$\displaystyle-2\int_{X}\frac{\mathrm{D}\lambda}{2\pi\mathrm{i}}\frac{\,\mathrm{tr}\,\hat{T}_{{p_{1}}}\ldots\hat{T}_{{p_{i-1}}}\mathsf{H}^{-1}(x,\lambda)\,\partial_{{\dot{\alpha}}}\mathsf{a}\,\mathsf{H}(x,\lambda)\hat{T}_{{p_{i}}}\ldots\hat{T}_{{p_{n}}}}{\langle
p_{1}p_{2}\rangle\ldots\langle p_{i-1}\lambda\rangle\langle\lambda
p_{i}\rangle\ldots\langle p_{n}p_{1}\rangle}\frac{\langle\lambda
j\rangle^{2}}{\langle\iota\lambda\rangle\langle\iota
j\rangle}\tilde{\kappa}_{j\,\dot{\beta}}$
$\displaystyle+\int_{X^{2}}\frac{\mathrm{D}\lambda\mathrm{D}\lambda^{\prime}}{(2\pi\mathrm{i})^{2}}\frac{1}{\langle
p_{1}p_{2}\rangle\ldots\langle p_{i-1}\lambda\rangle\langle\lambda
p_{i}\rangle\ldots\langle
p_{j-1}\lambda^{\prime}\rangle\langle\lambda^{\prime}p_{j}\rangle\ldots\langle
p_{n}p_{1}\rangle}\frac{\langle\lambda\lambda^{\prime}\rangle^{2}}{\langle\iota\lambda\rangle\langle\iota\lambda^{\prime}\rangle}\times$
$\displaystyle\times\,\mathrm{tr}\,\hat{T}_{{p_{1}}}\ldots\hat{T}_{{p_{i-1}}}\mathsf{H}^{-1}(x,\lambda)\,\partial_{{\dot{\alpha}}}\mathsf{a}\,\mathsf{H}(x,\lambda)\hat{T}_{{p_{i}}}\ldots\hat{T}_{{p_{j-1}}}\mathsf{H}^{-1}(x,\lambda^{\prime})\,\partial_{{\dot{\beta}}}\mathsf{a}\,\mathsf{H}(x,\lambda^{\prime})\hat{T}_{{p_{j}}}\ldots\hat{T}_{{p_{n}}}$
$\displaystyle+\text{perms.}\Biggr{\\}}\,,$

where
$\hat{T}_{j}\coloneqq\mathsf{H}^{-1}(x,\kappa_{j})T_{j}\mathsf{H}(x,\kappa_{j})$,
and $+\text{perms.}$ is a sum over the permutations of
$\\{p_{1},\ldots,p_{n}\\}$. Around a Cartan-valued background, the conjugation
on the colour factors reduces to
$\hat{T}^{\mathtt{a}_{j}}=T_{j}\exp(e_{j}g(x,\kappa_{j}))$, so that the
colour-ordered form factor is

$\displaystyle\mathscr{F}=$ $\displaystyle\frac{1}{\langle
12\rangle\ldots\langle
n1\rangle}\int_{\mathbb{M}}\mathrm{d}^{4}x\,e^{\mathrm{i}(Q-q)\cdot
x+\sum_{j}e_{j}g(x,\kappa_{j})}\iota_{\alpha}q^{\alpha\dot{\alpha}}\iota_{\beta}q^{\beta\dot{\beta}}\sum_{i,j}\Biggl{\\{}\frac{\langle
ij\rangle^{2}}{\langle\iota i\rangle\langle\iota
j\rangle}\tilde{\kappa}_{i\dot{\alpha}}\tilde{\kappa}_{j\dot{\beta}}$ (153)
$\displaystyle-2\int_{X}\frac{\mathrm{D}\lambda}{2\pi\mathrm{i}}\frac{\langle
i-1,i\rangle\langle\lambda j\rangle^{2}\tilde{\kappa}_{j\dot{\beta}}}{\langle
i-1,\lambda\rangle\langle\lambda
i\rangle\langle\iota\lambda\rangle\langle\iota
j\rangle}\left.\frac{\partial\mathsf{a}}{\partial\mu^{\dot{\alpha}}}\right|_{X}$
$\displaystyle+\int_{X^{2}}\frac{\mathrm{D}\lambda\mathrm{D}\lambda^{\prime}}{(2\pi\mathrm{i})^{2}}\frac{\langle
i-1,i\rangle\langle
j-1,j\rangle\langle\lambda\lambda^{\prime}\rangle^{2}}{\langle
i-1,\lambda\rangle\langle\lambda i\rangle\langle
j-1,\lambda\rangle\langle\lambda
j\rangle\langle\iota\lambda\rangle\langle\iota\lambda^{\prime}\rangle}\left.\frac{\partial\mathsf{a}}{\partial\mu^{\dot{\alpha}}}\right|_{X}\left.\frac{\partial\mathsf{a}}{\partial{\mu^{\prime}}^{\dot{\beta}}}\right|_{X}\Biggr{\\}}\,.$

We now consider the integral

$\mathcal{J}_{\alpha\beta\dot{\alpha}}(\ell)\coloneqq\int_{X}\frac{\mathrm{D}\lambda}{2\pi\mathrm{i}}\frac{\lambda_{\alpha}\lambda_{\beta}}{\langle\iota\lambda\rangle\langle\ell-1,\lambda\rangle\langle\lambda\ell\rangle}\left.\frac{\partial\mathsf{a}}{\partial\mu^{\dot{\alpha}}}\right|_{X}\,.$
(154)

Using the identity

$\displaystyle\frac{\lambda_{\alpha}\lambda_{\beta}}{\langle\iota\lambda\rangle\langle\ell-1\lambda\rangle\langle\ell\lambda\rangle}=$
$\displaystyle\frac{\kappa_{\ell-1\,\alpha}\kappa_{\ell-1\,\beta}}{\langle\ell-1,\ell\rangle^{2}\langle\iota,\ell-1\rangle}\left(\frac{\langle\iota\ell\rangle}{\langle\iota\lambda\rangle}-\frac{\langle\ell-1,\ell\rangle}{\langle\ell-1,\lambda\rangle}\right)$
(155)
$\displaystyle+\frac{\kappa_{\ell\,\alpha}\kappa_{\ell\,\beta}}{\langle\ell-1,\ell\rangle^{2}\langle\iota\ell\rangle}\left(\frac{\langle\iota,\ell-1\rangle}{\langle\iota\lambda\rangle}-\frac{\langle\ell,\ell-1\rangle}{\langle\ell\lambda\rangle}\right)$
$\displaystyle-\frac{\kappa_{\ell-1\,\alpha}\kappa_{\ell\,\beta}+\kappa_{\ell-1\,\beta}\kappa_{\ell\,\alpha}}{\langle\ell-1,\ell\rangle^{2}\langle\iota\lambda\rangle}\,,$

the integral can be evaluated explicitly and reads

$\displaystyle\mathcal{J}_{\alpha\beta\dot{\alpha}}(\ell)=-\mathrm{i}$
$\displaystyle\left(\frac{\kappa_{\ell-1\,\alpha}\kappa_{\ell-1\,\beta}}{\langle\ell-1,\iota\rangle\langle\ell-1,\ell\rangle}G_{\dot{\alpha}}(x,\kappa_{\ell-1})+\frac{\kappa_{\ell\,\alpha}\kappa_{\ell\,\beta}}{\langle\ell\iota\rangle\langle\ell,\ell-1\rangle}G_{\dot{\alpha}}(x,\kappa_{\ell})\right.$
(156)
$\displaystyle\left.+\frac{\iota_{\alpha}\iota_{\beta}}{\langle\iota,\ell-1\rangle\langle\iota\ell\rangle}G_{\dot{\alpha}}(x,\iota)\right)\,.$

