Unipolar components travelling at the speed of light in vacuo
Charles Varin
In this letter, we describe a physical situation where the time-integral of the electric field of an electromagnetic wave packet is not zero. More specifically, we demonstrate that the non-oscillating component of the Fourier spectrum, i.e. the component resulting from the Fourier transform of the electric field evaluated at $\omega=0$, can be associated to the area of an unipolar electromagnetic pulse that travels in free space at the speed of light. This feature offers the possibility for this field component to interact with matter over extended regions of space.
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\title{Unipolar components travelling at the speed of light \textit{in vacuo}}

\author{Charles Varin}\email{charles.varin@phy.ulaval.ca}
    \affiliation{Centre d'optique photonique et laser, Universit\'e
    Laval, Qu\'ebec, Qc, G1K 7P4, Canada}%

\author{Michel Pich\'e}%
    \affiliation{Centre d'optique photonique et laser, Universit\'e
    Laval, Qu\'ebec, Qc, G1K 7P4, Canada}%

\date{\today}

\begin{abstract}
In this letter, we describe a physical situation where the
time-integral of the electric field of an electromagnetic wave
packet is not zero. More specifically, we demonstrate that the
non-oscillating component of the Fourier spectrum, i.e. the
component resulting from the Fourier transform of the electric
field evaluated at $\omega=0$, can be associated to the area of an
unipolar electromagnetic pulse that travels in free space at the
speed of light. This feature offers the possibility for this field
component to interact with matter over extended regions of space.
\end{abstract}

\pacs{03.50.De, 03.30.+p}

\maketitle

The numerical value obtained by the integration of the electric
field of a wave packet from $t=-\infty$ to $t=\infty$ can be
associated to the amplitude of the non-oscillating component of
the Fourier spectrum $\mathbf{E}(\mathbf{r},\omega=0)$, that is~:
\begin{equation}\label{eq:fourier_static_component}
\mathbf{E}(\mathbf{r},\omega=0)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}
\mathbf{E}(\mathbf{r},t)\,dt,
\end{equation}
where $\mathbf{E}(\mathbf{r},t)$ is the electric field vector and
$\mathbf{r}$ is the position in space. From a geometrical point of
view, Eq.~(\ref{eq:fourier_static_component}) gives the area, or
the temporal average, of the signal. According to Kim \textit{et
al.}~\cite{kimPRL2000}, $\mathbf{E}(\mathbf{r},\omega=0)$
corresponds to a static field whose amplitude must fall off as
$1/|\mathbf{r}|^2$. It is thus believed that for a wave packet of
finite dimensions propagating far from its source the integral
given at Eq.~(\ref{eq:fourier_static_component}) must be zero
everywhere in space. However, we suspect that this interpretation
is incorrect.

The main doubt we have with the widely accepted interpretation of
Kim \textit{et al.}~\cite{kimPRL2000} concerns the fact that, in
free space, the electric field vector is proportional to the time
derivative of the vector potential, i.e.
$\mathbf{E}(\mathbf{r},t)=-\partial_{\,t}\mathbf{A}(\mathbf{r},t)$,
where $\partial_{\,t}\mathbf{A}(\mathbf{r},t)=\partial
\mathbf{A}(\mathbf{r},t)/\partial t$~\cite{jackson1999}.
Consequently, we can write~:
\begin{equation}\label{eq:vector_potential_definite_integral}
\mathbf{A}(\mathbf{r},-\infty)-\mathbf{A}(\mathbf{r},\infty)=\int_{-\infty}^{\,
\infty}\mathbf{E}(\mathbf{r},t')\,dt'.
\end{equation}
If we combine Eqs.~(\ref{eq:fourier_static_component})
and~(\ref{eq:vector_potential_definite_integral}), we immediately
see that~:
\begin{equation}\label{eq:vector_potential_infinity_vs_nonoscillating_component}
\mathbf{E}(\mathbf{r},\omega=0)=\frac{\mathbf{A}(\mathbf{r},-\infty)
-\mathbf{A}(\mathbf{r},\infty)}{\sqrt{2\pi}}.
\end{equation}
We hence observe that $\mathbf{E}(\mathbf{r},\omega=0)$ is related
to the net variation of the strength of the vector potential from
$t=-\infty$ to $t=\infty$. As far as we know, the vector potential
does not have to fall off as fast as $1/|\mathbf{r}|^2$ far from a
bounded source of radiation. Why should
$\mathbf{E}(\mathbf{r},\omega=0)$ be zero for all $\mathbf{r}$?
Let us now push this analysis a little further.

