THE MOTION OF A CHARGED PARTICLE IN THE ELECTROMAGNETIC FIELD OF A POLARIZATION-MODULATED WAVE Текст научной статьи по специальности «Физика»

UDC 539.12:537.63/.868

DOI: 10.18698/1812-3368-2022-6-106-122

THE MOTION OF A CHARGED PARTICLE

IN THE ELECTROMAGNETIC FIELD

OF A POLARIZATION-MODULATED WAVE

G.Ph. Kopytov1 D.I. Kudryavtsev2 V.A. Brazhko2

g137@mail.ru

dmitriy-kudryavtsev-2016@mail.ru vika.brazhko01@mail.ru

1 Razumovsky Moscow State University of Technology and Management (FCU), Moscow, Russian Federation

2 Kuban State University, Krasnodar, Russian Federation

Abstract

This article presents an exact solution of the equation of motion of a charged particle in the electromagnetic field of a high-intensity polarization-modulated wave. Expressions for the average kinetic energy of a particle without regard to its rest energy in the case of circular and linear polarization of a modulated wave are obtained. The motion of a charged particle in the field was analyzed and expressed in terms of dependences of its average kinetic energy on the electromagnetic wave intensity and on various types of modulation depths. The contribution of each type of modulation to the energy characteristics of a charged particle was demonstrated. Solving the equation of motion of a charged particle in the electromagnetic field of a plane wave opens up possibilities for various applications related, in particular, to various developments of multi-frequency lasers and laser modulation technology. This study was proposed due to the growing interest in experiments using high-intensity femtosecond laser radiation and high-temperature plasma

Keywords

Polarization, polarization modulation, charged particle, average kinetic energy, equation of motion

Received 14.02.2022 Accepted 14.09.2022 © Author(s), 2022

This work was financially supported by the Russian Foundation for Basic Research and an Administration of the Krasnodar Region (project no. 19-44-230026)

Introduction. Currently, modulation of electromagnetic waves finds a wide applications in the field of science and technology. One of the applied modulations stands out from the rest. This is about polarization modulation, which creates a two-dimensional signal, which is its distinguishing feature from frequency

and amplitude modulation. Its practical value is fully manifested in microscopy, emitter condition monitoring, diversity reception, interference suppression and more [1, 2]. The process of polarization modulation consists in changing the spatial orientation of the electric intensity vector of an electromagnetic wave according to a certain harmonic law. Thus, the essence of polarization modulation is the joint modulation of the amplitude and phase of an electromagnetic wave. This type of modulation provides good indicators of noise immunity and information transfer rate. Polarization modulation is carried out by rotating the plane of polarization [3] or by changing the polarization type[4], such modulation will be called polarization-optical [5]. Also, the same modulation of radiation can be performed by rotating the radiation source in space. In this case, the modulation is called spatially polarization. This type of modulation is used in laser therapy devices, which use amplitude and frequency modulation, and polarization as an additional modulation to them. As a result, laser radiation increases its efficiency of interaction with biological tissues, insofar as the required polarization is found for each molecule [6].

This article is devoted to the consideration of charged particle acceleration by an ultrashort laser pulse high order intensity of 1019 W/cm2. The relevance of this topic also leaves no questions, since particle acceleration obtained by the interaction of short laser radiation with plasma is one of the advanced problems of laser physics [7-9]. The results obtained can also be used in practical calculations of the corresponding experiments, in the development of modulation technology, the development of multifrequency lasers and the improvement of devices of the type [10].

The problem of the dynamics of a charged particle in the field of an external electromagnetic wave has already been analyzed quite deeply, which can be evidenced by several scientific papers. The first works considered this problem from the classical point of view belong to Ya.I. Frenkel and from the quantum to D.M. Volkov, and later to L.D. Landau and E.M. Lifshitz [11-13]. The solution of the motion equation of an elementary particle in the field of a plane monochromatic wave was obtained in [14-17]. More complex special cases, when a laser emits an electromagnetic wave modulated in frequency or amplitude were analyzed in subsequent papers [18-20]. However, there is a gap in solving the problem of the dynamics of a charged particle in an external electromagnetic field when the wave creating the field has a modulated polarization.

The aim of this work is to analyze the motion and energy properties of a charged particle in the field of a polarization-modulated electromagnetic wave.