We then see that the terms with residual integrals over $X$ and $X^{2}$ don’t
contribute to the form factor in (153). For the term with a single integral
over $X$, the sum over $i$ is

$\sum_{i}\left(\frac{\langle i-1,j\rangle^{2}}{\langle
i-1,\iota\rangle}G_{\dot{\alpha}}(x,\kappa_{i-1})-\frac{\langle
ij\rangle^{2}}{\langle
i\iota\rangle}G_{\dot{\alpha}}(x,\kappa_{i})+\frac{\langle\iota
j\rangle^{2}\langle i-1,i\rangle}{\langle\iota,i-1\rangle\langle\iota
i\rangle}G_{\dot{\alpha}}(x,\iota)\right)\,.$ (157)

The first two terms obviously cancel in the sum. The third one is telescopic
in $i$ as well, once we complete $\iota_{\alpha}$ to a basis
$\\{\iota_{\alpha},o_{\alpha}\\}$ of undotted spinors, introducing the spinor
$o_{\alpha}$ normalized such that $\langle\iota o\rangle=1$. The third term
then reduces to

$\sum_{i}\frac{\langle i-1,1\rangle}{\langle\iota,i-1\rangle\langle\iota
i\rangle}=\sum_{i}\left(\frac{\langle oi\rangle}{\langle\iota
i\rangle}-\frac{\langle o,i-1\rangle}{\langle\iota,i-1\rangle}\right)=0\,.$
(158)

Overall, the form factor reads

$\mathscr{F}=\frac{1}{\langle 12\rangle\ldots\langle
n1\rangle}\sum_{i,j}\frac{\langle
ij\rangle^{2}\langle\iota|q|i]\langle\iota|q|j]}{\langle\iota
i\rangle\langle\iota
j\rangle}\int_{\mathbb{M}}\mathrm{d}^{4}x\,e^{\mathrm{i}(Q-q)\cdot
x+\sum_{j}e_{j}g(x,\kappa_{j})}\,.$ (159)

Let $\mathcal{S}$ denote the sum over $i,j$

$\mathcal{S}\coloneqq
q^{\alpha\dot{\alpha}}q^{\beta\dot{\beta}}\sum_{i,j}\frac{\langle
ij\rangle^{2}}{\langle\iota i\rangle\langle\iota
j\rangle}\iota_{\alpha}\iota_{\beta}\tilde{\kappa}_{i\dot{\alpha}}\tilde{\kappa}_{j\dot{\beta}}\,.$
(160)

Using the Schouten identity and performing one of the sums, $\mathcal{S}$
reduces to

$\mathcal{S}=q^{\alpha\dot{\alpha}}q^{\beta\dot{\beta}}\left(Q_{\gamma\dot{\alpha}}\sum_{i}\frac{\iota_{\beta}\tilde{\kappa}_{i\dot{\beta}}\kappa_{i\alpha}\kappa^{\gamma}_{i}}{\langle\iota
i\rangle}+Q_{\gamma\dot{\beta}}\sum_{i}\frac{\iota_{\beta}\tilde{\kappa}_{i\dot{\alpha}}\kappa_{i\alpha}\kappa^{\gamma}_{i}}{\langle\iota
i\rangle}\right)\,,$ (161)

Finally, noticing the identity

$\displaystyle
q^{\alpha\dot{\alpha}}Q_{\gamma\dot{\alpha}}=\frac{1}{2}(q^{\alpha\dot{\alpha}}Q_{\gamma\dot{\alpha}}-q_{\gamma\dot{\alpha}}Q^{\alpha\dot{\alpha}})+\frac{1}{2}\delta^{\alpha}_{\gamma}q\cdot
Q\,,$ (162)

we can further simplify $\mathcal{S}$ down to

$\mathcal{S}=-(q\cdot Q)^{2}\,,$ (163)

and the form factor is finally (up to an overall numerical factor)

$\mathscr{F}=\frac{(q\cdot Q)^{2}}{\langle 12\rangle\ldots\langle
n1\rangle}\int_{\mathbb{M}}\mathrm{d}^{4}x\,e^{\mathrm{i}(Q-q)\cdot
x+\sum_{j}e_{j}g(x,\kappa_{j})}\,.$ (164)

The same procedure can be used to obtain (102) from the lifting of
$\,\mathrm{tr}\,\tilde{F}^{3}$ to twistor space. Such lifting reads

$\,\mathrm{tr}\,\tilde{F}^{3}=\int\mathrm{D}\lambda_{123}\,\langle\lambda_{1}\lambda_{2}\rangle\langle\lambda_{2}\lambda_{3}\rangle\langle\lambda_{3}\lambda_{1}\rangle\,\mathrm{tr}\,\partial_{{\dot{\alpha}}}\partial^{{\dot{\beta}}}\mathsf{a}_{1}\mathsf{U}_{12}\partial_{{\dot{\beta}}}\partial^{{\dot{\gamma}}}\mathsf{a}_{2}\mathsf{U}_{23}\partial_{{\dot{\gamma}}}\partial^{{\dot{\alpha}}}\mathsf{a}_{3}\mathsf{U}_{31}\\\
+6\int\mathrm{D}\lambda_{1234}\,\langle\lambda_{2}\lambda_{3}\rangle\langle\lambda_{3}\lambda_{4}\rangle\langle\lambda_{4}\lambda_{1}\rangle\,\mathrm{tr}\,\partial_{{(\dot{\alpha}}}\mathsf{a}_{1}\mathsf{U}_{12}\partial_{{\dot{\beta})}}\mathsf{a}_{2}\mathsf{U}_{23}\partial^{{\dot{\beta}}}\partial^{{\dot{\gamma}}}\mathsf{a}_{3}\mathsf{U}_{34}\partial_{{\dot{\gamma}}}\partial^{{\dot{\alpha}}}\mathsf{a}_{4}\mathsf{U}_{41}\\\
-12\int\mathrm{D}\lambda_{12345}\,\langle\lambda_{2}\lambda_{3}\rangle\langle\lambda_{4}\lambda_{5}\rangle\langle\lambda_{5}\lambda_{1}\rangle\,\mathrm{tr}\,\partial_{{(\dot{\alpha}}}\mathsf{a}_{1}\mathsf{U}_{12}\partial_{{\dot{\beta})}}\mathsf{a}_{2}\mathsf{U}_{23}\partial^{{(\dot{\beta}}}\mathsf{a}_{3}\mathsf{U}_{34}\partial^{{\dot{\gamma})}}\mathsf{a}_{4}\mathsf{U}_{45}\partial_{{\dot{\gamma}}}\partial^{{\dot{\alpha}}}\mathsf{a}_{5}\mathsf{U}_{51}\\\
+8\,\varepsilon^{\dot{\alpha}\dot{\delta}}\int\mathrm{D}\lambda_{123456}\,\langle\lambda_{2}\lambda_{3}\rangle\langle\lambda_{4}\lambda_{5}\rangle\langle\lambda_{6}\lambda_{1}\rangle\,\mathrm{tr}\,\partial_{{(\dot{\alpha}}}\mathsf{a}_{1}\mathsf{U}_{12}\partial_{{\dot{\beta})}}\mathsf{a}_{2}\mathsf{U}_{23}\partial^{{(\dot{\beta}}}\mathsf{a}_{3}\mathsf{U}_{34}\partial^{{\dot{\gamma})}}\mathsf{a}_{4}\mathsf{U}_{45}\partial_{{(\dot{\gamma}}}\mathsf{a}_{5}\mathsf{U}_{56}\partial_{{\dot{\delta})}}\mathsf{a}_{6}\mathsf{U}_{61}\,,$
(165)

