Ardavan et al. Vol. 25, No. 3 /March 2008/J. Opt. Soc. Am. A 543
Fundamental role of the retarded potential in theelectrodynamics of superluminal sources
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
4Process Modeling and Analysis, MS-F609, Los Alamos National Laboratory, Los Alamos,New Mexico 87545, USA
. INTRODUCTIONoving sources of electromagnetic radiation whose
peeds exceed the speed of light in vacuo have alreadyeen generated in the laboratory [1–4]. These sourcesrise from separation of charges: Their superluminallyoving distribution patterns are created by the coordi-
ated motion of aggregates of subluminally moving par-icles. A polarization current density is, however, on theame footing as the current density of free charges in thempére–Maxwell equation, so that the propagating dis-
ribution patterns of such polarization currents radiate,s would any other moving sources of the electromagneticeld [5–9].We have already shown, by means of an analysis based
n the classical expression for the retarded potential [Eq.6) below], that the radiation field of a superluminally ro-ating extended source at a given observation point Prises almost exclusively from those of its volume ele-ents that approach P, along the radiation direction, with
he speed of light and zero acceleration at the retardedime [8–10]. These elements comprise a filamentary partf the source whose radial and azimuthal widths become
−3, respectively), thearger the distance RP of the observer from the source,nd whose length is of the order of the length scale lz ofhe source parallel to the axis of rotation [10]. (Here r, �,nd z are the cylindrical polar coordinates of the sourceoints.)Once a source travels faster than its emitted waves, it
an make more than one retarded contribution to the fieldhat is observed at any given instant. This multivalued-ess of the retarded time [7–10] means that the wave-
ronts emitted by each of the contributing volume ele-ents of the source possess an envelope, which in this
ase consists of a two-sheeted, tubelike surface whoseheets meet tangentially along a spiraling cusp curve (seeigs. 1 and 4 of [10]). For moderate superluminal speeds,
he field inside the envelope receives contributions fromhree distinct values of the retarded time, while the fieldutside the envelope is influenced only by a single instantf emission time. Coherent superposition of the emittedaves on the envelope (where two of the contributing re-
arded times coalesce) and on its cusp (where all three ofhe contributing retarded times coalesce) results in not
008 Optical Society of America
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544 J. Opt. Soc. Am. A/Vol. 25, No. 3 /March 2008 Ardavan et al.
nly a spatial but also a temporal focusing of the waves:he contributions from emission over an extended periodf retarded time reach an observer who is located on theusp during a significantly shorter period of observationime.
The field of each contributing volume element of theource is strongest, therefore, on the cusp of the envelopef wavefronts that it emits. The bundle of cusps generatedy the collection of contributing source elements (i.e., byhe filamentary part of the source that approaches the ob-erver with the speed of light and zero acceleration) con-titutes a radiation subbeam whose widths in the polarnd azimuthal directions are of the order of ��P�RP
−1
nd ��P�RP−3, respectively [10]. (Here RP, �P, and �P are
he spherical polar coordinates of the observation point P.)he overall radiation beam generated by the source con-ists of a (necessarily incoherent [11]) superposition ofuch subbeams, a beam whose azimuthal width is theame as the azimuthal extent of the source and whose po-ar width arccos�c / �r����� ��P−� /2��arccos�c / �r���� isetermined by the radial extent 1� r�� r� r� of the su-erluminal part of the source [9,10]. (Here c is the speedf light in vacuo, � is the angular frequency of rotation ofhe source, and r�r� /c.)
Since the cusps only represent the loci of points athich the emitted spherical waves interfere construc-
ively (i.e., represent wave packets that are constantlyispersed and reconstructed out of other waves), the sub-eams generated by a superluminal source need not beubject to diffraction as are conventional radiation beams.evertheless, they have a decreasing angular width only
n the polar direction. Their azimuthal width ��P de-reases as RP
−3 with distance because they receive contri-utions from an azimuthal extent �� of the source thatikewise shrinks as RP
−3. They would have had a constantzimuthal width had the azimuthal extent of the contrib-ting part of the source been independent of RP. On thether hand, the solid angle occupied by the cusps has ahickness �zP in the direction parallel to the rotation axishat remains of the order of the height lz of the source dis-ribution at all distances (see Fig. 2 of [10]). Consequently,he polar width ��P of the particular subbeam that goeshrough the observation point decreases as RP
−1, insteadf being independent of RP [10].
Because it is of a constant linear width, parallel to theotation axis, an individual subbeam subtends an area ofhe order of RP, rather than RP
2. In order that the flux ofnergy remain the same across all cross sections of theubbeam, therefore, it is essential that the Poynting vec-or associated with this radiation correspondingly decayore slowly than that of a conventional, spherically de-
aying radiation: as RP−1, rather than RP
−2, within theundle of cusps that emanate from the constituent vol-me elements of the source and extend into the far zone.his result, which also follows from the superposition ofhe Liénard–Wiechert fields of the constituent volume el-ments of a rotating superluminal source [9,10], has noween demonstrated experimentally [2].The narrowing of the individual subbeams with dis-
ance suggests that the absolute value of the gradient ofhe radiation field described here should increase withistance, in contrast to that of a conventional, diffracting
adiation beam that decreases with distance. This is illus-rated by a simple example. Imagine a rotating radiationeam with the amplitude
A�RP,�P,tP� = A0RP1/2 exp�− �RP
3�P�2�,
here RP stands for the scaled distance RP� /c, �P��P�tP is the azimuthal angle in the rotating frame, tP is
he observation time, and A0 is a constant. This beamould be observed as a Gaussian pulse that has an azi-uthal width of the order of RP
−3 and carries a constantux of energy,
� A2RP2 sin �Pd�Pd�P = �2��1/2�c/��2A0
2,
cross any large sphere of radius RP. The gradient of themplitude of this pulse,
�A/��P = − 2A0RP7/2�RP
3�P�exp�− �RP3�P�2�,
ncreases in magnitude with distance as RP7/2 at the edges
f the pulse.In this paper, we derive the azimuthal (equivalently,
emporal) gradient �� /��P� of the radiation field that isenerated by a physically viable, rotating superluminalource directly from the retarded potential and show thathe absolute value of this gradient does increase as RP
7/2
ithin each subbeam. The spiky structure of the angularistribution of the emission from an accelerated superlu-inal source therefore follows not only from the geometry
f the emitted cusps (geometrical optics) that was consid-red in [10] but also from the calculation of the field dis-ribution (physical optics) that is presented here. This re-ult corroborates the earlier finding that the overalladiation beam consists of an incoherent superposition ofharply peaked subbeams that become narrower with dis-ance from the source [11].
There is, however, another, more significant implica-ion. The retarded solution to the wave equation that gov-rns the electromagnetic potential in the Lorenz gaugeEq. (2) below] generally entails three terms: an integralver the retarded value of the electric current density, anntegral over the boundary values of the potential and itsradient, and an integral over the initial values of the po-ential and its time derivative [see Eq. (3) below]. For aocalized distribution of electric current, the integral overhe retarded value of the source density is of the order ofˆ
P−1 in the far zone. If evaluated for a potential that is of
his order of magnitude in the far zone (i.e., decays asˆ
P−1), the integral over the boundary in this solutionould also be of the order of RP
−1 in the limit where theoundary tends to infinity. However, even potentials thatatisfy the Lorenz condition are arbitrary to within a so-ution of the homogeneous wave equation, so that one canlways use the gauge freedom in the choice of potential toet this boundary term identically equal to zero.
