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Coherent diffraction of electrons by standing light waves
Volume 4. number
COHERENT
November 1971
OPTICS COMMUNICATIONS
3
DIFFRACTION
OF
ELECTRONS
BY
STANDING
LIGHT
WAVES
H. DE LANG Philips
Research
Laboratories,
N. V. Philips’
Eindhoarn,
Glocilanlpenfabriekcn,
The hTetherlmds
Received 17 September 1971 It is shown that under common experimental conditions the diffraction of electrons by a standing light wave (Kapitza-Dirac scattering) contains a coherent part which although not recognized by existing theory is of major importance. New discussion of previous experimental results (in particular those of Schwarz) is indicated. New experiments are proposed where the diffracted electron beams interfere with the undiffracted beam.
The diffraction of electrons by standing light waves was predicted by Kapitza and Dirac in 1933 [l], but it was not until recently that the availability of very strong light sources made a meaningful experimental test possible. Several experiments have been carried out [2-51. In this paper we will question the adequacy of some of the theoretical arguments used in the discussion of these experiments, and will propose new experiments making use of the coherence properties of Kapitza-Dirac scattering. Let us first discuss a theoretical expression for the Bragg reflection of electrons at a standing light wave which in current literature is considered to be relevant under experimental conditions. This expression runs N/N,
=le4~oIoI~/16~2~~2h2~4~~oA~
,
(1)
where No = intensity of incident electron beam; N = intensity of Bragg reflection; 1 = thickness of standing light wave; 2, = velocity of electron; m = mass of electron; I = charge of electron; k = Planck’s constant; u = frequency of light; lo and 1; are the intensities of the two travelling light waves; IO =Jl(v)dv
;
Au = 1,Z&//‘(v)l’(v)
I; =Jl’(v)dv
;
dv ;
p. = permeability of free space; co = permittivity of free space The angular spread AT of the light in the plane containing the direction of the incident electrons has to satisfy the condition A71 << cAu/2vv where c = velocity of light. Although the light must be
quasi-monochromatic, a further condition for the validity of eq. (1) is that the time of flight of the electrons is much larger than the coherence time of the light, i.e. : lAv/ll’
X,1
.
(2)
In another case of experimental relevance considered by Bartell et al. [4] the light is again assumed to be quasi-monochromatic. but here the angular spread Au satisfies A?7
CAU,~~VU
and Arl __ c/21v
(3)
.
The reflection
probability
N,/N, = cle4 /_~,1,1;,/16rr~v~
is then calculated Aqm2h2 v5eo
as (4)
For the exact definition of Ag see ref. [4], p. 1495. Eqs. (1) and (4) are valid only if the standing light wave acts on the electron beam as a very weak phase structure i.e. if the right-hand sides of these equations are a very small fraction of unity. If this condition is not fulfilled the expressions take a more complicated form [S-S]. Confining ourselves to the approximation of a weak phase structure we will now discuss the coherence properties of the diffracted wave. Eqs. (1) and (3) are derived by summation of the electron intensities diffracted by all combinations of plane wave components of the two travelling light waves. In the case of eq. (1) these components are distributed over the frequency whereas for eq. (3) the distribution is over the direction. Since in both cases the components of an arbi191
Volume 4. nulnber 3
OPTICS
COMMUNICATIONS
trary combination have a random phase relation, the phase relation between the incident electron wave and the diffracted electron wave contributed by this combination is random too. The diffracted beam resulting from the summation of all these contributions will thus have no fixed phase relation with the incident beam. Further. even if the incident electron wave is fully collimated the diffracted wave has an angular spread. The angular spread due to the frequency spread of the light is of the order of lzAv/rr~)c whereas that due to the angular spread of the light is of the order of I~v(A~)~,/wIuc. The incoherence of the diffracted wave represented by eq. (I) seems to be in contradiction to the stationary position of the standing intensity pattern formed when a light beam with a spread in frequency or direction is reflected by a plane mirror. This intensity pattern is fully modulated in the vicinity of the mirror. In a theoretical paper [9], Bartell implicitly attributes the incoherent scattering to this standing intensity pattern by stating on p. 1565 that eq. (1) is valid only at a distance from the mirror which is small compared with the coherence length (./Au of the light. In a later publication [4], p. 1495, he withdraws this statement and declares eq. (1) to be valid at any distance from the mirror. The latter viewpoint sounds reasonable since in the