In the case of super form factors in $\mathcal{N}=4$ SYM, the same arguments
of Section 5.3 hold around gluonic backgrounds, so we briefly comment only on
the derivation of (112) from (109). We denote the background super-connection
as $\underline{\mathsf{a}}^{\text{(B)}}=\mathsf{a}+\mathsf{a}_{a}\chi^{a}$ and
a generic external state (80) as $\underline{\mathsf{a}}_{i}$. We first notice
that (109) is obtained from the super-field $\underline{\phi}_{ab}$ using the
supersymmetrized $K$-matrix

$\frac{1}{2}\,\mathrm{tr}\,\phi_{ab}\phi_{ab}=\frac{1}{2}\iota^{\alpha}\iota^{\beta}\iota^{\gamma}\iota^{\delta}\int\mathrm{d}^{8}\theta\,\prod_{\gamma,c}\theta^{\gamma
c}\,\,\mathrm{tr}\,(\partial_{\alpha a}\partial_{\beta
b}\underline{K}\,\partial_{\gamma a}\partial_{\delta b}\underline{K})\,,$
(166)

and integrating by parts twice. The super form factor is obtained by taking
the Fourier transform of

$\iota^{\alpha}\iota^{\beta}\left.\Bigg{(}\sum_{i,j}\int\frac{\mathrm{D}\lambda_{1}\ldots\mathrm{D}\lambda_{n}}{\langle\iota\lambda_{i}\rangle\langle\iota\lambda_{j}\rangle}\frac{\langle\lambda_{i}\lambda_{j}\rangle^{2}}{\langle\lambda_{1}\lambda_{2}\rangle\ldots\langle\lambda_{n}\lambda_{1}\rangle}\right.\\\
\left.\,\mathrm{tr}\,(\underline{\mathsf{H}}_{1}^{-1}\underline{a}_{1}\underline{\mathsf{H}}_{1}\ldots\underline{\mathsf{H}}_{i}^{-1}\partial_{a}\underline{a}_{i}\underline{\mathsf{H}}_{i}\ldots\underline{\mathsf{H}}_{j}^{-1}\partial_{b}\underline{a}_{j}\underline{\mathsf{H}}_{j}\ldots\underline{\mathsf{H}}_{n}^{-1}\underline{a}_{n}\underline{\mathsf{H}}_{n})\Bigg{)}\right|_{\theta^{\alpha
a}\theta^{\beta b}}\,,$ (167)

where we are extracting the $\theta^{\alpha a}\theta^{\beta b}$ component,
evaluating this expression on the on-shell state (80) and summing over
permutations. Around a Cartan-valued background, the holomorphic frame is
$\underline{\mathsf{H}}=\exp(-\underline{g})$, with

$\underline{g}(x,\theta,\lambda)=\frac{1}{2\pi\mathrm{i}}\int_{X}\frac{\mathrm{D}\lambda^{\prime}}{\langle\lambda\lambda^{\prime}\rangle}\frac{\langle\iota\lambda\rangle}{\langle\iota\lambda^{\prime}\rangle}\left.\underline{\mathsf{a}}^{(\text{B})}\right|_{X}=g(x,\lambda)+\theta^{\alpha
a}\mathcal{G}_{\alpha a}(x,\lambda)\,,$ (168)

where

$\mathcal{G}_{\alpha
a}(x,\lambda)=\frac{1}{2\pi\mathrm{i}}\int_{X}\frac{\mathrm{D}\lambda^{\prime}}{\langle\lambda\lambda^{\prime}\rangle}\frac{\langle\iota\lambda\rangle}{\langle\iota\lambda^{\prime}\rangle}\lambda^{\prime}_{\alpha}\left.\mathsf{a}_{a}\right|_{X}\,.$
(169)

Note that this function is related to (111) as

$\iota^{\alpha}\mathcal{G}_{\alpha a}(x,\lambda)=\langle\iota\lambda\rangle
g^{(f)}_{a}(x,\lambda)\,,$ (170)

so that the colour-ordered super form factor is finally given by

$\frac{1}{2}\frac{1}{\langle 12\rangle\ldots\langle
n1\rangle}\int_{\mathbb{M}}\mathrm{d}^{4}x\,e^{\mathrm{i}(Q-q)\cdot
x+\sum_{j}e_{j}g(x,\kappa_{j})}\\\ \sum_{i,j,k,\ell}\frac{\langle
ij\rangle^{2}}{\langle\iota i\rangle\langle\iota
j\rangle}\eta_{i\,a}\eta_{j\,b}\iota^{\alpha}\iota^{\beta}(\kappa_{k\,\alpha}\eta_{k\,a}+e_{k}\mathcal{G}_{\alpha
a}(x,\kappa_{k}))(\kappa_{\ell\,\beta}\eta_{\ell\,b}+e_{\ell}\mathcal{G}_{\beta
b}(x,\kappa_{\ell}))\,.$ (171)

With a computation completely analogous to the one carried out for the form
factor for $\,\mathrm{tr}\,\tilde{F}^{2}$, the last sums can be shown to be
independent of $o^{\alpha}$ and they coincide precisely with
$(\mathcal{Q}^{\alpha}_{a}\tilde{\mathcal{Q}}_{\alpha a})^{2}$.

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# Fields of an ultrashort tightly-focused laser pulse

Jian-Xing Li  Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69029
Heidelberg, Germany  Yousef I. Salamin  Max-Planck-Institut für Kernphysik,
Saupfercheckweg 1, 69029 Heidelberg, Germany  Department of Physics, American
University of Sharjah, POB 26666, Sharjah, United Arab Emirates  Karen Z.
Hatsagortsyan  Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, 69029
Heidelberg, Germany  Christoph H. Keitel  Max-Planck-Institut für Kernphysik,
Saupfercheckweg 1, 69029 Heidelberg, Germany

###### Abstract

Analytic expressions for the electromagnetic fields of an ultrashort, tightly
focused, linearly polarized laser pulse in vacuum are derived from scalar and
vector potentials, using a small parameter which assumes a small bandwidth of
the laser pulse. The derived fields are compared with those of the Lax series
expansion and the complex-source-point approaches and are shown to be well-
behaved and accurate even in the subcycle pulse regime. We further demonstrate
that terms stemming from the scalar potential and due to a fast varying pulse
envelope are non-negligible and may significantly influence laser-matter
interactions.

###### pacs:

(260.2110) Electromagnetic optics; (140.3538) Lasers, pulsed; (140.7090)
Ultrafast lasers.

##  I Introduction

With the rapid development of ultrashort and ultrastrong laser pulses krausz ;
wang ; eli ; piazza_rmp , laser-matter interactions involving such pulses have
been the subject of much interest brabec2 ; krushelnick ; kitagawa .
Theoretical efforts aimed at describing such pulses have also been evolving to
meet the need to model the electric and magnetic fields of the required pulses
horvath ; esarey ; esarey2 ; wang1 ; mora ; porras ; narozhny ; winful ; wang2
; lu ; hua ; yan ; becker ; sepke_pulse ; varin ; pukhov ; April ;
abdollahpour ; marceau ; gonoskov ; kaertner

In the study of many laser-matter interactions which employ a focused laser
beam, the laser fields are adequately described by solutions to the wave
equation in the paraxial approximation yariv_book . In the case of a tightly
focused laser beam, this description is improved upon further using higher-
order corrections to the paraxial solution. For example, in the Lax series
solution lax ; davis ; barton , the fields are given in ascending powers of
the square of the diffraction angle $\epsilon\equiv w_{0}/z_{r}$, and provide
a good description for a focused laser beam of long duration. Here $w_{0}$ and
$z_{r}$ are the waist radius at focus and the Rayleigh length, respectively.
It was further shown that high-order corrections to the paraxial solution hora
; sepke  can have non-negligible contributions for evaluations of direct laser
acceleration of particles sal-ol1 ; sal-apb ; salamin-prl .

