- Email: [email protected]
Response to comments on laser electron acceleration in vacuum
a.__
15 February
&I
1996
‘B
OPTICS COMMUNICATIONS
ELSEWlER
Optics Communications
124 (1996) 74-78
Response to comments on laser electron acceleration in vacuum * Carsten M. Haaland Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, TN 37831-6363, USA Received 6 September
1995; accepted 29 September 1995
Abstract Response is made to criticisms on Haaland’s paper on acceleration of electrons by laser beams in vacuum. All claims by critics are refuted.
Sprangle et al. [ 1] have presented critical comments on a paper by Haaland [ 21 on the subject of acceleration of electrons by laser beams in vacuum. Their critique is based on the following claims: ( 1) the fields in Haaland’s paper (Ref. [2] ) “are not physically realizable, i.e., V*E#O, because the field components E,, and E,, are neglected”. Here E,, and E,, are the longitudinal
components
of the electric vec-
tors in the reference frames of laser beams 1 and 2, respectively; (2) the “transformation of the fields and coordinates from the rotated frames to the accelerated frame is in error” in Ref. [ 21; (3) the results in Ref. [ 21 showing the electron-laser slippage distance to be essentially independent of laser spot size, as illustrated in Fig. 2 of Ref. [Z], are incorrect; (4) “the net energy change vanishes for infinite interaction distances in the crossed-beam configuration” in contrast to the finite and substantial energy gains presented in Ref. [ 21. * Research sponsored by the U.S. Department of Energy Office of High Energy Physics under Contract No. DE-AC05840R21400 with Oak Ridge National Laboratory, managed for the Department of Energy by Lockheed Martin Energy Systems, Inc. 0030-4018/96/$12.00 0 1996 Elsevier Science B.V. All rights reserved .S.S~10030-4018(95)00619-2
It will be shown in this response that these claims have no basis, that Sprangle et al. [ 1] have not correctly represented the physical nature of converging Gaussian beams, and that their results are flawed by errors in derivation. Furthermore, it will become evident that the Lawson-Woodward theorem cited in Ref. [ 1 ] cannot represent the electric field presented in Ref. [2] and therefore does not apply to this configuration. Regarding claim (l), it is true that the field components E,, and E,, were neglected in Ref. [ 21. However, from this neglect it does not follow that the fields are not physically realizable, or that V*E#O. These field components were neglected because they are not needed for the problem. The electric field components which are relevant to the problem are those longitudinal components which are in the reference frame of the electron beam, namely the E, components, and these can be determined correctly without calculating the corresponding laser beam vector components E,, and E,, as will be shown. But first, it appears that it is necessary to show that the fields used in Ref. [ 21 are physically realizable. For convenience of the reader, the equations representing these fields in Ref. [2] will be reproduced here, with the slight modification that “z”s will be replaced by “z,“s to correspond with the notation used by Ref. [ l] for the reference frame of laser beam # 1.
