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Two-dimensional solutions of the Dirac equation for the motion of an electron in a helical magnetic field
Volume 126, number 1
PHYSICS LETTERS A
14 December 1987
T W O - D I M E N S I O N A L S O L U T I O N S OF T H E DIRAC EQUATION FOR T H E M O T I O N OF AN E L E C T R O N IN A H E L I C A L M A G N E T I C FIELD Selguk SARITEPE and N.V.V.J. SWAMY Physics Department, Oklahoma State University, Stillwater, OK 74078, USA Received 28 July 1987; revised manuscript received 14 October 1987; accepted for publication 19 October 1987 Communicated by R.C. Davidson
Q u a n t u m theory o f a free electron in a helical wiggler magnetic field has been studied. In the previous studies the transverse m o m e n t u m operators in the Dirac hamiltonian were ignored. In this note, the 2-D solutions of the Dirac equation, based on a hamiltonian which includes one of the transverse m o m e n t u m operators (p, ¢ 0,/2v = 0) and the wiggler field as an external potential is derived. It is shown that there are solutions in terms of the Mathieu functions of fractional order.
I. Introduction
The helical wiggler magnetic field structure is used in the free electron laser (FEL) to generate synchrotron radiation with a narrow spectral bandwidth and to suppress the emission of harmonics on the axis. In the classical description of the FEL, the electrons are injected into the wiggler at relativistic speeds and radiate due to the wiggling motion. This spontaneous synchrotron radiation is amplified via stimulated emission and the coherence is established when the signal grows in intensity and the eletrons are bunched at the optical wavelengths due to the beating of the signal and the wiggler fields. In the classical analyses of the FEL interaction, the equations of motion for the relativistic electrons in a helical wiggler play an essential role and they have been studied exhaustively in the literature [ 1 ]. In the quantum theory of the FEL, one needs a wavefunction in place of the classical equation of motion. When the amplitude of the radiation field is small (small-signal regime) the radiation field can be treated as a perturbation. Therefore, one needs the eigenfunctions of a relativistic quantum mechanical hamiltonian which includes the helical wiggler field as an external potential since it cannot be treated as a perturbation. In the laboratory frame, one has the choice of using either the Dirac equation or the Klein-Gordon equation. The correct theory would have to use the Dirac equation in order to take the spin of the electron into account. Therefore, the solutions of the Dirac equation for the motion of an electron in a helical wiggler field is the starting point of a correct quantum theory of the FEL. In the previous attempts on this problem, the transverse m o m e n t u m operators in the Dirac hamiltonian were ignored. Becket and Mitter [ 2 ] gave the exact l-D solutions (/?x = 0, py = 0) for the case of a uniform wiggler. Later, 1-D exact solutions for the case of a tapered wiggler were derived [ 3 ]. In this Letter we shall derive the 2-D solutions using the hamiltonian which includes one of the transverse m o m e n t u m operators (t?x # 0, ~v = 0 ) .
2. 2-D solutions
We write the Dirac equation (time independent) in the direct product notation ~ = P l ~)o t Research supported by College of Arts and Sciences, Oklahoma State University.
28
0375-9601/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 126, number 1
PHYSICS LETTERS A
(cpl o'P+p3 mo ¢2 - E ) 7t = 0 .
14 December 1987
(1)
Here P is the kinetic momentum P = p - (e/c)d where
A=a cos(k~z) a~+a sin(k~z) 37
(2)
is the vector potential of the wiggler on the axis and kw= 27t/2w is the wavenumber of the wiggler. The hamiltonian (eq. (1)) commutes with the/?x and/?y operators. This forces the condition that the solutions must be the simultaneous eigenfunctions of/?x, ~ and the hamiltonian. Therefore we try the spinor forms
(z)l f3(z)|exp[ih-'(P.xx+P,,,Y)] ,
N (z)l |g3(z)lexp[ih-l(p2xx+ozyY)l ,
f4(z)d
Lg4(z)/
(3/
where PL,-,P~:.,Pzx and Pg.vare c-numbers. Operating on these spinors with the Dirac hamiltonian, one obtains two sets of operator equations for the first and second solutions respectively. Eliminatingfz (z), f3 (z) and f4 (z) from the first set of equations we obtain {c2~ 2 -eat|p._ exp(ikwz) +p,+ exp( -ikwz)] -2hck~cp:+q 2 + (hckw) z ) ×{c2~ 2 -eat|p._ exp(ik~z) +p~+ exp( -ikwz)] +q~}f~(z) = (eahck~)Zf~(z),
(4)
where
q~-mZc4-E2+e2a2+c2(p2x+p2v),
P,-=PL~--iPl.v, P,+--Plx+iP,.v.