In the case of the corresponding retarded solution ofhe wave equation for the electromagnetic field [Eq. (7)elow], on the other hand, one no longer has the freedomffered by a gauge transformation to render the boundaryerm equal to zero. Nor does this term always decay faster
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Ardavan et al. Vol. 25, No. 3 /March 2008/J. Opt. Soc. Am. A 545
han the source term, so that it could be neglected for aoundary that tends to infinity, as is commonly assumedn textbooks (e.g., p. 246 of [12]) and the published litera-ure [13–16]. The boundary contribution to the retardedolution of the wave equation governing the field entails aurface integral over the boundary values of both the fieldnd its gradient [see Eq. (8) below]. In the superluminalegime, where the gradient of the field increases as RP
7/2
ver a solid angle that decreases as RP−4, this boundary
ontribution turns out to be of the order of RP−1/2 (see Sec-
ion 4). Not only is this not negligible relative to the con-ribution from the source term, which decays as RP
−1
13–16], but the boundary term constitutes the dominantontribution toward the value of the radiation field in thisase.
Thus, if one ignores the boundary term in the retardedolution of the wave equation governing the field (as isone by Hannay [13–16]), one would obtain a result, inhe superluminal regime, that contradicts what is ob-ained by calculating the field via the retarded potential8–10]. However, the contradiction stems solely from hav-ng ignored a term in the solution to the wave equationhat is by a factor of the order of RP
1/2 greater than theerm that is normally kept in this solution. The contradic-ion disappears once the neglected term is taken into ac-ount: The solutions to both the wave equation that gov-rns the potential and the wave equation that governs theeld predict that the field of a rotating superluminalource decays as RP
−1/2 as RP tends to infinity.From a physical point of view, however, what one ob-
ains by including the boundary term in the retarded so-ution to the wave equation that governs the field is
erely a mathematical identity; it is not a solution thatould be used to calculate the field arising from a givenource distribution in free space. Unless its boundaryerm happens to be negligibly smaller than its sourceerm, a condition that cannot be known a priori, the solu-ion in question would require that one prescribe the fieldn the radiation zone (i.e., what one is seeking) as aoundary condition. The role played by the classical ex-ression for the retarded potential in radiation theory islearly much more fundamental than that played by theorresponding retarded solution of the wave equation gov-rning the field. The only way to calculate the free-spaceadiation field of an accelerated superluminal source is toalculate the retarded potential and differentiate the re-ulting expression to find the field (see also [17]).
We must emphasize that the contrast between the re-arded solutions for the potential and for the field is rel-vant only to extended superluminal sources. In the casef a point source, the Liénard–Wiechert field that one ob-ains from the classical expression for the retarded poten-ial [Eq. (6) below] is the same as that which one wouldbtain from the retarded solution for the field [Eq. (8) be-ow] by ignoring its boundary term (see [9,12]). As such,he present work is directly related neither to those en-ountered in the literature on superluminal point sourcestachyons) nor to those concerned with the solutions of theomogeneous wave equation describing nondiffracting lo-alized waves (see [18] for a review). The common featuref the localized signals appearing in the nonspherically
ecaying emission discussed here and the wave fields re-iewed in [18] is that they all have phases that are sta-ionary, or wave vectors that peak, in certain directions.owever, the sources we consider are extant polarization
urrents [2], rather than hypothetical tachyons, and theolutions of the inhomogeneous wave equation we discussescribe finite-energy signals that remain nondiffractingin one dimension) at all distances from their source, in-ependently of frequency [10], rather than having eitherfinite field depth or an infinite energy [18].This paper is organized as follows: Section 2 presents
he retarded solutions to the initial-boundary value prob-ems for the wave equations that govern the potential andhe field. We provide a detailed mathematical derivationf the gradient of the radiation field that is generated by aotating superluminal source in Section 3, with a brief ac-ount of the required background material in Subsection.A, the formulation of the problem in Subsection 3.B, theerivation of an integral representation of the gradient ofreen’s function in Subsection 3.C, the regularization of
he integral over the radial extent of the source in Sub-ection 3.D (and Appendix A), a description of contours ofteepest descent in Subsection 3.E, and the asymptoticvaluation of the gradient of the radiation field in Subsec-ion 3.F. Section 4 evaluates the boundary term in the re-arded solution to the wave equation governing the field,nd we conclude in Section 5.
. BOUNDARY TERM IN THE SOLUTION TOHE WAVE EQUATION
n the Lorenz gauge, the electromagnetic fields
E = − �PA0 − �A/��ctP�, B = �P A, �1�
re given by a four-potential A that satisfies the wavequation
�2A −1
c2
�2A
�t2 = −4�
cj, = 0, . . . ,3, �2�
here A0 /c and j0 /c are the electric potential and theharge density and A and j for =1,2,3 are the compo-ents of the magnetic potential A and the current density, respectively, in a Cartesian coordinate system [12]. Theolution to the initial-boundary value problem for Eq. (2)s given by
A�xP,tP� =1
c�0
tP
dt�V
d3xjG +1
4��
0
tP
dt��
dS · �G � A
− A � G� −1
4�c2�V
d3xA�G
�t− G
�A
�t t=0
,
�3�
n which G is the Green’s function and � is the surface en-losing the volume V (see, for example, p. 893 of [19]).
The potential that arises from a time-dependent local-zed source in unbounded space decays as RP
−1 when RP1, so that for an arbitrary free-space potential the sec-
nd term in Eq. (3) would be of the same order of magni-ude ��R−1� as the first term in the limit that the bound-
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546 J. Opt. Soc. Am. A/Vol. 25, No. 3 /March 2008 Ardavan et al.
ry � tends to infinity. However, even potentials thatatisfy the Lorenz condition � ·A+c−2�A0 /�t=0 are arbi-rary to within a solution of the homogeneous wave equa-ion: The gauge transformation
A → A + � , A0 → A0 − � /�t �4�
reserves the Lorenz condition if �2 −c−2�2 /�t2=0 (see12]). One can always use this gauge freedom in the choicef the potential to render the boundary contribution (theecond term) in Eq. (3) equal to zero, since this term, too,atisfies the homogenous wave equation. Under the nullnitial conditions �A�t=0= ��A /�t�t=0=0, assumed in thisaper, the contribution from the third term in Eq. (3) isdentically zero.
In the absence of boundaries, the retarded Green’sunction has the form
G�x,t;xP,tP� =��tP − t − R/c�
R, �5�
here � is the Dirac delta function and R is the magni-ude of the separation R�xP−x between the observationoint xP and the source point x. Irrespective of whetherhe radiation decays spherically or nonspherically, there-ore, the potential A due to a localized source distribu-ion, which is switched on at t=0 in an unbounded space,an be calculated from the first term in Eq. (3):
A�xP,tP� = c−1� d3xdtj�x,t���tP − t − R/c�/R, �6�
hat is, from the classical expression for the retarded po-ential. Whatever the Green’s function for the problemay be in the presence of boundaries, it would approach
hat in Eq. (5) in the limit where the boundaries tend tonfinity, so that one can also use this potential to calculatehe field on a boundary that lies at large distances fromhe source.
Next, let us consider the wave equation that governshe magnetic field
�2B −1
c2
�2B
�t2 = −4�
c� j. �7�
his may be obtained by simply taking the curl of theave equation for the vector potenial [Eq. (2) for 1,2,3]. We write the solution to the initial-boundaryalue problem for Eq. (7), in analogy with Eq. (3), as
Bk�xP,tP� =1
c�0
tP
dt�V
d3x�� j�kG
+1
4��
0
tP
dt��
dS · �G � Bk − Bk � G�
−1
4�c2�V
d3xBk
�G
�t− G
�Bk
�t t=0
, �8�
here k=1,2,3 designate the components of B and � jn a Cartesian coordinate system. Here we no longer havehe freedom, offered in the case of Eq. (3) by a gaugeransformation, to make the boundary term zero.
