# Supplementary Information for “Radiation by the superluminally moving
current sheet in the magnetosphere of a neutron star”

Houshang Ardavan
Institute of Astronomy, University of Cambridge,
Madingley Road, Cambridge CB3 0HA, United Kingdom
<EMAIL_ADDRESS>
###### Contents

  1. 1 Semi-analytic description of the magnetosphere
  2. 2 Formulation of the problem
    1. 2.1 The free-space solution of Maxwell’s equations that satisfies the required boundary conditions at infinity
    2. 2.2 The Green’s function for the problem and its loci of singularities
    3. 2.3 Bifurcation surface of an observation point
    4. 2.4 A uniform asymptotic approximation to the value of the Green’s function near the cusp locus of the bifurcation surface
    5. 2.5 Hadamard’s finite part of the divergent integral representing the field
  3. 3 Radiation field of the current sheet
    1. 3.1 The contribution from discontinuities of the Green’s function
    2. 3.2 Evaluation of the asymptotic values of the integrals over $\eta$ by the method of stationary phase
    3. 3.3 Dominance of the contribution from large values of $|k|$
    4. 3.4 Critical points of the phase functions $f_{lC}$ and ${\bar{f}}_{lC}$
    5. 3.5 A uniform asymptotic approximation to the integral over the colatitude of the source elements for large $|k|$
    6. 3.6 The remaining integration with respect to $k$
    7. 3.7 The divergence that arises from the vanishing width of the current sheet and its regularization
  4. 4 Characteristics of the resulting radiation
    1. 4.1 Pulse profiles and polarization position angles
    2. 4.2 Brightness temperature
    3. 4.3 Frequency spectrum
    4. 4.4 Flux density and its rate of decay with distance
  5. A Derivatives of the phase functions $f_{lC}$ and ${\bar{f}}_{lC}$

##  1 Semi-analytic description of the magnetosphere

In a reference frame marked by the spherical polar coordinates
$(r_{s},\theta,\varphi)$ for which $\theta=0$ coincides with the axis of
rotation, the distributions of the electric and magnetic fields in the
magnetosphere of an oblique rotator are, according to [1], given by

${\bf E}=-\frac{r_{s}\omega}{c}\sin\theta B_{r_{s}}{\hat{\bf e}}_{\theta},$
(1.1) $B_{r_{s}}=w_{3}\left(w_{1}B_{r_{s}}^{(1)}+w_{2}B_{r_{s}}^{(2)}\right)$
(1.2) $B_{\theta}=0,\qquad B_{\varphi}=-\frac{r_{s}\omega}{c}\sin\theta
B_{r_{s}},$ (1.3)

with

$B_{r_{s}}^{(1)}=B_{0}\left(\frac{r_{s0}}{r_{s}}\right)^{2}\left[1+0.02(1-{\cal
C}^{2})^{1/2}+0.22(|{\cal C}|-1)-0.07(|{\cal C}|-1)^{4}\right]{\rm sgn}({\cal
C}),$ (1.4)
$B_{r_{s}}^{(2)}=B_{0}\left(\frac{r_{s0}}{r_{s}}\right)^{2}\sin\theta\cos\left(\varphi-\omega
t+{\hat{r}}_{s}-{\hat{r}}_{s0}-\pi/6\right),$ (1.5) ${\cal
C}=\sin\alpha\sin\theta\cos\left(\varphi-\omega
t+{\hat{r}}_{s}-{\hat{r}}_{s0}\right)+\cos\alpha\cos\theta,$ (1.6)
$w_{1}=\left|1-2\alpha/\pi\right|,$ (1.7) $w_{2}=1+0.17|\sin(2\alpha)|-w_{1},$
(1.8) $w_{3}=1+0.2\sin^{2}\alpha,$ (1.9)
${\hat{r}}_{s}=r_{s}\omega/c,\qquad{\hat{r}}_{s0}=r_{s0}\omega/c,$ $None$

where $t$ ($\geq 0$) is time, $\omega$ is the angular frequency of rotation of
the star, $c$ is the speed of light in vacuum, $\alpha$ is the angle between
the rotation and magnetic axes of the star (henceforth referred to as the
inclination angle), $r_{s0}$ is the radius of the star and $B_{0}$ is the
magnitude of the star’s dipolar field at its magnetic pole. The caret on
$r_{s}$ and $r_{s0}$ (and any other variable with the dimension of a length)
is used in this paper to designate a variable that is rendered dimensionless
by being measured in units of the light-cylinder radius $c/\omega$.

Inserting these fields in Maxwell’s equations, we obtain the corresponding
distributions of the charge density $\rho$ and the current density ${\bf j}$
in the magnetosphere:

$\rho=\frac{1}{4\pi}{\bf\nabla\cdot E}=-\frac{\omega}{4\pi c}\left(2\cos\theta
B_{r_{s}}+\sin\theta\frac{\partial B_{r_{s}}}{\partial\theta}\right),$ (1.11)
${\bf j}=\frac{c}{4\pi}\left({\bf\nabla\times B}-\frac{1}{c}\frac{\partial{\bf
E}}{\partial t}\right)=\rho c\,{\hat{\bf e}}_{r_{s}}+\frac{c}{4\pi
r_{s}}\left(\frac{1}{\sin\theta}\frac{\partial
B_{r_{s}}}{\partial\varphi}{\hat{\bf e}}_{\theta}-\frac{\partial
B_{r_{s}}}{\partial\theta}{\hat{\bf e}}_{\varphi}\right),$ (1.12)

in which

$\frac{\partial
B_{r_{s}}^{(1)}}{\partial\theta}=\left[\sin\alpha\cos\theta\cos\left(\varphi-\omega
t+{\hat{r}}_{s}-{\hat{r}}_{s0}\right)-\cos\alpha\sin\theta\right]\frac{\partial
B_{r_{s}}^{(1)}}{\partial{\cal C}},$ (1.13) $\frac{\partial
B_{r_{s}}^{(1)}}{\partial\varphi}=-\sin\alpha\sin\theta\sin\left(\varphi-\omega
t+{\hat{r}}_{s}-{\hat{r}}_{s0}\right)\frac{\partial
B_{r_{s}}^{(1)}}{\partial{\cal C}},$ (1.14) $\frac{\partial
B_{r_{s}}^{(2)}}{\partial\theta}=\cot\theta B_{r_{s}}^{(2)},$ (1.15)
$\frac{\partial B_{r_{s}}^{(2)}}{\partial\varphi}=-\tan\left(\varphi-\omega
t+{\hat{r}}_{s}-{\hat{r}}_{s0}-\pi/6\right)B_{r_{s}}^{(2)},$ (1.16)
$\frac{\partial B_{r_{s}}^{(1)}}{\partial{\cal
C}}=B_{0}\left(\frac{r_{s0}}{r_{s}}\right)^{2}\left[-\frac{0.02\,|{\cal
C}|}{(1-{\cal C}^{2})^{1/2}}+0.22-0.28(|{\cal C}|-1)^{3}+1.46\,\delta({\cal
C})\right],$ (1.17)

the base vectors $({\hat{\bf e}}_{r_{s}},{\hat{\bf e}}_{\theta},{\hat{\bf
e}}_{\varphi})$ are those of the spherical coordinate system
$(r_{s},\theta,\varphi)$ and $\delta$ is the Dirac delta function. In deriving
(1.12), we have made use of the fact that here ${\bf E}$ depends on $\varphi$
and $t$ only in the combination $\varphi-\omega t$ and $r_{s}^{2}B_{r_{s}}$
depends on $\varphi$ and ${\hat{r}}_{s}$ only in the combination
$\varphi+{\hat{r}}_{s}$, i.e., that $\partial{\bf E}/\partial
t=-\omega\partial{\bf E}/\partial\varphi$ and
$\partial(r_{s}^{2}B_{r_{s}})/\partial
r_{s}=(\omega/c)\partial(r_{s}^{2}B_{r_{s}})/\partial\varphi$.

The magnetospheric current sheet shown in figure 1.1 occurs where the argument
of the delta function in (1.12), i.e., the function ${\cal C}$ defined by
(1.6), vanishes. Since the radiation whose frequency appreciably exceeds the
rotation frequency $\omega/2\pi$ is due entirely to this current sheet, we
base our analysis of the radiation field in the following sections only on
those terms in the above expressions for $\rho$ and ${\bf j}$ that involve the
Dirac delta function. We will see in § 4 that the radiation generated by this
sheet is in addition considerably more intense than that generated by the rest
of the magnetosphere. Disregarding the other terms in (1.11) and (1.12), we
obtain

$\rho^{(cs)}=-\frac{j_{0}}{c{\hat{r}}_{s}^{2}}\sin\theta\,h\,\delta({\cal
C}),$ (1.18)

and

${\bf j}^{(cs)}=\rho^{(cs)}c\,{\hat{\bf
e}}_{r_{s}}-\frac{j_{0}}{{\hat{r}}_{s}^{3}}\big{[}\sin\alpha\sin\left({\hat{\varphi}}+{\hat{r}}_{s}-{\hat{r}}_{s0}\right){\hat{\bf
e}}_{\theta}+h\,{\hat{\bf e}}_{\varphi}\big{]}\delta({\cal C}),$ (1.19)

for this sheet’s charge and current densities, where

${\hat{\varphi}}=\varphi-\omega t,$ (1.20)
$h=\sin\alpha\cos\theta\cos\left({\hat{\varphi}}+{\hat{r}}_{s}-{\hat{r}}_{s0}\right)-\cos\alpha\sin\theta,$
(1.21)

and

$j_{0}=0.365\omega B_{0}{\hat{r}}_{s0}^{2}w_{1}w_{3}/\pi.$ (1.22)

When the inclination angle $\alpha$ lies in the interval $[0,\pi/2]$, the
above source densities are non-zero only in

$\frac{\pi}{2}-\alpha\leq\theta\leq\frac{\pi}{2}+\alpha,$ (1.23)

as can be seen from the argument of the delta function that appears in (1.18)
and (1.19). It turns out that the symmetry of the current sheet with respect
to $\theta$ and $\alpha$ enables us to infer the results for
$\pi/2<\alpha<\pi$ from those for $0<\alpha<\pi/2$ (see § 3.4).

Figure 1.1: A single turn of the current sheet described by ${\cal C}=0$ for
$\alpha=\pi/3$ (see (1.6)). At the same time as propagating radially outward
with the speed of light in vacuum, $c$, the surface shown here rotates
uniformly with the angular frequency of rotation of the central neutron star,
i.e., with a linear speed $r\omega$ that exceeds $c$. The superluminal motion
of this current sheet is created by the coordinated movements of aggregates of
subluminally moving charged particles.

For the purposes of quoting the results derived in [2], it is convenient to
express the base vectors of the spherical polar coordinates used above in
terms of the base vectors $({\hat{\bf e}}_{r},{\hat{\bf
e}}_{\varphi},{\hat{\bf e}}_{z})$ of the cylindrical polar coordinates
$(r,\varphi,z)$ defined by

$r=r_{s}\sin\theta,\qquad z=r_{s}\cos\theta,$ $None$

i.e., to replace $({\hat{\bf e}}_{r_{s}},{\hat{\bf e}}_{\theta},{\hat{\bf
e}}_{\varphi})$ via

${\hat{\bf e}}_{r_{s}}=\sin\theta\,{\hat{\bf e}}_{r}+\cos\theta\,{\hat{\bf
e}}_{z},\qquad{\hat{\bf e}}_{\theta}=\cos\theta\,{\hat{\bf
e}}_{r}-\sin\theta\,{\hat{\bf e}}_{z}.$ $None$

Equations (1.18) and (1.19) can then be written as

$\rho^{(cs)}=-\frac{j_{0}}{{\hat{r}}_{s}^{3}}\,{\tilde{\rho}}\,\delta({\cal
C}),\qquad{\bf j}^{(cs)}=-\frac{j_{0}}{{\hat{r}}_{s}^{3}}\,{\tilde{\bf
j}}\,\delta({\cal C}),$ $None$

in which

${\tilde{\rho}}c={\hat{r}}_{s}\sin\theta\,h,$ (1.27)
${\tilde{j}}_{r}={\hat{r}}_{s}\sin^{2}\theta\,h+\sin\alpha\cos\theta\sin({\hat{\varphi}}+{\hat{r}}_{s}-{\hat{r}}_{s0}),$
(1.28) ${\tilde{j}}_{\varphi}=h,$ (1.29)
${\tilde{j}}_{z}={\hat{r}}_{s}\sin\theta\cos\theta\,h-\sin\alpha\sin\theta\sin({\hat{\varphi}}+{\hat{r}}_{s}-{\hat{r}}_{s0}),$
(1.30)

describe the dimensionless charge density ${\tilde{\rho}}$ and the cylindrical
components of the dimensionless current density ${\tilde{\bf j}}$ of the
magnetospheric current sheet as functions of
$({\hat{r}}_{s},\theta,{\hat{\varphi}})$ (see ($None$) and (1.20)).

##  2 Formulation of the problem

###  2.1 The free-space solution of Maxwell’s equations that satisfies the
required boundary conditions at infinity

To satisfy the required boundary conditions at infinity the free-space
radiation field of an accelerated superluminal source has to be calculated (in
the Lorenz gauge) by means of the retarded solution of the wave equation for
the electromagnetic potential. There is a fundamental difference between the
classical expression for the retarded potential and the corresponding retarded
solution of the wave equation that governs the electromagnetic field. While
the boundary contribution to the retarded solution of the wave equation for
the potential that appears in Kirchhoff’s surface-integral representation can
always be rendered equal to zero by means of a gauge transformation that
preserves the Lorenz condition, the corresponding boundary contribution to the
retarded solution of the wave equation (or any other equation) for the field
cannot be assumed to be zero a priori. Not to exclude emissions whose
intensity could decay more slowly than predicted by the inverse-square law, it
is essential that the radiation field is derived from the retarded potential
(see § 3 of [2] where this point is expounded).

Accordingly, we base the analysis in this paper on the classical expression

$\left[\matrix{{\mathbf{A}}\cr{\Phi}\cr}\right]=\frac{1}{c}\int{\rm d}^{3}{\bf
x}\,{\rm
d}t\,\left[\matrix{{\mathbf{j}^{(cs)}}\cr{\rho^{(cs)}c}\cr}\right]\frac{\delta(t-t_{P}+R/c)}{R}.$
(2.1)

for the retarded potentials that arise from the charge and current densities
described by ($None$) and insert this in

${\mathbf{E}}=-\nabla_{P}\Phi-\frac{1}{c}\frac{\partial{\mathbf{A}}}{\partial
t_{P}},\quad\quad{\mathbf{B}}=\nabla_{P}{\mathbf{\times A}},$ (2.2)

to obtain the corresponding expression for the generated fields:

$\left[\matrix{{\mathbf{E}}\cr{\mathbf{B}}\cr}\right]=\frac{1}{c^{2}}\int{\rm
d}^{3}{\bf x}\,{\rm
d}t\,\frac{\delta^{\prime}(t-t_{P}+R/c)}{R}\left[\matrix{{\mathbf{j}}^{(cs)}-\rho^{(cs)}c\,{\hat{\mathbf{n}}}\cr{\hat{\mathbf{n}}}{\mathbf{\times}}{\mathbf{j}}^{(cs)}\cr}\right],$
(2.3)

where $({\bf x},t)$ and $({\bf x}_{P},t_{P})$ are the space-time coordinates
of the source points and the observation point $P$, respectively, $R$ is the
magnitude of the separation ${\bf R}={\bf x}_{P}-{\bf x}$, ${\hat{\bf n}}={\bf
R}/R$ is a unit vector along the radiation direction and $\delta^{\prime}$
denotes the derivative of the Dirac delta function with respect to its
argument (see, e.g., [3]). Since the problem we will be analysing entails the
formation of caustics we cannot proceed to the far-field limit $|{\bf
x}_{P}|\to\infty$ before evaluating the above integral, as is customarily done
in radiation theory. The far-field approximation of the argument of the delta
function in (2.3) would replace spherical wave fronts by planar wave fronts
thereby relinquishing the possibility of their constructive interference. As
we will be applying our results to astronomical objects we can, however,
approximate the unit vector ${\hat{\bf n}}$ by its far-field value

${\hat{\bf n}}_{\infty}=\sin\theta_{P}{\hat{\bf
e}}_{r_{P}}+\cos\theta_{P}{\hat{\bf e}}_{z_{P}}$ (2.4)

at this stage, so that the magnetic field can be obtained from ${\bf
B}={\hat{\bf n}}_{\infty}{\bf\times}{\bf E}$ once the electric field is known.
In (2.4), $({\hat{\bf e}}_{r_{P}},{\hat{\bf e}}_{\varphi_{P}},{\hat{\bf
e}}_{z_{P}})$ are the cylindrical base vectors at the observation point $P$.

For the purposes of the present analysis, it is essential that the finiteness
of the duration of the source is taken into account (see appendix B of [2]).
If $\rho^{(cs)}$ and ${\bf j}^{(cs)}$ are turned on at $t=0$, then the
coordinates $t$ and $\varphi$ in ($None$) both range over $(0,\infty)$ but the
values of the combination ${\hat{\varphi}}=\varphi-\omega t$ in which they
occur has a limited range of length $2\pi$, e.g.,

$0\leq{\hat{\varphi}}<2\pi.$ (2.5)

As can be seen from the alternative form $\varphi={\hat{\varphi}}+\omega t$ of
(1.20), ${\hat{\varphi}}$ is a Lagrangian coordinate that labels the rotating
volume elements of the current distribution on each circle $r=$const,
$z=$const, by their azimuthal positions at the time $t=0$. This coordinate
cannot range over a wider interval because the aggregate of volume elements
that constitute a rotating source in its entirety can at most occupy an
azimuthal interval of length $2\pi$ at any given time (e.g., at $t=0$). In
this section, we mark the spatial coordinates of the source points by
cylindrical polar coordinates and eliminate $t$ in favour of
${\hat{\varphi}}$.

Thus changing the variables of integration in (2.3) to $({\bf
x},t)=(r,\varphi,z,{\hat{\varphi}})$ and introducing the dimensionless
coordinates ${\hat{r}}=r\omega/c$ and ${\hat{z}}=z\omega/c$, we obtain

${\bf
E}=\frac{1}{\omega}\sum_{m=1}^{\infty}\int{\hat{r}}{\textrm{d}}{\hat{r}}\,{\textrm{d}}{\hat{\varphi}}\,{\textrm{d}}{\hat{z}}\,\int_{{\hat{\varphi}}+2(m-1)\pi}^{{\hat{\varphi}}+2m\pi}{\textrm{d}}{\varphi}\,\frac{\delta^{\prime}(g-\phi)}{{\hat{R}}}({\mathbf{j}}^{(cs)}-\rho^{(cs)}c\,{\hat{\bf
n}}_{\infty}),$ (2.6)

where

${\hat{R}}=[({\hat{z}}-{\hat{z}}_{P})^{2}+{{\hat{r}}_{P}}^{2}+{\hat{r}}^{2}-2{\hat{r}}_{P}{\hat{r}}\cos(\varphi-\varphi_{P})]^{1/2},$
(2.7)

the function
$g({\hat{r}},\varphi,{\hat{z}};{\hat{r}}_{P},\varphi_{P},{\hat{z}}_{P})$ is
defined by

$g\equiv\varphi-\varphi_{P}+{\hat{R}},$ (2.8)

and the variable $\phi$ in the argument of the delta function stands for

$\phi\equiv{\hat{\varphi}}-{\hat{\varphi}}_{P}\qquad{\rm
with}\qquad{\hat{\varphi}}_{P}\equiv\varphi_{P}-\omega t_{P}.$ (2.9)

We have expressed the range of $\varphi$ integration as a sum of the intervals
of length $2\pi$ that the element initially located at ${\hat{\varphi}}$
traverses during each of its individual rotations: $m$ is a positive integer
enumerating successive rotation periods (the first rotation period being
designated by $m=1$) and the summation extends over the set of rotations
executed by the source over its lifetime.

