther implication of the above analysis is that the
enerated field decays nonspherically irrespective of whathe values of the frequencies �±� may be. There is no ap-roximation involved in introducing the transformation31), and the asymptotic expansion is for large �. Aserived here, therefore, the only condition for the validityf Eq. (57) is that the absolute value of 1
2�±RP sin4P sec2P should be large, a condition thats automatically satisfied in the far zone for all nonzerorequencies.
That the leading term in the asymptotic expansion ofhe integral in Eq. (32) is proportional to RP
−1/2, instead ofˆ
P−1, is a consequence of the particular features of thehase function �− described by Eqs. (26)–(30). These fea-ures originate in and reflect the particular properties ofhe time-domain phase t+R�t� /c; they are totally indepen-ent of both the wavelength of the radiation and the sizef the source. In contrast to all other nonspherically de-
cpstcglE
Bsp
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Twftod
EfTasBiwttsicn
4SDDWg
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cdavcrscdwt
cpa�cmpl�
Ipawwt
wc
Ardavan et al. Vol. 24, No. 8 /August 2007 /J. Opt. Soc. Am. A 2453
aying solutions of Maxwell’s equations reported in theublished literature (see, e.g, [27–29]), whose slowpreading and decay occur only within the Fresnel dis-ance from the source, the nonspherical decay that is dis-ussed here remains in force at all distances. In fact, thereater the distance RP from the source, the more theeading term dominates the asymptotic approximation inq. (57).The remaining z integration in the above expression for
ns amounts to a Fourier decomposition of the source den-ities sr,�,z�C with respect to z. Using Eqs. (26)–(29) to re-lace �C in Eq. (57) with its far-field value
�C RP − z cos P + 3�/2 �60�
nd using Eq. (23) to write out uj in terms of sr,�,z, we findhat the electric field Ens= n�Bns of the nonsphericallyecaying radiation is given by
Ens 4
3�2��1/2RP
−1/2�sec P�csc P exp�i��/����P + 3�/2��
� ��=�±
���1/2sgn���exp�i�
4sgn ��
�exp�− i��RP + �P + 3�/2����is� + �sr/��e
− ��isr − �s�/��cos P + �sz sin P/��e��, �61�
n which sr,�,z stand for the following Fourier transformsf sr,�,z�C with respect to z:
sr,�,z �−�
�
dz�sr,�,z�r, z��r=csc Pexp�i�z cos P�. �62�
his field is observable only at those polar angles Pithin the interval arccos�1/ r�� �P−� /2��arccos�1/ r��
or which sr,�,z�r=csc Pare nonzero; i.e., at those observa-
ion points (outside the plane of rotation) the cusp curvef whose bifurcation surface (Fig. 3) intersects the sourceistribution (Fig. 4).
. Relevance to Computational Models of the Emissionrom Superluminal Sourceshe asymptotic analysis outlined in this section providesbasis also for the computational treatment of the non-
pherically decaying field Bns. The original formulation ofns appearing in Eq. (19), in which the integral has a rap-
dly oscillating kernel, is not suitable for computing a fieldhose value in the radiation zone receives its main con-
ribution from such small fractions of the r and � integra-ion domains as �r�RP
−2 and ���RP−3. The above conver-
ion of the Fourier-type integrals into Laplace-typentegrals renders the selecting out and handling of theontributions from integrands with such narrow supportsumerically more feasible.
. COLLECTION OF NONDIFFRACTINGUBBEAMS DELINEATING THE OVERALLISTRIBUTION OF THE NONSPHERICALLYECAYING RADIATIONe have seen that the wavefronts that emanate from a
iven volume element of a rotating superluminal source
ossess an envelope consisting of two sheets that meetlong a cusp (Fig. 1). There is a higher-order focusing in-olved in the generation of the cusp than in that of thenvelope itself, so that the intensity of the radiation fromn extended source attains its maximum on the bundle ofusps that are emitted by various source elements. If aource element approaches an observeration point P withhe speed of light and zero acceleration along the radia-ion direction, then the cusp it generates passes through. The reason is that both the locus of source elementshat approach the observer with the speed of light andero acceleration [i.e., the cusp curve of the bifurcationurface (Fig. 3)] and the cusp that is generated by a givenource element are described by the same equation: theusp curve of the bifurcation surface resides in the spacef source points and so is given by Eq. (12) for a fixedrP,�P, zP�, while the cusp curve of the envelope resides inhe space of observation points and is given by Eq. (12) forfixed �r ,� ,z�.The collection of cusp curves that are generated by the
onstituent volume elements of an extended source thusefines what might loosely be termed a radiation beam,lthough its characteristics are distinct from those of con-entionally produced beams. The field decays nonspheri-ally only along the bundle of cusp curves embodying thisadiation beam. Since the cusp that is generated by aource element with the radial coordinate r lies on theone P=arcsin�1/ r� in the far zone, the nonsphericallyecaying radiation that arises from a source distributionith the radial extent r� r� r� is detectable only within
he conical shell arcsin�1/ r���P�arcsin�1/ r�.The field that is detected at a given point P within this
onical shell arises almost exclusively from a filamentaryart of the source parallel to the z axis whose radial andzimuthal extents are of the order of �r�RP
−2 and ��
RP−3, respectively [see Eqs. (58) and (59)]. The bundle of
usps emanating from this narrow filament occupies auch smaller solid angle than that described above. The
arametric equation zP= zP�rP�, �P=�P�rP� of the particu-ar cusp curve that emanates from a given source elementr , � ,z� can be written, using Eq. (12), as
zP = z ± �rP2 − 1�1/2�r2 − 1�1/2, �63�
�P = � − 2� + arccos�1/�rrP��. �64�
f we rewrite Eqs. (63) and (64) in terms of the sphericalolar coordinates �RP,P,�P� of the observation point Pnd solve them for P and �P as functions of �r ,z� and RP,e find that the cusp that is generated by a source pointith the coordinates �r , � ,z� passes through the following
wo points on a sphere of radius RP:
P = arccos� 1
rRP� z
r± �r2 − 1�1/2�RP
2 − 1 −z2
r2�1/2�� ,
�65�
�P = � − 2� + arccos�1/�RPr sin P��, �66�
here the � correspond to the two halves of this cuspurve above and below the plane of rotation (see Fig. 4).