C.M. Haaland/Optics
Communications
For a z,-directed Gaussian beam in the fundamental mode, solutions obtained by several independent methods for the magnitude of the field components (the scalar field) [ 3-81 (this list of references is not exhaustive) , using the paraxial approximation, is given by
(1) where A (amplitude) is a constant, represented by E,, in the comments by Ref. [ 11. The halfwidth of the beam, w(zi), or simply w,, is given by w, =w,,[l+ (Z,/ZR) *1I’*, in which w, is the minimum value of w1 at z, = 0, the radius of the waist of laser beam # 1, and zn= rrwglh is the Rayleigh range. For a beam with power P in watts produced within the beam radius r, = wO, the constant A, evaluated by integrating across the beam at zi = 0, with appropriate constants for magnetic permeability and dielectric permittivity in vacuum, becomes the peak electric field amplitude, given by A = Epeak= 16.7filwa Volt/m. The phase factor, @, in Eq. ( 1) is given by @=kz, + 2
-arctan(zi/z&
-4,
1 in which RI is the radius of the curvature of the spherical wave fronts, given by R,=z,(l+z;/z:)
(3)
and + is an arbitrary phase constant used to find the electron entry phase for maximum energy gain. According to the many references cited, these equations provide correct values for the fields at zi =O, even though the Fresnel approximation of the diffraction integral breaks down in this region. To compare with the electric field equations Eqs. (la) and (lb) in Sprangle’s paper, Eq. (1) may be rewritten as
El =J%
z exp( - ~:/w~>cos(h,),
(4)
where ~lh=~t-_z,-(~~)l(hR1)+tan-‘(z,/zR) - &,. It is easily shown, using Eq. (3), that (~8) / (AR,) = ( / (z,w:) , and with = radians added to the arbitrary phase constant so that 4, = & - r (to change the sign), the arguments of the trigonometric terms in Eqs. ( la) and ( lb) in Ref. [ 1] produce identical results as the argument of the cosine
124 (1996) 74-78
15
term in Eq. (4). Note that Eq. (4) provides the scalar electric field values of laser beam #l, whereas Ref. [ l] gives the E,, and E,, vector components of the field used by them. It will now be shown how the E,, and E,, vector components are correctly calculated from the scalar electric field given in Fq. (4)) even though these components will not be used in the calculation of the acceleration of the electrons. In each of the references cited for Eq. ( 1) , the physical importance of the radius of curvature, RI, is emphasized. The phase of the wave is constant on the spherical surface described by the radius of curvature, which has its origin on the axis of the laser beam. Consider a laser beam travelling in the positive zi direction, and converging to minimum waist at zi =O. For phase fronts on the negative z1 axis, the radius of curvature, given by Eq. (3)) will be negative and will have its origin on the positive z1 axis. The radius of curvature has minimum values at zi = TfTzn. For phase fronts within Rayleigh range from the minimum waist location, that is, for - zn < z1 < 0, the radius of curvature rapidly increases, becoming infinite at z1 = 0. Thus, at zi = 0 the phase front for fundamental-mode Gaussian beams is plane. In other words, there are no longitudinal components of the field for any value of xi at zi = 0. For beam phase fronts on the positive z1 axis,the radius of curvature will be positive and will have its origin on the negative z1 axis, and will rapidly decrease to its minimum at zi = zR, and then begin to increase again. As clearly described by Pierce (Ref. [ 41) , the flow of power in the laser beam moves in the direction that is normal to the spherical phase front specified by the radius of curvature R. The Poynting vector, S, = El X H,, for laser beam # 1, is therefore normal to the surface of the sphere described by R,, the radius of curvature. Consequently, the vector field components are tangent to the surface of this sphere. This important physical picture for the electric field vectors is illustrated in Fig. 1, from which the vector components E,, and E,, are calculated. For ilktrative purposes only, Fig. 1 is drawn in the xy plane, y = 0 and the angle f3 is drawn much larger than allowed by the-paraxial approximation, an approximation implicit in the derivation of Eq. ( 1). From Fig. 1, it is seen that I D2_ .2\11* E,, = E,cosS= E, “” ;‘I , 1
76
C.M. Haaland/Optics
Communications
Fig. 1.Illustration of the method of calculating the &, and E,, electric field vector components in the xz plane from the radius of curvature of the spherical phase fronts of a diverging laser beam that is linearly polarized, i.e., there are no I& components.
