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Absence of net acceleration of charged particles by a focused laser beam in free space
Physics Letters A 184 (1994) 440-444 North-Holland
PHYSICS LETTERS A
Absence of net acceleration of charged particles by a focused laser beam in free space Yu-Kun H o
.,b,~
and Liang Feng b
a CCAST (World Laboratory), P.O. Box 8 730, Beijing, China b T.D. Lee Physics Laboratory, Fudan University, Shanghai 200433, China
Received 20 July 1993; revised manuscript received 30 November 1993; accepted for publication 1 December 1993 Communicatedby B. Fricke
In the Born approximation it is shown that no net energy exchange can occur in a vacuum between a charged particle and a focused laser beam if the interaction length is unlimited.
This work is intended to answer an important question. The problem is whether or not it is possible to obtain a noticeable net energy gain as a charged particle passes through a focused laser beam in a vacuum, if the interaction length is unlimited. Knowledge of this problem is useful in many research areas, such as laser-based accelerators [ 1-12 ], the refraction of electrons by an electromagnetic wave [ 13,14 ], and laser-induced plasma heating [ 15 ]. A laser field may accelerate particles by two different mechanisms. One is to accelerate the particles by the longitudinal component of the laser electric field, which is parallel to the particle movement direction. The feature of this mechanism is that the electron energy gain is proportional to the square root of the intensity of the laser beam. In the other mechanism, the transverse electric field of light gives the particle a transverse velocity, and then the V × B force from the magnetic field causes longitudinal acceleration. It is evident that here the electron energy gain is proportional to the intensity of the laser beam. The latter is often called the ponderomotive force. In this Letter we restrict ourself to the study of the former mechanism. Generally the transverse field components will result in a complex electron motion. For simplicity, we start with examining the configuration shown in fig. 1. Two laser beams with an identical frequency symmetrically converge onto the z-axis at an angle 0o to form a crossing region. A charged particle, which is initially relativistic, is assumed to move right along the z-axis and pass through the crossing region. For simplicity it is assumed that the two waves are uniform in the y direction, linearly polarized with the electric fields lying in the x - z plane and the magnetic field in the y direction. The two waves have the same amplitude and circular frequency Ogo=koc (c is the velocity of light in vacuum). The waists of both beams are at z = 0. Every beam may be Fourier expanded with components all of which are plane waves traveling in different directions. The corresponding vector potentials of the two beams, A~ and A2, are given as [7] Oo+n/2
Ai =Ao
I
~¢(0-- 0o) ( -a2 cos 0+~ sin O) sin(kox sin O + k o z cos O - o g o t + ~Uo)dO,
(1)
O0--n/2
t Mailing address: Physics Department 2, Fudan University, Shanghai 200433, China.
440
0375-9601/94/$ 07.00 © 1994 ElsevierScienceB.V. All rights reserved. SSDI 0375-9601 (93)E095 1-6
Volume 184, number 6
PHYSICSLETTERSA
24 January 1994
X
-"eo~C ~ " --'-
f
J
I-
_i
iOee r interaction
electron
beorn 2
region
Fig. 1. The configuration suggested in ref. [ 6 ] for electron acceleration by two laser beams.
-0o+~/2
A2=Ao
d ( 0 + 0 o ) ( - ~ c o s 0+£ sin 0) sin(koxsinO+kozcosO-coot+~to+q)o) dO,
j
(2)
-0o-~/2
where ~, ~ and ~ denote the unit vectors along the x, y and z directions, respectively, d ( 0 ) is the angular spectrum, which represents the Fourier transform of the transverse spatial distribution of the light beam. ~'o is determined by the initial phase of the injected particle in the field. ~o designates the phase difference of the two waves. Generally, ~t(0) is an even function of 0. The composite fields are given by At =AI +A2 O0+rc/2
=Ao
~
~(0-0o)
0o - x / 2
X { - ~ cos 0 [sin (kox sin 0 +koz cos O-oJot+ ~Uo)+ sin(koz cos O-kox sin O-o~ot+ ~Uo+~o) ] +£ sin 0 [sin (kox sin O+koz cos O-coot+~Uo)- sin(koz cos O-kox sin O-o~ot+~,o +~o) ]} dO.