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2454 J. Opt. Soc. Am. A/Vol. 24, No. 8 /August 2007 Ardavan et al.
The Taylor expansion of the expressions on the right-and sides of Eqs. (65) and (66) in powers of RP
−1 yields
P = arcsin�1/r� − �z/r�RP−1 ± 1
2 �r2 − 1�1/2RP−2 + ¯ ,
�67�
�P = � − 3�/2 − RP−1 + ¯ . �68�
hese show that incremental changes �r, �z, and �� inhe position of the source element �r , � ,z� result in the fol-owing changes in the �P,�P� coordinates of the point athich the cusp arising from that element intersects a
phere of radius RP in the far zone:
�P = − r−1�r2 − 1�−1/2�r − r−1�zRP−1 + ¯ �69�
nd ��P=��+¯. Because �r is of the order of RP−2, while
z is of the order of unity for the filamentary source of theeld that is detected at P, the dominant term in the ex-ression on the right-hand side of Eq. (69) is that propor-ional to the z extent of the filament. Given the observa-ion point P, and hence a set of fixed values for theimensions (�r, ��, �z) of the filamentary source of theeld that is detected at P, it therefore follows that theundle of cusps generated by such a filament occupies aolid angle with the dimensions
�P − �z sin PRP−1, ��P �� �70�
n the far zone. [Here, we have made use of the fact that
Parcsin�1/ r� to within the zeroth order in RP−1 to ex-
ress r in Eqs. (70) in terms of P.]The bundle of cusp curves occupying the solid angle
Eqs. (70)] embodies a subbeam that does not diverge inhe direction of P. The polar width �P of this subbeamecreases with the distance RP in such a way that thehickness RP�P of the subbeam in the polar direction re-ains constant: it equals the projection of the z extent, �z,
f the contributing filamentary source onto a directionormal to the line of sight at all RP. The azimuthal widthf the subbeam, on the other hand, diverges as does anyther radiation beam: ��P is independent of RP (see Fig.).Thus, the bundle of cusps that emanates from the fila-entary locus of the set of source elements responsible for
he nonspherically decaying field at P intersects a largephere of radius RP (enclosing the source) along a striphe thickness of whose narrow side is independent of RP.ccording to Eqs. (70), the area RP
2 sin P�P��P sub-ended by this subbeam increases as RP, rather than RP
2,ith the radius of the sphere enclosing the source. Con-
ervation of energy demands, therefore, that the Poyntingector associated with this radiation should correspond-ngly decrease as RP
−1 instead of RP−2, in order that the flux
f energy remain the same across various cross sections ofhe subbeam. This requirement is, of course, automati-ally met by the (nonspherical) rate of decay of the inten-ity of the radiation that propagates along the subbeam.
The nondiffracting subbeams that are detected at twoistinct observation points within the solid anglercsin�1/ r �� �arcsin�1/ r � arise from two distinct
� P
lamentary parts of the source with essentially no com-on elements [see Eqs. (58) and (59)]. The subbeam that
asses through an observation point P�, though sharinghe same general properties as that which passes through, arises from those elements of the source, located at r�csc P�, ��=�P�, that approach P�, rather than P, with
he speed of light and zero acceleration at the retardedime. Not only are the focused wave packets that embodyhe cusp constantly dispersed and reconstructed out ofther (spherically spreading) waves [9], but also the fila-ents that act as sources of these focused waves each oc-
upy a vanishingly small ��RP−5� disjoint volume of the
verall source distribution and so are essentially indepen-ent of one another. Unlike conventional radiation beams,hich have fixed sources, the subbeam that passes
hrough an observation point P arises from a sourcehose location and extent depend on P.It would be possible to identify the individual nondif-
racting subbeams only in the case of a source whoseength scale of spatial variations is comparable with RP
−2
e.g., in the case of a turbulent plasma with a superlumi-ally rotating macroscopic distribution). The overalleam within which the nonspherically decaying radiations detectable would then consist of an incoherent superpo-ition of coherent, nondiffracting subbeams with widelyiffering amplitudes and phases. The individual coherentubbeams decay nonspherically, but the incoherence ofheir phase relationships ensures that the integrated fluxf energy associated with their superposition across thisnite solid angle remains independent of RP. Note thathe individual subbeams constituting the overall beamould be narrower and more distinguishable, the farther
he observer is from the source.