###  2.2 The Green’s function for the problem and its loci of singularities

To put the current density
${\mathbf{j}}^{(cs)}=j^{(cs)}_{r}{\hat{\mathbf{e}}}_{r}+j^{(cs)}_{\varphi}{\hat{\mathbf{e}}}_{\varphi}+j^{(cs)}_{z}{\hat{\mathbf{e}}}_{z}$
into a form suitable for performing the integration with respect to $\varphi$,
we need to express the $\varphi$-dependent base vectors
$({\hat{\mathbf{e}}}_{r},{\hat{\bf e}}_{\varphi},{\hat{\mathbf{e}}}_{z})$
associated with the source point $(r,\varphi,z)$ in terms of the constant base
vectors
$({\hat{\mathbf{e}}}_{r_{P}},{\hat{\mathbf{e}}}_{\varphi_{P}},{\hat{\mathbf{e}}}_{z_{P}})$
at the observation point $(r_{P},\varphi_{P},z_{P})$:

$\left[\matrix{{\hat{\mathbf{e}}}_{r}\cr{\hat{\mathbf{e}}}_{\varphi}\cr{\hat{\mathbf{e}}}_{z}\cr}\right]=\left[\matrix{\cos(\varphi-\varphi_{P})&\sin(\varphi-\varphi_{P})&0\cr-\sin(\varphi-\varphi_{P})&\cos(\varphi-\varphi_{P})&0\cr
0&0&1\cr}\right]\left[\matrix{{\hat{\bf e}}_{r_{P}}\cr{\hat{\bf
e}}_{\varphi_{P}}\cr{\hat{\bf e}}_{z_{P}}\cr}\right].$ (2.10)

Once the resulting expression,

$\displaystyle{\bf j}^{(cs)}$ $\displaystyle=$
$\displaystyle[j^{(cs)}_{r}\cos(\varphi-\varphi_{P})-j^{(cs)}_{\varphi}\sin(\varphi-\varphi_{P})]{\hat{\mathbf{e}}}_{r_{P}}$
(2.11)
$\displaystyle+[j^{(cs)}_{r}\sin(\varphi-\varphi_{P})+j^{(cs)}_{\varphi}\cos(\varphi-\varphi_{P})]{\hat{\mathbf{e}}}_{\varphi_{P}}+j^{(cs)}_{z}{\hat{\mathbf{e}}}_{z_{P}},$

is inserted in (2.6) and $\delta^{\prime}(g-\phi)$ is written as
$-\partial\delta(g-\phi)/\partial{\hat{\varphi}}$ (see (2.9)), we arrive at

${\mathbf{E}}=-\frac{1}{\omega}\sum_{j=1}^{3}\int_{\mathcal{S}}{\hat{r}}{\textrm{d}}{\hat{r}}\,{\textrm{d}}{\hat{\varphi}}\,{\textrm{d}}{\hat{z}}\,\frac{\partial
G_{j}}{\partial{\hat{\varphi}}}{\bf u}_{j},$ (2.12)

in which

$\left[\matrix{G_{1}\cr G_{2}\cr
G_{3}\cr}\right]=\sum_{m=1}^{\infty}\int_{{\hat{\varphi}}+2(m-1)\pi}^{{\hat{\varphi}}+2m\pi}{\rm
d}\varphi\,{\delta(g-\phi)\over{\hat{R}}}\left[\matrix{\cos(\varphi-\varphi_{P})\cr\sin(\varphi-\varphi_{P})\cr
1\cr}\right]$ (2.13)

denotes the outcome of the remaining integration with respect to $\varphi$ and

${\bf u}_{j}=-\frac{j_{0}}{{\hat{r}}_{s}^{3}}\,{\tilde{\bf
u}}_{j}\,\delta({\cal C})$ (2.14)

with

${\tilde{\bf u}}_{1}={\tilde{j}}_{r}{\hat{\bf
e}}_{r_{P}}+{\tilde{j}}_{\varphi}{\hat{\bf e}}_{\varphi_{P}},$ (2.15)
${\tilde{\bf u}}_{2}={\tilde{j}}_{r}{\hat{\bf
e}}_{\varphi_{P}}-{\tilde{j}}_{\varphi}{\hat{\bf e}}_{r_{P}},$ (2.16)
${\tilde{\bf u}}_{3}=-{\tilde{\rho}}c\sin\theta_{P}{\hat{\bf
e}}_{r_{P}}+({\tilde{j}}_{z}-{\tilde{\rho}}c\cos\theta_{P}){\hat{\bf
e}}_{z_{P}}$ (2.17)

(see (1.27)–(1.30)). Note that since ${\hat{\varphi}}+2(m-1)\pi$ designates
the same source element as ${\hat{\varphi}}+2m\pi$ the dependence on
${\hat{\varphi}}$ of the limits of integration in (2.13) does not contribute
toward the values of the derivatives of $G_{j}$ with respect to
${\hat{\varphi}}$.

The function
$G_{j}({\hat{r}},{\hat{\varphi}},{\hat{z}};{\hat{r}}_{P},{\hat{\varphi}}_{P},{\hat{z}}_{P})$
here acts as the Green’s function for the present problem. It describes the
Liénard-Wiechert field that arises from an individual volume element of the
rotating distribution pattern of the source. If we specialize the current
distribution to a rotating point charge $q$, i.e., let $j_{r}=j_{z}=0$ and
$j_{\varphi}=r^{\prime}\omega
q\delta(r-r^{\prime})\delta({\hat{\varphi}})\delta(z)$ with a constant
$r^{\prime}$, then (2.13) at an observation point in the far zone would
describe the familiar field of synchrotron radiation when
$r^{\prime}<c/\omega$ and a synergic field combining attributes of both
synchrotron and Čerenkov emissions when $r^{\prime}>c/\omega$.

Depending on the value of

$\Delta=({{\hat{r}}_{P}}^{2}-1)({\hat{r}}^{2}-1)-({\hat{z}}-{\hat{z}}_{P})^{2}$
(2.18)

for a given source point $(r,{\hat{\varphi}},z)$ with $r\omega>c$, the
$\varphi$-dependence of the function $g$ that appears in the definition of the
Green’s function $G_{j}$ in (2.13) has one of the generic forms shown in
figure 2.1. As can be seen from the curve labelled $\Delta>0$ in this figure,
there are values,

$\varphi_{\pm}=\varphi_{P}+2m\pi-\arccos\left(\frac{1\mp\Delta^{1/2}}{{\hat{r}}{\hat{r}}_{P}}\right),$
(2.19)

of the retarded position of the source point at which

$\frac{\partial
g}{\partial\varphi}=1+\frac{{\hat{r}}{\hat{r}}_{P}\sin(\varphi-\varphi_{P})}{{\hat{R}}}$
(2.20)

vanishes and so $G_{j}$ diverges. These turning points of $g$ occur at source
points for which $\partial(R|_{\varphi={\hat{\varphi}}+\omega t})/\partial
t=-c$, i.e., the source points that approach the observer, along the radiation
direction ${\hat{\bf n}}$, with the speed of light at the retarded time. The
inflection point of $g$ (see the curve labelled $\Delta=0$ in figure 2.1), at
which

$\frac{\partial^{2}g}{\partial\varphi^{2}}\Big{|}_{\varphi=\varphi_{\pm}}=\mp\frac{\Delta^{1/2}}{{\hat{R}}_{\pm}}$
(2.21)

in addition vanishes, occurs at source points that approach the observer not
only with the wave speed but also with zero acceleration at the retarded time,
i.e., for which both $\partial(R|_{\varphi={\hat{\varphi}}+\omega t})/\partial
t=-c$ and $\partial^{2}(R|_{\varphi={\hat{\varphi}}+\omega t})/\partial
t^{2}=0$ at the time when $g|_{\varphi={\hat{\varphi}}+\omega t}=\phi$ and
$\partial g/\partial\varphi=\partial^{2}g/\partial\varphi^{2}=0$. In (2.21),

${\hat{R}}_{\pm}=[({\hat{z}}-{\hat{z}}_{P})^{2}+{\hat{r}}^{2}+{{\hat{r}}_{P}}^{2}-2(1\mp\Delta^{1/2})]^{1/2}$
(2.22)

is the value of ${\hat{R}}$ at the extrema $\varphi_{\pm}$ of $g$.

The envelope of the wave fronts emanating from a given rotating source element
$({\hat{r}},{\hat{\varphi}},{\hat{z}})$, on which $\partial g/\partial\varphi$
vanishes, consists of the rigidly rotating two-sheeted surface
${\hat{\varphi}}-{\hat{\varphi}}_{P}=g(\varphi_{\pm})$ in the space
$({\hat{r}}_{P},{\hat{\varphi}}_{P},{\hat{z}}_{P})$ of observation points.
This surface, which is shown in figures. 2.2 and 2.3, is described by

$\phi_{\pm}\equiv{\hat{\varphi}}_{\pm}-{\hat{\varphi}}_{P}=\varphi_{\pm}-\varphi_{P}+{\hat{R}}_{\pm}$
(2.23)

(see (2.8), (2.9), (2.19) and (2.22)). The two sheets of this surface
tangentially meet along a cusp on which $\partial^{2}g/\partial\varphi^{2}$ as
well as $\partial g/\partial\varphi$ vanishes (see figures 2.3 and 2.4). Three
distinct wave fronts, emitted at three differing values of the retarded time,
pass through any given observation point inside the envelope. At an
observation point located on the envelope or its cusp, respectively two or all
three of these waves coalesce and interfere constructively (see figure 2.2).

Figure 2.1: Generic forms of the function $g(\varphi)$ for source points whose
$({\hat{r}},{\hat{z}})$ coordinates lie across the boundary $\Delta=0$
delineating the projection of the cusp curve of the bifurcation surface onto
the $({\hat{r}},{\hat{z}})$ plane. Depending on whether $\phi$ lies outside or
inside the interval $(\phi_{-},\phi_{+})$, contributions are made toward the
observed field (i.e., the argument $g(\varphi)-\phi$ of the Dirac delta
function in (2.13) vanishes) at either one or three retarded positions of the
source. For a horizontal line $g=\phi$ that either approaches an extremum of
$g(\varphi)$ from inside the interval $(\phi_{-},\phi_{+})$ or passes through
an inflection point of $g(\varphi)$, two or all three of the retarded
positions in question coalesce and so their contributions interfere
constructively to form caustics. This figure is for ${\hat{r}}=3$ and only
shows two rotation periods. At higher speeds, the difference between the
values of $\phi_{+}$ and $\phi_{-}$ can be large enough for a horizontal line
$g=\phi$ to intersect $g(\varphi)$ over more than one rotation period (see
figure 36 in [2]). Contributions toward the observed field can thus arise, not
only from one or three, but from any odd number of retarded positions of the
source. There are contributions from more than three retarded times whenever
the rotation period of the source is shorter than the time taken by the
collapsing sphere $|{\bf x}-{\bf x}_{P}|=c(t-t_{P})$, centred on the
observation point $P$, to cross the orbit of the source.

Figure 2.2: Cross sections with the plane ${\hat{z}}_{P}={\hat{z}}$ of the
spherical wave fronts emanating from a rotating source point. This source has
an angular frequency of rotation, $\omega$, that is constant and a speed,
$r\omega$, that exceeds the speed of light $c$ in vacuum. The larger circle
depicts the orbit of the source and the smaller circle the light cylinder
$r=c/\omega$. The heavier curves show the intersection of the envelope of
these wave fronts (see figure 2.3) with the plane of rotation.

Figure 2.3: Three-dimensional view (in the space
$({\hat{r}}_{P},{\hat{\varphi}}_{P},{\hat{z}}_{P})$ of observation points) of
the envelope of wave fronts emanating from the rotating source point
$({\hat{r}},{\hat{\varphi}},{\hat{z}}$). This envelope consists of two sheets
that tangentially meet along a cusp (see figure 2.4). The singular sheet,
i.e., the sheet that issues from the source point with an initial conical
shape, is that described by
${\hat{\varphi}}_{P}={\hat{\varphi}}-\phi_{-}({\hat{r}}_{P},{\hat{z}}_{P};{\hat{r}},{\hat{z}})$.

Figure 2.4: The cusp along which the two sheets of the envelope of wave fronts
meet and are tangent to one another. This cusp touches and is tangent to the
light cylinder ${\hat{r}}_{P}=1$ on the plane ${\hat{z}}_{P}={\hat{z}}$ and
spirals outward into the far field on the hyperboloid generated by the
revolution of the curve
$\Delta({\hat{r}}_{P},{\hat{z}}_{P};{\hat{r}},{\hat{z}})=0$.

###  2.3 Bifurcation surface of an observation point

Reciprocally, the locus in the space of source points
$({\hat{r}},{\hat{\varphi}},{\hat{z}})$ on which $\partial g/\partial\varphi$
vanishes is a two-sheeted cusped surface issuing from the fixed observation
point $P$ (see figure 2.5). We refer to this locus, which is described by
(2.23) for fixed values of $({\hat{r}}_{P},{\hat{\varphi}}_{P},{\hat{z}}_{P})$
rather than fixed values of $({\hat{r}},{\hat{\varphi}},{\hat{z}})$, as the
bifurcation surface of the observation point $P$. The two sheets
$\phi=\phi_{+}$ and $\phi=\phi_{-}$ of this surface meet along the following
cusp:

$\displaystyle
C:\left\\{\begin{array}[]{c}{\hat{r}}={\hat{r}}_{C}({\hat{z}})=[1+({\hat{z}}-{\hat{z}}_{P})^{2}/({{\hat{r}}_{P}}^{2}-1)]^{1/2},\\\
\varphi=\varphi_{C}({\hat{z}})=\varphi_{P}+2m\pi-\arccos[1/({\hat{r}}{\hat{r}}_{P})],\end{array}\right.$
(2.26)

where $m$ is the same integer as that appearing in (2.13). In this paper we
refer to both $C$ and its projection onto the $(r_{s},\theta)$ plane as the
_cusp locus of the bifurcation surface_ ; whether it is $C$ itself or its
projection that is referred to will be clear from the context.

The source points inside the bifurcation surface, close to its cusp, make
their contributions toward the observed value of the field at three distinct
retarded positions in their trajectory (where a horizontal line $g=\phi$ in
figure 2.1 intersects the curve $\Delta>0$ between its extrema), while those
outside the bifurcation surface make their contributions at a single retarded
position (where the curve $\Delta<0$ is intersected by $g=\phi$ in figure
2.1). For the source points on the bifurcation surface (i.e., those for which
$g=\phi_{\pm}$ in figure 2.1), two of the contributing retarded positions
coalesce at the extrema of the curve $\Delta>0$ in figure 2.1 giving rise to a
divergent value of the Green’s function at $P$ (see figures 9 and 10 of [2]).
For the source points located on the cusp locus $C$ of the bifurcation surface
(i.e., those for which $\Delta=0$ in figure 2.1), all three of the
contributing retarded positions coalesce at the inflection point of the curve
$\Delta=0$ in figure 2.1 giving rise to a higher-order singularity in $G_{j}$.

Figure 2.5: The two sheets $\phi=\phi_{\pm}$ of the bifurcation surface
issuing from the observation point $P$, the cusp $C$ of this surface and the
light cylinder ${\hat{r}}=1$. In contrast to the envelope of wave fronts which
resides in the space of observation points, the surface shown here resides in
the space $(r,{\hat{\varphi}},z)$ of source points: it is the locus of source
points that approach $P$, along the radiation direction, with the speed of
light at the retarded time. The two sheets of this surface meet along a cusp
that tangentially touches the light cylinder at ${\hat{z}}={\hat{z}}_{P}$ and
moves outward spiralling around the rotation axis on the hyperboloid generated
by the revolution of the curve
$\Delta({\hat{r}},{\hat{z}};{\hat{r}}_{P},{\hat{z}}_{P})=0$ (see figure 2.4).
The source points on this cusp approach the observer along the radiation
direction not only with the speed of light but also with zero acceleration at
the retarded time. If the position of the observation point is such that the
cusp shown here intersects the current sheet, there will be wave fronts with
differing emission times that are received simultaneously: while the source
points outside the bifurcation surface make their contributions toward the
value of the observed field at a single instant of retarded time, the source
points inside this surface make their contributions at $3$ (or $5,7,\cdots$)
distinct instants of retarded time.

###  2.4 A uniform asymptotic approximation to the value of the Green’s
function near the cusp locus of the bifurcation surface

The time-domain version [4] of the method of Chester et al. [5] can be
employed to derive a uniform asymptotic approximation to the value of $G_{j}$
for the source points close to the cusp $C$ of the bifurcation surface. The
result, which corresponds to that derived in §4.5 of [2] for the case $n=1$,
is

$G_{j}=\left\\{\begin{array}[]{lll}G_{j}^{\rm{in}}&\Delta>0,\,\,|\chi|<1\\\
G_{j}^{\rm{out}}&\Delta\geq 0,\,\,|\chi|\geq 1\\\
G_{j}^{\rm{sub}}&\Delta<0,\,\,|\chi^{\prime}|>1\end{array}\right.$ (2.27)

with

$G_{j}^{\rm{in}}\simeq\sum_{m=1}^{\infty}2{\mathcal{H}}_{\infty}c_{1}^{-2}(1-\chi^{2})^{-1/2}[p_{j}\cos(\textstyle{\frac{1}{3}}\arcsin\chi)-c_{1}q_{j}\sin(\textstyle{\frac{2}{3}}\arcsin\chi)],\quad|\chi|<1,\quad$
(2.28) $\displaystyle
G_{j}^{\rm{out}}\simeq\sum_{m=1}^{\infty}{\mathcal{H}}_{\infty}c_{1}^{-2}(\chi^{2}-1)^{-1/2}[p_{j}\sinh(\textstyle{\frac{1}{3}}{\textrm{arccosh}}|\chi|)+c_{1}q_{j}{\textrm{sgn}}(\chi)\sinh(\textstyle{\frac{2}{3}}{\textrm{arccosh}}|\chi|)],$
$\displaystyle|\chi|>1,\qquad\qquad$ (2.29)
$G_{j}^{\rm{sub}}\simeq\sum_{m=1}^{\infty}{\mathcal{H}}_{\infty}c_{1}^{-2}({\chi^{\prime}}^{2}+1)^{-1/2}[p_{j}\cosh(\textstyle{\frac{1}{3}}{\textrm{arcsinh}}\chi^{\prime})+|c_{1}|q_{j}\sinh(\textstyle{\frac{2}{3}}{\textrm{arcsinh}}\chi^{\prime})],\quad|\chi^{\prime}|>1,$
(2.30)

where

$\chi=\frac{3(\phi-c_{2})}{2{c_{1}}^{3}},\qquad\chi^{\prime}=\frac{3(\phi-
c_{2})}{2|c_{1}|^{3}},$ $None$
$c_{1}=[{\textstyle\frac{3}{4}}(\phi_{+}-\phi_{-})]^{1/3},\qquad
c_{2}={\textstyle\frac{1}{2}}(\phi_{+}+\phi_{-}),$ $None$
$\phi_{\pm}={\hat{\varphi}}_{\pm}-{\hat{\varphi}}_{P}\quad{\rm
with}\quad{\hat{\varphi}}_{P}=\varphi_{P}-\omega t_{P},$ $None$
$\displaystyle\left[\matrix{p_{1}\cr p_{2}\cr p_{3}\cr}\right]$
$\displaystyle=\frac{1}{{\hat{r}}{\hat{r}}_{P}}\left(\frac{c_{1}}{2\Delta^{1/2}}\right)^{1/2}\left[\matrix{{\hat{R}}_{+}^{-{\textstyle\frac{1}{2}}}+{\hat{R}}_{-}^{-{\textstyle\frac{1}{2}}}+\Delta^{1/2}({\hat{R}}_{-}^{-{\textstyle\frac{1}{2}}}-{\hat{R}}_{+}^{-{\textstyle\frac{1}{2}}})\cr-({\hat{R}}_{-}^{{\textstyle\frac{1}{2}}}+{\hat{R}}_{+}^{{\textstyle\frac{1}{2}}})\cr{\hat{r}}{\hat{r}}_{P}({\hat{R}}_{-}^{-{\textstyle\frac{1}{2}}}+{\hat{R}}_{+}^{-{\textstyle\frac{1}{2}}})\cr}\right],\quad$
(2.34) $\displaystyle\left[\matrix{q_{1}\cr q_{2}\cr
q_{3}\cr}\right]=\frac{1}{{\hat{r}}{\hat{r}}_{P}(2c_{1}\Delta^{1/2})^{1/2}}\left[\matrix{{\hat{R}}_{-}^{-{\textstyle\frac{1}{2}}}-{\hat{R}}_{+}^{-{\textstyle\frac{1}{2}}}+\Delta^{1/2}({\hat{R}}_{-}^{-{\textstyle\frac{1}{2}}}+{\hat{R}}_{+}^{-{\textstyle\frac{1}{2}}})\cr{\hat{R}}_{+}^{{\textstyle\frac{1}{2}}}-{\hat{R}}_{-}^{{\textstyle\frac{1}{2}}}\cr{\hat{r}}{\hat{r}}_{P}({\hat{R}}_{-}^{-{\textstyle\frac{1}{2}}}-{\hat{R}}_{+}^{-{\textstyle\frac{1}{2}}})\cr}\right],$
(2.35) ${\mathcal{H}}_{\infty}={\rm H}[{\hat{R}}_{P}-\omega t_{P}+2m\pi]-{\rm
H}[{\hat{R}}_{P}-\omega t_{P}+2(m-1)\pi],$ (2.36)

and ${\rm H}(x)$ denotes the Heaviside step function.

The two-dimensional loci $\chi=\pm 1$ across which the above expression for
the Green’s function $G_{j}$ changes form correspond to the two sheets
$\phi_{\pm}$ of the bifurcation surface, respectively. As a source point
$(r,{\hat{\varphi}},z)$ in the vicinity of the cusp $C$ approaches the
bifurcation surface from inside, i.e., as $\chi\to 1-$ or $\chi\to-1+$,
$G_{j}^{\rm in}$ diverges. However, as a source point approaches either one of
the sheets of the bifurcation surface from outside, the numerator and the
denominator in (2.29) vanish simultaneously and $G_{j}^{\rm out}$ tends to a
finite limit:

$G_{j}^{\rm out}\big{|}_{\phi=\phi_{\pm}}=G_{j}^{\rm out}\big{|}_{\chi=\pm
1}=\textstyle{\frac{1}{3}}{\cal H}_{\infty}c_{1}^{-2}\left(p_{j}\pm
2c_{1}q_{j}\right).$ (2.37)

Note that $c_{1}$, and hence $p_{j}$ and $q_{j}$, are independent of $m$ (see
(2.19), (2.23) and ($None$)). Thus the Green’s function $G_{j}$ is singular
only on the inner side of the bifurcation surface (see figures 9 and 10 of
[2]).

###  2.5 Hadamard’s finite part of the divergent integral representing the
field

It follows from (2.27) and (2.28) that the factor $\partial
G_{j}/\partial{\hat{\varphi}}$ in the integrand of the integral (2.12)
diverges as $(1-\chi^{2})^{-3/2}$ and so has a non-integrable singularity on
the bifurcation surface where $\chi^{2}$ equals 1. This singularity has arisen
because we differentiated the retarded potential (2.1) under the integral sign
when calculating the field. Had we evaluated the integral in (2.1) prior to
differentiating it we would have found a singularity-free expression.
Interchanging the orders of integration and differentiation is mathematically
permissible when the integrand is discontinuous only if one treats the
resulting integral as a generalized function and so one handles any non-
integrable singularities that consequently arise by means of Hadamard’s
regularization technique (see [6], [7] and the illustrative example in
appendix A of [2]).