and E,, = E,sinG= E, 2. 1
(6)
Note that at z, =O, R, is infinite, and E,, =0 for all values of x1, as expected when the spherical wave fronts of a converging fundamental-mode Gaussian beam momentarily become a plane wave as they pass through the minimum waist at z, = 0. This behavior is in accordance with behavior described by authors of references cited in connection with Eq. ( 1). In contrast,. Eq. ( 1b) by Sprangle et al. shows E,, to be zero at 6, = 0 only for x1 = 0. This behavior, and the significant differences between their Eq. (lb) and Eq. (6) abose. does not agree with behavior described in references cited in connection with Eq. ( 1). One may uell question how Eq. (lb) in Ref. [ l] was obtained. The authors cite no previous work and show no derivation for this equation. The exprersions for Exi and Ez, in Eqs. (5) and (6) ma? be used to show that C.E, + 0 in the paraxial limit, in refutation of claim (I J by Sprangle et al. In the review article by Kogelnik and Li [ 61, the authors write *‘Gaussian beams of this kind are produced by many lasers that oscillate in the fundamental mode”, referring to actual beams that are described by Eq. ( 1). Thus the claim ( 1J by Sprangle et al., that the fields in Ref. [ 21 are physically unrealizable, is evidently in error. In regard to claim (2) by Sprangle et al., it is curious that the equations presented by them contain terms in
124 (1996) 74-78
the trigonometric arguments to correct for phase shift due to the curvature of the phase fronts, but the authors never mention the radius of curvature in their comments, and they do not consider the effect of the radius of curvature in determining the correct orientation of the electric vectors. The simple rectilinear transformation which they use to transform the fields and coordinates from the rotated frame to the accelerated frame leads to two errors, namely, (a) incorrect phase due to curvature of the phase front, and (b) incorrect determination of the electric field vectors, especially in the crucial z range of approximately - 5zR to + 5zR, where the radius of curvature of the phase fronts undergoes rapid and large changes. A correct method for determining the electric field vectors on the electron beam axis is described by referring to Fig. 2. Note that the rotation of the laser beam reference frame results in placement of the origin of the radius of curvature on the opposite side of the electron beam axis from the laser axial point under consideration. From Fig. 2 it may be seen that the correct determination of the electric vector components from laser beam # 1 along the electron beam axis results in E,= E,cosa
(7)
and, using E, = - E,sincw = - E,sin( 8-- +), the approximations COS~=R,/(R~+~~)“~, sin+=x,/ (R:+xf)“’
and Eq. (3), one obtains
Fig. 2. Illustration of the method of calculating the electric field vector components along an electron-beam axis from the radius of curvature of a laser beam with axis rotated Odegrees from the electron beam axis.
C.M. HaalandlOptics
Communications
E,= -El
=
&sin 8
--El
(z~/cos*e+2z:z~+z4R)1’*~
Field components for laser beam #2 are calculated in a similar manner. Because of the assumption that the two laser beams are n radians out of phase with each other and linearly polarized, the E, terms cancel and the E, terms add. This method was used in Ref. [ 21, but the details and illustration were not given. It should be evident from this presentation that claim (2) by Sprangle et al. is incorrect, since their transformation is incorrect by neglecting the effect of the curvature of the phase fronts on the direction of the electric field vectors. To refute claims (3) and (4) of Sprangle et al., the summed electric field components along the electron beam axis will be presented in a format for direct comparison with Eqs. (3a) and (3b) in Ref. [ 11. The E, field given by Eq. (8) may be transformed to the electron beam reference frame by the simple transformations z, =zcose, r, =x, =zsin@ e,= w,/z,, &=zlzR and using Eq. (4) for E,. The contributions from the two laser beams introduce a factor of two, and with the small-angle approximation applied where appropriate, we obtain for the summed longitudinal electric field component along the electron beam axis (9)
E,= where
&=
22”
3
cost)-
( -tan-Y+
1
p
+ 1
&.
