The motion of a charged particle in the field of a frequency-modulated electromagnetic wave and in the constant magnetic field

Accepted Manuscript

The motion of a charged particle in the field of a frequency-modulated electromagnetic wave and in the constant magnetic field Nikolay S. Akintsov , Vladislav A. Isaev , Gennadii F. Kopytov , Alexander A. Martynov PII: DOI: Reference:

S2405-7223(15)30067-0 10.1016/j.spjpm.2015.12.010 SPJPM 57

To appear in:

St. Petersburg Polytechnical University Journal: Physics and Mathematics

Received date: Accepted date:

24 December 2015 24 December 2015

Please cite this article as: Nikolay S. Akintsov , Vladislav A. Isaev , Gennadii F. Kopytov , Alexander A. Martynov , The motion of a charged particle in the field of a frequency-modulated electromagnetic wave and in the constant magnetic field, St. Petersburg Polytechnical University Journal: Physics and Mathematics (2015), doi: 10.1016/j.spjpm.2015.12.010

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ACCEPTED MANUSCRIPT

Nikolay S. Akintsov, Vladislav A. Isaev, Gennadii F. Kopytov, Alexander A. Martynov Kuban State University

The motion of a charged particle in the field of a frequency-modulated electromagnetic wave and in the constant magnetic field

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In this article the problem on the motion of a charged particle in the field of frequencymodulated electromagnetic wave and in the external uniform static magnetic field has been analyzed; the exact solutions of the corresponding equations have been presented. This problem is of great importance to study the interaction of high-intensity laser pulses with solid targets and to develop practically multifrequency lasers and the laser-modulation emission technology. The formulae for the mean kinetic energy of a relativistic charged particle as a function of initial conditions, electromagnetic wave amplitude, wave intensity and its polarization parameter were obtained. The different cases of initial conditions of a charged particle motion and of a wave polarization were investigated. The obtained results can be put to use when studying the hightemperature plasma formed on the surface of the target and when searching for new modes of laser- plasma interaction.

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Plane electromagnetic wave; Charged particle; Ultrashort laser pulse.

Introduction

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The trailblazing study by Tajima and Dawson [1] has attracted a wide interest in laser-induced particle acceleration from researchers all over the world.

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Currently, the focus of theoretical and practical studies is on accelerating the motion of charged plasma particles by ultra-short laser pulses of high intensity [2–

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5]. The advances in laser technologies have allowed to create terawatt and petawatt laser pulses [6 –10] that can be used to study the interaction between the strong

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fine-focused light pulses and the charged particles in plasma. The development of such areas of physics and engineering as plasma physics, astrophysics, powerful

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relativistic high-frequency electronics, and accelerating machines pave the way for studying the interaction of charged particles with frequency-modulated electromagnetic waves. Relativistic charged particles in strong electromagnetic fields play a special role in these interactions. Obtaining the energy characteristics of a charged particle in the field of a frequency-modulated electromagnetic wave is necessary for designing multi-frequency lasers that can be then used in practice and for developing various laser modulation techniques.

ACCEPTED MANUSCRIPT

In the present paper we discuss the electron dynamics in an intense frequency-modulated electromagnetic field of elliptical polarization with a constant uniform magnetic field. Studying the interaction of charged particles with ultrashort femtosecond laser pulses with radiation intensities up to 1022 W/cm2 is currently one of the main areas in laser physics. The problem of a charged particle moving in the field of a plane frequency-

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modulated electromagnetic wave was formulated and solved for the cases of linear and circular polarization in Ref. [11]. However, the authors did not average the speed, the momentum, and the kinetic energy of the particle over the oscillation period in the field of the plane frequency-modulated electromagnetic wave in the presence of the constant uniform magnetic field, which is, without a doubt, is both

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of theoretical and practical interest.

The goal of this study is to analyze the motion of a charged particle in the external field of a randomly-polarized frequency-modulated electromagnetic wave of high intensity in the presence of a constant uniform magnetic field. In particular,

M

the equations for the mean kinetic particle energy averaged over its oscillation

PT

ED

period need to be formulated.

