# Decomposing User-APP Graph into Subgraphs for Effective APP and User
Embedding Learning

Tan Yu, Jun Zhi, Yufei Zhang, Jian Li, Hongliang Fei, Ping Li
10900 NE 8th St. Bellevue, Washington 98004, USA
No. 10 Xibeiwang East Road, Beijing 100193, China

{tanyu01, zhijun, zhangyufei, lijian26, hongliangfei<EMAIL_ADDRESS>Cognitive Computing Lab, Baidu Research  Baidu Search Ads (Phoenix Nest),
Baidu Inc.

###### Abstract

APP-installation information is helpful to describe the user’s
characteristics. The users with similar APPs installed might share several
common interests and behave similarly in some scenarios. In this work, we
learn a user embedding vector based on each user’s APP-installation
information. Since the user APP-installation embedding is learnable without
dependency on the historical intra-APP behavioral data of the user, it
complements the intra-APP embedding learned within each specific APP. Thus,
they considerably help improve the effectiveness of the personalized
advertising in each APP, and they are particularly beneficial for the cold
start of the new users in the APP. In this paper, we formulate the APP-
installation user embedding learning into a bipartite graph embedding problem.
The main challenge in learning an effective APP-installation user embedding is
the imbalanced data distribution. In this case, graph learning tends to be
dominated by the popular APPs, which billions of users have installed. In
other words, some niche/specialized APPs might have a marginal influence on
graph learning. To effectively exploit the valuable information from the niche
APPs, we decompose the APP-installation graph into a set of subgraphs. Each
subgraph contains only one APP node and the users who install the APP. For
each mini-batch, we only sample the users from the same subgraph in the
training process. Thus, each APP can be involved in the training process in a
more balanced manner. After integrating the learned APP-installation user
embedding into our online personal advertising platform, we obtained a
considerable boost in CTR, CVR, and revenue.

###### Index Terms:

advertising, search, cross-modal

##  I Introduction

For different users, a personalized advertising system feeds different ads
based on the estimated relevance between the ad and the user’s interest.
Normally, the relevance between a user and an ad is measured by the similarity
between their embeddings, which are learned jointly from the users’ historical
behaviors on the ads. Nevertheless, for new users, there are no historical
user-ad behaviors for learning effective user embedding. This issue of
modeling new users is normally defined as the cold start problem. To solve the
cold start problem, we usually exploit the user’s demographic attributes, such
as age, region, and gender. The attribute embedding has been effectively
learned based on the ordinary users’ rich experience accumulated in the past
and can readily generalize well to the new users. Since the attribute
embedding does not rely on the historical user behaviors, they are useful for
tackling the cold start problem.

This work explores a new type of attribute embedding learned from the APP-
installation information. The users who install the same APP might share some
common interests and tend to behave similarly. Meanwhile, a user’s installed
APP lists might encode much richer fine-grained information about a user than
basic demographic information, such as age, gender, and location. Thus, if
exploiting the users’ APP-installation information effectively, we might
significantly boost the performance of our personalized advertising platform
for the new users. In fact, the APP-installation information benefits not only
the new users but also the regular users. This is because the learned user’s
APP-installation embedding complements the user’s behavior embedding. Thus,
when incorporating the APP-installation embedding into our model, we also
observed improvements for the regular users’ personalized advertising
performance.

Figure 1: The visualization of a user-APP undirected bipartite graph. A node
with a blue dot denotes a user and a node with a green box denotes an APP. An
edge exists between an APP and a user if the user installs the APP.

We formulate the APP-installation embedding as a bipartite graph embedding
problem. The bipartite graph consists of two types of nodes, including the
user nodes and the APP nodes as visualized in Figure 1. An edge exists between
a user node and an APP node if the user has installed the APP in his/her
mobile phone. Straightforwardly, we could utilize any existing graph learning
methods such as graph convolutional neural network (GCN) to learn the user
node embedding and the APP node embedding. Nevertheless, a serious issue
caused by imbalanced data distribution makes the training of the graph
learning model extremely challenging. Specifically, for a popular
APP111https://www.businessofapps.com/data/most-popular-apps/, it is installed
by billions of users, generating billions of edges in the graph. In contrast,
a niche APP installed by millions of users can only create millions of edges
in the graph. In this case, graph learning is dominated by the billions of
edges created by the popular APPs, and the edges from the niche APPs might be
swamped. But the edges from the popular APPs might not encode useful
discriminating information since everyone almost installs them. In contrast,
the edges from the niche APPs might be very useful for describing a user’s
characteristics, but that useful information might not gain enough attention
when training the graph embedding.

In this work, we propose a novel sampling approach to tackle the imbalanced
data distribution issue for learning effective APP-installation user
embeddings. Specifically, we decompose the user-APP graph into a set of sub-
graphs. Each subgraph contains only a single APP and the users who install the
APP. In the training process, we sample a subgraph for each iteration and
construct training triplets based on users within the subgraph for embedding
learning. In this manner, the popular APPs and the niche APPs will be involved
in the training process in a fair manner. The offline and online experiments
demonstrate the excellence of our method.

##  II Related Work

Factorization based methods. Factorization-based methods rely on an affinity
matrix encoding the connections between nodes in the graph. They factorize the
affinity matrix to obtain the embedding vectors for nodes. A pioneering work,
Laplacian Eigenmaps [3] aims to keep the embedding of two nodes close when the
weight of the edge connecting these two nodes is high. It seeks to minimize
the weighted summation of squares of distance between nodes while the weight
of each item in the summation corresponds to the weight of the edge. It is
formulated into an eigenproblem. Nevertheless, it is extremely slow when
solving the eigen problem in the scenario when the number of nodes is huge.
Ahmed _et al._ [1] propose a framework for large-scale graph decomposition.
They partition a graph based on minimizing the number of neighboring vertices.
GraRep [5] integrates global structural information of the graph into the
graph learning process. HOPE [17] learns the graph embedding for nodes with
the asymmetric transitivity, which is a critical property of the directed
graph. [22] develop a graph kernel methods based on the iterations of power
method applied on the adjacency matrix.

Random walks based methods. Random walks are very useful when we have only
access to a part of the graph or the graph is too large to be modeled
globally. DeepWalk [18] creates multiple random walks, and maximizes the sum
of log-likelihoods for each random walk. It preserves higher-order proximity
between nodes in the graph. _node2vec_ [10] also encodes higher-order
proximity between nodes by maximizing the probability of occurrence of
subsequent nodes. It conducts a trade-off between breadth-first searches (BFS)
and depth-first searches (DFS) on the graph to generate a more effective graph
embedding than DeepWalk. Walklets [19] additionally incorporates explicit
modeling in random walks. Hierarchical Representation Learning for Networks
(HARP) [7] proposes a better initialization strategy to avoid the local optima
in optimization.

Neural network based methods. SDNE [24] stacks multiple layers of non-linear
functions to preserve highly non-linear network structure. It adopts an auto-
encoder structure which uses the embedding to reconstruct its neighbors. DNGR
[6] feeds the positive point-wise mutual information matrix into a stacked
denoising autoencoder to capture higher-order proximity in the learned graph
embedding. Nevertheless, SDNE and DNGR consider the whole graph and take as
input the global neighborhood of each node, which are not efficient for large-
scale graphs. Recently, graph convolution neural network (GCN) provides an
effective and efficient solution by adopting a configuration with local
constraints. These methods can be categorizes into spatial-based methods [16,
2, 11, 23, 8, 25] and spectral-based methods [4, 12, 9, 13, 14, 15]. Spatial-
based methods directly conduct convolution on the original graph. In contrast,
spectral-based methods conduct convolution on the spectrum of the adjacent
matrix of the graph. In [20, 21], it was shown that the incorporation of the
node representation vectors computed by a random walk based method in GCN can
effectively boost the performance of the GCN.

##  III Method

In this section, we introduce graph-based embedding learning for modeling the
APP-installation information of users.

###  III-A Graph Decomposition

Definition. We denote the set of APPs used for building the graph by
$\\{{s}_{i}\\}_{i=1}^{N}$, and denote the set of users by
$\\{{t}_{j}\\}_{j=1}^{M}$. They constitute the node set
$\mathcal{V}=\\{{p}_{1},\cdots,{p}_{N},{u}_{1},\cdots,{u}_{M}\\}$. Meanwhile,
the edge set $\mathcal{E}$ contains all edges connecting two nodes
$\\{e_{s_{i},t_{i}}\\}_{i=1}^{L}$, where $s_{i}$ denotes the index of the user
and $t_{i}$ denotes the index of the user in the $i$-th edge,
$e_{s_{i},t_{i}}$. That is, the existence of edge $e_{s_{i},t_{i}}$ means that
the user $u_{s_{i}}$ has installed the APP $a_{t_{i}}$. The user-APP graph
$\mathcal{G}$ is constructed based on the node set and the edge set
$\\{\mathcal{E},\mathcal{V}\\}$. We further define the subgraph
$\mathcal{G}_{i}$ with the node set $\mathcal{V}_{i}$ and the edge set
$\mathcal{E}_{i}$. $\mathcal{V}_{i}$ contains only one APP node $p_{i}$ and
the user nodes connected to $p_{i}$. $\mathcal{E}_{i}$ contains all edges
which connect the APP node $p_{i}$. We visualize the process of decomposing a
graph into a set of subgraphs in Figure 2.

Figure 2: The visualization of decomposing a user-APP graph into subgraphs.
The user-APP graph $\mathcal{G}$ consists of the node set
$\mathcal{V}=\\{p_{i}\\}_{i=1}^{3}\cup\\{u_{i}\\}_{i=1}^{8}$ where $p_{i}$
denotes an APP node and $u_{i}$ is a user node. $\mathcal{G}$ is decomposed
into three subgraphs $\\{\mathcal{G}_{i}\\}_{i=1}^{3}$. $\mathcal{G}_{i}$
consists of the node set $\mathcal{V}_{i}$. In this example,
$\mathcal{V}_{1}=\\{p_{1},u_{4},u_{5},u_{6}\\}$,
$\mathcal{V}_{2}=\\{p_{2},u_{1},u_{2},u_{3}\\}$, and
$\mathcal{V}_{3}=\\{p_{3},u_{7},u_{8},u_{1}\\}$. Note that, the user $u_{1}$
is connected with two APPs $p_{2}$ and $p_{3}$. Thus, $u_{1}$ is included in
two subgraphs, $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$.

###  III-B Graph Learning

Initialization. We denote the embedding of the user $u_{i}$ by
$\mathbf{u}_{i}$ and the embedding of an APP $p_{j}$ by $\mathbf{p}_{j}$. We
denote the indices of users installing the APP $u_{i}$ by $\mathcal{I}_{i}$.
The user embeddings are randomly initialized. In parallel, an APP embedding
$\mathbf{p}_{j}$ is initialized by averaging the embeddings of users
installing the APP:

$\mathbf{p}_{j}=\frac{\sum_{k\in\mathcal{P}_{i}}\mathbf{u}_{k}}{|\mathcal{P}_{i}|},$
(1)

where $|\mathcal{P}_{i}|$ denotes the cardinality of the set
$\mathcal{P}_{i}$, _i.e._ , the number of users installing the APP $p_{i}$.

Subgraph sampling. As we mentioned, for a subgraph $\mathcal{G}_{i}$, it
contains an APP node ($p_{i}$) and the users installing the APP $p_{i}$. Let
us denote the probability of sampling the subgraph $\mathcal{G}_{i}$ as
$P(\mathcal{G}_{i})$. A native sampling approach is sampling the sub-graph
with a probability proportional to the number of user nodes in the subgraph.
That is,

$P(\mathcal{G}_{i})=\frac{|\mathcal{P}_{i}|}{\sum_{j=1}^{N}|\mathcal{P}_{j}|},\forall
i\in[1,N],$ (2)

where $N$ is total number of APPs and $|\mathcal{P}_{i}|$ denotes the number
of users in the subgraph $\mathcal{G}_{i}$. In this case, each edge connecting
a user and an APP will be involved in the training process with an equal
probability. Nevertheless, this strategy will make the embedding learning
dominated by the popular APPs with a huge number of users and the
contributions from some niche APPs with a small number of users will be
underestimated. To make the contributions from different APPs balanced, we can
devise that the sampling probability of each sub-graph to be equal. That is,

$P(\mathcal{G}_{i})=\frac{1}{N},\forall i\in[1,N].$ (3)

In this case, the edges based on niche APPs with a small number of users will
be over-sampled, and the edges based on the popular APPs with a huge number of
users will be under-sampled. Nevertheless, it might lead to repeatedly
sampling for edges from niche APPs, and some edges from the popular APPs might
have little chance to be involved in the training process. It tends to make
the learned embedding prone to over-fitting due to a lack of diversity in the
training samples. To achieve a balanced sampling and meanwhile suppress over-
fitting, we adopt a trade-off sampling approach. It devises the probability as

$P(\mathcal{G}_{i})=\frac{|\mathcal{P}_{i}|^{\tau}}{\sum_{j=1}^{N}|\mathcal{P}_{j}|^{\tau}},\forall
i\in[1,N],$ (4)

where $\tau$ is a pre-defined positive constant. Normally, we set $0<\tau<1$.
It assigns a higher sampling probability to the subgraph containing more nodes
for suppressing over-fitting and meanwhile achieving a good balance among
different APPs. When $\tau=1$, it degenerates to the naive sampling approach
defined in Eq. (2). On the other hand, when $\tau=0$, it degenerates to the
balanced sampling approach defined in Eq. (3). By default, we set $\tau=0.5$
in our experiments.