Hadamard’s procedure consists of performing an integration by parts and
discarding the divergent (integrated) term in the resulting expression. The
remaining finite part is the value that Hadamard’s regularization assigns to
the integral; in the present case, it is the value we would have obtained if
we had first evaluated the finite integral representing the retarded potential
and had differentiated the result $[\Phi({\textbf{x}}_{P},t_{P}),{\bf
A}({\textbf{x}}_{P},t_{P})]$ of that evaluation subsequently. (The more direct
approach, in which the potential is first evaluated and then differentiated,
cannot of course be carried out for any realistic source distribution
analytically.)

The ${\hat{\varphi}}$ coordinates ${\hat{\varphi}}_{\pm}$ of the two sheets of
the bifurcation surface depend on the observation time $t_{P}$ [see ($None$)
and (2.9)], so that these two sheets move across the ${\hat{\varphi}}$ extent
of the source distribution as $t_{P}$ elapses. If the position of the
observation point is such that the cusp locus of the bifurcation surface
intersects the source distribution, the two sheets of this surface (which
tangentially meet at the cusp) will divide the volume of the source into a
part that lies inside and a part that lies outside the bifurcation surface.
The Lagrangian coordinates ${\hat{\varphi}}$ designating the initial azimuthal
positions of the constituent volume elements of a source that fully occupies
an annular region range over the interval $0\leq{\hat{\varphi}}<2\pi$. The
$({\hat{r}},{\hat{z}})$ coordinates of these source elements either fall in
$\Delta\geq 0$ or in $\Delta<0$. The elements in $\Delta\geq 0$ are always
divided into two sets: a set inside the bifurcation surface for which
${\hat{\varphi}}_{-}\leq{\hat{\varphi}}\leq{\hat{\varphi}}_{+}$ and so the
Green’s function $G_{j}$ has the form $G_{j}^{\rm in}$ and a set outside for
which ${\hat{\varphi}}$ lies either in $(0,{\hat{\varphi}}_{-})$ or in
$({\hat{\varphi}}_{+},2\pi)$ and so $G_{j}$ has the form $G_{j}^{\rm out}$
[see (2.27)]. On the other hand, if the position of the observation point is
such that $\Delta<0$ for all values of $({\hat{r}},{\hat{z}})$ within the
magnetosphere, then the source lies entirely outside the bifurcation surface
and $G_{j}$ has the form $G_{j}^{\rm sub}$. Note that, for certain space-time
coordinates of the observation point $P$, the values of ${\hat{\varphi}}_{-}$
and ${\hat{\varphi}}_{+}$ that lie in the interval $(0,2\pi)$ could correspond
to different rotation periods, i.e., to different values of $m$ [see (2.19),
(2.22) and (2.23)]. To simplify the notation, here we adopt an observation
time $t_{P}$ at which the values of ${\hat{\varphi}}_{-}$ and
${\hat{\varphi}}_{+}$ that lie in the interval $(0,2\pi)$ correspond to the
same rotation period $m$.

Breaking up the volume of integration in the expression for the radiation
field ${\mathbf{E}}$ into the domains of validity of $G_{j}^{\rm in}$,
$G_{j}^{\rm out}$ and $G_{j}^{\rm sub}$, we can therefore write the
${\hat{\varphi}}$-integral over ${\bf u}_{j}$ in (2.12) as

$\displaystyle{\mathbf{I}}_{\hat{\varphi}}$ $\displaystyle\equiv$
$\displaystyle\int_{0}^{2\pi}{\textrm{d}}{\hat{\varphi}}\,{\mathbf{u}}_{j}\frac{\partial
G_{j}}{\partial{\hat{\varphi}}}$ (2.38) $\displaystyle=$ $\displaystyle{\rm
H}(\Delta)\left[\left(\int_{0}^{{\hat{\varphi}}_{-}}+\int_{{\hat{\varphi}}_{+}}^{2\pi}\right){\textrm{d}}{\hat{\varphi}}\,{\mathbf{u}}_{j}\frac{\partial
G_{j}^{\rm
out}}{\partial{\hat{\varphi}}}+\int_{{\hat{\varphi}}_{-}}^{{\hat{\varphi}}_{+}}{\textrm{d}}{\hat{\varphi}}\,{\mathbf{u}}_{j}\frac{\partial
G_{j}^{\rm in}}{\partial{\hat{\varphi}}}\right]$ $\displaystyle+{\rm
H}(-\Delta)\int_{0}^{2\pi}{\textrm{d}}{\hat{\varphi}}\,{\mathbf{u}}_{j}\frac{\partial
G_{j}^{\rm sub}}{\partial{\hat{\varphi}}}.$

If we now integrate every term of the above expression by parts, recall that
${\hat{\varphi}}=0$ labels the same source point as does
${\hat{\varphi}}=2\pi$, and use the fact that the exact version of $G_{j}$
given in (2.13) is periodic in ${\hat{\varphi}}$ as well as in $\varphi$ (with
the same period $2\pi$), we arrive at

$\displaystyle{\mathbf{I}}_{\hat{\varphi}}$ $\displaystyle=$
$\displaystyle{\rm H}(\Delta)\Bigg{\\{}\left[{\mathbf{u}}_{j}\left(G_{j}^{\rm
in}-G_{j}^{\rm
out}\right)\right]_{{\hat{\varphi}}={\hat{\varphi}}_{-}}^{{\hat{\varphi}}={\hat{\varphi}_{+}}}-\left(\int_{0}^{{\hat{\varphi}}_{-}}+\int_{{\hat{\varphi}}_{+}}^{2\pi}\right){\textrm{d}}{\hat{\varphi}}\,\frac{\partial{\mathbf{u}}_{j}}{\partial{\hat{\varphi}}}G_{j}^{\rm
out}$ (2.39)
$\displaystyle-\int_{{\hat{\varphi}}_{-}}^{{\hat{\varphi}}_{+}}{\textrm{d}}{\hat{\varphi}}\,\frac{\partial{\mathbf{u}}_{j}}{\partial{\hat{\varphi}}}G_{j}^{\rm
in}\Bigg{\\}}-{\rm
H}(-\Delta)\int_{0}^{2\pi}{\textrm{d}}{\hat{\varphi}}\,\frac{\partial{\mathbf{u}}_{j}}{\partial{\hat{\varphi}}}G_{j}^{\rm
sub},$

an expression that reduces to

$\displaystyle{\mathbf{I}}_{\hat{\varphi}}$ $\displaystyle=$
$\displaystyle{\rm H}(\Delta)\left[{\mathbf{u}}_{nj}\left(G_{j}^{\rm
in}-G_{j}^{\rm
out}\right)\right]_{{\hat{\varphi}}={\hat{\varphi}}_{-}}^{{\hat{\varphi}}={\hat{\varphi}_{+}}}-\int_{0}^{2\pi}{\textrm{d}}{\hat{\varphi}}\,\frac{\partial{\mathbf{u}}_{j}}{\partial{\hat{\varphi}}}G_{j},$

once the integrals over $G_{j}^{\rm in}$, $G_{j}^{\rm out}$ and $G_{j}^{\rm
sub}$ are combined in the light of (2.27).

We have seen in the last paragraph of § 2.4 that $G_{nj}^{\rm in}$ diverges at
${\hat{\varphi}}={\hat{\varphi}}_{\pm}$. The Hadamard finite part of
${\mathbf{I}}_{\hat{\varphi}}$ is therefore given by the right-hand side of
(LABEL:3E38) without the divergent terms involving $G_{j}^{\rm
in}|_{{\hat{\varphi}}={\hat{\varphi}}_{-}}$ and $G_{j}^{\rm
in}|_{{\hat{\varphi}}={\hat{\varphi}}_{+}}$:

${\rm Fp}\\{{\mathbf{I}}_{\hat{\varphi}}\\}=-{\rm
H}(\Delta){\mathbf{u}}_{j}G_{j}^{\rm
out}\big{|}_{{\hat{\varphi}}={\hat{\varphi}}_{-}}^{{\hat{\varphi}}={\hat{\varphi}_{+}}}-\int_{0}^{2\pi}{\textrm{d}}{\hat{\varphi}}\,\frac{\partial{\mathbf{u}}_{j}}{\partial{\hat{\varphi}}}G_{j},$
(2.41)

where ${\rm Fp}\\{{\mathbf{I}}_{\hat{\varphi}}\\}$ denotes the Hadamard finite
part of the divergent integral ${\mathbf{I}}_{\hat{\varphi}}$ (see [6, 7]).

Once the integral with respect to ${\hat{\varphi}}$ in (2.12) is equated to
the expression on the right-hand side of (2.41), we find that

${\mathbf{E}}={\mathbf{E}}^{\rm v}+{\mathbf{E}}_{+}^{\rm
b}-{\mathbf{E}}_{-}^{\rm b}$ (2.42)

with

${\mathbf{E}}^{\rm
v}=\frac{1}{\omega}\sum_{j=1}^{3}\int\,{\hat{r}}{\textrm{d}}{\hat{r}}\,{\textrm{d}}{\hat{\varphi}}\,{\textrm{d}}{\hat{z}}\,G_{j}\frac{\partial{\mathbf{u}}_{j}}{\partial{\hat{\varphi}}},$
(2.43)

and

${\mathbf{E}}_{\pm}^{\rm
b}=\frac{1}{\omega}\sum_{j=1}^{3}\int\,{\hat{r}}{\textrm{d}}{\hat{r}}\,{\textrm{d}}{\hat{z}}\,{\rm
H}(\Delta)\,G_{j}^{\rm
out}{\mathbf{u}}_{j}\Big{|}_{{\hat{\varphi}}={\hat{\varphi}}_{\pm}}.$ (2.44)

The term ${\bf E}^{\rm v}$ constitutes the contribution from the entire volume
of the source while the terms ${\bf E}_{\pm}^{\rm b}$ denote the contributions
from the discontinuities of the Green’s function on the two sheets
$\phi=\phi_{\pm}$ of the bifurcation surface, respectively. We will see that
the terms ${\bf E}_{\pm}^{\rm b}$ describe an unconventional radiation field
with characteristics that turn out to differ from any previously known
radiation fields.

##  3 Radiation field of the current sheet

###  3.1 The contribution from discontinuities of the Green’s function

Since the description of the current sheet given in § 1 is in terms of
spherical polar coordinates, it is more convenient, for the purposes of
evaluating the contributions ${\bf E}^{b}_{\pm}$ to the radiation field ${\bf
E}$, to change the integration variables in (2.44) from
$({\hat{r}},{\hat{z}})$ to $({\hat{r}_{s}},\theta)$ while continuing to
designate the orientations of the current density and the field by means of
the cylindrical base vectors $({\hat{\bf e}}_{r_{P}},{\hat{\bf
e}}_{\varphi_{P}},{\hat{\bf e}}_{z_{P}})$. Equations (1.27)–(1.30), (2.14),
(2.37) and (2.44) then jointly yield

${\bf E}_{\pm}^{\rm
b}=-\frac{j_{0}}{3\omega}\sum_{j=1}^{3}\int_{\frac{\pi}{2}-\alpha}^{\frac{\pi}{2}+\alpha}{\rm
d}\theta\int{\rm d}{\hat{r}}_{s}\,{\rm H}(\Delta)\frac{\sin\theta(p_{j}\pm
2c_{1}q_{j})}{{\hat{r}}_{s}c_{1}^{2}}{\tilde{\bf u}}_{j}\delta({\cal
C})\Big{|}_{{\hat{\varphi}}={\hat{\varphi}}_{\pm}},$ (3.1)

in which $\Delta$, $c_{1}$, $p_{j}$, $q_{j}$ and ${\hat{\varphi}}_{\pm}$ are
expressed as functions of $({\hat{r}}_{s},\theta)$ by the insertion of
($None$) in (2.18), (2.19) ,(2.22) and ($None$)–(2.36).

It can be seen from (1.6) that the argument of the delta function in (3.1)
vanishes when

$\cos({\hat{\varphi}}+{\hat{r}}_{s}-{\hat{r}}_{s0})=-\cot\alpha\cot\theta,$
(3.2)

i.e., when ${\hat{\varphi}}$ assumes one of the following values

${\hat{\varphi}}_{l}=(-1)^{l}\arccos(\cot\alpha\cot\theta)-{\hat{r}}_{s}+{\hat{r}}_{s0}+(2n+1)\pi,$
(3.3)

where $l$ is either $1$ or $2$ and $n$ is the integer for which the
requirement $0\leq{\hat{\varphi}}_{l}\leq 2\pi$ set by (2.5) is met. Hence, an
alternative form of this delta function, suitable for first performing the
integration with respect to ${\hat{r}}_{s}$ in (3.1), is

$\delta({\cal
C})=\frac{1}{\sin\alpha\sin\theta(1-\cot^{2}\alpha\cot^{2}\theta)^{1/2}}\sum_{l=1}^{2}\delta({\hat{\varphi}}-{\hat{\varphi}}_{l}),$
(3.4)

an expression that holds true for any given values of ${\hat{r}}_{s}$ and
$\theta$.

Inserting (2.15)–(2.17) and (3.4) in (3.1) and using (2.15)–(2.17) and (3.3)
to evaluate ${\tilde{\bf u}}_{j}$ at ${\hat{\varphi}}={\hat{\varphi}}_{l}$, we
obtain

${\bf E}_{\pm}^{\rm
b}=\frac{j_{0}}{3\omega\sin\alpha}\sum_{l=1}^{2}\sum_{j=1}^{3}\int_{\frac{\pi}{2}-\alpha}^{\frac{\pi}{2}+\alpha}{\rm
d}\theta\int{\rm d}{\hat{r}}_{s}\,{\rm H}(\Delta)\frac{{\cal
H}_{\infty}(p_{j}\pm 2c_{1}q_{j}){\bf
v}_{lj}}{{\hat{r}}_{s}c_{1}^{2}(1-\cot^{2}\alpha\cot^{2}\theta)^{1/2}}\delta({\hat{\varphi}}_{\pm}-{\hat{\varphi}}_{l}),$
(3.5)

where

${\bf
v}_{l1}=[{\hat{r}}_{s}\cos\alpha\sin\theta+(-1)^{l}\sin\alpha\cos\theta(1-\cot^{2}\alpha\cot^{2}\theta)^{1/2}\,]{\hat{\bf
e}}_{r_{P}}+\cos\alpha\csc\theta\,{\hat{\bf e}}_{\varphi_{P}},$ (3.6) ${\bf
v}_{l2}=-\cos\alpha\csc\theta\,{\hat{\bf
e}}_{r_{P}}+[{\hat{r}}_{s}\cos\alpha\sin\theta+(-1)^{l}\sin\alpha\cos\theta(1-\cot^{2}\alpha\cot^{2}\theta)^{1/2}\,]{\hat{\bf
e}}_{\varphi_{P}},$ (3.7)

and

${\bf
v}_{l3}=[{\hat{r}}_{s}\cos\alpha\cos\theta-(-1)^{l}\sin\alpha\sin\theta(1-\cot^{2}\alpha\cot^{2}\theta)^{1/2}\,]{\hat{\bf
e}}_{z_{P}}-{\hat{r}}_{s}\cos\alpha\,{\hat{\bf n}}_{\infty}$ (3.8)

(see also (1.21), (1.27)–(1.30) and (2.4)). The step function ${\cal
H}_{\infty}$ in (3.5) ensures that the contribution from the $m$th rotation
cycle reaches a far-field observer at $({\hat{R}}_{P},\theta_{P},\varphi_{P})$
during the interval ${\hat{R}}_{P}+2(m-1)\pi\leq\omega
t_{P}\leq{\hat{R}}_{P}+2m\pi$ of observation time (see (2.36)). If the
observation time is set at midpoint of this interval, i.e.,

$\omega t_{P}={\hat{R}}_{P}+(2m-1)\pi,$ (3.9)

the argument of the delta function in (3.5) assumes the form

$\displaystyle f_{l}^{\pm}$ $\displaystyle=$
$\displaystyle{\hat{\varphi}}_{\pm}-{\hat{\varphi}}_{l}\big{|}_{\omega
t_{P}={\hat{R}}_{P}+(2m-1)\pi}$ $\displaystyle=$
$\displaystyle{\hat{R}}_{\pm}-{\hat{R}}_{P}-\arccos\left(\frac{1\mp\Delta^{1/2}}{{\hat{r}}{\hat{r}}_{P}}\right)+{\hat{r}}_{s}-{\hat{r}}_{s0}-(-1)^{l}\arccos(\cot\alpha\cot\theta)+\varphi_{P}-2n\pi$

(see (2.9), (2.19), (2.23) and (3.3)). Note that there is no loss of
generality in fixing the observation time: because $t_{P}$ only appears in the
combination $\varphi_{P}-\omega t_{P}$, the temporal variation of the
radiation is equally well described by its dependence on $\varphi_{P}$.

Replacing the delta function in (3.5) by its Fourier representation

$\delta\left({\hat{\varphi}}_{\pm}-{\hat{\varphi}}_{l}\big{|}_{\omega
t_{P}={\hat{R}}_{P}+(2m-1)\pi}\right)=\delta(f_{l}^{\pm})=\frac{1}{2\pi}\int_{-\infty}^{\infty}{\rm
d}k\exp({\rm i}kf_{l}^{\pm}),$ (3.11)

we therefore obtain

$\displaystyle{\bf E}_{\pm}^{\rm b}$ $\displaystyle=$
$\displaystyle\frac{j_{0}}{6\pi\omega\sin\alpha}\sum_{l=1}^{2}\sum_{j=1}^{3}\int_{\frac{\pi}{2}-\alpha}^{\frac{\pi}{2}+\alpha}{\rm
d}\theta\int_{-\infty}^{\infty}{\rm d}k$ (3.12) $\displaystyle\times\int{\rm
d}{\hat{r}}_{s}\,{\rm H}(\Delta)\frac{(p_{j}\pm 2c_{1}q_{j}){\bf
v}_{lj}}{{\hat{r}}_{s}c_{1}^{2}(1-\cot^{2}\alpha\cot^{2}\theta)^{1/2}}\exp({\rm
i}kf_{l}^{\pm}),$

in which we have interchanged the orders of integration with respect to
${\hat{r}}_{s}$ and $k$. Note that the range of integration over
${\hat{r}}_{s}$ is set by $\Delta\geq 0$.

A pair of integration variables more suitable than $({\hat{r}}_{s},\theta)$
for evaluating (3.12) are $\eta$ and $\tau$ defined by

$\eta=\Delta^{1/2}=\left[{\hat{r}}_{s}^{2}({\hat{r}}_{P}^{2}\sin^{2}\theta-1)+2{\hat{r}}_{s}{\hat{z}}_{P}\cos\theta-{\hat{R}}_{P}^{2}+1\right]^{1/2}$
(3.13)

(see (2.18)) and

$\tau=\arccos(\csc\alpha\cos\theta)$ (3.14)

since the Jacobian

$\frac{\partial({\hat{r}}_{s},\theta)}{\partial(\eta,\tau)}=\frac{\eta\sin\alpha(1-\cot^{2}\alpha\cot^{2}\theta)^{1/2}}{[({\hat{r}}_{P}^{2}-1)({\hat{R}}_{P}^{2}\sin^{2}\theta-1)+({\hat{r}}_{P}^{2}\sin^{2}\theta-1)\eta^{2}]^{1/2}}$
(3.15)

of this transformation removes the singularities of the integrand that occur
on the boundaries ($\theta=\pi/2\pm\alpha$ and $\Delta=0$ where $c_{1}=0$) of
the integration domain. If we now express ${\hat{r}}_{s}$ and $\theta$
everywhere in terms of $\eta$ and $\tau$ by inverting (3.13) and (3.14) we
obtain

$\theta=\arccos(\sin\alpha\cos\tau),$ (3.16)
${\hat{r}}_{s}=\frac{[({\hat{r}}_{P}^{2}-1)({\hat{R}}_{P}^{2}\sin^{2}\theta-1)+({\hat{r}}_{P}^{2}\sin^{2}\theta-1)\eta^{2}]^{1/2}-{\hat{z}}_{P}\cos\theta}{{\hat{r}}_{P}^{2}\sin^{2}\theta-1},$
(3.17)

and so can rewrite (3.12) as

$\displaystyle{\bf E}_{\pm}^{\rm b}$ $\displaystyle=$
$\displaystyle\frac{j_{0}}{6\pi\omega}\sum_{l=1}^{2}\sum_{j=1}^{3}\int_{0}^{\pi}{\rm
d}\tau\int_{-\infty}^{\infty}{\rm d}k$ (3.18)
$\displaystyle\times\int_{0}^{\infty}{\rm d}\eta\,\frac{\eta(p_{j}\pm
2c_{1}q_{j}){\bf
v}_{lj}}{{\hat{r}}_{s}c_{1}^{2}[({\hat{r}}_{P}^{2}-1)({\hat{R}}_{P}^{2}\sin^{2}\theta-1)+({\hat{r}}_{P}^{2}\sin^{2}\theta-1)\eta^{2}]^{1/2}}\exp({\rm
i}kf_{l}^{\pm}),\qquad$

where ${\hat{r}}_{s}$ and $\theta$ henceforth stand for the functions of
$(\eta,\tau)$ given by (3.16) and (3.17). We shall see below that the limiting
value of the ratio $\eta/c_{1}$ at $\eta=0$, where $c_{1}$ vanishes, is
finite.

The task of the rest of this section is to evaluate the right-hand side of
(3.18) by treating it as a repeated integral.

