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Explanation of the diffraction fine-structure in overexposed thick holograms
Volume 15, number 2
OPTICS COMMUNICATIONS
October 1975
EXPLANATION OF THE DIFFRACTION FINE-STRUCTURE IN OVEREXPOSED THICK HOLOGRAMS M.R.B. FORSHAW EMI Electronics Ltd., Feltham, Middlesex, TW13 7DZ, UK
Received 23 June 1975
The far-field diffraction patterns of overexposed thick holographic gratings often contain circular or arcuate structures. The geometrical properties of these patterns were explained earlier. This letter explains the fine-structure of the rings using an extension of the kinematical theory of X-ray diffraction.
1. Introduction A number of authors have described the unexpected appearance of diffraction effects which occur when thick holographic recording media are "over-exposed" to coherent radiation. In 1970 Biedermann [1] reported unexplained maxima in the light scattered from photographic emulsions. In 1972 the angular selectivity of the effect was demonstrated in LiNbO 3 [2]. In 1973 a similar effect was demonstrated in DCG and used in the design of holographic viewing screens [3]. Also in 1973, Moran and Kaminow first showed the characteristic feature of the effect, viz., the appearance of circular and arcuate maxima in the far-field diffraction pattern [4]. These authors used PMMA exposed to a single intense laser beam. The geometrical structure of these maxima was explained shortly afterwards [5] in terms of the kinematical theory of X-ray or optical diffraction [6] and it was shown that similar patterns were also obtained in overexposed twobeam gratings in photopolymers. In 1974 Magnusson and Gaylord [7] extended the theory of ref. [5] to two-beam gratings, and a further example of the phenomenon has recently been illustrated [8]. Although the geometrical properties of the scattered light have been explained satisfactorily, the detailed fine-structure of the maxima has not been accounted for. It was suggested in ref. [5] that the explanation lay in the use of the dynamical theory of optical or X-ray diffraction, since some features of the fine structure resembled those of Kossel lines [9]. 218
However, it appears that an explanation can be provided in terms of the simpler kinematical theory.
2. Qualitative theory It is proposed that light is scattered from randomly distributed inhomogeneities in a thick recording medium when it is illuminated with one or more coherent plane waves. These inhomogeneities may be present in the medium prior to exposure: for example, they may be silver halide grains in a photographic emulsion, or impurities and mixing defects in a photopolymer or ferroelectric crystal. Alternatively, they may be optically-induced defects due to high-intensity regions of an exposing speckle pattern. Each defect or inhomo. geneity will scatter some of the incident light and the scattered light interferes with the unperturbed component to form a complex fringe pattern. If the material is of a self-developing type (e.g. a photopolymer) then the induced fringes may diffract additional light, leading to a reinforcement of the induced fringe pattern. If the material is an emulsion or DCG then the effects of the induced fringes will only be noticeable when the medium is overdeveloped. The diffraction patterns of interest are observed when the "over-exposed" medium is illuminated with a reconstructing wave which is not identical with one or all of the original recording waves. It is proposed that the details of the fine-structure can be explained by assuming that there are at least three sets of wave-
Volume 15, number 2
OPTICS COMMUNICATIONS
fields present during reconstruction. The first is the normal set of diffracted waves, i.e. those expected in the absence of scattering. The second set is generated by diffraction from the induced fringe pattern mentioned above. This wavefield set contains only a limited number of plane-wave components, travelling mainly in the directions of the rings and arcs as described in [ 5 - 7 ] . The third set of waves is produced through scattering of the reconstructing light by the original scattering defects or inhomogeneities, which are still present in the medium. This last set of waves forms a background speckle pattern in the far field, but the amplitude and phase of this background is correlated with the amplitude and phase of the scattered light originally present during recording.
3. Quantitative theory It is assumed that the total amount of light scattered from the recording or reconstructing waves by the inhomogeneities and the induced fringe patterns is small in comparison with the incident intensity, so that the kinematical theory of optical scattering is applicable [10]. Only non-absorbing phase media are considered. Polarization effects are ignored and a single recording wave is considered, whose amplitude at any point r in the medium may be expressed as Er(r) = EroeX p [ik r ' r ] ,
(1)
where k r is the reference wave vector inside the medium; Ikrl = koK~/2 where K o is the mean dielectric constant and k 0 is the vacuum wavevector. The original wavefield scattered from a single in. homogeneity can be expanded in terms of plane waves. To simplify the notation only one scattering centre will be considered for the moment. If this is located at r = r n then the scattered plane-wave component travelling in any direction k s may be written in the form
X exp [ik s- (r - r n)] asl exp [iCsl ].
slowly varying functions of (k s - kr). The total scattered amplitude is obtained by integrating (2) over all possible values of k s and summing over the scattering centres. This total scattered wavefield then interferes with the reference wave (1) to produce a modulation of the dielectric constant K(r). It is assumed that each cisoidal component of the fringe pattern has an amplitude which is linearly proportional to the amplitude (and ptaase) of the corresponding scattered wave (2). The resultant multiple grating is illuminated by a reconstructing wave E c(r) = Ec0 exp Ilk c "r].