Another solution of Maxwell’s equations for a focused laser beam can be
obtained using the complex-source-point approach (CSPA) deschamps ; agrawal ;
couture . The fields via CSPA, when expanded in terms of the parameter
$\epsilon$ near the propagation axis of the focused beam, coincide with those
via the Lax series expansion approximation (LSEA) sal-ol . For a Gaussian beam
it was explicitly shown that the summation of an infinite Lax series reduces
exactly to the solution of CSPA couture . The CSPA solution for a propagating
spherical wave is singular at the point $r=\pm i$($z+iz_{r}$), with the
coordinate in the propagation direction $z$ and the radius $r$, which limits
its utility far from the propagation axis and causes problems in numerical
simulations. However, a method for the regularization of CSPA solutions was
found sheppard ; ulanowski . Rather than a propagating spherical wave, one may
use the field of a simple point source and a point sink placed at the origin,
which constitutes a standing spherical wave. Then the focused beam, according
to CSPA, is free from singularities sheppard ; ulanowski ; april1 ; april2 ;
april3 ; April . The latter field was shown to form when using a laser beam
focusing with a parabolic mirror.

However, the use of LSEA, or analogously CSPA, is limited in that the pulse
temporal shape is factorized in the field expressions and, consequently, it
does not take into account distortions of the pulse during propagation. While
this may work well for rather long multi-cycle laser pulses, in which the
change in pulse-shape is not appreciable, yet for many important current and
future developments involving ultrashort and tightly-focused pulses of laser
light, the space-time evolution and distortion of the pulse-shape ought to be
taken into account esarey ; esarey2 ; wang1 ; mora ; porras ; narozhny ;
winful ; wang2 ; lu ; hua ; yan ; becker ; sepke_pulse ; varin ; pukhov . An
exact analytical solution of Maxwell’s equations is known which describes a
focused laser beam of a special form with a cosine-squared pulse shape of
arbitrary length sepke_pulse . However, this solution in the form of a double
sum is inconvenient for analytical treatments. The 3D vectorial Maxwell’s
equations are solved in pukhov  for a laser pulse, yet, only in the paraxial
approximation. The extension of CSPA to describe pulsed laser light has also
been implemented heyman ; winful ; April ; kaertner . Nonetheless, analytical
results have been obtained mostly for isodiffracting laser pulses winful ;
marceau .

The LSEA approach was generalized to describe the spatio-temporal evolution of
an ultrashort and tightly-focused laser beam by solving the wave equation with
the use of two small parameters, namely, the spatial diffraction angle
$1/k_{0}w_{0}$ and the temporal spreading parameter $1/\omega_{0}\tau_{0}$,
where $\tau_{0}$ is the laser pulse duration, $\omega_{0}$ the frequency, and
$k_{0}$ the wave number ($\omega_{0}=ck_{0}$ in vacuum) wang2 ; lu ; hua ; yan
. The corrections to the paraxial and slowly varying envelope approximations
for the fields of the focused radially polarized laser pulse of the $\rm
TM_{01}$ Gaussian mode were derived in varin  with a purely time-domain
approach. It was shown that due to those corrections the longitudinal field
decreases quickly along the propagation direction.

About two decades ago, Esarey et. al esarey  proposed a way to describe the
spatio-temporal evolution of an ultrashort and tightly-focused laser beam by
solving the wave equation with the use of a single small parameter, namely,
the relative bandwidth of the laser pulse
$\varpi\equiv\Delta\omega/\omega_{0}\ll 1$. This solution describes the field
of an ultrashort laser pulse when the duration of the pulse fulfils the
condition $\tau_{0}\lesssim z_{r}/c$. The advantage here is that the
ultrashort and tightly-focused laser beam is described by a compact expression
in terms of an expansion over a single small parameter, in contrast to the
approach of wang2 ; lu ; hua ; yan ; varin  which uses expansions over two
small parameters. In esarey , the laser pulse propagating in a plasma was
considered and the transverse fields (with respect to the propagation
direction) were calculated from a transverse vector potential alone. However,
the description of the longitudinal components of the laser field, which
become very significant for tightly focused laser beams lax ; varin , requires
the introduction of a scalar potential. Moreover, when the laser field is
described by scalar and vector potentials, the scalar potential will alter the
transverse fields, as well.

In this paper we derive analytic expressions for the electromagnetic fields in
vacuum of an ultrashort and tightly focused propagating laser pulse of linear
polarization and of a Poisson-like spectrum. The fields are derived using the
approach of esarey , but combining both contributions of vector and scalar
potentials to describe properly the longitudinal field of the ultrashort and
tightly focused laser beam. The field expressions include the first order
corrections with respect to the small bandwidth parameter $\varpi$.

The structure of the paper is as follows. In Sec. II the calculation of the
fields is outlined. Comparison of the obtained field structure with the CSPA
and LSEA solutions for the pulses are discussed in Sec. III. And, our
conclusions will be given in Sec. IV.

##  II Calculation of the fields

Expressions for the electromagnetic fields propagating in vacuum may be found
from vector and scalar potentials $\bm{A}$ and $\Phi$, respectively, which
satisfy the equations

$\displaystyle\nabla^{2}\bm{A}-\frac{1}{c^{2}}\frac{\partial^{2}\bm{A}}{\partial
t^{2}}=0,$ (1)
$\displaystyle\nabla^{2}\Phi-\frac{1}{c^{2}}\frac{\partial^{2}\Phi}{\partial
t^{2}}=0,$ (2)

provided the Lorentz condition (SI units are used),

$\displaystyle\bm{\nabla}\cdot\bm{A}+\frac{1}{c^{2}}\frac{\partial\Phi}{\partial
t}=0,$ (3)

is satisfied, simultaneously. They are equivalent to the full set of Maxwell’s
equations in the absence of charges and currents jackson . Note that, assuming

$\displaystyle\Phi=\phi_{0}\phi(x,y,z,t)\exp[i(k_{0}z-\omega_{0}t)],$ (4)

where $\phi_{0}$ is a constant, and $k_{0}=2\pi/\lambda_{0}$ a central wave
number, corresponding to the wavelength $\lambda_{0}$, the Lorentz condition
yields

$\displaystyle\Phi=\frac{c^{2}\bm{\nabla}\cdot\bm{A}}{i\omega_{0}-(\partial\phi/\partial
t)/\phi}.$ (5)

In the special case of a vector potential polarized linearly, say along the
$x$-direction, the vector and scalar potentials satisfy identical (scalar)
wave equations. As such, the potentials can only differ by multiplicative
constants. We may, therefore, write

$\displaystyle\bm{A}=\hat{x}a_{0}a(x,y,z,t)\exp[i(k_{0}z-\omega_{0}t)],$ (6)

where $\hat{x}$ is a unit vector in the polarization direction and $a_{0}$ a
constant. Now, setting $\phi=a$ the scalar potential is turned into

$\displaystyle\Phi=\frac{c^{2}\bm{\nabla}\cdot\bm{A}}{i\omega_{0}-(\partial
a/\partial t)/a}.$ (7)

Thus, the problem of finding the scalar and vector potentials reduces to
finding a solution for $a(x,y,z,t)$. Then, expressions for the fields $\bm{E}$
and $\bm{B}$ may, respectively, be found from

$\bm{E}=-\frac{\partial\bm{A}}{\partial
t}-\bm{\nabla}\Phi,\quad\text{and}\quad\bm{B}=\bm{\nabla}\times\bm{A}.$ (8)

In the absence of a scalar potential function $\Phi$, the electric field
$\bm{E}$ will have the same polarization as the vector potential $\bm{A}$,
whereas the magnetic field $\bm{B}$ will not be altered by introduction of the
scalar potential.