###  3.2 Evaluation of the asymptotic values of the integrals over $\eta$ by
the method of stationary phase

It follows from (LABEL:4E10) in conjunction with (3.16) and (3.17) that
$\partial f_{l}^{\pm}/\partial\eta$ vanishes at $\eta=0$, so that the main
contribution towards the asymptotic value for large $|k|$ of the integral over
$\eta$ in (3.18) comes from the vicinity of the cusp locus $C$ of the
bifurcation surface at which $\eta$ is zero (see, e.g., [8]). We can therefore
approximate the functions $f_{l}^{\pm}$ in the phase of the integrand of
(3.18) by the following leading terms in their Taylor expansions in powers of
$\eta$:

$f_{l}^{\pm}\simeq
f_{lC}+\textstyle{\frac{1}{2}}a\,\xi^{2}\pm\textstyle{\frac{1}{3}}\xi^{3},$
(3.19)

where

$\xi=\frac{\eta}{({\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}\sin^{2}\theta-1)^{1/2}},$
(3.20) $\displaystyle f_{lC}$ $\displaystyle=$
$\displaystyle({\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}\sin^{2}\theta-1)^{1/2}-{\hat{R}}_{P}+{\hat{r}}_{sC}-\arccos\left(\frac{1}{{\hat{r}}_{P}{\hat{r}}_{sC}\sin\theta}\right)$
(3.21)
$\displaystyle-(-1)^{l}\arccos(\cot\alpha\cot\theta)+\varphi_{P}-{\hat{r}}_{s0}-2n\pi,$
$a=({\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}\sin^{2}\theta-1)\left[\frac{({\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}\sin^{2}\theta-1)^{1/2}+{\hat{r}}_{sC}}{{\hat{r}}_{sC}({\hat{r}}_{P}^{2}-1)^{1/2}({\hat{R}}_{P}^{2}\sin^{2}\theta-1)^{1/2}}-\frac{1}{({\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}\sin^{2}\theta-1)^{1/2}}\right],$
(3.22)

and

${\hat{r}}_{sC}=\frac{({\hat{r}}_{P}^{2}-1)^{1/2}({\hat{R}}_{P}^{2}\sin^{2}\theta-1)^{1/2}-{\hat{z}}_{P}\cos\theta}{{\hat{r}}_{P}^{2}\sin^{2}\theta-1}$
(3.23)

denotes the value of ${\hat{r}}_{s}$ at the cusp locus $C$ (see (3.17)). Note
that the third-order term has to be included in the above expansion to take
account of the difference between $f_{l}^{+}$ and $f_{l}^{-}$. Note, moreover,
that according to ($None$$a$) and (LABEL:4E10), $c_{1}$ can be written as
$[\textstyle{\frac{3}{4}}(f_{l}^{+}-f_{l}^{-})]^{1/3}$ which implies, in
conjunction with (3.19), that $c_{1}\simeq 2^{-1/3}\xi$ near $\eta=0$.

If we now apply the method of stationary phase to the $\eta$-integral, i.e.,
insert (3.19) in (3.18) and replace the amplitude of the exponential in the
integrand of (3.18) by its value at the stationary point $\eta=0$ (see, e.g.,
[8]), we obtain the following expression for the leading term in the
asymptotic expansion of this integral for large $|k|$:

${\bf E}_{\pm}^{\rm
b}\simeq\frac{j_{0}}{3\pi\omega}\sum_{l=1}^{2}\int_{0}^{\pi}{\rm
d}\tau\int_{-\infty}^{\infty}{\rm d}k\,b\exp({\rm
i}kf_{lC})\int_{0}^{\infty}{\rm d}\xi\,(\xi^{-1}{\bf P}_{l}\pm{\bf
Q}_{l})\exp[{\rm
i}k(\textstyle{\frac{1}{2}}a\,\xi^{2}\pm\textstyle{\frac{1}{3}}\xi^{3})],$
(3.24)

where

$b=\frac{{\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}\sin^{2}\theta-1}{{\hat{r}}_{sC}({\hat{r}}_{P}^{2}-1)^{1/2}({\hat{R}}_{P}^{2}\sin^{2}\theta-1)^{1/2}},$
(3.25) ${\bf P}_{l}=2^{-1/3}\sum_{j=1}^{3}p_{j}{\bf v}_{lj}\Big{|}_{\eta=0},$
(3.26) ${\bf Q}_{l}=2^{1/3}\sum_{j=1}^{3}q_{j}{\bf v}_{lj}\Big{|}_{\eta=0},$
(3.27)

and use has been made of the fact that
$\eta/c_{1}=2^{1/3}({\hat{r}}_{P}^{2}{\hat{r}}_{sC}^{2}\sin^{2}\theta-1)^{1/2}$in
the limit $\eta\to 0$. (Note that $b$ tends to $1$ as ${\hat{R}}_{P}$ tends to
infinity.) The above asymptotic approximation for large $|k|$ is justified
since the frequency of the radiation we are interested in has a much higher
value than the rotation frequency $\omega/2\pi$ of the central neutron star.

Hence, the unconventional contribution ${\bf E}_{+}^{b}-{\bf E}_{-}^{b}$ to
the radiation field ${\bf E}$ that was encountered in (2.42) can be written as

${\bf E}^{\rm uc}={\bf E}_{+}^{b}-{\bf
E}_{-}^{b}\simeq\frac{j_{0}}{3\pi\omega}\sum_{l=1}^{2}\int_{0}^{\pi}{\rm
d}\tau\int_{-\infty}^{\infty}{\rm d}k\,b\exp({\rm i}kf_{lC})(I_{P}{\bf
P}_{l}+I_{Q}{\bf Q}_{l}),$ (3.28)

once the integration with respect to $\xi$ has been carried out analytically
to obtain

$\displaystyle I_{Q}$ $\displaystyle=$ $\displaystyle 2\int_{0}^{\infty}{\rm
d}\xi\,\exp(\textstyle{\frac{1}{2}}{\rm
i}ka\xi^{2})\cos\left(\textstyle{\frac{1}{3}}k\xi^{3}\right)$ (3.29)
$\displaystyle=$
$\displaystyle\frac{2\pi}{|k|^{1/3}}\exp\left(\textstyle{\frac{1}{12}}{\rm
i}ka^{3}\right){\rm Ai}\left(-\textstyle{\frac{1}{4}}k^{2/3}a^{2}\right),$

in terms of the Airy function ${\rm Ai}$ and

$\displaystyle I_{P}$ $\displaystyle=$ $\displaystyle 2{\rm
i}\int_{0}^{\infty}{\rm d}\xi\,\xi^{-1}\exp(\textstyle{\frac{1}{2}}{\rm
i}ka\xi^{2})\sin\left(\textstyle{\frac{1}{3}}k\xi^{3}\right)$ (3.30)
$\displaystyle=$ $\displaystyle{\rm sgn}(k)\bigg{[}\frac{{\rm
i}\pi}{3}-\frac{3^{1/6}}{2}\Gamma\left(\textstyle{\frac{2}{3}}\right)a\,k^{1/3}\,{}_{3}F_{4}\left(\matrix{1/6&5/12&11/12&{}\cr
1/3&1/2&5/6&7/6\cr};-\frac{k^{2}a^{6}}{144}\right)$ $\displaystyle+\frac{{\rm
i}}{2^{2}\times
3^{7/6}}\Gamma\left(\textstyle{-\frac{2}{3}}\right)a^{2}k^{2/3}\,{}_{3}F_{4}\left(\matrix{1/3&7/12&13/12&{}\cr
1/2&2/3&7/6,&4/3\cr};-\frac{k^{2}a^{6}}{144}\right)$
$\displaystyle+\frac{5\pi{\rm i}}{2^{6}\times
3^{1/3}\Gamma\left(-2/3\right)}a^{4}k^{4/3}\,{}_{3}F_{4}\left(\matrix{2/3&11/12&17/12&{}\cr
5/6&4/3&3/2&5/3\cr};-\frac{k^{2}a^{6}}{144}\right)$
$\displaystyle-\frac{7\Gamma\left(-2/3\right)}{2^{5}\times 3^{13/6}\times
5}a^{5}k^{5/3}\,{}_{3}F_{4}\left(\matrix{5/6&13/12&19/12&{}\cr
7/6&3/2&5/3&11/6\cr};-\frac{k^{2}a^{6}}{144}\right)\bigg{]}$

in terms of the generalized hypergeometric function ${}_{3}F_{4}$ (see, e.g.,
[9]). Mathematica has been used to perform the above integrations by first
employing a change of integration variable to cast $I_{Q}$ and $I_{P}$ in the
forms of sine and cosine Fourier transforms and using the relations in §5.5 of
[9] to simplify the gamma functions in the resulting expressions.

###  3.3 Dominance of the contribution from large values of $|k|$

In this section we assess the expectation that the main contribution toward
the value of the radiation field should come from large values of $|k|$. We
will do this by evaluating the $k$-integral in (3.28) once with the exact
value of its integrand and another time with the asymptotic value of its
integrand for large $|k|$ and comparing the outcomes of these two evaluations.

The functions ${\bf P}_{l}$, ${\bf Q}_{l}$ and $b$ in the integrand of (3.28)
are independent of $k$. Once the Airy function that appears in the expression
for $I_{Q}$ is expressed in terms of Bessel functions, the $k$-integral
multiplying ${\bf Q}_{l}$ assumes the form of a tabulated Fourier transform
(cf., [10]) and so can be evaluated exactly to obtain

$\int_{-\infty}^{\infty}{\rm d}k\,\exp\left({\rm
i}kf_{lC}\right)I_{Q}=\frac{8\pi}{a^{2}}{\cal F}_{1}^{\rm
e}\left(\frac{12}{a^{3}}f_{lC}+1\right),$ (3.31)

where

$\displaystyle{\cal F}_{1}^{\rm e}(x)$ $\displaystyle=$
$\displaystyle\frac{{\rm
H}(|x|-1)}{2(x^{2}-1)^{1/2}}\left[\left(|x|+\sqrt{x^{2}-1}\right)^{1/3}-\left(|x|+\sqrt{x^{2}-1}\right)^{-1/3}\right]$
(3.32) $\displaystyle+\frac{2{\rm
H}(1-|x|)}{(1-x^{2})^{1/2}}\cos\left(\frac{1}{3}\arcsin|x|\right).$

The $k$-integral multiplying ${\bf P}_{l}$ in (3.28) also consists of the sum
of five Fourier transforms each of which can be evaluated explicitly by means
of Mathematica. The result is

$\int_{-\infty}^{\infty}{\rm d}k\,\exp\left({\rm
i}kf_{lC}\right)I_{P}=\frac{12}{a^{3}}{\cal F}_{2}^{\rm
e}\left(\frac{12}{a^{3}}f_{lC}\right),$ (3.33)

where

$\displaystyle{\cal F}_{2}^{\rm e}(x)$ $\displaystyle=$
$\displaystyle-\frac{2\pi}{3x}+\frac{2^{2/3}\pi}{3x^{4/3}}{\rm
H}(|x|-2)\bigg{[}{}_{4}F_{3}\left(\matrix{1/6&5/12&2/3&11/12\cr
1/3&1/2&5/6&{}\cr};\frac{4}{x^{2}}\right)$
$\displaystyle+\frac{1}{(2x)^{1/3}}{}_{4}F_{3}\left(\matrix{1/3&7/12&5/6&13/12\cr
1/2&2/3&7/6&{}\cr};\frac{4}{x^{2}}\right)$
$\displaystyle-\frac{5}{6x}{}_{4}F_{3}\left(\matrix{2/3&11/12&7/6&17/12\cr
5/6&4/3&3/2&{}\cr};\frac{4}{x^{2}}\right)$
$\displaystyle-\frac{7}{3(2x)^{4/3}}{}_{4}F_{3}\left(\matrix{5/6&13/12&4/3&19/12\cr
7/6&3/2&5/3&{}\cr};\frac{4}{x^{2}}\right)\bigg{]}$
$\displaystyle-\frac{\pi}{3}{\rm H}(2-|x|)\bigg{[}\frac{1-{\rm
sgn}(x)}{\sqrt{6|x|}}{}_{4}F_{3}\left(\matrix{5/12&7/12&11/12&13/12\cr
1/2&3/4&5/4&{}\cr};\frac{x^{2}}{4}\right)$
$\displaystyle+\frac{35}{108}\sqrt{\frac{|x|}{6}}[1-{\rm
sgn}(x)]{}_{4}F_{3}\left(\matrix{11/12&13/12&17/12&19/12\cr
5/4&3/2&7/4&{}\cr};\frac{x^{2}}{4}\right)$
$\displaystyle-\frac{2}{x}\,{}_{4}F_{3}\left(\matrix{1/6&1/3&2/3&5/6\cr
1/4&1/2&3/4&{}\cr};\frac{x^{2}}{4}\right)+\frac{4}{9}{}_{4}F_{3}\left(\matrix{2/3&5/6&7/6&4/3\cr
3/4&5/4&3/2&{}\cr};\frac{x^{2}}{4}\right)\bigg{]}.$

Note that repeated use has to be made of the relations in §5.5 of [9] to cast
the result in the above simplified form.

The function ${\cal F}_{1}^{\rm e}$ diverges as $(1-x^{2})^{-1/2}$ at
$|x|=1-$. Moreover, the generalized hypergeometric functions appearing in
(LABEL:4E34) are singular in the limit where their arguments approach unity.
While the singularities of the individual terms of (LABEL:4E34) at $x=2$,
$x=-2-$ and $x=0$ cancel out when added together, the function ${\cal
F}_{2}^{\rm e}$ diverges as $(x+2)^{-1/2}$ at $x=-2+$ (see figure 3.1).

Figure 3.1: The solid (red) lines in this figure depict the exact values
(${\cal F}_{1}^{\rm e}$ and ${\cal F}_{2}^{\rm e}$) of the functions derived
in (3.32) and (LABEL:4E34). The dashed (blue) lines depict the approximate
versions (${\cal F}_{1}^{\rm a}$ and ${\cal F}_{2}^{\rm a}$) of these
functions given by (3.41) and (3.42) for $k_{1}=3/(5a^{3})$ and
$k_{2}=12/(5a^{3})$.

The factor $a$ that multiplies $k$ in the arguments of the Airy function in
(3.29) and the generalized hypergeometric functions in (3.30) is non-zero and
positive everywhere and approaches the finite value

$a\simeq\csc\theta_{P}\csc\theta(1-\cos\theta_{P}\cos\theta)$ (3.35)

as ${\hat{R}}_{P}$ tends to infinity (see(3.22)). To calculate the
contribution of large $|k|$ toward the values of the integrals in (3.31) and
(3.33) we can therefore replace the functions $I_{P}$ and $I_{Q}$ that appear
in the integrands of these integrals by the following leading terms

$I_{P}^{\infty}=\lim_{k\to\pm\infty}I_{P}=\left(\frac{2\pi}{a^{3}|k|}\right)^{1/2}\exp\left(\pm{\textstyle\frac{3}{4}}{\rm
i}\pi\right)\left[\exp\left({\textstyle\frac{1}{6}}{\rm
i}a^{3}k\right)\pm{\textstyle\frac{1}{3}}{\rm i}\right],$ (3.36)
$I_{Q}^{\infty}=\lim_{k\to\pm\infty}I_{Q}=\left(\frac{2\pi}{a|k|}\right)^{1/2}\exp\left(\mp{\textstyle\frac{1}{4}}{\rm
i}\pi\right)\left[\exp\left({\textstyle\frac{1}{6}}{\rm i}a^{3}k\right)\pm{\rm
i}\right],$ (3.37)

in their asymptotic expansions for large $|k|$ before performing the
integrations. (Note that (3.37) is found more easily by a direct asymptotic
evaluation of the integral over $\xi$ in the first member of (3.30) than by
the calculation of the limiting value of the second member of this equation.)
To calculate the corresponding contribution from small values of $|k|$ we
should replace $I_{P}$ and $I_{Q}$ by the following leading terms

$I_{P}^{0}=\lim_{|k|\to 0}I_{P}=\frac{{\rm i}\pi}{3}{\rm sgn}(k),\qquad
I_{Q}^{0}=\lim_{|k|\to 0}I_{Q}=\frac{2\pi}{3^{2/3}\Gamma(2/3)|k|^{1/3}},$
(3.38)

in their expansions about $k=0$ (see [9]).

It turns out that the functions ${\cal F}_{1}^{\rm e}$ and ${\cal F}_{2}^{\rm
e}$, defined in (3.31)–(LABEL:4E34), can be accurately approximated by

${\cal F}_{1}^{\rm
a}\left(\frac{12}{a^{3}}f_{lC}+1\right)=\frac{a^{2}}{4\pi}\Re\int_{k_{1}}^{\infty}{\rm
d}k\,\exp\left({\rm i}kf_{lC}\right)I^{\infty}_{Q}$ (3.39)

and

${\cal F}_{2}^{\rm
a}\left(\frac{12}{a^{3}}f_{lC}\right)=\frac{a^{3}}{6\pi}\Re\left\\{\int_{0}^{k_{2}}{\rm
d}k\,\exp\left({\rm i}kf_{lC}\right)I^{0}_{P}+\int_{k_{2}}^{\infty}{\rm
d}k\,\exp\left({\rm i}kf_{lC}\right)I^{\infty}_{P}\right\\},$ (3.40)

respectively, with appropriate values of the free parameters $k_{1}$ and
$k_{2}$. We have here replaced the integrals over $0\leq|k|\leq k_{1,2}$ and
$k_{1,2}\leq|k|<\infty$ by twice the real parts of the integrals over $0\leq
k\leq k_{1,2}$ and $k_{1,2}\leq k<\infty$ because the contributions from $k<0$
toward the values of the $k$-integrals in (3.31) and (3.33) equal the complex
conjugates of the contributions from $k>0$. Evaluation of the integrals in
(3.39) and (3.40) results in

$\displaystyle{\cal F}_{1}^{\rm a}(x)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{3}{2}}\left\\{\frac{1}{|x+1|^{1/2}}\left[\frac{1}{2}-C\left(\sqrt{\frac{a^{3}k_{1}}{6\pi}|x+1|}\right)\right]\right.$
(3.41) $\displaystyle\left.+\frac{{\rm
sgn}(x+1)}{|x+1|^{1/2}}\left[\frac{1}{2}-S\left(\sqrt{\frac{a^{3}k_{1}}{6\pi}|x+1|}\right)\right]\right.$
$\displaystyle\times\left.\frac{1}{|x-1|^{1/2}}\left[\frac{1}{2}-C\left(\sqrt{\frac{a^{3}k_{1}}{6\pi}|x-1|}\right)\right]\right.$
$\displaystyle\left.-\frac{{\rm
sgn}(x-1)}{|x+1|^{1/2}}\left[\frac{1}{2}-S\left(\sqrt{\frac{a^{3}k_{1}}{6\pi}|x-1|}\right)\right]\right\\},$

and

$\displaystyle{\cal F}_{2}^{\rm a}(x)$ $\displaystyle=$
$\displaystyle-\frac{4\pi}{3x}\sin^{2}\left(\frac{a^{3}k_{2}}{24}x\right)-\sqrt{\frac{2}{3}}\pi\left\\{\frac{1}{|x+2|^{1/2}}\left[\frac{1}{2}-C\left(\sqrt{\frac{a^{3}k_{2}}{6\pi}|x+2|}\right)\right]\right.$
(3.42) $\displaystyle\left.+\frac{{\rm
sgn}(x+2)}{|x+2|^{1/2}}\left[\frac{1}{2}-S\left(\sqrt{\frac{a^{3}k_{2}}{6\pi}|x+2|}\right)\right]+\frac{1}{3|x|^{1/2}}\left[\frac{1}{2}-C\left(\sqrt{\frac{a^{3}k_{2}}{6\pi}|x|}\right)\right]\right.$
$\displaystyle\left.-\frac{{\rm
sgn}(x)}{3|x|^{1/2}}\left[\frac{1}{2}-S\left(\sqrt{\frac{a^{3}k_{2}}{6\pi}|x|}\right)\right]\right\\}$

where $C$ and $S$ are the Fresnel cosine and sine integrals, respectively.

The exact (${\cal F}_{1}^{\rm e},{\cal F}_{2}^{\rm e}$) and approximate
(${\cal F}_{1}^{\rm a},{\cal F}_{2}^{\rm a}$) versions of the functions ${\cal
F}_{1}$ and ${\cal F}_{2}$ are compared in figure 3.1 for
$k_{1}={\textstyle\frac{3}{5}}a^{-3}$ and
$k_{2}={\textstyle\frac{12}{5}}a^{-3}$. The goodness of the fit in this figure
and the shortness of the intervals $0\leq k\leq k_{1,2}$ relative to
$k_{1,2}\leq k<\infty$ show that (i) the difference in values of the exact and
approximate versions of these two functions is negligibly small in the
intervals over which ${\cal F}_{1}$ and ${\cal F}_{2}$ make their main
contributions toward the value of the integrand of the $\tau$-integral in
(3.28) and (ii) the contributions of large $|k|$ toward the values of ${\cal
F}_{1}^{\rm e}$ and ${\cal F}_{2}^{\rm e}$ is by far greater than those of
small $|k|$. In order that we can derive an analytic expression for the
radiation field ${\bf E}^{\rm uc}$, we will accordingly base the analysis in
the rest of this section on the following approximate values of $I_{P}$ and
$I_{Q}$

$I^{\rm a}_{P}={\rm H}(k_{2}-|k|)I_{P}^{0}+{\rm H}(|k|-k_{2})I_{Q}^{\infty}$
(3.43)

and

$I^{\rm a}_{Q}={\rm H}(|k|-k_{1})I_{Q}^{\infty}$ (3.44)

that were derived in (3.36)–(3.38) and let $k_{1}$ and $k_{2}$ have the values
that were used in plotting figure 3.1.