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2. Problem statement The equation for the motion of the charged particle with the mass m and the

AC

charge q in a high-frequency laser electromagnetic field in the presence of a constant uniform magnetic field H 0 has the following form [9]: dp q  qE  V  H Σ  . dt c

(1)

where p is the momentum of the charged particle; E is the strength of the electric laser field of radiation; H Σ  H 0  H is the strength of the combined magnetic

ACCEPTED MANUSCRIPT

field, including the uniform constant magnetic field H 0 and the magnetic component of the laser field H ; q is the particle charge. Eq. (1) is complemented by the initial conditions for the velocity and the position of the particle: V  0   V0 , r  0   r0 . The particle momentum and its velocity are connected by the following

p

mV V2 1 2 c

The change in the particle energy

is determined by the equation

.

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

mc 2

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equality [9]:

V2 1 2 c

 m 2c 4  p 2c 2

M

d  qEV . dt

(2)

(3)

(4)

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the relationship

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The energy, the momentum, and the velocity of the particle are connected by

p

V c2

.

(5)

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It is assumed in this paper that the frequency of the electromagnetic wave is

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modulated by the harmonic law:

   sin    ,

where    /  is the modulation index equal to the ratio between the

frequency deviation  and the frequency of the modulating wave   ;  is the constant phase;

  t  z / c. Let us assume that the plane frequency-modulated wave propagates along the z axis, while the strength H 0  kH 0 of the constant uniform magnetic field is also

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directed along the z axis (k is the basis vector of the z axis). In this case the vector components of the electric (E) and the magnetic (H) fields for the plane frequencymodulated electromagnetic waves are determined by the expressions [11]:





 Ex  H y  bx exp i      sin       ;   (6)  E y   H x  fby exp i      sin      ;  Ez  H z  0,  where  is carrier wave frequency;  is the constant phase; the x and the y axes coincide with the direction of the bx and the by axes of the wave polarization ellipse, with bx  by  0 ; f  1 is the polarization parameter (the upper and the lower signs in the expression for E y correspond to the right and the left polarization, respectively [14, 15]). If we apply the Jacobi–Anger expansion then the real part of the expressions (6) takes the form   E  H  b y x  J n    cos  n ,  x n     (7)  E y   H x  fby  J n    cos  n , n    Ez  H z  0,   where J n    is the n-th-order Bessel function; n    n    n . As can be seen from Eqs. (7), the spectrum of the frequency-modulated wave



PT

ED

M

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

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is symmetric in frequency:

n    n

 , the Bessel function become

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and is not limited. However, for n

negligibly low, and therefore the spectral width can be limited. The practical spectral width is determined from the expression

  2    1 , i.e. in the expansions (7) the index n can be varied in the range from from  N to N , where the number N    1 . For example, if  spectral width

1 and N  1, then the

  2 coincides with that of a harmonic amplitude-modulated

wave [11], i.e. the frequency-modulated wave is in this case transformed into an

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1 and N   , the spectral width is equal to

amplitude-modulated one. When  twice the frequency deviation:

  2 m . 3. The solution for the charge motion equation The solution of Eqs. (1) and (4) with E and H from the expressions (7) has the form:

py 

fqby



n  N



N



n  N

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px 



J n    sin  n q  H0 y  x , c 1  n 

N

qbx

J n    sin  n q  H0 x   y. c 1  n 

where    /  .

(8)

x



 1  n 

n  N

fqby с



J n    sin  n

N

N



M

y

qbx с

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The equations (8) show the differentiation with respect to  :

n  N

 c y 

c



x,

J n    sin  n c  c x   y ,  1  n 

(9)

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where с  qH 0 /  is the cyclotron frequency. The constants  x ,  y , and  in Eqs. (9) taking into account the formulae (3)

PT

and (7) are determined by the initial phase of the wave n  0  n0   1  n  kz0    n

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(k is the wavenumber) and by the initial velocity of the particle V0  0 : of (3)

AC

and (7) we find

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mVx 0

x 

V02 1 2 c mVy 0

y 

V2 1  02 c





qbx



fqby



J n    sin  n

N

 1  n 

n  N

N



n  N



q H 0 y0 , c

J n    sin  n q  H 0 x0 , c 1  n 

(10)

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 V  mc 1  z 0  c    . V02 1 2 c

By transforming the system of differential equations (9), we obtain the following form for the equations:



y  c2 y 

N



n N

fqcby



J n    cos  n 

fqcby

J n    sin  n

N

 1  n 

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qcbx

x  c2 x 

k

n  N



c c  ,  y

qcbx N J n    sin  n c c J  cos    x.     n n  k 1  n     n N n N N

(11)