Embedding learning within a subgraph. Let denote the app embedding with a
subgraph by $\mathbf{p}$, the embedding of a user installing the APP by
$\mathbf{u}^{+}_{i}$ and that of a user who does not install the APP by
$\mathbf{u}^{-}_{j}$. The user and APP embedding learning seeks to keep a
large similarity between $\mathbf{p}$ and $\mathbf{u}^{+}_{i}$. At the same
time, it seeks to maintain a small similarity between $\mathbf{p}$ and
$\mathbf{u}^{-}_{j}$. Straightforwardly, we can learn the user and the APP
embedding through a pairwise loss:

$\mathcal{L}_{\mathrm{pair}}=\frac{1}{n^{+}}\sum_{i=1}^{n^{+}}\mathrm{log}(1+e^{-\beta
s(\mathbf{p},\mathbf{u}^{+}_{i})})-\frac{1}{n^{-}}\sum_{i=1}^{n^{-}}\mathrm{log}(1+e^{-\beta
s(\mathbf{p},\mathbf{u}^{-}_{i})}),$ (5)

where $n_{+}$ denotes the number of users installing the APP and $n_{-}$
denotes the number of users who do not install, $\beta$ is a pre-defined
positive constant controlling the softness, and $s(\cdot,\cdot)$ measures the
cosine similarity between two vectors, as visualized in Figure 3.

Figure 3: The visualization of user and APP embedding learning within a
subgraph. In this example, the APP embedding is $\mathbf{p}$ (green box).
There are four users installing the APP, $\\{\mathbf{u}^{+}_{i}\\}_{i=1}^{4}$
(blue dots) with the centroid $\mathbf{u}^{c}$ (yellow dot) and four users not
installing the APP, $\\{\mathbf{u}^{-}_{i}\\}_{i=1}^{4}$ (red dots).

In parallel to the pairwise loss defined above, we devise an additional
centroid loss to further enhance the effectiveness of the learned embedding.
To be specific, we first compute the centroid of the embeddings of users
installing the APP:

$\mathbf{u}_{c}=\frac{1}{n_{+}}\sum_{i=1}^{n^{+}}\mathbf{u}^{+}_{i}.$ (6)

Then the centroid loss is computed by

$\mathcal{L}_{\mathrm{centroid}}=\mathrm{log}(1+e^{-\beta
s(\mathbf{p},\mathbf{u}^{+}_{c})}).$ (7)

To stabilize the training, we update the user embedding and the APP embedding
in an alternating manner:

  1. 1)

Fix user embedding $\\{\mathbf{u}^{+}_{i}\\}_{i=1}^{N_{+}}$ and
$\\{\mathbf{u}^{-}_{i}\\}_{i=1}^{N_{-}}$, and update the APP embedding
$\mathbf{p}$ using the centroid loss $\mathcal{L}_{\mathrm{centroid}}$.

  2. 2)

Fix the APP embedding $\mathbf{p}$, and update the positive user embedding
$\\{\mathbf{u}^{+}_{i}\\}_{i=1}^{N_{+}}$ using the pairwise loss
$\mathcal{L}_{\mathrm{pair}}$.

To improve the training efficiency, we achieve this iterative training manner
in a parallel way by utilizing the stop-gradient trick. That is, we devise the
final loss
$\mathcal{L}=\mathcal{L}_{\mathrm{centroid}}+\mathcal{L}_{\mathrm{pair}}$. In
the meanwhile, we stop the gradient derived by
$\mathcal{L}_{\mathrm{centroid}}$ back-propagating to
$\\{\mathbf{u}^{+}_{i}\\}_{i=1}^{N_{+}}$ and
$\\{\mathbf{u}^{-}_{i}\\}_{i=1}^{n_{-}}$ and meanwhile stop the gradient from
$\mathcal{L}_{\mathrm{pair}}$ back-propagating to $\mathbf{p}$ and
$\\{\mathbf{u}^{-}_{i}\\}_{i=1}^{n_{-}}$.

##  IV Experiments

Dataset. To train the model, we collect the information of 80 million users
and 50 thousand APPs. On average, each user installs around $30$ APPs.

###  IV-A Offline experiments

Memory. For each APP, we randomly sample $96$ users who have already installed
the APP and $96$ users not installing the APP. Note that these APP
installation has been involved in the training process. For each APP-user
pair, we compute the cosine similarity between their embeddings. Then we
threshold the cosine similarity to $0$ or $1$ to predict whether the user has
installed the APP or not. In Table I, we show the experimental result. As
shown in the table, in the training data, the learned embedding can achieve a
$0.953$ precision and $0.981$ AUC, which demonstrates the powerful fitting
capability of the learned embeddings.

TABLE I: The memory performance of the learned user and APP embeddings. We report the prediction precision and area-under-curve (AUC) for the APP installation.  Precision | 0.953
---|---
AUC | 0.981
TABLE II: The inference performance of the learned user and APP embeddings. AUC+ denotes the AUC excluding APPs with more than $8\%$ users and AUC∗ denotes the AUC excluding APPs with more than $2.5\%$ users.  | AUC | AUC+ | AUC∗
---|---|---|---
APP-side | 0.797 | 0.840 | 0.854
User-side | 0.786 | 0.829 | 0.844

Inference. To evaluate the inference performance of the learned user and APP
embedding, we report the classification AUC on the user side and that on the
APP side. The user-side AUC is averaged over users. For each user, we test the
prediction accuracy using several APPs the user has installed and several APPs
the user does not install. The APP-side AUC is measured in a similar manner
but is averaged over APPs. Note that the testing cases for inference are not
involved in the training process. To be specific, our whole data is collected
during $N$ days. We use the data in the first $N-5$ days for training and that
from the last $5$ days for testing. Meanwhile, we also report the AUC without
excluding APPs with a huge number of users. To be specific, we report AUC∗,
which excludes APPs with more than $2.5\%$ users. We also report AUC+, which
excludes that with more than $8\%$ users. As shown in Table II, the AUC
achieved in the inference is lower than that in Table I. In the meanwhile, by
excluding some APPs with a huge number of users, AUC+ and AUC∗ are larger than
AUC.

Ablation study Here, we investigate the influence of removing
$\mathcal{L}_{\mathrm{centroid}}$ or the stop-gradient strategy through
ablation study. As shown in Table III, when removing
$\mathcal{L}_{\mathrm{centroid}}$, the AUC drops from $0.981$ to $0.977$ and
the precision decreases from $0.953$ to $0.948$. Meanwhile, without the stop-
gradient strategy, both the AUC and the precision decrease considerably.

TABLE III: The influence of removing $\mathcal{L}_{\mathrm{centroid}}$ or the stop-gradient strategy in inference.  | Ours | w/o $\mathcal{L}_{\mathrm{centroid}}$ | w/o stop-gradient
---|---|---|---
Precision | 0.953 | 0.948 | 0.942
AUC | 0.981 | 0.977 | 0.973

###  IV-B Online experiments

We have integrated the user embedding learned from the APP-installation
information as a feature which complements the existing user embedding learned
from historical behaviors. After launching it in our online personalized
advertising platform, we achieved a $+1.1\%$ CTR improvement, a $+1.7\%$ CVR
boost, a +$2.6\%$ increase in revenue as shown in Table IV.

TABLE IV: The online experiments in our online advertising platform during one week.  CTR | CVR | Revenue
---|---|---
$+1.1\%$ | $+1.7\%$ | $+2.6\%$

##  V Conclusion

In this paper, we exploit the APP-installation information to assist in
modeling the user’s characteristics for personalized advertising. To this end,
we build a user-APP bipartite graph and adopt a graph convolution network to
learn the user embedding. We use the learned user embedding from our user-APP
graph as the complementary information to the existing user representation
learned from the user profile and the user’s historical behaviors. After
deploying it in our advertising platform, both CTR and CVR improve
considerably.

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# Light-front approach to relativistic electrodynamics

Gaetano Fiore1,2
1 Dip. di Matematica e Applicazioni, Università di Napoli “Federico II”,
Complesso Universitario M. S. Angelo, Via Cintia, 80126 Napoli, Italy
2 INFN, Sez. di Napoli, Complesso MSA, Via Cintia, 80126 Napoli, Italy

###### Abstract

We illustrate how our recent light-front approach simplifies relativistic
electrodynamics with an electromagnetic (EM) field $F^{\mu\nu}$ that is the
sum of a (even very intense) plane travelling wave
$F_{t}^{\mu\nu}(ct\\!-\\!z)$ and a static part $F_{s}^{\mu\nu}(x,y,z)$; it
adopts the light-like coordinate $\xi=ct\\!-\\!z$ instead of time $t$ as an
independent variable. This can be applied to several cases of extreme
acceleration, both in vacuum and in a cold diluted plasma hit by a very short
and intense laser pulse (slingshot effect, plasma wave-breaking and laser
wake-field acceleration, etc.)

##  1 Introduction

The equation of motion of a particle of rest mass $m$, electric charge $q$ in
an external EM field

$\displaystyle\begin{array}[]{l}\displaystyle\dot{\bf p}(t)=q{\bf E}[ct,{\bf
x}(t)]+\frac{{\bf p}(t)}{\sqrt{m^{2}c^{2}\\!+\\!{\bf p}^{2}(t)}}\wedge q{\bf
B}[ct,{\bf x}(t)],\\\\[6.0pt] \displaystyle\dot{\bf x}(t)=\frac{c{\bf
p}(t)}{\sqrt{m^{2}c^{2}\\!+\\!{\bf p}^{2}(t)}}\end{array}$ (3)

in its general form is non-autonomous and highly nonlinear in the unknowns
${\bf x}(t),{\bf p}(t)$. Here ${\bf x},{\bf p}$ are the position and
relativistic momentum of the particle, while ${\bf E},{\bf B}$ are the
electric and magnetic field; we use Gauss CGS units. In the situations
characterizing many standard applications (3) can be solved or simplified
making one or more of the following assumptions:

  1. 1.

${\bf E},{\bf B}$ are constant or vary “slowly” in space/time;

  2. 2.

the motion of the particle keeps non-relativistic;

  3. 3.

${\bf E},{\bf B}$ are so small that nonlinear effects in ${\bf E},{\bf B}$ are
negligible;

  4. 4.

${\bf E},{\bf B}$ are monochromatic waves, or very slow modulations thereof.

If the role of the charges (isolated or in the form of a continuum) as sources
of the EM field cannot be neglected, these equations are to be coupled with
the Maxwell equations; the resulting system is much more complicated. In many
interesting situations none of the above assumptions is satisfied. For
instance, many violent astrophysical electrodynamic processes occur in the
presence of extremely intense and rapidly varying electromagnetic fields (see
e.g. [1] and references therein). On the other hand, the on-going, amazing
developments of laser technologies leading to compact sources of extremely
intense and short coherent EM waves111The technique of Chirped Pulse
Amplification [2, 3], awarded with the Nobel Prize in physics in 2018, yields
pulses of intensity up to $10^{23}$ Watt per square centimeter and duration
down to femtoseconds. Huge investments (e.g. for the Extreme Light
Infrastructure program within the EU ESFRI roadmap) in new technologies (thin
film or relativistic mirror compression, etc. [4, 1]) will yield even more
intense/short pulses, at a lower cost. allow extreme accelerations through the
Laser Wake Field Acceleration (LWFA) mechanism [5] in plasmas222In fact, huge
investments aiming at the construction of table-top particle accelerators
based on Wake Field Acceleration are being made all over the world. For
instance, in Europe the large network of research centers “European Plasma
Research Accelerator with eXcellence In Applications” (EUPRAXIA) has been
created to develop the associated technologies and has just issued its
expected Conceptual Design Report of reliable accelerators of this kind [6, 7,
8]. Among the present and future applications of accelerators we mention:  •
Medicine: diagnostics (PET,…), cancer therapy by accelerated particles
(electrons, protons, ions) or locally induced radioisotope production,…; •
Research: elementary particle physics, materials science, structural biology,
inertial nuclear fusion, electron beams for X-ray free electron laser,…; •
Industry: atomic scale lithography, surface treatment of materials,
sterilization, additive-layer (i.e. 3D print) manufacturing, detection systems
across shields,…; • Environmental remediation: flue gas cleanup, petroleum
cracking, transmutation of nuclear wastes,…. . To deal with such complex
problems it is is almost unavoidable to resort to numerical resolution
methods, e.g. particle-in-cell (PIC) techniques for plasmas. PIC or other
codes in general involve huge and costly computations for each choice of the
free parameters; exploring the parameter space blindly to pinpoint the
interesting regions is prohibitive, if not accompanied by some analytical
insight that can simplify the work, at least in special cases or in a limited
space-time region. It is therefore important to look for new approaches
simplifying the study of the dynamics induced by (3) when conditions (i-iv)
are not fulfilled.

Here we summarize an approach [9] that systematically applies the light-front
formalism [10]; it is especially fruitful if in the spacetime region of
interest ${\bf E},{\bf B}$ are the sum of static parts and a plane transverse
travelling wave propagating in a fixed ($z$, say) direction:

${\bf E}(t,{\bf
x})=\\!\\!\underbrace{{\bm{\epsilon}}^{\scriptscriptstyle\perp}(ct\\!-\\!z)}_{travelling\,wave}\\!+\underbrace{{\bf
E}_{s}({\bf x})}_{static},\qquad{\bf B}(t,{\bf
x})=\underbrace{\mathbf{k}\wedge{\bm{\epsilon}}^{\scriptscriptstyle\perp}(ct\\!-\\!z)}_{travelling\,wave}+\underbrace{{\bf
B}_{s}({\bf x})}_{static}.$ (4)

We decompose 3-vectors such as ${\bf x}$ in the form ${\bf
x}\\!=\\!x\mathbf{i}\\!+\\!y\mathbf{j}\\!+\\!z\mathbf{k}\\!={\bf
x}^{\scriptscriptstyle\perp}\\!+\\!z\mathbf{k}$ ($x,y,z$ are the cartesian
coordinates) and denote as ${\rm x}\\!\equiv\\!(ct,{\bf x})$ the set of
coordinates of Minkowski spacetime points with respect to the laboratory
frame. We assume only that ${\bm{\epsilon}}^{\scriptscriptstyle\perp}(\xi)$ is
piecewise continuous and

$\displaystyle\begin{array}[]{ll}&\mbox{ {\bf a})
}\quad{\bm{\epsilon}}^{\scriptscriptstyle\perp}\mbox{ has a compact support
}[0,l],\\\\[10.0pt] \mbox{ or}&\mbox{ {\bf
a'})}\quad{\bm{\epsilon}}^{\scriptscriptstyle\perp}\in
L^{1}(\mathbb{R}),\end{array}$ (7)

and treat on the same footing all ${\bm{\epsilon}}^{\scriptscriptstyle\perp}$
fulfilling (7) regardless of their Fourier analysis, in particular:

  1. 1.

