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Remarks on coulomb corrections in scattering and ionization in a laser field
Volume 77, number 4
OPTICS COMMUNICATIONS
1 July 1990
REMARKS ON COULOMB CORRECTIONS IN SCATYERING AND IONIZATION IN A LASER FIELD F. E H L O T Z K Y
Institutefor TheoreticalPhysics, Universityof lnnsbruck, A-6020 Innsbruck, Austria Received 22 January 1990
The Coulomb effects in scattering or ionization in a laser field have recently been taken into account by using an approximate wave function consisting of a Coulomb-wave multiplied by a Gordon-Volkov factor. We shall comment on the limits of validity of this approximation.
Recently much work has been devoted to scattering and ionization in a laser field and we refer to some o f the existing reviews [ 1-4] and to the book by Mittleman [5], In some o f these investigations it turned out to be o f importance to take care of the long range Coulomb effects. These effects were taken into account for the first time by Jain and Tzoar [ 6 ] in a paper on C o m p t o n scattering in a laser field. In this paper an approximate wave function has been suggested, consisting o f a Coulomb wave multiplied by a G o r d o n - V o l k o v factor describing the interaction o f the particle with the laser field. This so-called Volkov-Coulomb (VC) wave function has then been applied to ionizing collisions in a laser field by Cavaliere et al. [ 7 ], by Banerji and Mittleman [ 8 ], by Martin et al. [ 9 ] and by C h e n [ 10 ] and to laser assisted scattering by Rosenberg [ 11 ]. More recently, the same type of wave function has been used to account for the Coulomb corrections in multiphoton ionization by Basile et al. [ 12 ], by Potvliege and Shakeshaft [ 13 ] and by Leone et al. [ 14 ]. In all of these investigations comparatively little has been said about the limits o f validity o f the above mentioned laser corrected Coulomb wave. It is the purpose of the present work to elaborate somewhat on these limits. We start from the Schrrdinger equation for a charged particle in an electromagnetic wave and in a potential field
[ (P--eA/c)2/2m+ V]~=ihOtct,
(1)
where p = - i h V and we assume the laser field to be represented in dipole approximation by A ( t ) = Ao~ cos cot. With this field we may rewrite ( 1 ) in the form
[ -h2A/2m+ihc/~ cos cot~. V + a ( 1 + c o s 2cot)+ V(x) ] ¢,=ih0tct.
(2)
Here we have introduced the intensity parameter
/l=eAo/mc 2 and the laser induced Stark shift a= mc2 lt2 / 4. For solving (2) we make the ansatz [ 15 ] ~,(x, t ) = ~ ~'E(P, t) f(p, t) exp(ip.x/h) d3p, (3a) where
9'e(P, t ) = u e ( P ) exp ( - iEt / h )
(3b)
is a solution of (2) in m o m e n t u m space in the absence of the laser field and f ( p , t) accounts for the interaction with the radiation. Inserting (3a, b) into (2) we get
(p2 / 2 m - E)ue(p ) +fV(p')uE(p--p')d3p')f(p,t) = {ihf(p, t) + [c/~o-¢ cos cot - t r ( 1 + c o s 2cot) ]f(p, t)} uCo) .
0030-4018/90/$03.50 © Elsevier Science Publishers B.V. (North-Holland)
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Volume 77, number 4
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In the second term on the lhs of (4) we have app r o x i m a t e d f ( p - p ' , t) b y f ( p , t), which is the value of this function at V(p' = 0). If f (p, t) depends very little on p, we may approximately separate (4). Then, on account of (3b), the lhs o f (4) vanishes and we can thus evaluate for the rhs
f(p, t) =exp{i[ ( c lt/ og)p.~ sin ~ot -- (a/2co) sin 2 o t - t r t ] / h } .