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The solution of the second-order differential equations (10) is obtained as a sum of the solutions of the homogeneous equation and the particular solution of

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the inhomogeneous equation with the initial conditions taken into account. We obtain the following solutions for the x and y coordinates:

AC

CE

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 y  fqby n N  qb  n N x   x  n  cosΦ   n sin Φ c    c  k   k  k n  N n  N c   fqbyc n N  n  y qbx n N  sin Φ   n cosΦ n  ;   n  k n N 1  n   k n N c k  fqby N  x  qbx N y   n sin Φ c   n    k  cosΦc   k n  N c   k n N  fqby N qbxc N n  x   cosΦ  sin Φ  ; n n n  k n  k n c k  N  N 1  n 

where k   / c ; n is the order of the Bessel function;





Φc  ct ; n  J n    /  2 1  n   c2 . 2

(12)

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Using (8) and (11), we obtain the expressions for the px and p y components of the particle momentum: px 

N

 An sin Φn  B cosΦc  C sin Φc 

n  N

py 

N

K

N

n

n

n  N

sin Φ n  F cosΦc  G sin Φc 

n

 D cosΦ ,

n  N

(13)

N

I

n

cosΦ n ,

n  N

An 

qbx

N

Z n 1  n  ; B  fqbyc 2



Z

n  N

Dn   fqbyc Z n ; K n 

fqby



n

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where   x ; C  qbxc

Z n 1  n  ; F   y  qbxc

Z ; n

n  N N

Z ; n

(14)

n  N

n N

 Z ;I

 qbxc Z n .

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G   fqbyc

2

N

n

n

n  N

From the formulae (3) and (4) we find the z-component of the particle momentum:

(15)

M

pz   g ,

where

1 N D 2  An2  I n2  K n2  cos  2Φ n   2   n 4 n N

ED

g h





n  nB



1

N

F cos Φ c  G sin Φ c   K n sin Φ n  2  n  N

 A D N

AC



1

CE



PT

N N 1 1  2  An AnB  K n K nB sin Φ n sin Φ nB  2  B cosΦ c  C sin Φ c   An sin Φ n  2 n ,nB  N  n  N

 2 n ,n

B  N n  nB

n

nB



 K n I nB sin Φ n sin Φ nB 

1 N   An Dn  I n K n  sin  2Φ n   2 2 n N 1 B 2  C 2  F 2  G 2  cos  2Φ c   2  4

N 1 1  2  BC  FG  sin  2Φ c   2  B cosΦ c  C sin Φ c   Dn cosΦ n  2  n  N



1



F cosΦ c  G sin Φ c  2 

N

 I n cosΦ n 

n  N

N 1   Dn Dl  I n Il  cosΦ n cosΦ nB , 2 2 n ,nB  N n  nB

(16) and

ACCEPTED MANUSCRIPT 1  m 2c 2 1  N  h   2  1  2    An2  Dn2  K n2  I n2   B 2  C 2  F 2  G 2   , 2  2  n N 

(17)

where k    nB    nB ; nB is the order of the Bessel function. From (3) and (4) we find the expression for the energy of the particle:

  с 1  g .

(18)

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Using the expressions (5), (13) and (15), we obtain a parametric representation of the particle velocity in the parameter  :

N  N  dx c Vx   An sin Φ n  B cosΦ c  C sin Φ c   Dn cosΦ n  ,   dt 1  g    n N n  N 

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N  N  dy c Vy   En sin Φ n  F cosΦ c  G sin Φ c   I n cosΦ n  , (19)   dt 1  g    n N n  N  dz cg Vz   . dt 1  g

It follows from Eqs. (11) and (18) that the motion of the particle in the

M

external field of a frequency-modulated electromagnetic wave in a constant

ED

uniform magnetic field directed along the z axis is the superposition of the movement with a constant velocity Vz and the vibrational motion with the

PT

frequency

   1  n  / 1  h 

CE

that is different from the frequency of the field  , the modulation frequency   , and the cyclotron frequency c .