Following Esarey et. al esarey , a change of variables is first applied to
$\zeta=z-ct$, $\eta=(z+ct)/2$. Consequently, the wave equation, Eq. (1),
becomes

$\displaystyle\left(\nabla_{\perp}^{2}+2\frac{\partial^{2}}{\partial\eta\partial\zeta}\right)\bm{A}=0.$
(9)

Inserting Eq. (6) into Eq. (9), it reads

$\displaystyle\left(\nabla_{\perp}^{2}+2ik_{0}\frac{\partial}{\partial\eta}+2\frac{\partial^{2}}{\partial\zeta\partial\eta}\right)a\left(r,\zeta,\eta\right)=0,$
(10)

where $r=\sqrt{x^{2}+y^{2}}$. This equation is exact, i.e., no approximations
have been assumed yet. When the last term is neglected in Eq. (10), it becomes
similar to the paraxial equation yariv_book , which has an analytical
solution. Using the scaled variables $\tilde{r}=r/w_{0}$,
$\tilde{\eta}=\eta/z_{r}$, and $\tilde{\zeta}=\zeta/\tau_{0}$, Eq. (10) reads

$\displaystyle\left(\tilde{\nabla}_{\perp}^{2}+4i\frac{\partial}{\partial\tilde{\eta}}+4\varpi\frac{\partial^{2}}{\partial\tilde{\zeta}\partial\tilde{\eta}}\right)a\left(\tilde{r},\tilde{\zeta},\tilde{\eta}\right)=0.$
(11)

The last term is proportional to a small parameter
$\varpi\equiv(\omega_{0}\tau_{0})^{-1}$. Taking into account this term
perturbatively, the final solution can be derived as a series with powers of
$\varpi$. Most easily the latter can be carried out by Fourier transforming
Eq. (10) with regard to $\zeta$, which yields

$\displaystyle\left[\nabla_{\perp}^{2}+2i\left(k_{0}+k\right)\frac{\partial}{\partial\eta}\right]a_{k}\left(r,k,\eta\right)=0,$
(12)

where

$\displaystyle
a_{k}\left(r,k,\eta\right)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}a\left(r,\zeta,\eta\right)\exp(-ik\zeta)d\zeta$
(13)

is the Fourier transformation of $a\left(r,\zeta,\eta\right)$. Equation (12)
has the same mathematical structure as that of the paraxial wave equation,
which has explicit solutions for different modes. For instance, taking the
lowest-order Gaussian mode, the Fourier transform of the laser envelope reads
lax

$\displaystyle
a_{k}\left(r,k,\eta\right)=f_{k}\exp\psi_{k}\left(r,k_{0}+k,\eta\right),$ (14)

with

$\displaystyle\psi_{k}\left(r,k_{0}+k,\eta\right)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\ln(1+\alpha_{k}^{2})-\frac{\rho^{2}}{1+i\alpha_{k}}$
(15) $\displaystyle-i\tan^{-1}\alpha_{k},$

where $\alpha_{k}=\eta/z_{rk}$, $z_{rk}=(k_{0}+k)w_{0}^{2}/2$, $\rho=r/w_{0}$,
$w_{0}$ is the laser focal radius, and $f_{k}$ the initial axial envelope
profile with the focus taken at $\eta=0$.

The laser pulse envelope $a\left(r,\zeta,\eta\right)$ is calculated as the
inverse transform of Eq. (14):

$\displaystyle
a(r,\zeta,\eta)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f_{k}\exp(\psi_{k})\exp(ik\zeta)dk.$
(16)

The inverse Fourier transform can be carried out analytically if
$\psi_{k}(r,k_{0}+k,\eta)$, viewed as a function of $k^{\prime}=k+k_{0}$, is
Taylor-series expanded around the central wavenumber $k_{0}$, as in

$\displaystyle\psi_{k}(r,k_{0}+k,\eta)$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\psi_{0}^{(n)}(r,k_{0},\eta)\frac{k^{n}}{n!},$
(17) $\displaystyle\psi_{0}^{(n)}(r,k_{0},\eta)$ $\displaystyle\equiv$
$\displaystyle\frac{\partial^{n}\psi_{0}(r,k_{0},\eta)}{\partial k_{0}^{n}},$
(18)

where $\psi_{0}(r,k_{0},\eta)\equiv\psi_{k}(r,k_{0}+k,\eta)|_{k=0}$. For
instance, the first two terms in the series expansion above are

$\displaystyle\psi_{0}^{(0)}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\ln(1+\alpha^{2})-\frac{\rho^{2}}{1+i\alpha}-i\tan^{-1}\alpha,$
(19) $\displaystyle\psi_{0}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{i\alpha}{k_{0}}\left[\frac{1}{1+i\alpha}-\frac{\rho^{2}}{(1+i\alpha)^{2}}\right],$
(20)

where $\alpha=\eta/z_{r}$, and $z_{r}$ is the Rayleigh length of the laser
beam.

The expansion of Eq. (17) is valid as long as $k/k_{0}\ll 1$. From the
uncertainty relation $\left(\Delta k\right)\left(\Delta\zeta\right)\sim 1$,
with $\Delta k\sim k$ and $\Delta\zeta\sim c\tau_{0}$, one obtains $k\sim
1/c\tau_{0}$. Therefore, the fields derived from this approach are valid when
$\tau_{0}\gg 1/\omega_{0}$, which is fulfilled for a pulse length of at least
half a cycle. Thus, Eq. (17) represents the sought series expansion over the
small parameter $\varpi$.

The expansion Eq. (17) is applicable in the space-time region where
$\alpha\lesssim 1$. Taking into account that
$\alpha=\eta/z_{r}=(z+ct)/2=z/z_{r}-\zeta/(2z_{r})$, the range of
applicability of the obtained solution may be estimated by $z\lesssim z_{r}$
and $c\tau_{0}\lesssim z_{r}$, while the latter reads, equivalently,

$\displaystyle\omega_{0}\tau_{0}\lesssim(k_{0}w_{0})^{2}.$ (21)

The laser pulse envelope $a(r,\zeta,\eta)$ can, in principle, be obtained by
an inverse Fourier transform, i.e., by inserting Eq. (17) into Eq. (16).
Furthermore, the fields $\bm{E}$ and $\bm{B}$ may then be calculated via Eqs.
(7) and (8).

###  II.1 Laser pulse with Gaussian spectrum

For calculating the fields of an ultrashort tightly focused laser pulse one
needs to specify the spectral distribution of the field $f_{k}$ in Eq. (16).
Simple analytic expressions will be obtained when the spectrum of the laser
pulse is a Gaussian. Although, in this case it is not possible to avoid
unphysical negative and zero frequency components, it still can be applied
when the pulse’s bandwidth is narrow enough, i.e.,
$\Delta\omega\ll\omega_{0}$. More complex solutions based on Poissonian
spectra are free from the problem above and discussed in the next section.