Our task in this paper is to demonstrate that theoundary contribution in Eq. (8) is, in fact, by a factor ofhe order of RP
1/2 larger than the source term of this equa-ion in the far zone when the source is superluminal andccelerated. For this purpose, we need to know how theradient �Bk in the second term in Eq. (8) decays in thear zone. We shall calculate, in the following section, theeld B and its gradient directly from the classical expres-ion for the retarded potential [Eq. (6)] and use the result-ng expressions to evaluate the second term in Eq. (8) for
boundary that lies in the far zone.
. GRADIENT OF THE RADIATION FIELDENERATED BY A ROTATINGUPERLUMINAL SOURCE. Background: Exact Expression for the Radiationielde base our analysis on the generic superluminal source
istribution considered in [9,10], which has already beenreated in the laboratory [2]. This source comprises a po-arization current density 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 in Eq. (9) along each circle of radius r within theource is the same as that of a sinusoidal wave train, ofavelength 2�r /m, whose m cycles fit around the circum-
erence of the circle smoothly. As time elapses, this waverain both propagates around each circle with the velocity� and oscillates in its amplitude with the frequency �.his is a generic source: One can construct any distribu-
ion with a uniformly rotating pattern, Pr,�,z�r , � ,z�, byhe superposition over m of terms of the formr,�,z�r ,z ,m�cos�m��.
To find the retarded field that follows from Eq. (6) forhe source described in Eq. (9), we first calculated in [9]he Liénard–Wiechert field of a circularly moving pointource with a speed r��c, i.e., a generalization of theynchrotron radiation to the superluminal regime. Wehen evaluated the integral representing the retardedeld (rather than the retarded potential) of the extendedource in Eq. (9) by superposing the fields generated byhe constituent volume elements of this source, i.e., by us-ng the generalization of the synchrotron field as thereen’s function for the problem (see also [17]). In the su-erluminal regime, this Green’s function has extendedingularities that arise from the coherent superposition ofhe emitted waves on the envelope of wavefronts and itsusp.
Inserting the expression for j=�P /�t from Eq. (9) intoq. (6), and changing the variables of integration from
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Ardavan et al. Vol. 25, No. 3 /March 2008/J. Opt. Soc. Am. A 547
x , t�= �r ,� ,z , t� to �r ,� ,z , ��, we found in Eq. (20) of [9]hat the magnetic field B of the generated radiation isiven by
B = −1
2i��/c�2 �
=±
�V
rdrd�dz exp�− i���j=1
3
uj�Gj/��,
�11�
here ±��� /��±m,
u1 � sr cos �Pe� + s�e�, u2 � − s� cos �Pe� + sre�,
u3 � − sz sin �Pe�, �12�
nd Gj �j=1,2,3� are the functions resulting from the re-aining integration with respect to �:
ith R�R� /c, �� is the interval of azimuthal angle tra-ersed by the source, and V is the volume occupied by theource in the �r , � ,z� space. The unit vector e� ��ezn� / �ez n� (which is parallel to the plane of rotation),
ˆ �� n e�, and the radiation direction n�R /R togetherorm an orthonormal triad (ez is the base vector associ-ted with the coordinate z). The corresponding expressionor the electric field in the limit RP��xP�→�, where n
xP/ �xP�, is given by E= nB, as in any other radiation.A distinctive feature of the emission from a superlumi-
al source is the multivaludeness of the retarded time7–10]. At any given observation time, at least three dis-inct contributions, arising from three differing retardedimes, are made toward the value of the radiation field byhe part of the source that lies within the following vol-me of the �r , � ,z� space:
his volume is bounded by a two-sheeted surface, the so-alled bifurcation surface, whose two sheets �=�±�r ,z�eet tangentially along a cusp (see Figs. 3 and 4 of [9]).he strongest contributions are made by the source ele-ents that lie close to the cusp curve �=0, ��=�±��=0,here the two sheets of the bifurcation surface meet tan-entially. For R �1, the filamentary locus of these con-
P
ributing source elements is essentially parallel to the ro-ation axis and has exceedingly narrow radial andzimuthal widths, of the orders of RP
−2 and RP−3, respec-
ively (see Fig. 2 of [10])The asymptotic values of the Green’s functions Gj close
o the cusp curve of the bifurcation surface (where ��1)re given by
Gj = � Gjin ��� � 1
Gjout ��� � 1� , �20a�
ith
Gjin � 2c1
−2�1 − �2�−1/2�pj cos� 13 arcsin ��
− c1qj sin� 23 arcsin ��� , �20b�
nd
Gjout � c1
−2��2 − 1�−1/2�pj sinh� 13 arccosh����
+ c1qj sgn���sinh� 23 arccosh����� , �20c�
here
� � 3�� − c2�/�2c13�, �21�
ith
c1 � � 34�1/3��+ − �−�1/3, c2 � 1
2 ��+ + �−�, �22�
nd the symbol � denotes asymptotic approximation. Theerivation of these asymptotic values, together with thexact expressions for the coefficients pj�r ,z� and qj�r ,z�ay be found in the appendixes of [8,9]. Here we onlyeed the following limiting values of these coefficients for
ˆP�1:
p1 � 21/3��/c�RP−2 exp�i��c/��, �23�
p2 � − RPp1, p3 � − p2, �24�
nd
q1 � 22/3��/c�RP−1 exp�i��c/��, �25�
q2 � − q3 � − i��/��q1, �26�
here �c��P+3� /2 in this limit. Note that, in these ex-ressions, Gj
in,out represent the different forms assumedy the Green’s functions Gj inside and outside the bifur-ation surface, i.e., for � inside and outside the interval�−,�+�, respectively (see Fig. 6 of [9]).
The above results show that as a source point �r , � ,z� inhe vicinity of the cusp curve �=0, ��=�±��=0, approacheshe bifurcation surface from inside, i.e., as �→1− or �→1+, Gj
in and hence Gj diverge. However, as a source pointpproaches one of the sheets of the bifurcation surfacerom outside, Gj tends to a finite limit:
�Gjout��=�±
= �Gjout��=±1 � �pj ± 2c1qj�/�3c1
2�; �27�
or the numerator of Gjout is also zero when ���=1. The
reen’s function Gj is singular, in other words, only on thenner side of the bifurcation surface (see Fig. 6 of [9]).
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548 J. Opt. Soc. Am. A/Vol. 25, No. 3 /March 2008 Ardavan et al.
. Formulation of the Problemt turns out that none of the components of the gradient of
can be evaluated for the source distribution in Eq. (9)ithout a lengthy calculation. However, we shall see inection 4 that the radial component of �Bk is of the samerder of magnitude in the far zone as the azimuthal (or,quivalently, temporal) component �Bk /��P of the gradi-nt of Bk�rP, �P,zP�. Since this component of the field gra-ient is both algebraically simpler to calculate and moreirectly related to the observeable characteristics of theenerated subbeams (Section 1), it will be the only com-onent that we shall here evaluate in detail. The relation-hip between the far-field values of this and the otheromponents of the field gradient is not difficult to estab-ish (Section 4).