derivation of this expression the distance from the mirror does not appear as a parameter. Moreover, the standing intensity pattern can exist only by virtue of the coherent properties of the two running waves. In the derivations of eq. (1). however, such coherence properties have not been accounted for. The conclusion is apparently that eq. (1). although correct in itself. represents only a part of the diffraction. viz. the incoherent part due to time fluctuations in the interference pattern of the travelling waves: apart from the incoherent part a coherent COW tribution may be expected owing to the phase grating represented by the time-stationary part of the interference pattern. Incidentally, this time-stationary part of the interference is the only one manifest when using slow optical square-law detection. Now in order to analyse the role of the standing intensity pattern in Kapitza-Dirac scattering, let us again consider the case of a quasi-monochromatic plane light wave incident on a plane mirror. As explained above, eq. (1) is the result of a summation of the intensities diffracted by combinations of independent frequency components. However, among these combinations there are special ones, viz. those where a frequency 192
component of the incident wave is combined with its own reflection. The diffracted beams contributed by such combinations are all in phase if the diffraction takes place within a distance from the mirror which is small compared with the coherence length c,/Au of the light. which in practice can be of the order of several centimetres or more. The total intensity diffracted by these special combinations (which together form the stationary part of the interference pattern) is obtained by squaring the sum of the diffracted amplitudes. Making use of a formula for purely monochromatic light (see e.g. ref. [4]. p. 1495, eq. (2)) the relative amplitude (apart from a constant phase factor) contributed by an infinitesimal frequency interval dv can be written
where Y is the amplitude reflectance of the mirror: .f( f_Y)= (sincu)/a; 01= 2iiZ(li - Og)jh; (r- fig = deviation from the Bragg angle. Actually the Bragg angle 8B is frequency dependent. The corresponding variation of cy is of the order of IX, UAU‘c2 which in realistic cases is very much less than unity: hence the frequency dependence of _f( a) is negligible. Integration of (5) over the frequency and squaring gives the relative intensity of the coherent Bragg reflection close to the mirror: N
cob
‘No = I2 e4 pi0 I, I;,;l6;;
2
2242 Wl h v v E
0’
(6)
yhere210 is the intensity of the incident wave and lo -2Y I, is the intensity of the reflected wave. If the distance n of the electron beam from the mirror is not very small compared with the coherence length. the differential contributions of eq. (5) get out of phase because of the variation of the phase factor exp (4riva/c) over the frequency range. Instead of (6) we then find
where F = (s I( u) exp (4ni W/C) d u)/1, Incidentally. F is a measure of the coherence between the two travelling waves; in optics IF, is known as the reduction factor for the contrast of a two-beam interference pattern as caused by frequency spread and path difference. If we compare eqs. (1) and (6) we see that near the mirror where F = 1 we have h’coh,‘Nincoh = lAU/V which according to eq. (2)
Volume 4, number 3
OPTICS COMMUNICATIONS
[a necessary condition for the validity of eq. (l)] is much larger than unity. This indicates the importance of the coherent diffraction if the electron beam crosses the light beam near the mirror and the time of flight is much larger than the coherence time. The question remains whether an electron beam with a width well within the coherence length of the light is a broad beam in the sense that it does not substantially broaden by Fresnel diffraction after travelling over a distance I through the light beam. A criterion for that is that the Fresnel number should be much larger than unity. Taking the width of the electron beam as one tenth of the coherence length of the light, we find the Fresnel number to be about It is easily verified that in 10-3c2/(Av)2ZXe. relevant cases this is a very large number. Let us now turn to the case of a light beam with angular spread. If such a light beam is incident on a plane mirror there will be a standing intensity pattern with a modulation depth which goes from unity to zero as the distance from the mirror increases from zero to infinity. The standing intensity pattern formed when a light beam with angular spread is reflected by a plane mirror can be formally considered to be caused by a virtual frequency spread. Designating the deviation from normal incidence by y (where 0 C y cx 1) we define the total intensity of the incident light between y and y +dy by H(y) ydy. The angular deviation y is associated with a virtual frequency shift according to (v- vo)/vo
= - r2/2
.