(3)
In the y-z plane ( x = 0 ) , let ~o=~, the composite fields reduce to
Bt =0,
(4) 0o+~/2 i,
Et=2Eo•
_} d ( 0 - - 0 o ) sinOcos(kozcosO-coot+~o) dO.
(5)
00--~/2
These results indicate that in the y-z plane the transverse electric field and magnetic field vax{ish. A particle moving along the z-axis feels only the longitudinal electric field. Thus this example is especially appropriate to the present study. As we carry out numerical calculations, the following rectangular distribution function is adopted for the angular spectrum, 101 > ~/2kod,
~¢(0) =0,
=kod/X,
101
(6)
where d is a measure of the beam width. Here the electron motion is treated classically, and the dynamic equation of motion for an electron with mass mo and charge qo in an electromagnetic wave field described by the four-vector potential A,, is given by
moc
=qo ~ v f i°,
F,~=O,Aw-O~Am,
(7)
i=1
where vt is the electron velocity, and y-~ = x ~ - v Z / c 2. For the specific field given by eqs. (4) and (5), and 441
Volume 184, n u m b e r 6
PHYSICS LETTERS A
24 January 1994
assuming that the electron is initially moving along the z-axis, eq. (7) may be rewritten as O0+n/2
dfl_2qoEo (1_fl2)3/2 dt
moc
~
~(O-Oo) sinOcos~'dO,
(8)
Oo--n/2
where fl= v/c and
(9)
q/= koz cos O-tOot+~o
is the particle phase relative to the radiative wave. Taking account o f f l = ( 1 / c ) d z / d t and integrating eq. (8) numerically with respect to a specific set of parameters, one can find the instantaneous particle location as a function of time, from which the particle energy versus time or moving distance is obtained. The solid lines in figs. 2 and 3 show the calculated electron energies (7) as a function of the distance traveled along its trajectory. In those calculations, the angular spectrum is eq. (6), the electron initial energy is 7o=316.2 (flo=0.999995), 2Aoqo/moc=0.6, kod= 104, 0o=0.02. In fig. 2 the initial phase is taken to be q/o=~, the electron is mainly accelerated in the crossing region ~ and then gradually loses the gained energy. Figure 3 is for q/o= 0, the electron is first decelerated in the region ~, and then slowly goes back to its initial energy. From that it can be seen that depending on the initial phase the electron may undergo acceleration or deceleration in the region ~. However, in any case the net energy gain is zero, provided we compare the electron initial energy and final energy at locations far away from the interaction zone. N o w we prove the above by analytical calculations. Compared with the above numerical approach, the analytical approach does not refer to any specific parameter value and beam configuration. Consequently, the conclusion reached there is more general. First, it is assumed that as one calculates the phase slippage of a moving particle in the wave, the electron velocity may be taken to be uniform (Born approximation). Then, one gets from eq. (9) that
(10)
~,) =tOo(rio cos O- 1 ) cos ~dt.
d(sin
400
400
350
30O
300
25O
250 ,
-50
-q,F
-25 kz
0 (10 4)
,
,
25
50
Fig. 2. The energies of an electron, moving in the light configuration of fig. 1, versus the distance traveled along its trajectory. In the calculations the following representative parameters were used: 2qoAo/moc=0.6 (the maximum electric field strength in the crossing region is about 10 li V / c m ) , flo=0.999995, /Cod= 104, 0o = 0.02, ~Vo= n, the angular spectrum of the light beam is given by eq. (6). The solid line is obtained by numerically solving the dynamic equation (8), so that it may be viewed as the exact solution. The solid line with squares corresponds to eq. ( 11 ), the Born approximation.
442
200 -50
-25
0
kz (lO4)
25
Fig. 3. The same as in fig. 2, but for q/o= 0.