The component �B /��P of the gradient of B may be cal-ulated by differentiating the right-hand side of Eq. (11)nder the integral sign and using the fact that �Gj /��P=�Gj /��. It follows from an argument identical to thativen in [9] (in connection with calculating B itself) thathe contribution ��B /��P���0 arising from the source ele-ents in ��0 toward the value of �B /��P can be written
s
��B/��P���0 = ��B/��P�in + ��B/��P�out �28�
ith
��B/��P�in,out =1
2i��/c�2�
j=1
3 ���0
rdrdzujLjin,out,
�29a�
here
Ljin = �
=±
��−
�+
d� exp�− i����2Gj/��2�in, �29b�
nd
Ljout = �
=±
�−�−�P
�−
+��+
�−�P d� exp�− i����2Gj/��2�out.
�29c�
nce it is integrated by parts, the integral in Eq. (29b) inurn splits into three terms:
Ljin = �
=±� exp�− i������Gj/���in + iGj
in���−
�+
− 3��−
�+
d� exp�− i��Gjin� , �30�
f which the first two (integrated) terms are divergentsee Eq. (20b)]. Hadamard’s finite part of Lj
in and hence of�B /��P�in, here designated by the prefix F, is obtained byiscarding this divergent contribution toward the value ofin (see [9,20]):
j
F�Ljin� = − 3 �
=±
��−
�+
d� exp�− i��Gjin. �31�
ote that the singularity of the kernel of this integral,.e., the singularity of Gj
in, is like that of ��±− ��−1/2 and sos integrable.
The boundary contributions from �=�± that resultrom the integration of the right-hand side of Eq. (29c) byarts are well-defined automatically:
Ljout = − �
=±� exp�− i������Gj/���out + iGj
out���−
�+
+ �−�−�P
�−
+��+
�−�P d�3 exp�− i��Gjout� ,
�32�
ince ��Gj /���out (like Gjout) tends to a finite limit as the
ifurcation surface is approached from outside (see Sub-ection 3.C). In deriving Eq. (32), we have made use of theact that ���Gj /���out��=�−�P
equals ���Gj /���out��=−�−�Phen �±� ±�− �P. The integral representing Lj
out, inther words, is finite by itself and needs no regulariza-ion.
If we now insert F�Ljin� and Lj
out from Eqs. (31) and32) into Eq. (29a) and combine ��B /��P�in and�B /��P�out, we arrive at an expression for the Hadamardnite part of ��B /��P���0 that entails both a volume and aurface integral:
F���B/��P���0� = ��B/��P�s + ��B/��P�ns. �33�
he volume integral
��B/��P�s = −1
2i��/c�2 �
=±
3���0
rdrdz�−�
�
d� exp�− i��
�j=i
3
ujGj �34�
as the same form as the familiar integral representationf the field of a subluminal source [12] and decays spheri-ally (as RP
−1 for RP�1).The surface integral
��B/��P�ns � −1
2i��/c�2�
j=1
3 ���0
rdrdzujLjedge �35�
tems from the boundary contribution
Ljedge � �
=±
exp�− i������Gj/���out + iGjout���−
�+
�36�
n Eq. (32). It is this contribution that turns out to in-rease, rather than decay, in the limit RP→�. To see this,e need to know the values of ��Gj /���out at �=�±, in ad-ition to those of �G out� , which are given in Eq. (27).
j �=�±
CFGafs
ww
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Ardavan et al. Vol. 25, No. 3 /March 2008/J. Opt. Soc. Am. A 549
. Azimuthal (or Temporal) Gradient of the Green’sunctionreen’s function (13) depends on � only through the vari-ble �, which appears in the argument of the Dirac deltaunction, so that the differentiation of Eq. (13) with re-pect to � simply yields
�Gj/�� = −���
d�hj������g − ��, �37�
here �� stands for the derivative of the delta functionith respect to its argument, and
h1
h2
h3� =
exp�i��/��
R cos�� − �P�
sin�� − �P�
1� . �38�
ntegrating the right-hand side of Eq. (37) by parts, webtain
�Gj
��= −�
��
d�hj���
�g/��
d
d���g − ��
=���
d�d
d�� hj���
�g/�����g − ��, �39�
hen the source trajectory intersects the bifurcation sur-ace of the observation point (i.e., the argument of theelta function vanishes within ��). A uniform asymptoticpproximation to this integral, for small �, can be foundy the method of Chester et al. in the time domain [21,22].Where it is analytic (i.e., for all x�xP), the function
��� transforms to the cubic function
g��� = 13�3 − c1
2� + c2, �40�
here � is a new variable of integration replacing � andhe coefficients c1 and c2 [defined in Eq. (22)] are suchhat the values of the two functions on opposite sides ofq. (40) coincide at their extrema. Insertion of Eq. (40)nd its derivative,
�g
��=
�2 − c12
d�/d��41�
n Eq. (39) results in
�Gj
��=�
��
d��−Fj
��2 − c12�2
+F�j
�2 − c12�
�1
3�3 − c1
2� + c2 − � , �42a�
here
Fj � d�
d�3 �2g
��2hj, �42b�
F�j � d�
d�2�hj
��, �42c�
nd �� is the image of �� under transformation (40).
As in the evaluation of Gj in [8,9], the leading term inhe asymptotic expansion of the integral (42a) for small1, which corresponds to small � [see Eq. (53) below], canow be obtained by replacing the functions F and F� in its
ntegrand with Pj+Qj� and P�j+Q�j�, respectively, and ex-ending its range �� to �−� ,��:
�Gj
����
−�
�
d��−Pj + Qj�
��2 − c12�2
+P�j + Q�j�
�2 − c12 �
�1
3�3 − c1
2� + c2 − � , �43a�
here
Pj = 12 ��Fj��=c1
+ �Fj��=−c1�, �43b�
Qj = 12c1
−1��Fj��=c1− �Fj��=−c1
�, �43c�
P�j = 12 ��F�j��=c1
+ �F�j��=−c1�, �43d�
nd
Q�j = 12c1
−1��F�j��=c1− �F�j��=−c1
�. �43e�
ote that the extrema
�± = 2� − arccos��1 � �1/2�/�rrP�� �44�
f the function g��� transform into �= �c1, respectively.The derivatives �d� /d���=±c1
that appear in the defini-ions of the coefficients �Pj ,Qj ,P�j ,Q�j� are indeterminate.heir values must be found by repeated differentiation ofqs. (15) and (40) with respect to �:
�dg/d���d�/d�� = �2 − c12, �45a�
�d2g/d�2��d�/d��2 + �dg/d���d2�/d�2� = 2�, �45b�
nd so forth, and the evaluation of the resulting relationst �= ±c1. This procedure, which amounts to applying’Hôpital’s rule, yields
d�/�d���=±c1= �2c1R��1/2/�1/4. �46�
sing ��2g /��2��±= ��1/2 /R± and Eq. (46), we find from
q. (42b) that
�Fj��=±c1= ± 2c1�fj��=±c1
, �47�
n which fj= �d� /d��hj are the functions earlier encoun-ered in the evaluation of Gj in [8,9]. Hence, Pj=2c1
2qjnd Qj=2pj, where pj and qj are precisely the same as theoefficients in Eqs. (20) that are approximated in Eqs.23)–(26) (see [9]).
We now need to evaluate �Gj /�� only outside the bifur-ation surface, i.e., for ����1 [see Eqs. (21) and (36)]. Inhis region, the argument of the � function in Eq. (43a)as a single zero at
see Appendix A of [8]). Outside the bifurcation surface,herefore, Eq. (43a) yields
Kd
iE
ns
Timst
i
b
[E
Aeitt
DoTbe
etr
E
Wdl
osifF(
=
w
Fptv
Ipidsil�
itssnaim
i
550 J. Opt. Soc. Am. A/Vol. 25, No. 3 /March 2008 Ardavan et al.
�Gj
��out
�1
��2 − c12���−
Pj + Qj�
��2 − c12�2
+P�j + Q�j�
�2 − c12 ��
�*
.