A modulated monochromatic wave (here $a_{1}^{2}\\!+\\!a_{2}^{2}\\!=\\!1$):

${\bm{\epsilon}}^{\scriptscriptstyle\perp}\\!(\xi)\\!=\\!\underbrace{\epsilon(\xi)}_{\mbox{modulation}}\underbrace{[\mathbf{i}a_{1}\cos(k\xi\\!+\\!\varphi)\\!+\\!\mathbf{j}a_{2}\sin(k\xi)]}_{\mbox{carrier
wave ${\bm{\epsilon}}_{o}^{{\scriptscriptstyle\perp}}\\!(\xi)$}}.$ (8)

  2. 2.

A superposition of a finite number of waves of type 1.

  3. 3.

An ‘impulse’ (few, one, or even a fraction of oscillation) [11, 4].

The starting point of our approach is: since no particle can reach the speed
of light $c$, then $\tilde{\xi}(t)\\!=\\!ct\\!-\\!z(t)$ is strictly growing,
and we can recast the eq. of motion (3) in an equivalent form
($\widehat{\ref{EOM}}$) making the change of independent variable
$t\mapsto\xi\\!=\\!ct\\!-\\!z$; the term
${\bm{\epsilon}}^{\scriptscriptstyle\perp}[ct\\!-\\!z(t)]$, where the unknown
$z(t)$ is in the argument of the highly nonlinear and rapidly varying
${\bm{\epsilon}}^{\scriptscriptstyle\perp}$, becomes the known forcing term
${\bm{\epsilon}}^{\scriptscriptstyle\perp}(\xi)$, what makes (12-19) more
manageable than (3). More radically, we can apply Hamilton’s principle to the
action functional $S(\lambda)$ of the particle parametrizing the worldline
$\lambda$ (fig. 1) by $\xi$ instead of $t$; the associated Euler-Lagrange and
Hamilton equations are equivalent and have the same features as (12-19). The
Hamilton equations involve also a more convenient set of canonically
conjugated momenta as unknown (dependent) variables.

We apply the approach first to an isolated particle (section 2), then to a
cold diluted plasma initially at rest and hit by a plane EM wave (section 3).

##  2 Electrodynamics of a single particle

Figure 1: Every worldline $\lambda$ and hyperplane $\xi\\!=$const in Minkowski
space intersect once. The wave-particle interaction occurs only along the
intersection of $\lambda$ with the support of the EM wave (painted pink),
which assuming (7) is delimited by the $\xi\\!=\\!0$ and $\xi\\!=\\!l$
hyperplanes.

###  2.1 Set-up and general results

Given points spacetime ${\rm x}_{0},{\rm x}_{1}$ with ${\rm x}_{1}$ in the
causal cone of ${\rm x}_{0}$, let $\Lambda$ be the set of time-like curves
from ${\rm x}_{0}$ to ${\rm x}_{1}$. Given a $\lambda\in\Lambda$, we can
equivalently use either $t$ or $\xi$ as a parameter on $\lambda$ in the
corresponding action functional of the particle:

$\displaystyle S(\lambda)=-\\!\int_{\lambda}\\!mc^{2}d\tau+qA({\rm
x})=-\\!\int\limits_{t_{0}}^{t_{1}}\\!\\!dt\,\underbrace{\frac{mc^{2}\\!\\!+\\!qu^{\mu}A_{\mu}}{\gamma}}_{L[{\bf
x},\dot{\bf
x},t]}=-\\!\int\limits_{\xi_{0}}^{\xi_{1}}\\!\\!\frac{d\xi}{c}\,\underbrace{\frac{mc^{2}\\!\\!+\\!q\hat{u}^{\mu}\hat{A}_{\mu}}{\hat{s}}}_{{\cal
L}[\hat{\bf x},\hat{\bf x}^{\prime},\xi]}.$ (9)

Here $A({\rm x})=A_{\mu}({\rm x})d{\rm x}^{\mu}=A^{0}({\rm x})cdt\\!-\\!{\bf
A}({\rm x})\cdot d{\bf x}$ is the EM potential 1-form (the dot is the scalar
product in Euclidean ${\mathbb{R}}^{3}$), which is related to the EM fields by
${\bf E}=-\partial_{t}{\bf A}/c-\nabla A^{0}$ and ${\bf
B}=\nabla\\!\wedge\\!{\bf A}$ ($E^{i}\\!=\\!F^{i0}$, $B^{1}\\!=\\!F^{32}$, etc
in terms of the EM tensor $F^{\mu\nu}$). For any given function $f(ct,{\bf
x})$ we denote $\hat{f}(\xi,\hat{\bf x})\equiv f(\xi\\!+\\!\hat{z},\hat{\bf
x})$, abbreviate $\dot{f}\\!\equiv\\!df/dt$,
$\hat{f}^{\prime}\\!\equiv\\!d\hat{f}/d\xi$ (total derivatives). In
particular, $\hat{\bf x}(\xi)$ is the particle position seen as a function of
$\xi$; it is determined by $\hat{\bf x}(\xi)={\bf x}(t)$. We raise and lower
greek letter indices by the Minkowski metric
$\eta_{\mu\nu}\\!=\\!\eta^{\mu\nu}$, with $\eta_{00}\\!=\\!1$,
$\eta_{11}\\!=\\!-1$, etc.); $(cd\tau)^{2}=(cdt)^{2}\\!-\\!d{\bf x}^{2}$ is
the square of the infinitesimal Minkowski distance ($\tau$ is the proper time
of the particle),
$dt/d\tau\\!=\\!\gamma\\!=\\!1/\sqrt{1\\!-\\!{\bm{\beta}}^{2}}$ (with
${\bm{\beta}}\equiv\dot{\bf x}/c$) is the Lorentz relativistic factor,
$u\\!=\\!(u^{0}\\!,{\bf
u})\\!\equiv\\!(\gamma,\\!\gamma{\bm{\beta}})\\!=\\!\left(\\!\frac{p^{0}}{mc^{2}},\\!\frac{{\bf
p}}{mc}\\!\right)$ is the 4-velocity, i.e. the dimensionless version of the
4-momentum, and

$s\equiv\frac{d\xi}{d(c\tau)}=u^{0}\\!-u^{z}=u^{-}=\gamma(1-\beta^{z})>0$ (10)

is the light-like component $u^{-}$ of $u$, as well as the Doppler factor
experienced by the particle, and is positive-definite [the first $=$ in (10)
follows from $\gamma\\!=\\!dt/d\tau$, $p^{z}\\!=\\!mdz/d\tau$]. We name $s$
the light-like relativistic factor, or shortly the $s$-factor. In terms of the
“hatted” coordinates and their derivatives333In fact, since
$c\hat{t}(\xi)=\xi+\hat{z}$ implies
$c\hat{t}^{\prime}(\xi)=1+\hat{z}^{\prime}(\xi)>0$, we find
$\displaystyle\frac{1}{\hat{s}}=\frac{d\tau}{dt}\frac{d(ct)}{d\xi}=\frac{1}{\gamma}\frac{d(ct)}{d\xi}=\sqrt{1\\!-\\!\left(\frac{d{\bf
x}}{cdt}\right)^{2}}\frac{d(ct)}{d\xi}=\sqrt{\left(\frac{cdt}{d\xi}\right)^{2}\\!-\\!\left(\frac{d{\bf
x}}{d\xi}\right)^{2}}=\sqrt{\left(1\\!+\\!\hat{z}^{\prime}\right)^{2}\\!-\\!\hat{\bf
x}^{\prime}{}^{2}}=\sqrt{1\\!+\\!2\hat{z}^{\prime}\\!-\\!\hat{\bf
x}^{\scriptscriptstyle\perp}{}^{\prime}{}^{2}}.$

$\hat{\bf u}\\!=\\!\hat{s}\hat{\bf
x}^{\prime},\qquad\frac{1}{\hat{s}}=\sqrt{1\\!+\\!2\hat{z}^{\prime}\\!-\\!\hat{\bf
x}^{\scriptscriptstyle\perp}{}^{\prime}{}^{2}}.$

$\hat{\gamma},\hat{u}^{z},\hat{\bm{\beta}},\hat{\bf x}^{\prime}$ can be
expressed as rational functions of $\hat{\bf
u}^{{\scriptscriptstyle\perp}},\hat{s}$:

$\displaystyle\hat{\gamma}\\!=\\!\frac{1\\!+\\!\hat{\bf
u}^{{\scriptscriptstyle\perp}}{}^{2}\\!\\!+\\!\hat{s}^{2}}{2\hat{s}},\quad\hat{u}^{z}\\!=\\!\hat{\gamma}\\!-\\!\hat{s},\quad\hat{\bm{\beta}}\\!=\\!\frac{\hat{\bf
u}}{\hat{\gamma}},\quad$ (11) $\displaystyle\hat{\bf
x}^{\scriptscriptstyle\perp}{}^{\prime}=\displaystyle\frac{\hat{\bf
u}^{\scriptscriptstyle\perp}}{\hat{s}},\qquad\quad\hat{z}^{\prime}=\displaystyle\frac{1\\!+\\!\hat{\bf
u}^{\scriptscriptstyle\perp}{}^{2}}{2\hat{s}^{2}}\\!-\\!\frac{1}{2}$ (12)

By Hamilton’s principle, any extremum $\lambda$ of $S$ is the worldline of a
possible motion of the particle with initial position ${\bf x}_{0}$ at time
$t_{0}$ and final position ${\bf x}_{1}$ at time $t_{1}$. Hence it fulfills
Euler-Lagrange equations in both forms $\frac{d}{dt}\frac{\partial
L}{\partial\dot{\bf x}}=\frac{\partial L}{\partial{\bf x}}$ and
$\frac{d}{d\xi}\frac{\partial{\cal L}}{\partial\hat{\bf
x}^{\prime}}=\frac{\partial{\cal L}}{\partial\hat{\bf x}}$, equivalent to (3).
The Legendre transforms yield the Hamiltonians $H\\!\equiv\\!\dot{\bf
x}\\!\cdot\\!\frac{\partial L}{\partial\dot{\bf x}}\\!-\\!L\\!=\\!\gamma
mc^{2}\\!+\\!qA^{0}$ and $\hat{H}\\!\equiv\\!\hat{\bf
x}^{\prime}\\!\cdot\\!\frac{\partial{\cal L}}{\partial\hat{\bf
x}^{\prime}}\\!-\\!{\cal L}\\!=\\!\hat{\gamma}mc^{2}\\!+\\!q\hat{A}^{0}$
respectively. In fact, the change of ‘time’ $t\mapsto\xi$ induces a
generalized canonical (i.e. contact) transformation mapping $H\mapsto\hat{H}$.
The latter is a rational function of $\hat{\bf
x},\hat{\bf\Pi}\\!\equiv\\!\frac{\partial{\cal L}}{\partial\hat{\bf
x}^{\prime}}$, or, equivalently, of $\hat{s},\hat{\bf
u}^{{\scriptscriptstyle\perp}}$:

$\displaystyle\hat{H}(\hat{\bf
x},\\!\hat{\bf\Pi};\\!\xi)=mc^{2}\frac{1\\!+\\!\hat{s}^{2}\\!\\!+\\!\hat{\bf
u}^{{\scriptscriptstyle\perp}2}\\!}{2\hat{s}}\\!+\\!q\hat{A}^{0}\\!(\xi,\hat{\bf
x}),\qquad$ (13)
$\displaystyle\mbox{where}\>\>\left\\{\\!\\!\begin{array}[]{l}\displaystyle
mc^{2}\hat{\bf
u}^{{\scriptscriptstyle\perp}}\\!\\!=\\!\hat{\bf\Pi}^{{\scriptscriptstyle\perp}}\\!\\!-\\!q\hat{\bf
A}^{{\scriptscriptstyle\perp}}(\xi,\hat{\bf x}),\\\\[6.0pt] \displaystyle
mc^{2}\hat{s}\\!=\\!-\hat{\Pi}^{z}\\!\\!-\\!q[\hat{A}^{0}\\!-\\!\hat{A}^{z}](\xi,\hat{\bf
x}),\end{array}\right.\qquad$ (16)

while $H({\bf x},{\bf P},t)\\!=\\!\sqrt{\\!m^{2}c^{4}\\!+\\!(c{\bf
P}\\!-\\!q{\bf A})^{2}}+\\!qA^{0}$ (where ${\bf P}\\!\equiv\\!\frac{\partial
L}{\partial\dot{\bf x}}\\!=\\!{\bf p}\\!+\\!\frac{q}{c}{\bf A}$) is not. Eq.
(3) are also equivalent to the Hamilton equations $\hat{\bf
x}^{\prime}\\!=\\!\frac{\partial\hat{H}}{\partial\hat{\bf\Pi}},\>\hat{\bf\Pi}^{\prime}\\!=\\!-\frac{\partial\hat{H}}{\partial\hat{\bf
x}}$. These amount to (12) and

$\displaystyle\begin{array}[]{l}\displaystyle\hat{\bf
u}^{\scriptscriptstyle\perp}{}^{\prime}=\frac{q}{mc^{2}\,\hat{s}}\\!\left[\hat{\gamma}\hat{\bf
E}\\!+\\!\hat{\bf u}\\!\wedge\\!\hat{\bf
B}\right]^{\scriptscriptstyle\perp}\\!,\\\\[12.0pt]
\displaystyle\hat{s}^{\prime}=\frac{q}{mc^{2}}\\!\left[\frac{\hat{\bf
u}^{\scriptscriptstyle\perp}}{\hat{s}}\\!\cdot\\!\hat{\bf
E}^{\scriptscriptstyle\perp}\\!-\\!\hat{E}^{z}\\!-\\!\frac{(\hat{\bf
u}^{\scriptscriptstyle\perp}\\!\wedge\\!\hat{\bf
B}^{\scriptscriptstyle\perp})^{z}}{\hat{s}}\right]\end{array}$ (19)

with $\hat{\gamma}$ as given in (11). All the new equations, in particular
these ones, can be also obtained more directly from the old ones by putting a
caret on all dynamical variables and replacing $d/dt$ by
$(c\hat{s}/\hat{\gamma})d/d\xi$. Along the solutions $\hat{H}$ gives the
particle energy as a function of $\xi$. Once solved (12-19), analytically or
numerically, we just need to invert
$\hat{t}(\xi)\\!=\\!\xi\\!+\\!\hat{z}(\xi)$ and set ${\bf x}(t)=\hat{\bf
x}[\xi(t)]$,… to obtain the solution as a function of $t$.