Problem statement. The article [21] presents a problem formulation of the following form. Assuming that the amplitude of the electromagnetic wave is modulated according to the harmonic law:

b(£) = b±o {1 + 5am cos [<B(£ + 5pm sin (c>qO£, + ) + Co ]}, (1)

and we apply the Jacobi — Anger expansion [22], then we obtain components of the electromagnetic wave vectors:

Ex — H y — h

0 x

1 + ÔAM X Jl (Ôpм)C0SФl

l

(

Ey - ~Hx - fb0 y

X Jn(5fm)cos Фп,

n = -ce

Л „

1 + 5 AM E Jl (Ôpм)C0SФl

l = -cc

E Jn (5fm) sin Фп , (2)

Ez = Hz = 0.

The z-axis is the direction of the electromagnetic wave propagation, when the x and y axes coincide with the direction of the semiaxes of the wave polarization ellipse b0x u b0y, moreover b0x > b0y > 0; the relativistic case corresponds to the condition L, = t - (z / c); œ is wave carrier frequency; f = ±1 is polarization parameter: the upper and lower signs in the expressions for Ey correspond to right and left polarization; ôAM is amplitude modulation depth and ôAM g [0,1]; ra0 is modulation frequency; 5PM is polarization modulation depth; a = raj / œ0> a g [0,1], is modulation coefficient; ôFM is frequency modulation depth; y0 is initial phase of the wave y0 e[0, 2tc]; Jn (5fm), Jl (5pm) are n-th order and l-th order Bessel function; On = (co + nra')^ + a + n^0; Oi =(co +1&'0 )£ + + a + l^o.

Motion of a charged particle in the field of a polarization-modulated electromagnetic wave. The equation of motion of a particle of mass m and charge q has the form:

Î - q (E+Г [v *HH

(З)

The solution of equation (3) applying equation system (2) gives particle momentum components:

p, = ^

ш

I ZnÎËF^ sin Фп +—AM

n = -N 1 + n<x 2

£ £ Jn (5fm )Jl (5pm ) x

n = -Nl = -N ПСХ - lâo

sin (Фп - Фl )

ôAM I I Jn (5fm )Ji (5pm ) sin 2 n = -Nl = -N 2 + nix + lâo

(Фп + Ф1 )

+ X x

qbo y

Py = +

ш

x cos

N Jn (5fm ) 5ам N £ Jn (5fm )Jl (5pm)

X —Г—T^ cos Фп + —— X X —"-Г^ТГ--X

n = -n 1 + nu 2 n = _N l = _N na -l

5am N N Jn (sfm )Ji (5pm)

(Фп - Ф1 z X °fm; JlKcos (On + Oi)

v 7 2 n = _Nl = _N 2 + na + lao

Pz = lg,

+ X y>

(4)

where a = ro' /ro; a0 = ©0/ro; y = rnc (i - v0z /c- vQ /c2; X*, Xy are the integration constants.

Note that the expression was replaced by the component pz, which will be used in further calculations. Thus, the value of g will have the following form:

h q2 (C -bly) i " J2 (Sfm)

g = h--\ 2 2 i Z , . .o cos 20„ +

4y2ra2 [n=—N (1 + ncx

X2 NN

-ЧМ

4 n=—Nl=—N

+ Z Z Jn (Sfm )Jl2 (Spm )

cos 2 (фп + Ol) cos 2 (фп -Ol)

(2 + ncx + lcxo )2 (ncx - lcxo )2

q2 I Sam N j2 (5fm )Jl (Spm ) г, 2 . . /■=- д \

y2ra2 I 2 n = _N (1 + ncxдпа-lcx0JL v ;

+ bo2y cos фп cos (фп - Ol v]^ z ^ ^ ^ X

v /J 2 n=_N (1 + пад2 + ncx + lcx0 j

x [b02x sin Фп sin (Фп + Фi ) + b02y cos Фп cos (Фп + Фl)] +

+ sam £ £ J (sfm J (spm ) x 4 n = -Nl = -N (n<X-lcx0 )(2 + ПСХ + lcx0 )

b(^x sin (Фп - Ol )sin (Фп + <Ф l) + b02y cos (Фп - ф l )cos (Фп + ф l)

N

z

Jn (Sfm )

y2ra I n= — n 1 + ncx

(Xxbox sinOn + Xyboy cosOn ) +

Sam n n Jn (Sfm)Jl (Spm) г , . /-¡- д \_ , (ж JkM

+- z Z -1-г-77--1 Xxbox sin (Фп + Ol 1 + X ybo y cos ( Фп + Ф l ll +

2 n=-Ni=-N 2 + ncx + loco l v / v /J

^ £ £ Jn (Shx )j;x(SpM) , sin (Фп - Ol) + Xyboy cos (Ф„-Ol )]} . 2 n = — N l = — N ncx — lexo I