(5)
However, the classical Lorentz equation of motion m£c(t) =eE(t) has the solution
x~(t)=OCosin~ot,
O~o=-aoE,
ao=ClZ/o),
(6)
and hence, by substituting (3b), (5) and (6) into (3a), we get
~,(x, t) = u E [ x - x ~ ( t) l × e x p [ - i ( a / 2 f i o ) ) sin 2 o ) t - i ( E + a ) t / h ] ,
(7)
which is identical to the space-translated approximation presented by Jain and Tzoar [6]. Our deviation first of all requires t h a t f ( p , t) depends very little on p or, according to ( 5 ) and (6), we must have (clzp/ho)) =c~oK<< I. With p=hK~-h/ao (ao is the first Bohr radius) we thus get p<<2~zao/2, 2 being the wavelength of the laser light. Hence, for a Nd: YAG laser we find/z << 6 × 10- 4 corresponding to an intensity 1 << 3.6 × 10 l~ W / c m 2. Consequently (7) is a low intensity approximation. Moreover, our ansatz (3a) requires that the interaction with the radiation field, described by f ( p , t), alters very little the initial atomic state qJE(X, t) and causes no transitions. This will be the case, i f f ( p , t) is almost constant over many periods r~-h/IEI of the atomic state ~UE(X, t). Consequently we conclude from ( 5 ), ~oz<< 1 or riog<< [El (and a<< [El ). Therefore, (7) is also a low frequency approximation. Let us now consider Coulomb scattering in a laser field. Putting x - X c ( t ) = x ' , the essential part o f (7) is UE(X'). Introducing spherical polar coordinates x-= ( r, 0, ~0) and x' - ( r', 0', ~0' ), the Coulomb wave in the asymptotic region r'-,oo (0' ¢ 0 ) reads [ 16]
1 July 1990
Here v is the velocity of scattered particles, K = p / h their wave number and 7= - aZ/fl ( a = e2/hc, fl= v~ c) describes the strength of the Coulomb interaction (for electrons). Since we are in the asymptotic region r-,oo, where we can assume ao<
r'=lx-xc(t)l~-r-n'xc(t)+
....
n=x/r.
(9)
Consequently we may write for r-~oo
UE(r, 0, t) : r'-l/z{ [1 +72/2iKrsin2(O/2) ] ×exp[i(K(z--zc(t) ) + 7 In K ( r - z ) ) ] + [f(O)/r] exp [i ( K r - K . x ~ ( t ) - 7 In 2Kr) ] },
(10) where f ( 0 ) is the scattering amplitude in the absence of the laser field [ 16 ] riO)=
7F(1+i7) exp[-iTlnsin2(O/2)], 2K_F(1 - i 7 ) sin2(0/2) (10a)
and we have taken account of the fact that the logarithmic terms in (8) yield only corrections o f the order a o / r < < 1 in the laser field. Hence, if we consider the scattered wave, in particular, in C o m p t o n scattering and in ionization processes in a laser field, we may write on a c c o u r ' o f (10) for the space-translated Coulomb wave in ( 7 ) with r >> ao UE[X--Xc(/) ] '~UE(X) exp[-ip'Xc(t)/h]
.
(11)
This is the desired approximate solution of (2) with V(x) = - e 2 Z / r , which essentially has been used by all authors mentioned in the introduction considering laser-assisted and laser-induced processes in a Coulomb field. As we have seen in the preceding paragraph, an approximate solution o f the Schr6dinger equation for a charged particle simultaneously embedded in a laser field and in a Coulomb potential can be found, which on account of eqs. (6), (7) and ( 11 ) is of the form ~u(x, t) =UE(X) exp{--i[p-O~oSin cot
UE(r',O')=V ~/2{[1+72/2iKr ' sin2(O ' / 2 ) ]
+ (cr/2co) sin 209t+ ( E + a ) t ] / f i } ,
(12)
×exp[i(Kz'+71n K(r'-z' ) ) ]
+[['(O')/r'] e x p [ i ( K r ' - 7 1 n 2 K r ' ) ] } .
310
(8)
where UE(X) is the Coulomb wave unperturbed by the laser radiation. We have found (12) to be a low frequency approximation with hto << E, as well as a
Volume 77, number 4
OPTICS COMMUNICATIONS
low intensity a p p r o x i m a t i o n with ppc<> Oto, where, according to ( 6 ) , ao is the a m p l i t u d e o f the classical electron oscillations in the laser field. By c o m b i n i n g the low frequency a n d the low intensity c o n d i t i o n s we find E >> ppc which m a y be rewritten in the form v >> Vosc [7 ], where v is the average electron velocity a n d Vosc= pc is the velocity a m p l i t u d e o f electron oscillations in the laser field. F r o m this last condition, however, again follows E>> a. The same restriction on the admissible laser intensity as before follows from the a s y m p t o t i c c o n d i t i o n ao/r=l~/kr<
I July 1990
the potential V(x) into V(x+xc(t)). As we have seen, at low intensities a n d low frequencies one may a p p a r e n t l y write V(x+xc(t) ) ~ V(x) as long as ao/ r<< 1 and we obtain the a s y m p t o t i c VC wave (12). Recently, a relativistic generalization o f the spacetranslated solution (7) has been presented by Rashid [19], but since this a u t h o r does not transform the potential as well, his solution o f the Dirac equation will have a similar restricted range o f validity as ind i c a t e d here for the non-relativistic case. This p a p e r has been written while the author was a visitor at the Theoretical Physics Institute o f the U n i v e r s i t y o f Alberta, E d m o n t o n , Canada, and the a u t h o r wishes to acknowledge the kind hospitality extended to him there. In particular he wishes to thank Prof. A.Z. Capri for his interest and support.