AC

Then, by integrating the equality (15), we obtain the equation of motion along the z axis:

z  t   z  zt    t     t  ,

(20)

where z and Vz are constant;

 t  

N

 n  t  ,   t  

n  N

N

  t 

n  N

n

,

  t  Tn     t  ,   t  Tс     t  ,

(21)

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and θ(t), η(t) are periodic functions with the periods Tn  2 / n , Tс  2 / c .

In the formula (20) Vz 

ch . 1 h

(22)

It follows from the expressions (19) that g is also a sum of 2 N  1 periodic

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functions with the periods Tn and Tc . The period Tn of the oscillation of the particle in the field of a plane frequency-modulated wave and the period Tc of the particle oscillation in a magnetic field are determined by the formulae









Φ t  Tn  Φ  t  , Φc t  Tc  Φc  t  ,

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from which, taking into account the expressions (6), (19) and the equation (20), it follows that

Tn 

2 (1  h) (1  h) 2 T ; Tc  .  (1  n ) (1  n ) c

(23)

M

Thus, the motion of the particle is a superposition of several harmonic

ED

oscillations with different periods: Tn and Tc . When the modulation frequency   is equal to zero, we can obtain the oscillation periods (the expressions for these

PT

have been obtained in Ref. [13]).

CE

4. The motion of the particle averaged over the oscillation period

AC

In this section we will perform the averaging of the momentum p and the

energy  of the particle over the periods of its oscillations (23) in the field of the frequency-modulated electromagnetic wave and the constant magnetic field. Let us introduce new variables to substitute t: Φn is the full phase of an n-th harmonic oscillation, Φc is the full phase of the cyclotron oscillation;

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Φn  Φ n  t  ; dt  

dΦn 1 1 g  dΦ ;  1  n  1  Vz  t  / c  1  n  n

Φc  Φc  t  ; dt  

dΦc

c

(24)

.

Since the motion of the particle is a superposition of harmonic oscillations with the frequencies n and c , the averaging will be carried out according to the

1 f t   2

Φ t 

1 T Φ t  n

Φc  t 



Φc  t 

f  t

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formula

1 g dΦ dΦ ,  1  n  n c

where f  t '  is an arbitrary function.

(25)

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By averaging the components (18) of the particle velocity, we obtain:

Vx  0; Vy  0; Vz 

ch . 1 h

(26)

As expected, the speed of the particle Vz in Eqs. (26) corresponds to the Vz

M

quantity described by the formula (22).

It follows from Eqs. (26) that the average transverse components of the

ED

particle momentum equal zero. For the average value of the longitudinal component of the particle momentum, we obtain the expression:

 

PT

1 N 2 2 2 2 2 pz  h  h  D  A  I  K   n n n n  1  h  32 4 n N 2



CE

N 1   An AnB  K n K nB 16 4 n ,nB  N



2

N 1 2 2  4  B  C    An2  Dn2   4 n  N

AC

n  nB



N 1 1 N 2 2 2 2 2 F  G I  K  A D  In Kn      n n  4  4   n n 4 8 n N n  N



N 1  4  An DnB  K n I nB 4 n ,nB  N



2



1 2 2 2 2 2 B  C  F  G   32 4

n  nB



N 1 1 2  2  BC  FG    Dn DnB  I n I nB 8 16 4 n ,nB  N n  nB



2

  .  

(27)

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The average energy  of the particle is determined by the formula:

 

2 c  1 N 2 1  h  D 2  An2  I n2  K n2      4   n 1 h  32 n N

N N 1 1 2 2 2  A A  K n K k   4  B  C    An2  Dn2   4   n k 16 n ,k  N 4 n  N nk N 1 1 N 2 2 2 2 2 F  G I  K  A D  I K          n n n n n n 4 4 4 8  n  N n  N

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

1 N 1 2 2 2 2 2 2  4   An Dk  K n I k   B  C  F  G   4 n ,k  N 32 4 n k

 1 1 2 2  2  BC  FG   D D  InIk   4   n k 8 16 n ,k  N  n k 

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N

(28)

It is obvious from this formula, taking into account the expression (14), that

 depends on the intensity of the wave, its initial phase and polarization, the frequency of the carrier wave  , the modulation frequency   , the cyclotron

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frequency c , and the initial velocity of the particle.