(8)
Hence the distribution function Z(V) of the intensity over the frequency, to be inserted in the expression for Fin eq. (7), is defined by I(v) = H({-2(v-
=o
vo)/vo}1’2)/vo
if
v G v. ;
if
u>
v. .(9)
From eqs. (8) and (9) we see that the virtual relative spread in frequency is roughly given by the square of the angular spread. The coherence length is therefore given by c/(A~)~ v. The reason why an angular spread can be formally described by a spread in frequency here is that the nodal planes of the intensity pattern formed by an incident plane wave and its reflection by a plane mirror are always parallel to the mirror surface, independent of the angle of incidence. If however the reflector is a retrodirective one (e.g. a Porro prism) then the direction of the nodal planes is a linear function of y. causing the interference pattern to fade out even
November
1971
near the reflector and hence causing the coherent Kapitza-Dirac scattering to be much less than that obtained using a plane mirror. We shall now discuss some new viewpoints raised by the existence of coherent diffraction in previous experimental situations. It is of historical interest to consider the experiment originally proposed by Kapitza and Dirac [l]. Here the coherence length of the light is several centimetres, so if the electron beam crosses the light beam within say 0.5 cm from the mirror we have Ncoh’Nincoh = lAV/u which is about 300 for this case. However. we must remember that in order to keep f(o) [see eq. (5)] close to unity, the angular spread of the electron beam would have to be no more than about A,‘41 = 10m6 radians. If the angular spread would be of the order of 10e3 radians, which is just sufficient to discriminate the diffracted beam from the direct beam, the average value off2(cr) would be of the order of 2.5 x 10m3% reducing the coherent intensity to the same order of magnitude as the incoherent one. As a more recent example we shall now discuss a critical remark by Bartell et ai. (ref. [4], p. 1503) concerning an experimental result claimed by Schwarz [3]. As one of the results of scattering experiments with an electron beam close to the mirror, Schwarz reports to have discriminated between upward scattering and downward scattering. This result is questioned by Bartell et al. with the argument that the large frequency spread of the light in this experiment leads to an extremely diffuse distribution of effective Bragg plane orientations and should make the expected value of the up-down difference less than the scattering in either direction alone by several orders of magnitude. This argument certainly holds as far as the incoherent scattering is concerned. For the coherent part of the diffraction. however. the variation in effective Bragg plane orientation is of the order of AI&,/C = 4 X 10e7 rad’ ians which is much smaller than the tolerance in the direction of incidence for keeping f( LY)= 1, which is X,/41 Y 10m4 radians. As to the spectral width of the laser used by Schwarz. this is reported to be 10 A resulting in the impractically small coherence length of only 1 mm. Presumably this width is the envelope of the spectral distribution of all spikes. As, however, the width of the individual spikes can be expected to be considerable smaller the effective coherence length might be large enough. Concerning the experimental work of Bartell et al. [4], we would like to comment on the kind 193
Volume 4. number 3
OPTICS COMMUNICATIONS
of reflector used. As mentioned above this reflector, a Porro prism, does not possess the favourable property of a plane mirror that the orientation of the nodal planes is independent of the direction of incidence of the light. This causes the coherent scattering when using a Porro prism to be considerably smaller than that when using a plane mirror. Finally, we would like to propose a novel kind of Kapitza-Dirac experiment making use of the coherence properties of the electron diffraction. The idea is to bring the diffracted electron beams to interference with the undiffracted one. The modulation depth of the resulting periodic electron intensity pattern, which can be recorded on a photographic plate or made visible with a fluorescent screen, is a measure of the relative amplitude of the diffracted beams. This offers a double advantage. Firstly, since the relative amplitude is the square root of the relative intensity the detectability of very weak diffracted beams is greatly enhanced. Secondly, the detection of a periodic intensity pattern with a period corresponding to that of the standing light wave would provide convincing evidence of the reality of Kapitza-Dirac scattering. Since even a low diffraction probability, say 10W3, will give a detectable modulation in the electron interference pattern the use of a continuous light source becomes feasible, in particular if a long wavelength is chosen (e.g. the 10.6 ~1 radiation of a CO2 laser). As a first possibility, let us suppose the thickness I of the light beam to be large enough to ensure that pure Bragg reflection can occur, the condition for this being that 1 N c2/4v2Xe or larger. In this case there are only two electron beams emerging from the standing light wave, viz. the undiffracted one and the Bragg reflection. These two beams give a sinusoidal intensity pattern. In order to record this pattern the standing light wave is imaged on the recording target with the aid of a simple electron-optical system.