( 81
e,W
(l+Z*) (10)
Comparison of Eqs. (9) and ( 10) with Eqs. (3a) and (3b) in Ref. [l] reveals four differences with Sprangle’s equations, namely, the factor multiplying the exponential in Eq. (9) and the first three terms of &, in Eq. (10). The first terms of Eq. ( 10) and Eq. (3b) of Ref. [ 11 are pertinent to claim (3). The first term in Eq. (3b) of Ref. [l] for the phase angle results in errors of several orders of magnitude because of inappropriate assumptions applied in combining terms where the dif-
124 (1996) 74-78
71
ference between two numbers near unity is involved. The first term of Eq. ( 10) is correct and illustrates the terms involved. For p corresponding to an electron energy of, for example, 1000 MeV, the assumption that cose=l,asmadeinRef. [l],resultsin -1.3X10W7 for the difference (case - 1lp), whereas the use of the actual value of co& results in - 0.005. This erroneous term in Eq. (3b) of Ref. [l] is responsible for the significant variation in slip length with spot size which is reported by Sprangle et al. Their claim (3) is incorrect, and Haaland’s results showing small change in slip distance with spot size is correct. The factors multiplying the exponential term in Eq. (9) and Eq. (3a) of Ref. [ 1 ] are pertinent to claim (4). The additional factor ( 1 +$) - “* in Eq. (9)) which does not appear in Eq. (3a) of Ref. [ 11, results from using the radius of curvature to obtain the correct orientation of the electric field vectors to the axis of the electron beam. Sprangle et al. did not consider this important effect. This factor results in an identical acceleration field strength calculated by both Haaland and Sprangle et al. at one location only, at z”=O, but results in a smaller acceleration field, and, most importantly, a lower dece&tion field in regions outside the crossover region, for Haaland’s results. Visual inspection of Fig. 2 in Ref. [l] shows that the area representing deceleration (below the zero line) appears to be approximately equal to the area representing acceleration (above the zero line), whereas in Fig. 2 in Ref. [ 21, the area representing deceleration appears to be less than the acceleration area (as it is). The lower deceleration area in Haaland’s Fig. 2 results primarily from the additional factor in Eq. (9). The net effect of this important factor is that the electron will obtain a substantial gain in energy when the acceleration effect is integrated over the path length from minus to plus infinity. This gain negates claim (4) by Sprangle et al. Moreover, the second term in Eq. ( 10) shows a $ factor in the numerator where Eq. (3b) of Ref. [l] shows only .Z The cube has inadvertently been dropped in Eq. (3b) in Ref. [ 11, as may be readily seen by examining Eq. (2a) in Ref. [ 1] and going through the steps leading to Eq. (3b). If this cube had been carried through to the arguments of the sine term in Eq. (6) of Ref. [ 11, it is evident that as z. + a, A W would not go to zero, as claimed by Sprangle et al. Thus, their claim (4) that “the net energy change vanishes for infinite interaction distances in the crossed beam contigura-
78
CM. Haaland/Optics
Communications
tion”, in contrast to the finite and substantial energy gains resulting from Haaland’s calculations, is not substantiated by their Eq. (6) because this equation is incorrect. The third term of Eq. ( 10) does not contain the factor two which appears in Eq. (3a) of Sprangle et al. If this factor is introduced into Eq. (lo), the slip length is found to vary with spot size. This factor arises from Eq. ( lb) which Sprangle et al. use for the E,, field component. As stated above, this equation does not correctly describe the laser beam properties at zi = 0, is not referenced by them, nor is the derivation indicated, and it does not appear in any of the references cited in connection with Eq. ( 1) of this response. Because of the errors in Eqs. (3a) and (3b) obtained by Sprangle et al., their derivative Eqs. (4), (5) and (6) are also incorrect, thus invalidating their claims against Haaland’s paper. The Lawson-Woodward theorem described in Ref. [ 11, which Haaland referred to as the Palmer theorem because Palmer was the first to completely define it [ 91, is a neat theorem that applies to plane electromagnetic waves of infinite lateral extent, and the conclusion that electrons cannot be accelerated by such waves over
124 (1996) 74-78
an infinite interaction length is almost self-evident. The theorem does not extend to the physical case of converging Gaussian beams with curved phase fronts with constantly changing radius of curvature. The results obtained in Ref. [ 21 provide evidence of a case where the Lawson-Woodward-Palmer theorem does not apply.
References [ l] P. Sprangle, E. Esarey, .I. KralI and A. Ting, Optics Comm. 124 (1996) 69. [ ] C.M. Haaland, Optics Comm. 114 ( 1995) 280. [3] G.D. Boyd and J.P. Gordon, Bell System Technical Journal 40 (1961) 489. [4] J.R. Pierce, Proc. National Academy of Science47 ( 1961) 1808. [5] S. Ramo, J.R. Whinnery and T. von Duzer, Fields and Waves in Communications Electronics (John Wiley and Sons, New York, NY, 1965) p. 572. [6] H. Kogelnik and T. Li, Appl. Optics 5 (1966) 1550. [7] D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold Company, New York, NY, 1972) p. 234. [ 81 A.E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986) p. 663. [ 91 R.B. Palmer, Lecture Notes in Physics 296, Frontiers of Particle Beams (Springer, Berlin, 1988) p. 607.