(2)
Expression (2) assumes that the scatterer size is very small, so that there is a definite phase relationship between the incident wave (the first two terms) and the scattered wave (the last three terms). It is assumed that the modulus asl and phase ~sl of the "envelope" are
(3)
It is assumed here that ]kcl = [kr[ but the theory is readfly extended to incorporate wavelength changes [5,7]. The amplitude of the diffracted wave generated by the fringe component corresponding to the original scattered component (2) is most easily derived by taking the corresponding dynamical theory formula (e.g. [11], formula 42) and letting the fringe modulation tend to zero. It can be shown [12] that the reconstructed "image" amplitude at the rear surface of the medium is
Ei(mD ) -"- Ec0 - iaiex p [i(q~sl + (k r - ks).rn) ] X exp[i(k c + k s - k r ) . m D ] e x p [ i x ] s i n [ x ] / x
,
(4)
where a i ~" koKs/K 1/2, m = outward unit vector normal to rear surface of hologram, D = hologram thickness,
x = ½koK1/2D A0 sin[kr, ks] ,
(5)
[k r, ks] = angle between the recording vectors, A0 = deviation from Bragg angle for the particular fringe pattern, measured in the plane containing k r and k s. Ks, the dielectric constant amplitude modulus of the particular fringe pattern, is assumed to be proportional to asl. Expression (4) describes a diffracted plane wave, travelling approximately in the direction k i = k c + k s --
Esl (r) = Er0eX p [ik r.rn]
October 1975
k r .
In addition to the diffracted wave (4), there is another wave travelling in the direction k i. This is the wave scattered from the original inhomogeneity by the reconstructing wave: Es2 (mD) = E 0exp [ik c .r n ] X exp [ik i.(mD - rn) ] as2 exp [i¢s2 ].
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Volume 15, number 2
OPTICS COMMUNICATIONS
October 1975
If the scattering centres are randomly distributed then the far field intensity in the direction of k i can be approximated by calculating the intensity due to the effect o f one scatterer alone and then applying a speckle pattern modulation term. From (4) and (6), the intensity in the direction k i becomes
o
I(x) = E 2c0(as22 + a2(sin [x]/x) 2 - 2as2aisin Ix] X sin[x + (~bsl - - ~bs2) + A k ' ( r n + roD)]/x),
(7)
where Ak = k c - k i + k s - k r.
(8)
Expression (7) applies to reconstructed waves generated from the "primary image surface" [6]. A different expression must be used for waves generated from the "conjugate image surface", in which Ak is replaced by Ak' = k c - k i - k s + k r.
(9)
It can be shown that the dot product involving Ak or Ak' is negligible in comparison with x and it is assumed that the scattering centres are so small that 0sl - 0s2 ~ 0. Fig. 1 is a plot of (7) for various values of the ratio r = (ai/as2) 2 = (diffracted intensity)/(scattered intensity). It can be seen that the intensity is an asymmetrical function of x and exhibits a characteristic crest-and-trough over a wide range of values of r.
•
IO0
.
,
,
,i
.
,
Fig. 2. Far-field pattern for over-exposed thick two-beam grating with no displacement or rotation. Parameters given in text.
4. Experiment A two-beam thick holographic grating was made using a variant of the styrene-based 4128 photopolymer described in [ 13]. The grating thickness was 1.38 mm, its mean refractive index was 1.5, the angle between the recording beams was 17.6 ° and the recording wavelength was 633 nm. Fig. 2 is a photograph taken through the grating and shows the "farfield" diffraction pattern obtained when the grating was reconstructed with a single reconstructing source. The bright spot on the right is the zero-order beam, while the spot on the left is the first-order beam. The diffraction efficiency was approximately 50%. Fig. 3 is a photograph
IO
)( v
!
~i=.