By eliminating g~ (z), g3(z) and g4(z) from the second set of equations a similar operator equation is obtained for g2(z), the only difference being the replacement off~ (z) in eq. (4) by g2(z). At this stage we change the variable kwz=2~ ;
(5)
then the eq. (4) becomes (pL~#0, p~:,=O) [(d~2 -4i~-2q
cos 2 ~ - (22 + 4 ) ) ( ~
- 2q cos 2 ~ - 2 2 ) - 2 3 ]f~ (~) = 0 ,
(6)
where
41elacpL,. q-
(~/Ckw)2 '
4q2 ~2 --~ ( ~ c k w ) 2 ,
( 4ea ~2 A3~--~-C-~. ]
and ¢ is a dimensionless variable. Let f~ (~) = ce~(~, q) + r se~(~, q ) ,
(7)
where ce~ and se~ are the Mathieu functions (even and odd respectively) of fractional order v and r is an unknown constant to be determined. Since u is not an integer in general, the Mathieu functions ce~ and se~ are linearly independent therefore even and odd Mathieu functions are needed to represent the most general solution. The defining equation [4] for the Mathieu functions is -2qcos2~+a o se(~,q)=O,
(8)
where 29
Volume 126, number 1
1 aq=V~-+2(v2-1)
PHYSICS LETTERS A
q2+
5V2+7 p2 32(v2-1)3(
4)
q4 +""
14 December 1987
(9)
with the following condition qZ
(10)
<
2(v2--1)
This c o n d i t i o n is satisfied in the case o f F E L as we shall see shortly when we relate v to the longitudinal mom e n t u m and q to the transverse m o m e n t u m . Since the initial m o m e n t u m c o m p o n e n t s are the design parameters (hence v a n d q are given n u m b e r s ) , aq will be treated as a constant. Therefore
- 2q cos 2 ~ - 22 (ce. + r se.) = - (aq + 22 ) (ce. + r se.) .
( 1 1)
Using this result eq. (6) reduces to [2(2+4)-23](ce.+rse.)+4i2
dse. r d~
dce.]= d~ ]
0
'
(12)
where 2 = au+22. F o r eq. (12) to be equal to zero, considering the evenness and the oddness o f the M a t h i e u functions, we must have [ 2 ( 2 + 4 ) - 2 3 ] ce~ - 4 i 2 r F = 0 ,
(13)
[ 2 ( 2 + 4 ) - 2 3 ] r se~ - 4 i 2 G = 0 .
(14)
Here we d e n o t e d F=d se./d~,
G=d ce./d(.
(15)
We obtain the same conditions (eqs. (13 ) and (14)) from the second set o f equations by letting g2 (~) = ce~ + r se.. The next step is to solve these equations simultaneously for r a n d 2. We obtain X r= - 4i2F/ce~
(16)
where X=2(2+4)
-23,
(17)
X 2 + 1622 ( F / e e . ) ( G / s e . ) = 0 .
(18)
Eq. (18) is the most crucial equation towards the solution o f the Dirac equation. This is a d e t e r m i n i n g equation for v since X, 2, F, G, se. and ce. are all functions o f v. I f we could d e t e r m i n e v exactly from this equation we would have exact solutions o f the Dirac equation for an electron in a wiggler (t?x ~ 0,/?v = 0). But the M a t h i e u functions eel, se~ and their derivatives F a n d G are only known as power series [4] therefore explicit exact solutions do not exist. Exact solutions exist only in the sense that v can be expressed in terms o f M a t h i e u functions algebraicly. Recalling the c o n d i t i o n (10) a n d e x a m i n i n g the expansions o f M a t h i e u functions, we ~dn easily observe that F/cev~v
,
G/se, ~ - v.