. CONCLUDING REMARKShe analysis we have presented here was motivated byuestions encountered in the course of the design, con-truction, and testing of practical machines for investigat-ng the emission from superluminal sources [6]. The origi-al mathematical treatment of the nonsphericallyecaying radiation [13], in which the integral over the vol-me of the source has a rapidly oscillating kernel, is notuitable for the computational modeling of the emissionrom such machines. We have seen that the nonspheri-ally decaying radiation detected in the radiation zone re-eives its main contribution from such small fractions ofhe radial and azimuthal integration domains as �rRP
−2 and ���RP−3. The above conversion of the Fourier-
ype integrals into Laplace-type integrals renders the se-ecting out and handling of the contributions from inte-rands with such narrow supports numerically moreeasible.
Not only the nonspherical decay of its intensity but alsohe narrowness of both the beam into which it propagatesnd the region of the source from which it arises are fea-ures that are unique to the emission from a rotating su-erluminal source. These features are not shared by anyther known emission mechanism. On the other hand,hey are remarkably similar to the observed features ofn emission that has long been known to radio astrono-
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Ardavan et al. Vol. 24, No. 8 /August 2007 /J. Opt. Soc. Am. A 2455
ers: the (as yet unexplained) extreme properties of theiant pulses that are received from pulsars (see, e.g., Refs.16–20]). The giant radio pulses from the Crab pulsarave a temporal structure of the order of a nanosecond18]. Under the assumption that they decay sphericallyike other conventional emissions, the observed values ofhese pulses’ fluxes imply that their energy densities gen-rally exceed the energy densities of both the magneticeld and the plasma in the magnetosphere of a pulsar byany orders of magnitude [19]. “The plasma structures
esponsible for these emissions must be smaller than oneeter in size, making them by far the smallest objects
ver detected and resolved outside the Solar System, andhe brightest transient radio sources in the sky.” [18]
The highly stable periodicity of the mean profiles of thebserved pulses [16], i.e., the rigidly rotating distributionf the radiation from pulsars, can arise only from a sourcehose distribution pattern correspondingly rotates rig-
dly, a source whose average density depends on the azi-uthal angle � in the combination �−�t only: Maxwell’s
quations demand that the charge and current densitieshat give rise to this radiation should have exactly theame symmetry �� /�t=−�� /��� as that of the observed ra-iation fields E and B. On the other hand, the domain ofpplicability of such a symmetry cannot be localized; a so-ution of Maxwell’s equations that satisfies this symmetrypplies either to the entire magnetosphere or to a regionhose boundary is an expanding wavefront. Unless there
s no plasma outside the light cylinder, therefore, the mac-oscopic distribution of electric current in the magneto-phere of a pulsar should have a superlumially rotatingattern in r�c /�. The superluminal source described byq. (15) captures the essential features of the macroscopicharge-current distribution that is present in the mag-etosphere of a pulsar and is thus an inevitable implica-ion of the observational data on these objects.
Once it is acknowledged that the source of the observediant pulses should have a superluminally rotating distri-ution pattern, the extreme values of their brightnessemperature ��1039 K�, temporal width ��1 ns�, andource dimension ��1 m� are all explained by the resultsf the above analysis. The nonspherical decay of the re-ulting radiation would imply that the energy density ando the brightness temperature of the observed pulses arey a factor of the order of R / �r�− r�2 smaller than thosehat are normally estimated by using an inverse-squareaw [13], a factor that ranges from 1015 to 1025 in the casef known pulsars [16].
The nondiffracting nature of this nonspherically decay-ng radiation [Eqs. (70)], together with its arising onlyrom the filamentary part of the source that approacheshe observer with the speed of light and zero accelerationEqs. (58) and (59)], likewise explains the values of itsemporal width and source dimension. Furthermore, thathe overall beam within which the nonspherically decay-ng radiation is detectable should in general consist of anncoherent superposition of coherent, nondiffracting sub-eams (Section 4) is consistent with the conclusioneached by Popov et al. [20] that “the radio emission of therab pulsar at the longitudes of the main pulse and inter-ulse consists entirely of giant pulses.”[20]Two other features of the emission from a rotating su-
erluminal source that were derived elsewhere [12,30]re also consistent with the observational data on pulsars16]: the occurrence of concurrent orthogonal polarizationodes with swinging position angles [12] and a broad-
and frequency spectrum [30].
CKNOWLEDGMENThis work is supported by U.S. Department of Energyrant LDRD 20050540ER.
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