�49�
eeping only the first term in this expresssion, which isominant when c1�1, we obtain
�Gj
��out
� −2 sinh3� 1
3 arccosh����c1
5��2 − 1�3/2
�c1qj + 2pj sgn���cosh� 13 arccosh����� ,
�50�
n which pj�r ,z� and qj�r ,z� assume the values given inqs. (23)–(26) when RP�1.Evaluation of the right-hand side of Eq. (50) at �=�±
ow yields the following term that appears in the expres-ion for Lj
edge:
exp�− i�����Gj/���out��−
�+
� − � 23�3c1
−5 exp�− i��P + c2��
�pj cos� 23c1
3� − 12ic1qj sin� 2
3c13�� . �51�
he asymptotic expansions of Gjout and ��Gj /���out given
n Eqs. (20c) and (50) are for small c1. To be consistent, weust likewise replace the expression that is found by in-
erting Gjout and ��Gj /���out in Eq. (36) with the leading
erm in its expansion in powers of c1. The result is
Ljedge � − 21/3� 2
3�3RP−1pjc1
−5 exp�i��c/��
�=±
exp�− i��P + �−��, �52�
n which pj��RP−1−1+1� and �c��P+3� /2.
The far-field value of c1 close to the cusp curve of theifurcation surface (where �=0) is given by
c1 � 2−1/3RP−1�1/2 �53�
see Eq. (22)]. Inserting Eq. (52) into Eq. (35) and usingq. (53), we finally arrive at
��B/��P�ns � 22
33
RP4 exp�i���c/� − �/2�� �
=±
exp�− i�P� �j=1
3
pj���0
rdrdz�−5/2uj
exp�− i�−�. �54�
s in [10], the integral over r in this expression may bevaluated by contour integration. Since the singularity ofts integrand at �=0 is not integrable, however, the con-our integral that passes through this singularity needso be in addition regularized.
. Regularization of the Integral over the Radial Extentf the Sourcehe kernel of the integral in Eq. (54) has the same phaseut a different amplitude compared to that of the integralncountered in Eq. 19 of [10]. Hence, the asymptotic
valuation of integral (54) entails the same techniques ashose used before, but the regularization of this integralequires an extension of the procedure followed in [10].
The function �−�r , z� in the phase of the integrand inq. (54) 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.
�55�
hen the observer is located in the far zone, this one-imensional locus of stationary points coincides with theocus,
r = rS � �1 + �z − zP�2/�rP2 − 1��1/2, �56�
f source points that approach the observer with thepeed of light and zero acceleration at the retarded time,.e., with the projection �=0 of the cusp curve of the bi-urcation surface onto the �r ,z� plane (see Fig. 4 of [10]).or RP�1, the separation rC− rS vanishes as RP
−2 [see Eq.69) below] and both rC and rS assume the value csc �P.
It follows from Eq. (18) that at the stationary point rrC,
��−�r=rC� �C = RC + �C − �P, �57�
���−/�r�r=rC=0, and
�2�−/��r2�r=rC� a = − RC
−1��rP2 − 1��rC
2 − 1�−1 − 2�, �58�
here
�C = �P + 2� − arccos�rC/rP�, RC = rC�rP2 − rC
2 �1/2.
�59�
or observation points of interest to us (the observationoints located outside the plane of rotation, �P�� /2, inhe far zone, RP�1), the parameter a in Eq. (58) has aalue whose magnitude increases with increasing RP:
a � − RP sin4 �P sec2 �P. �60�
n other words, the phase function �− is more sharplyeaked at its maximum the farther the observation points from the source. This property of the phase function �−istinguishes the asymptotic analysis that will be pre-ented in this section from those commonly encounteredn radiation theory. What turns out to play the role of aarge parameter in this asymptotic expansion is distanceRP�, not frequency �±��.
The first step in the asymptotic analysis of the integraln Eq. (54) is to introduce a change of variable �=��r , z�hat replaces the original phase �− of the integrand by asimple a polynomial as possible [23]. This transformationhould be one to one and should preserve the number andature of the stationary points of the phase. Since �− hassingle isolated stationary point at r= rC�z�, it can be cast
nto a canonical form by means of the following transfor-ation:
�−�r, z� = �C�z� + 12a�z��2, �61�
n which a is the coefficient given in Eqs. (58) and (60).The integral in Eq. (54) can thus be written as
i
w
aa
w
Ttd
wtvf
awacmwF
icifml
sboadn
ETmo
agiehioa(oe
p
b(
faauuaa
Fcurtso
Ardavan et al. Vol. 25, No. 3 /March 2008/J. Opt. Soc. Am. A 551
���0
rdrdz�−5/2uj exp�− i�−� =����S
dzd�F��, z�exp�i��2�,
�62�
n which
F��, z� � r�−5/2uj��r/���exp�− i�C�, �63�
ith
�r/�� = a�rR−�r2 − 1 − �1/2�−1, �64�
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, �65�
here
�S � ��−�r=rS= 2� − arccos�1/�rSrP�� + �rS
2 rP2 − 1�1/2.
�66�
he upper limit of integration in Eq. (62) is determined byhe image of the support of the source density (s in uj) un-er transformation (61).By substituting the value of rC from Eq. (55) in Eq. (17),
e find that �1/2= r2−1 at C. Thus, the Jacobian �r /�� ofhe above transformation is indeterminate at �=0. Itsalue at this critical point must be found by repeated dif-erentiation of Eq. (61) with respect to �,
���−/�r���r/��� = a�, �67�
��2�−/�r2���r/���2 + ���−/�r���2r/��2� = a, �68�
nd the evaluation of the resulting relation (68) at r= rCith the aid of Eq. (58). This procedure, which amounts topplying l’Hôpital’s rule, yields ��r /����=0=1: A result weould have anticipated from the coincidence of transfor-ation (61) with the Taylor expansion of �− about r= rC toithin the leading order. Correspondingly, the amplitude��� in Eq. (63) has the value �rC�rC
2 −1�−5uj�r=rCexp�−i�C� at the critical point C.To an observer in the far field �RP�1�, the phase of the
ntegrand on the right-hand side of Eq. (62) is rapidly os-illating, irrespective of how low the harmonic numbers± (i.e., the radiation frequencies ±�) may be. The lead-
ng contribution to the asymptotic value of integral (62)rom the stationary point �=0 can therefore be deter-ined by the method of stationary phase. However, in the
imit RP→�, �S reduces to
�S � − 3−1/2 cos4 �P csc5 �PRP−2, �69�
o that the stationary point �=0 is separated from theoundary point �=�S by an interval of the order of RP
−2
nly. We therefore need to employ a technique for thesymptotic analysis of integral (62) that is capable of han-ling the contributions from both rC and rS simulta-eously.
. Contours of Steepest Descenthe technique we shall employ for this purpose is theethod of steepest descents [23]. We regard the variable
f integration in
J�z� � ��S
��
d�F��, z�exp�i��2� �70�
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 (61), i.e., the boundaryf the support of the source term uj that appears in themplitude F�� , z�. We shall treat only the case in which and hence �) is positive; J�z� for negative can then bebtained by taking the complex conjugate of the derivedxpression and replacing �C with −�C [see Eq. (63)].