194
November 1971
If the thickness 1 of the standing light wave is about c2/16v2Ae or smaller there will be little difference in the intensities of upward and downward scattering. There are then three electron beams emerging from the standing light wave, viz. the undiffracted beam and two equally strong diffracted ones. In order to obtain phase contrast, i.e. and intensity modulated interference pattern, it is necessary to shift the phase of the diffracted beams over 7r/2 with respect to that of the undiffracted one. This can be achieved either by imaging the standing light wave with the aid of a defocussed electron-optical system or using no optical system at all and placing the target surface at a suitable distance behind the standing light wave. It is also possible to obtain the required phase difference with the aid of an electron-optical system with a suitable amount of spherical aberration. In conclusion, we consider the role of the coherent Kapitza-Dirac scattering to have been underestimated in literature, and suggest that existing experimental results should be rediscussed in this light; a number of experiments with coherent detection are proposed.
REFERENCES [l] P. L. Kapitza and P. A. M. Dirac, Proc. Cambridge Phil. Sot. 29 (1933) 297. [2] II. Schwarz, H. A. Tourtelotte and it’. 14”.Gaertner, Phys. Letters 19 (1965) 202. [3] H. Schwarz, Z. Physik 204 (1967) 276. [4] L. S. Bartell, R. R. Roskos and H. Bradford Thomson, Phys. Rev. 166 (1968) 1494. [5] H. Chr. Pfeiffer, Phys. Letters 26A (1968) 362. [6] H.Schoenebeck, Phys. Letters 27A (1968) 286. [7] M. V. Fedorov, Soviet Phys. JETP 25 (1967) 952. [8] R. Gush and H. P. Gush, Phys. Rev. D3 (1971) 1712. [9] L. S. Bartell, J.Appl. Phys. 38 (1967) 1561.
COHERENT
November 1971
OPTICS COMMUNICATIONS
3
DIFFRACTION
OF
ELECTRONS
BY
STANDING
LIGHT
WAVES
H. DE LANG Philips
Research
Laboratories,
N. V. Philips’
Eindhoarn,
Glocilanlpenfabriekcn,
The hTetherlmds
Received 17 September 1971 It is shown that under common experimental conditions the diffraction of electrons by a standing light wave (Kapitza-Dirac scattering) contains a coherent part which although not recognized by existing theory is of major importance. New discussion of previous experimental results (in particular those of Schwarz) is indicated. New experiments are proposed where the diffracted electron beams interfere with the undiffracted beam.
The diffraction of electrons by standing light waves was predicted by Kapitza and Dirac in 1933 [l], but it was not until recently that the availability of very strong light sources made a meaningful experimental test possible. Several experiments have been carried out [2-51. In this paper we will question the adequacy of some of the theoretical arguments used in the discussion of these experiments, and will propose new experiments making use of the coherence properties of Kapitza-Dirac scattering. Let us first discuss a theoretical expression for the Bragg reflection of electrons at a standing light wave which in current literature is considered to be relevant under experimental conditions. This expression runs N/N,
=le4~oIoI~/16~2~~2h2~4~~oA~
,
(1)
where No = intensity of incident electron beam; N = intensity of Bragg reflection; 1 = thickness of standing light wave; 2, = velocity of electron; m = mass of electron; I = charge of electron; k = Planck’s constant; u = frequency of light; lo and 1; are the intensities of the two travelling light waves; IO =Jl(v)dv
;
Au = 1,Z&//‘(v)l’(v)
I; =Jl’(v)dv
;
dv ;
p. = permeability of free space; co = permittivity of free space The angular spread AT of the light in the plane containing the direction of the incident electrons has to satisfy the condition A71 << cAu/2vv where c = velocity of light. Although the light must be
quasi-monochromatic, a further condition for the validity of eq. (1) is that the time of flight of the electrons is much larger than the coherence time of the light, i.e. : lAv/ll’
X,1
.