References [ 1] F. Ehlotzky, ed., Fundamentals of laser interactions II, Proc. Seminar at Obergurgl, Austria, 1989, Lect. Notes Phys. 339 (1989). [ 2 ] F. Ehlotzky, ed., Fundamentals of laser interactions, Proc. Seminar at Obergurgl, Austria, 1985, Lect. Notes Phys. 229 (1985). [ 3] F. Ehlotzky, Can. J. Phys. 63 (1985) 907. [4] L. Rosenberg, Adv. At. Mol. Phys. 18 (1982) 1. [ 5 ] M.H. Mittleman, Introduction to the theory of laser-atom interactions (Plenum Press, New York, 1982). [6 ] M. Jain and N. Tzoar, Phys. Rev. A 18 ( 1978 ) 538. [ 7 ] P. Cavaliere, G. Ferrante and C. Leone, J. Phys. B 13 (1980) 4495. [ 8 ] J. Banerji and M.H. Mittleman, J. Phys. B 14 ( 1981 ) 3717. [9] P. Martin, V. Veniard, A. Maquet, P. Franken and C.J. Joachain, Phys. Rev. A 39 (1989) 6178. [ 101 X.J. Chen, Phys. Rev. A 40 ( 1989 ) 1795. [ 11 ] L. Rosenberg, Phys. Rev. A 34 (1986) 4567; A 26 (1982) 132. [ 12 ] S. Basile, F. Trombetta, G. Ferrante, R. Burlon and C. Leone, Phys. Rev. A 37 (1988) 1050. [ 13 ] R.M. Potvliege and R. Shakeshaft, Phys. Rev. A 38 ( 1988 ) 4597. [ 14] C. Leone, S. Bivona, R. Burlon, F. Morales and G. Ferrante, Phys. Rev. A 40 (1989) 1828. [ 15 ] F. Ehlotzky, Can. J. Phys. 59 ( 1981 ) 1200. [16] C.J. Joachain, Quantum collision theory (North Holland, Amsterdam, 1975). [ 17 ] G. Ferrante and C. Leone, Phys. Rev. A 26 ( 1982 ) 3101. l 18 ] W.C. Henneberger, Phys. Rev. Lett. 21 ( 1968 ) 838. [19] S. Rashid, Phys. Rev. A 40 (1989) 4242.
311
OPTICS COMMUNICATIONS
1 July 1990
REMARKS ON COULOMB CORRECTIONS IN SCATYERING AND IONIZATION IN A LASER FIELD F. E H L O T Z K Y
Institutefor TheoreticalPhysics, Universityof lnnsbruck, A-6020 Innsbruck, Austria Received 22 January 1990
The Coulomb effects in scattering or ionization in a laser field have recently been taken into account by using an approximate wave function consisting of a Coulomb-wave multiplied by a Gordon-Volkov factor. We shall comment on the limits of validity of this approximation.
Recently much work has been devoted to scattering and ionization in a laser field and we refer to some o f the existing reviews [ 1-4] and to the book by Mittleman [5], In some o f these investigations it turned out to be o f importance to take care of the long range Coulomb effects. These effects were taken into account for the first time by Jain and Tzoar [ 6 ] in a paper on C o m p t o n scattering in a laser field. In this paper an approximate wave function has been suggested, consisting o f a Coulomb wave multiplied by a G o r d o n - V o l k o v factor describing the interaction o f the particle with the laser field. This so-called Volkov-Coulomb (VC) wave function has then been applied to ionizing collisions in a laser field by Cavaliere et al. [ 7 ], by Banerji and Mittleman [ 8 ], by Martin et al. [ 9 ] and by C h e n [ 10 ] and to laser assisted scattering by Rosenberg [ 11 ]. More recently, the same type of wave function has been used to account for the Coulomb corrections in multiphoton ionization by Basile et al. [ 12 ], by Potvliege and Shakeshaft [ 13 ] and by Leone et al. [ 14 ]. In all of these investigations comparatively little has been said about the limits o f validity o f the above mentioned laser corrected Coulomb wave. It is the purpose of the present work to elaborate somewhat on these limits. We start from the Schrrdinger equation for a charged particle in an electromagnetic wave and in a potential field
[ (P--eA/c)2/2m+ V]~=ihOtct,
(1)
where p = - i h V and we assume the laser field to be represented in dipole approximation by A ( t ) = Ao~ cos cot. With this field we may rewrite ( 1 ) in the form
[ -h2A/2m+ihc/~ cos cot~. V + a ( 1 + c o s 2cot)+ V(x) ] ¢,=ih0tct.
