22C04
Theoretical Study of Nonlocal Effect of the Ponderomotive Force in HighIntensity Laser Fields Using the Noncanonical Lie Perturbation Theory
非正準Lie摂動論を用いた高強度レーザー場中での動重力の非局所効果に関する理論研究
Natsumi Iwata, Yasuaki Kishimoto and Kenji Imadera
岩田夏弥, 岸本泰明，今寺賢志
Department of Fundamental Energy Science, Graduate School of Energy Science, Kyoto University,Gokasho, Uji, Kyoto 611-0011, Japan
京都大学大学院エネルギー科学研究科 〒611-0011 京都府宇治市五ヶ庄
We investigate the relativistic ponderomotive force in a tightly focused high intensity laser field which takes into
account the higher order structure of the laser field amplitude. An extended formula for the ponderomotive force
was derived by using the noncanonical Lie perturbation theory, which is based on the phase space Lagrangian
formalism. The force is found to depend not only on the gradient of the field amplitude, but also on the curvatureand its derivative in the higher order originating from the nonlocal particle motion during a laser period.

1. Introduction
The intensity of ultra-short high power lasers
has reached the range of 1022W/cm2, whereelectrons irradiated by such lasers exhibit
relativistic characteristics. Recently, efforts
aiming at higher intensities of 1023-26W/cm2 havebeen made, which is expected to open up an
entirely new scientific regime [1]. In order toachieve such high intensities, reduction of thepulse length and/or the spot size is necessary sothat the ponderomotive force (light pressure)
becomes of critical importance in determining thelaser-plasma interaction.
The ponderomotive force has been derived
based on the averaging method [2] and isexplained as the force proportional to the field
gradient resulting from the first order
perturbation to the uniform field. In this method,terms related to the higher-order derivatives havebeen neglected. However, they could become
important in the case of strong focusing where thelaser field amplitude varies within the excursion
length of the oscillatory particle motion. Namely,
since the particle experiences nonlocal fieldstructure around the oscillation center, theaverage field strength and/or its gradient over theoscillation cycle varies in the existence of
higher-order derivatives of the laser field
amplitude. In order to investigate such nonlocal
effects, here, we introduce the noncanonical Lie
perturbation theory [3]. Based on the perturbed
phase space Lagrangian, we derive the equationsof motion of the particle taking into account thehigher order field structure, i.e., the second andthird derivatives of the field amplitude.

2. Noncanonical Lie Perturbation Theory
We introduce the extended phase space expressedby canonical variables, zcμ = (t; x,pc) = (t; x, y, z, pcx,
pcy, pcz), and the corresponding covariant vector, γcμ= ( − h; pc,0), where h is the relativistic Hamiltonian.
In this paper, we use Latin indices that run from 1
to 6 whereas Greek run from 0 to 6. Using thesenotations, the variational principle is expressed as δ∫ γcμ dzcμ = 0, where γcμ dzcμ is referred to as afundamental 1-form. Under an arbitrary coordinatetransformation zcμ  zμ, the new covariant vector is
obtained by the scalar relationship γcμdzcμ = γμdzμ.
Then, the equation of motion in an arbitrary
noncanonical coordinate is given by the variational
principle as dzi/dz0 = Jij (∂γj /∂z0 − ∂γ0 /∂z j), where J ij
is the Poisson tensor, Jij = (∂i γj − ∂j γi) -1. The Lietransformation is characterized by the near-identity
transformation operator L as zμ  z’ μ = exp(εL) zμ[3]. The corresponding covariant vector istransformed as γμ  γ’μ = exp( − εL) γμ+∂μ S, where S
is the gauge function. In the Lie perturbation
method, we utilize the coordinate transformation tosimply describe the perturbed motion in each orderby carefully choosing the generator of thetransformation and the gauge function.
3. Preparatory Transformation
We consider a single particle motion in therelativistic regime irradiated by transversely
non-uniform high intensity laser field. We express
the laser field by the normalized vector potential, a= qA/mc2, as a = ax(x) sin η ex, where q and m are
the charge and rest mass of the particle, respectively,
c is the speed of light, η ≡ ωt − kz z is the laser phaseand ex is unit vector in the x-direction. We introduce
a smallness parameter, l /L ~ ε, l 2/R ~ ε2, l 3/T ~ ε3,where l is the transverse excursion length of theparticle and L-1 ≡ ax-1∂x ax, R -1 ≡ ax-1∂x2ax andT -1 ≡ ax-1∂x3ax are scaling factors of the field
amplitude variation.
Since the particle in a uniform laser field exhibits
the figure-eight motion in the period of η keeping
the quantity pη ≡ pz − γmc constant [4], we at firsttransform the canonical coordinate zcμ to a new one,zμ = (η; x,y,z,px, py, pη), which is noncanonical. Here,γ is the relativistic factor and p = pc − mca is themechanical momentum. By taking both η and pη ascoordinate variables, the new Hamiltonian K isfound to be expressed in a simple form, K= − (2kz pη) -1(m2c2+ p⊥2+ pη2).
4. Perturbation Analysis
Here, we expand the amplitude of the vector
potential a around the oscillation center of the
figure-eight orbit, X, as~
~~

