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Effects of intrinsic non-linearities on efficiency and image contrast of bleached holograms
Volume
2, number
November
OPTICS COMMUNICATIONS
6
EFFECTS AND
OF
INTRINSIC
IMAGE
NON-LINEARITIES
CONTRAST
OF
ON
BLEACHED
1970
EFFICIENCY
HOLOGRAMS
C. CLAUSEN and H. DAMMANN Philips
Forschungslaboratorium
Hamburg
Received
GmbH,
Hamburg
54,
Germany
15 June 1970
Efficiency and image contrast of bleached holograms from diffusely illuminated binary objects have The maximum efficiency is 18% for the thin and about 25% for been measured for various exposures. the thick holograms. A distinct minimum of the contrast versus exposure curves is clearly explained by a corresponding maximum phase modulation.
1.
INTRODUCTION
In recent years several papers have been published on bleaching processes of a photographically recorded holographic interference pattern [l]. Of main interest was the increase in reconstruction efficiency, and therefore first of all diffraction efficiencies have been measured and maximized. Accordingly, almost all of the authors have investigated only holographic diffraction gratings and the influence of various recording and bleaching parameters on their diffraction efficiencies. Less attention has been given to the image quality or image contrast [2-41. In reconstruction from bleached holograms, numerous factors cause a decrease in image quality. The main factors are: a) Non-linearities in the phase response of the recording medium and intrinsic non-linearities of the reconstruction characteristic of the phase hologram itself. b) Scattering from random phase variations which are due to the non-uniform amplitude of the signal wave front, particularly in the case of diffuse objects. c) Inherent scattering effects and imperfections of the recording medium. Image degradation due to c) can be avoided only by improvements of the recording medium, which is a task of material research and shall not be considered here. Factors a) and b) may be suitably discussed in mathematical terms. In a holographic recording process, using a constant-amplitude reference wave, the intensity distribution I(X) in the hologram plane may generally be written
Z(x) = K2+.2(x)+2Ku(x)
COS[~(X)+~~YX]
,
(1)
where K and a(x) are the amplitudes and 2nc~x and $(h.) are the phases of the reference and object wave, respectively *. By suitable processing, e.g. development and bleaching, a phase distribution is produced, which, assuming constant modulation transfer functions, may be written as a function of intensity. Rather than the phase distribution itself the most interesting part is the phase variation P(X) due to the variation of intensity. This intensity variation, denoted by s(x) , is given by s(x) = [a2(x) - a2(x)] + 2Ku(x) cos [4(x) + 211~~x1 = s,(X) + Si(X) ,
(2)
where a2(x) is the mean value of a2(x). The response of the recording medium phase variation V(X) = g[a(x)]
is the (3)
,
where g denotes the response function. The ultimate object of concern, namely the amplitude transparency t(x) of the hologram, is thus given by t(x) = e-iq(X)
= 1 - icp(x) -+~2(x)+~icp3(x)+.
.. . (4)
Image degradation
is caused by the term sn(x) in
* In all cases,
2-D spatial variables will be replaced by 1-D variables for simplicity of notation.
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Volume 8. number (i
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COMMUNICA’rIOKS
(2)) by non-linearities possibly involved in the response function &S(X)] in (3) and by the intrinsic non-linear terms in (4). It is the “noise” term S,(x) in (2) which causes the random phase variation mentioned in b). However, S,(X) can be neglected for lar e reference-object irradiance ratios E2__K 2,+ ,~n (x), since ; sn(x)i is of the order +a= K2 j262 and T;-i(,Y) j is of the order K(~r=)l/~ = K~,/E. Upatnieks and Leonard [4] have recently investigated image degradation due to the random phase variations caused by the “noise” term sn(x) in the range 1 -‘r c 2 -r: 1000. In their theoretical considerations, they have assumed the linear phase response o?(x) =cs(x) (c = real constant). i.e. g[s(x)] =cs(,Y) in (3). Their results will be taken into consideration in the discussion of our results (section 3). Regarding the response function g[s(?c)], it will be shown in the detailed discussions below that the linear form ,L~[s(x)] =CS(X) may be a satisfying approximation in the low exposure region corresponding to optical densities below about 3. For higher exposures, non-linearities of g will no longer be insignificant. The intrinsic non-linear terms in (4) will be of main importance in the present paper. It will be shown that the reconstruction properties of the bleached holograms of diffuse objects obtained in our experiments can essentially be derived from these intrinsic non-linearities. In particular, a characteristic minimum of the image contrast, plotted as a function of exposure (or, equivalently of mean density before bleaching), is conclusively explained by a corresponding maximum mean phase modulation of the hologram. A “curious” minimum in this contrast function reported by Upatnieks and Leonard [4] and also observed by McMahon and Franklin [2], which heretofore remained unexplained, is believed to be identical with the described minimum.