The path of steepest descent through the stationaryoint C at which �=0 is given, according to
i�2 = − 2uv + i�u2 − v2�, �71�
y u=v when � is positive. If we designate this path by C1see Fig. 1), then
�C1
d�F��, z�exp�i��2� = �1 + i��−�
�
�dvF��=�1+i�v exp�− 2�v2�
� �2�/�1/2 exp�− i��C − �/4��
�uj�C sin7 �P�sec �P�9RP−1/2, �72�
or RP�1. Here we have obtained the leading term in thesymptotic expansion of the above integral for large RP bypproximating �F��=�1+i�v by its value at C, where v=0, andsing Eqs. (55) and (60) to replace rC and � with their val-es in the far zone. Note that the next term in thissymptotic expansion is smaller than this leading term byfactor of order RP
−1/2.
ig. 1. Integration contours in the complex plane �=u+iv. Theritical point C lies at the origin, and uS and u� are the imagesnder transformation (61) of the radial boundaries r= rS�z� and
ˆ = r��z� of the part of the source that lies within ��0. The con-ours C1, C2, and C3 are the paths of steepest descent through thetationary point C and through the lower and upper boundariesf the integration domain, respectively.
p(i
ip+
t
�
Tbm
Hi
vcimptTwg(Hs
i
p+T
ip+
t
Ti�fr
[l
t
FLTttCphttCi(al
552 J. Opt. Soc. Am. A/Vol. 25, No. 3 /March 2008 Ardavan et al.
The path of steepest descent through the boundaryoint S, at which u�uS=�S and v=0 [see Eqs. (65) and69)], is given by u=−�v2+uS
2�1/2, i.e., by the contour des-gnated as C2 in Fig. 1. The real part of
�i�2�C2= 2v�v2 + uS
2�1/2 + iuS2, �73�
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�−�2, then it follows from
���C2= − �uS
2 + i�2�1/2 �74�
hat
C2
d�F��, z�exp�i��2� = exp�i��uS2 − �/2��
�0
�
�d���uS2 + i�2�−1/2
F��=−�uS2 + i�2�1/2 exp�− ��2�. �75�
he function �F�C2in this expression has to be determined
y inverting the following version of the original transfor-ation (61):
�−�r, z� − �S�z� = 12ia�2. �76�
ere we have used Eqs. (65) and (74) to rewrite Eq. (61)n terms of �.
Since the dominant contribution toward the asymptoticalue of the above integral for RP�1 comes from the vi-inity of the boundary point S, where �=0, the requirednversion of transformation (76) can be carried out by
eans of a Taylor expansion of the phase function �− inowers of � (see Appendix A). We find in Appendix A thathe resulting expression for �F����C2
diverges at �=0 as �−4.herefore, as in the case of the integral over � in Eq. (11),e must regard the divergent integral in Eq. (75) as aeneralized function that equals its Hadamard finite partsee, for example, [17]). The procedure for finding theadamard finite part of this integral, though lengthy, is
n the limit RP�1 (see Appendix A).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. 1.
he real part of the exponent
�i�2�C3= − 2v�v2 + u�
2�1/2 + iu�2 �78�
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 �79�
hat
�C3
d�F��, z�exp�i��2� =1
2exp�i��u�
2 − �/2��
�0
�
�d��u�2 + i��−1/2
F��=�u�
2 + i��1/2 exp�− ���.
�80�
he asymptotic value of this integral for RP�1 receivests dominant contribution from �=0. Because the functionF�C3
is regular, on the other hand, its value at �=0 can beound by simply evaluating the expression in Eq. (63) atˆ = r�. The result, for RP→�, is
�F�C3,�=0 � r�2 RP
−4 sin4 �P sec2 �P�r�2 sin2 �P − 1�−3�uj�r=r�
exp�− i�C�u� �81�
see Eqs. (17) and (71)]. This in conjunction with Watson’semma therefore implies that
�C3
d�F��, z�exp�i��2� � r�2 �r�
2 sin2 �P − 1�−3�uj�r=r�
exp�− i���−�r=r�+ �/2��−1RP
−5,
�82�
o within the leading order in RP−1.
. Asymptotic Value of the Gradient of the Field forarge Distanceshe integral in Eq. (70) equals the sum of the three con-our integrals in Eqs. (72), (77), and (82); the contribu-ions of the contours that connect C1 and C2, and C1 and3, at infinity (see Fig. 1) are exponentially small com-ared to those of C1, C2, and C3 themselves. On the otherand, the leading term in the asymptotic value of the in-egral over C3 decreases (with increasing RP) much fasterhan those in the asymptotic values of the integrals over
1 and C2: The integral over C3 decays as RP−5, while the
ntegrals over C1 and C2 decay as RP−1/2. According to Eqs.
54), (62), (72), and (77), the leading term in thesymptotic expansion of the contribution ��B /��P�ns, forarge RP, is therefore given by
��B/��P�ns �1
37 �35 − 64i�RP7/2 sin7 �P�sec �P�9 exp�i��C/��
�=±
�2����1/2 sgn��expi�
4sgn
�3
pj��
dz exp�− i��C + �P���uj�C, �83�
j=1 −�
io
�s(
at
w�
Bpcoa�(ftces
4IWLlf�so
ast
si�(dtc
Ef�
w
wtt
wgp
c
p
w
wdt
Ardavan et al. Vol. 25, No. 3 /March 2008/J. Opt. Soc. Am. A 553
n which ± can also be negative (see the first paragraphf Subsection 3.E).
The remaining z integration in the above expression for�B /��P�ns amounts to a Fourier decomposition of theource densities �sr,�,z�C with respect to z. Using Eqs.57)–(59) to replace �C in Eq. (83) with its far-field value
�C � RP − z cos �P + 3�/2, �84�
nd using Eq. (12) to write out uj in terms of sr,�,z, we findhat
��B/��P�ns �1
37 �35 − 64i�RP7/2 sin7 �P�sec �P�9
exp�i��/����P + 3�/2��
�=±
�2����1/2 sgn��
exp�i��
4sgn�� − �RP + �P + 3�/2���
��s� cos �P − sz sin �P�e� − sre��, �85�
here sr,�,z stand for the following Fourier transforms ofsr,�,z�C with respect to z:
sr,�,z ��−�
�
�dzsr,�,z�r, z��r=csc �Pexp�iz cos �P�. �86�
eing the contribution from the source elements that ap-roach the observer with the speed of light and zero ac-eleration at the retarded time, this expression is validnly at those polar angles �P within the intervalrccos�1/ r��� ��P−� /2��arccos�1/ r�� for which
sr,�,z�r=csc �Pis nonzero, i.e., at those observation points
outside the plane of rotation) the cusp curve of whose bi-urcation surface intersects the source distribution. Athese polar angles, the above expression for ��B /��P�ns
onstitutes the dominant contribution toward the gradi-nt �B /��P of the magnetic field of the radiation (see Sub-ection 3.B).
. EVALUATION OF THE BOUNDARY TERMN THE RETARDED SOLUTION TO THE
AVE EQUATION GOVERNING THE FIELDet the boundary � in the second term of Eq. (8) be a
arge sphere enclosing the source. The element dS of areaor this boundary then has the form �2 sin �d�d�e�, where� ,� ,�� are the spherical polar coordinates in the space ofource points, i.e., are related to the cylindrical polar co-rdinates �r ,� ,z� we have been using by
� � �r2 + z2�1/2, � � arctan�r/z�, �87�
nd e� is a unit vector in the direction of increasing �. In-erting this into the integrand of the boundary contribu-ion in Eq. (8), we obtain
Bboundary = �2� dt��
d�d� sin ���G�B/�RP�RP=� − B�G/���,
�88�
ince �e� ·��B=�B /��= ��B /�RP�RP=�. We will be identify-
ng the magnetic field B and its gradient on the boundarywith those of the radiation field that arises from source
9). The terms �B / ��RP�� and �B�� in Eq. (88), which act asensities of two-dimensional sources, both have rigidly ro-ating distribution patterns, i.e., are functions of t in theombination �=�−�t only [see Eq. (85)].