Replacing $I_{P}$ and $I_{Q}$ in (3.28) with their approximate values $I^{\rm
a}_{P}$ and $I^{\rm a}_{Q}$ and interchanging the orders of integration with
respect to $k$ and $\tau$, we write (3.28) as

$\displaystyle{\bf E}^{\rm uc}$ $\displaystyle\simeq$
$\displaystyle\frac{2j_{0}}{3\omega}\sum_{l=1}^{2}\int_{0}^{\infty}{\rm
d}k\int_{0}^{\pi}{\rm d}\tau\,b\,\bigg{\\{}\bigg{[}\frac{{\rm i}}{3}{\rm
H}(k_{2}-k){\bf P}_{l}+\sqrt{\frac{2}{\pi ak}}\exp\left(\frac{{\rm
i}\pi}{4}\right)$ (3.45) $\displaystyle\times\left({\rm H}(k-k_{1}){\bf
Q}_{l}-\frac{1}{3a}{\rm H}(k-k_{2}){\bf P}_{l}\right)\bigg{]}\exp\left({\rm
i}kf_{lC}\right)$ $\displaystyle+\sqrt{\frac{2}{\pi ak}}\exp\left(-\frac{{\rm
i}\pi}{4}\right)\left({\rm H}(k-k_{1}){\bf Q}_{l}-\frac{1}{a}{\rm
H}(k-k_{2}){\bf P}_{l}\right)\exp\left({\rm
i}k{\bar{f}}_{lC}\right)\bigg{\\}},$

where

${\bar{f}}_{lC}=f_{lC}+{\textstyle\frac{1}{6}}a^{3},$ (3.46)

and proceed to evaluate the integral over $\tau$ first. That the
unconventional radiation field ${\bf E}^{\rm uc}$ is given by the real part of
the above expression (in which the integrals over $-\infty<k<\infty$ are
written as twice the integrals over $0\leq k<\infty$) is understood.

###  3.4 Critical points of the phase functions $f_{lC}$ and ${\bar{f}}_{lC}$

The first two derivatives with respect to $\tau$ of the functions $f_{lC}$ and
${\bar{f}}_{lC}$–defined by (3.21) and (3.43)–which appear in (3.42) are given
in appendix A.

It follows from (A.1)–(A.5) that the nature of the critical points of $f_{lC}$
is determined by the value of the coordinate $\theta_{P}$ of the observation
point: this function can have two turning points (a maximum and a minimum),
can have a single inflection point, or can be monotonic. As indicated by
(3.21), the changes $\theta\to\pi-\theta$, $\theta_{P}\to\pi-\theta_{P}$ and
$\varphi_{P}\to\varphi_{P}+\pi$ transform $f_{2C}(\tau)$ into $f_{1C}(\tau)$.
This means that $f_{2C}$ has the same kind of critical points as
$f_{1C}(\tau)$ but in a different hemisphere (in $\theta>\pi/2$ and
$\theta_{P}>\pi/2$ instead of $\theta<\pi/2$ and $\theta_{P}<\pi/2$ and vice
versa). Moreover, the function $f_{lC}$ for $\alpha>\pi/2$ follows from that
for $\alpha<\pi/2$ if we replace $\theta$ by $\pi-\theta$ at the same time as
replacing $\alpha$ by $\pi-\alpha$ (see (3.21)). It is sufficient, therefore,
to consider only the cases in which $\alpha<\pi/2$. Note that, owing to the
presence of the factor $w_{1}$ in the expression for the density of the
current sheet, the field ${\bf E}^{\rm uc}$ is zero for $\alpha=\pi/2$ (see
(1.7)).

Figure 3.2: The relative positions of the critical angles $\theta_{P1S}$ and
$\theta_{P2S}$ along the $\theta_{P}$-axis for $\alpha<\pi/3$ and
$\alpha>\pi/3$. The ranges of values of $\theta_{P}$ for which $f_{1C}$ and
$f_{2C}$ have two turning points as functions of $\tau$ are also shown.
Outside the shown intervals, both $f_{1C}$ and $f_{2C}$ vary monotonically
with $\tau$.

Figure 3.3: Dependence of the phase functions $f_{1C}$ and $f_{2C}$ on the
integration variable $\tau$ for $\alpha=\pi/4$ in frames (i) and (ii) and for
$\alpha=3\pi/8$ in frames (iii) and (iv). In frames (i) and (ii),
$\theta_{P}>\theta_{P1S}>\theta_{P2S}$ for the blue curves $a$,
$\theta_{P}=\theta_{P1S}>\theta_{P2S}$ for the red curves $b$ and
$\theta_{P2S}<\theta_{P}<\theta_{P1S}$ for the green curves $c$. In frames
(iii) and (iv), $\theta_{P1S}<\theta_{P}<\theta_{P2S}$ for the blue curves
$a$, $\theta_{P}=\theta_{P1S}<\theta_{P2S}$ for the red curves $b$ and
$\theta_{P}<\theta_{P1S}<\theta_{P2S}$ for the green curves $c$. Note that a
change in the coordinate $\varphi_{P}$ of the observation point shifts the
above curves (which have here been plotted for ${\hat{R}}_{P}=10^{6}$)
vertically without changing their shapes. These curves illustrate that while
in the case of $\pi/3<\alpha<\pi/2$ both $f_{1C}$ and $f_{2C}$ have maxima,
minima or inflection points, in the case of $0<\alpha<\pi/3$ either $f_{1C}$
or $f_{2C}$ is a monotonic function of $\tau$.

In this paper, we denote the values of $\tau$, $\theta_{P}$ and $\varphi_{P}$
at which

$f_{lC}=\frac{\partial
f_{lC}}{\partial\tau}=\frac{\partial^{2}f_{lC}}{\partial\tau^{2}}=0$ (3.47)

by $\tau_{lS}$, $\theta_{PlS}$, and $\varphi_{PlS}$, respectively. It turns
out that $\theta_{P1S}$ and $\theta_{P2S}$ always lie on opposite sides of
$\theta_{P}=\pi/2$ (see figure 3.2): they satisfy
$\theta_{P2S}<\pi/2<\theta_{P1S}$ for $0<\alpha<\pi/3$ and
$\theta_{P1S}<\pi/2<\theta_{P2S}$ for $\pi/3<\alpha<\pi/2$ when the
observation point lies at infinity. For $\alpha=\pi/3$ they have the values
$\theta_{P1S}=\pi/2-1/{\hat{R}}_{P}$ and $\theta_{P2S}=\pi/2+1/{\hat{R}}_{P}$
when ${\hat{R}}_{P}\gg 1$.

Plots of $f_{lC}$ versus $\tau$ for various values of the free parameters
$\theta_{P}$ and $\alpha$ display the following features. When $\alpha<\pi/3$,
(i) $f_{1C}$ is monotonic for $0<\theta_{P}<\theta_{P1S}$ and has a maximum
and a minimum for $\theta_{P1S}<\theta_{P}<\pi$, (ii) $f_{2C}$ has a maximum
and a minimum for $0<\theta_{P}<\theta_{P2S}$ and is monotonic for
$\theta_{P2S}<\theta_{P}<\pi$ and (iii) the extrema of $f_{1C}$ and $f_{2C}$
coincide at an inflection point for $\theta_{P}=\theta_{P1S}$ and
$\theta_{P}=\theta_{PS2}$, respectively. Hence, both $f_{1C}$ and $f_{2C}$ are
monotonic for $\theta_{P2S}<\theta_{P}<\theta_{P1S}$ when $0<\alpha<\pi/3$. On
the other hand, when $\pi/3<\alpha<\pi/2$, (i) $f_{1C}$ is monotonic in
$0<\theta_{P}<\theta_{P1S}$ and has two turning points in
$\theta_{P1S}<\theta_{P}<\pi$, (ii) $f_{2C}$ has a maximum and a minimum in
$0<\theta_{P}<\theta_{P2S}$ and is monotonic in $\theta_{P2S}<\theta_{P}<\pi$
and (iii) once again the extrema of $f_{1C}$ and $f_{2C}$ coincide at an
inflection point for $\theta_{P}=\theta_{P1S}$ and $\theta_{P}=\theta_{PS2}$,
respectively. Hence $f_{1C}$ and $f_{2C}$ each have a maximum and a minimum
for $\theta_{P2S}<\theta_{P}<\theta_{P1S}$ when $\pi/3<\alpha<\pi/2$. Figure
3.3 illustrates some of the generic forms assumed by $f_{1C}$ and $f_{2C}$ as
functions of $\tau$ inside the integration domain $0<\tau<\pi$.

Changes in values of the free parameter $\varphi_{P}$ simply shift the curve
representing $f_{lC}$ versus $\tau$ up or down without altering its shape (see
(3.21)). But changes in values of the remaining free parameter ${\hat{R}}_{P}$
does alter the relative positions of the two turning points of $f_{lC}$ when
$\theta_{P}$ is close to $\theta_{PlS}$. The length of the interval separating
the $\tau$ coordinates of the maximum and minimum of $f_{lC}$ decreases with
increasing ${\hat{R}}_{P}$ in a case where $\theta_{P}$ is close to
$\theta_{PlS}$ and so this interval is small. If we denote the $\tau$
coordinates of the maximum and minimum of $f_{lC}$ by $\tau_{l{\rm max}}$ and
$\tau_{l{\rm min}}$, respectively, then it turns out that $|\tau_{l{\rm
max}}-\tau_{l{\rm min}}|\propto{\hat{R}}_{P}^{-1/2}$ for ${\hat{R}}_{P}\gg 1$
when $\theta_{P}$ has the value $\theta_{PlS}|_{{\hat{R}}_{P}\to\infty}$ for
any given $\alpha$, i.e., when $\theta_{P}$ is such that $|\tau_{l{\rm
max}}-\tau_{l{\rm min}}|$ shrinks to zero as ${\hat{R}}_{P}$ tends to
infinity. We shall see in § 4.4 that this property of $f_{lC}$ results in a
decay of the radiation’s intensity with distance in the direction of
$\theta_{PlS}$ that is slower than predicted by the inverse-square law.

Note that $f_{1C}$ and $f_{2C}$ for a given value of $n$ are equal at $\tau=0$
and differ by $2\pi$ at $\tau=\pi$ (see (3.21) and (3.14)). In cases where the
absolute value of $f_{lC}|_{\tau=\pi}-f_{lC}|_{\tau=0}$ for either $l=1$ or
$l=2$ is greater than $2\pi$ (e.g., when $\alpha=15^{\circ}$ and
$\theta_{P}=65^{\circ}$) or $f_{lC}|_{\tau_{l{\rm min}}}$ and
$f_{lC}|_{\tau_{l{\rm max}}}$ differ by more than $2\pi$ (e.g., when
$\alpha=85^{\circ}$ and $\theta_{P}=5^{\circ}$), ordinates of the points on
$f_{1C}$ and $f_{2C}$ in figure 3.3 also span intervals whose lengths exceed
$2\pi$. In such cases, therefore, $f_{1C}$ and $f_{2C}$ for several values of
$n$ (i.e., several cycles of retarded time) simultaneously contribute toward
the intensity of the pulse that is observed during a single period.

The above discussion applies also to the modified functions ${\bar{f}}_{1C}$
and ${\bar{f}}_{2C}$: their critical points differ from those of $f_{1C}$ and
$f_{2C}$ only in their positions, not in their nature. The generic forms
assumed by ${\bar{f}}_{1C}$ and ${\bar{f}}_{2C}$ are the same as those
illustrated in figure 3.3 except that the role of $\alpha=\pi/3$ in figure 3.2
is played by $\alpha=0.8707129958$.

Depending on relative positions of the coordinate $\theta_{P}$ of the
observation point and the inclination angle $\alpha$, therefore, the four
phase functions $f_{1C}$, $f_{2C}$, ${\bar{f}}_{1C}$ and ${\bar{f}}_{2C}$ in
the integrand of the $\tau$-integral in (3.45) can jointly have a set of
isolated stationary points with one to eight members (where $\partial
f_{lC}/\partial\tau$ and/or $\partial{\bar{f}}_{lC}/\partial\tau$ for $l=1$
and/or $l=2$ vanish) or can have one or two degenerate stationary points
(where $\partial^{2}f_{lC}/\partial\tau^{2}$ and/or
$\partial^{2}{\bar{f}}_{lC}/\partial\tau^{2}$ simultaneously vanish with
$\partial f_{lC}/\partial\tau$ and/or $\partial{\bar{f}}_{2C}/\partial\tau$).
The number of contributing stationary points is higher in cases where the
ordinates of the curves depicted in figure 3.3 span intervals whose lengths
exceed $2\pi$ and so the contributions from several cycles of retarded time
are received during a single period of observation time.

###  3.5 A uniform asymptotic approximation to the integral over the
colatitude of the source elements for large $|k|$

Since this paper is concerned with determining the radiation field ${\bf
E}^{\rm uc}$ also at observation points for which the turning points of the
phase functions $f_{lC}$ and ${\bar{f}}_{lC}$ are close to one another or
coalescent, we need to obtain an asymptotic approximation to the value of the
$\tau$-integral in (3.45) that is uniform with respect to the interval
separating the nearby saddle points of these phase functions (see [5, 8]).

In cases where $f_{lC}$ and ${\bar{f}}_{lC}$ each have two turning points, one
at their maxima $\tau_{l{\rm max}}$ and ${\bar{\tau}}_{l{\rm max}}$ and one at
their minima $\tau_{l{\rm min}}$ and ${\bar{\tau}}_{l{\rm min}}$, each of
these functions can be tansformed into a cubic function via

$f_{lC}=\textstyle{\frac{1}{3}}\lambda^{3}-\sigma_{l1}^{2}\lambda+\sigma_{l2},\qquad{\rm
and}\qquad{\bar{f}}_{lC}=\textstyle{\frac{1}{3}}\lambda^{3}-{\bar{\sigma}}_{l1}^{2}\lambda+{\bar{\sigma}}_{l2},$
(3.48)

where

$\sigma_{l1}=\left[\textstyle{\frac{3}{4}}\left(f_{lC}\left|{}_{\tau=\tau_{l{\rm
max}}}-f_{lC}\left|{}_{\tau=\tau_{l{\rm
min}}}\right.\right.\right)\right]^{1/3},$ (3.49)
${\bar{\sigma}}_{l1}=\left[\textstyle{\frac{3}{4}}\left({\bar{f}}_{lC}\left|{}_{\tau={\bar{\tau}}_{l{\rm
max}}}-{\bar{f}}_{lC}\left|{}_{\tau={\bar{\tau}}_{l{\rm
min}}}\right.\right.\right)\right]^{1/3},$ (3.50)

and

$\sigma_{l2}=\textstyle{\frac{1}{2}}\left(f_{lC}\left|{}_{\tau=\tau_{l{\rm
max}}}+f_{lC}\left|{}_{\tau=\tau_{l{\rm min}}}\right.\right.\right),$ (3.51)
${\bar{\sigma}}_{l2}=\textstyle{\frac{1}{2}}\left({\bar{f}}_{lC}\left|{}_{\tau={\bar{\tau}}_{l{\rm
max}}}+{\bar{f}}_{lC}\left|{}_{\tau={\bar{\tau}}_{l{\rm
min}}}\right.\right.\right).$ (3.52)

The transformation of the integration variable in (3.45) from $\tau$ to
$\lambda$ thus results in

$\displaystyle{\bf E}^{\rm uc}$ $\displaystyle\simeq$
$\displaystyle\frac{2j_{0}}{3\omega}\sum_{l=1}^{2}\int_{0}^{\infty}{\rm
d}k\int_{0}^{\pi}{\rm d}\lambda\,b\,\bigg{\\{}\bigg{[}\frac{{\rm i}}{3}{\rm
H}(k_{2}-k){\bf P}_{l}+\sqrt{\frac{2}{\pi ak}}\exp\left(\frac{{\rm
i}\pi}{4}\right)$ (3.53) $\displaystyle\times\left({\rm H}(k-k_{1}){\bf
Q}_{l}-\frac{1}{3a}{\rm H}(k-k_{2}){\bf P}_{l}\right)\bigg{]}\frac{{\rm
d}\tau}{{\rm d}\lambda}\bigg{|}_{f_{lC}}\exp\left[{\rm
i}k\left(\textstyle{\frac{1}{3}}\lambda^{3}-\sigma_{l1}^{2}\lambda+\sigma_{l2}\right)\right]$
$\displaystyle+\sqrt{\frac{2}{\pi ak}}\exp\left(-\frac{{\rm
i}\pi}{4}\right)\left({\rm H}(k-k_{1}){\bf Q}_{l}-\frac{1}{a}{\rm
H}(k-k_{2}){\bf P}_{l}\right)$ $\displaystyle\times\frac{{\rm d}\tau}{{\rm
d}\lambda}\bigg{|}_{{\bar{f}}_{lC}}\exp\left[{\rm
i}k\left(\textstyle{\frac{1}{3}}\lambda^{3}-{\bar{\sigma}}_{l1}^{2}\lambda+{\bar{\sigma}}_{l2}\right)\right]\bigg{\\}},$

whose asymptotic value for large $k$ can be written, as shown by [5], as

$\displaystyle{\bf E}^{\rm uc}$ $\displaystyle\simeq$
$\displaystyle\frac{2j_{0}}{3\omega}\sum_{l=1}^{2}\int_{0}^{\infty}{\rm
d}k\bigg{\\{}\exp({\rm i}k\sigma_{l2})\bigg{[}\mbox{{\boldmath${\cal
P}$}}_{l}\int_{-\infty}^{\infty}{\rm d}\lambda\,\exp[{\rm
i}k(\textstyle{\frac{1}{3}}\lambda^{3}-\sigma_{l1}^{2}\lambda)]$
$\displaystyle+\mbox{{\boldmath${\cal Q}$}}_{l}\int_{-\infty}^{\infty}{\rm
d}\lambda\,\lambda\exp[{\rm
i}k(\textstyle{\frac{1}{3}}\lambda^{3}-\sigma_{l1}^{2}\lambda)]\bigg{]}+\exp({\rm
i}k{\bar{\sigma}}_{l2})$
$\displaystyle\times\bigg{[}\mbox{{\boldmath${\bar{\cal
P}}$}}_{l}\int_{-\infty}^{\infty}{\rm d}\lambda\,\exp[{\rm
i}k(\textstyle{\frac{1}{3}}\lambda^{3}-{\bar{\sigma}}_{l1}^{2}\lambda)]+\mbox{{\boldmath${\bar{\cal
Q}}$}}_{l}\int_{-\infty}^{\infty}{\rm d}\lambda\,\lambda\exp[{\rm
i}k(\textstyle{\frac{1}{3}}\lambda^{3}-{\bar{\sigma}}_{l1}^{2}\lambda)]\bigg{]}\bigg{\\}},$

where

$\mbox{{\boldmath${\cal P}$}}_{l}=\frac{1}{2}\left(\mbox{{\boldmath${\cal
K}$}}_{l}\left|{}_{\tau=\tau_{l{\rm min}}}+\mbox{{\boldmath${\cal
K}$}}_{l}\left|{}_{\tau=\tau_{l{\rm max}}}\right.\right.\right),$ (3.55)
$\mbox{{\boldmath${\cal
Q}$}}_{l}=\frac{1}{2\sigma_{l1}}\left(\mbox{{\boldmath${\cal
K}$}}_{l}\left|{}_{\tau=\tau_{l{\rm min}}}-\mbox{{\boldmath${\cal
K}$}}_{l}\left|{}_{\tau=\tau_{l{\rm max}}}\right.\right.\right),$ (3.56)
$\mbox{{\boldmath${\bar{\cal
P}}$}}_{l}=\frac{1}{2}\left(\mbox{{\boldmath${\bar{\cal
K}}$}}_{l}\left|{}_{\tau={\bar{\tau}}_{l{\rm
min}}}+\mbox{{\boldmath${\bar{\cal
K}}$}}_{l}\left|{}_{\tau={\bar{\tau}}_{l{\rm max}}}\right.\right.\right),$
(3.57) $\mbox{{\boldmath${\bar{\cal
Q}}$}}_{l}=\frac{1}{2{\bar{\sigma}}_{l1}}\left(\mbox{{\boldmath${\bar{\cal
K}}$}}_{l}\left|{}_{\tau={\bar{\tau}}_{l{\rm
min}}}-\mbox{{\boldmath${\bar{\cal
K}}$}}_{l}\left|{}_{\tau={\bar{\tau}}_{l{\rm max}}}\right.\right.\right),$
(3.58)

with

$\displaystyle\mbox{{\boldmath${\cal K}$}}_{l}$ $\displaystyle=$
$\displaystyle b\frac{{\rm d}\tau}{{\rm
d}\lambda}\bigg{|}_{f_{lC}}\bigg{[}\frac{{\rm i}}{3}{\rm H}(k_{2}-k){\bf
P}_{l}+\sqrt{\frac{2}{\pi ak}}\exp\left(\frac{{\rm i}\pi}{4}\right)$ (3.59)
$\displaystyle\times\left({\rm H}(k-k_{1}){\bf Q}_{l}-\frac{1}{3a}{\rm
H}(k-k_{2}){\bf P}_{l}\right)\bigg{]},$ $\mbox{{\boldmath${\bar{\cal
K}}$}}_{l}=b\sqrt{\frac{2}{\pi ak}}\exp\left(-\frac{{\rm
i}\pi}{4}\right)\frac{{\rm d}\tau}{{\rm
d}\lambda}\bigg{|}_{{\bar{f}}_{lC}}\left[{\rm H}(k-k_{1}){\bf
Q}_{l}-\frac{1}{a}{\rm H}(k-k_{2}){\bf P}_{l}\right]$ (3.60)

(see also [8]). Evaluating the integrals over $\lambda$ (cf. [9]), we obtain

$\displaystyle{\bf E}^{\rm uc}$ $\displaystyle\simeq$ $\displaystyle\frac{4\pi
j_{0}}{3\omega}\sum_{l=1}^{2}\int_{0}^{\infty}{\rm
d}k\,k^{-1/3}\bigg{\\{}\exp({\rm i}k\sigma_{l2})$ (3.61)
$\displaystyle\times\left[\mbox{{\boldmath${\cal P}$}}_{l}{\rm
Ai}(-k^{2/3}\sigma_{l1}^{2})-{\rm i}k^{-1/3}\mbox{{\boldmath${\cal
Q}$}}_{l}{\rm Ai}^{\prime}(-k^{2/3}\sigma_{l1}^{2})\right]+\exp({\rm
i}k{\bar{\sigma}}_{l2})$ $\displaystyle\times\left[\mbox{{\boldmath${\bar{\cal
P}}$}}_{l}{\rm Ai}\left(-k^{2/3}{\bar{\sigma}}_{l1}^{2}\right)-{\rm
i}k^{-1/3}\mbox{{\boldmath${\bar{\cal Q}}$}}_{l}{\rm
Ai}^{\prime}\left(-k^{2/3}{\bar{\sigma}}_{l1}^{2}\right)\right]\bigg{\\}},$

where ${\rm Ai}$ and ${\rm Ai}^{\prime}$ are the Airy function and the
derivative of the Airy function with respect to its argument, respectively.
Note that $(\mbox{{\boldmath${\cal P}$}}_{l},\mbox{{\boldmath${\cal
Q}$}}_{l})$ are different from the vectors $({\bf P}_{l},{\bf Q}_{l})$ defined
in (3.26)–(3.27).