2. EXPERIMENTS RESULTS
AND EXPERIMENTAL
In a. commonly used holographic arrangement, we have produced interference patterns according to (l), with a He-Ne-laser. The object consisted of part of a usual, binary resolution card and was placed directly behind a ground glass diffusor. Except for the speckle patterns, uniform intensity of the object wave was achieved in the hologram plane. We have used a collimated reference wave, the spatial frequency N 264
November 1970
in the hologram plane being 1000 and 200 lp,‘mm, respectively. The low carrier frequency 200 lp ‘mm was chosen to facilitate direct inspection of the hologram phase distribution by an interference microscope. Moreover, in this case the holograms can be considered thin, so that eq. (4) is valid. We use the well known [7] discriminator Q = 2rXL ?zor12in order to decide whether the hologram is thin or thick. X is the wavelength of light, L the thickness and ?zo the refractive index of the emulsion, and d the fringe spacing of the holographic interference pattern. A hologram is thick if Q ) 1. In our case, L - 7 p , N 1.5, and X =0.63 p, one obtains Q = 15 for 120 1000 lp :‘mm and 6 N 0.5 for 200 lp ‘mm. Thus [7] the hologram is thin in the latter case and thick for 1000 lp ‘mm. The reference-object irradiance ratio was 4 : 1 in both cases and additionally 10 : 1 for 200 lp ‘mm. We have used Agfa-Gevaert 8375 plates, which were processed according to the following directions for use * : 1. Stress-relieve the emulsion in a water vapor atmosphere. 2. Expose the hologram. 3. Preharden the emulsion for 10 minutes in SH-5 bath**. 4. Wash 2 minutes. 5. Develop in Agfa-Gevaert G3P 5 minutes. 6. Stop bath. 7. Fix. 8. Wash. 9. Bleach. 10. Wash 5 minutes. 11. Successive baths 50%, ‘75?&, 90% alcohol, each for 2 minutes. 12. Dry in air. The bleach bath consists of two components: Part A: 120 g CuSO4, 7.5 g KBr and 150 g citric acid in 1 1 water. Part B: 1 part H202 (30% solution) in 7 parts water. For total exposure below 100 PWsec cmm2, the plates are bleached for 12 minutes in a mixture 1 : 1 for A and B. For exposure beyond 100 nWsec cm-2 the plates are bleached for 6 minutes in solution A and for 6 minutes in the 1 : 1 mixture A and B. This precaution is necessary because otherwise, for high exposure levels, H202 removes the emulsion or parts of it from the glass base. The net effect of the bleach is that the metallic silver image, which is obtained after fixing, is converted into an AgBr image. The dielectric constant and hence the refractive index of AgBr is considerably larger than that of the emulsion, * The essentials of this procedure are proposed by Pennington and Harper [a]. ** SH-5 bath: 10 g NaZC03, 50 g NazS04, 40 cm3 benzotriazole (0.5% alcoholic solution), Hz0 to make 1 1. Add formaline at 5 cm3/1 before use.