50
Volume 184, number 6
PHYSICS LETTERSA
24 January 1994
By integrating eq. (8) one finds 0o+7t/2
?fl=
_ - 2qoAo ~ moc
sin 0
Oo - n / 2
~¢(0- 0o) 1 - flo cos 0
sin ~ d 0 + ~ ,
(11)
where :,~ is a constant of motion determined by the initial conditions. We notice that 7> 1, and ),fl= x/y 2 - 1 is a monotone function of 7. By comparing eq. (11 ) with (5), it may be found that the right hand side of eq. (11 ) represents a vector potential of a light field, where the original angular spectrum ~¢(0-0o) is replaced by ~¢'(0) =
~¢(0-0o) 1 - B o cos 0 '
(12)
and ~o, the initial phase of the particle with respect to the field, is changed by ½~. Since d o ( 0 - 0o) is confined in a narrow 0 space around 0o the new field given by d ' ( 0 ) is similar to the field given by eq. (5). It can be seen from eq. ( 11 ) that the electron looks like moving in a potential field. Its energy is determined by the field strength at the instantaneous location. Assume that t = 0 and t = to correspond to the electron initial and final states, i.e. the states before and after transiting the interaction zone, respectively. Since the fields vanish at those two locations, we have 00+~/2
~'fll~°=- 2q°A-----9-° "1 moc
-
80--n/2
sin0 d ( 0 - 0 o ) s i n V l ~ o O d O = O 1 - flo cos 0 "
(13)
Equation (13) demonstrates that there is no net energy exchange between the particle and the beam field. Now we make a straightforward extension of the above study. Let us consider a more general case, where an electron initially moving along the z-axis is accelerated by a single laser beam, described by eq. ( 1 ). I f the electron trajectory is assumed not to be altered by the transverse electric component and the Lorentz force of the laser field when it passes through the focal spot of the laser beam (which might be regarded as a Born approximation [ 7 ] ), the dynamic equation of electron motion can be given as Oo+n/2
dB dt
qoEo ( 1 - f l z ) s/2 moc
j
d ( O - O o ) sin O cos ~.udO
(14)
oo-~/2
which is of a form similar to eq. (8). Evidently a process similar to the above derivation would reproduce eq. (13), which means that no net energy exchange can occur when an electron passes through a focused laser beam if only the longitudinal electric field is taken into account. Furthermore, in the rest frame of a particle which transits a laser beam, the laser field experienced by the particle looks like a laser pulse. Thus we reach the interesting result that electrons should not be accelerated by the longitudinal electric field of a laser pulse in vacuum. This conclusion is important, since in practice the laser beam is finite in time [ 16 ]. A few more comments may be made concerning the laser acceleration scheme of fig. 1. The result obtained in ref. [ 6 ] is not correct, because the study of ref. [ 6 ] was based on an artificial field rather than a Maxwell field. The author incorrectly assumed the field for each beam to be a plane wave, infinite in extent in the propagation direction, but sharply delimited in intensity in the transverse direction. Thus an electron can only be accelerated (or decelerated) when it passes through the whole focused laser beam. Obviously, this assumed field does not satisfy the Maxwell equation. In fact, a radiation field with a finite width must take the pattern of eq. ( 1 ), and in intensity it should spread over a large region in the transverse direction. Because of phase slippage, an electron passing through this laser beam would undergo alternately acceleration and deceleration. Although the energy absorbed by the electron in a selected acceleration phase may not be equal to that lost during the successive deceleration phase, no net acceleration can occur if the interaction length is unlimited. In summary, in terms of both the numerical and analytical calculations it has been proved that the coupling 443
Volume 184, number 6
PHYSICS LETTERS A
24 January 1994
o f focused laser b e a m s to free charged p a n i c l e s in a v a c u u m c a n n o t give rise to a net acceleration o v e r a n u n l i m i t e d i n t e r a c t i o n length. I n those c a l c u l a t i o n s the B o r n a p p r o x i m a t i o n is a d o p t e d a n d o n l y the l o n g i t u d i n a l electric field has b e e n t a k e n i n t o a c c o u n t . T h i s c o m p o n e n t is parallel to the d i r e c t i o n o f m o v e m e n t o f the particle. I f the p a n i c l e b e a m o b l i q u e l y intersects the focused laser b e a m , the energy o f the particle before e n t e r i n g the i n t e r a c t i o n r e g i o n will be e q u a l to the energy o f the p a n i c l e after t r a n s i t i n g the region. I n the case that the p a n i c l e c o - m o v e s w i t h the laser b e a m , the p a n i c l e energy d e p e n d s s i n u s o i d a l l y o n the phase, a n d the p a n i c l e will u n d e r g o alternately acceleration a n d deceleration. E x t e n d i n g the p r e s e n t study b y t a k i n g a c c o u n t o f the acceleration m e c h a n i s m o f the p o n d e r o m o t i v e p o t e n t i a l [ 1 l, 12 ] will be o u r next research object. O n e o f us ( Y . H . ) is i n d e b t e d to Professor M. Scully for e n l i g h t e n i n g discussions a n d for d r a w i n g his att e n t i o n to this i n t e r e s t i n g p r o b l e m .