The indeterminate quantities $({\rm d}\tau/{\rm
d}\mu)_{f_{lC}}|_{\tau={\tau_{l{\rm min}}}}$ and $({\rm d}\tau/{\rm
d}\mu)_{f_{lC}}|_{\tau={\tau_{l{\rm max}}}}$ that appear in (3.61) have to be
found by repeated differentiation of (3.48) with respect to $\lambda$ and the
evaluation of the resulting relations

$\frac{\partial f_{lC}}{\partial\tau}\frac{{\rm d}\tau}{{\rm
d}\lambda}=\lambda^{2}-\sigma_{l1}^{2},$ (3.62)

and

$\frac{\partial^{2}f_{lC}}{\partial\tau^{2}}\left(\frac{{\rm d}\tau}{{\rm
d}\lambda}\right)^{2}+\frac{\partial f_{lC}}{\partial\tau}\frac{{\rm
d}^{2}\tau}{{\rm d}\lambda^{2}}=2\lambda,$ (3.63)

at $\tau={\tau_{l{\rm min}}}$ and $\tau={\tau_{l{\rm max}}}$. This procedure
results in

$\left(\frac{{\rm d}\tau}{{\rm
d}\lambda}\right)_{f_{lC}}\Bigg{|}_{\lambda=\pm\sigma_{l1}}=\left(\frac{2\lambda}{\partial^{2}f_{lC}/\partial\tau^{2}}\Big{|}_{\lambda=\pm\sigma_{l1}}\right)^{1/2},$
(3.64)

in which $\lambda=\pm\sigma_{l1}$ are the images of $\tau={\tau_{l{\rm min}}}$
and $\tau={\tau_{l{\rm max}}}$, respectively. Likewise,

$\left(\frac{{\rm d}\tau}{{\rm
d}\lambda}\right)_{{\bar{f}}_{lC}}\Bigg{|}_{\lambda=\pm{\bar{\sigma}}_{l1}}=\left(\frac{2\lambda}{\partial^{2}{\bar{f}}_{lC}/\partial\tau^{2}}\Big{|}_{\lambda=\pm{\bar{\sigma}}_{l1}}\right)^{1/2},$
(3.65)

a result that can be obtained by applying the procedure described above to the
function ${\bar{f}}_{lC}$. In this expression, too,
$\lambda=\pm{\bar{\sigma}}_{l1}$ are the images of $\tau={{\bar{\tau}}_{l{\rm
min}}}$ and $\tau={{\bar{\tau}}_{l{\rm max}}}$, respectively.

Note that every one of the terms appearing in (3.61) would contribute toward
the value of the radiation field only when the phase functions $f_{1C}$,
$f_{2C}$, ${\bar{f}}_{1C}$ and ${\bar{f}}_{2C}$ each have two turning points
(see figure 3.2). If any one of these functions varies monotonically with
$\tau$, for the prescribed values of $\alpha$ and $\theta_{P}$, then the terms
entailing the (non-existent) $\tau$ coordinates of its maximum and minimum
should be omitted from (3.61).

###  3.6 The remaining integration with respect to $k$

To perform the $k$-integration in (3.61) we first need to render the
$k$-dependence of the coefficients $\mbox{{\boldmath${\cal P}$}}_{l}$,
$\mbox{{\boldmath${\cal Q}$}}_{l}$, $\mbox{{\boldmath${\bar{\cal P}}$}}_{l}$
and $\mbox{{\boldmath${\bar{\cal Q}}$}}_{l}$ that appear in this equation
explicit. This can be done by rewriting (3.55)–(3.60) as

$\left[\matrix{\mbox{{\boldmath${\cal P}$}}_{l}\cr\mbox{{\boldmath${\cal
Q}$}}_{l}\cr}\right]={\rm H}(k_{2}-k)\left[\matrix{\mbox{{\boldmath${\cal
P}$}}^{(0)}_{l}\cr\mbox{{\boldmath${\cal Q}$}}^{(0)}_{l}\cr}\right]+\left({\rm
H}(k-k_{1})\left[\matrix{\mbox{{\boldmath${\cal
P}$}}^{(1)}_{l}\cr\mbox{{\boldmath${\cal Q}$}}^{(1)}_{l}\cr}\right]+{\rm
H}(k-k_{2})\left[\matrix{\mbox{{\boldmath${\cal
P}$}}^{(2)}_{l}\cr\mbox{{\boldmath${\cal
Q}$}}^{(2)}_{l}\cr}\right]\right)\,k^{-1/2},$ (3.66)
$\left[\matrix{\mbox{{\boldmath${\bar{\cal
P}}$}}_{l}\cr\mbox{{\boldmath${\bar{\cal Q}}$}}_{l}\cr}\right]=\left({\rm
H}(k-k_{1})\left[\matrix{\mbox{{\boldmath${\bar{\cal
P}}$}}^{(1)}_{l}\cr\mbox{{\boldmath${\bar{\cal
Q}}$}}^{(1)}_{l}\cr}\right]+{\rm
H}(k-k_{2})\left[\matrix{\mbox{{\boldmath${\bar{\cal
P}}$}}^{(2)}_{l}\cr\mbox{{\boldmath${\bar{\cal
Q}}$}}^{(2)}_{l}\cr}\right]\right)\,k^{-1/2},$ (3.67)

where

$\mbox{{\boldmath${\cal
P}$}}^{(j)}_{l}=\frac{1}{2}\left(\mbox{{\boldmath${\cal
K}$}}^{(j)}_{l}\left|{}_{\tau=\tau_{l{\rm min}}}+\mbox{{\boldmath${\cal
K}$}}^{(j)}_{l}\left|{}_{\tau=\tau_{l{\rm max}}}\right.\right.\right),\qquad
j=0,1,2,$ (3.68) $\mbox{{\boldmath${\cal
Q}$}}^{(j)}_{l}=\frac{1}{2\sigma_{l1}}\left(\mbox{{\boldmath${\cal
K}$}}^{(j)}_{l}\left|{}_{\tau=\tau_{l{\rm min}}}-\mbox{{\boldmath${\cal
K}$}}^{(j)}_{l}\left|{}_{\tau=\tau_{l{\rm max}}}\right.\right.\right),\qquad
j=0,1,2,$ (3.69) $\mbox{{\boldmath${\bar{\cal
P}}$}}^{(j)}_{l}=\frac{1}{2}\left(\mbox{{\boldmath${\bar{\cal
K}}$}}^{(j)}_{l}\left|{}_{\tau={\bar{\tau}}_{l{\rm
min}}}+\mbox{{\boldmath${\bar{\cal
K}}$}}^{(j)}_{l}\left|{}_{\tau={\bar{\tau}}_{l{\rm
max}}}\right.\right.\right),\qquad j=1,2,$ (3.70) $\mbox{{\boldmath${\bar{\cal
Q}}$}}^{(j)}_{l}=\frac{1}{2{\bar{\sigma}}_{l1}}\left(\mbox{{\boldmath$\bar{{\cal
K}}$}}^{(j)}_{l}\left|{}_{\tau={\bar{\tau}}_{l{\rm
min}}}-\mbox{{\boldmath${\bar{\cal
K}}$}}^{(j)}_{l}\left|{}_{\tau={\bar{\tau}}_{l{\rm
max}}}\right.\right.\right),\qquad j=1,2,$ (3.71)

with

$\mbox{{\boldmath${\cal K}$}}^{(0)}_{l}=\frac{{\rm i}}{3}b\,\frac{{\rm
d}\tau}{{\rm d}\lambda}\bigg{|}_{f_{lC}}{\bf P}_{l},$ (3.72)
$\left[\matrix{\mbox{{\boldmath${\cal K}$}}^{(1)}_{l}\cr\mbox{{\boldmath${\cal
K}$}}^{(2)}_{l}\cr}\right]=b\,\sqrt{\frac{2}{\pi a}}\frac{{\rm d}\tau}{{\rm
d}\lambda}\bigg{|}_{f_{lC}}\exp\left(\frac{{\rm
i}\pi}{4}\right)\left[\matrix{{\bf Q}_{l}\cr-\frac{1}{3a}{\bf
P}_{l}\cr}\right],$ (3.73) $\left[\matrix{\mbox{{\boldmath${\bar{\cal
K}}$}}^{(1)}_{l}\cr\mbox{{\boldmath${\bar{\cal
K}}$}}^{(2)}_{l}\cr}\right]=b\,\sqrt{\frac{2}{\pi a}}\frac{{\rm d}\tau}{{\rm
d}\lambda}\bigg{|}_{{\bar{f}}_{lC}}\exp\left(-\frac{{\rm
i}\pi}{4}\right)\left[\matrix{{\bf Q}_{l}\cr-\frac{1}{a}{\bf
P}_{l}\cr}\right].$ (3.74)

The quantities appearing in (3.68)–(3.74) are all independent of $k$, so that
the remaining integrals in (3.61) are–according to (3.66)–of two types: those
that extend over a finite interval $0\leq k\leq k_{i}$ and those that extend
over a semi-infinite interval $k_{i}\leq k<\infty$, where $i$ is either $1$ or
$2$.

The rapid oscillations of the Airy functions for large $k$ (cf. [9]) ensure
that the integrals over $0\leq k\leq k_{i}$ in (3.61) receive their main
contributions from the vicinity of $k=0$. The ranges of these integrals can
therefore be extended to $0<k<\infty$ without introducing an appreciable
error. Moreover, each of the integrals over $k_{i}<k<\infty$ is accurately
approximated (according to numerical integrations) if it is written as the
difference between two integrals with the same integrands but with the ranges
$0\leq k<\infty$ and $0\leq k\leq k_{i}$ and the Airy function in the
integrand of the integral over $0\leq k\leq k_{i}$ is replaced by its value at
$k=0$, as in (3.80) below.

Once these approximations are applied to (3.61) and the Airy functions that
appear in the resulting equation are expressed in terms of Bessel functions,
the $k$-integrals in question assume the form of tabulated Fourier transforms
(cf., [9, 10]) and can be evaluated analytically to arrive at

${\bf E}^{\rm uc}={\bf E}^{\rm uc}_{\cal P}+{\bf E}^{\rm uc}_{\cal Q}$ (3.75)

with

$\displaystyle{\bf E}^{\rm uc}_{\cal P}$ $\displaystyle\simeq$
$\displaystyle\frac{2^{4/3}\pi
j_{0}}{3^{1/3}\omega}\sum_{l=1}^{2}\left\\{\sigma_{l1}^{-2}{\cal
G}_{1}(\eta_{l})\mbox{{\boldmath${\cal
P}$}}^{(0)}_{l}+\left(\frac{2}{3\sigma_{l1}}\right)^{1/2}\left[{\cal
G}_{3}(\eta_{l},\kappa_{1})\mbox{{\boldmath${\cal P}$}}^{(1)}_{l}+{\cal
G}_{3}(\eta_{l},\kappa_{2})\mbox{{\boldmath${\cal
P}$}}^{(2)}_{l}\right]\right.$ (3.76)
$\displaystyle\left.+\left(\frac{2}{3{\bar{\sigma}}_{l1}}\right)^{1/2}\left[{\cal
G}_{3}({\bar{\eta}}_{l},{\bar{\kappa}}_{1})\mbox{{\boldmath${\bar{\cal
P}}$}}^{(1)}_{l}+{\cal
G}_{3}({\bar{\eta}}_{l},{\bar{\kappa}}_{2})\mbox{{\boldmath${\bar{\cal
P}}$}}^{(2)}_{l}\right]\right\\}$

and

$\displaystyle{\bf E}^{\rm uc}_{\cal Q}$ $\displaystyle\simeq$
$\displaystyle\frac{2^{5/3}\pi j_{0}}{3^{2/3}{\rm
i}\,\omega}\sum_{l=1}^{2}\left\\{\sigma_{l1}^{-1}{\cal
G}_{2}(\eta_{l})\mbox{{\boldmath${\cal
Q}$}}^{(0)}_{l}+\left(\frac{2\sigma_{l1}}{3}\right)^{1/2}\left[{\cal
G}_{4}(\eta_{l},\kappa_{1})\mbox{{\boldmath${\cal Q}$}}^{(1)}_{l}+{\cal
G}_{4}(\eta_{l},\kappa_{2})\mbox{{\boldmath${\cal
Q}$}}^{(2)}_{l}\right]\right.$ (3.77)
$\displaystyle\left.+\left(\frac{2{\bar{\sigma}}_{l1}}{3}\right)^{1/2}\left[{\cal
G}_{4}({\bar{\eta}}_{l},{\bar{\kappa}}_{1})\mbox{{\boldmath${\bar{\cal
Q}}$}}^{(1)}_{l}+{\cal
G}_{4}({\bar{\eta}}_{l},{\bar{\kappa}}_{2})\mbox{{\boldmath${\bar{\cal
Q}}$}}^{(2)}_{l}\right]\right\\}$

where

$\displaystyle{\cal G}_{1}(x)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}{\rm d}\kappa\exp({\rm i}\kappa
x)\kappa^{-1/3}{\rm
Ai}\left[-\left({\textstyle\frac{3}{2}}\kappa\right)^{2/3}\right]$
$\displaystyle=$
$\displaystyle\left({\textstyle\frac{2}{3}}\right)^{2/3}(1-x^{2})^{-1/2}[{\rm
H}(x+1)-{\rm H}(x-1)]\cos\left({\textstyle\frac{1}{3}}\arcsin
x\right)+18^{-1/3}{\rm i}(x^{2}-1)^{-1/2}$ $\displaystyle\times\left\\{{\rm
H}(x-1)\left[\exp({\rm i}\pi/6)\left(\sqrt{x^{2}-1}+x\right)^{-1/3}+\exp(-{\rm
i}\pi/6)\left(\sqrt{x^{2}-1}+x\right)^{1/3}\right]\right.$
$\displaystyle\left.-{\rm H}(-x-1)\left[\exp(-{\rm
i}\pi/6)\left(\sqrt{x^{2}-1}-x\right)^{-1/3}+\exp({\rm
i}\pi/6)\left(\sqrt{x^{2}-1}-x\right)^{1/3}\right]\right\\},$
$\displaystyle{\cal G}_{2}(x)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}{\rm d}\kappa\exp({\rm i}\kappa
x)\kappa^{-2/3}{\rm
Ai}^{\prime}\left[-\left({\textstyle\frac{3}{2}}\kappa\right)^{2/3}\right]$
$\displaystyle=$ $\displaystyle\left({\textstyle\frac{2}{3}}\right)^{1/3}{\rm
i}(1-x^{2})^{-1/2}[{\rm H}(x+1)-{\rm
H}(x-1)]\sin\left({\textstyle\frac{2}{3}}\arcsin x\right)+12^{-1/3}{\rm
i}(x^{2}-1)^{-1/2}$ $\displaystyle\times\left\\{{\rm H}(x-1)\left[\exp({\rm
i}\pi/3)\left(\sqrt{x^{2}-1}+x\right)^{-2/3}-\exp(-{\rm
i}\pi/3)\left(\sqrt{x^{2}-1}+x\right)^{2/3}\right]\right.$
$\displaystyle\left.-{\rm H}(-x-1)\left[\exp(-{\rm
i}\pi/3)\left(\sqrt{x^{2}-1}-x\right)^{-2/3}-\exp({\rm
i}\pi/3)\left(\sqrt{x^{2}-1}-x\right)^{2/3}\right]\right\\},$
$\displaystyle{\cal G}_{3}(x,\kappa_{0})$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}{\rm d}\kappa\,\exp({\rm i}\kappa
x)\kappa^{-5/6}{\rm
Ai}\left[-\left({\textstyle\frac{3}{2}}\kappa\right)^{2/3}\right]-{\rm
Ai}(0)\int_{0}^{\kappa_{0}}{\rm d}\kappa\,\exp({\rm i}\kappa x)\kappa^{-5/6}$
(3.80) $\displaystyle=$ $\displaystyle{\rm
H}(x-1)\left[\frac{\Gamma(5/6)}{6^{2/3}x^{5/6}\Gamma(4/3)}\exp\left(\frac{5{\rm
i}\pi}{12}\right)\,{}_{2}F_{1}\left(\frac{11}{12},\frac{5}{12};\frac{4}{3};\frac{1}{x^{2}}\right)\right.$
$\displaystyle\left.+\frac{\Gamma(1/6)}{3^{2/3}x^{1/6}\Gamma(2/3)}\exp\left(\frac{{\rm
i}\pi}{12}\right)\,{}_{2}F_{1}\left(\frac{7}{12},\frac{1}{12};\frac{2}{3};\frac{1}{x^{2}}\right)\right]$
$\displaystyle+\frac{2^{2/3}\pi}{3^{1/6}}[{\rm H}(x+1)-{\rm
H}(x-1)]\left[\frac{1}{\Gamma(7/12)\Gamma(11/12)}\,{}_{2}F_{1}\left(\frac{5}{12},\frac{1}{12};\frac{1}{2};x^{2}\right)\right.$
$\displaystyle\left.+\frac{2{\rm
i}x}{\Gamma(5/12)\Gamma(1/12)}\,{}_{2}F_{1}\left(\frac{11}{12},\frac{7}{12};\frac{3}{2};x^{2}\right)\right]$
$\displaystyle+{\rm
H}(-x-1)\left[\frac{\Gamma(5/6)}{6^{2/3}|x|^{5/6}\Gamma(4/3)}\exp\left(-\frac{5{\rm
i}\pi}{12}\right)\,{}_{2}F_{1}\left(\frac{11}{12},\frac{5}{12};\frac{4}{3};\frac{1}{x^{2}}\right)\right.$
$\displaystyle\left.+\frac{\Gamma(1/6)}{3^{2/3}|x|^{1/6}\Gamma(2/3)}\exp\left(-\frac{{\rm
i}\pi}{12}\right)\,{}_{2}F_{1}\left(\frac{7}{12},\frac{1}{12};\frac{2}{3};\frac{1}{x^{2}}\right)\right]$
$\displaystyle-{\rm Ai}(0)\exp\left[\frac{{\rm i}\pi}{12}{\rm
sgn}(x)\right]|x|^{-1/6}\left[\Gamma\left(\frac{1}{6}\right)-\Gamma\left(\frac{1}{6},-{\rm
i}\kappa_{0}x\right)\right],$ $\displaystyle{\cal G}_{4}(x,\kappa_{0})$
$\displaystyle=$ $\displaystyle\left\\{\int_{0}^{\infty}{\rm
d}\kappa\,\exp({\rm i}\kappa x)\kappa^{\nu-1}{\rm
Ai}^{\prime}\left[-\left({\textstyle\frac{3}{2}}\kappa\right)^{2/3}\right]-{\rm
Ai}^{\prime}(0)\int_{0}^{\kappa_{0}}{\rm d}\kappa\,\exp({\rm i}\kappa
x)\kappa^{\nu-1}\right\\}_{\nu=-1/6}$ (3.81) $\displaystyle=$
$\displaystyle{\rm
H}(x-1)\left[\frac{\Gamma(7/6)}{48^{1/3}x^{7/6}\Gamma(5/3)}\exp\left(\frac{7{\rm
i}\pi}{12}\right)\,{}_{2}F_{1}\left(\frac{13}{12},\frac{7}{12};\frac{5}{3};\frac{1}{x^{2}}\right)\right.$
$\displaystyle\left.-\frac{x^{1/6}\Gamma(-1/6)}{3^{1/3}\Gamma(1/3)}\exp\left(-\frac{{\rm
i}\pi}{12}\right)\,{}_{2}F_{1}\left(\frac{5}{12},-\frac{1}{12};\frac{1}{3};\frac{1}{x^{2}}\right)\right]$
$\displaystyle+12^{1/6}\pi[{\rm H}(x+1)-{\rm
H}(x-1)]\left[\frac{1}{\Gamma(13/12)\Gamma(5/12)}\,{}_{2}F_{1}\left(\frac{7}{12},-\frac{1}{12};\frac{1}{2};x^{2}\right)\right.$
$\displaystyle\left.-\frac{2{\rm
i}x}{\Gamma(7/12)\Gamma(-1/12)}\,{}_{2}F_{1}\left(\frac{13}{12},\frac{5}{12};\frac{3}{2};x^{2}\right)\right]$
$\displaystyle+{\rm
H}(-x-1)\left[\frac{\Gamma(7/6)}{48^{1/3}|x|^{7/6}\Gamma(5/3)}\exp\left(-\frac{7{\rm
i}\pi}{12}\right)\,{}_{2}F_{1}\left(\frac{13}{12},\frac{7}{12};\frac{5}{3};\frac{1}{x^{2}}\right)\right.$
$\displaystyle\left.-\frac{|x|^{1/6}\Gamma(-1/6)}{3^{1/3}\Gamma(1/3)}\exp\left(\frac{{\rm
i}\pi}{12}\right)\,{}_{2}F_{1}\left(\frac{5}{12},-\frac{1}{12};\frac{1}{3};\frac{1}{x^{2}}\right)\right]$
$\displaystyle-{\rm Ai}^{\prime}(0)\exp\left[-\frac{{\rm i}\pi}{12}{\rm
sgn}(x)\right]|x|^{1/6}\left[\Gamma\left(-\frac{1}{6}\right)-\Gamma\left(-\frac{1}{6},-{\rm
i}\kappa_{0}x\right)\right],$

and

$\eta_{l}=\frac{3\sigma_{l2}}{2\sigma_{l1}^{3}}=\frac{f_{lC}|_{\tau=\tau_{l{\rm
max}}}+f_{lC}|_{\tau=\tau_{l{\rm min}}}}{f_{lC}|_{\tau=\tau_{l{\rm
max}}}-f_{lC}|_{\tau=\tau_{l{\rm min}}}},$ (3.82)
${\bar{\eta}}_{l}=\frac{3{\bar{\sigma}}_{l2}}{2{\bar{\sigma}}_{l1}^{3}}=\frac{{\bar{f}}_{lC}|_{\tau={\bar{\tau}}_{l{\rm
max}}}+{\bar{f}}_{lC}|_{\tau={\bar{\tau}}_{l{\rm
min}}}}{{\bar{f}}_{lC}|_{\tau={\bar{\tau}}_{l{\rm
max}}}-{\bar{f}}_{lC}|_{\tau={\bar{\tau}}_{l{\rm min}}}}.$ (3.83)

In the above expressions, ${}_{2}F_{1}$ is the hypergeometric function,
$\Gamma(\nu,x)$ is the incomplete gamma function and the integration variable
$\kappa$ is related to $k$ via
$\kappa={\textstyle\frac{2}{3}}\sigma_{l1}^{3}k$. Note that ${\cal G}_{4}$ is
here found by performing the integrations in the first line of (3.81) for
$\nu>0$ and evaluating the analytic continuation of the resulting expression
at $\nu=-1/6$. This yields a value for ${\cal G}_{4}$ that exactly agrees with
the outcome of the numerical evaluation of the integrals defining this
function.