Constant h has the form

1 m2c2 + yj2x +l2y _ 1 + q2 (b0x + b0y )

h = ■

У

2 2

у2ю2

N T 2 (R^A Я2 NN v Jn{ "FM^ , О AM ^ v T2 (r \T2(R

L —-—2+—- L L Jn (ofm )Ji (opM )x

i = -n 2 (1 + na) 4 n = -N l = -N

x ¡П (5fm ) J2 (5PM )

(1 + nä )2 +(1 + la0 )2 (1 + nä )2 -(1 +1d0 )2

(5)

From equation system (4) we can obtain a parametric representation of the particle velocity:

= ( 1 _ М/qchyx_ I £ MSfm) ^

c Д юу I n = _N 1 + na

§AM N N

—M E E Jn (Sfm )Ji (SPM )

2 n = -N l = -N

sin (On -ФI) sin (On + Ol )

na -la0 2 + «a + l a0

c

X x

Y

Vy =| 1 -

1 _ qcboyy j N j^s^ cosOn

юу I n = _N 1 + na

5 дм

N N

EE Jn (Sfm )Ji (SPM )

2 n = -Nl = -N

cos (On - Oi) cos (On + Фl )

na -la0 2 + na + la0

■X y

V. =

cg 1+g

The next step is to find the coordinates of a charged particle from equation system:

qbox I N Jn (Sfm )/ ж ж \

x = xo— i E , „2 (cosOn -cosOon)

yfc® [ n = -N (1 + nd )

5дм N N Jn (SFM )Jl (Spm )

= -Nl = -N [na — la0)

[cos (On - Ol) - cos (O0n - O0l)]

5дм N N Jn (Sfm )Jl (5pm )г гж , ж \ /ж ,ж м

- Е Е —-"- 2 ^cos (On + Ol )- cos (O0n+ O0l )J

i = -Nl = -N (2 + net + ld0)

clx

V

_qboy I N Jn (Sfm)/.;=: • ж \ y = yo +—T2-1 Z —--2(sinOn -sinOon) +

yk& [ n = -N (1 + ncx

^ £ £ Jn (p) J (л) [sin(фп _Ol sin(öo^Ool)]H

2 n = -N l = -N (n(X-l(Xo )

—АМ I I Jn (P) Jl (П)2 [«in (Фп + Ol )" sin (öon + öol )]

2 n = -Nl = -N (2 + ПСХ + la0 )

К

+CJüL (SSo), у

^ ^ q2 (bQx - bo2 y) i £ Jj (Sfm ), . . ч

z = zo + ch (^-^o)--2 1 Z —1-3 isin2On - sin 2Oon H

8y2ko2 [ n = -N (1 + ncx

. L lJnJ) [ s'"2 (Ф"+ Ф l)" s'"2 ^ Ool H"

^ . L InJMJm1 [ s"2 (Фп - Ф l)-S'"2 K-öol И}

q2 s N Jj (5fm )Jl (5/m )

5 am z —^—-

2у2кю2 = _N 2(1 + ncx)(ncx-1cxo) К + bo2 y JSSCzpSL - bo2 y)

sin CDl - sin Ool iu2 u2 ^sin (2Фп - Фl)- sin (2Con - «bol)

2 (1 + ncx )-(1 +1 (xo)

l a0

q2 - £ Jj (Sfm )Jl (§pm )

„ —2öam ^ ^-* w„-ä-—\ x

2у2ш2 n = _N 2 (1 + ncx Д2 + ncx + lcxo j

(b„2x+bo2, -(,

>2 - b2 'o x bo y

sin ( 2Фп + Фl)- sin ( 2Ф0п + O0l)

2 (1 + na ) + (1 + la0)

2y 2k

q2 52am £ £ Jj (Sfm )jf (Spm )

ю

i=-Nl=-N4

(1 + na )2 -(1 +1ao

(bo2x + b0y )

sin 2Фl - sin 2<I>0l

1 + lao

"(box " bo2y)

sin 2Фn - sin 2Ф,

0n

1 + ncx

у 2k

ra

£ Jn (Sfm )

n = -N (1 + ncx )2

[ Xxbox ( cosön - cos öon )±1 yboy ( sin Фп - sin öon ) ] "

£ £ Jn (Sfm )Ji (Spm )

i = -Nl = -N (2 + na + lcx0 )2

X [xxb0x (cos (On + Ol) - cos (O0n + «O0l )) ± ± Xyb0y (sin (On + Ol) - sin (O0n + <O0l ))] +

5дм N N Jn (Sfm )Ji (5pm )

+—- z z — л л 2—x

2 n = -Nl = -N (net -la0) x [гxb0x (cos (On - Ol ) - cos (Oon - <O0l ) ) ±

±Xyb0y I sin (On -CDl) -sin (O0n -Ool)

Motion of a particle averaged over one period of oscillations. Let us rewrite the coordinates of a charged particle as functions depending on time

x(t) = X + vxt + £(t),

y(t) = y + v yt + T|(t),

z(t) = z + vzt + C(t).