15 February
&I
1996
‘B
OPTICS COMMUNICATIONS
ELSEWlER
Optics Communications
124 (1996) 74-78
Response to comments on laser electron acceleration in vacuum * Carsten M. Haaland Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, TN 37831-6363, USA Received 6 September
1995; accepted 29 September 1995
Abstract Response is made to criticisms on Haaland’s paper on acceleration of electrons by laser beams in vacuum. All claims by critics are refuted.
Sprangle et al. [ 1] have presented critical comments on a paper by Haaland [ 21 on the subject of acceleration of electrons by laser beams in vacuum. Their critique is based on the following claims: ( 1) the fields in Haaland’s paper (Ref. [2] ) “are not physically realizable, i.e., V*E#O, because the field components E,, and E,, are neglected”. Here E,, and E,, are the longitudinal
components
of the electric vec-
tors in the reference frames of laser beams 1 and 2, respectively; (2) the “transformation of the fields and coordinates from the rotated frames to the accelerated frame is in error” in Ref. [ 21; (3) the results in Ref. [ 21 showing the electron-laser slippage distance to be essentially independent of laser spot size, as illustrated in Fig. 2 of Ref. [Z], are incorrect; (4) “the net energy change vanishes for infinite interaction distances in the crossed-beam configuration” in contrast to the finite and substantial energy gains presented in Ref. [ 21. * Research sponsored by the U.S. Department of Energy Office of High Energy Physics under Contract No. DE-AC05840R21400 with Oak Ridge National Laboratory, managed for the Department of Energy by Lockheed Martin Energy Systems, Inc. 0030-4018/96/$12.00 0 1996 Elsevier Science B.V. All rights reserved .S.S~10030-4018(95)00619-2
It will be shown in this response that these claims have no basis, that Sprangle et al. [ 1] have not correctly represented the physical nature of converging Gaussian beams, and that their results are flawed by errors in derivation. Furthermore, it will become evident that the Lawson-Woodward theorem cited in Ref. [ 1 ] cannot represent the electric field presented in Ref. [2] and therefore does not apply to this configuration. Regarding claim (l), it is true that the field components E,, and E,, were neglected in Ref. [ 21. However, from this neglect it does not follow that the fields are not physically realizable, or that V*E#O. These field components were neglected because they are not needed for the problem. The electric field components which are relevant to the problem are those longitudinal components which are in the reference frame of the electron beam, namely the E, components, and these can be determined correctly without calculating the corresponding laser beam vector components E,, and E,, as will be shown. But first, it appears that it is necessary to show that the fields used in Ref. [ 21 are physically realizable. For convenience of the reader, the equations representing these fields in Ref. [2] will be reproduced here, with the slight modification that “z”s will be replaced by “z,“s to correspond with the notation used by Ref. [ l] for the reference frame of laser beam # 1.