References [ 1 ] T.C. Katsouless, SPIE Vol. 664. High intensity laser processes, 2 ( 1986 ), and references therein. [2] M.O. Scully, Appl. Phys. B 51 (1990) 238. [ 3 ] A.A. Chernikov, G. Sehmidt and A.I. Neishtadt, Phys. Rev. Lett. 68 (1992) 1507. [ 4 ] M.S. Hussein and M.P. Pato, Phys. Rev. Lett. 68 ( 1992 ) 1136. [ 5 ] L. Cicchitelli, H. Hora and R. Postle, Phys. Rev. A 41 (1990) 3727. [ 6 ] S. Sakai, Japan. J. Appl. Phys. 30 ( 1991 ) L 1887. [ 7 ] J.A. Edighoffer and R.H. Pantell, J. Appl. Phys. 50 ( 1979 ) 6120. [ 8 ] A. Loeb and L. Friedland, Phys. Rev. A 33 (1986) 1828. [9 ] Y. Nishida, N. Yugami, H. Onihashi, T. Taura and K. Otsuka, Phys. Rev. Lett. 66 ( 1991 ) 1854. [ 10] S. Kawata, T. Maruyama, H. Watanabe and I. Takahashi, Phys. Rev. Lett. 66 ( 1991 ) 2072. [ 11 ] I. Wernick and T.C. Marshall, Phys. Rev. A 46 (1992) 3566. [ 12] E.J. Sternbach and H. Ghalila; Nucl. Instrum. Methods A 304 ( 1991 ) 691. [ 13] K.W.B. Kibble, Phys. Rev. Lett. 16 (1966) 1054; Phys. Rev. 150 (1966) 1060. [ 14 ] P.H. Bucksbaum, M. Bashkansky and T.J. McIlrath, Phys. Rev. Lett. 59 ( 1987 ) 349. [ 15 ] B.M. Penetrante and J.N. Bardsley, Phys. Rev. A 43 ( 1991 ) 3100. [ 16 ] P.R. Freeman and P.H. Bucksbaum, J. Phys. B 24 ( 1991 ) 325.
444
PHYSICS LETTERS A
Absence of net acceleration of charged particles by a focused laser beam in free space Yu-Kun H o
.,b,~
and Liang Feng b
a CCAST (World Laboratory), P.O. Box 8 730, Beijing, China b T.D. Lee Physics Laboratory, Fudan University, Shanghai 200433, China
Received 20 July 1993; revised manuscript received 30 November 1993; accepted for publication 1 December 1993 Communicatedby B. Fricke
In the Born approximation it is shown that no net energy exchange can occur in a vacuum between a charged particle and a focused laser beam if the interaction length is unlimited.