Using these results we o b t a i n 30
(19) (20)
Volume 126,number 1
PHYSICS LETTERSA
14 December 1987
r~i.
(21)
The 2-D solutions are required to be consistent with the I-D solutions derived in refs. [3] and [2], i.e., when q = 0 the 2-D solutions must go over into the I-D solutions. When q=0, Mathieu functions reduce to ce~(~, q = 0 ) = c o s v~,
(22)
se~(~, q = 0 ) =sin v~,
(23)
and fl (~) becomes f~ (~) Iq=o =ce~ + r se~ =cos v~ + i sin z,~= exp(iv~) .
(24)
This must be equal to the exponential in the 1-D solution exp(iv~) - exp[iv(kwz/2)] =,exp (ih -~pl z ) . This leads us to the identity (25)
~,=-2p,/hk~ .
Here p] is the effective longitudinal momentum of the electron (up-spin, spin parallel to the direction of motion) inside the wiggler. Using eqs. (19) and (20), eq. (18) becomes [2(2+4) -23] 2 - 1622u 2 = 0 .
(26)
Further expanding, using eq. (9) and 2=aq+22 we obtain 24 --823 "~-(16-223)22 +8(222
--23)2+22 = 0 .
(27)
The solution of this equation leads us to cp~ ~ - ½hck~ + [ c2(po_- + ½hkw) 2 - e Z a 2 +c2pg± -c2p2x] ,/2.
(28)
This is the determining equation for the effective longitudinal momentum in terms of the design parameters "a", kw, the initial momentum components Po~, Pol; and the transverse momentum correction P~x which is going to be determined from a condition given by the normalization procedure. The determination ofp~ a n d p ~ in terms of the given quantities completes the solution of the Dirac equation. Once we know fj (z) then f2 (z), f3 (z) and f4 (z) can be determined from the first set of operator equations we mentioned in the beginning of this section. The 2-D up-spin solution of the Dirac equation for an electron in a wiggler, satisfying all the conditions (commutes with/?~ and/?y, reduces to the I-D solution when p~x=0) will be
~r/t = N l
K1 exp(ik~z) [Cplx e x p ( i k ~ z ) - e a l K j +cp, E+moC 2
[cPL~ e x p ( - i k ~ z ) - e a ] - c ( p l E + moc z
where K~ =
+hkw)Kl exp(ik~z)
1
(ce~+i se~) exp(ih-lp~.,x) ,
(29)
c~p~ +tl2 + ½e2a2(pL~/p ~)2 eahck,,,
The normalization constants N] and P~x are determined from the normalization condition ~T ~ = 1, where ~T is the Dirac conjugate of the solution 7tr. We find 31
Volume 126, number 1
PHYSICS LETTERS A
14 December 1987
N 1~ 1/,~,
(30)
c2 p~ , ~., 7( moc2)( hck.~ )( y / K + t ) ,
(31)
where 7 is the relativistic factor of the electron and K = - ea/moc 2 is the wiggler strength. Similarly the down-spin solution will be K2 e x p ( - i k w z ) 1 [ cp2, exp(ikwz) - e a ] K 2 + ( cp2 - h c k ~ ) exp(-ikwz) E + moc 2
~=N2
( c e . + i se.) e x p ( i h - ~ p 2 x x )
,
(32)
[ cP2,- e x p ( - i k , ~ z ) - e a ] - c P 2 K 2 E + moc 2
where K2 = -
c2p~ +rl~_ + ~e2a2(p2,/p2)2 eahcl~ '
rl~-m2c4-E2
cp2 "~ ½hck,, + [c2(Po_- - ~ 1 k w ) -" _ e Z a 2 + c 2 p g l - c 2 p L ]
N,~
1/x/~,
+ e 2 a 2 +c2p2, , ''2 ,
c2p~,,,~y(moc2)(hckw)(7/K-1).