C.M. Haaland/Optics
Communications
For a z,-directed Gaussian beam in the fundamental mode, solutions obtained by several independent methods for the magnitude of the field components (the scalar field) [ 3-81 (this list of references is not exhaustive) , using the paraxial approximation, is given by
(1) where A (amplitude) is a constant, represented by E,, in the comments by Ref. [ 11. The halfwidth of the beam, w(zi), or simply w,, is given by w, =w,,[l+ (Z,/ZR) *1I’*, in which w, is the minimum value of w1 at z, = 0, the radius of the waist of laser beam # 1, and zn= rrwglh is the Rayleigh range. For a beam with power P in watts produced within the beam radius r, = wO, the constant A, evaluated by integrating across the beam at zi = 0, with appropriate constants for magnetic permeability and dielectric permittivity in vacuum, becomes the peak electric field amplitude, given by A = Epeak= 16.7filwa Volt/m. The phase factor, @, in Eq. ( 1) is given by @=kz, + 2
-arctan(zi/z&
-4,
1 in which RI is the radius of the curvature of the spherical wave fronts, given by R,=z,(l+z;/z:)
(3)
and + is an arbitrary phase constant used to find the electron entry phase for maximum energy gain. According to the many references cited, these equations provide correct values for the fields at zi =O, even though the Fresnel approximation of the diffraction integral breaks down in this region. To compare with the electric field equations Eqs. (la) and (lb) in Sprangle’s paper, Eq. (1) may be rewritten as
El =J%
z exp( - ~:/w~>cos(h,),
(4)
where ~lh=~t-_z,-(~~)l(hR1)+tan-‘(z,/zR) - &,. It is easily shown, using Eq. (3), that (~8) / (AR,) = (
124 (1996) 74-78
15
term in Eq. (4). Note that Eq. (4) provides the scalar electric field values of laser beam #l, whereas Ref. [ l] gives the E,, and E,, vector components of the field used by them. It will now be shown how the E,, and E,, vector components are correctly calculated from the scalar electric field given in Fq. (4)) even though these components will not be used in the calculation of the acceleration of the electrons. In each of the references cited for Eq. ( 1) , the physical importance of the radius of curvature, RI, is emphasized. The phase of the wave is constant on the spherical surface described by the radius of curvature, which has its origin on the axis of the laser beam. Consider a laser beam travelling in the positive zi direction, and converging to minimum waist at zi =O. For phase fronts on the negative z1 axis, the radius of curvature, given by Eq. (3)) will be negative and will have its origin on the positive z1 axis. The radius of curvature has minimum values at zi = TfTzn. For phase fronts within Rayleigh range from the minimum waist location, that is, for - zn < z1 < 0, the radius of curvature rapidly increases, becoming infinite at z1 = 0. Thus, at zi = 0 the phase front for fundamental-mode Gaussian beams is plane. In other words, there are no longitudinal components of the field for any value of xi at zi = 0. For beam phase fronts on the positive z1 axis,the radius of curvature will be positive and will have its origin on the negative z1 axis, and will rapidly decrease to its minimum at zi = zR, and then begin to increase again. As clearly described by Pierce (Ref. [ 41) , the flow of power in the laser beam moves in the direction that is normal to the spherical phase front specified by the radius of curvature R. The Poynting vector, S, = El X H,, for laser beam # 1, is therefore normal to the surface of the sphere described by R,, the radius of curvature. Consequently, the vector field components are tangent to the surface of this sphere. This important physical picture for the electric field vectors is illustrated in Fig. 1, from which the vector components E,, and E,, are calculated. For ilktrative purposes only, Fig. 1 is drawn in the xy plane, y = 0 and the angle f3 is drawn much larger than allowed by the-paraxial approximation, an approximation implicit in the derivation of Eq. ( 1). From Fig. 1, it is seen that I D2_ .2\11* E,, = E,cosS= E, “” ;‘I , 1
76
C.M. Haaland/Optics
Communications
Fig. 1.Illustration of the method of calculating the &, and E,, electric field vector components in the xz plane from the radius of curvature of the spherical phase fronts of a diverging laser beam that is linearly polarized, i.e., there are no I& components.