Here

, Xx i ~\ x = x0 H--(z0 - z) +

У

qbox I N J" (5fm )

I

cos O0n +

§дм N N

+ —— Z Z Jn (sfm )Jl (5pm )

2 n=-nl=-N

уки [ n = -N (1 + na У

cos (Oon -Ool) cos (Oon + Ool)

Л Л 2 ^ Л Л 2

(net -ld0) (2 + nä + la0)

Vx =

c%x 1

у 1 + h

m=-Ъ±Ф)-iI z +

У УкЮ [ n = -N (1 + nä )2

cos (On - Ol ) cos (On + Ol )

§дм N N

+ —— Z Z Jn (sfm )Jl (5pm )

2 n=-Nl=-N

(nä-ld0) (2 + nä + ld0)

У = yo + — (Zo - z)± . ^ яч у ук® [ n = -n (1 + not)

ib0y \ N Jn (5FM) . ^ ^ J Z —-"T sinOon +

5 AM N N

+ —— z z Jn (sfm )Jl (5pm )

2 n=-Nl=-N

sin (Oon - <bol) sin (Oon + Ool )

л Л 2 ^ л Л 2

(nä -la0) (2 + nä + la0)

V y =

CX y 1 у 1 + h

xy qboy i N Jn (sfm) . ^

У Ук® [n = -N (1 + na)2

sin Фп +

§АМ N N

+ —— x x Jn (sfm )Jl (5pm)

2 n=-Nl=-N

sin(On - CDl) + sin(On + Фl)

(na -ld0 )2 (2 + na + la0 )2

z = zo +

q2 (bo2x - boy)

2y 2k&z

£ Jj (Sfm ) . _■=■

X -1—T7 sin 2Oon

n = -N 4 (1 + na)

, SA— I £ Jj (Sfm J (Spm ) x 4 n = -N l = -N 4 (2 + ncx +1cx0 )3

x sin 2 (Con + Col) + % i I Jj2 <8fm J <8p3m > sin 2 (Con - Col)

4 n = -N l = -N 4 (ncx - lcx0 )

q2 L £ Jj (Sfm )Jl (Spm ) Sam x - • - -—

4 у 2кю2

[ = _N (1 + ncx)(ncx-lcx0)

' b2 _ b2 ч sin (2Oon - C°l ) _(b2 + b2 \SincD0l ^0x boy' 2(1 + ncx)-(1 + lcxo) ^ + boy

s N Jj (Sfm )Ji (Spm ) ■ о am l -

-lcxr

1 = _N (1 + ncx)(2 + ncx + lcx0)

л2 _b2 \ s1"(2°on + Col) , b2

^0x boy' 2(1 + ncx) + (1 + lcxo) ^ + boy ,Sam £ I Jj (Sfm J (Spm ) ч

sin (Col l CXn

4

i = -N l = -N (1 + ncx) -(1 + lcx0 )

(bix - boy -(bUx + fy

v 7/ 1 + na v 7/ 1

sin 2Ool

+1 do

X Jn ^5fM] (xxbox cos Con ±Xyboy sin Con ) +

y2k® [ n = -N (1 + nd )2

| Sam £ £ Jn (Sfm )Ji (Spm ) x 2 n =-N i = -n (2 + na +1a0 )2

(1+h )C(i) = -

Xxbox cos ( Фоп + Фо1) ± Xyboy sin ( Фоп + <bol ) 5ам N £ Jn (5рм)Ji (5pm )

4--A, A, --- 2- X

2 n = -N l = -N (na -1do )

xxbox cos (Фоп -Ф01 )±Xyboy sin (Фоп - ФФо1 )

ch 1 + h'

£ J2 (5fm ) .

X —^-Г-Т sin 2Фп +

vz =■

q2 (bQx - b2y)

2у2кш2

_ n = -n 4 (1 + na)

.Sjm £ £ Jn <SfmJ (Spm) sin2(Ф,,+Ф1V

4 n = -Nl = -N 4(2 + na + lao)3 v '

-am £ £ Ш^тм!sta2Ф0

4 n = -Ni = -n 4(na-lao)3

q2 R N J2n (Sfm )Jl (5pm )

+-:-rÖAM L —:-~—:-

2y2 kш2 sin Фl

(b»2x + bo2y )гПФ-(bo2x -bo2y)

N 2 (1 + na)(na -la0)

sin (2Фп - Фl)

2 (1 + na )-(1 + la0)