This work is intended to answer an important question. The problem is whether or not it is possible to obtain a noticeable net energy gain as a charged particle passes through a focused laser beam in a vacuum, if the interaction length is unlimited. Knowledge of this problem is useful in many research areas, such as laser-based accelerators [ 1-12 ], the refraction of electrons by an electromagnetic wave [ 13,14 ], and laser-induced plasma heating [ 15 ]. A laser field may accelerate particles by two different mechanisms. One is to accelerate the particles by the longitudinal component of the laser electric field, which is parallel to the particle movement direction. The feature of this mechanism is that the electron energy gain is proportional to the square root of the intensity of the laser beam. In the other mechanism, the transverse electric field of light gives the particle a transverse velocity, and then the V × B force from the magnetic field causes longitudinal acceleration. It is evident that here the electron energy gain is proportional to the intensity of the laser beam. The latter is often called the ponderomotive force. In this Letter we restrict ourself to the study of the former mechanism. Generally the transverse field components will result in a complex electron motion. For simplicity, we start with examining the configuration shown in fig. 1. Two laser beams with an identical frequency symmetrically converge onto the z-axis at an angle 0o to form a crossing region. A charged particle, which is initially relativistic, is assumed to move right along the z-axis and pass through the crossing region. For simplicity it is assumed that the two waves are uniform in the y direction, linearly polarized with the electric fields lying in the x - z plane and the magnetic field in the y direction. The two waves have the same amplitude and circular frequency Ogo=koc (c is the velocity of light in vacuum). The waists of both beams are at z = 0. Every beam may be Fourier expanded with components all of which are plane waves traveling in different directions. The corresponding vector potentials of the two beams, A~ and A2, are given as [7] Oo+n/2
Ai =Ao
I
~¢(0-- 0o) ( -a2 cos 0+~ sin O) sin(kox sin O + k o z cos O - o g o t + ~Uo)dO,
(1)
O0--n/2
t Mailing address: Physics Department 2, Fudan University, Shanghai 200433, China.
440
0375-9601/94/$ 07.00 © 1994 ElsevierScienceB.V. All rights reserved. SSDI 0375-9601 (93)E095 1-6
Volume 184, number 6
PHYSICSLETTERSA
24 January 1994
X
-"eo~C ~ " --'-
f
J
I-
_i
iOee r interaction
electron
beorn 2
region
Fig. 1. The configuration suggested in ref. [ 6 ] for electron acceleration by two laser beams.
-0o+~/2
A2=Ao
d ( 0 + 0 o ) ( - ~ c o s 0+£ sin 0) sin(koxsinO+kozcosO-coot+~to+q)o) dO,
j
(2)
-0o-~/2
where ~, ~ and ~ denote the unit vectors along the x, y and z directions, respectively, d ( 0 ) is the angular spectrum, which represents the Fourier transform of the transverse spatial distribution of the light beam. ~'o is determined by the initial phase of the injected particle in the field. ~o designates the phase difference of the two waves. Generally, ~t(0) is an even function of 0. The composite fields are given by At =AI +A2 O0+rc/2
=Ao
~
~(0-0o)
0o - x / 2
X { - ~ cos 0 [sin (kox sin 0 +koz cos O-oJot+ ~Uo)+ sin(koz cos O-kox sin O-o~ot+ ~Uo+~o) ] +£ sin 0 [sin (kox sin O+koz cos O-coot+~Uo)- sin(koz cos O-kox sin O-o~ot+~,o +~o) ]} dO.
(3)
In the y-z plane ( x = 0 ) , let ~o=~, the composite fields reduce to
Bt =0,
(4) 0o+~/2 i,
Et=2Eo•
_} d ( 0 - - 0 o ) sinOcos(kozcosO-coot+~o) dO.