Once the free-space Green’s function (5) is inserted inq. (88), we can therefore cast this equation in the same
orm as Eq. (11) by changing the integration variable t toˆ ; this results in
Bboundary = �2��
d�d� sin ���Gb�B/�RP�RP=� − B�Gb/���,
�89�
ith
Gb �� d�R−1��g − ��, �90�
here �=�� /c and g and � are the same functions ashose appearing in Eqs. (13)–(15). Equation (90) implieshat
Gb = ��=�j
1
R��g/���= �
�=�j
�R + �RP sin � sin �P sin��j − �P��−1,
�91�
here �j are the solutions of the transcendental equation���=� [see Eq. (15)]. For ��1, the number of retardedositions �j of the rapidly rotating distribution patterns of
�B�� and �B / ��RP�� that contribute toward the value of Gban be appreciably larger than three (see [7]).
The expression in Eq. (11) for the magnetic field B de-ends on RP through �Gj /�� only, so that
�B/�RP = −1
2i��/c�2 �
=±
�V
rdrd�dz exp�− i��
�j=1
3
uj�2Gj/�RP��, �92�
ith
�2Gj/�RP�� = − ��/c����
d�hj���
R−1����g − �� − R−1���g − ����R/�RP,
�93�
here hj are the functions defined in Eq. (38) and a primeenotes differentiation of the delta function with respecto its argument [see Eq. (37)].
sowstf
[rwtm�
t3
g�aduoea
fstd�ti
Gte
Twdoo
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toTtwbHp
mfts
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554 J. Opt. Soc. Am. A/Vol. 25, No. 3 /March 2008 Ardavan et al.
It follows from a comparison of the calculations de-cribed in Subsections 3.A and 3.C that the contributionf the second term on the right-hand side of Eq. (93) to-ard the value of �B /�RP is by a factor of the order of RP
−4
maller than that of the first term. Ignoring this smallerm, we obtain an expression for −�2Gj /�RP�� that dif-ers from the expression for �2Gj /��2 only by the factor of
�R/�RP = R−1�RP − z cos �P − r sin �P cos�� − �P�� �94�
see Eq. (14)], which reduces to 1 in the limit RP�1. Cor-espondingly, the leading contribution, ��B /�RP�ns, to-ard the value of �B /�RP is given by an expression iden-
ical to that in Eq. (85) for ��B /��P�ns, except that it isultiplied by −1 (see Section 3). The absolute value of
B /�RP is therefore of the same order of magnitude ashat of �B /��P and so increases as RP
7/2 (see Subsection.F).We already know that the radiation subbeams that are
enerated by the superluminal source (9) have the widths�P�RP
−1 and ��P�RP−3 (see [10]), so that the extents ��
nd �� of the source distribution in Eq. (89) are of the or-ers �−1 and �−3, respectively. In the limit where the val-es of � and RP (i.e., the positions of the boundary and thebserver, respectively) tend to infinity independently ofach other, the value of Green’s function Gb on the bound-ry reduces to
�Gb�� � ��=�j
��RP sin � sin �P�sin��j − �P���−1; �95�
or �RP is much greater than the separation R of the ob-ervation point and the source points in this limit. Hence,he absolute values of �Gb�� and �Gb /�� �� diminish withistance as ��RP�−1 and �−2RP
−1, respectively. Since�B /�RP� increases as RP
7/2 while �B� decreases as RP−1/2,
his means that, of the two terms inside the paranthesesn Eq. (89), the second is negligibly smaller than the first.
Inserting the orders of magnitude of the factors ��, ��,
b, and ��B /�RP�RP=�into Eq. (89), in the order in which
hey appear in the first term on the right-hand side of thisquation, we obtain
�Bboundary� � �2 �−3 �−1 ��RP�−1 �7/2 � �1/2RP−1.
�96�
hus, the absolute value of the boundary contribution to-ard the value of the field decays as RP
−1/2 when the ra-ius � of the spherical boundary � and the coordinate RPf the observation point P are both large and of the samerder of magnitude.
. CONCLUDING REMARKShe unaviodably lengthy calculation that we have pre-ented in both Sections 3 and 4 lends support to the con-lusions of [10] on the morphology of the radiation beamhat is generated by a polarization current with a super-uminally rotating distribution pattern and clarifies aundamental issue concerning the method of calculating
he radiation field of such a polarization current, whichas been the source of a long-standing controversy13–17]. This calculation establishes
(i) that the absolute values of both the radial compo-ent �B /�RP and the azimuthal or temporal componentB /��P of the gradient of the radiation field that is gen-rated by the superluminal source distribution (9) in thear zone are of the order of RP
7/2 at any observation pointithin the overall radiation beam arising from this source
24],(ii) that the angular distribution of the emitted field
ontains sharply focused structures, i.e., that the overalladiation beam is composed of an incoherent superposi-ion of rapidly narrowing subbeams [11],
(iii) that the boundary contribution toward the solutionf the wave equation governing the field decays as RP
−1/2
s the boundary tends to infinity, i.e., that the seconderm in Eq. (8) is by a factor of the order of RP
1/2 greaterhan the first term in this equation for a � that lies inˆ
P�1,(iv) that the discrepancy between the predictions of
qs. (2) and (7) disappears once one includes the bound-ry term that is normally neglected in solution (8), and(v) that Hannay’s erroneous contention that the field ofrotating superluminal source should diminish as RP
−1,s does a conventional radiation field [13–16], stems fromis having neglected the boundary term in the solutionEq. (8)] to the wave equation governing the field [Eq. (7)].
The sharply focused radiation pulses encountered inhe present analysis are in fact observed in astronomicalbjects that are thought to contain superluminal sources.he radio emission received from pulsars is composed (of-
en entirely [25]) of a collection of so-called giant pulseshose widths are as narrow as 1 ns [26] and whoserightness temperatures are as high as 1039 K [27].ankins et al. [26] note the puzzling brightness of theseulses:The plasma structures responsible for these emissionsust be smaller than one meter in size, making them by
ar the smallest objects ever detected and resolved outsidehe Solar System, and the brightest transient radioources in the sky.
The small size of the emitting structures reflects, in theresent context, the narrowing (as RP
−2 and RP−3, respec-
ively) of the radial and azimuthal dimensions of the fila-entary part of the source that approaches the observer
t P with the speed of light and zero acceleration at theetarded time [10]. This, together with the nonsphericalecay of the individual subbeams generated by such fila-ents (as RP
−1/2 instead of RP−1), easily accounts for the
bservationally inferred values of the brightness tem-erature of the giant pulses.The azimuthal (or temporal) gradient of the intensity of
hese pulses often appears infinitely sharp at either theireading or trailing edges (see Fig. 1 of [26]). Correspond-ngly, the emission mechanism discussed in this paperets no upper limit on the gradient � /��P of the radiationeld (i.e., on the sharpness of the leading or trailing edgef the pulse), if the length scale of spatial or temporalariations of its source are comparable with R−3. Accord-
P
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ayrc
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a
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Tt
Ardavan et al. Vol. 25, No. 3 /March 2008/J. Opt. Soc. Am. A 555
ng to the superluminal model of pulsars [10,28,29] (tohich the present findings apply), the more distant a pul-
ar is, the narrower and brighter its giant pulses shoulde.