>. I.(/)
z
IM I,-
r
O-I
z
4:1
I00: 0.01
h
J
-5
I I
+5
o ANGLE
PARAMETER
( x )
Fig. 1. Variation of intensity l(x) with angle parameter x as delrmed by expressions (5) and (7).
220
Fig. 3. Far-field pattern for same grating rotated through 0.8 ° about horizontal axis and 4.5 ° about vertical axis.
Volume 15, number 2
OPTICS COMMUNICATIONS
of the farfield pattern which was obtained when the grating was rotated about its vertical and horizontal inplane axes by approximately 0.8 ° and 4.5 ° respectively.
October 1975
three correlated sets of waves present near the intersection, viz., one set of scattered waves and two sets of diffracted waves, all originating from the same scattering centres.
5. Comparison of theory and experiment 6. Discussion From [ 5 - 7 ] the right half of the pattern in fig. 2 is produced by the zero-order wave being diffracted from sets of fringes produced by interference between the original "object" (left hand) wave and light scattered from the original "reference" (right hand) wave. A complementary description applies to the left half of fig. 2. The circle is generated by intersection of the respective conjugate image surfaces and the tangent arcs are generated from the primary image surfaces [6]. From fig. 1 the difference in x from crest to trough is nearly independent o f r and equal to 2.5. Inserting this in (5) and correcting for refraction effects yields 0ext, the external angle, measured normal to the arc or circle, between crest and trough: A0ext =
5K1/2/koDsin [kr' ks] ext"
(10)
Putting [kr, ks] = 17.6 ° in (10) gives A0ex t = 1.8 mrad. From fig. 2 the measured value is 1.9 -+ 0.2 mrad. In fig. 3 the bright spot on the left is the weakly reconstructed image of the "direct" source. On the right, the large circle and the near-vertical arc are generated, as before, by diffraction of the zero-order light from fringes due to interference between the original object wave and light scattered from the original reference wave. The small circle is part of the conjugate image surface for interference between the reference wave and light scattered from the reference wave, while the broad near-horizontal arc is part of the primary image surface from the same source. Putting [kr, ks] = 17.6 ° - 0.8 ° = 16.8 ° in (10) gives A0ex t = 1.9 mrad for the angular difference between peak and trough. The experimental values are 1.9 -+ 0.2 mrad for the near-vertical arc and 2.0 -+ 0.4 mrad for the large circle. For the small circle and horizontal arc the theoretical value is A0ex t = 7.0 mrad and the experimental values are 6.8 -+ 1.0 mrad and 7.7 -+ 1.5 mrad respectively. In fig. 3 it can be seen that there is a "phase change" in the relative positions of peak and trough on the large circle where it intersects the small circle. The present theory accounts for this by assuming that there are
The present theory accounts in part for the difference which is observed between the scattering patterns from different media. When the scattered and diffracted components are comparable in intensity then fine-structures like those in figs. 2 and 3 will be observed. On the other hand, when the diffracted component is much stronger than the scattered component, as appears to be the case with photographic emulsions, then a nearlysymmetrical fine-structure will be observed [8] as indicated by the top curve of fig. I. The present theory predicts subsidiary diffraction maxima which are rarely observed: it is not clear whether this is due to experimental effects which mask the subsidiary maxima or whether the lack of maxima would be predicted by a more sophisticated theory. The theory is unable to handle the self-developing behaviour of certain media, the effect of strong scattering, or the existence of large scattering centres. Despite these limitations, it appears to provide a reasonable explanation and a quantitative description of the diffraction fine-structure in overexposed thick holograms.
References [1] K. Biedermann, Optik 31 (1970) 367. [2] W. Phillips, J.J. Amodei and D.L. Staebler, RCA Rev. 33 (1972) 94. [3] D. Meyerhofer, Appl. Opt. 12 (1973) 2180. [4] J.M. Moran and I.P. Kaminow, Appl. Opt. 12 (1973) 1974. [5] M.R.B. Forshaw, Appl. Opt. 13 (1974) 2. [6] M.R.B. Forshaw, Opt. Commun. 8 (1973) 201. [7] L. Magnusson and T.K. Gaylord, Appl. Opt. 13 (1974) 1545. [8] K. Biedermann, Optica Acta 22 (1975) 103. [9] R.W. James, The Optical Principles of the Diffraction of X-rays, Bell and Sons (London) 1967. [10] M.R.B. Forshaw, Optica Acta 20 (1973) 669. [11] H. Kogelnik, Bell Syst. Tech. Journ. 48 (1969) 2909. [12] To be published. [13] M.R.B. Forshaw, Opt. Laser Teeh. 6 (1974) 28.