3. Conclusions
The solutions presented in this paper can form the basis for a perturbation theory of the free electron laser operating in the small-signal regime. Such studies [2,3] using the 1-D solutions have shown that one obtains the same results (apart from small quantum mechanical corrections) as linear or quasi-linear analyses based on classical mechanics. 2-D solutions would change this conclusion slightly in the favor of quantum mechanical analysis. It is obvious from the results of this paper that the quantum mechanical correction Plx (or P2~-) to the transverse m o m e n t u m ea/c beomes more significant when the electrons are highly relativistic (7 > 103). Also, the rate of spin-flipping processes should increase due to the effect of an oscillating transverse m o m e n t u m component. Therefore, one expects the purely quantum mechanical spin and transverse m o m e n t u m effects to become more important for the short wavelength FELs which use highly relativistic electron beams (1 GeV). Such an analysis has been completed and the results will be published in a forthcoming paper [ 5 ]. More importantly, the 2-D Dirac solutions derived in this paper could provide guidance in the development of a Dirac theory of the free electron laser in a helical wiggler field including transverse momentum, based on a hamiltonian which includes the radiation field ab initio. Such a result would represent substantial progress in the quantum theory of the free electron lasers with applications to experimental problems not accessible by classical analyses.
References [ 1 ] B.M. Kincaid, J. Appl. Phys. 48 (1977) 2684; J.P. Blewet and R. Chasman, J. Appl. Phys. 48 (1977) 2692. [2] W. Becket and H. Mitter, Z. Phys. B 35 (1979) 399. [3] S. Saritepe and N.V.V.J. Swamy, Phys. Lett. A 113 (1985) 69. [4] N.W. MaeLaehlan, Theory and application of Mathieu functions (Dover, New York, 1964). [ 5 ] S. Saritepe and N.V.V.J. Swamy, to be published.
32
PHYSICS LETTERS A
14 December 1987
T W O - D I M E N S I O N A L S O L U T I O N S OF T H E DIRAC EQUATION FOR T H E M O T I O N OF AN E L E C T R O N IN A H E L I C A L M A G N E T I C FIELD Selguk SARITEPE and N.V.V.J. SWAMY Physics Department, Oklahoma State University, Stillwater, OK 74078, USA Received 28 July 1987; revised manuscript received 14 October 1987; accepted for publication 19 October 1987 Communicated by R.C. Davidson
Q u a n t u m theory o f a free electron in a helical wiggler magnetic field has been studied. In the previous studies the transverse m o m e n t u m operators in the Dirac hamiltonian were ignored. In this note, the 2-D solutions of the Dirac equation, based on a hamiltonian which includes one of the transverse m o m e n t u m operators (p, ¢ 0,/2v = 0) and the wiggler field as an external potential is derived. It is shown that there are solutions in terms of the Mathieu functions of fractional order.
I. Introduction
The helical wiggler magnetic field structure is used in the free electron laser (FEL) to generate synchrotron radiation with a narrow spectral bandwidth and to suppress the emission of harmonics on the axis. In the classical description of the FEL, the electrons are injected into the wiggler at relativistic speeds and radiate due to the wiggling motion. This spontaneous synchrotron radiation is amplified via stimulated emission and the coherence is established when the signal grows in intensity and the eletrons are bunched at the optical wavelengths due to the beating of the signal and the wiggler fields. In the classical analyses of the FEL interaction, the equations of motion for the relativistic electrons in a helical wiggler play an essential role and they have been studied exhaustively in the literature [ 1 ]. In the quantum theory of the FEL, one needs a wavefunction in place of the classical equation of motion. When the amplitude of the radiation field is small (small-signal regime) the radiation field can be treated as a perturbation. Therefore, one needs the eigenfunctions of a relativistic quantum mechanical hamiltonian which includes the helical wiggler field as an external potential since it cannot be treated as a perturbation. In the laboratory frame, one has the choice of using either the Dirac equation or the Klein-Gordon equation. The correct theory would have to use the Dirac equation in order to take the spin of the electron into account. Therefore, the solutions of the Dirac equation for the motion of an electron in a helical wiggler field is the starting point of a correct quantum theory of the FEL. In the previous attempts on this problem, the transverse m o m e n t u m operators in the Dirac hamiltonian were ignored. Becket and Mitter [ 2 ] gave the exact l-D solutions (/?x = 0, py = 0) for the case of a uniform wiggler. Later, 1-D exact solutions for the case of a tapered wiggler were derived [ 3 ]. In this Letter we shall derive the 2-D solutions using the hamiltonian which includes one of the transverse m o m e n t u m operators (t?x # 0, ~v = 0 ) .