2

q2 ~ N J2n (§fm )Jl (5pm )

+-:-rö am L —:-~-:-

2у2кш2

(b»2x+ bo2y )ггФ-К - bo2y)

J_N 2 (1 + na)(2 + na + lao)

sin (2Фп + Фl)

2 (1 + nix ) + (1 + lao)

4 "AM у у 2у2кш2 2 ^ ^

N N J2n (5fm )Jl2 (5pm )

n = -nl = -n 4 ^(1 + na)2 -(1 +1ao )2 j (b2 + b2 \^п2Ф l _ib2 _ b2 \^П2Фп

lbox + boy J1 + ldo (b0x boy ) 1

. + na

+ i X Jn (xxboxC os Фп ±Xyboy SinФn ) +

У2кш [ n = -n (1 + na )2

2

| 5дм £ £ Jn (5fm )Ji (5pM) x 2 n = -Nl = -N (2 + HCL + lä0 )2

xxbox cos (On + Ф) ± Xyboy sin (On + Ol)

5дм N N Jn (SFM )Jl (5pm ) 4--Z Z --- 2- X

2 n = -Nl = -N (h(X — let0 ) xxbox cos (On - Ol) ± Xyboy sin (On - Ol)

Now, when the particle coordinates are presented as functions depending on time, let us average the obtained expressions over one period of oscillations:

x = x+vx

( T Л t+— 2

q2c% yboxbo y

2y3o3 (1 + h)

I J2(5fm) +

= — N (1 + hO, )3

JÄM

N N т2

X X

JH(5fm)J2 (5pm) 5^ N N J" (5fmJ (5pm)

n = -Nl = -N (h(X-l(X0

XX д д 3

n = -Nl = -N (2 + H<X + l(X0)3

f

y = y + V y

T t + — 2

+

q2cxxboxboy

2у3ш3 (1 + h)

£ JH (5fm) +

3

h = -N (1 + H(X )

;дм

■4M

N N JH (5fm J (5pm )

^ ^ -Г^-ГТэ--

= -Nl = -n [na -la0)

N N JH (5fm)Jf (5pm)"

4 h = -n l = -n (2 + na + la0)

z = z + vz

f TT ^ t + —

V 2 У

The next step is to obtain averaged expressions for the momentum components of a charged particle:

Px =Xx I1"

q2b.

'0x

2у2ю2 (1 + h)

N JH (Sfm ) ^ -ГТ

n = -N (1 + net)

Si— £ £ JH (5fm )Jl2 (5pm ) + 5i— n £ JH (Sfm J (5pm )

t = -N l = -N (n(X — lct0 )

i = -Nl = -N (2 + na + la0)

Py =гу -p-

q2b0

0 y

2у2ю2 (1 + h)

N Jj (Sfm ) ^ -ГТ

i = -N (1 + n(X )

Sam £ £ Jn2 (Sfm J (5pm )

4 2 = -N l = -N (ncx -lcxo )2

5А— £ £ J2 (Sfm J (Spm )

4 n = -Nl = -N (2 + ncx + lcx0 )2

Pz =

1 + h

(h + h2 )-

q4 (bo2x " bo2y )2

4,, 4

32y 4 ®

" Jn4 (Sfm )

n = - N (1 + 2Cx J

'am

- £ Jn4 (5FM )Ji4 (5PM )

16 n = _Nl = _N (2 + ncx + lcxo)4

n n J4 (5fm )ji4 (5pm )

am z I

16 n = -Nl = -N (na - lcxo)4

q4 (bo4x + bo4y)

16 у 4 ®

4^4

8 4 N "am ^

4 n = -N

J4 (Sfm ) J2 (5pm )

(1 + ncx )2 - (1 +1cx0 )2

N J4 (Sfm )J2 (Spm ) (1 + net) +(1 + lao ) 20 am l ----

n = -N

(1 + ncx )

(1 + ncx )2 - (1 + lcx0 )2

2 у 4 ш2

(xxbo2x +xyb02y )] I

£ J2n (sFM ) + 2 SAM

N N

x Z Z Jj (SFM )J2 (spm )

2 = -N l = -N

n = -N (1 + 2Cx ) 4

(1 + ncx )2 +(1 + lcx0 )2

(1 + ncx )2 -(1 + lcx0 )2

And finally, an expression for the average kinetic energy of a charged particle moving in the field of a polarization-modulated electromagnetic wave was obtained:

1 + h

(1 + h )2

(b02x - b02y )2

4^4

32y 4 ю

£ Jn4 (Sfm )

n = -N (1 + 4(x )4

2

+ 5ам £ £ /" (5fm / (5pm ) +

16

r = -nt = -n (2 + nä +1 a0 )