Volume
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OPTICS
COMMUNICATIONS
so that a phase variation p(x) is generated in the hologram. The processing steps l., 3. and 11. are inserted in order to remove inherent scattering effects of the emulsion [c) in section
F, is the total area and F the actually illuminated area of the object frame. The estimate (7) is sufficiently accurate for our purposes. (In fact, it is rather good: R, of course, increases to the edge of the image frame, but this effect is approximately compensated by the distortion flux outside the frame.) From (6) and (7) we thus have:
We have measured the efficiency and the contrast of the reconstructed images. These data do not change remarkably if the holograms, in reconstruction, are inserted into a liquid gate. The contrast R in the binary reconstructed image is defined as the ratio (5)
,
where I is the mean intensity in the “luminous” part of the image and In is that in the “dark” part (fig. 1). This contrast was measured in the centre of the image.
Fig.
1. Ideal
binary
image
(hatched
area)
and image
It was observed that the “noise”-intensity I, has an almost gaussian distribution across the area of the image and somewhat outside of it. This is very plausible if the noise is due to nonlinearities [a) in section 11, because in this case it is the higher correlations of the image which cause the image degradation [5,6]. These higher correlations approach very soon a gaussian form for the aperiodic object used. Regarding the reconstruction efficiency, we have to distinguish between total diffraction flux and ideal image flux [5,6]. It is: total diffraction flux= ideal image flux+ distortion flux or equivalently nt=n+nn
9
1970
where
11.
R = (I-In)/In
November
(‘3
where qt, n and vn denote total efficiency, ideal image efficiency, and distortion efficiency, respectively. The distortion flux is defined as that radiant flux of the image, which is supplied by the noise intensity In. The distortion efficiency can be derived from the image contrast R [eq. (5), fig. 11:
We have measured nt, F,/F and R, and then calculated 77. We have varied the mean total exposure
distortion.
The mean
E =
height
7[K2+a2(x)]
of the “noise”
is In.
(7 = exposure
time)
in the range 10 nWsec
cmm2 G E s 2600 PWsec
cm -2 .
The corresponding mean densities before bleaching according to these mean exposures are shown in fig. 2. The resulting ideal image efficiencies and the corresponding central image contrasts are shown in fig. 3. All values are mean values of several experiments (about 5 - 10). Considerable variations of the measurements are observed for high exposures (beyond about 200 PWsec cm-2, i.e. densities of 3, see fig. 2), and the best results in image contrast in this region are better by a factor of 2 and more. Therefore, distinct conclusions can be based on the form of the curves in this region, rather than on their actual height.
265
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b’
i:
2. number
2 hlenn density
OPTICS COMMUiYICATIONS
6
of the emulsions
before
3. DISCUSSION OF EXPERIMENTAL
hicaching versus exposure. usual D =Lqlog (E)] curve.
RESULTS
Let us first discuss the curves for the carrier frequency 200 lp,‘mm, in which case the hologram can be treated as thin and thus can be described by (4). The main feature of the two efficiency curves (200 lp mm, fig. 3a) are the two maxima, looking like the two humps of a camel. In each curve, the first maximum value is urnax = (18 i l)‘[. This is in very good agreement with theory 161, which predicts a maximum ideal image efficiency of 18.4”; for thin phase holograms of diffuse obIrcts. the phase response (3) of which is ,,l(S) (,a( \-) ((, real constant). In theory this maximum is reached for a mean phase variation ‘:;:ic;~&,r _ 1. As is well known also for phase gratings, the first order diffraction efficiency will decrease beyond this optimum phase variation. Accordingly, the decrease of the efficiency curves in fig. 3a is caused by an increase of the mean phase modulation (,91beyond this optimum value. This last statement we have proven experimentally (fig. 4). In fig. 4, some interferograms are shown, taken with an interference microscope from bleached phase holograms. The corresponding exposure rates E of the holograms 266
November
The lower
abscissa
corresponds
1970
to the
are indicated in fig. 3 by the corresponding characters at the bottom. The characters a -e correspond to the 10: 1 curves while f corresponds to the 4 : 1 curves. The interference lines in the photographs of fig. 4 are produced in the microscope and are perpendicular to the mean direction of the holographic interference lines. A variation of the optical path length across the holographic interference lines causes a corresponding fringe shift of the interference lines produced in the microscope. In fig. 4a, for instance, these phase variations across the holographic interference lines are very small, so that the interference lines produced by the microscope are essentially straight lines. The fringe shifts are proportional to the phase variations ,,v(x) of the hologram. A fringe shift of one line width (e.g. from dark to dark) corresponds to a phase variation / U(X / = 2n. G (x))“’ The mean phase modulations ~1 = (@ can be obtained from the photographs of fig. 4. One can clearly recognize that this mean phase modulation increases from a to d and decreases from cl to e. The mean phase modulation thus passes a maximum in the region around d. The first maximum efficiency, however, is obtained closely before r. and the corresponding ~71 is the
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OPTICS
6
-E
COMMUNICATIOISS
pWs cd
b) -E
Fig. 3. Ideal image efficiency17 (fig. 3a) and image -referencecontrast R (fig. 3b) versus exposure. object irradiance ratio c2 =lO : 1, spatial frequency of
carrier wave 200 lp/mm; - .- E2= 4 : 1, spatial frequency 200 Ip/mm; ---- E2= 4 : 1, spatial frequency 1000 lp/mm. The characters a-f at the bottom correspond to fig. 4.