Under the EM field (4) eqs (19) amount to

$\displaystyle\begin{array}[]{l}\displaystyle\hat{\bf
u}^{\scriptscriptstyle\perp}{}^{\prime}\\!=\frac{q}{mc^{2}}\\!\left[(1\\!+\\!\hat{z}^{\prime})\hat{\bf
E}^{\scriptscriptstyle\perp}_{s}\\!+\\!(\hat{\bf
x}^{\prime}\\!\wedge\\!\hat{\bf
B}_{s})^{\scriptscriptstyle\perp}\\!+\\!{\bm{\epsilon}}^{\scriptscriptstyle\perp}(\xi)\right]\\!,\\\\[10.0pt]
\displaystyle\hat{s}^{\prime}=\frac{-q}{mc^{2}}\left[\hat{E}^{z}_{s}-\hat{\bf
x}^{\scriptscriptstyle\perp}{}^{\prime}\\!\cdot\\!\hat{\bf
E}^{\scriptscriptstyle\perp}_{s}\\!+(\hat{\bf
x}^{\scriptscriptstyle\perp}{}^{\prime}\\!\wedge\\!\hat{\bf
B}^{\scriptscriptstyle\perp}_{s})^{z}\right]\\!,\end{array}\qquad$ (22)

while the energy gain (normalized to $mc^{2}$) in the interval
$[\xi_{0},\xi_{1}]$ is

${\cal
E}\equiv\frac{\hat{H}(\xi_{1})\\!-\\!\hat{H}(\xi_{0})}{mc^{2}}=\int^{\xi_{1}}_{\xi_{0}}\\!\\!\\!d\xi\,\frac{q{\bm{\epsilon}}^{\scriptscriptstyle\perp}}{mc^{2}}\\!\cdot\\!\frac{\hat{\bf
u}^{\scriptscriptstyle\perp}}{\hat{s}}.$ (23)

We can obtain the fields (4) from an EM potential of the same form,
$A^{\mu}({\rm x})=\alpha^{\mu}(ct\\!-\\!z)+A_{s}^{\mu}({\bf x})$; in the
Landau gauges ($\partial_{\mu}A^{\mu}\\!=\\!0$) ${\bf A}_{s}$ must fufill the
Coulomb gauges ($\nabla\\!\cdot\\!{\bf A}_{s}\\!=\\!0$), and it must be
$\alpha^{z}{}^{\prime}=\alpha^{0}{}^{\prime}$,
${\bm{\epsilon}}^{\scriptscriptstyle\perp}\\!=\\!-{\bm{\alpha}}^{\scriptscriptstyle\perp}{}^{\prime}$,
${\bf E}_{s}\\!=\\!-\nabla\\!A_{s}^{0}$, ${\bf
B}_{s}\\!=\\!\nabla\\!\wedge\\!{\bf A}_{s}$. We shall set
$\alpha^{z}=\alpha^{0}=0$, as $\alpha^{z},\alpha^{0}$ appear neither in the
observables ${\bf E},{\bf B}$ nor in the equations of motion. We fix
${\bm{\alpha}}^{\scriptscriptstyle\perp}(\xi)$ uniquely by requiring that it
vanish as $\xi\to-\infty$; this leads to

$\displaystyle{\bm{\alpha}}^{\scriptscriptstyle\perp}(\xi)\\!\equiv\\!-\\!\\!\int^{\xi}_{-\infty}\\!\\!\\!\\!\\!\\!\\!dy\,{\bm{\epsilon}}^{\scriptscriptstyle\perp}\\!(y).$
(24)

Under rather general assumptions on $\epsilon$ (8) implies

${\bm{\alpha}}^{\scriptscriptstyle\perp}(\xi)=-\frac{\epsilon(\xi)}{k}\>{\bm{\epsilon}}^{\scriptscriptstyle\perp}_{p}\\!(\xi)+O\left(\frac{1}{k^{2}}\right)\simeq-\frac{\epsilon(\xi)}{k}\,{\bm{\epsilon}}^{\scriptscriptstyle\perp}_{p}\\!(\xi)$
(25)

where
${\bm{\epsilon}}^{\scriptscriptstyle\perp}_{p}\\!(\xi)\\!:=\\!{\bm{\epsilon}}^{\scriptscriptstyle\perp}_{o}(\xi\\!+\\!\pi/2k)$;
in the appendix of [9] we recall upper bounds for the remainder $O(1/k^{2})$.
For very slow modulations (i.e. $|\epsilon^{\prime}|\\!\ll\\!|k\epsilon|$,
namely the modulating amplitude $\epsilon$ does not vary significantly over
the wavelength $\lambda\\!\equiv\\!2\pi/k$) \- like the ones characterizing
most conventional applications (radio broadcasting, ordinary laser beams,
etc.) - the right estimate is very good. Consequently, if $\epsilon(\xi)$ goes
to zero also as $\xi\\!\to\\!\infty$, then
${\bm{\alpha}}\\!^{{\scriptscriptstyle\perp}}(\xi)$ approximately does as
well. In particular, in case a) eq. (24) implies
${\bm{\alpha}}^{\scriptscriptstyle\perp}(\xi)\\!=\\!0$ if $\xi\\!\leq\\!0$,
${\bm{\alpha}}^{\scriptscriptstyle\perp}(\xi)\\!=\\!{\bm{\alpha}}^{\scriptscriptstyle\perp}(l)\simeq
0$ if $\xi\\!\geq\\!l$.

While the usual Hamilton equations $\dot{\bf P}=-\partial H/\partial{\bf x}$,
$\dot{\bf x}=\partial H/\partial{\bf P}$, which amount to

$\displaystyle\dot{\bf u}(t)=\frac{q}{mc}\\!\left\\{{\bf
E}_{s}\\!\\!+\\!\left(\\!\frac{\dot{\bf x}}{c}\\!\wedge\\!{\bf
B}_{s}\\!\right)\\!\\!+\\!\left(\\!\frac{\dot{\bf
x}^{\scriptscriptstyle\perp}}{c}\cdot{\bm{\epsilon}}^{\scriptscriptstyle\perp}[ct\\!-\\!z(t)]\\!\right)\\!\mathbf{k}\\!-\\!\frac{1}{c}\frac{d}{dt}{\bm{\alpha}}^{\scriptscriptstyle\perp}[ct\\!-\\!z(t)]\right\\}\\!,$
(26)

and $\dot{\bf x}={\bf u}/\sqrt{1\\!+\\!{\bf u}^{2}}$, have no rational form
and contain the unknown $ct\\!-\\!z(t)$ in the argument of the rapidly varying
function
${\bm{\epsilon}}^{\scriptscriptstyle\perp},{\bm{\alpha}}^{\scriptscriptstyle\perp}$,
(12-22) on the contrary are rational in the unknowns and contain
${\bm{\epsilon}}^{\scriptscriptstyle\perp}(\xi)$ as a known forcing term. This
simplifies their study and the determination of ${\cal E}$.

###  2.2 Dynamics under an EM potential of the form
$A^{\mu}\\!=\\!A^{\mu}(t,\\!z)$

Eq. (19) are further simplified if $A^{\mu}$ does not depend on transverse
coordinates. This applies in particular if ${\bf
E}_{s}=E_{s}^{z}(z)\mathbf{k}$, ${\bf B}_{s}={\bf
B}_{s}^{\scriptscriptstyle\perp}(z)$, choosing $A^{0}=-\int^{z}\\!d\zeta
E_{s}^{z}(\zeta)$, ${\bf
A}^{\scriptscriptstyle\perp}\\!={\bm{\alpha}}^{\scriptscriptstyle\perp}\\!-\\!\mathbf{k}\wedge\\!\int^{z}\\!d\zeta{\bf
B}^{\scriptscriptstyle\perp}(\zeta)$, $A^{z}\\!\equiv\\!0$. As
$\partial\hat{H}/\partial\hat{\bf x}^{\scriptscriptstyle\perp}\\!=\\!0$, we
find $\hat{\bf\Pi}^{\scriptscriptstyle\perp}\\!=\\!q{\bf
K}^{\scriptscriptstyle\perp}\\!=\\!\mbox{const}$, i.e. the known result
$\frac{mc^{2}}{q}\hat{\bf u}^{\scriptscriptstyle\perp}\\!=\\!{\bf
K}^{\scriptscriptstyle\perp}\\!-\\!\hat{\bf
A}^{{\\!\scriptscriptstyle\perp}}\\!(\xi,\\!\hat{z})$. Setting
$v\\!:=\\!\hat{\bf u}^{\scriptscriptstyle\perp}{}^{2}$ and replacing in (22)
we obtain the first order system in the unknowns $\hat{z},\hat{s}$

$\displaystyle\hat{z}^{\prime}=\displaystyle\frac{1\\!+\\!\hat{v}}{2\hat{s}^{2}}\\!-\\!\frac{1}{2},\qquad\hat{s}^{\prime}=\frac{-q}{mc^{2}}E_{s}^{z}(\hat{z})\\!-\\!\frac{1}{2\hat{s}}\frac{\partial\hat{v}}{\partial\hat{z}}.$
(27)

Once solved system (27) for $\hat{z}(\xi),\hat{s}(\xi)$, the other unknowns
are obtained integrating (12):

$\displaystyle\hat{\bf x}(\xi)={\bf
x}_{0}+\\!\displaystyle\int^{\xi}_{\xi_{0}}\\!\\!\\!dy\,\frac{\hat{\bf
u}(y)}{\hat{s}(y)}$ (28)

[the $z$-component of (28) amounts to (27a) with initial condition
$\hat{z}(\xi_{0})\\!=\\!z_{0}$]. If in addition ${\bf B}_{s}\\!\equiv\\!0$,
then ${\bf A}_{s}\\!\equiv\\!0$ (in the Coulomb gauge), implying that
$\hat{\bf
u}^{\scriptscriptstyle\perp}\\!(\xi)\\!=\\!\frac{q}{mc^{2}}\left[{\bf
K}^{\scriptscriptstyle\perp}\\!-{\bm{\alpha}}^{\scriptscriptstyle\perp}(\xi)\right]$
and $\hat{v}\\!=\\!\hat{\bf u}^{{\scriptscriptstyle\perp}2}$ in (27) are
already known. Thus the system (27) to be solved simplifies to

$\displaystyle\hat{z}^{\prime}=\frac{1\\!+\\!\hat{v}}{2\hat{s}^{2}}\\!-\\!\frac{1}{2},\qquad\hat{s}^{\prime}=\frac{-q}{mc^{2}}E_{s}^{z}(\hat{z}).$
(29)

Remarks. Some noteworthy properties of the corresponding solutions are [9]:

  1. 1.

Where ${\bm{\epsilon}}^{\scriptscriptstyle\perp}(\xi)\\!=\\!0$ then
$\hat{v}(\xi)\\!=\\!v_{c}\\!=$const, $\hat{H}$ is conserved, (29) is solved by
quadrature.

  2. 2.

In case (7a) the final transverse momentum is $mc\hat{\bf
u}^{\scriptscriptstyle\perp}(l)$. If $\epsilon$ of (8) varies very slowly and
$\hat{\bf u}^{\scriptscriptstyle\perp}\\!(0)\\!=\\!{\bf 0}$, then by (25)
$\hat{\bf u}^{\scriptscriptstyle\perp}(l)\\!\simeq\\!0$.

  3. 3.

Fast oscillations of ${\bm{\epsilon}}^{\scriptscriptstyle\perp}$ make
$\hat{z}(\xi)$ oscillate much less than $\hat{\bf
x}^{\scriptscriptstyle\perp}(\xi)$, and $\hat{s}(\xi)$ even less: as
$\hat{s}\\!>\\!0$, $\hat{v}\\!=\\!\hat{\bf
u}^{\scriptscriptstyle\perp}{}^{2}\\!\geq\\!0$, integrating (29a) averages the
fast oscillations of ${\bf u}^{\scriptscriptstyle\perp}$ to yield much smaller
relative oscillations of $\hat{z}$, while integrating (29b) averages the
residual small oscillations of $E_{s}^{z}[\hat{z}(\xi)]$ to yield an
essentially smooth $\hat{s}(\xi)$. On the contrary,
$\hat{\gamma}(\xi),\hat{\bm{\beta}}(\xi),\hat{\bf u}(\xi),...$, which are
recovered via (11), oscillate fast, and so do also
$\gamma(t),{\bm{\beta}}(t),{\bf u}(t),..$. See e.g. fig. 2,4,6.

  4. 4.