(2)
In another case of experimental relevance considered by Bartell et al. [4] the light is again assumed to be quasi-monochromatic. but here the angular spread Au satisfies A?7
CAU,~~VU
and Arl __ c/21v
(3)
.
The reflection
probability
N,/N, = cle4 /_~,1,1;,/16rr~v~
is then calculated Aqm2h2 v5eo
as (4)
For the exact definition of Ag see ref. [4], p. 1495. Eqs. (1) and (4) are valid only if the standing light wave acts on the electron beam as a very weak phase structure i.e. if the right-hand sides of these equations are a very small fraction of unity. If this condition is not fulfilled the expressions take a more complicated form [S-S]. Confining ourselves to the approximation of a weak phase structure we will now discuss the coherence properties of the diffracted wave. Eqs. (1) and (3) are derived by summation of the electron intensities diffracted by all combinations of plane wave components of the two travelling light waves. In the case of eq. (1) these components are distributed over the frequency whereas for eq. (3) the distribution is over the direction. Since in both cases the components of an arbi191
Volume 4. nulnber 3
OPTICS
COMMUNICATIONS
trary combination have a random phase relation, the phase relation between the incident electron wave and the diffracted electron wave contributed by this combination is random too. The diffracted beam resulting from the summation of all these contributions will thus have no fixed phase relation with the incident beam. Further. even if the incident electron wave is fully collimated the diffracted wave has an angular spread. The angular spread due to the frequency spread of the light is of the order of lzAv/rr~)c whereas that due to the angular spread of the light is of the order of I~v(A~)~,/wIuc. The incoherence of the diffracted wave represented by eq. (I) seems to be in contradiction to the stationary position of the standing intensity pattern formed when a light beam with a spread in frequency or direction is reflected by a plane mirror. This intensity pattern is fully modulated in the vicinity of the mirror. In a theoretical paper [9], Bartell implicitly attributes the incoherent scattering to this standing intensity pattern by stating on p. 1565 that eq. (1) is valid only at a distance from the mirror which is small compared with the coherence length (./Au of the light. In a later publication [4], p. 1495, he withdraws this statement and declares eq. (1) to be valid at any distance from the mirror. The latter viewpoint sounds reasonable since in the derivation of this expression the distance from the mirror does not appear as a parameter. Moreover, the standing intensity pattern can exist only by virtue of the coherent properties of the two running waves. In the derivations of eq. (1). however, such coherence properties have not been accounted for. The conclusion is apparently that eq. (1). although correct in itself. represents only a part of the diffraction. viz. the incoherent part due to time fluctuations in the interference pattern of the travelling waves: apart from the incoherent part a coherent COW tribution may be expected owing to the phase grating represented by the time-stationary part of the interference pattern. Incidentally, this time-stationary part of the interference is the only one manifest when using slow optical square-law detection. Now in order to analyse the role of the standing intensity pattern in Kapitza-Dirac scattering, let us again consider the case of a quasi-monochromatic plane light wave incident on a plane mirror. As explained above, eq. (1) is the result of a summation of the intensities diffracted by combinations of independent frequency components. However, among these combinations there are special ones, viz. those where a frequency 192
component of the incident wave is combined with its own reflection. The diffracted beams contributed by such combinations are all in phase if the diffraction takes place within a distance from the mirror which is small compared with the coherence length c,/Au of the light. which in practice can be of the order of several centimetres or more. The total intensity diffracted by these special combinations (which together form the stationary part of the interference pattern) is obtained by squaring the sum of the diffracted amplitudes. Making use of a formula for purely monochromatic light (see e.g. ref. [4]. p. 1495, eq. (2)) the relative amplitude (apart from a constant phase factor) contributed by an infinitesimal frequency interval dv can be written
where Y is the amplitude reflectance of the mirror: .f( f_Y)= (sincu)/a; 01= 2iiZ(li - Og)jh; (r- fig = deviation from the Bragg angle. Actually the Bragg angle 8B is frequency dependent. The corresponding variation of cy is of the order of IX, UAU‘c2 which in realistic cases is very much less than unity: hence the frequency dependence of _f( a) is negligible. Integration of (5) over the frequency and squaring gives the relative intensity of the coherent Bragg reflection close to the mirror: N