optimum value for maximum efficiency. This optimum value (cpl = 1 in theory, see above) is again reached on the opposite side of the ‘plmaximum, thus causing the second maximum in the efficiency curves. The actual height of this second maximum for the 10 : 1 curve is somewhat lower than that of the other maxima. This is probably caused by a non-linear phase response and, additionally, by the considerable variations of the measured values in this region mentioned above.
November
1970
The mean phase modulation in f (fig. 4) is similar to that in d and hence larger than the optimum modulation. Accordingly, the exposure rate corresponding to f lies beyond the first maximum efficiency for the 4 : 1 curve, see fig. 3a. This first maximum is reached at lower exposure rates than for the 10 : 1 curve, because for constant mean exposure E, the signal is larger for the 4 : 1 curve. Turning now to the contrast curves in fig. 3b, the most striking characteristics are the distinct minima for moderate exposures. The exposure rates for these minima coincide with those for the corresponding efficiency curves. These contrast minima are clearly explained by the corresponding maximum phase modulations ~1, which cause maximum image degradation due to the intrinsic non-linearities in (4). In theory [6], the image contrast is proportional to @I4 for the linear-phase response P(X) = CS(X) in (3). Let us now briefly discuss the curves for the carrier frequency 1000 lp,/mm. The overall appearance of these curves is the same as that for the thin holograms, and the dependence of the phase modulation on exposure is quite similar. For low exposures, the 1000 lp,‘mm efficiency curve nearly coincides with the corresponding 200 lp/mm curve. This corresponds to the well known fact, that for low phase modulations the efficiency of a thick phase grating is essentially the same as that for a thin one. In the exposure region 20 - 100 PWsec cmm2, the thick holograms are more advantageous in both efficiency and contrast. However, the increase in maximum efficiency * is not so dramatic, as perhaps has been expected, and the increase in image contrast is much more pronounced in increasing the reference-object irradiance ratio from 4 : 1 to 10: 1 rather than increasing the spatial frequency from 200 to 1000 lpi/mm. In the above discussions, image degradation due to factors b) and c) in section 1 and nonlinearities involved in the phase response function g[s(x)] in (3) have not been taken into ac count. However, these effects will also be more or less present, and will affect, in particular, the actual height of the contrast curves. In our opinion, factor C) of section 1 can be largely neglected in our investigations because * We have made very few holograms with 10 : 1 reference-object irradiance ratio and 1000 tp/mm spatial frequency of the carrier wave. The efficiency was in no case greater than about 25%. Pennington and Harper [8] have reported 28% efficiency, and Burckhardt and Doherty [3] have obtained 25% for bleached thick holograms of diffuse objects. 267
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November
1970
b
e Fig.
taken with an mterference microscope from phase holograms with different exposures. The interference lines are perpendicular to those in the interferofframs. The corresponding exposure values are indicated by the corresponding characters in fig. 3.