- - /4 (5fm )/f (5pmУ

16 n = -ni = -n (na-lao)4

94 + Ky)

4„4

16y 4ffl

5Лм £ /n4 (5fm)/2 (5pm) _ +

n = -N

(1 + nä )2-(1 + ld0 )2

2Я2 n /4 (5fm )/2 (5pm ) (1 + nd)2 +(1 + lao )2

+ АМ Ъ -

n = -N

(1 + nek)

(1 + nä )2 -(1 + la0 )2

} +

2y4 ш2

S2 N N + 2 °ЛМ

(xibo2x +x2ybo2yH I

n /" (5pm )

и e e /" (5fm )/? (5pm )

4 n=-Nl=-N

n = -n (1 + nä )2

(1 + nä )2 +(1 + la0 )2 (1 + nä )2 -(1 + la0 )2

Average kinetic energy of a particle moving in the field of a polarization-modulated electromagnetic wave of circular and linear polarization in the absence of the initial particle velocity. Let us consider the values of the average kinetic energy of a particle moving in the field of a polarization-modulated wave in circular and linear polarization in the absence of the initial particle velocity:

v o = 0, On ( 0 ) = Oo„ =-o(1 + nd ) Zo / c, 6I (0) = 6oi = - «(1 +1do)zo/c, then constant ix, Xy has this form:

X V

qbo v

ш

N /n (5fm ) . л

n = -N

1 + na

, 5лм д д /n (5fm )/i (5pm ) . /ф ф ч

+- X X —1—т^-г-1 Sin (Фоп- Фо1 )

2 n = -Nl = -N na-lao

+ -Л2М £ £ J" (8fm>-fl ^) Sin (Фоп +ФоО ^

2 ..... 2 + na + lao v '

(6)

n=-Nl=-N

2

X y —

qbo y

ш

N Jn (§fm ) O X —r—cosOon +

n = — N

1 + HCL

, 5дм N N Jh (5fm ) Jl (5pm ) (o o )^ +- X X —-—r^r^-- cos(oo^ ool) +

2 H = -ni = -n na-lao

(6)

5 дм

N N

X X

Jn (5fm )Jl (5pm )

cos (Oon +Ool )

2 n = -ni = -n 2 + na + lao

Substituting constants (6) into equation (5), the constant takes the following form:

h =

q2 (bo2x+bo2 y)

2^2

4 y2®

N t2

JH (5fm ) ^ -ГТ

n = -N (1 + HCl)

52дм £ £ J2 rs w2 rs 4 (1 + на)2 +(1 +lao )2 —— ^ L Jn (OFM )Jl (Opm )-

n=-Nl=-N

(1 + на )2 -(1 +1d0 )2

Having obtained all necessary constants, let us consider the case when the particle moves in the field of a polarization-modulated electromagnetic wave of circular polarization.

For circular polarization have bo x = bo y = and

2 ц/ 320(1 + Эц) + 16ц62 ц

¥ = mc2 — -1--- 1

16 (1 + Эц) 16 (1 + Эц)

51— £ JH (SfmJ (SPM) + (1 + ncx )2-(1 + la0 )2

4 n = — N l = -N

оя2 n Jh (5fm)J2 (5pm) (1 + nd)2 +(1 +1do)2 + 20дм Ъ ~

n = -N l = -N

(1 + HCL )

(1 + H(X) -(1 + ld0

(7)

where ¥ = s - mc2 is the energy of a particle without considering its rest energy; = q2b2/( y2®2) = 2q21 X2/(nm2c 5).

The next step is to consider the case when the particle moves in the field of a polarization-modulated electromagnetic wave of linear polarization. For linear polarization have box = b, boy = o, and

w =

= mc2 -4 120-

Here

цТ

цМ

160(ц0 + 2 )

2^0 + 4^0 + 4 -72^6 + 4 л/6^0 + 4 V2^6 + 4 V6^0 + 4 / '

(8)

0 = £ /п2 (Sfm )

n=-N 2 (1 + па )

52

АМ

N N

л Z Z /2 (Sfm )/2 (5pm )

4 п=-Nl=-N

(1 + net )2 +(1 + l(X0 )2 (1 + net )2 - (1 +1â0 )2

N 74 X4 N N

T = Z + z Z /4 (Sfm)/4 (5pm)x

n = -N (1 + 4a) 8 4 = -Nl = -N

^ (1 + 4<X )4 + 6 (1 + 4<X )2 (1 + la0 )2 + (1 + la0 )4 (1 + 4(k )2 -(1 + la0 )2

From the obtained formulas (7), (8), we will construct graphs of the dependences of the average kinetic energy of a particle on the radiation intensity and the depth of the amplitude, frequency and polarization modulation of an electromagnetic wave to trace the tendency towards a change in the energy characteristics of particles from these parameters (Figure) (graphs created in Mathcad 15).