(2)
Here we have introduced the intensity parameter
/l=eAo/mc 2 and the laser induced Stark shift a= mc2 lt2 / 4. For solving (2) we make the ansatz [ 15 ] ~,(x, t ) = ~ ~'E(P, t) f(p, t) exp(ip.x/h) d3p, (3a) where
9'e(P, t ) = u e ( P ) exp ( - iEt / h )
(3b)
is a solution of (2) in m o m e n t u m space in the absence of the laser field and f ( p , t) accounts for the interaction with the radiation. Inserting (3a, b) into (2) we get
(p2 / 2 m - E)ue(p ) +fV(p')uE(p--p')d3p')f(p,t) = {ihf(p, t) + [c/~o-¢ cos cot - t r ( 1 + c o s 2cot) ]f(p, t)} uCo) .
0030-4018/90/$03.50 © Elsevier Science Publishers B.V. (North-Holland)
(4) 309
Volume 77, number 4
OPTICS COMMUNICATIONS
In the second term on the lhs of (4) we have app r o x i m a t e d f ( p - p ' , t) b y f ( p , t), which is the value of this function at V(p' = 0). If f (p, t) depends very little on p, we may approximately separate (4). Then, on account of (3b), the lhs o f (4) vanishes and we can thus evaluate for the rhs
f(p, t) =exp{i[ ( c lt/ og)p.~ sin ~ot -- (a/2co) sin 2 o t - t r t ] / h } .
(5)
However, the classical Lorentz equation of motion m£c(t) =eE(t) has the solution
x~(t)=OCosin~ot,
O~o=-aoE,
ao=ClZ/o),
(6)
and hence, by substituting (3b), (5) and (6) into (3a), we get
~,(x, t) = u E [ x - x ~ ( t) l × e x p [ - i ( a / 2 f i o ) ) sin 2 o ) t - i ( E + a ) t / h ] ,
(7)
which is identical to the space-translated approximation presented by Jain and Tzoar [6]. Our deviation first of all requires t h a t f ( p , t) depends very little on p or, according to ( 5 ) and (6), we must have (clzp/ho)) =c~oK<< I. With p=hK~-h/ao (ao is the first Bohr radius) we thus get p<<2~zao/2, 2 being the wavelength of the laser light. Hence, for a Nd: YAG laser we find/z << 6 × 10- 4 corresponding to an intensity 1 << 3.6 × 10 l~ W / c m 2. Consequently (7) is a low intensity approximation. Moreover, our ansatz (3a) requires that the interaction with the radiation field, described by f ( p , t), alters very little the initial atomic state qJE(X, t) and causes no transitions. This will be the case, i f f ( p , t) is almost constant over many periods r~-h/IEI of the atomic state ~UE(X, t). Consequently we conclude from ( 5 ), ~oz<< 1 or riog<< [El (and a<< [El ). Therefore, (7) is also a low frequency approximation. Let us now consider Coulomb scattering in a laser field. Putting x - X c ( t ) = x ' , the essential part o f (7) is UE(X'). Introducing spherical polar coordinates x-= ( r, 0, ~0) and x' - ( r', 0', ~0' ), the Coulomb wave in the asymptotic region r'-,oo (0' ¢ 0 ) reads [ 16]
1 July 1990
Here v is the velocity of scattered particles, K = p / h their wave number and 7= - aZ/fl ( a = e2/hc, fl= v~ c) describes the strength of the Coulomb interaction (for electrons). Since we are in the asymptotic region r-,oo, where we can assume ao<
r'=lx-xc(t)l~-r-n'xc(t)+
....
n=x/r.
(9)
Consequently we may write for r-~oo
UE(r, 0, t) : r'-l/z{ [1 +72/2iKrsin2(O/2) ] ×exp[i(K(z--zc(t) ) + 7 In K ( r - z ) ) ] + [f(O)/r] exp [i ( K r - K . x ~ ( t ) - 7 In 2Kr) ] },
(10) where f ( 0 ) is the scattering amplitude in the absence of the laser field [ 16 ] riO)=
7F(1+i7) exp[-iTlnsin2(O/2)], 2K_F(1 - i 7 ) sin2(0/2) (10a)
and we have taken account of the fact that the logarithmic terms in (8) yield only corrections o f the order a o / r < < 1 in the laser field. Hence, if we consider the scattered wave, in particular, in C o m p t o n scattering and in ionization processes in a laser field, we may write on a c c o u r ' o f (10) for the space-translated Coulomb wave in ( 7 ) with r >> ao UE[X--Xc(/) ] '~UE(X) exp[-ip'Xc(t)/h]
.