cob
‘No = I2 e4 pi0 I, I;,;l6;;
2
2242 Wl h v v E
0’
(6)
yhere210 is the intensity of the incident wave and lo -2Y I, is the intensity of the reflected wave. If the distance n of the electron beam from the mirror is not very small compared with the coherence length. the differential contributions of eq. (5) get out of phase because of the variation of the phase factor exp (4riva/c) over the frequency range. Instead of (6) we then find
where F = (s I( u) exp (4ni W/C) d u)/1, Incidentally. F is a measure of the coherence between the two travelling waves; in optics IF, is known as the reduction factor for the contrast of a two-beam interference pattern as caused by frequency spread and path difference. If we compare eqs. (1) and (6) we see that near the mirror where F = 1 we have h’coh,‘Nincoh = lAU/V which according to eq. (2)
Volume 4, number 3
OPTICS COMMUNICATIONS
[a necessary condition for the validity of eq. (l)] is much larger than unity. This indicates the importance of the coherent diffraction if the electron beam crosses the light beam near the mirror and the time of flight is much larger than the coherence time. The question remains whether an electron beam with a width well within the coherence length of the light is a broad beam in the sense that it does not substantially broaden by Fresnel diffraction after travelling over a distance I through the light beam. A criterion for that is that the Fresnel number should be much larger than unity. Taking the width of the electron beam as one tenth of the coherence length of the light, we find the Fresnel number to be about It is easily verified that in 10-3c2/(Av)2ZXe. relevant cases this is a very large number. Let us now turn to the case of a light beam with angular spread. If such a light beam is incident on a plane mirror there will be a standing intensity pattern with a modulation depth which goes from unity to zero as the distance from the mirror increases from zero to infinity. The standing intensity pattern formed when a light beam with angular spread is reflected by a plane mirror can be formally considered to be caused by a virtual frequency spread. Designating the deviation from normal incidence by y (where 0 C y cx 1) we define the total intensity of the incident light between y and y +dy by H(y) ydy. The angular deviation y is associated with a virtual frequency shift according to (v- vo)/vo
= - r2/2
.
(8)
Hence the distribution function Z(V) of the intensity over the frequency, to be inserted in the expression for Fin eq. (7), is defined by I(v) = H({-2(v-
=o
vo)/vo}1’2)/vo
if
v G v. ;
if
u>
v. .(9)
From eqs. (8) and (9) we see that the virtual relative spread in frequency is roughly given by the square of the angular spread. The coherence length is therefore given by c/(A~)~ v. The reason why an angular spread can be formally described by a spread in frequency here is that the nodal planes of the intensity pattern formed by an incident plane wave and its reflection by a plane mirror are always parallel to the mirror surface, independent of the angle of incidence. If however the reflector is a retrodirective one (e.g. a Porro prism) then the direction of the nodal planes is a linear function of y. causing the interference pattern to fade out even
November
1971
near the reflector and hence causing the coherent Kapitza-Dirac scattering to be much less than that obtained using a plane mirror. We shall now discuss some new viewpoints raised by the existence of coherent diffraction in previous experimental situations. It is of historical interest to consider the experiment originally proposed by Kapitza and Dirac [l]. Here the coherence length of the light is several centimetres, so if the electron beam crosses the light beam within say 0.5 cm from the mirror we have Ncoh’Nincoh = lAV/u which is about 300 for this case. However. we must remember that in order to keep f(o) [see eq. (5)] close to unity, the angular spread of the electron beam would have to be no more than about A,‘41 = 10m6 radians. If the angular spread would be of the order of 10e3 radians, which is just sufficient to discriminate the diffracted beam from the direct beam, the average value off2(cr) would be of the order of 2.5 x 10m3% reducing the coherent intensity to the same order of magnitude as the incoherent one. As a more recent example we shall now discuss a critical remark by Bartell et ai. (ref. [4], p. 1503) concerning an experimental result claimed by Schwarz [3]. As one of the results of scattering experiments with an electron beam close to the mirror, Schwarz reports to have discriminated between upward scattering and downward scattering. This result is questioned by Bartell et al. with the argument