For the lowest order solution,

$\displaystyle a^{(0)}(r,\zeta,\eta)$ $\displaystyle=$
$\displaystyle\exp(\psi_{0}^{(0)})\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f_{k}\exp(ik\zeta)dk$
(22) $\displaystyle=$ $\displaystyle\exp(\psi_{0}^{(0)})f(\zeta),$

since $\psi_{0}^{(0)}$ is independent of $k$. The laser envelope $f(\zeta)$ is
the inverse Fourier transform of $f_{k}$. The solution of Eq. (15) is singular
at $k=-k_{0}$. To avoid the physically uninteresting situation arising from
$k=-k_{0}$, when the laser field is axially uniform, see Eq. (12), the
following Gaussian solution of $f_{k}$ is considered esarey

$\displaystyle
f_{k}=\frac{\sigma}{k_{0}}\left(1+\frac{k}{k_{0}}\right)\exp\left(-\frac{\sigma^{2}}{2}\frac{k^{2}}{k_{0}^{2}}\right),$
(23)

where, $\sigma=\frac{k_{0}L}{2\sqrt{2\ln 2}}$, and $L=c\tau_{0}$ equals the
full-width-at-half-maximum (FWHM) of the laser pulse. The inverse Fourier
transform of $f_{k}$ gives

$\displaystyle
f(\zeta)=\left(1+\frac{ik_{0}\zeta}{\sigma^{2}}\right)\exp\left(-\frac{k_{0}^{2}\zeta^{2}}{2\sigma^{2}}\right).$
(24)

With Eqs. (6), (22) and (24), the vector potential of the zeroth-order
Gaussian solution reads

$\displaystyle\bm{A}=\hat{x}a_{0}\beta\left(1+\frac{ik_{0}\zeta}{\sigma^{2}}\right)\exp\left(ik_{0}\zeta-\frac{k_{0}^{2}\zeta^{2}}{2\sigma^{2}}-\beta\rho^{2}\right),$
(25)

where we have used the ansatz

$\displaystyle\exp(\psi_{0}^{(0)})$ $\displaystyle=$
$\displaystyle\beta\exp\left(-\beta\rho^{2}\right),$ (26) $\displaystyle\beta$
$\displaystyle=$ $\displaystyle\frac{1}{1+i\alpha}.$ (27)

Following are the electric and magnetic field components, calculated via Eq.
(8) from Eq. (25)

$\displaystyle E_{x}$ $\displaystyle=$
$\displaystyle\frac{E_{1}}{C_{1}}\left[-C_{1}^{2}+\frac{\beta^{2}x^{2}}{z_{r}^{2}}-\frac{\beta}{k_{0}z_{r}}+\frac{i\beta
x^{2}}{C_{1}z_{r}\left(C_{2}-\zeta/2\right)^{2}}\right],$ $\displaystyle
E_{y}$ $\displaystyle=$
$\displaystyle\frac{E_{1}fxy}{C_{1}z_{r}^{2}}\left[\beta+\frac{iz_{r}}{C_{1}(z+iz_{r}-\zeta/2)^{2}}\right],$
$\displaystyle E_{z}$ $\displaystyle=$ $\displaystyle\frac{E_{1}\beta
x}{C_{1}z_{r}^{2}}\left[z_{r}\left(i+\frac{i\beta^{2}r^{2}}{4z_{r}^{2}}+\frac{k_{0}\zeta}{\sigma^{2}}\right)-\frac{i\beta}{k_{0}}\right.$
$\displaystyle\left.+\frac{iz_{r}}{\left(\sigma^{2}-ik_{0}\zeta\right)}+\frac{C_{2}z_{r}}{C_{1}}\right],$
$\displaystyle B_{x}$ $\displaystyle=$ $\displaystyle 0,$ $\displaystyle
B_{y}$ $\displaystyle=$ $\displaystyle
E_{1}\left(-i+\frac{i\beta}{2k_{0}zr}-\frac{i\beta^{2}r^{2}}{4z_{r}^{2}}-\frac{k_{0}\zeta}{\sigma^{2}}+\frac{1}{i\sigma^{2}+k_{0}\zeta}\right),$
$\displaystyle B_{z}$ $\displaystyle=$ $\displaystyle\frac{E_{1}\beta
y}{z_{r}},$ (28)

where

$\displaystyle E_{1}$ $\displaystyle=$
$\displaystyle\frac{E_{0}\beta}{F_{N}}\left(1+\frac{ik_{0}\zeta}{\sigma^{2}}\right)\exp\left(ik_{0}\zeta+i\varphi_{0}-\frac{k_{0}^{2}\zeta^{2}}{2\sigma^{2}}-\beta\rho^{2}\right),$
$\displaystyle C_{1}$ $\displaystyle=$ $\displaystyle
i+\frac{ik_{0}^{2}r^{2}-C_{3}}{C_{3}^{2}}-\frac{1}{k_{0}\zeta+i\sigma^{2}}+\frac{k_{0}\zeta}{\sigma^{2}},$
$\displaystyle C_{2}$ $\displaystyle=$
$\displaystyle\frac{C_{3}-2ik_{0}^{2}r^{2}}{C_{3}^{3}}+\frac{1}{(k_{0}\zeta+i\sigma^{2})^{2}}+\frac{1}{\sigma^{2}}$
$\displaystyle C_{3}$ $\displaystyle=$ $\displaystyle
2k_{0}\left(z+iz_{r}-\zeta/2\right),$ (29)

$\varphi_{0}$ is the carrier envelope phase and $E_{0}=k_{0}a_{0}$ the field
amplitude. The various components are scaled to make $E/E_{0}=1$, at
$x=y=z=t=0$, employing a normalization factor
$F_{N}=i/\left(k_{0}z_{r}c_{0}\right)-ic_{0}$, with
$c_{0}=1+1/\left(2k_{0}z_{r}\right)+1/\sigma^{2}$.

The fields calculated from a vector potential alone esarey  are clearly
transverse, with

$E_{x}=-E_{1}C_{1},$ (30)

and $E_{y}=E_{z}=0$. The magnetic field components, however, would be the same
as in Eqs. (II.1). Thus we see that the scalar potential, which is necessary
for describing the longitudinal fields, also modifies the transverse fields
and, in this case, the normalization factor is
$F_{N}=i[1+1/(2kz_{r})+1/\sigma^{2}]$.

###  II.2 Zeroth-order Poisson solution

For ultrashort laser pulses the Gaussian spectrum $f_{k}$ contains unphysical
negative frequencies. A spectrum which does not contain negative frequencies
has to be chosen to adequately describe ultrashort pulses. Here, we choose the
Poissonian caron ; feng

$\displaystyle
f_{k}=\left(\frac{s}{k_{0}}\right)^{s+1}\frac{k^{s}e^{-sk/k_{0}}}{\Gamma(s+1)}H(k),$
(31)

where $s$ is a real positive parameter, $\Gamma(x)$ the gamma function, and
$H$($k$) the unit step function which ensures that the pulse does not exhibit
negative frequencies. Note that the propagation term $e^{ik\zeta}$ is included
therein. The parameter $s$ determines the laser pulse duration $\tau_{0}$ (the
root mean square of the temporal distribution $|f(t)|^{2}$) via April

$\displaystyle\tau_{0}=\frac{2s}{\omega_{0}\sqrt{2s-1}}.$ (32)

Taking the inverse Fourier transform of $f_{k}$ given by Eq. (31), the laser
envelope is obtained as

$\displaystyle f(\zeta)=\left(1-\frac{ik_{0}\zeta}{s}\right)^{-s-1}.$ (33)

With this, the zeroth-order vector potential takes the form

$\displaystyle\bm{A}^{(0)}=\hat{x}a_{0}\beta\left(1-\frac{ik_{0}\zeta}{s}\right)^{-s-1}\exp(-\beta\rho^{2}).$
(34)