According to (3.82), the variable $\eta_{l}$ that appears in the arguments of
the functions ${\cal G}_{1},\cdots,{\cal G}_{4}$ equals 1 when
$f_{lC}|_{\tau=\tau_{l{\rm min}}}=0$ and equals $-1$ when
$f_{lC}|_{\tau=\tau_{l{\rm max}}}=0$. This holds true, as indicated by (3.83),
also for ${\bar{\eta}}_{l}$ when ${\bar{f}}_{lC}$ vanishes at its maximum or
minimum. Moreover, $\eta_{l}$ assumes an infinitely large value at the point
where the maximum and minimum of $f_{lC}$ coalesce and so $\sigma_{l1}$
vanishes, an unbounded upper limit that is also approached by
${\bar{\eta}}_{l}$ as the turning points of ${\bar{f}}_{lC}$ coalesce to form
an inflection point.

Figure 3.4: Real part (in red) and imaginary part (in blue) of the functions
${\cal G}_{1},\cdots,{\cal G}_{4}$ given by (LABEL:4E78)–(3.82) for
$\kappa_{1}=10^{-2}$ and $\kappa_{2}=4\times 10^{-2}$. The limiting values of
these functions across their discontinuities at $x=\pm 1$ are given by
(3.84)–(3.87).

###  3.7 The divergence that arises from the vanishing width of the current
sheet and its regularization

The functions ${\cal G}_{1},\cdots,{\cal G}_{4}$ that appear in (3.76) and
(3.77) approach the following divergent values as their argument $x$ tends to
$\pm 1$ from inside and outside the interval $-1<x<1$ (see figure 3.4):

$\lim_{x\to\pm 1}{\cal G}_{1}=(2^{2}\times 3)^{-1/6}\left[\pm{\rm i}\,{\rm
H}(\pm x-1)(x^{2}-1)^{-1/2}+{\rm H}(1\mp x)(1-x^{2})^{-1/2}\right],$ (3.84)
$\lim_{x\to\pm 1}{\cal G}_{2}=(2^{-4}\times 3)^{1/6}\left[-{\rm H}(\pm
x-1)(x^{2}-1)^{1/2}\pm{\rm i}\,{\rm H}(1\mp x)(1-x^{2})^{-1/2}\right],$ (3.85)
$\lim_{x\to\pm 1}{\cal G}_{3}=(2^{5}\times
3)^{-1/6}\pi^{-1/2}\exp\left(\mp\frac{3{\rm i}\pi}{4}\right)\left[{\rm H}(\pm
x-1)\ln(x^{2}-1)+{\rm H}(1\mp x)\ln(1-x^{2})\right],$ (3.86) $\lim_{x\to\pm
1}{\cal G}_{4}=\frac{3^{1/6}}{2^{7/6}\pi^{1/2}}\exp\left(\mp\frac{{\rm
i}\pi}{4}\right)\left[{\rm H}(\pm x-1)\ln(x^{2}-1)+{\rm H}(1\mp
x)\ln(1-x^{2})\right]$ (3.87)

(cf., §15.4 of [9]). The radiation field ${\bf E}^{\rm uc}$ correspondingly
diverges when $\eta_{l}=\pm 1$ (or ${\bar{\eta}}_{l}=\pm 1$), i.e., when
$f_{lC}$ (or ${\bar{f}}_{lC}$) vanishes at one of its turning points.

The above singularities in the expression for the radiation field stem from
assigning a zero width to the current sheet. Because its charge and current
densities are proportional to a Dirac delta function, the current sheet
described by (1.18) and (1.19) has a vanishing thickness. The vanishing
thickness of the current sheet in turn results in an infinitely wide range of
values for the variable $k$ that appears in its Fourier representation (see
(3.11)). But, given that it is created by the coordinated motion of aggregates
of subluminally moving particles, a superluminally moving source is
necessarily volume-distributed: it can neither be point-like nor be
distributed over a line or a surface [11]. In a physically more realistic
model of the magnetosphere, where the processes that occur on plasma scales
within the current sheet are taken into account, this sheet would have a non-
zero thickness and the singularities in question would not occur.

To circumvent the divergence that arises from overlooking the finite width of
the current sheet, we will here replace the integration domain $0\leq
k<\infty$ in (3.61) with $0\leq k\leq k_{\rm u}$ and treat the upper limit
$k_{\rm u}$ $(\gg 1)$ on the range of values of $|k|$ as a free parameter. The
thickness of the current sheet is dictated by microphysical processes that are
not well understood: the standard Harris solution of the Vlasov-Maxwell
equations that is commonly used in analysing a current sheet [12] is not
applicable in the present case because the current sheet in the magnetosphere
of a neutron star moves faster than light and so has no rest frame.
Introducing the upper limit $k_{\rm u}$ is tantamount to assuming that the
(unknown) thickness of the current sheet is of the order of $1/k_{\rm u}$ in
units of the light-cylinder radius $c/\omega$.

Figure 3.5: Real part (in red) and imaginary part (in blue) of the regularized
versions (${\cal G}^{\rm r}_{1},\cdots,{\cal G}^{\rm r}_{4}$) of the functions
${\cal G}_{1},\cdots,{\cal G}_{4}$ for $\kappa_{1}=10^{-2}$,
$\kappa_{2}=4\times 10^{-2}$ and $\kappa_{\rm u}=10^{2}$. The higher the value
of $\kappa_{\rm u}$, the larger are the absolute values of the maxima and
minima of these functions and the shorter are the wavelengths of their
modulations (microstructure). In the limit $\kappa_{\rm u}\to\infty$, these
functions approach those depicted in figure 3.4.

The singularities that arise from $k=\infty$ can thus be regularized by (i)
changing the ranges of those integrals in (LABEL:4E78)–(3.82) that extend over
$0\leq k<\infty$ to $0\leq k\leq k_{\rm u}$, (ii) writing the resulting
integrals over $0\leq k\leq k_{\rm u}$ as

$\int_{0}^{k_{\rm u}}{\rm d}k\cdots=\left(\int_{0}^{\infty}-\int_{k_{\rm
u}}^{\infty}\right){\rm d}k\,\cdots,$ (3.88)

(iii) equating the integrals over $(0,\infty)$ on the right-hand side of
(3.88) to the expressions found in (LABEL:4E78)–(3.82) and (iv) replacing the
Airy functions in the integrands of the integrals over $k_{\rm u}\leq
k<\infty$ by the leading terms in their asymptotic expansions for large $k$
(cf., [9]). The integrals over $k_{\rm u}\leq k<\infty$ can then be performed
analytically to arrive at the following regularized versions, ${\cal G}^{\rm
r}_{1},\cdots,{\cal G}^{\rm r}_{4}$, of ${\cal G}_{1},\cdots,{\cal G}_{4}$:

${\cal G}^{\rm r}_{i}={\cal G}_{i}-{\cal G}^{\rm u}_{i},\qquad i=1,\cdots,4,$
(3.89)

in which the subtracted contributions arising from $k_{\rm u}\leq k<\infty$
are given by

$\displaystyle{\cal G}_{1}^{\rm u}$ $\displaystyle=$
$\displaystyle(2^{5}\times 3)^{-1/6}\pi^{-1/2}\left[\exp\left(\frac{{\rm
i}\pi}{4}\left[{\rm sgn}(x+1)-1\right]\right)\Gamma\left(\frac{1}{2},-{\rm
i}\kappa_{\rm u}(x+1)\right)|x+1|^{-1/2}\right.$ (3.90)
$\displaystyle\left.+\exp\left(\frac{{\rm i}\pi}{4}\left[{\rm
sgn}(x-1)+1\right]\right)\Gamma\left(\frac{1}{2},-{\rm i}\kappa_{\rm
u}(x-1)\right)|x-1|^{-1/2}\right],$ $\displaystyle{\cal G}_{2}^{\rm u}$
$\displaystyle=$ $\displaystyle-\frac{3^{1/6}{\rm
i}}{2^{7/6}\pi^{1/2}}\left[\exp\left(\frac{{\rm i}\pi}{4}\left[{\rm
sgn}(x+1)-1\right]\right)\Gamma\left(\frac{1}{2},-{\rm i}\kappa_{\rm
u}(x+1)\right)|x+1|^{-1/2}\right.$ (3.91)
$\displaystyle\left.-\exp\left(\frac{{\rm i}\pi}{4}\left[{\rm
sgn}(x-1)+1\right]\right)\Gamma\left(\frac{1}{2},-{\rm i}\kappa_{\rm
u}(x-1)\right)|x-1|^{-1/2}\right],$ $\displaystyle{\cal G}_{3}^{\rm u}$
$\displaystyle=$ $\displaystyle(2^{5}\times
3)^{-1/6}\pi^{-1/2}\exp\left(\frac{3{\rm i}\pi}{4}\right)\left\\{{\rm
Ci}\left(\kappa_{\rm u}|x+1|\right)-{\rm i}\,{\rm
sgn}(x+1)\left[\frac{\pi}{2}-{\rm Si}\left(\kappa_{\rm
u}|x+1|\right)\right]\right.$ (3.92) $\displaystyle\left.+{\rm i}\,{\rm
Ci}\left(\kappa_{\rm u}|x-1|\right)+{\rm sgn}(x-1)\left[\frac{\pi}{2}-{\rm
Si}\left(\kappa_{\rm u}|x-1|\right)\right]\right\\},$ $\displaystyle{\cal
G}_{4}^{\rm u}$ $\displaystyle=$
$\displaystyle\frac{3^{1/6}}{2^{7/6}\pi^{1/2}}\exp\left(-\frac{3{\rm
i}\pi}{4}\right)\left\\{-{\rm Ci}\left(\kappa_{\rm u}|x+1|\right)+{\rm
i}\,{\rm sgn}(x+1)\left[\frac{\pi}{2}-{\rm Si}\left(\kappa_{\rm
u}|x+1|\right)\right]\right.$ (3.93) $\displaystyle\left.+{\rm i}\,{\rm
Ci}\left(\kappa_{\rm u}|x-1|\right)+{\rm sgn}(x-1)\left[\frac{\pi}{2}-{\rm
Si}\left(\kappa_{\rm u}|x-1|\right)\right]\right\\}.$

Here, $\kappa_{\rm u}={\textstyle\frac{2}{3}}\sigma_{l1}^{3}k_{\rm u}$ and
${\rm Ci}$ and ${\rm Si}$ are the cosine and sine integrals, respectively. It
can be easily verified that the singularities of the functions ${\cal G}^{\rm
u}_{i}$ at $x=\pm 1$ are the same as those of the functions ${\cal G}_{i}$
that were derived in (LABEL:4E78)–(3.82) and so cancel out in ${\cal
G}_{i}^{\rm r}$ when ${\cal G}_{i}^{\rm u}$ are subtracted from ${\cal G}_{i}$
(see figure 3.5).

The following singularity-free expressions for the two parts ${\bf E}^{\rm
uc}_{\cal P}$ and ${\bf E}^{\rm uc}_{\cal P}$ of the radiation field ${\bf
E}^{\rm uc}$ are thus obtained by replacing the functions ${\cal G}_{i}$ in
(3.76) and (3.77) by their regularized versions ${\cal G}_{i}^{\rm r}$:

$\displaystyle{\bf E}^{\rm uc}_{\cal P}$ $\displaystyle\simeq$
$\displaystyle\frac{2^{4/3}\pi
j_{0}}{3^{1/3}\omega}\sum_{l=1}^{2}\left\\{\sigma_{l1}^{-2}{\cal G}^{\rm
r}_{1}(\eta_{l},\kappa_{\rm u})\mbox{{\boldmath${\cal
P}$}}^{(0)}_{l}+\left(\frac{2}{3\sigma_{l1}}\right)^{1/2}\left[{\cal G}^{\rm
r}_{3}(\eta_{l},\kappa_{1},\kappa_{\rm u})\mbox{{\boldmath${\cal
P}$}}^{(1)}_{l}+{\cal G}^{\rm r}_{3}(\eta_{l},\kappa_{2},\kappa_{\rm
u})\mbox{{\boldmath${\cal P}$}}^{(2)}_{l}\right]\right.$ (3.94)
$\displaystyle\left.+\left(\frac{2}{3{\bar{\sigma}}_{l1}}\right)^{1/2}\left[{\cal
G}^{\rm r}_{3}({\bar{\eta}}_{l},{\bar{\kappa}}_{1},{\bar{\kappa}}_{\rm
u})\mbox{{\boldmath${\bar{\cal P}}$}}^{(1)}_{l}+{\cal G}^{\rm
r}_{3}({\bar{\eta}}_{l},{\bar{\kappa}}_{2},{\bar{\kappa}}_{\rm
u})\mbox{{\boldmath${\bar{\cal P}}$}}^{(2)}_{l}\right]\right\\}$

and

$\displaystyle{\bf E}^{\rm uc}_{\cal Q}$ $\displaystyle\simeq$
$\displaystyle\frac{2^{5/3}\pi j_{0}}{3^{2/3}{\rm
i}\,\omega}\sum_{l=1}^{2}\left\\{\sigma_{l1}^{-1}{\cal G}^{\rm
r}_{2}(\eta_{l},\kappa_{\rm u})\mbox{{\boldmath${\cal
Q}$}}^{(0)}_{l}+\left(\frac{2\sigma_{l1}}{3}\right)^{1/2}\left[{\cal G}^{\rm
r}_{4}(\eta_{l},\kappa_{1},\kappa_{\rm u})\mbox{{\boldmath${\cal
Q}$}}^{(1)}_{l}+{\cal G}^{\rm r}_{4}(\eta_{l},\kappa_{2},\kappa_{\rm
u})\mbox{{\boldmath${\cal Q}$}}^{(2)}_{l}\right]\right.$ (3.95)
$\displaystyle\left.+\left(\frac{2{\bar{\sigma}}_{l1}}{3}\right)^{1/2}\left[{\cal
G}^{\rm r}_{4}({\bar{\eta}}_{l},{\bar{\kappa}}_{1},{\bar{\kappa}}_{\rm
u})\mbox{{\boldmath${\bar{\cal Q}}$}}^{(1)}_{l}+{\cal G}^{\rm
r}_{4}({\bar{\eta}}_{l},{\bar{\kappa}}_{2},{\bar{\kappa}}_{\rm
u})\mbox{{\boldmath${\bar{\cal Q}}$}}^{(2)}_{l}\right]\right\\},$

where ${\bar{\kappa}}_{\rm
u}={\textstyle\frac{2}{3}}{\bar{\sigma}}_{l1}^{3}k_{\rm u}$. The right-hand
sides of (3.94) and (3.95) are indeterminate at points where the maximum and
minimum of $f_{lC}$ (or ${\bar{f}}_{lC}$) coalesce and $\sigma_{l1}$ and
$\eta_{l}$ (or ${\bar{\sigma}}_{l1}$ and ${\bar{\eta}}_{l}$) respectively
assume the values zero and infinity simultaneously. The limiting values of
${\bf E}^{\rm uc}_{\cal P}$ and ${\bf E}^{\rm uc}_{\cal P}$ at such points,
given by l’Hôpital’s rule, are however finite and regular. Note that in the
limit $k_{\rm u}\to\infty$, the functions ${\cal G}^{\rm r}_{1}$ and ${\cal
G}^{\rm r}_{2}$ diverge as $k_{\rm u}^{1/2}$ while the functions ${\cal
G}^{\rm r}_{3}$ and ${\cal G}^{\rm r}_{4}$ diverge as $\ln(k_{\rm u})$.

##  4 Characteristics of the resulting radiation

###  4.1 Pulse profiles and polarization position angles

As a result of receiving contributions from multiple stationary points of the
phase functions $f_{lC}$ and ${\bar{f}}_{lC}$, the unconventional component
${\bf E}^{\rm uc}$ of the radiation field ${\bf E}$ has an amplitude that
exceeds that of the conventional component ${\bf E}^{\rm v}$ of this field by
many orders of magnitude (see (2.42)–(2.44) and (3.28) and the figures below).
We can therefore calculate the Stokes parameters of the present radiation from
the expressions

$I=|E_{\varphi_{P}}^{\rm uc}|^{2}+|E_{\theta_{P}}^{\rm uc}|^{2},\quad
Q=|E_{\varphi_{P}}^{\rm uc}|^{2}-|E_{\theta_{P}}^{\rm uc}|^{2},$ (4.1)
$U=2\Re\left(E_{\varphi_{P}}^{\rm uc}{E_{\theta_{P}}^{\rm
uc}}^{*}\right),\quad V=-2\Im\left(E_{\varphi_{P}}^{\rm
uc}{E_{\theta_{P}}^{\rm uc}}^{*}\right),$ (4.2)
$L=(Q^{2}+U^{2})^{1/2},\quad\psi=\frac{1}{2}\arctan\frac{U}{Q},$ (4.3)

based on the spherical components $E_{\varphi_{P}}^{\rm uc}$ and
$E_{\theta_{P}}^{\rm uc}$ of ${\bf E}^{\rm uc}$ alone. (The superscript $*$ in
the above expressions denotes complex conjugation.)

It can be seen from (3.75) and (3.94)–(3.95) that the unconventional radiation
field ${\bf E}^{\rm uc}$ consists of the sum of two distinct parts: one part,
${\bf E}^{\rm uc}_{\cal P}$, depending on the vectors $\mbox{{\boldmath${\cal
P}$}}^{(j)}_{l}$ and $\mbox{{\boldmath${\bar{\cal P}}$}}^{(j)}_{l}$ and
another part, ${\bf E}^{\rm uc}_{\cal Q}$, depending on the vectors
$\mbox{{\boldmath${\cal Q}$}}^{(j)}_{l}$ and $\mbox{{\boldmath${\bar{\cal
Q}}$}}^{(j)}_{l}$. As suggested by the occurrence of the factor ${\rm i}$ in
(3.95), these two parts turn out to be out of phase with one another by
approximately $\pi/2$. By calculating the Stokes parameters for ${\bf E}^{\rm
uc}_{\cal P}$ and ${\bf E}^{\rm uc}_{\cal Q}$ separately, we will show here
that the polarization position angles associated with these two fields are
approximately orthogonal to one another in general. Accordingly, the two
distinct parts of the radiation field defined by ${\bf E}^{\rm uc}_{\cal P}$
and ${\bf E}^{\rm uc}_{\cal Q}$ are respectively referred to in this paper as
the $P$ and $Q$ polarization modes.

In this section we begin with evaluating the Stokes parameters $(I,V,L)$ of
the radiation analysed in §4 in units of

$I_{0}=\left(\frac{B_{0}{\hat{r}}_{s0}^{2}}{{\hat{R}}_{P}}\right)^{2}$ (4.4)

(see (1.22) and (3.94)–(3.95)) as functions of the longitude $\varphi_{P}$ at
${\hat{R}}_{P}\gg 1$ for various values of the colatitude $\theta_{P}$, the
inclination angle $\alpha$ and the lower bound $k_{\rm u}^{-1}$ on the width
(in units of the light-cylinder radius) of the current sheet. We will evaluate
the exact expression for the radiated field at a suitably large value of
${\hat{R}}_{P}$, rather than proceeding to the far-field limit
$R_{P}\to\infty$, because as we have already pointed out in § 3.4 the relative
positions of the stationary points of the phase functions $f_{lC}$ and
${\bar{f}}_{lC}$ depend on ${\hat{R}}_{P}$ at colatitudes for which the maxima
and minima of these functions are close to one another. However, away from
such colatitudes the shapes of the pulse profiles and position-angle
distributions we will be discussing do not change perceptibly with distance
once the value of ${\hat{R}}_{P}$ exceeds $10^{6}$. We will therefore use
${\hat{R}}_{P}=10^{6}$ for plotting most of the figures in this section.