that the large frequency spread of the light in this experiment leads to an extremely diffuse distribution of effective Bragg plane orientations and should make the expected value of the up-down difference less than the scattering in either direction alone by several orders of magnitude. This argument certainly holds as far as the incoherent scattering is concerned. For the coherent part of the diffraction. however. the variation in effective Bragg plane orientation is of the order of AI&,/C = 4 X 10e7 rad’ ians which is much smaller than the tolerance in the direction of incidence for keeping f( LY)= 1, which is X,/41 Y 10m4 radians. As to the spectral width of the laser used by Schwarz. this is reported to be 10 A resulting in the impractically small coherence length of only 1 mm. Presumably this width is the envelope of the spectral distribution of all spikes. As, however, the width of the individual spikes can be expected to be considerable smaller the effective coherence length might be large enough. Concerning the experimental work of Bartell et al. [4], we would like to comment on the kind 193
Volume 4. number 3
OPTICS COMMUNICATIONS
of reflector used. As mentioned above this reflector, a Porro prism, does not possess the favourable property of a plane mirror that the orientation of the nodal planes is independent of the direction of incidence of the light. This causes the coherent scattering when using a Porro prism to be considerably smaller than that when using a plane mirror. Finally, we would like to propose a novel kind of Kapitza-Dirac experiment making use of the coherence properties of the electron diffraction. The idea is to bring the diffracted electron beams to interference with the undiffracted one. The modulation depth of the resulting periodic electron intensity pattern, which can be recorded on a photographic plate or made visible with a fluorescent screen, is a measure of the relative amplitude of the diffracted beams. This offers a double advantage. Firstly, since the relative amplitude is the square root of the relative intensity the detectability of very weak diffracted beams is greatly enhanced. Secondly, the detection of a periodic intensity pattern with a period corresponding to that of the standing light wave would provide convincing evidence of the reality of Kapitza-Dirac scattering. Since even a low diffraction probability, say 10W3, will give a detectable modulation in the electron interference pattern the use of a continuous light source becomes feasible, in particular if a long wavelength is chosen (e.g. the 10.6 ~1 radiation of a CO2 laser). As a first possibility, let us suppose the thickness I of the light beam to be large enough to ensure that pure Bragg reflection can occur, the condition for this being that 1 N c2/4v2Xe or larger. In this case there are only two electron beams emerging from the standing light wave, viz. the undiffracted one and the Bragg reflection. These two beams give a sinusoidal intensity pattern. In order to record this pattern the standing light wave is imaged on the recording target with the aid of a simple electron-optical system.
194
November 1971
If the thickness 1 of the standing light wave is about c2/16v2Ae or smaller there will be little difference in the intensities of upward and downward scattering. There are then three electron beams emerging from the standing light wave, viz. the undiffracted beam and two equally strong diffracted ones. In order to obtain phase contrast, i.e. and intensity modulated interference pattern, it is necessary to shift the phase of the diffracted beams over 7r/2 with respect to that of the undiffracted one. This can be achieved either by imaging the standing light wave with the aid of a defocussed electron-optical system or using no optical system at all and placing the target surface at a suitable distance behind the standing light wave. It is also possible to obtain the required phase difference with the aid of an electron-optical system with a suitable amount of spherical aberration. In conclusion, we consider the role of the coherent Kapitza-Dirac scattering to have been underestimated in literature, and suggest that existing experimental results should be rediscussed in this light; a number of experiments with coherent detection are proposed.
REFERENCES [l] P. L. Kapitza and P. A. M. Dirac, Proc. Cambridge Phil. Sot. 29 (1933) 297. [2] II. Schwarz, H. A. Tourtelotte and it’. 14”.Gaertner, Phys. Letters 19 (1965) 202. [3] H. Schwarz, Z. Physik 204 (1967) 276. [4] L. S. Bartell, R. R. Roskos and H. Bradford Thomson, Phys. Rev. 166 (1968) 1494. [5] H. Chr. Pfeiffer, Phys. Letters 26A (1968) 362. [6] H.Schoenebeck, Phys. Letters 27A (1968) 286. [7] M. V. Fedorov, Soviet Phys. JETP 25 (1967) 952. [8] R. Gush and H. P. Gush, Phys. Rev. D3 (1971) 1712. [9] L. S. Bartell, J.Appl. Phys. 38 (1967) 1561.