(11)
This is the desired approximate solution of (2) with V(x) = - e 2 Z / r , which essentially has been used by all authors mentioned in the introduction considering laser-assisted and laser-induced processes in a Coulomb field. As we have seen in the preceding paragraph, an approximate solution o f the Schr6dinger equation for a charged particle simultaneously embedded in a laser field and in a Coulomb potential can be found, which on account of eqs. (6), (7) and ( 11 ) is of the form ~u(x, t) =UE(X) exp{--i[p-O~oSin cot
UE(r',O')=V ~/2{[1+72/2iKr ' sin2(O ' / 2 ) ]
+ (cr/2co) sin 209t+ ( E + a ) t ] / f i } ,
(12)
×exp[i(Kz'+71n K(r'-z' ) ) ]
+[['(O')/r'] e x p [ i ( K r ' - 7 1 n 2 K r ' ) ] } .
310
(8)
where UE(X) is the Coulomb wave unperturbed by the laser radiation. We have found (12) to be a low frequency approximation with hto << E, as well as a
Volume 77, number 4
OPTICS COMMUNICATIONS
low intensity a p p r o x i m a t i o n with ppc<
I July 1990
the potential V(x) into V(x+xc(t)). As we have seen, at low intensities a n d low frequencies one may a p p a r e n t l y write V(x+xc(t) ) ~ V(x) as long as ao/ r<< 1 and we obtain the a s y m p t o t i c VC wave (12). Recently, a relativistic generalization o f the spacetranslated solution (7) has been presented by Rashid [19], but since this a u t h o r does not transform the potential as well, his solution o f the Dirac equation will have a similar restricted range o f validity as ind i c a t e d here for the non-relativistic case. This p a p e r has been written while the author was a visitor at the Theoretical Physics Institute o f the U n i v e r s i t y o f Alberta, E d m o n t o n , Canada, and the a u t h o r wishes to acknowledge the kind hospitality extended to him there. In particular he wishes to thank Prof. A.Z. Capri for his interest and support.
References [ 1] F. Ehlotzky, ed., Fundamentals of laser interactions II, Proc. Seminar at Obergurgl, Austria, 1989, Lect. Notes Phys. 339 (1989). [ 2 ] F. Ehlotzky, ed., Fundamentals of laser interactions, Proc. Seminar at Obergurgl, Austria, 1985, Lect. Notes Phys. 229 (1985). [ 3] F. Ehlotzky, Can. J. Phys. 63 (1985) 907. [4] L. Rosenberg, Adv. At. Mol. Phys. 18 (1982) 1. [ 5 ] M.H. Mittleman, Introduction to the theory of laser-atom interactions (Plenum Press, New York, 1982). [6 ] M. Jain and N. Tzoar, Phys. Rev. A 18 ( 1978 ) 538. [ 7 ] P. Cavaliere, G. Ferrante and C. Leone, J. Phys. B 13 (1980) 4495. [ 8 ] J. Banerji and M.H. Mittleman, J. Phys. B 14 ( 1981 ) 3717. [9] P. Martin, V. Veniard, A. Maquet, P. Franken and C.J. Joachain, Phys. Rev. A 39 (1989) 6178. [ 101 X.J. Chen, Phys. Rev. A 40 ( 1989 ) 1795. [ 11 ] L. Rosenberg, Phys. Rev. A 34 (1986) 4567; A 26 (1982) 132. [ 12 ] S. Basile, F. Trombetta, G. Ferrante, R. Burlon and C. Leone, Phys. Rev. A 37 (1988) 1050. [ 13 ] R.M. Potvliege and R. Shakeshaft, Phys. Rev. A 38 ( 1988 ) 4597. [ 14] C. Leone, S. Bivona, R. Burlon, F. Morales and G. Ferrante, Phys. Rev. A 40 (1989) 1828. [ 15 ] F. Ehlotzky, Can. J. Phys. 59 ( 1981 ) 1200. [16] C.J. Joachain, Quantum collision theory (North Holland, Amsterdam, 1975). [ 17 ] G. Ferrante and C. Leone, Phys. Rev. A 26 ( 1982 ) 3101. l 18 ] W.C. Henneberger, Phys. Rev. Lett. 21 ( 1968 ) 838. [19] S. Rashid, Phys. Rev. A 40 (1989) 4242.
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