(5)
00--~/2
These results indicate that in the y-z plane the transverse electric field and magnetic field vax{ish. A particle moving along the z-axis feels only the longitudinal electric field. Thus this example is especially appropriate to the present study. As we carry out numerical calculations, the following rectangular distribution function is adopted for the angular spectrum, 101 > ~/2kod,
~¢(0) =0,
=kod/X,
101
(6)
where d is a measure of the beam width. Here the electron motion is treated classically, and the dynamic equation of motion for an electron with mass mo and charge qo in an electromagnetic wave field described by the four-vector potential A,, is given by
moc
=qo ~ v f i°,
F,~=O,Aw-O~Am,
(7)
i=1
where vt is the electron velocity, and y-~ = x ~ - v Z / c 2. For the specific field given by eqs. (4) and (5), and 441
Volume 184, n u m b e r 6
PHYSICS LETTERS A
24 January 1994
assuming that the electron is initially moving along the z-axis, eq. (7) may be rewritten as O0+n/2
dfl_2qoEo (1_fl2)3/2 dt
moc
~
~(O-Oo) sinOcos~'dO,
(8)
Oo--n/2
where fl= v/c and
(9)
q/= koz cos O-tOot+~o
is the particle phase relative to the radiative wave. Taking account o f f l = ( 1 / c ) d z / d t and integrating eq. (8) numerically with respect to a specific set of parameters, one can find the instantaneous particle location as a function of time, from which the particle energy versus time or moving distance is obtained. The solid lines in figs. 2 and 3 show the calculated electron energies (7) as a function of the distance traveled along its trajectory. In those calculations, the angular spectrum is eq. (6), the electron initial energy is 7o=316.2 (flo=0.999995), 2Aoqo/moc=0.6, kod= 104, 0o=0.02. In fig. 2 the initial phase is taken to be q/o=~, the electron is mainly accelerated in the crossing region ~ and then gradually loses the gained energy. Figure 3 is for q/o= 0, the electron is first decelerated in the region ~, and then slowly goes back to its initial energy. From that it can be seen that depending on the initial phase the electron may undergo acceleration or deceleration in the region ~. However, in any case the net energy gain is zero, provided we compare the electron initial energy and final energy at locations far away from the interaction zone. N o w we prove the above by analytical calculations. Compared with the above numerical approach, the analytical approach does not refer to any specific parameter value and beam configuration. Consequently, the conclusion reached there is more general. First, it is assumed that as one calculates the phase slippage of a moving particle in the wave, the electron velocity may be taken to be uniform (Born approximation). Then, one gets from eq. (9) that
(10)
~,) =tOo(rio cos O- 1 ) cos ~dt.
d(sin
400
400
350
30O
300
25O
250 ,
-50
-q,F
-25 kz
0 (10 4)
,
,
25
50
Fig. 2. The energies of an electron, moving in the light configuration of fig. 1, versus the distance traveled along its trajectory. In the calculations the following representative parameters were used: 2qoAo/moc=0.6 (the maximum electric field strength in the crossing region is about 10 li V / c m ) , flo=0.999995, /Cod= 104, 0o = 0.02, ~Vo= n, the angular spectrum of the light beam is given by eq. (6). The solid line is obtained by numerically solving the dynamic equation (8), so that it may be viewed as the exact solution. The solid line with squares corresponds to eq. ( 11 ), the Born approximation.
442
200 -50
-25
0
kz (lO4)
25
Fig. 3. The same as in fig. 2, but for q/o= 0.