If ${\bf u}^{\scriptscriptstyle\perp}\\!(0)\\!=\\!{\bf 0}$ and the EM wave is
a very slowly modulated (8)-(7a), integrating (23) by parts across $[0,l]$ and
using (25) we find that the final energy gain is given by

${\cal
E}_{f}=\int^{l}_{0}\\!d\xi\>\frac{\hat{v}^{\prime}(\xi)}{2\hat{s}(\xi)}\simeq\int^{l}_{0}\\!d\xi\>\frac{\hat{v}(\xi)\hat{s}^{\prime}(\xi)}{2\hat{s}^{2}(\xi)};$
(30)

this will be automatically positive (resp. negative) if $\hat{s}(\xi)$ is
growing (resp. decreasing) in all of $[0,l]$. Correspondingly, the interaction
with the EM wave can be used to accelerate (resp. decelerate) the particle.

###  2.3 Dynamics under travelling waves and static fields ${\bf E}_{s},{\bf
B}_{s}$

If ${\bf E}_{s},{\bf B}_{s}\\!=$const then eq. (22) are immediately integrated
to yield

$\displaystyle\begin{array}[]{l}\hat{\bf
u}^{\scriptscriptstyle\perp}=\displaystyle\frac{q}{mc^{2}}\left[{\bf
K}^{\scriptscriptstyle\perp}\\!-{\bm{\alpha}}^{\scriptscriptstyle\perp}(\xi)\\!+(\xi\\!+\\!\hat{z}){\bf
E}^{\scriptscriptstyle\perp}_{s}+(\hat{\bf x}\\!\wedge\\!{\bf
B}_{s})^{\scriptscriptstyle\perp}\\!\right],\\\\[12.0pt]
\hat{s}=\displaystyle\frac{-q}{mc^{2}}\left[K^{z}\\!+\xi E^{z}_{s}-\hat{\bf
x}^{\scriptscriptstyle\perp}\\!\cdot\\!{\bf
E}^{\scriptscriptstyle\perp}_{s}\\!+(\hat{\bf
x}^{\scriptscriptstyle\perp}\\!\wedge\\!{\bf B}_{s})^{z}\right]\end{array}$
(33)

(the integration constants $K^{j}$ are fixed by the initial conditions), or
more explictly

$\displaystyle\begin{array}[]{l}\hat{u}^{x}=w^{x}(\xi)\\!+\\!(e^{x}\\!-\\!b^{y})\hat{z}\\!+\\!b\hat{y},\\\\[10.0pt]
\hat{u}^{y}=w^{y}(\xi)\\!+\\!(e^{y}\\!+\\!b^{x})\hat{z}\\!-\\!b\hat{x},\\\\[10.0pt]
\hat{s}=w^{z}(\xi)\\!+\\!(e^{x}\\!-\\!b^{y})\hat{x}\\!+\\!(e^{y}\\!+\\!b^{x})\hat{y};\end{array}$
(37)

here we have introduced the dimensionless functions
${\bm{w}}^{\scriptscriptstyle\perp}(\xi)\\!:=\\!q\left[{\bf
K}^{\scriptscriptstyle\perp}\\!\\!-\\!{\bm{\alpha}}^{\scriptscriptstyle\perp}(\xi)\\!+\xi{\bf
E}^{\scriptscriptstyle\perp}_{s}\right]/mc^{2}$,
$w^{z}(\xi)\\!:=\\!-q(K^{z}\\!\\!+\\!\xi E^{z}_{s})/mc^{2}$ and the constants
${\bf e}^{\scriptscriptstyle\perp}:=\\!q{\bf
E}^{\scriptscriptstyle\perp}_{s}/mc^{2}$, ${\bf
b}^{\scriptscriptstyle\perp}\\!+\\!b\mathbf{k}\\!:=\\!q{\bf B}_{s}/mc^{2}$.
Hence, it remains to solve the system of three first order equations in
rational form in the unknowns $\hat{x},\hat{y},\hat{z}$ that is obtained
replacing (37) in (12).

Proposition 9 in [9]. If ${\bf E}_{s},{\bf B}_{s}$ are constant fulfilling the
only condition ${\bf
B}^{\scriptscriptstyle\perp}_{s}\\!=\\!\mathbf{k}\wedge{\bf
E}^{\scriptscriptstyle\perp}_{s}$ then, setting
$\kappa\\!:=\\!qE_{s}^{z}/mc^{2}$, $x_{0}\\!:=\\!\hat{x}(\xi_{0})$,
$y_{0}\\!:=\\!\hat{y}(\xi_{0})$, $z_{0}\\!:=\\!\hat{z}(\xi_{0})$,
$s_{0}\\!:=\\!\hat{s}(\xi_{0})$, we can put the solutions of (12) in the
compact form

$\displaystyle\begin{array}[]{l}\displaystyle(\hat{x}+i\hat{y})(\xi)=(s_{0}\\!-\\!\kappa\xi)^{ib/\kappa}\left[\frac{(x_{0}+iy_{0})}{(s_{0}\\!-\\!\kappa\xi_{0})^{ib/\kappa}}+\int^{\xi}_{\xi_{0}}\\!\\!\\!\\!d\zeta\,\frac{(w^{x}+iw^{y})(\zeta)}{(s_{0}\\!-\\!\kappa\zeta)^{1+ib/\kappa}}\right],\\\\[12.0pt]
\displaystyle\hat{s}(\xi)=s_{0}\\!-\\!\kappa\xi,\qquad\hat{u}^{z}(\xi)\\!=\\!\frac{1}{2(s_{0}\\!-\\!\kappa\xi)}+(s_{0}\\!-\\!\kappa\xi)\,\frac{\hat{\bf
x}^{\scriptscriptstyle\perp}{}^{\prime}{}^{2}(\xi)-1}{2},\\\\[12.0pt]
\displaystyle\hat{\bf
u}^{\scriptscriptstyle\perp}\\!(\xi)\\!=\\!(s_{0}\\!-\\!\kappa\xi)\,\hat{\bf
x}^{\scriptscriptstyle\perp}{}^{\prime}(\xi),\qquad\quad\hat{\gamma}(\xi)\\!=s_{0}\\!-\\!\kappa\xi\\!+\\!\hat{u}^{z}(\xi),\\\\[12.0pt]
\displaystyle\hat{z}(\xi)\\!=\\!z_{0}+\int^{\xi}_{\xi_{0}}\frac{d\zeta}{2}\left[\frac{1}{(s_{0}\\!-\\!\kappa\zeta)^{2}}\\!+\\!\hat{\bf
x}^{\scriptscriptstyle\perp}{}^{\prime}{}^{2}(\zeta)\\!-\\!1\right].\end{array}$
(42)

Up to our knowledge such general solutions have not appeared in the literature
before Ref. [9], but reduce to known ones under additional assumptions on
${\bf E}_{s},{\bf B}_{s}$. Now we briefly review some of them, adopting for
simplicity the initial conditions ${\bf x}_{0}\\!=\\!{\bf 0}\\!=\\!{\bf
u}_{0}$, whence $s_{0}=1$.

####  2.3.1 Zero static fields case: ${\bf E}_{s}\\!=\\!{\bf B}_{s}\\!=\\!{\bf
0}$.

Then (42) becomes [12, 13]:

$\displaystyle\begin{array}[]{l}\displaystyle\hat{s}\\!\equiv\\!1,\quad\hat{\bf
u}^{\scriptscriptstyle\perp}\\!\\!=\\!\frac{-q{\bm{\alpha}}^{\scriptscriptstyle\perp}}{mc^{2}},\quad\hat{u}^{z}\\!=\\!\frac{\hat{\bf
u}^{\scriptscriptstyle\perp}{}^{2}}{2},\quad\hat{\gamma}\\!=\\!1\\!+\\!\hat{u}^{z}\\\\[16.0pt]
\displaystyle\hat{z}(\xi)\\!=\\!\int^{\xi}_{\xi_{0}}\\!\\!\\!\\!dy\,\frac{\hat{\bf
u}^{{\scriptscriptstyle\perp}2}(y)}{2},\quad\hat{\bf
x}^{\scriptscriptstyle\perp}\\!(\xi)\\!=\\!\int^{\xi}_{\xi_{0}}\\!\\!\\!\\!dy\,\hat{\bf
u}^{\scriptscriptstyle\perp}\\!(y).\end{array}$ (45)

Figure 2: Solutions (45) and $e^{-}$ trajectories in the $zx$ plane induced by
two $x$-polarized pulses with carrier wavelength
$\lambda\\!\equiv\\!2\pi/k\\!=\\!0.8\mu$m, gaussian modulation
$\epsilon(\xi)=E^{\scriptscriptstyle\perp}_{\scriptscriptstyle
M}\exp[-\xi^{2}/2\sigma]$, $\sigma\\!=\\!20\mu$m2,
$a_{0}\\!\equiv\\!eE^{\scriptscriptstyle\perp}_{\scriptscriptstyle
M}/kmc^{2}\\!=\\!0.8$ (left) or $a_{0}\\!=\\!3.3$ (right).

The solutions (45) induced by two $x$-polarized pulses and the corresponding
$e^{-}$ trajectories in the $zx$ plane are shown in fig. 2. Note that:

  * •

The maxima of $u^{z}$, ${\bm{\alpha}}^{\scriptscriptstyle\perp}{}^{2}$ (and
approximately also of $\epsilon(\xi)$, if $\epsilon(\xi)$ is slowly varying)
coincide.

  * •

Since $u^{z}\\!\geq\\!0$, the $z$-drift is positive-definite. Rescaling
${\bm{\epsilon}}^{\scriptscriptstyle\perp}\mapsto
a{\bm{\epsilon}}^{\scriptscriptstyle\perp}$, we find that $\hat{\bf
x}^{\scriptscriptstyle\perp},\hat{\bf u}^{\scriptscriptstyle\perp}$ scale like
$a$, while $\hat{z},\hat{u}^{z}$ scale like $a^{2}$; hence the trajectory goes
to a straight line in the limit $a\\!\to\\!\infty$. This is due to magnetic
force $q{\bm{\beta}}\wedge{\bf B}$ in (3).

  * •

Corollary The final 4-velocity $u$ and energy gain read

${\bf u}^{\scriptscriptstyle\perp}_{f}\\!=\\!\hat{\bf
u}^{\scriptscriptstyle\perp}(\infty),\quad u^{z}_{f}={\cal
E}_{f}=\frac{1}{2}{\bf
u}^{\scriptscriptstyle\perp}_{f}{}^{2}=\gamma_{f}\\!-\\!1;$ (46)

Both are very small if the pulse modulation $\epsilon$ is slow [extremely
small if $\epsilon\\!\in\\!{\cal S}(\mathbb{R})$ or
$\epsilon\\!\in\\!C^{\infty}_{c}(\mathbb{R})$].

We compare this result with the socalled Lawson-Woodward or General
Acceleration Theorem [14, 15, 16, 17]. This states that, in spite of large
energy variations during the interaction, the final energy gain ${\cal E}_{f}$
of a charged particle ${\cal P}$ interacting with an EM field is zero if:

  1. 1.

the interaction occurs in ${\mathbb{R}}^{3}$ vacuum (no boundaries);

  2. 2.

${\bf E}_{s}={\bf B}_{s}={\bf 0}$ and
${\bm{\epsilon}}^{\scriptscriptstyle\perp}$ is very slowly modulated;

  3. 3.

$v^{z}\simeq c$ along the whole acceleration path;

  4. 4.

nonlinear (in ${\bm{\epsilon}}^{\scriptscriptstyle\perp}$) effects
$q{\bm{\beta}}\\!\wedge\\!{\bf B}$ are negligible;

  5. 5.

the power radiated by ${\cal P}$ is negligible.

Our Corollary, as Ref. [18], states that the final energy gain is zero also if
we relax iii), iv), but the EM field is a plane travelling wave. To obtain a
non-zero ${\cal E}_{f}$ one has to violate some other conditions of the
theorem, as e.g. we see in next cases.

####  2.3.2 Case ${\bf E}_{s}=0$, ${\bf B}_{s}=B^{z}_{s}\mathbf{k}$, and
cyclotron autoresonance.

Figure 3: The electron motion (49) (up) and the $zx$-projection of the
corresponding trajectory (down) induced in a longitudinal magnetic field
$B^{z}\\!=\\!10^{5}$G by a circularly polarized modulated EM wave (8) with
wavelength $\lambda\\!\equiv\\!2\pi/k\\!=\\!1$mm, $b\\!=\\!k=\\!58.6$cm-1,
gaussian enveloping amplitude
$\epsilon(\xi)=E^{\scriptscriptstyle\perp}_{\scriptscriptstyle
M}\exp[-\xi^{2}/2\sigma]$ with $\sigma\\!=\\!3$cm2 and
$a_{0}\\!\equiv\\!eE^{\scriptscriptstyle\perp}_{\scriptscriptstyle
M}/kmc^{2}\\!=\\!0.15$, trivial initial conditions (${\bf x}_{0}\\!=\\!{\bf
u}_{0}\\!=\\!0$), giving ${\cal E}_{f}\\!\simeq\\!28.5$.