PPENDIX A: HADAMARD’S FINITE PARTF THE DIVERGENT CONTRIBUTIONROM THE INTEGRAL OVER THEONTOUR C2
ur first task in this appendix is to make the dependencef the integrand of integral (75) on the integration vari-ble � explicit. This entails (i) inverting Eq. (76) in the vi-inity of the critical point �=0 to obtain r as a function offor a fixed value of z and (ii) expanding the function
�F�C2that appears in the integrand of integral (75) in
owers of �. Next, we calculate the Hadamard finite partf the resulting integral (whose integrand turns out to di-erge as �−4 at �=0) by following the standard proceduresed in the literature on generalized functions [20].Because it contains the factor
�1/2 = �rP2 − 1�1/2�r2 − rS
2�1/2, �A1�
he function �−�r , z� is not analytic at r= rS [see Eqs. (17),18), and (56)]. If, however, we eliminate r in �− in favor of
� � �r2 − rS2�1/2, �A2�
hen the resulting function
�−��, z� = �rP2�rS
2 + �2� − �1 + �rP2 − 1�1/2��2�1/2 + 2�
− arccos�rP−1�rS
2 + �2�−1/2�1 + �rP2 − 1�1/2���
�A3�
an be expanded into a Taylor series about �=0 to obtain
�− = �S +1
2RP
−1 cos2 �P�2 −1
3sin3 �P�3
+1
8RP
−3 cos2 �P�5 sin2 �P − 1��4 +1
5sin5 �P�5 + ¯ .
�A4�
ere the coefficients in this series are approximated byheir dominant values for RP�1, and the coordinate rShat appears in them is replaced with its value csc �P athe radius from which the main contributions toward theeld in the far zone arise [see Eq. (56)]. Equation (A4), inonjunction with Eq. (76), provides us with an analytic ex-ression for ���� that we can invert to find � (and hence r)s a function of �.Repeated differentiations of Eq. (76) with respect to �
nd so on, which when evaluated at S (where �=�=0),ield ��� /���S, ��2� /��2�S, etc., in terms of the known de-ivatives ���−/���S, ��2�−/��2�S, etc., that constitute theoefficients in Eq. (A4). Using these derivatives of � at S,
e can therefore write down the Taylor expansion of � inowers of �:
� = � + �2 +5
2�3 + 8�4 +
231
8�5 + ¯ , �A6a�
here
� �1
3RP sin3 �P sec2 �P�, �A6b�
nd
� �1
3exp�− i�/4�RP
2 sin5 �P sec4 �P�. �A6c�
he dependence of r on � now follows from Eqs. (A2) andA6).
According to Eqs. (63), (64), (74), (A1), and (A2), the ex-licit form of the function �F�C2
is given by
�F��=−�uS2 + i�2�1/2 �
1
3RP
−5 csc5 �Puj�−1��S
2 + i�2�1/2
exp�− i��C − �/2�����−3�/����C2
�A7�
or RP�1. Insertion of this expression into Eq. (75) yields
�C2
d�F��, z�exp�i��2� =1
3RP
−8 cot6 �P csc5 �P
exp�− i��S + �/4��
uj��3I1 − I2��C2, �A8�
n which
I1 ��0
�
d��−4 exp�− ��2�����, �A9�
nd
I2 ��0
�
d��−3 exp�− ��2��d�/d��, �A10�
ith ���� / ��−3. Here we have used the fact that �C�uS
2 =�S, where �S stands for the value of �− at S.That the integrals I1 and I2 have turned out to diverge
s a consequence of our having interchanged the orders ofntegration and differentiation in Eq. (37) [see also Eq.13)]. The standard technique for regularizing such diver-ent integrals is to treat them as generalized functionshose physically significant values (i.e., the values thate would have found had we not interchanged the ordersf integration and differentiation) are given by their Had-mard finite parts [20].To apply the technique to I1, one begins by appealing to
aylor’s theorem to represent the continuously differen-iable function � as
ws
Tee
it
E
blt
Htt
tEi�
apE
wdr
Tf
w
ws
a
AHcGdFo
R
556 J. Opt. Soc. Am. A/Vol. 25, No. 3 /March 2008 Ardavan et al.
���� = ��0� + ���0�� +1
2���0��2 +
1
3!���0��3 +
1
4!�������4,
�A11�
here � is a number lying between 0 and 1. One then in-erts Eq. (A11) into Eq. (A9) to rewrite I1 as
I1 = lim�→0���0��
�
�
d��−4 exp�− ��2� + ���0�
��
�
d��−3 exp�− ��2� +1
2���0��
�
�
d��−2 exp�− ��2�
+1
3!���0��
�
�
d��−1 exp�− ��2�
+1
4!��
�
d�������exp�− ��2�� . �A12�
he first four integrals inside the square brackets in thisxpression can be easily evaluated as functions of �� ,��;.g.,
��
�
d��−4 exp�− ��2� =1
3�−3�1 − 2��2�exp�− ��2�
+2
3�1/2�3/2 erfc��1/2��, �A13�
n which the error function erfc��1/2�� approaches unity inhe limit �→0.
The remaining fifth integral on the right-hand side ofq. (A12) equals
��
�
d�������exp�− ��2� = 4!��
�
d��−4����� − ��0� − ���0��
−1
2���0��2 −
1
3!���0��3�
exp�− ��2� �A14�
y virtue of Eq. (A11). For ��1 (i.e., RP�1) and �=0, theeading term in the asymptotic value of the right-hand in-egral in Eq. (A14) is given by
�0
�
d�������exp�− ��2� � ���0��0
�
d� exp�− ��2�
=1
2��/��1/2���0�. �A15�
ere we have applied l’Hôpital’s rule to remove the inde-erminacy in the value of the kernel of the right-hand in-egral in Eq. (14) at �=0.
Hadamard’s finite part of the limiting version of each ofhe integrals that appear inside the square brackets inq. (A12) is obtained by simply discarding those terms in
ts representation as a function of �� ,�� that diverge whentends to zero; e.g.,
F��0
�
d��−4 exp�− ��2�� =2
3�1/2�3/2 �A16�
ccording to Eq. (A13). Thus, Eq. (A15) and the finitearts of the divergent integrals on the right-hand side ofq. (A12) jointly yield
F�I1� �2
3�1/2��0��3/2 +
1
2���0��ln � + ��� −
1
2�1/2���0��1/2
−1
12���0��ln � + �� +
1
48�1/2���0��−1/2, � � 1,
�A17�
here �=0.57721 is Euler’s constant. The same proce-ure, when applied to the integral defined in Eq. (A10),esults in
F�I2� �1
2���0��ln � + ��� − �1/2���0��1/2 −
1
4���0��ln � + ��
+1
12�1/2���0��−1/2, � � 1. �A18�
he required derivatives of � at �=0 can be read off theollowing expansion of �� / ��−3:
� = 1 − 3� −3
2�2 − 4�3 −
105
8�4 + ¯ , �A19�
hich follows from Eq. (A6a) [see also Eq. (A6c)].Evaluating the right-hand sides of Eqs. (A17) and (A18)
ith the aid of Eqs. (A6c) and (A19), and inserting the re-ulting expressions into Eq. (A8), we finally arrive at
CKNOWLEDGMENTS. Ardavan thanks Boris Bolotovskii for his help and en-
ouragement, and Alexander Schekochihin and Januszil for their stimulating questions and comments. A. Ar-avan is supported by the Royal Society. J. Singleton, J.asel, and A. Schmidt are supported by U.S. Departmentf Energy grant LDRD 20050540ER.
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