2. 2-D solutions
We write the Dirac equation (time independent) in the direct product notation ~ = P l ~)o t Research supported by College of Arts and Sciences, Oklahoma State University.
28
0375-9601/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 126, number 1
PHYSICS LETTERS A
(cpl o'P+p3 mo ¢2 - E ) 7t = 0 .
14 December 1987
(1)
Here P is the kinetic momentum P = p - (e/c)d where
A=a cos(k~z) a~+a sin(k~z) 37
(2)
is the vector potential of the wiggler on the axis and kw= 27t/2w is the wavenumber of the wiggler. The hamiltonian (eq. (1)) commutes with the/?x and/?y operators. This forces the condition that the solutions must be the simultaneous eigenfunctions of/?x, ~ and the hamiltonian. Therefore we try the spinor forms
(z)l f3(z)|exp[ih-'(P.xx+P,,,Y)] ,
N (z)l |g3(z)lexp[ih-l(p2xx+ozyY)l ,
f4(z)d
Lg4(z)/
(3/
where PL,-,P~:.,Pzx and Pg.vare c-numbers. Operating on these spinors with the Dirac hamiltonian, one obtains two sets of operator equations for the first and second solutions respectively. Eliminatingfz (z), f3 (z) and f4 (z) from the first set of equations we obtain {c2~ 2 -eat|p._ exp(ikwz) +p,+ exp( -ikwz)] -2hck~cp:+q 2 + (hckw) z ) ×{c2~ 2 -eat|p._ exp(ik~z) +p~+ exp( -ikwz)] +q~}f~(z) = (eahck~)Zf~(z),
(4)
where
q~-mZc4-E2+e2a2+c2(p2x+p2v),
P,-=PL~--iPl.v, P,+--Plx+iP,.v.
By eliminating g~ (z), g3(z) and g4(z) from the second set of equations a similar operator equation is obtained for g2(z), the only difference being the replacement off~ (z) in eq. (4) by g2(z). At this stage we change the variable kwz=2~ ;
(5)
then the eq. (4) becomes (pL~#0, p~:,=O) [(d~2 -4i~-2q
cos 2 ~ - (22 + 4 ) ) ( ~
- 2q cos 2 ~ - 2 2 ) - 2 3 ]f~ (~) = 0 ,
(6)
where
41elacpL,. q-
(~/Ckw)2 '
4q2 ~2 --~ ( ~ c k w ) 2 ,
( 4ea ~2 A3~--~-C-~. ]
and ¢ is a dimensionless variable. Let f~ (~) = ce~(~, q) + r se~(~, q ) ,
(7)
where ce~ and se~ are the Mathieu functions (even and odd respectively) of fractional order v and r is an unknown constant to be determined. Since u is not an integer in general, the Mathieu functions ce~ and se~ are linearly independent therefore even and odd Mathieu functions are needed to represent the most general solution. The defining equation [4] for the Mathieu functions is -2qcos2~+a o se(~,q)=O,
(8)
where 29
Volume 126, number 1
1 aq=V~-+2(v2-1)
PHYSICS LETTERS A
q2+
5V2+7 p2 32(v2-1)3(
4)
q4 +""
14 December 1987
(9)
with the following condition qZ
(10)
<
2(v2--1)
This c o n d i t i o n is satisfied in the case o f F E L as we shall see shortly when we relate v to the longitudinal mom e n t u m and q to the transverse m o m e n t u m . Since the initial m o m e n t u m c o m p o n e n t s are the design parameters (hence v a n d q are given n u m b e r s ) , aq will be treated as a constant. Therefore
- 2q cos 2 ~ - 22 (ce. + r se.) = - (aq + 22 ) (ce. + r se.) .