In principle, any electromagnetic waveform can be
generated~\cite{{feynman1977},{jackson1999}}. For example, if an
arbitrary current $J(t)$ circulates into a conducting wire, a wave
packet whose electric field is proportional to the temporal
variations of the driving signal will be emitted. Assuming that
the center of such a wire (of finite length $\Delta l$ and radius
$a$) is located at the origin $\mathbf{r}=(0,0,0)$ of a set of
cylindrical coordinates $\mathbf{r}=(r,\theta,z)$ with a uniform
current flowing along the $z$-axis, the vector potential
propagating at a great distance from the source is written as
follows~\cite{jackson1999}~:
\begin{equation}\label{eq:radiatied_potential_conducting_wire}
\mathbf{A}(r,z,t)\simeq\Pi(z)\frac{a^2\mu_0\Delta
l}{4\,r}\,[J(t)\,]_{\mathrm{ret}}\,\mathbf{a}_z,
\end{equation}
where $\Pi(z)=1$ for $-\Delta l \leq z \leq \Delta l$ and 0
otherwise, $\mu_0$ is the permeability of free space,
$\mathbf{a}_z$ is a unit vector oriented along the wire (lying on
the $z$ axis), and $[\ldots]_{\mathrm{ret}}$ means that the
quantity within the brackets must be expressed in terms of the
retarded time $t_\mathrm{r}=t-r/c$, with $c$ being the speed of
light \textit{in vacuo}. Combining
Eqs.~(\ref{eq:vector_potential_infinity_vs_nonoscillating_component})
and~(\ref{eq:radiatied_potential_conducting_wire}), we calculated
the non-oscillating component of the field and we obtained this~:
\begin{equation}\label{eq:non_oscillating_component_wire}
\mathbf{E}(r,z,\omega=0)=\Pi(z)\frac{a^2\mu_0\Delta
l}{4\sqrt{2\pi}}\left(\frac{J(-\infty)
-J(\infty)}{r}\right)\mathbf{a}_z.
\end{equation}
It thus appears that the time integral of the electric field
vanishes when there is no net variation of current in the wire,
that is when $J(-\infty)=J(\infty)$. Globally, this is what we
expect to happens. However, if the current fluctuations are
sufficiently separated, the radiation field appears as a series of
isolated pulses (see Fig.~\ref{Fig}). These electromagnetic wave
packets of finite energy carry a non-oscillating component.
Concordantly, if $J(t)=J_0-\Delta J\, \mathrm{erf}[\,t/\Delta
t\,]$ -- corresponding to the decrease of a steady current from
the value $J_0+\Delta J$ to a new value $J_0-\Delta J$ within a
certain time interval $\Delta t$ -- we find (far from the
source)~:
\begin{equation}\label{eq:radiatied_field_conducting_wire_example}
\mathbf{E}(r,z,t)\simeq\Pi(z)\frac{a^2\mu_0\Delta l\Delta
J}{2\sqrt{\pi}\Delta t }\,\frac{\exp\left(-\,t_\mathrm{r}^2/\Delta
t^2\right)}{r}\,\mathbf{a}_z,
\end{equation}
and~:
\begin{equation}\label{eq:radiatied_field_conducting_wire_example_integral}
\mathbf{E}(r,z,\omega=0)=\Pi(z)\frac{a^2\mu_0\Delta l\Delta
J}{2\sqrt{2\pi}\,r}\,\mathbf{a}_z.
\end{equation}
Hence, the non-oscillating Fourier component of the field radiated
by a bounded conducting wire does not vanish in the far field,
locally (see
Eq.~(\ref{eq:radiatied_field_conducting_wire_example_integral})).
It seems here that $\mathbf{E}(r,z,\omega=0)$ corresponds to the
area of an unipolar electromagnetic pulse that travels in free
space at the speed of light and whose amplitude falls off as $1/r$
far from the source. We Recall that the radiated electric field is
proportional to $\partial_{\,t}J(t)$ -- \textit{via}
$\mathbf{E}(\mathbf{r},t)=-\partial_{\,t}\mathbf{A}(\mathbf{r},t)$,
the field given at
Eq.~(\ref{eq:radiatied_field_conducting_wire_example}) is thus
free of any static field, i.e. a field whose amplitude is constant
in time (see also Brabec and Krausz~\cite{brabecRMP2000}).