50
Volume 184, number 6
PHYSICS LETTERSA
24 January 1994
By integrating eq. (8) one finds 0o+7t/2
?fl=
_ - 2qoAo ~ moc
sin 0
Oo - n / 2
~¢(0- 0o) 1 - flo cos 0
sin ~ d 0 + ~ ,
(11)
where :,~ is a constant of motion determined by the initial conditions. We notice that 7> 1, and ),fl= x/y 2 - 1 is a monotone function of 7. By comparing eq. (11 ) with (5), it may be found that the right hand side of eq. (11 ) represents a vector potential of a light field, where the original angular spectrum ~¢(0-0o) is replaced by ~¢'(0) =
~¢(0-0o) 1 - B o cos 0 '
(12)
and ~o, the initial phase of the particle with respect to the field, is changed by ½~. Since d o ( 0 - 0o) is confined in a narrow 0 space around 0o the new field given by d ' ( 0 ) is similar to the field given by eq. (5). It can be seen from eq. ( 11 ) that the electron looks like moving in a potential field. Its energy is determined by the field strength at the instantaneous location. Assume that t = 0 and t = to correspond to the electron initial and final states, i.e. the states before and after transiting the interaction zone, respectively. Since the fields vanish at those two locations, we have 00+~/2
~'fll~°=- 2q°A-----9-° "1 moc
-
80--n/2
sin0 d ( 0 - 0 o ) s i n V l ~ o O d O = O 1 - flo cos 0 "
(13)
Equation (13) demonstrates that there is no net energy exchange between the particle and the beam field. Now we make a straightforward extension of the above study. Let us consider a more general case, where an electron initially moving along the z-axis is accelerated by a single laser beam, described by eq. ( 1 ). I f the electron trajectory is assumed not to be altered by the transverse electric component and the Lorentz force of the laser field when it passes through the focal spot of the laser beam (which might be regarded as a Born approximation [ 7 ] ), the dynamic equation of electron motion can be given as Oo+n/2
dB dt
qoEo ( 1 - f l z ) s/2 moc
j
d ( O - O o ) sin O cos ~.udO
(14)
oo-~/2
which is of a form similar to eq. (8). Evidently a process similar to the above derivation would reproduce eq. (13), which means that no net energy exchange can occur when an electron passes through a focused laser beam if only the longitudinal electric field is taken into account. Furthermore, in the rest frame of a particle which transits a laser beam, the laser field experienced by the particle looks like a laser pulse. Thus we reach the interesting result that electrons should not be accelerated by the longitudinal electric field of a laser pulse in vacuum. This conclusion is important, since in practice the laser beam is finite in time [ 16 ]. A few more comments may be made concerning the laser acceleration scheme of fig. 1. The result obtained in ref. [ 6 ] is not correct, because the study of ref. [ 6 ] was based on an artificial field rather than a Maxwell field. The author incorrectly assumed the field for each beam to be a plane wave, infinite in extent in the propagation direction, but sharply delimited in intensity in the transverse direction. Thus an electron can only be accelerated (or decelerated) when it passes through the whole focused laser beam. Obviously, this assumed field does not satisfy the Maxwell equation. In fact, a radiation field with a finite width must take the pattern of eq. ( 1 ), and in intensity it should spread over a large region in the transverse direction. Because of phase slippage, an electron passing through this laser beam would undergo alternately acceleration and deceleration. Although the energy absorbed by the electron in a selected acceleration phase may not be equal to that lost during the successive deceleration phase, no net acceleration can occur if the interaction length is unlimited. In summary, in terms of both the numerical and analytical calculations it has been proved that the coupling 443
Volume 184, number 6
PHYSICS LETTERS A
24 January 1994
o f focused laser b e a m s to free charged p a n i c l e s in a v a c u u m c a n n o t give rise to a net acceleration o v e r a n u n l i m i t e d i n t e r a c t i o n length. I n those c a l c u l a t i o n s the B o r n a p p r o x i m a t i o n is a d o p t e d a n d o n l y the l o n g i t u d i n a l electric field has b e e n t a k e n i n t o a c c o u n t . T h i s c o m p o n e n t is parallel to the d i r e c t i o n o f m o v e m e n t o f the particle. I f the p a n i c l e b e a m o b l i q u e l y intersects the focused laser b e a m , the energy o f the particle before e n t e r i n g the i n t e r a c t i o n r e g i o n will be e q u a l to the energy o f the p a n i c l e after t r a n s i t i n g the region. I n the case that the p a n i c l e c o - m o v e s w i t h the laser b e a m , the p a n i c l e energy d e p e n d s s i n u s o i d a l l y o n the phase, a n d the p a n i c l e will u n d e r g o alternately acceleration a n d deceleration. E x t e n d i n g the p r e s e n t study b y t a k i n g a c c o u n t o f the acceleration m e c h a n i s m o f the p o n d e r o m o t i v e p o t e n t i a l [ 1 l, 12 ] will be o u r next research object. O n e o f us ( Y . H . ) is i n d e b t e d to Professor M. Scully for e n l i g h t e n i n g discussions a n d for d r a w i n g his att e n t i o n to this i n t e r e s t i n g p r o b l e m .
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