( 1 1)
Using this result eq. (6) reduces to [2(2+4)-23](ce.+rse.)+4i2
dse. r d~
dce.]= d~ ]
0
'
(12)
where 2 = au+22. F o r eq. (12) to be equal to zero, considering the evenness and the oddness o f the M a t h i e u functions, we must have [ 2 ( 2 + 4 ) - 2 3 ] ce~ - 4 i 2 r F = 0 ,
(13)
[ 2 ( 2 + 4 ) - 2 3 ] r se~ - 4 i 2 G = 0 .
(14)
Here we d e n o t e d F=d se./d~,
G=d ce./d(.
(15)
We obtain the same conditions (eqs. (13 ) and (14)) from the second set o f equations by letting g2 (~) = ce~ + r se.. The next step is to solve these equations simultaneously for r a n d 2. We obtain X r= - 4i2F/ce~
(16)
where X=2(2+4)
-23,
(17)
X 2 + 1622 ( F / e e . ) ( G / s e . ) = 0 .
(18)
Eq. (18) is the most crucial equation towards the solution o f the Dirac equation. This is a d e t e r m i n i n g equation for v since X, 2, F, G, se. and ce. are all functions o f v. I f we could d e t e r m i n e v exactly from this equation we would have exact solutions o f the Dirac equation for an electron in a wiggler (t?x ~ 0,/?v = 0). But the M a t h i e u functions eel, se~ and their derivatives F a n d G are only known as power series [4] therefore explicit exact solutions do not exist. Exact solutions exist only in the sense that v can be expressed in terms o f M a t h i e u functions algebraicly. Recalling the c o n d i t i o n (10) a n d e x a m i n i n g the expansions o f M a t h i e u functions, we ~dn easily observe that F/cev~v
,
G/se, ~ - v.
Using these results we o b t a i n 30
(19) (20)
Volume 126,number 1
PHYSICS LETTERSA
14 December 1987
r~i.
(21)
The 2-D solutions are required to be consistent with the I-D solutions derived in refs. [3] and [2], i.e., when q = 0 the 2-D solutions must go over into the I-D solutions. When q=0, Mathieu functions reduce to ce~(~, q = 0 ) = c o s v~,
(22)
se~(~, q = 0 ) =sin v~,
(23)
and fl (~) becomes f~ (~) Iq=o =ce~ + r se~ =cos v~ + i sin z,~= exp(iv~) .
(24)
This must be equal to the exponential in the 1-D solution exp(iv~) - exp[iv(kwz/2)] =,exp (ih -~pl z ) . This leads us to the identity (25)
~,=-2p,/hk~ .
Here p] is the effective longitudinal momentum of the electron (up-spin, spin parallel to the direction of motion) inside the wiggler. Using eqs. (19) and (20), eq. (18) becomes [2(2+4) -23] 2 - 1622u 2 = 0 .
(26)
Further expanding, using eq. (9) and 2=aq+22 we obtain 24 --823 "~-(16-223)22 +8(222
--23)2+22 = 0 .
(27)
The solution of this equation leads us to cp~ ~ - ½hck~ + [ c2(po_- + ½hkw) 2 - e Z a 2 +c2pg± -c2p2x] ,/2.