\begin{figure}
\includegraphics{Fig1.eps}
\caption{Momentary variations of current in a bounded conducting
wire (top) and the corresponding radiated electric field (bottom).
Globally, the temporal average of the radiated field is zero. But,
when the delay between two successive variations of current is
sufficiently large ($t_2-t_1>>\Delta t$), the field appears as two
isolated unipolar pulses, separated in space by a distance equal
to $c\,(t_2-t_1)$. \label{Fig}}
\end{figure}

Thus, in this letter we have shown that the non-oscillating
component of the Fourier spectrum
$\mathbf{E}(\mathbf{r},\omega=0)$ is proportional to the net
change in the strength of the vector potential, from $t=-\infty$
to $t=\infty$. This relation reveals that the temporal average of
an electromagnetic signal can vanish or not, depending on how it
has been produced. Consequently, we observe that the derivation
provided by Belenov and Nazarkin~\cite{belenovJOSAA1994}
corresponds to a special family of wave packets, namely zero-area
pulses, whose vector potential has the same strength at
$t=\pm\infty$. Here, we have described the case of a bounded
conducting wire. We have shown that the variations of current
inside the wire produce a series of unipolar electromagnetic
pulses that travel in free space at the speed of light ; the
amplitude of these pulses falls off as $1/r$ far from the wire.
Locally, the temporal average of an isolated unipolar pulse is not
zero. Consequently, we are convinced that the non-oscillating
component of an electromagnetic field can interact with matter
over extended regions of space.

\begin{acknowledgments}
C. Varin and M. Pich\'e thank Les fonds de recherche sur la nature
et les technologies (Qu\'ebec), the Natural Sciences and
Engineering Research Council (Canada), and the Canadian Institute
for Photonic Innovations for their financial support.
\end{acknowledgments}

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\bibitem[{\citenamefont{Kim et~al.}(2000)\citenamefont{Kim, McDonald, and
  Stupakov}}]{kimPRL2000}
\bibinfo{author}{\bibfnamefont{K.~J.} \bibnamefont{Kim}},
  \bibinfo{author}{\bibfnamefont{K.~T.} \bibnamefont{McDonald}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{G.~V.}
  \bibnamefont{Stupakov}}, \bibinfo{journal}{Phys. Rev. Lett.}
  \textbf{\bibinfo{volume}{84}}, \bibinfo{pages}{3210} (\bibinfo{year}{2000}).

\bibitem[{\citenamefont{Jackson}(1999)}]{jackson1999}
\bibinfo{author}{\bibfnamefont{J.~D.} \bibnamefont{Jackson}},
  \emph{\bibinfo{title}{Classical Electrodynamics}}
  (\bibinfo{publisher}{Wiley}, \bibinfo{address}{New York, NY},
  \bibinfo{year}{1999}), \bibinfo{edition}{3rd} ed.

\bibitem[{\citenamefont{Feynman et~al.}(1977)\citenamefont{Feynman, Leighton,
  and Sands}}]{feynman1977}
\bibinfo{author}{\bibfnamefont{R.~P.} \bibnamefont{Feynman}},
  \bibinfo{author}{\bibfnamefont{R.~B.} \bibnamefont{Leighton}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Sands}},
  \emph{\bibinfo{title}{Lectures on Physics}}
  (\bibinfo{publisher}{Addison-Wesley}, \bibinfo{address}{Reading, MA},
  \bibinfo{year}{1977}).

\bibitem[{\citenamefont{Brabec and Krausz}(2000)}]{brabecRMP2000}
\bibinfo{author}{\bibfnamefont{T.}~\bibnamefont{Brabec}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{F.}~\bibnamefont{Krausz}},
  \bibinfo{journal}{Rev. Mod. Phys.} \textbf{\bibinfo{volume}{72}},
  \bibinfo{pages}{545} (\bibinfo{year}{2000}).

\bibitem[{\citenamefont{Belenov and Nazarkin}(1994)}]{belenovJOSAA1994}
\bibinfo{author}{\bibfnamefont{E.~M.} \bibnamefont{Belenov}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{A.~V.} \bibnamefont{Nazarkin}},
  \bibinfo{journal}{J. Opt. Soc. Am. A} \textbf{\bibinfo{volume}{11}},
  \bibinfo{pages}{168} (\bibinfo{year}{1994}).

\end{thebibliography}