Eq. (42) becomes $\hat{s}\equiv 1$ and, since
$(1\\!-\\!\kappa\xi)^{ib/\kappa}=\exp\left[\frac{ib}{\kappa}\log(1\\!-\\!\kappa\xi)\right]=\exp\left[-\frac{ib}{\kappa}\kappa\xi+O(k)\right]$
reduces to $e^{-ib\xi}$ as $\kappa\to 0$,

$\displaystyle\begin{array}[]{l}\displaystyle(\hat{x}\\!+\\!i\hat{y})(\xi)\\!=\\!\\!\int^{\xi}_{0}\\!\\!\\!\\!d\zeta\,e^{ib(\zeta\\!-\\!\xi)}(w^{x}\\!\\!+\\!iw^{y})(\\!\zeta\\!),\qquad\hat{\bf
u}^{\scriptscriptstyle\perp}\\!=\\!\hat{\bf
x}^{\scriptscriptstyle\perp}{}^{\prime}\\!,\\\\[12.0pt]
\displaystyle\hat{u}^{z}\\!=\\!\hat{z}^{\prime}\\!=\\!\frac{\hat{\bf
u}^{\scriptscriptstyle\perp}{}^{2}}{2}\\!=\\!{\cal
E}\\!=\\!\hat{\gamma}\\!-\\!1,\qquad\hat{z}(\xi)\\!=\\!\\!\int^{\xi}_{0}\\!\\!\\!\\!d\zeta\>\frac{\hat{\bf
u}^{\scriptscriptstyle\perp}{}^{2}\\!(\zeta)}{2}.\end{array}$ (49)

The solution (49) reduces to that of [19, 20] if
${\bm{\epsilon}}^{\scriptscriptstyle\perp}$ is monochromatic. This leads to
cyclotron autoresonance if
$-b\\!=\\!k\\!=\\!\frac{2\pi}{\lambda}\\!\gg\\!\frac{1}{l}$. In fact, this
conditions ensures that the cyclotron frequency associated to $B^{z}_{s}$
equals the EM wave frequency, implying an accelerated transverse motion as in
a cyclotron (with a spiral trajectory); since also the transverse magnetic
field oscillates with the same frequency, the associated (longitudinal)
magnetic force keeps always the same sign and thus accelerates the particle in
the $z$-direction during the whole EM pulse. If for simplicity
${\bm{\epsilon}}^{\scriptscriptstyle\perp}$ is slowly modulated, circularly
polarized then $w^{x}(\xi)\\!+\\!iw^{y}(\xi)\\!\simeq\\!e^{ik\xi}{\rm
w}(\xi)$, where ${\rm w}(\xi)\\!\equiv\\!q\epsilon(\xi)/kmc^{2}$, and

$\displaystyle(\hat{x}\\!+\\!i\hat{y})(\xi)\simeq
iW\\!(\xi)e^{ik\xi},\qquad(\hat{u}^{x}\\!+\\!i\hat{u}^{y})(\xi)\simeq-
ke^{ik\xi}W(\xi),$

where $W(\xi)\equiv\\!\int^{\xi}_{0}\\!\\!d\zeta\,{\rm w}(\zeta)\\!>\\!0$
grows with $\xi$. In particular if
${\bm{\epsilon}}^{\scriptscriptstyle\perp}(\xi)\\!=\\!{\bf 0}$ for
$\xi\\!\geq\\!l\\!\equiv$, then for such $\xi$

$\hat{z}^{\prime}(\xi)\\!\simeq\\!\frac{k^{2}}{2}W^{2}(l)\\!\simeq\\!2{\cal
E}_{f},\qquad\frac{|\hat{\bf
x}^{\scriptscriptstyle\perp}{}^{\prime}(\xi)|}{\hat{z}^{\prime}(\xi)}\\!\simeq\\!\frac{2}{kW\\!(l)}\\!\ll\\!1;$

the final energy gain and collimation are noteworthy resp. by the first,
second relation.

####  2.3.3 Case ${\bf E}_{s}=E_{s}^{z}\mathbf{k}$, ${\bf B}_{s}={\bf 0}$

Figure 4: Left: the motion (50) induced by a linearly polarized modulated EM
wave (8) with wavelength $\lambda\\!=\\!2\pi/k\\!=\\!0.8\mu$m, gaussian
enveloping amplitude
$\epsilon(\xi)\\!=\\!E^{\scriptscriptstyle\perp}_{\scriptscriptstyle
M}\exp[-\xi^{2}/2\sigma]$ with $\sigma\\!=\\!20\mu$m2 and
$|q|E^{\scriptscriptstyle\perp}_{\scriptscriptstyle M}/kmc^{2}\\!=\\!6.6$,
trivial initial conditions, ${\bf B}_{s}\\!=\\!{\bf 0}$, ${\bf
E}_{s}\\!=\\!\mathbf{k}E^{z}_{\scriptscriptstyle M}$, where
$E^{z}_{\scriptscriptstyle M}q\\!\simeq\\!37$GeV/m (with such a wave this
yields the maximum energy gain, ${\cal E}_{f}\\!\simeq\\!1.5$). Right: the
corresponding trajectory in the $zx$ plane within an hypothetical acceleration
device based on a laser pulse and metallic gratings $G,P$ at potentials
$V\\!=\\!0,V_{p}$, with $qV_{p}/z_{p}\\!\simeq\\!37$GeV/m.

Then the solution (42) reduces to444Clearly the domain of definition is
$\xi\in[0,\xi_{f}[$, where $\xi_{f}\leq 1/\kappa$; however
$\hat{t}(\xi_{f})=\infty$, so that as a function of $t$ the solution is
defined in all of $[0,\infty[$, as required [9].
$\hat{s}(\xi)=1\\!-\\!\kappa\,\xi$,

$(\hat{x}+i\hat{y})(\xi)=\\!\\!\int\limits^{\xi}_{0}\\!\\!\\!dy\,\frac{(w^{x}\\!\\!+\\!iw^{y})(y)}{1\\!-\\!\kappa
y},\quad\hat{z}(\xi)=\\!\\!\int\limits^{\xi}_{0}\\!\\!\\!\frac{dy}{2}\left\\{\\!\frac{1\\!+\\!\hat{v}(y)}{[1\\!-\\!\kappa
y]^{2}}\\!-\\!1\right\\}\\!;$ (50)

if ${\bm{\epsilon}}^{\scriptscriptstyle\perp}$ is slowly modulated then by
(30) the final energy gain ${\cal
E}_{f}\simeq\int^{l}_{0}\\!d\xi\>\frac{-\kappa\hat{v}(\xi)}{2\hat{s}^{2}(\xi)}$
is negative if $\kappa\\!>\\!0$, positive if $\kappa\\!\leq\\!0$ and has a
unique maximum point $\kappa_{\scriptscriptstyle M}<0$ if $\epsilon(\xi)$ has
a finite support with a unique maximum. An acceleration device based on this
solution could be as follows: at $t=0$ the particle lies at rest with
$z_{0}\\!\lesssim\\!0$, just at the left of a metallic grating $G$ contained
in the $z\\!=\\!0$ plane and set at zero electric potential; another metallic
plate $P$ contained in a plane $z\\!=\\!z_{p}\\!>\\!0$ is set at electric
potential $V=V_{p}$. A short laser pulse
${\bm{\epsilon}}^{\scriptscriptstyle\perp}$ hitting the particle boosts it
beyond $G$ through the ponderomotive force; choosing $qV_{p}\\!>\\!0$ implies
$\kappa\\!=\\!-qV_{p}/z_{p}mc^{2}\\!<\\!0$, and a backward longitudinal
electric force $qE^{z}_{s}$. If $qV_{p}$ is large enough, then $z(t)$ reaches
a maximum smaller than $z_{p}$, then is accelerated backwards and exits the
grating with energy ${\cal E}_{f}$ and negligible transverse momentum.
However, a large ${\cal E}_{f}$ requires extremely large $|V_{p}|$, far beyond
the material breakdown threshold, what prevents its realization as a static
field (namely, sparks between $G,P$ would arise and rapidly reduce $|V_{p}|$).
Alternatively, one can make the pulse itself generate such large $|E_{s}^{z}|$
within a plasma at the right time, so as to induce the slingshot effect [21,
22, 23]; this is briefly explained at the end of next section.

##  3 Impact of a short laser pulse onto a cold diluted plasma

Figure 5: Left: a plane EM wave of finite length approaching normally a plasma
in equilibrium. Right: normalized EM wave
${\bm{\epsilon}}^{\scriptscriptstyle\perp}$ (blue) with carrier wavelength
$\lambda\\!=\\!0.8\mu$m, linear polarization, gaussian modulation
$\epsilon(\xi)=E^{\scriptscriptstyle\perp}_{\scriptscriptstyle
M}\exp[-\xi^{2}/2\sigma]$ with $\sigma\\!=\\!20\mu$m2,
$a_{0}\\!\equiv\\!eE^{\scriptscriptstyle\perp}_{\scriptscriptstyle
M}/kmc^{2}\\!=\\!3.3$ (whence the average pulse intensity is $10^{19}$W/cm2);
the associated ${\bf u}^{\scriptscriptstyle\perp}_{e}$ is painted purple. $l$
is the length of the $z$-interval where the amplitude $\epsilon$ overcomes all
ionization thresholds of the atoms of the gas yielding the plasma; here we
have chosen helium, whence $l\\!\simeq\\!40\lambda\\!=\\!32\mu$m, and the
thresholds for $1^{st}$ and $2^{nd}$ ionization are overcome almost
simultaneously.

###  3.1 Our simplified hydrodynamic model and its range of validity

We distinguish the types of particles (electrons and ions, possibly of various
kinds) composing the plasma by an index $h$, and denote by $q_{h},m_{h}$ their
charges, masses. Assume that the plasma is initially in hydrodynamic
conditions with all initial data not depending on ${\bf
x}^{\scriptscriptstyle\perp}$; these data consist of EM fields of the form (4)
as well as the set of initial values of the Eulerian velocity ${\bf v}_{h}$
and density $n_{h}$ of each fluid consisting of the same type of particles.
Consequently also the solutions of the Lorentz-Maxwell and continuity
equations for ${\bf B},{\bf E},{\bf u}_{h},n_{h}$ do not depend on ${\bf
x}^{\scriptscriptstyle\perp}$, nor do the displacements $\Delta{\bf
x}_{h}\equiv{\bf x}_{h}(t,{\bf X})\\!-\\!{\bf X}$ on ${\bf
X}^{\scriptscriptstyle\perp}$. Here ${\bf x}_{h}(t,\\!{\bf X})$ is the
position at time $t$ of the material element of the $h$-th fluid with initial
position ${\bf X}\\!\equiv\\!(X,\\!Y,\\!Z)$; ${\bf X}_{h}(t,{\bf x})$ is the
inverse of ${\bf x}_{h}(t,\\!{\bf X})$ (at fixed $t$);
${\bm{\beta}}_{h}\\!=\\!{\bf v}_{h}/c$, etc. For brevity we refer to the
particles (of the $h$-th type) contained: in this material element, as the
‘${\bf X}$ particles’ (e.g. ‘${\bf X}$ electrons’); in a region $\Omega$, as
the ‘$\Omega$ particles’; in the layer between $Z,Z\\!+\\!dZ$, as the ‘$Z$
particles’. More specifically, we consider (fig. 5) a very short and intense
EM plane wave (laser pulse) (7a) hitting normally a cold plasma initially in
equilibrium, possibly immersed in a static and uniform magnetic field ${\bf
B}_{s}$ (actually, the plasma may be locally obtained from a gas ionized by
the very high electric field of the pulse itself). The initial conditions are:

$\displaystyle\begin{array}[]{l}n_{h}(0,{\bf x})\\!=\\!0\quad\mbox{if
}\>z\\!\leq\\!0,\qquad{\bf u}_{h}(0,{\bf x})\\!=\\!{\bf 0},\qquad j^{0}(0,{\bf
x})\\!=\\!\sum\limits_{h}\\!q_{h}n_{h}(0,{\bf x})\\!\equiv\\!0,\\\\[8.0pt]
{\bf E}(0,{\bf
x})\\!=\\!{\bm{\epsilon}}^{\scriptscriptstyle\perp}(-z),\qquad\quad{\bf
B}(0,{\bf
x})\\!=\\!\mathbf{k}\wedge{\bm{\epsilon}}^{\scriptscriptstyle\perp}(-z)+{\bf
B}_{s},\end{array}$ (53)

whence the 4-current density
$j\\!=\\!(j^{0},\mathbf{j})\\!=\\!\sum_{h}q_{h}n_{h}(1,{\bm{\beta}}_{h})$ is
zero at $t=0$. Then the Maxwell equations $\nabla\\!\cdot\\!{\bf E}\\!=\\!4\pi
j^{0}$, $\partial_{t}E^{z}\\!/c+\\!4\pi
j^{z}\\!\\!=\\!(\\!\nabla\\!\\!\wedge\\!{\bf B})^{z}\\!\\!=\\!0$ imply [13]

$E^{{\scriptscriptstyle
z}}(t,\\!z)=4\pi\sum_{h}q_{h}\widetilde{N}_{h}[Z_{h}(t,z)],\qquad\widetilde{N}_{h}(Z)\\!:=\\!\int^{Z}_{0}\\!\\!d\zeta\,n_{h}(0,\\!\zeta);$
(54)

using (54) to express $E^{z}$ in terms of the (still unknown) longitudinal
motion [$Z_{h}(t,\cdot)$ is the inverse of $z_{h}(t,\cdot)$] we reduce the
number of unknowns by one.