(28)
This is the determining equation for the effective longitudinal momentum in terms of the design parameters "a", kw, the initial momentum components Po~, Pol; and the transverse momentum correction P~x which is going to be determined from a condition given by the normalization procedure. The determination ofp~ a n d p ~ in terms of the given quantities completes the solution of the Dirac equation. Once we know fj (z) then f2 (z), f3 (z) and f4 (z) can be determined from the first set of operator equations we mentioned in the beginning of this section. The 2-D up-spin solution of the Dirac equation for an electron in a wiggler, satisfying all the conditions (commutes with/?~ and/?y, reduces to the I-D solution when p~x=0) will be
~r/t = N l
K1 exp(ik~z) [Cplx e x p ( i k ~ z ) - e a l K j +cp, E+moC 2
[cPL~ e x p ( - i k ~ z ) - e a ] - c ( p l E + moc z
where K~ =
+hkw)Kl exp(ik~z)
1
(ce~+i se~) exp(ih-lp~.,x) ,
(29)
c~p~ +tl2 + ½e2a2(pL~/p ~)2 eahck,,,
The normalization constants N] and P~x are determined from the normalization condition ~T ~ = 1, where ~T is the Dirac conjugate of the solution 7tr. We find 31
Volume 126, number 1
PHYSICS LETTERS A
14 December 1987
N 1~ 1/,~,
(30)
c2 p~ , ~., 7( moc2)( hck.~ )( y / K + t ) ,
(31)
where 7 is the relativistic factor of the electron and K = - ea/moc 2 is the wiggler strength. Similarly the down-spin solution will be K2 e x p ( - i k w z ) 1 [ cp2, exp(ikwz) - e a ] K 2 + ( cp2 - h c k ~ ) exp(-ikwz) E + moc 2
~=N2
( c e . + i se.) e x p ( i h - ~ p 2 x x )
,
(32)
[ cP2,- e x p ( - i k , ~ z ) - e a ] - c P 2 K 2 E + moc 2
where K2 = -
c2p~ +rl~_ + ~e2a2(p2,/p2)2 eahcl~ '
rl~-m2c4-E2
cp2 "~ ½hck,, + [c2(Po_- - ~ 1 k w ) -" _ e Z a 2 + c 2 p g l - c 2 p L ]
N,~
1/x/~,
+ e 2 a 2 +c2p2, , ''2 ,
c2p~,,,~y(moc2)(hckw)(7/K-1).
3. Conclusions
The solutions presented in this paper can form the basis for a perturbation theory of the free electron laser operating in the small-signal regime. Such studies [2,3] using the 1-D solutions have shown that one obtains the same results (apart from small quantum mechanical corrections) as linear or quasi-linear analyses based on classical mechanics. 2-D solutions would change this conclusion slightly in the favor of quantum mechanical analysis. It is obvious from the results of this paper that the quantum mechanical correction Plx (or P2~-) to the transverse m o m e n t u m ea/c beomes more significant when the electrons are highly relativistic (7 > 103). Also, the rate of spin-flipping processes should increase due to the effect of an oscillating transverse m o m e n t u m component. Therefore, one expects the purely quantum mechanical spin and transverse m o m e n t u m effects to become more important for the short wavelength FELs which use highly relativistic electron beams (1 GeV). Such an analysis has been completed and the results will be published in a forthcoming paper [ 5 ]. More importantly, the 2-D Dirac solutions derived in this paper could provide guidance in the development of a Dirac theory of the free electron laser in a helical wiggler field including transverse momentum, based on a hamiltonian which includes the radiation field ab initio. Such a result would represent substantial progress in the quantum theory of the free electron lasers with applications to experimental problems not accessible by classical analyses.
References [ 1 ] B.M. Kincaid, J. Appl. Phys. 48 (1977) 2684; J.P. Blewet and R. Chasman, J. Appl. Phys. 48 (1977) 2692. [2] W. Becket and H. Mitter, Z. Phys. B 35 (1979) 399. [3] S. Saritepe and N.V.V.J. Swamy, Phys. Lett. A 113 (1985) 69. [4] N.W. MaeLaehlan, Theory and application of Mathieu functions (Dover, New York, 1964). [ 5 ] S. Saritepe and N.V.V.J. Swamy, to be published.
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