Finally, the following field components may be obtained

$\displaystyle E_{x}$ $\displaystyle=$
$\displaystyle\tilde{E}_{1}\left[\frac{2\beta}{iD_{1}w_{0}^{2}}\left(\frac{2\beta
x^{2}}{w_{0}^{2}}-1-\frac{\beta^{2}x^{2}}{D_{1}w_{0}^{2}z_{r}}\right)-iD_{1}\right],$
$\displaystyle E_{y}$ $\displaystyle=$
$\displaystyle\frac{\tilde{E}_{1}\beta^{2}xy}{iD_{1}w_{0}^{4}}\left(4-\frac{2\beta}{D_{1}z_{r}}\right),$
$\displaystyle E_{z}$ $\displaystyle=$ $\displaystyle\frac{\tilde{E}_{1}2\beta
x}{D_{1}w_{0}^{2}}\left(-D_{1}-\frac{2k_{0}s^{\prime}}{s-ik_{0}\zeta}-\frac{D_{2}}{D_{1}}\right),$
$\displaystyle B_{y}$ $\displaystyle=$
$\displaystyle\tilde{E}_{1}\left(iD_{1}+\frac{2ik_{0}s^{\prime}}{s-ik_{0}\zeta}\right),$
$\displaystyle B_{z}$ $\displaystyle=$ $\displaystyle\frac{2\tilde{E}_{1}\beta
y}{w_{0}^{2}},$ (35)

where

$\displaystyle\tilde{E}_{1}$ $\displaystyle=$
$\displaystyle\frac{E_{0}\beta}{\tilde{F}_{N}}\left(1-\frac{ik_{0}\zeta}{s}\right)^{-s-1}\exp(-\beta\rho^{2}+i\varphi_{0}),$
$\displaystyle D_{1}$ $\displaystyle=$
$\displaystyle\frac{\beta}{2z_{r}}-\frac{\rho^{2}\beta^{2}}{2z_{r}}-\frac{k_{0}s^{\prime}}{s-ik_{0}\zeta},$
(36) $\displaystyle D_{2}$ $\displaystyle=$
$\displaystyle-\frac{\beta^{2}}{4z_{r}^{2}}+\frac{\rho^{2}\beta^{3}}{2z_{r}^{2}}+\frac{k_{0}^{2}s^{\prime}}{\left(s-ik_{0}\zeta\right)^{2}},$
(37)

and, $s^{\prime}=1+s$. Also, the normalization factor
$\tilde{F}_{N}=-d_{0}-2/d_{0}w_{0}^{2}$, and
$d_{0}=i/(2z_{r})-ik_{0}\left(1+s\right)/s$.

###  II.3 First-order Poisson solution

The first-order solution of $a(r,\zeta,\eta)$, using Eqs. (16) and (17), can
be written as

$\displaystyle a(r,\zeta,\eta)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f_{k}\exp(\psi_{0}^{(0)}+k\psi_{0}^{1})\exp(ik\zeta)dk$
(38) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f_{k}\exp(\psi_{0}^{(0)})\exp[ik(\zeta-i\psi_{0}^{(1)})]dk$
$\displaystyle=$ $\displaystyle
f(\zeta)(\zeta-i\psi_{0}^{(1)})\exp(\psi_{0}^{(0)}).$

Accordingly, the first-order Poisson-based vector potential $\bm{A}$ reads

$A^{(1)}=\hat{x}a_{0}\beta\left[1-\frac{ik_{0}\left(\zeta-i\psi_{0}^{(1)}\right)}{s}\right]^{-s-1}\exp(-\beta\rho^{2}),$
(39)

with $\psi_{0}^{(1)}$ given by Eq. (20). Using Eq. (39), the electromagnetic
fields are

$\displaystyle E_{x}$ $\displaystyle=$
$\displaystyle\frac{i\bar{E}_{1}}{2z_{r}}\left(\beta-\rho^{2}\beta^{2}+\frac{H_{1}s^{\prime}}{H_{2}s}\right)-\frac{2\tilde{E}_{1}\beta}{H_{3}w_{0}^{2}}\left(1+\frac{i\beta
s^{\prime}\alpha}{H_{2}s}\right)$ $\displaystyle+H_{4}x,$ $\displaystyle
E_{y}$ $\displaystyle=$ $\displaystyle H_{4}y,$ $\displaystyle E_{z}$
$\displaystyle=$
$\displaystyle\frac{\bar{E}_{1}x\beta}{H_{3}w_{0}^{2}z_{r}}\left[2i\beta-i\rho^{2}\beta^{2}-\frac{\beta
s^{\prime}(i+3\beta\alpha-\rho^{2}\beta^{2}\alpha)}{H_{2}s}-\right.$
$\displaystyle\left.\frac{is^{\prime}(4kz_{r}-H_{1})}{H_{2}s}+\frac{\beta(2+s)s^{\prime}(4kz_{r}-H_{1})\alpha}{H_{2}^{2}s^{2}}\right]-$
$\displaystyle\frac{\bar{E}_{1}x\beta^{3}}{2H_{3}^{2}z_{r}^{2}w_{0}^{2}}\left(1+\frac{i\beta
s^{\prime}\alpha}{H_{2}s}\right)\left[\frac{s^{\prime}(2\beta-
H_{1})(4kz_{r}-H_{1})}{H_{2}^{2}s^{2}\beta^{2}}\right.$
$\displaystyle\left.+1-2\rho^{2}\beta-\frac{2s^{\prime}(1-2\rho^{2}\beta-i\beta\alpha+3i\rho^{2}\beta^{2}\alpha)}{H_{2}s}\right],$
$\displaystyle B_{y}$ $\displaystyle=$
$\displaystyle\frac{i\bar{E}_{1}\beta}{2z_{r}}\left[\rho^{2}\beta-1+\frac{s^{\prime}(4kz_{r}-H_{1})}{H_{2}\beta
s}\right],$ $\displaystyle B_{z}$ $\displaystyle=$
$\displaystyle\frac{2\bar{E}_{1}\beta y}{w_{0}^{2}}\left(1+\frac{i\beta
s^{\prime}\alpha}{H_{2}s}\right),$ (40)

where

$\displaystyle\bar{E}_{1}$ $\displaystyle=$
$\displaystyle\frac{E_{0}\beta}{\bar{F}_{N}}\left[1-ik_{0}\left(\zeta+i\psi_{0}^{(1)}\right)/s\right]^{-s-1}\times$
(41) $\displaystyle\exp(-\beta\rho^{2}+i\varphi_{0}),$ $\displaystyle H_{1}$
$\displaystyle=$ $\displaystyle
2kz_{r}-\beta+\rho^{2}\beta^{2}-2i\rho^{2}\beta^{3}\alpha+i\beta^{2}\alpha,$
(42) $\displaystyle H_{2}$ $\displaystyle=$ $\displaystyle
1-ik_{0}\left(\zeta+i\psi_{0}^{(1)}\right)/s,$ (43) $\displaystyle H_{3}$
$\displaystyle=$ $\displaystyle\frac{\partial a/\partial ct}{a}$ (44)
$\displaystyle=$
$\displaystyle\frac{i\beta}{2z_{r}}(\rho^{2}\beta-1)-\frac{iH_{1}s^{\prime}}{2H_{2}z_{r}s},$
$\displaystyle H_{4}$ $\displaystyle=$
$\displaystyle\frac{4\tilde{E}_{1}\beta^{2}x}{H_{3}w_{0}^{4}}\left[1+\frac{2i\beta
s^{\prime}\alpha^{2}}{H_{2}s}-\frac{\beta^{2}s^{\prime}\alpha(2+s)}{H_{2}^{2}s^{2}}-\right.$
(45) $\displaystyle\left.\frac{\beta(iH_{2}^{2}s^{2}-H_{1}s^{\prime}\alpha-
iH_{2}s^{\prime}s-2H_{2}s^{\prime}s\beta\alpha)}{2H_{2}^{3}H_{3}z_{r}s^{3}}\times\right.$
$\displaystyle\left.(H_{2}s+is^{\prime}\beta\alpha)\right],$

$s^{\prime}=s+1$, and the normalization constant is

$\displaystyle\bar{F}_{N}=\frac{i}{2z_{r}}\left[1+\frac{s^{\prime}(2kz_{r}-1)}{s}\right]+\frac{4iz_{r}s}{w_{0}^{2}(1-2kz_{r}s^{\prime})}.$
(46)

Figure 1: (Color online) Contour plots of electric field components normalized
by $E_{0}$ at $t=0$. (a)–(c) are based on the Gaussian field of Eqs. (II.1).
(a) and (b) show $E_{x}$ and $E_{z}$ in the $xz-$plane ($y=0$) and (c) shows
$E_{y}$ in the $xy-$plane ($z=0$). (d) Same as (a) but based on the Gaussian
field, which is derived solely from the vector potential, see Eq. (30). The
parameters used are $w_{0}=L=\lambda_{0}=1~{}\mu$m.