and E,, = E,sinG= E, 2. 1
(6)
Note that at z, =O, R, is infinite, and E,, =0 for all values of x1, as expected when the spherical wave fronts of a converging fundamental-mode Gaussian beam momentarily become a plane wave as they pass through the minimum waist at z, = 0. This behavior is in accordance with behavior described by authors of references cited in connection with Eq. ( 1). In contrast,. Eq. ( 1b) by Sprangle et al. shows E,, to be zero at 6, = 0 only for x1 = 0. This behavior, and the significant differences between their Eq. (lb) and Eq. (6) abose. does not agree with behavior described in references cited in connection with Eq. ( 1). One may uell question how Eq. (lb) in Ref. [ l] was obtained. The authors cite no previous work and show no derivation for this equation. The exprersions for Exi and Ez, in Eqs. (5) and (6) ma? be used to show that C.E, + 0 in the paraxial limit, in refutation of claim (I J by Sprangle et al. In the review article by Kogelnik and Li [ 61, the authors write *‘Gaussian beams of this kind are produced by many lasers that oscillate in the fundamental mode”, referring to actual beams that are described by Eq. ( 1). Thus the claim ( 1J by Sprangle et al., that the fields in Ref. [ 21 are physically unrealizable, is evidently in error. In regard to claim (2) by Sprangle et al., it is curious that the equations presented by them contain terms in
124 (1996) 74-78
the trigonometric arguments to correct for phase shift due to the curvature of the phase fronts, but the authors never mention the radius of curvature in their comments, and they do not consider the effect of the radius of curvature in determining the correct orientation of the electric vectors. The simple rectilinear transformation which they use to transform the fields and coordinates from the rotated frame to the accelerated frame leads to two errors, namely, (a) incorrect phase due to curvature of the phase front, and (b) incorrect determination of the electric field vectors, especially in the crucial z range of approximately - 5zR to + 5zR, where the radius of curvature of the phase fronts undergoes rapid and large changes. A correct method for determining the electric field vectors on the electron beam axis is described by referring to Fig. 2. Note that the rotation of the laser beam reference frame results in placement of the origin of the radius of curvature on the opposite side of the electron beam axis from the laser axial point under consideration. From Fig. 2 it may be seen that the correct determination of the electric vector components from laser beam # 1 along the electron beam axis results in E,= E,cosa
(7)
and, using E, = - E,sincw = - E,sin( 8-- +), the approximations COS~=R,/(R~+~~)“~, sin+=x,/ (R:+xf)“’
and Eq. (3), one obtains
Fig. 2. Illustration of the method of calculating the electric field vector components along an electron-beam axis from the radius of curvature of a laser beam with axis rotated Odegrees from the electron beam axis.
C.M. HaalandlOptics
Communications
E,= -El
=
&sin 8
--El
(z~/cos*e+2z:z~+z4R)1’*~
Field components for laser beam #2 are calculated in a similar manner. Because of the assumption that the two laser beams are n radians out of phase with each other and linearly polarized, the E, terms cancel and the E, terms add. This method was used in Ref. [ 21, but the details and illustration were not given. It should be evident from this presentation that claim (2) by Sprangle et al. is incorrect, since their transformation is incorrect by neglecting the effect of the curvature of the phase fronts on the direction of the electric field vectors. To refute claims (3) and (4) of Sprangle et al., the summed electric field components along the electron beam axis will be presented in a format for direct comparison with Eqs. (3a) and (3b) in Ref. [ 11. The E, field given by Eq. (8) may be transformed to the electron beam reference frame by the simple transformations z, =zcose, r, =x, =zsin@ e,= w,/z,, &=zlzR and using Eq. (4) for E,. The contributions from the two laser beams introduce a factor of two, and with the small-angle approximation applied where appropriate, we obtain for the summed longitudinal electric field component along the electron beam axis (9)
E,= where
&=
22”
3
cost)-
( -tan-Y+
1
p
+ 1
&.