221
OPTICS COMMUNICATIONS
October 1975
EXPLANATION OF THE DIFFRACTION FINE-STRUCTURE IN OVEREXPOSED THICK HOLOGRAMS M.R.B. FORSHAW EMI Electronics Ltd., Feltham, Middlesex, TW13 7DZ, UK
Received 23 June 1975
The far-field diffraction patterns of overexposed thick holographic gratings often contain circular or arcuate structures. The geometrical properties of these patterns were explained earlier. This letter explains the fine-structure of the rings using an extension of the kinematical theory of X-ray diffraction.
1. Introduction A number of authors have described the unexpected appearance of diffraction effects which occur when thick holographic recording media are "over-exposed" to coherent radiation. In 1970 Biedermann [1] reported unexplained maxima in the light scattered from photographic emulsions. In 1972 the angular selectivity of the effect was demonstrated in LiNbO 3 [2]. In 1973 a similar effect was demonstrated in DCG and used in the design of holographic viewing screens [3]. Also in 1973, Moran and Kaminow first showed the characteristic feature of the effect, viz., the appearance of circular and arcuate maxima in the far-field diffraction pattern [4]. These authors used PMMA exposed to a single intense laser beam. The geometrical structure of these maxima was explained shortly afterwards [5] in terms of the kinematical theory of X-ray or optical diffraction [6] and it was shown that similar patterns were also obtained in overexposed twobeam gratings in photopolymers. In 1974 Magnusson and Gaylord [7] extended the theory of ref. [5] to two-beam gratings, and a further example of the phenomenon has recently been illustrated [8]. Although the geometrical properties of the scattered light have been explained satisfactorily, the detailed fine-structure of the maxima has not been accounted for. It was suggested in ref. [5] that the explanation lay in the use of the dynamical theory of optical or X-ray diffraction, since some features of the fine structure resembled those of Kossel lines [9]. 218
However, it appears that an explanation can be provided in terms of the simpler kinematical theory.
2. Qualitative theory It is proposed that light is scattered from randomly distributed inhomogeneities in a thick recording medium when it is illuminated with one or more coherent plane waves. These inhomogeneities may be present in the medium prior to exposure: for example, they may be silver halide grains in a photographic emulsion, or impurities and mixing defects in a photopolymer or ferroelectric crystal. Alternatively, they may be optically-induced defects due to high-intensity regions of an exposing speckle pattern. Each defect or inhomo. geneity will scatter some of the incident light and the scattered light interferes with the unperturbed component to form a complex fringe pattern. If the material is of a self-developing type (e.g. a photopolymer) then the induced fringes may diffract additional light, leading to a reinforcement of the induced fringe pattern. If the material is an emulsion or DCG then the effects of the induced fringes will only be noticeable when the medium is overdeveloped. The diffraction patterns of interest are observed when the "over-exposed" medium is illuminated with a reconstructing wave which is not identical with one or all of the original recording waves. It is proposed that the details of the fine-structure can be explained by assuming that there are at least three sets of wave-
Volume 15, number 2
OPTICS COMMUNICATIONS
fields present during reconstruction. The first is the normal set of diffracted waves, i.e. those expected in the absence of scattering. The second set is generated by diffraction from the induced fringe pattern mentioned above. This wavefield set contains only a limited number of plane-wave components, travelling mainly in the directions of the rings and arcs as described in [ 5 - 7 ] . The third set of waves is produced through scattering of the reconstructing light by the original scattering defects or inhomogeneities, which are still present in the medium. This last set of waves forms a background speckle pattern in the far field, but the amplitude and phase of this background is correlated with the amplitude and phase of the scattered light originally present during recording.
3. Quantitative theory It is assumed that the total amount of light scattered from the recording or reconstructing waves by the inhomogeneities and the induced fringe patterns is small in comparison with the incident intensity, so that the kinematical theory of optical scattering is applicable [10]. Only non-absorbing phase media are considered. Polarization effects are ignored and a single recording wave is considered, whose amplitude at any point r in the medium may be expressed as Er(r) = EroeX p [ik r ' r ] ,
(1)
where k r is the reference wave vector inside the medium; Ikrl = koK~/2 where K o is the mean dielectric constant and k 0 is the vacuum wavevector. The original wavefield scattered from a single in. homogeneity can be expanded in terms of plane waves. To simplify the notation only one scattering centre will be considered for the moment. If this is located at r = r n then the scattered plane-wave component travelling in any direction k s may be written in the form
X exp [ik s- (r - r n)] asl exp [iCsl ].
slowly varying functions of (k s - kr). The total scattered amplitude is obtained by integrating (2) over all possible values of k s and summing over the scattering centres. This total scattered wavefield then interferes with the reference wave (1) to produce a modulation of the dielectric constant K(r). It is assumed that each cisoidal component of the fringe pattern has an amplitude which is linearly proportional to the amplitude (and ptaase) of the corresponding scattered wave (2). The resultant multiple grating is illuminated by a reconstructing wave E c(r) = Ec0 exp Ilk c "r].