Define ${\bm{\alpha}}^{\scriptscriptstyle\perp}$ as in (24). In the Landau
gauges (53) are compatible with the following initial conditions for the gauge
potential:

$\displaystyle{\bf A}(0,{\bf
x})={\bm{\alpha}}^{\scriptscriptstyle\perp}(-z)\\!+\\!{\bf
B}_{s}\\!\wedge\\!{\bf x}/2,\qquad\partial_{t}{\bf A}(0,{\bf
x})=-c{\bm{\epsilon}}^{\scriptscriptstyle\perp}(-z),$ (55)

$A^{0}(0,\\!{\bf x})=\partial_{t}A^{0}(0,\\!{\bf x})=0$. Eq (53-55) and
causality imply that ${\bf x}_{h}(t,{\bf X})\\!=\\!{\bf X}$, ${\bf
A}^{\scriptscriptstyle\perp}(t,{\bf x})\\!\equiv\\!{\bf B}_{s}\\!\wedge\\!{\bf
x}/2$ if $ct\\!\leq\\!z$, $\mathbf{j}\\!\equiv\\!{\bf 0}$ if
$ct\\!\leq\\!|z|$. ${\bf A}^{\scriptscriptstyle\perp}$ is coupled to the
current through $\Box{\bf A}^{\scriptscriptstyle\perp}=4\pi{\bf
j}^{\scriptscriptstyle\perp}$. Equipped with (55) the latter amounts to the
integral equation

${\bf
A}\\!^{{\scriptscriptstyle\perp}}\\!\\!-\\!{\bm{\alpha}}\\!^{{\scriptscriptstyle\perp}}\\!-\\!\frac{1}{2}\left({\bf
B}_{s}\\!\wedge\\!{\bf
x}\right)^{\perp}\\!=\\!2\pi\\!\\!\int\\!\\!\\!d\\!sd\zeta\,\theta(ct\\!\\!-\\!\\!s\\!-\\!|z\\!\\!-\\!\\!\zeta|)\theta(\\!s\\!)\,{\bf
j}^{\scriptscriptstyle\perp}\\!\left(\frac{s}{c},\zeta\right);$ (56)

here we have used the Green function of the d’Alembertian
$\partial_{t}^{2}/c^{2}\\!-\\!\partial_{z}^{2}$ in dimension 2 ($\theta$
stands for the Heaviside step function). The right-hand side (rhs) is zero for
$t\leq 0$ ($t=0$ is the beginning of the laser-plasma interaction). Within
short time intervals $[0,t^{\prime}]$ (to be determined a posteriori) we can
thus: approximate ${\bf
A}^{\scriptscriptstyle\perp}(t,z)\simeq{\bm{\alpha}}^{\scriptscriptstyle\perp}(ct\\!-\\!z)\\!+\\!\left(\frac{{\bf
B}_{s}}{2}\\!\wedge\\!{\bf x}\right)^{\perp}$; also neglect the motion of ions
with respect to the motion of the (much lighter) electrons. Hence we set
$z_{p}(t,Z)\\!\equiv\\!Z$, and the proton density $n_{p}$ (due to ions of all
kinds) equals the initial one and therefore the initial electron density
$\widetilde{n_{0}}(z)\\!:=\\!n_{e}(0,z)$, by the initial electric neutrality
of the plasma.

We now adopt $\xi$ instead of $t$ as the independent variable. Setting
$\widetilde{N}(Z)\\!:=\\!\int^{Z}_{0}\\!\\!d\zeta\,\widetilde{n_{0}}(\zeta)$,
the equations (19) & initial conditions for the electron fluid amount to

$\begin{array}[]{l}mc^{2}\hat{s}_{e}^{\prime}(\xi,Z)=4\pi
e^{2}\left[\widetilde{N}(\hat{z}_{e})\\!-\\!\widetilde{N}(Z)\right]+e(\Delta\hat{\bf
x}^{\scriptscriptstyle\perp}_{e}{}^{\prime}\\!\wedge\\!\hat{\bf
B}^{\scriptscriptstyle\perp}_{s})^{z}\\!,\\\\[10.0pt] mc^{2}\hat{\bf
u}^{\scriptscriptstyle\perp}_{e}{}^{\prime}(\xi,Z)=e{\bm{\alpha}}^{\scriptscriptstyle\perp}{}^{\prime}-e(\Delta\hat{\bf
x}^{\prime}_{e}\\!\wedge\\!\hat{\bf
B}_{s})^{\scriptscriptstyle\perp}\\!,\qquad\Delta\hat{\bf
x}^{\prime}_{e}=\frac{\widehat{\bf
u}_{e}(\xi,Z)}{\hat{s}_{e}(\xi,Z)}\end{array}$ (57) $\Delta\hat{\bf
x}_{e}(0,\\!{\bf X})\\!=\\!0,\qquad\hat{\bf u}_{e}(0,\\!{\bf X})\\!=\\!{\bf
0}\qquad\Rightarrow\quad\hat{s}_{e}(0,\\!{\bf X})\\!=\\!1.$ (58)

This makes eq. (57) a family parametrized by $Z$ of decoupled ODEs (of the
type considered in section 2) in the unknowns $\Delta\hat{\bf
x}_{e},\hat{s}_{e}$, $\hat{\bf u}^{\scriptscriptstyle\perp}_{e}$, rather than
a set of PDEs. After solving these equations and inverting $\hat{z}_{e}(x,Z)$,
all Eulerian fields $f(ct,z)$ will be obtained from $\hat{f}(\xi,Z)$ by the
replacement $(\xi,Z)\mapsto\left(ct\\!-\\!z,\hat{Z}_{e}(ct\\!-\\!z,z)\right)$.

If ${\bf B}_{s}\\!=\\!{\bf 0}$, then as in section 2.2 (57b) is solved by
$\hat{\bf
u}^{\scriptscriptstyle\perp}_{e}(\xi,Z)\\!=\\!e{\bm{\alpha}}^{\scriptscriptstyle\perp}\\!(\xi)/mc^{2}$,
while, abbreviating $\hat{\Delta}\\!\equiv\\!\Delta\hat{z}_{e}$ and
$v\\!\equiv\\!\hat{\bf u}^{{\scriptscriptstyle\perp}2}$, (57a) and the
$z$-component of (57c) take [22, 23] the form of (29),

$\displaystyle\hat{\Delta}^{\prime}\\!=\displaystyle\frac{1\\!+\\!v}{2\hat{s}^{2}}\\!-\\!\frac{1}{2},\qquad\hat{s}^{\prime}_{e}\\!=\frac{4\pi
e^{2}}{mc^{2}}\left\\{\widetilde{N}[\hat{z}_{e}]\\!-\\!\widetilde{N}(Z)\right\\}.$
(59)

For every $Z$ (59) have the form of Hamilton equations
$q^{\prime}=\partial\hat{H}/\partial p$, $p^{\prime}=-\partial\hat{H}/\partial
q$ of a 1-dim system: $\xi,\hat{\Delta},-\hat{s}_{e}$ play the role of
$t,q,p$, while the Hamiltonian is rational in $\hat{s}_{e}$ and reads

$\displaystyle\begin{array}[]{l}\displaystyle\hat{H}(\hat{\Delta},\hat{s}_{e},\xi;Z)\equiv\frac{\hat{s}_{e}^{2}+1\\!+\\!v(\xi)}{2\hat{s}_{e}}+{\cal
U}(\hat{\Delta};Z),\\\\[8.0pt] {\cal U}(\Delta;Z)\\!\equiv\\!\frac{4\pi
e^{2}}{mc^{2}}\left[\widetilde{{\cal
N}}\\!\left(Z\\!+\\!\Delta\right)\\!-\\!\widetilde{{\cal
N}}\\!(Z)\\!-\\!\widetilde{N}\\!(Z)\Delta\right],\\\\[8.0pt]
\displaystyle\widetilde{{\cal
N}}(Z)\equiv\int^{Z}_{0}\\!\\!\\!d\zeta\,\widetilde{N}(\zeta)\\!=\\!\int^{Z}_{0}\\!\\!\\!d\zeta\,\widetilde{n_{0}}(\zeta)\,(Z\\!-\\!\zeta);\end{array}$
(63)

we have defined the potential energy ${\cal U}$ fixing the free additive
constant so that ${\cal U}(0,Z)\\!\equiv\\!0$. Eq.s (59-58) can be solved
numerically, or by quadrature where
${\bm{\epsilon}}^{\scriptscriptstyle\perp}(\xi)\\!=\\!0$ (i.e. after the
pulse). Finally, the equation $\hat{\bf x}_{e}^{\prime}\\!=\\!\hat{\bf
u}_{e}/\hat{s}_{e}$ with initial conditions (58) is solved by

$\hat{\bf x}^{\scriptscriptstyle\perp}_{e}(\xi,{\bf X})-{\bf
X}^{\scriptscriptstyle\perp}=\\!\int^{\xi}_{0}\\!\\!\\!d\eta\,\frac{\hat{\bf
u}^{\scriptscriptstyle\perp}_{e}(\eta)}{\hat{s}(\eta,Z)},\qquad\hat{z}_{e}(\xi,Z)-Z=\hat{\Delta}(\xi,Z).$
(64)

By derivation we obtain several useful relations, in particular

$\frac{\partial Z_{e}}{\partial
z}(t,z)=\left.\frac{\hat{\gamma}_{e}}{\hat{s}_{e}\,\partial_{Z}\hat{z}_{e}}\right|_{(\xi,Z)=\big{(}ct\\!-z,Z_{e}(t,z)\big{)}}.$
(65)

Hence the maps $\hat{\bf x}_{e}(\xi,\cdot)\\!:\\!{\bf X}\\!\mapsto\\!{\bf x}$,
${\bf x}_{e}(t,\cdot)\\!:\\!{\bf X}\\!\mapsto\\!{\bf x}$ are invertible as
long as $\partial_{Z}\hat{z}_{e}\\!\equiv\\!\partial\hat{z}_{e}/\partial Z$ is
positive. The approximation on ${\bf A}^{\scriptscriptstyle\perp}\\!(t,\\!z)$
is acceptable as long as the so determined motion makes
$|\mbox{rhs}(\ref{inteq1})|\\!\ll\\!|{\bm{\alpha}}^{\scriptscriptstyle\perp}\\!+\frac{{\bf
B}_{s}}{2}\\!\wedge{\bf x}|$; otherwise rhs(56) determines the first
correction to ${\bf A}^{\scriptscriptstyle\perp}$; and so on.

Summarizing, the present hydrodynamic model is justified in a sufficiently
short time interval $[0,t^{\prime}]$ where $\partial_{Z}\hat{z}_{e}\\!>\\!0$,
$|\mbox{rhs}(\ref{inteq1})|\\!\ll\\!|{\bm{\alpha}}^{\scriptscriptstyle\perp}\\!+\frac{{\bf
B}_{s}}{2}\\!\wedge{\bf x}|$, and the motion of ions is negligible.

###  3.2 Some general features of the motions ruled by (59-58)

$\widetilde{N}(Z)$ grows with $Z$, and so does the rhs(59b) with
$\hat{\Delta}$. As soon as $v(\xi)$ becomes positive for $\xi\\!>\\!0$, then
so do also $\hat{\Delta}$ and $\hat{s}_{e}\\!-\\!1$: by (64), all electrons
reached by the pulse start to oscillate transversely and drift forward (pushed
by the ponderomotive force); the $Z\\!\simeq\\!0$ electrons leave behind
themselves a layer of ions $L_{t}$ of finite thickness
$\zeta(t)\\!=\\!\Delta(t,0)\\!=\\!\hat{\Delta}[\tilde{\xi}(t,0),0]$ completely
deprived of electrons, see fig. 7. $\hat{s}_{e}$ keeps growing as long as
$\hat{\Delta}\\!\geq\\!0$, making the rhs(59a) vanish at the first
$\bar{\xi}(Z)\\!>\\!0$ where
$\hat{s}_{e}^{2}(\bar{\xi},\\!Z)\\!=\\!1\\!+\\!v(\bar{\xi})$ and become
negative for $\xi\\!>\\!\bar{\xi}$. Hence $\hat{\Delta}(\xi,\\!Z)$ reaches a
positive maximum at $\xi\\!=\\!\bar{\xi}(Z)$ and then starts decreasing
towards negative values (electrons are attracted back by ions in $L_{t}$). For
$\xi\\!>\\!l$ energy is conserved, the paths in $\big{(}\Delta,s_{e}\big{)}$
phase space are the level curves (parametrized by $Z$) fulfilling
$\hat{H}(\Delta,s_{e};Z)\\!=\\!h(Z)\\!=\\!1\\!+\\!\\!\int^{l}_{0}\\!d\xi\>v^{\prime}(\xi)/\hat{s}(\xi,Z)$
[by (30)]; the dependence of
$\hat{P}\\!\equiv\\!\big{(}\hat{\Delta},\hat{s}_{e}\big{)}$ on $\xi$ is
obtained by quadrature. Since ${\cal U}(\Delta;0)\\!=\\!0$ for
$\Delta\\!\leq\\!0$, then $\hat{\Delta}(\xi,0)\\!\to\\!-\\!\infty$ as
$\xi\\!\to\\!\infty$: the $Z\\!=\\!0$ electrons escape to
$z_{e}\\!=\\!-\infty$ infinity. Whereas if $Z\\!>\\!0$ then ${\cal
U}(\Delta;\\!Z)\\!\to\\!\infty$ as $|\Delta|\\!\to\\!\infty$, the path is a
cycle, and $P(\xi;Z)$ is periodic in $\xi$, with period

$\displaystyle c\,T(Z)\equiv\Xi(Z)=\\!\int^{\Delta_{{\scriptscriptstyle
M}}}_{\Delta_{m}}\\!\\!\frac{2\>d\Delta}{\sqrt{1-\mu^{2}/[h(Z)\\!-\\!{\cal
U}(\Delta;Z)]^{2}}},\qquad\mu^{2}\\!\equiv\\!1\\!+\\!v(l):$ (66)

all $\hat{\Delta}(\cdot,Z)$ oscillate around zero, i.e. all $Z\\!>\\!0$
electrons do $T(Z)$-periodic longitudinal oscillations about their initial
positions $z=Z$.

Figure 6:  a) Normalized gaussian EM pulse of full width at half maximum
$l_{fwhm}\\!=\\!10.5\lambda$, linear polarization, peak amplitude
$a_{0}\\!\equiv\\!2\pi eE^{\scriptscriptstyle\perp}_{\scriptscriptstyle
M}/mc^{2}k\\!=\\!2$ (leading to a peak intensity
$I\\!=\\!1.7\\!\times\\!10^{19}$W/cm2 if $\lambda=0.8\mu$m). b) Corresponding
solution of (67-68) for $Z\\!\geq\\!Z_{d}$, with
$\widetilde{n_{0}}(Z)\\!=\\!n_{0}\theta(Z)$, $n_{0}\\!\equiv\\!10^{18}$cm-3
(whence $h\\!=\\!1.36$); as anticipated, $\hat{s}$ is indeed insensitive to
fast oscillations of ${\bm{\epsilon}}^{\scriptscriptstyle\perp}$. c)
Corresponding normalized electron density inside the bulk as a function of $z$
at $ct=200\lambda$. d) Phase portraits for the same $\widetilde{n_{0}}(Z)$,
$h\\!=\\!1.36$, $\mu=1$.