( 81
e,W
(l+Z*) (10)
Comparison of Eqs. (9) and ( 10) with Eqs. (3a) and (3b) in Ref. [l] reveals four differences with Sprangle’s equations, namely, the factor multiplying the exponential in Eq. (9) and the first three terms of &, in Eq. (10). The first terms of Eq. ( 10) and Eq. (3b) of Ref. [ 11 are pertinent to claim (3). The first term in Eq. (3b) of Ref. [l] for the phase angle results in errors of several orders of magnitude because of inappropriate assumptions applied in combining terms where the dif-
124 (1996) 74-78
71
ference between two numbers near unity is involved. The first term of Eq. ( 10) is correct and illustrates the terms involved. For p corresponding to an electron energy of, for example, 1000 MeV, the assumption that cose=l,asmadeinRef. [l],resultsin -1.3X10W7 for the difference (case - 1lp), whereas the use of the actual value of co& results in - 0.005. This erroneous term in Eq. (3b) of Ref. [l] is responsible for the significant variation in slip length with spot size which is reported by Sprangle et al. Their claim (3) is incorrect, and Haaland’s results showing small change in slip distance with spot size is correct. The factors multiplying the exponential term in Eq. (9) and Eq. (3a) of Ref. [ 1 ] are pertinent to claim (4). The additional factor ( 1 +$) - “* in Eq. (9)) which does not appear in Eq. (3a) of Ref. [ 11, results from using the radius of curvature to obtain the correct orientation of the electric field vectors to the axis of the electron beam. Sprangle et al. did not consider this important effect. This factor results in an identical acceleration field strength calculated by both Haaland and Sprangle et al. at one location only, at z”=O, but results in a smaller acceleration field, and, most importantly, a lower dece&tion field in regions outside the crossover region, for Haaland’s results. Visual inspection of Fig. 2 in Ref. [l] shows that the area representing deceleration (below the zero line) appears to be approximately equal to the area representing acceleration (above the zero line), whereas in Fig. 2 in Ref. [ 21, the area representing deceleration appears to be less than the acceleration area (as it is). The lower deceleration area in Haaland’s Fig. 2 results primarily from the additional factor in Eq. (9). The net effect of this important factor is that the electron will obtain a substantial gain in energy when the acceleration effect is integrated over the path length from minus to plus infinity. This gain negates claim (4) by Sprangle et al. Moreover, the second term in Eq. ( 10) shows a $ factor in the numerator where Eq. (3b) of Ref. [l] shows only .Z The cube has inadvertently been dropped in Eq. (3b) in Ref. [ 11, as may be readily seen by examining Eq. (2a) in Ref. [ 1] and going through the steps leading to Eq. (3b). If this cube had been carried through to the arguments of the sine term in Eq. (6) of Ref. [ 11, it is evident that as z. + a, A W would not go to zero, as claimed by Sprangle et al. Thus, their claim (4) that “the net energy change vanishes for infinite interaction distances in the crossed beam contigura-
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tion”, in contrast to the finite and substantial energy gains resulting from Haaland’s calculations, is not substantiated by their Eq. (6) because this equation is incorrect. The third term of Eq. ( 10) does not contain the factor two which appears in Eq. (3a) of Sprangle et al. If this factor is introduced into Eq. (lo), the slip length is found to vary with spot size. This factor arises from Eq. ( lb) which Sprangle et al. use for the E,, field component. As stated above, this equation does not correctly describe the laser beam properties at zi = 0, is not referenced by them, nor is the derivation indicated, and it does not appear in any of the references cited in connection with Eq. ( 1) of this response. Because of the errors in Eqs. (3a) and (3b) obtained by Sprangle et al., their derivative Eqs. (4), (5) and (6) are also incorrect, thus invalidating their claims against Haaland’s paper. The Lawson-Woodward theorem described in Ref. [ 11, which Haaland referred to as the Palmer theorem because Palmer was the first to completely define it [ 91, is a neat theorem that applies to plane electromagnetic waves of infinite lateral extent, and the conclusion that electrons cannot be accelerated by such waves over
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an infinite interaction length is almost self-evident. The theorem does not extend to the physical case of converging Gaussian beams with curved phase fronts with constantly changing radius of curvature. The results obtained in Ref. [ 21 provide evidence of a case where the Lawson-Woodward-Palmer theorem does not apply.
References [ l] P. Sprangle, E. Esarey, .I. KralI and A. Ting, Optics Comm. 124 (1996) 69. [ ] C.M. Haaland, Optics Comm. 114 ( 1995) 280. [3] G.D. Boyd and J.P. Gordon, Bell System Technical Journal 40 (1961) 489. [4] J.R. Pierce, Proc. National Academy of Science47 ( 1961) 1808. [5] S. Ramo, J.R. Whinnery and T. von Duzer, Fields and Waves in Communications Electronics (John Wiley and Sons, New York, NY, 1965) p. 572. [6] H. Kogelnik and T. Li, Appl. Optics 5 (1966) 1550. [7] D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold Company, New York, NY, 1972) p. 234. [ 81 A.E. Siegman, Lasers (University Science Books, Mill Valley, CA, 1986) p. 663. [ 91 R.B. Palmer, Lecture Notes in Physics 296, Frontiers of Particle Beams (Springer, Berlin, 1988) p. 607.