(2)
Expression (2) assumes that the scatterer size is very small, so that there is a definite phase relationship between the incident wave (the first two terms) and the scattered wave (the last three terms). It is assumed that the modulus asl and phase ~sl of the "envelope" are
(3)
It is assumed here that ]kcl = [kr[ but the theory is readfly extended to incorporate wavelength changes [5,7]. The amplitude of the diffracted wave generated by the fringe component corresponding to the original scattered component (2) is most easily derived by taking the corresponding dynamical theory formula (e.g. [11], formula 42) and letting the fringe modulation tend to zero. It can be shown [12] that the reconstructed "image" amplitude at the rear surface of the medium is
Ei(mD ) -"- Ec0 - iaiex p [i(q~sl + (k r - ks).rn) ] X exp[i(k c + k s - k r ) . m D ] e x p [ i x ] s i n [ x ] / x
,
(4)
where a i ~" koKs/K 1/2, m = outward unit vector normal to rear surface of hologram, D = hologram thickness,
x = ½koK1/2D A0 sin[kr, ks] ,
(5)
[k r, ks] = angle between the recording vectors, A0 = deviation from Bragg angle for the particular fringe pattern, measured in the plane containing k r and k s. Ks, the dielectric constant amplitude modulus of the particular fringe pattern, is assumed to be proportional to asl. Expression (4) describes a diffracted plane wave, travelling approximately in the direction k i = k c + k s --
Esl (r) = Er0eX p [ik r.rn]
October 1975
k r .
In addition to the diffracted wave (4), there is another wave travelling in the direction k i. This is the wave scattered from the original inhomogeneity by the reconstructing wave: Es2 (mD) = E 0exp [ik c .r n ] X exp [ik i.(mD - rn) ] as2 exp [i¢s2 ].
(6) 219
Volume 15, number 2
OPTICS COMMUNICATIONS
October 1975
If the scattering centres are randomly distributed then the far field intensity in the direction of k i can be approximated by calculating the intensity due to the effect o f one scatterer alone and then applying a speckle pattern modulation term. From (4) and (6), the intensity in the direction k i becomes
o
I(x) = E 2c0(as22 + a2(sin [x]/x) 2 - 2as2aisin Ix] X sin[x + (~bsl - - ~bs2) + A k ' ( r n + roD)]/x),
(7)
where Ak = k c - k i + k s - k r.
(8)
Expression (7) applies to reconstructed waves generated from the "primary image surface" [6]. A different expression must be used for waves generated from the "conjugate image surface", in which Ak is replaced by Ak' = k c - k i - k s + k r.
(9)
It can be shown that the dot product involving Ak or Ak' is negligible in comparison with x and it is assumed that the scattering centres are so small that 0sl - 0s2 ~ 0. Fig. 1 is a plot of (7) for various values of the ratio r = (ai/as2) 2 = (diffracted intensity)/(scattered intensity). It can be seen that the intensity is an asymmetrical function of x and exhibits a characteristic crest-and-trough over a wide range of values of r.
•
IO0
.
,
,
,i
.
,
Fig. 2. Far-field pattern for over-exposed thick two-beam grating with no displacement or rotation. Parameters given in text.
4. Experiment A two-beam thick holographic grating was made using a variant of the styrene-based 4128 photopolymer described in [ 13]. The grating thickness was 1.38 mm, its mean refractive index was 1.5, the angle between the recording beams was 17.6 ° and the recording wavelength was 633 nm. Fig. 2 is a photograph taken through the grating and shows the "farfield" diffraction pattern obtained when the grating was reconstructed with a single reconstructing source. The bright spot on the right is the zero-order beam, while the spot on the left is the first-order beam. The diffraction efficiency was approximately 50%. Fig. 3 is a photograph
IO
)( v
!
~i=.
>. I.(/)
z
IM I,-
r
O-I
z
4:1
I00: 0.01
h
J
-5
I I
+5
o ANGLE
PARAMETER
( x )
Fig. 1. Variation of intensity l(x) with angle parameter x as delrmed by expressions (5) and (7).