5. The case of arbitrary wave polarization for a particle initially at rest

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Let us discuss the case when the particle is initially at rest (V0  0 ) and is located in  0,0, z0  . Let us express the  x ,  y ,  constants from the formula (13), taking into account that n  0  n0   1  n  kz0    n , Φс  0   Φc0  0 : N

N

n  N

n  N

 x    An sin  n 0   Dn 1  cos  n 0 , N

N

n  N

n  N

 y    K n sin  n 0   I n 1  cos  n 0 .

(29)

For a wave with arbitrary polarization we obtain the following equality [14]: bx2  by2   2b2 ,

(30)

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where  is the ellipticity parameter (   1 corresponds to the linear, and   1/ 2 to the circular polarization, while in other cases,  0    1 corresponds

to the elliptical polarization). From the expression (17) we obtain the value of h at the initial time:





q 2  2b 2  N 2 2 h Z n  2 1  n   c2 sin 2  n 0  2 2   4m c  n N





Let us say that

с   ,

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2   2 1  n   3c2  . 

(31)

(32)

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where  is the frequency ratio between с and  , and  0;1  1;   . Since within this problem we examine the acceleration of the charged particle in a high-frequency laser field with a constant uniform magnetic field but the radiation reaction is not taken into account, the particle energy should become

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infinitely large, since for   1 the cyclotron authoresonance condition is satisfied. However, an infinitely large energy is impossible in real conditions, so this case

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can be excluded from consideration.

Substituting the ration (32) into the expression (31), we obtain that

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where

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h



N

  J    Z

4 n N

n

n

sin 2 Φn 0  Z n2Tn ,



(33)



n  J n    / 1  n    2 ; 2

Tn  1  n   3 2 ; 2

q 2  2b 2 2q 2  2 2 2 I2 , 2 5 mc m c

(34)

and I  c 2b2 / 4 is the intensity of the elliptically polarized electromagnetic wave, while   2 c /  is the wavelength.

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By substituting the expressions (29) – (34) into the formula (28), we obtain the average energy of the particle initially at rest in the wave of arbitrary polarization:

 

N с  2 2 32 1  h   Z n4  Sn2  8 2 N n       32 1  h   n  N

 4 2  Z n4  Qn  2Sn N n  sin 2  n 0   2  Z n4 Sn2 sin 4 Φ n 0  . n  N n N  N

(35)

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N

(35)

As can be seen from this expression, the average energy of the particle depends on the initial phases, the amplitude, the intensity and polarization of an electromagnetic wave, on the carrier wave frequencies, the modulation and the

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cyclotron frequency.

Further averaged over the initial phase  n 0 , the energy 

of the charged

particle in the field of a plane frequency-modulated electromagnetic wave and in the constant uniform magnetic field is given by

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  Rn  J n    Z n  Pn  4 Rn    1  8 2 Rn  Μn    2 Z n4Gn  Z n3 H n  Rn     1    , Μ n   8 Rn n 2 J n    

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  mс 2  mс 2 

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where

 n   J n    Z n   Z n2Tn  4 ,



(36)



Rn   Z n2Tn  4 , Pn  Z n J n     2Z nTn ,

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Gn  S  8 Nn , H n  Qn  2Sn Nn . The resulting formulae (28), (29), (33), (35) and (36) for the average 2 n

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kinetic energy of the particle contain an explicit dependence on the initial particle velocity, the amplitude of the electromagnetic wave, the frequency modulation index, the frequencies of the carrier wave and the modulation, the cyclotron frequency, the intensity and its polarization, which allows making practical calculations. When 

1 , N = 1, the formulae (28), (29), (33), (35) and (36) take

the form obtained in Ref. [13].

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6. Conclusions This article offers the exact analytic solution of the equations for the charged particle motion in the external field of a frequency-modulated electromagnetic wave and in the constant magnetic field. We have formulated the dependence of the charged particle velocity on the intensity of a plane frequency-modulated

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electromagnetic wave with arbitrary polarization. The velocity in question depends on the amplitude and the polarization parameter of the electromagnetic wave, on the carrier, the modulation and the cyclotron frequencies. In the frequencymodulated electromagnetic wave (7) the fields E and H are periodic with their average values equaling zero. It could be assumed that the frequency-modulated

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electromagnetic wave and the constant uniform filed have an alternating effect on the charged particle, and that the average deviation caused by this effect is also zero. However, this assumption is incorrect. In particular, the particle in the field of a plane frequency-modulated electromagnetic wave systematically drifts in the