Assume now that the electron density has an upper bound and becomes a constant
well inside the bulk: $0\\!<\\!\widetilde{n_{0}}(z)\\!\leq\\!n_{b}$ if
$z\\!>\\!0$ and $\widetilde{n_{0}}(z)\\!=\\!n_{0}$ if $z\\!\geq\\!z_{s}$, for
some constant $n_{b}\\!\geq\\!n_{0}$, $z_{s}\\!\geq\\!0$. Then there exist
$Z_{b}\\!>\\!0$ and $Z_{d}\\!>\\!Z_{b},z_{s}$ such that:

  1. 1.

the $Z\\!\in]0,\\!Z_{b}[$ electrons exit and re-enter the bulk;

  2. 2.

the $Z\\!\geq\\!Z_{b}$ electrons remain inside the bulk, i.e.
$\hat{z}_{e}\\!\geq\\!0$ for all $\xi$;

  3. 3.

the $Z\\!\geq\\!Z_{d}$ electrons fulfill $\hat{z}_{e}\\!\geq\\!z_{s}$ for all
$\xi$.

For the latter $\widetilde{n_{0}}(\\!\hat{z}_{e}\\!)\\!\equiv\\!n_{0}$, ${\cal
U}(\Delta,Z)\\!\equiv\\!M\Delta^{2}\\!/2$, so that (59-58) no longer depends
on $Z$ and reduces to the same Cauchy problem for all $Z\geq Z_{d}$:

$\displaystyle\Delta^{\prime}=\displaystyle\frac{1+v}{2s^{2}}\\!-\\!\frac{1}{2},\qquad\quad
s^{\prime}=M\Delta,\qquad M\\!:=\\!\frac{4\pi
e^{2}n_{0}}{mc^{2}}\\!\equiv\\!\frac{\omega_{p}^{2}}{c^{2}},$ (67)
$\displaystyle\Delta(0)\\!=\\!0,\qquad\qquad\qquad s(0)\\!=\\!1.$ (68)

Eq. (67) is the equation of motion of a relativistic harmonic oscillator (with
a forcing term $v$ for $0<\xi<l$); consequently, also the final energy $h$ is
the same for all $Z\\!\geq\\!Z_{d}$. We denote as $\Xi_{{\scriptscriptstyle
H}}(n_{0})\\!=\\!cT_{{\scriptscriptstyle H}}(n_{0})$ the corresponding plasma
period (66) (recall that $T_{{\scriptscriptstyle
H}}\\!\geq\\!T_{{\scriptscriptstyle H}}^{{\scriptscriptstyle
nr}}\\!\equiv\\!2\pi/\omega_{p}\\!=\\!\sqrt{\\!\pi m\\!/n_{0}e^{2}}$, which is
the non-relativistic limit of $T_{{\scriptscriptstyle H}}$555When
$\hat{v}\\!=\\!0$ then (67) implies
$\Delta^{\prime\prime}\\!=\\!-M\Delta/\hat{s}^{3}$. In the nonrelativistic
regime $\hat{s}\\!\simeq\\!1$, $\tilde{\xi}(t)\\!\simeq\\!ct$,
$cd/d\xi\\!\simeq\\!d/dt$, and this becomes the nonrelativistic harmonic
equation $\ddot{\Delta}\\!=\\!-\omega_{p}^{2}\Delta$.). Thus for all $Z>Z_{d}$
electrons it is $\hat{z}_{e}(\xi,Z)=Z\\!+\\!\Delta(\xi)$ and hence
$z_{e}(t,Z)=Z\\!+\\!\Delta(ct\\!-\\!z)$ for all $\xi$. The restriction of
$z_{e}(t,\cdot)$ to $[Z_{d},\infty[$ is invertible [with inverse
$Z_{e}(t,z)=z-\Delta(ct\\!-\\!z)$]: no $Z$-electron layer with
$Z\\!\geq\\!Z_{d}$ can collide with another one. This justifies there the
hydrodynamical picture used so far. All Eulerian fields are found to depend on
$t,z$ only through $ct\\!-\\!z$, e.g. ${\bf u}(t,z)=\hat{\bf
u}(ct\\!-\\!z)$,…: a plasma travelling-wave with spacial period
$\Xi_{{\scriptscriptstyle H}}(n_{0})$ and phase velocity $c$ trails the pulse
for $z\geq Z_{d}$ [24]. On the other hand, plasma wave breakings [25], i.e.
collisions among (sufficiently close) $Z$-electron layers with
$Z\\!<\\!Z_{d}$, occur only at sufficiently large times, due to the nontrivial
$Z$-dependence of (66) in the regions where $\widetilde{n_{0}}(Z)$ is non-
homogenous (in particular near the vacuum-plasma interface $Z\\!\sim\\!0$).
There our hydrodynamic description is globally self-consistent for
$t\\!<\\!t_{c}$ ($t_{c}$ stands for the time of the first wave-breaking) and
allows to determine [24] $t_{c}$ and where wave-breakings occur; the use of
kinetic theory (a statistical description of the plasma in phase space that
takes collisions into account) is necessary after the first wave-breaking. As
known, a moderate wave-breaking is actually welcome as a possible injection
mechanism in the plasma wave of a bunch of electrons fast enough and in phase
to be accelerated ’surfing’ the wave, according to the LWFA mechanism [5]; a
phase of the right sign arises where $\widetilde{n_{0}}(Z)$ decreases.

Figure 7: Conditions as in fig. 6
($\widetilde{n_{0}}(Z)\\!=\\!n_{0}\theta(Z)$). Left: Normalized total charge
density $1\\!-\\!n_{e}/n_{0}$ at $ct=20\lambda$; the yellow part denotes the
positive ion layer $L_{t}$, the thin blue stripes denote the negative parts
(due to an excess of electrons). Right: The corresponding worldlines of
$Z$-electrons; for $Z\geq Z_{d}\simeq 10\lambda$ they do not intersect,
whereas the $Z<Z_{d}$ ones that first intersect do about after $5/4$
oscillations induced by the pulse, here at
$t=t_{c}\\!\simeq\\!84\lambda/c\\!=\\!224$fs. In pink the support $0\leq
ct\\!-\\!z\leq l$ of the pulse, in red the ’effective support’
$\frac{l\\!-\\!l_{fwhm}}{2}\leq ct\\!-\\!z\leq\frac{l\\!+\\!l_{fwhm}}{2}$ of
the pulse,in yellow the evolution of the pure-ion layer $L_{t}$.

Let $T_{{\scriptscriptstyle H}}(n_{b})$ is the plasma period (66) associated
to the electron density $\widetilde{n_{0}}(Z)\equiv n_{b}$. If

$2l\lesssim\Xi_{{\scriptscriptstyle H}}(n_{b})\equiv cT_{{\scriptscriptstyle
H}}(n_{b})$ (69)

(i.e. the pulse is sufficiently short) then $\hat{\Delta}(l,\\!Z)\\!\geq\\!0$
for all $Z$: the pulse overcomes all electrons before they overshoot their
initial $z$-coordinate; in particular, it has completely entered the bulk
before any small-$Z$ electron gets out of it, namely before $L_{t}$ is
refilled. Condition (69) also secures that the spacial period of the plasma
wave is larger that the pulse length.

In fig. 6 we plot the solution corresponding to the pulse of fig. 5-right
(with $l\\!\simeq\\!40\mu$m, $l_{fwhm}\\!\simeq\\!10.5\mu$m) and to
$n_{0}\\!=\\!10^{18}$cm-3; $s(\xi)$ is indeed insensitive to the fast
oscillations of ${\bm{\epsilon}}^{\scriptscriptstyle\perp}$ (see remark
2.2.3), $\Delta(\xi)$ grows positive for small $\xi$. The other unknowns are
obtained through (28). After the pulse is passed the solution becomes periodic
with period $\Xi_{\scriptscriptstyle H}\\!\simeq\\!47\mu$m. These $l_{fwhm}$,
$n_{0}$ fulfill (69). Replacing these solutions in the rhs(56) we find that
${\bf
A}^{\scriptscriptstyle\perp}\\!\simeq\\!{\bm{\alpha}}^{\scriptscriptstyle\perp}$
is indeed verified at least for
$t\\!<\\!t_{c}\\!\simeq\\!5\xi_{\scriptscriptstyle H}/c$. In fig. 7 we plot
the charge density just after the pulse has hit the plasma and the worldlines
of the $Z$-electrons for $0<Z<30\lambda$ and $0\leq t\leq 100\lambda/c\simeq
267$fs.

###  3.3 Comparison with the ‘comoving frame’ approach

The Lorentz-Maxwell and continuity equations in a (homogeneous) plasma admit
[26] travelling wave solutions where the EM field (plane wave laser pulse) has
the form ${\bf E}(t,{\bf
x})\\!=\\!{\bm{\epsilon}}^{\scriptscriptstyle\perp}(v_{l}t\\!-\\!z)$, ${\bf
B}(t,{\bf
x})=\mathbf{k}\wedge{\bm{\epsilon}}^{\scriptscriptstyle\perp}(v_{l}t\\!-\\!z)$;
i.e. the pulse travels with a (constant) velocity ${\bf
v}_{l}=v_{l}\mathbf{k}$, $v_{l}\leq c$. If $v_{l}\\!<\\!c$, in studying the
electrodynamics (plasma waves, LWFA, …) induced by such a pulse it is common
and useful to adopt the ‘comoving frame’ approach. This consists in describing
the physics not w.r.t. the inertial reference frame $\cal{R}$ of the
laboratory, but w.r.t. a one $\cal{R}^{\prime}$ comoving with the pulse, i.e.
moving with velocity ${\bf v}_{l}$ with respect to (w.r.t.) $\cal{R}$. Setting
$\beta_{l}=v_{l}/c$, $\gamma_{l}^{-1}=\sqrt{1\\!-\\!\beta_{l}^{2}}$, the
spacetime coordinates w.r.t. $\cal{R},\cal{R}^{\prime}$ are related by the
Lorentz transformation

$ct^{\prime}=\gamma_{l}(ct-\beta_{l}z),\qquad{\bf
x}^{\scriptscriptstyle\perp}{}^{\prime}={\bf
x}^{\scriptscriptstyle\perp},\qquad z^{\prime}=\gamma_{l}(z-\beta_{l}ct),$
(70)

and more generally the components of all 4-tensors [in particular the
canonically conjugated 4-momentum $(H,c{\bf P})$ and the EM tensor
$(F^{\mu\nu})$] w.r.t. $\cal{R},\cal{R}^{\prime}$ are related by this Lorentz
transformation and can be expressed in terms of ${\rm x}^{\prime}$; the pulse
part ${\bm{\epsilon}}^{\scriptscriptstyle\perp}{}^{\prime}$ of the EM field is
a function of $t^{\prime}$ only. Therefore adopting $t^{\prime}$ as the
independent variable has the same advantage described before, i.e.
${\bm{\epsilon}}^{\scriptscriptstyle\perp}{}^{\prime}(t^{\prime})$ appears in
the equations of motion of all charged particles (including the plasma
constituents) as a known forcing term; correspondingly, the known expression
${\bf A}^{\scriptscriptstyle\perp}{}^{\prime}({\rm
x}^{\prime})={\bm{\alpha}}^{\scriptscriptstyle\perp}{}^{\prime}(t^{\prime})$
appears in the Hamiltonian of all charged particles. However, this approach is
no more applicable when $v=c$, for which the transformation (70) is ill-
defined.

On the contrary, although $\xi=ct\\!-\\!\beta_{l}z\propto t^{\prime}$ can no
more be interpreted as the time coordinate w.r.t. an inertial reference frame,
in our approach adopting $\xi$ instead of $t^{\prime}$ as the independent
variable keeps this advantage and allows to take the limit $v\to c$.

###  3.4 Finite laser spot radius corrections, slingshot effect, and
discussion

The above predictions are based on idealizing the laser pulse as a plane EM
wave. In a more realistic picture the laser pulse is cylindrically symmetric
around the $\vec{z}$-axis and has a finite spot radius $R$. The first rough
correction to the above predictions is that only the $Z\\!\simeq\\!0$
electrons inside a cylinder of radius $R$ are pushed forward and leave behind
themselves a cylinder $c_{t}$ of ions of the same radius and finite height
$\zeta(t)\\!=\\!\Delta(t,0)\\!=\\!\hat{\Delta}[\tilde{\xi}(t,0),0]$ completely
deprived of electrons. Using causality and heuristic arguments we can compute
[22] further (rough) corrections, in particular estimate the motion of the
lateral electrons (LE), just outside $c_{t}$, toward the axis of $c_{t}$,
attracted by the ions. If $R$ is not too small the $Z$-electrons with smallest
$Z$ and closest to the axis succeed in exiting the bulk before the LE reach
the axis and close their way out, and proceed indefinitely in the negative $z$
direction. As a result, for carefully tuned $R,\widetilde{n_{0}}(Z)$
[fulfilling (69), in particular] the impact of a very short and intense laser
pulse on the surface of a cold low-density plasma (or gas, ionized into a
plasma by the pulse itself), as considered e.g. in fig. 5-right, may induce,
beside a plasma traveling-wave trailing the pulse, also the slingshot effect
[22, 23, 21], i.e. the backward acceleration and expulsion from the plasma of
some surface electrons (those with smallest $Z$ and closest to the
$\vec{z}$-axis) with remarkable energy. For reviews see also [27, 28]. On the
other hand, if $R$ is smaller the LE may close the rear part of $c_{t}$ and
make it into a ion bubble (completely deprived of electrons), before any
electron gets out of the bulk. Depending on the conditions, the bubble may
then disappear or trail the pulse; in the latter case the LWFA takes place in
the particularly favourable bubble regime [29, 30, 31, 32].

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