220
Fig. 3. Far-field pattern for same grating rotated through 0.8 ° about horizontal axis and 4.5 ° about vertical axis.
Volume 15, number 2
OPTICS COMMUNICATIONS
of the farfield pattern which was obtained when the grating was rotated about its vertical and horizontal inplane axes by approximately 0.8 ° and 4.5 ° respectively.
October 1975
three correlated sets of waves present near the intersection, viz., one set of scattered waves and two sets of diffracted waves, all originating from the same scattering centres.
5. Comparison of theory and experiment 6. Discussion From [ 5 - 7 ] the right half of the pattern in fig. 2 is produced by the zero-order wave being diffracted from sets of fringes produced by interference between the original "object" (left hand) wave and light scattered from the original "reference" (right hand) wave. A complementary description applies to the left half of fig. 2. The circle is generated by intersection of the respective conjugate image surfaces and the tangent arcs are generated from the primary image surfaces [6]. From fig. 1 the difference in x from crest to trough is nearly independent o f r and equal to 2.5. Inserting this in (5) and correcting for refraction effects yields 0ext, the external angle, measured normal to the arc or circle, between crest and trough: A0ext =
5K1/2/koDsin [kr' ks] ext"
(10)
Putting [kr, ks] = 17.6 ° in (10) gives A0ex t = 1.8 mrad. From fig. 2 the measured value is 1.9 -+ 0.2 mrad. In fig. 3 the bright spot on the left is the weakly reconstructed image of the "direct" source. On the right, the large circle and the near-vertical arc are generated, as before, by diffraction of the zero-order light from fringes due to interference between the original object wave and light scattered from the original reference wave. The small circle is part of the conjugate image surface for interference between the reference wave and light scattered from the reference wave, while the broad near-horizontal arc is part of the primary image surface from the same source. Putting [kr, ks] = 17.6 ° - 0.8 ° = 16.8 ° in (10) gives A0ex t = 1.9 mrad for the angular difference between peak and trough. The experimental values are 1.9 -+ 0.2 mrad for the near-vertical arc and 2.0 -+ 0.4 mrad for the large circle. For the small circle and horizontal arc the theoretical value is A0ex t = 7.0 mrad and the experimental values are 6.8 -+ 1.0 mrad and 7.7 -+ 1.5 mrad respectively. In fig. 3 it can be seen that there is a "phase change" in the relative positions of peak and trough on the large circle where it intersects the small circle. The present theory accounts for this by assuming that there are
The present theory accounts in part for the difference which is observed between the scattering patterns from different media. When the scattered and diffracted components are comparable in intensity then fine-structures like those in figs. 2 and 3 will be observed. On the other hand, when the diffracted component is much stronger than the scattered component, as appears to be the case with photographic emulsions, then a nearlysymmetrical fine-structure will be observed [8] as indicated by the top curve of fig. I. The present theory predicts subsidiary diffraction maxima which are rarely observed: it is not clear whether this is due to experimental effects which mask the subsidiary maxima or whether the lack of maxima would be predicted by a more sophisticated theory. The theory is unable to handle the self-developing behaviour of certain media, the effect of strong scattering, or the existence of large scattering centres. Despite these limitations, it appears to provide a reasonable explanation and a quantitative description of the diffraction fine-structure in overexposed thick holograms.
References [1] K. Biedermann, Optik 31 (1970) 367. [2] W. Phillips, J.J. Amodei and D.L. Staebler, RCA Rev. 33 (1972) 94. [3] D. Meyerhofer, Appl. Opt. 12 (1973) 2180. [4] J.M. Moran and I.P. Kaminow, Appl. Opt. 12 (1973) 1974. [5] M.R.B. Forshaw, Appl. Opt. 13 (1974) 2. [6] M.R.B. Forshaw, Opt. Commun. 8 (1973) 201. [7] L. Magnusson and T.K. Gaylord, Appl. Opt. 13 (1974) 1545. [8] K. Biedermann, Optica Acta 22 (1975) 103. [9] R.W. James, The Optical Principles of the Diffraction of X-rays, Bell and Sons (London) 1967. [10] M.R.B. Forshaw, Optica Acta 20 (1973) 669. [11] H. Kogelnik, Bell Syst. Tech. Journ. 48 (1969) 2909. [12] To be published. [13] M.R.B. Forshaw, Opt. Laser Teeh. 6 (1974) 28.
221