W, MeV

MeV

0

4 6

a

Ikz, EW 0

4 6 b

IX2, EW

Energy dependences of a charged particle:

a is on the electromagnetic wave intensity for linear polarization (1), and circular polarization (2); b is on different types of modulations of a circularly (1'-3r) and linearly (1-3) polarized wave (1 — amplitude; 2 — polarization; 3 — frequency)

The figure shows that the energy of a charged particle depends linearly on the wave intensity and quadratically on the amplitude modulation depth, therefore, radiation modulation contributes more to the energy characteristics of the particle than the intensity, which is one of the reasons for the importance of the problem of achieving high modulation depths, as presented in [23].

Thus, one can resort to increasing the intensity or amplitude modulation depth of an electromagnetic wave to obtain high-energy particles; however, one can control the particle energy by manipulating the frequency and polarization characteristics of the accelerating radiation.

Conclusion. Thus, in this work, a detailed analysis of the motion of a charged particle in the field of a polarization-modulated electromagnetic wave was presented. Expressions for the momentum, velocity, coordinates were obtained and then the obtained expressions were averaged over the oscillation period. Expressions for the average kinetic energy for circular and linear polarization were also obtained. The graphs of the dependences of the average kinetic energy on the radiation intensity and the amplitude modulation depth are presented. The energy properties of a charged particle depend on several parameters of radiation, namely on its intensity, the depth of amplitude modulation, the polarization type, etc. Note that in the absence of amplitude and frequency modulation (5 AM = 0, Jn (Sfm ) = 1, a = 0), formulas (7), (8) are transformed to the form of particle motion in the field of a plane monochromatic wave presented in [15].

REFERENCES

[1] Gupta V.K., Kornfield J.A., Ferencz A., et al. Controlling molecular order in "Hairy-Rod" Langmuir — Blodgett films: a polarization-modulation microscopy study. Science, 1994, vol. 265, no. 5174, pp. 940-942.

DOI: https://doi.org/10.1126/science.265.5174.940

[2] Popovsky V.V., Kuzminich I.V. Method of polarization modulation and multiplexing in fiber-optic communication lines. Vestnik nauchnykh konferentsiy [Bulletin of Scientific Conferences], 2016, no. 10-2, pp. 94-100 (in Russ.).

[3] Knyazkov A.V. Polarize-optical light modulation. Nauchno-tekhnicheskie vedomosti SPbGPU. Fiziko-matematicheskie nauki [St. Petersburg State Polytechnical University Journal. Physics and Mathematics], 2013, no. 3, pp. 156-161 (in Russ.).

[4] López-Ortega A., Zapata-Herrera M., Maccaferri N., et al. Enhanced magnetic modulation of light polarization exploiting hybridization with multipolar dark plasmons in magnetoplasmonic nanocavities. Light Sci. Appl., 2020, vol. 9, art. 49.

DOI: https://doi.org/10.1038/s41377-020-0285-0

[5] Lyu Z., Wang C. All-optically phase-induced polarization modulation by means of holographic method. Sci. Rep., 2020, vol. 10, no. 1, art. 5657.

DOI: https://doi.org/10.1038/s41598-020-62549-z

[6] Kovalev A.A., Yakunin V.V., Glazkov Yu.B. Spacial-polarization modulation of laser electromagnetic irradiation. Lazernaya meditsina [Laser Medicine], 2011, vol. 15, no. 2, pp. 120-121 (in Russ.).

[7] Li S., Li G., Ain Q., et al. A laser-plasma accelerator driven by two-color relativistic femtosecond laser pulses. Sci. Adv., 2019, vol. 5, no. 11.

DOI: https://doi.org/10.1126/sciadv.aav7940

[8] Malka V., Fritzlerer S., Lefebvre E., et al. Electron acceleration by a wake field forced by an intense ultrashort laser pulse. Science, 2002, vol. 298, no. 5598, pp. 1596-1600. DOI: https://doi.org/10.1126/science.1076782

[9] Pompili R., Anania M., Bisesto F., et al. Femtosecond dynamics of energetic electrons in high intensity laser-matter interactions. Sci. Rep., 2016, vol. 6, art. 35000.

DOI: https://doi.org/10.1038/srep35000

[10] Orlov A.B., Lutin E.A., Zhelyaeva L.E., et al. Mikropoloskovaya antennaya reshetka s polyarizatsionnoy adaptatsiey [Adaptive-polarization microstrip antenna array]. Patent RU 2156526. Appl. 27.11.1998, publ. 20.09.2000 (in Russ.).