The number of components of a pulse profile is determined by the total number
of stationary points of the four phase functions $f_{1C}$, $f_{2C}$,
${\bar{f}}_{1C}$ and ${\bar{f}}_{2C}$ described in § 3.4 and the number of
values of $n$ (i.e., the number of cycles of retarded time) that contribute to
the radiation received during a single period of observation time. The
longitudinal interval occupied by various components of a pulse profile is
determined by the separation between the $\tau$ coordinates of the stationary
points of these phase functions. To be able to depict the pulse profiles over
the entire longitudinal intervals occupied by their various components while
displaying the finite width of each component, we will plot the profiles of
high-intensity pulses for $k_{\rm u}=10^{2}$. It should be borne in mind,
however, that intensity at the peaks of the main pulses in these figures is a
linearly increasing function of $k_{\rm u}$ (see the last paragraph of § 3.7).
The higher the value of $k_{\rm u}$ is, the narrower are the rapid low-
amplitude modulations (microstructure) of the pulse profile. For $k_{\rm
u}=10^{4}$ and higher, these modulations are too sharp and dense to show up in
most of the figures plotted here. We will also adopt those branches of the
multi-valued function arctan appearing in (4.3) that yield continuous
polarization position-angle distributions across various components of a given
pulse.

In most of the examples given below, the inclination angle of the magnetic
axis and the colatitude of the observation point are set in the upper
hemisphere $0<\theta_{P}<90^{\circ}$. At any given value of the inclination
angle $\alpha$, the pulse observed at $180^{\circ}-\theta_{P}$ differs from
that observed at $\theta_{P}$ only in that the intensity $V$ of its circularly
polarized part is replaced by $-V$ and the longitude $\varphi_{P}$ is replaced
by $\varphi_{P}+180^{\circ}$. Moreover, the results for $\alpha>90^{\circ}$
follow from those for $\alpha<90^{\circ}$ by replacing $\theta_{P}$,
$\varphi_{P}$ and $V$ by $180^{\circ}-\theta_{P}$, $\varphi_{P}+180^{\circ}$
and $-V$, respectively (see § 3.4). Note also that in these examples the range
of values of the azimuthal coordinate $\varphi_{P}$ that is spanned by the
observation point across the pulse window differs from that given by
(3.94)–(3.95) for any $n$ (see also (3.21)): the origin of this coordinate is
shifted in each case to place the starting point of the plotted pulse profile
at $\varphi_{P}=0$.

Figure 4.1: The Stokes parameters $I$, $V$ and the position angles $\psi$ of
the polarization modes $P$ and $Q$ at an observation point with the colatitude
$\theta_{P}=5^{\circ}$ for the inclination angle $\alpha=5^{\circ}$ and
$k_{\rm u}=10^{2}$. In this case, only the stationary points of $f_{C2}$ and
${\bar{f}}_{C2}$ contribute toward the field. The large value of the intensity
and the short duration of this pulse stems from the proximity of the maximum
and minimum of $f_{2C}$. At its peak, the right-hand component of this pulse
has the intensity $I=1.05\times 10^{12}I_{0}$ and the longitudinal width
$6.76\times{10^{-9}}$ second when $k_{\rm u}$ is $10^{7}$. This is the only
pulse occurring in the entire pulse window.

Figure 4.2: The Stokes parameters $I$, $V$ and the position angles $\psi$ of
the polarization modes $P$ and $Q$ at an observation point with the colatitude
$\theta_{P}=2.5^{\circ}$ for the inclination angle $\alpha=5^{\circ}$ and
$k_{\rm u}=10^{4}$. In this case, only the stationary points of $f_{C2}$ and
${\bar{f}}_{C2}$ contribute toward the field. The pulse window encompasses
another similar pulse at a longitudinal distance of about $140^{\circ}$ from
this one. Note that not only does $V$ change sign across the pulse profile but
also the position angle of the $Q$ mode swings through $180^{\circ}$ across
the depicted interval and differs from that of the $P$ mode by approximately
$90^{\circ}$ on the right and $45^{\circ}$ on the left of the pulse.

Figure 4.3: The Stokes parameters $I$, $V$ and the position angles $\psi$ of
the polarization modes $P$ and $Q$ at an observation point with the colatitude
$\theta_{P}=20^{\circ}$ for the inclination angle $\alpha=5^{\circ}$ and
$k_{\rm u}=10^{2}$. In this case, the stationary points of ${\bar{f}}_{C2}$
alone contribute toward the field. Although not easily discernible because of
the low value of its intensity $V$, circular polarization reverses sense
across the right-hand component of this pulse. In contrast to those shown in
the previous figures, on the other hand, the position angles of the two
polarization modes are essentially coincident in this case. This is the only
pulse occurring in the entire pulse window.

Figure 4.4: The Stokes parameters $(I,V,L)$ and the position angles $\psi$ of
the polarization modes $P$ and $Q$ at an observation point with the colatitude
$\theta_{P}=55^{\circ}$ for the inclination angle $\alpha=45^{\circ}$ and
$k_{\rm u}=10^{2}$. In this case, only the stationary points of $f_{C2}$ and
${\bar{f}}_{C2}$ contribute toward the field. Note that the polarization of
this pulse changes from linear to circular across it and the position angle of
the Q mode swings through $180^{\circ}$ across each one of its two components.
The pulse window encompasses another pulse at a longitudinal distance of about
$115^{\circ}$ from this one

Figure 4.5: The Stokes parameters $(I,V,L)$ and the position angles $\psi$ of
the polarization modes $P$ and $Q$ at an observation point with the colatitude
$\theta_{P}=35^{\circ}$ for the inclination angle $\alpha=45^{\circ}$ and
$k_{\rm u}=10^{2}$. In this case, only the stationary points of $f_{C2}$ and
${\bar{f}}_{C2}$ contribute toward the field. Position angle of the $Q$ mode
turns through $360^{\circ}$ across the narrow gap at longitude $70^{\circ}$ in
the curve depicting the distribution of the position angle of this
polarization mode. These are the only pulses occurring in the entire pulse
window.

Figure 4.6: The Stokes parameters $(I,V,L)$ and the position angles $\psi$ of
the polarization modes $P$ and $Q$ at an observation point with the colatitude
$\theta_{P}=30^{\circ}$ for the inclination angle $\alpha=65^{\circ}$ and
$k_{\rm u}=10^{2}$. In this case, only the stationary points of $f_{C2}$ and
${\bar{f}}_{C2}$ contribute toward the field. Note that the position angles of
both modes swing through $180^{\circ}$ across the right-hand component of the
above pulse. Note also the high degree of circular polarization of the pulse
throughout the depicted longitudes. The pulse window encompasses in addition a
weaker pulse at a longitudinal distance of about $20^{\circ}$ from these ones.

Figure 4.7: The Stokes parameters $(I,V,L)$ and the position angles $\psi$ of
the polarization modes $P$ and $Q$ at an observation point with the colatitude
$\theta_{P}=77.5^{\circ}$ for the inclination angle $\alpha=65^{\circ}$ and
$k_{\rm u}=10^{7}$. In this case, the stationary points of all of the phase
functions ($f_{C1}$, ${\bar{f}}_{C2}$, ${\bar{f}}_{C1}$, and ${\bar{f}}_{C2}$)
contribute toward the field. Position angle of the $Q$ mode turns through
$360^{\circ}$ across the narrow gap at longitude $100^{\circ}$ in the curve
depicting the distribution of the position angle of this polarization mode.
Note the approximate orthogonality of the position angles of the two modes
across the right-hand components of the pulse and their approximate
coincidence across the left-hand components.

Figure 4.8: The Stokes parameters $(I,V,L)$ and the position angles $\psi$ of
the polarization modes $P$ and $Q$ at an observation point with the colatitude
$\theta_{P}=110^{\circ}$ for the inclination angle $\alpha=80^{\circ}$,
$k_{\rm u}=10^{4}$ (pulse profile) and $k_{\rm u}=10^{7}$ (position angle). In
this case, the stationary points of all of the phase functions ($f_{C1}$,
${\bar{f}}_{C2}$, ${\bar{f}}_{C1}$, and ${\bar{f}}_{C2}$) contribute toward
the field. This is an example of a case in which the Stokes parameters are
comparable in magnitude over some longitudinal intervals and the pulse profile
entails several widely separated components. It is an example also of a multi-
component pulse the position angles of whose different components have
differing longitudinal variations.

Figure 4.9: (a) The Stokes parameters $(I,V,L)$ at an observation point with
the coordinates ${\hat{R}}_{P}=10^{13}$ and $\theta_{P}=90^{\circ}$ for the
inclination angle $\alpha=60^{\circ}$ and $k_{\rm u}=10^{7}$. In this case,
the stationary points of all of the phase functions ($f_{C1}$,
${\bar{f}}_{C2}$, ${\bar{f}}_{C1}$, and ${\bar{f}}_{C2}$) contribute toward
the field. (b) The right-hand component of the pulse depicted in a is here
plotted over a sufficiently short longitudinal interval to resolve its peak
and width. The values of $\alpha$, $\theta_{P}$, $k_{\rm u}$ and distance in b
are the same as in a but the origin of longitude is shifted in b for clarity.
The shape of the pulse depicted in b is the same in all cases of this type.
The extraordinary values of the amplitudes and widths of such pulses,
illustrated by the above example, are what underpin the high brightness
temperatures and broad frequency spectra of the radiation generated by the
current sheet. (For the value of the azimuthal coordinate $\varphi_{P}$ of the
peak, see the text.)

Some examples of the linearly and circularly polarized intensity and
polarization position angle distributions of the pulses described by (3.75),
(3.94) and (3.95) are shown in figures 4.1–4.8.

As suggested by figure 4.9, a radically different type of pulse is detected
when the colatitude $\theta_{P}$ of the observation point has a value close to
(or equal to) $\theta_{PlS}({\hat{R}}_{P},\alpha)$ or
${\bar{\theta}}_{PlS}({\hat{R}}_{P},\alpha)$ for which the extrema of one (or
more) of the phase functions $f_{lC}$ and ${\bar{f}}_{lC}$ coalesce into an
inflection point (see § 3.4). In the example plotted in figure 4.9, the
critical angle $\theta_{P2S}$ happens to lie within a distance of the order of
$1/{\hat{R}}_{P}$ radians from the longitude $\pi/2$ of the observation point.
When sampled over a wide longitudinal interval, the profile that is shown in
part $a$ of figure 4.9 does not radically differ from those of the pulses
shown in other figures. The exceptionally high intensity and narrow width of
the right-hand component of this pulse at its peak, i.e., what is depicted in
part $b$ of figure 4.9, shows up only when the exact position of this peak is
resolved. Here, we have set ${\hat{R}}_{P}=10^{13}$ and have determined the
value $\varphi_{P}=134.80887902374020168766219^{\circ}$ (where $n=2$ in
(3.21)) of the azimuthal coordinate of this peak graphically: by plotting the
distribution of the Stokes parameter $I$ over successively shorter
longitudinal intervals centred on the peak of the distribution until the
maximum value of $I$ stops growing. This zooming in procedure reveals not only
the pulse shown in figure 4.9(b) but also another as narrow and as intense
pulse at a longitudinal distance of $4\times 10^{-18}$ deg from it. In
addition, the coincidence of the limiting values of $\theta_{P1S}$ and
${\bar{\theta}}_{P2S}$ for $R_{P}\to\infty$ results, in this case, in two
identical pulses whose longitudes are separated by $180^{\circ}$.

The features exhibited by the pulse in figure 4.9 can be inferred from (3.75),
(3.94) and (3.95) also analytically. The only variables in the expression for
${\bf E}^{\rm uc}$ that depend on the observer’s longitude are $\sigma_{l2}$
and ${\bar{\sigma}}_{l2}$ which vary linearly with $\varphi_{P}$ (see (3.51)
and (3.52) and note that, according to (3.21), $\varphi_{P}$ drops out of the
expressions for $\sigma_{l1}$ and ${\bar{\sigma}}_{l1}$). The variables
$\sigma_{l2}$ and ${\bar{\sigma}}_{l2}$, on the other hand, appear in
(3.94)–(3.95) only in the combinations $\sigma_{l2}/\sigma_{l1}^{3}$ and
${\bar{\sigma}}_{l2}/{{\bar{\sigma}}_{l1}}^{3}$. Hence, in cases where the
turning points of the phase functions are sufficiently close to one another
for $\sigma_{l1}$ or ${\bar{\sigma}}_{11}$ to be appreciably smaller than $1$,
as in figure 4.9, the arguments of the functions ${\cal G}^{\rm r}_{i}$ that
appear in the expression for ${\bf E}^{\rm uc}$ are highly sensitive functions
of $\varphi_{P}$. Not only the widths but, as indicated by (LABEL:4E78)–(3.81)
and (3.89)–(3.93), also the amplitudes of ${\cal G}^{\rm r}_{i}$ vary sharply
with $\varphi_{P}$ when $\sigma_{l1}$ or ${\bar{\sigma}}_{11}$ assume values
that are close to zero.

###  4.2 Brightness temperature

The brightness temperature $T_{\rm b}$ of the present radiation can be
calculated by equating the magnitude of the Poynting flux of this radiation
($c|{\bf E}^{\rm uc}|^{2}/4\pi$) to the Rayleigh-Jeans law ($2k_{\rm B}T_{\rm
b}\nu^{2}\Delta\nu/c^{2}$) for the energy that a black body of the same
temperature would emit per unit time per unit area into the frequency band
$\Delta\nu$ centred on the frequency $\nu$, where $k_{\rm B}$ is the Boltzmann
constant. The resulting equation can then be solved for $T_{\rm b}$ to obtain

$T_{\rm b}=\frac{c^{3}}{8\pi k_{\rm B}\nu^{2}\Delta\nu}|{\bf E}^{\rm
uc}|^{2}.$ (4.5)

This in conjunction with (4.1) shows that $T_{\rm b}$ is related to the
dimensionless Stokes parameter ${\hat{I}}=I/I_{0}$ by

$T_{\rm b}=\frac{5.37\times 10^{-4}\,c\,\omega^{2}r_{s0}^{4}B_{0}^{2}}{k_{\rm
B}{R}_{P}^{2}\nu^{2}\Delta\nu}{\hat{I}},$ (4.6)

where ${r}_{s0}$ and $B_{0}$ are the radius of the star and the magnitude of
the star’s dipolar field at its magnetic pole, respectively (see the first
paragraph of § 1).

Replacing the parameters that appear in (4.6) by

$B_{0}=10^{12}{\hat{B}}_{0}\,\,{\rm Gauss},\qquad R_{P}=D\,\,{\rm
kpc}=3.085\times 10^{21}D\,\,{\rm cm},$ (4.7)
$\omega=10^{2}{\hat{P}}^{-1}\,\,{\rm rad}/{\rm sec},\qquad
r_{s0}=10^{6}d\,\,{\rm cm},\qquad\nu=10^{8}{\hat{\nu}}\,\,{\rm
Hz},\qquad\Delta\nu=10^{6}\Delta{\hat{\nu}}\,\,{\rm Hz},$ (4.8)

we obtain

$T_{\rm b}=1.23\times 10^{10}\,{\hat{I}}\,{\hat{T}}_{\rm
b}\,\,\,\,{}^{\circ}{\rm K}\qquad{\rm with}\qquad{\hat{T}}_{\rm
b}=\frac{{\hat{B}}_{0}^{2}d^{4}}{{\hat{P}}^{2}D^{2}{\hat{\nu}}^{2}\Delta{\hat{\nu}}}$
(4.9)

in which the value of ${\hat{I}}$ is specified, as in the case of figures
4.1–4.9, by the numerical evaluation of the Stokes parameter $I$ in units of
$I_{0}$ at the highest peak of the pulse detected at ${\hat{R}}_{P}=10^{13}$
(i.e., at ${\hat{R}}_{P}=1.028\times 10^{13}D{\hat{P}}^{-1}$ when the factor
$1.028D{\hat{P}}^{-1}$ equals unity).

The brightness temperature implied by (4.9) and $k_{\rm u}=10^{7}$ is listed
in Table 4.1 for the pulses depicted in figures 4.1, 4.4 and 4.9 and for a
pair of examples of the pulses that are detected at the critical colatitudes
$\lim_{R_{P}\to\infty}\theta_{PlS}$. Table 4.1 also shows the full width at
half maximum $\delta\varphi_{P}$ of the listed pulses (see figure 4.9(b)).
Once resolved, the longitudinal distributions of the narrow pulses that stem
from the focusing of the radiation when $\sigma_{l1}$ and
${\bar{\sigma}}_{l1}$ are small all have the same shape as that of the pulse
shown in figure 4.9(b). Note that, as indicated by the last column of Table
4.1, the pulse profiles depicted in figures 4.1–4.8 have to be plotted on
considerably shorter longitudinal scales before their peaks assume the shape
shown in figure 4.9(b) and the maximum values of their dimensionless intensity
${\hat{I}}$ can be discerned graphically (see § 4.1). In general, as one
reduces the longitudinal interval over which $I$ is plotted, the peak of the
pulse splits in two before the finite widths of either of the partitioned
pulses are visible.

Values of $T_{\rm b}$ higher than those listed in Table 4.1 are predicted by
(4.9) when the colatitude of the observation point lies closer to one of the
critical angles $\theta_{PlS}$ or ${\bar{\theta}}_{PlS}$. In the case of
$\alpha=60^{\circ}$, for example, the listed value ($1.17\times
10^{40}\,{}^{\circ}$K) of $T_{\rm b}$ corresponds to
$\theta_{P}=\lim_{R_{P}\to\infty}\theta_{P1S}=\lim_{R_{P}\to\infty}\theta_{P2S}=\pi/2$.
For an observation point whose colatitude is closer to the critical angles in
question than $\pi/2$ is, e.g., for $\theta_{P}=\theta_{P1S}+10^{-20}$ rad,
$T_{\rm b}$ and $\delta\varphi_{P}$ have the values $1.17\times
10^{54}\,{}^{\circ}$K and $2.26\times 10^{-34}$ deg, respectively.

$\alpha\,\,({\rm deg})$ | $\theta_{P}\,\,({\rm deg})$ | $T_{\rm b}/{\hat{T}}_{\rm b}\,\,({}^{\circ}{\rm K})$ | $\delta\varphi_{P}\,\,({\rm deg})$
---|---|---|---
$5$ | $5$ | $1.29\times 10^{22}$ | $6.76\times 10^{-9}$
$30$ | $\lim_{R_{P}\to\infty}\theta_{P2S}$ | $6.89\times 10^{40}$ | $2.39\times 10^{-24}$
$45$ | $55$ | $2.34\times 10^{17}$ | $3.81\times 10^{-7}$
$60$ | $90$ | $1.17\times 10^{40}$ | $6.94\times 10^{-25}$
$75$ | $\lim_{R_{P}\to\infty}\theta_{P1S}$ | $4.06\times 10^{38}$ | $5.73\times 10^{-25}$
Table 4.1: Brightness temperature $T_{\rm b}$ and full width at half maximum
$\delta\varphi_{P}$ of the pulse detected at colatitude $\theta_{P}$ for the
inclination angle $\alpha$ and ${\hat{R}}_{P}=10^{13}$, $k_{\rm u}=10^{7}$.
The dimensionless factor ${\hat{T}}_{\rm b}$, defined by (4.7)–(4.9), is of
the order of unity in the case of most radio pulsars. The limiting values of
$\theta_{P1S}$ and $\theta_{P2S}$ in the second column are
$48.533945294618400228^{\circ}$ and $33.932818533330613261^{\circ}$,
respectively.

###  4.3 Frequency spectrum

Given that the radiation field ${\bf E}^{\rm uc}$ depends on the observation
time $t_{P}$ only in the combination $\varphi_{P}-\omega t_{P}$, the frequency
spectrum of the present radiation is equally well described by the Fourier
decomposition of the field ${\bf E}^{\rm uc}$ with respect to the azimuthal
angle $\varphi_{P}$ of the observation point. In the present case, the content
of this spectrum stems from two factors. One factor is the thickness of the
current sheet ($\simeq c/(k_{\rm u}\omega$) which is manifested in the sharp
small-amplitude modulations (microstructure) of the pulse profile (see figures
4.1 and 4.5): the wavelengths of such modulations are proportional to $k_{\rm
u}^{-1}$. The other factor is the full width at half maximum
($\delta\varphi_{P}$) of the pulse with the highest peak in the pulse profile
(see figure 4.9(b)): the fraction $\delta\varphi_{P}/2\pi$ of a period during
which such narrow pulses propagate past a detector is by many orders of
magnitude smaller than the fraction $k_{\rm u}^{-1}$ of the light-cylinder-
radius in cases where the colatitude of the observation point is close to or
coincident with one of the critical angles described in § 3.4 (see Table 4.1).
While the Fourier decomposition of the fluctuations associated with the first
factor yields a frequency spectrum centred on radio waves when $k_{\rm
u}\gtrsim 10^{5}$, that of the fluctuations associated with the second factor
yields a wide spectral distribution extending to gamma rays: the value
$\delta\varphi_{P}=1.21\times 10^{-26}$ radian appearing in the last column of
Table 4.1 corresponds to a frequency spectrum that extends as far as
$\nu\simeq\omega/(2\pi\delta\varphi_{P})\simeq 1.31\times
10^{27}{\hat{P}}^{-1}$ Hz.

Our replacing the Dirac delta function in (3.5) by its Fourier representation
(3.11) is tantamount to Fourier analysing the fluctuations of the radiation
field that arise from the short thickness of the current sheet with respect to
$\varphi_{P}$ since the argument of that delta function depends on
$\varphi_{P}$ linearly. The spectral distribution of the part of the radiation
that stems from the thickness of the current sheet is therefore given by the
$k$ dependence of

$\displaystyle{\tilde{I}}$ $\displaystyle\simeq$
$\displaystyle\left(\frac{4\pi
j_{0}}{3\omega}\right)^{2}\sum_{l=1}^{2}k^{-2/3}\bigg{|}\exp({\rm
i}k\sigma_{l2})$ (4.10) $\displaystyle\times\left[\mbox{{\boldmath${\cal
P}$}}_{l}{\rm Ai}(-k^{2/3}\sigma_{l1}^{2})-{\rm
i}k^{-1/3}\mbox{{\boldmath${\cal Q}$}}_{l}{\rm