xx2
x3
a x ( x ) = a x ( X )1 + ε + ε 2+ε3+ ,
2! R
3!TL(1)where ~x ≡ X − x . Here, L, R and T are evaluatedat X. In the zeroth order of ε, Eq. (1) denotes auniform field. In this order, we treat ax = ax (X) as aconstant. From the zeroth order 1-form in the
noncanonical coordinate z μ, we can derive theequations of motion that yield the analyticalsolution denoting the unperturbed figure-eightmotion. From the solution, the excursion length l isobtained as l = ax /kz ζ0, where ζ0 is the constantdefined by the initial condition, pη (η = 0) = − mcζ0.
Next, to investigate the motion of the oscillation
center, we again transform the coordinate to thenew one,Z µ = (η ; X , Y , Z , Px , Py , pη ) ,(2)by using the relation Z μ = z μ − [z (0) μ]os., where[z (0) μ]os. is the oscillatory part of the zeroth orderorbit. Note that hereafter we treat ax (X) not as aconstant but as a variable depending on X. In thecoordinate Eq. (2), we derive the perturbed
equations of motion up to the third order of ε basedon the Lie perturbation method. It is found that thecovariant vector in the Lie-transformed coordinateZ’ μ does not depend on the variables Y’ and Z’ sothat the corresponding variables Py’ and pη’ remain
constant in all order. For the x-direction, we obtain
the equation of motion,dPx′mca x  l ε 3  3 l l 2 l 3 3 l 3 =−+ −
ε + dη2  L 8  2 L R T 2 L3  (2)l+ εmca x cos 2η .
LThis equation determines the averaged motion ofthe oscillation center up to the order of ε3. We see
that, in addition to the first-order ponderomotiveforce proportional to the field gradient, thethird-order terms related to R -1 and T -1 appear in Eq.
(3). According to the equation, the ponderomotive
force is modified to become larger when the second
and/or third derivative is positive. This is because
the positive second derivative increases the averagevalue of the field strength so that the excursion
length becomes larger, and positive third derivative
gives steeper gradient at the oscillation center inaverage, respectively. Note that the second-order
average force which depends on the second
derivative, l 2/R , does not appear. This result
indicates that, due to the symmetric nature of thecurvature, the nonlocal effect to the field gradient iscancelled during single laser period. We confirmedthat the same can be said for all the following
even-order derivatives.
5. Summary
Based on the noncanonical oscillation-center
coordinate, we derived the equations of motion up
to the third order of ε by using the noncanonical Lieperturbation theory. We found that, in the thirdorder, terms related to the second- andthird-derivatives of the laser field amplitude areadded as a modulation to the first-order
ponderomotive force. These additional forces
originate from the nonlocal particle motion aroundthe oscillation center. The result suggests that thecontrol of the higher order field structure isimportant in the case of tight focusing.
Acknowledgments
This study was supported by a Grant-in-Aid from
JSPS (No. 21340171).
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