[11] Frenkel Ya.I. Sobranie izbrannykh trudov. T. 1. Elektrodinamika [Collection of selected works. Vol. 1. Electrodynamics]. Moscow, Leningrad, ONTI-GTTI Publ., 1934.

[12] Volkov D.M. Electron in the field of plane polarized electromagnetic waves from the point of view of the Dirac equation. ZhETF, 1937, vol. 7, no. 11, pp. 1286-1289 (in Russ.).

[13] Landau L.D., Lifshitz E.M. The classical theory of fields. Course of theoretical physics. Vol. 2. The classical theory of fields. Butterworth-Heinemann, 1975.

[14] Boldyrev E.M. Motion of a particle in a static magnetic field and the field of a monochromatic electromagnetic plane wave. Tech. Phys., 1999, vol. 44, no. 5, pp. 575-579. DOI: https://doi.org/10.1134/L1259386

[15] Chikhachev A.S., Chulkov V.V. Dependence of the motion of a charged particle on the phase of flight in the field of a powerful electromagnetic wave. J. Commun. Technol. Electron., 2007, vol. 52, no. 12, pp. 1373-1376.

DOI: https://doi.org/10.1134/S106422690712011X

[16] Andreyev S.N., Makarov V.P., Rukhadze A.A. On the motion of a charged particle in a plane monochromatic electromagnetic wave. Quantim Electron., 2009, vol. 39, no. 1, pp. 68-72.

[17] Kopytov G.F., Martynov A.A., Akintsov N.S. A charged particle moves in the field of the plane elliptically polarized electromagnetic wave. Fundamentalnye issledovaniya [Fundamental Research], 2014, no. 9-5, pp. 1013-1018 (in Russ.).

[18] Kopytov G.F., Martynov A. A., Akintsov N.S. Motion of a charged particle in the field of a circularly polarized amplitude-modulated electromagnetic wave. Ekolo-gicheskiy vestnik nauchnykh tsentrov Chernomorskogo ekonomicheskogo sotrudnichestva [Ecological Bulletin of Research Centers of the Black Sea Economic Cooperation], 2014, vol. 11, no. 2, pp. 39-43 (in Russ.).

[19] Akintsov N.S., Kopytov G.F., Martynov A.A., et al. Motion of a charged particle in the field of an arbitrarily polarized amplitude-modulated electromagnetic wave. Fizi-cheskoe obrazova4ie v vuzakh [Physical Education in Universities], 2015, vol. 21, no. S1, pp. 14-15 (in Russ.).

[20] Kopytov G.F., Martynov A.A., Akintsov N.S. The motion of a charged particle in the field by a frequency-modulated electromagnetic wave. Na4osistemy: fizika, khimiya, matematika [Nanosystems: Physics, Chemistry, Mathematics], 2015, vol. 6, no. 5, pp. 661-671. DOI: https://doi.org/10.17586/2220-8054-2015-6-5-661-671

[21] Akintsov N.S. Spectral characteristics of polarization-modulated electromagnetic wave in vacuum. Fizicheskoe obrazova4ie v vuzakh [Physical Education in Universities], 2016, vol. 22, no. S1, pp. 35-37 (in Russ.).

[22] Bateman G., Erdelyia. Higher transcendental functions. McGraw-Hill, 1953.

[23] Sim S., Jang H., Koirala N., et al. Ultra-high modulation depth exceeding 2,400 % in optically controlled topological surface plasmons. Nat. Commu4., 2015, vol. 6, art. 8814. DOI: https://doi.org/10.1038/ncomms9814

Kopytov G.Ph. — Dr. Sc. (Phys.-Math.), Head of Department of Physics, Razumov-sky Moscow State University of Technology and Management (FCU) (ul. Zemlyanoy val 73, Moscow, 109004 Russian Federation).

Kudryavtsev D.I. — Master's Student, Faculty of Physics and Technology, Kuban State University (Stavropolskaya ul. 149, Krasnodar, 350040 Russian Federation).

Brazhko V.A. — Student, Faculty of Physics and Technology, Kuban State University (Stavropolskaya ul. 149, Krasnodar, 350040 Russian Federation).

Please cite this article as:

Kopytov G.Ph., Kudryavtsev D.I., Brazhko V.A. The motion of a charged particle in the electromagnetic field of a polarization-modulated wave. Herald of the Bauma4 Moscow State Techmcal Umversity, Series Natural Scieuces, 2022, no. 6 (105), pp. 106122. DOI: https://doi.org/10.18698/1812-3368-2022-6-106-122