##  III Discussion

Typical distributions of the electric field components are shown in Fig. 1.
Figs. 1 (a)–(c) are based on Eqs. (II.1), derived from combined vector and
scalar potentials (VSP) while Fig. 1(d) is based on Eq. (30) or fields derived
from a vector potential (VP) alone. Note that Figs. 1 (a) and (d) are hardly
distinguishable for the parameter set used, on account of the fact that the
additional terms, except the first one in the $E_{x}$ component of Eq. (II.1),
constitute a very small correction to the first term, i.e., the right side of
Eq. (30). Easily visible deviations may be reached when subcyle pulses are
employed instead. On the other hand, $E_{z}$ and $E_{y}$, illustrated by Figs.
1 (b) and (c) are solely due to the VSP approach and are absent from the VP
analysis. Those components, although more than two orders of magnitude weaker
than their associated $E_{x}$ component, yet play a significant role in the
treatment of interaction of matter with such a laser pulse in vacuum.

Figure 2 compares and contrasts our Gaussian solutions for ultrashort focused
laser pulses, see Eq. (II.1), with those via CSPA becker  and LSEA sal-apb .
In Fig. 2 (a) the pulse duration is relatively long $L=5\lambda_{0}$, while in
Fig. 2 (b) the pulse is of subcyle duration. In both cases the laser beam is
tightly focused with a waist size of $w_{0}=\lambda_{0}$. It is hard to
distinguish the fields stemming from all three approaches for the case of a
long laser pulse ($L=5\lambda_{0}$). The slight difference between the CSPA
and LSEA solutions, on the one hand, and ours, on the other, in this case is
due to the fact that the applicability condition Eq. (21) of our solution is
at its limit, namely, $\omega\tau_{0}/(kw_{0})^{2}\approx 0.8$. In the case of
a subcycle pulse ($L=\lambda_{0}/2$), the variations of $E_{x}$ at focus with
time based on the CSPA and our Gaussian solution are almost identical, whereas
those based on the LSEA are clearly different due to temporal diffraction
effects (the LSEA solution is valid only for pulses of infinitely long
duration).

Figure 2: (Color online) The time variation of the normalized transverse laser
field $E_{x}$ at focus. The pulse lengths are (a) $L=5\lambda_{0}$ and (b)
$L=\lambda_{0}/2$. The other parameters are the same as in Fig. 1. The blue
dashed curves show our solution of Eq. (II.1), while the solid red and dot-
dashed black curves are based on fields of CSPA and LSEA, respectively. Please
note that: (a) all three curves are close to each other with merely slight
differences; (b) the blue dashed and solid red curves are overlapping.  Figure
3: (Color online) Time variation of the normalized $E_{x}$ at focus. The pulse
lengths are (a) $L\approx 0.6\lambda_{0}$ ($s=2$) and (b) $L\approx
2.7\lambda_{0}$ ($s=50$) with $w_{0}=5\lambda_{0}$, while the remaining
parameters are the same as in Fig. 1. The red solid and blue dashed curves
present the fields of zeroth-order Gaussian and zeroth-order Poisson solution,
respectively.

In Fig. 3, we compare the fields of pulses with different spectral
distributions, namely, Gaussian and Poissonian. Note that the red solid and
blue dashed curves present the fields of zeroth-order Gaussian and zeroth-
order Poisson solution, respectively. For a subcycle laser pulse of $L\approx
0.6\lambda_{0}$ ($s=2$), see Fig. 3(a), the field with Gaussian spectrum is
significantly different from the Poisson solution. Therefore, the Gaussian
spectral profile for the pulse is not suitable anymore, and the Poisson field
should be employed to simulate ultrashort pulses. In Fig. 3(b), for a several-
cycle laser pulse with $L\approx 2.7\lambda_{0}$ ($s=50$), the Gaussian’s
field overlaps with the Poisson solution near the pulse center, and they very
slightly differ from each other around the pulse’s tail $ct/\sigma\gtrsim 8$.
Therefore, the Gaussian spectrum is assumed to be an acceptable solution for
this case.

Figure 4: (Color online) Time variation of the normalized $E_{x}$ at focus.
(a) $w_{0}=\lambda_{0}$, and (b) $w_{0}=0.5\lambda_{0}$ with $L\approx
0.6\lambda_{0}$ ($s=2$), while the remaining parameters are the same as in
Fig. 1. The blue solid and red dotted curves present the fields of zeroth- and
first-order Poisson solutions, respectively.

We compare the zeroth- and first-order Poisson solutions for a subcycle laser
pulse with different laser focusing sizes in Fig. 4. Note that the blue solid
and red dotted curves present the fields of zeroth- and first-order Poisson
solutions, respectively. For a tightly focused laser pulse of
$w_{0}=\lambda_{0}$, see Fig. 4(a), the first-order solution of the Poisson
field overlaps with the zeroth-order solution very well. Therefore, the
zeroth-order Poisson field suffices to simulate such pulses. In Fig. 4(b), for
an ultra-tightly focused laser pulse with $w_{0}=0.5\lambda_{0}$, the first-
order solution of the Poisson field overlaps with the zeroth-order solution
near the pulse center as well, however, they are distinguishably different
from each other around the pulse’s tail $ct/\sigma\gtrsim 1$. Consequently, in
this case the zeroth-order solution is not sufficient anymore, and the first-
order correction should be taken into account.

##  IV Conclusion

Following the approach of esarey , we have derived the field expressions of an
ultrashort tightly focused laser pulse propagating in vacuum, using an
expansion over a single small parameter which assumes that the relative
bandwidth of the laser pulse is not large. The basic solution is derived using
a Poissonian initial spectral distribution for the pulse, and it is applicable
for pulse lengths not larger than the Rayleigh length. The accuracy of the
solution is verified in comparison with that of CSPA of becker . It is
demonstrated, with the introduction of a scalar potential, that the fields do
not only develop new axial components, but also modify the transverse ones
quite significantly. Moreover, deviations of the derived fields from those
based on the LSEA are shown to be non-negligible when the pulse duration is
less than a few cycles, and substantial for subcycle pulses. It has been shown
that the Gaussian spectral distribution cannot be applied in the case of
subcycle laser pulses, because they yield pulses differing significantly from
those via the Poissonion-spectrum-based ones, with the same bandwidth.
Compared with solutions of the wave equation expanded in power-series of two
parameters, the diffraction parameter ($1/kw_{0}$) and the time dilation
parameter ($1/\omega\tau_{0}$), respectively, our solution is more compact,
cf., e.g., yan , and may be of utility for current and future applications in
laser acceleration and other ultrafast physics.

###### Acknowledgements.

YIS has been partially supported by an American University of Sharjah Faculty
Research Grant (FRG-III).

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