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direction of the electromagnetic field propagation. This is confirmed by an

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analytical calculation of the velocity and the momentum components, as well as of the average kinetic energy of a particle. With an increase in field intensity,

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according to the formula (23), the frequency of the oscillatory motion of the particle, the modulation frequency and the cyclotron frequency tend to zero. It was

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shown that the motion of the particle averaged over the oscillation periods Tn and Tc is a superposition of the constant motion and the vibrational motion with the

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carrier frequency, the cyclotron frequency and an n-th vibrational motion with the frequency n . In the absence of the frequency modulation, all the formulae are transformed into the respective formulae given in Ref. [13]. The solutions have been obtained in the explicit dependence on the initial data, the amplitude of the electromagnetic wave, the carrier wave frequency, the modulation frequency, the wave intensity and its polarization parameter, which allows using the obtained solutions in practical calculations. as well as the drift direction of wave

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propagation. The values of the momentum and energy of the particle, averaged over the period of vibration, were calculated. The oscillation period of the particle differs from that of the field. As the field intensity is increased, the frequency of the oscillatory motion of the particle tends to zero according to (23). It was shown that motion of the particle is the superposition of motion at a constant velocity and vibrational motion with the frequency of the electromagnetic field, cyclotron

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frequency and the frequency modulation different from the field frequency. In the absence of the frequency modulation, all the formulae go to the appropriate formulae given in [13]. The solutions obtained are presented in the explicit dependence on the initial data, the amplitude of the electromagnetic wave, the wave intensity and its polarization parameter that allows everyone to apply the

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solutions in practice. The practical value of the study we carried out is that the results obtained can be used for designing various relativistic electronics devices. Furthermore, our results may be of interest for astrophysical research, and may also be used for interpreting the experiments with plasma in an external frequency-

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modulated electromagnetic field in the presence of a uniform magnetic field.

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Acknowledgment

The study has been conducted with the financial support of the state task of the Ministry of

REFERENCES

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Education and science of the Russian Federation (project no. 1269).

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[4] Y. Oishi, T. Nayuki, T. Fujii, Measurement of source profile of proton beams generated by ultraintense laser pulses using a Thomson mass spectrometer, Phys. Plasmas, 12 (2005) 073102. [5] A. Pukhov, S. Gordienko, S. Kiselev, The bubble regime of laser – plasma acceleration: monoenergetic electrons and scalability, Rep. Prog. Phys. 46 () (2004) 179.

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[8] D.J. Umstadter, Relativistic laser – plasma interactions, Phys. D: Appl. Phys. 36()(2003) 152.

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[10] S.C. Wilks, A.B. Landon, T.E. Cowan, Energetic proton generation in

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ultra-intense laser-solid interactions, Phys. Plasmas. 8 ()(2001) 542. [11] G.F. Kopytov, S.S. Oksuzyan, V.B. Tlyachev, K voprosu o

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harakteristikah izlucheniya elektrona v modulirovannom elektromagnitnom pole, Izvestiya Vuzov. Fizika (1987)15 p. Dep. VINITI 14.09.85, No. 7353.

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[12] L.D. Landau, E.M. Lifshits, Teoriya polya [The Field Theory], Moscow, Nauka, 2004. The motion of a

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[13] G.F. Kopytov, A.A. Martynov, N.S. Akintsov,

charged particle in the field of an electromagnetic wave and the constant magnetic field, St. Petersburg State Polytechnical University Journal. Physics and Mathematics. No. 4 (206) (2014) 55–63. [14] R.M. Azzam, N.M. Bashara, Ellipsometriya i polyarizovannyi svet [Ellipsometry and polarized light], Moscow, Mir, 1981. [15] Newton R. Scattering Theory of Waves and Particles. Moscow, Mir, 1967.

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THE AUTHORS Nikolay S. Akintsov Kuban State University 149 Stavropolskaya St., Krasnodar, 350040, Russian Federation [email protected]

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Vladislav A. Isaev Kuban State University 149 Stavropolskaya St., Krasnodar, 350040, Russian Federation [email protected]

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Gennadii F. Kopytov Kuban State University 149 Stavropolskaya St., Krasnodar, 350040, Russian Federation [email protected]

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Alexander A. Martynov Kuban State University 149 Stavropolskaya St., Krasnodar, 350040, Russian Federation [email protected]