4. lnterferograms
holographic
of the precautions in the plate processings mentioned above. We have confirmed this experimentally in a wide exposure region. However, the decrease of the ~2 = 10 : 1 contrast curve for low exposures may be caused by this effect, which comes out here because of the very low efficiencies. The same effect is probably present also at extremely high exposures for all curves. Factor b) has been investigated by Upatnieks and Leonard [4]. Although their results cannot be compared directly, it can be derived from their results that image degradation due to b) is relatively negligible in the minimum region of our contrast curves. For high contrasts, on the other hand, this factor will affect the actual height of the curves, in particular for the lower reference-object irradiance ratio. The non-linearities involved in the phase response function q(x) =g[s(X)] will cause similar effects. From the bleaching process, which converts Ag into AgBr, it may be reasoned that the 268
phase response q(x) is roughly proportional to density before bleaching, if we assume q(x) to be proportional to the AgBr concentration. In fig. 2, density before bleaching is plotted versus exposure. This somewhat unusual density curve is approximately linear in the low exposure region (say below 100 yWsec cm-2), so that the linear-phase characteristic Q(X) = es(x) may be a sufficient approximation. In fact, as shown above, the assumption q(x) = es(x) leads to a satisfying agreement of theoretical and experimental results in this region. For higher exposure rates, a deeper understanding of the properties of bleached holograms will be possible only if the phase response function g[s(x)] is better known. A contribution of Chang and George in their recent paper [9] may be a fruitful step in this direction.
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OPTICS COMMUNICATIONS
REFERENCES [l] J. N. Latta, Appl. Opt. 7 (1968) 2409; J. Upatnieks and C. Leonard, Appl. Opt. 8 (1969) 85; and references therein; L.H.Lin, Appl. Opt. 8 (1969) 963. [2] D.H.McMahon and A.R.Franklin: Appl. Opt. 8 (1969) 1927. [3] C. B. Burckhardt and E. T. Doherty, Appl. Opt. 8 (1969) 2479.
November
1970
[4] J. Upatnieks and C.Leonard, J. Opt. Sot. Am. 60 (1970) 297. [5] J. W.Goodman and G.R. Knight, J. Opt. Sot. Am. 58 (1968) 1276. [6] H.Dammann, to be published. [7] W.R.Klein, Proc. IEEE 54 (1966) 803. [S] K. S. Pennington and J. S. Harper, J. Opt. Sot. Am. 69 (1969) 481A and private communication. [9] M.Chang and N.George, Appl. Opt. 9 (1970) 713.
269
2, number
November
OPTICS COMMUNICATIONS
6
EFFECTS AND
OF
INTRINSIC
IMAGE
NON-LINEARITIES
CONTRAST
OF
ON
BLEACHED
1970
EFFICIENCY
HOLOGRAMS
C. CLAUSEN and H. DAMMANN Philips
Forschungslaboratorium
Hamburg
Received
GmbH,
Hamburg
54,
Germany
15 June 1970
Efficiency and image contrast of bleached holograms from diffusely illuminated binary objects have The maximum efficiency is 18% for the thin and about 25% for been measured for various exposures. the thick holograms. A distinct minimum of the contrast versus exposure curves is clearly explained by a corresponding maximum phase modulation.
1.
INTRODUCTION
In recent years several papers have been published on bleaching processes of a photographically recorded holographic interference pattern [l]. Of main interest was the increase in reconstruction efficiency, and therefore first of all diffraction efficiencies have been measured and maximized. Accordingly, almost all of the authors have investigated only holographic diffraction gratings and the influence of various recording and bleaching parameters on their diffraction efficiencies. Less attention has been given to the image quality or image contrast [2-41. In reconstruction from bleached holograms, numerous factors cause a decrease in image quality. The main factors are: a) Non-linearities in the phase response of the recording medium and intrinsic non-linearities of the reconstruction characteristic of the phase hologram itself. b) Scattering from random phase variations which are due to the non-uniform amplitude of the signal wave front, particularly in the case of diffuse objects. c) Inherent scattering effects and imperfections of the recording medium. Image degradation due to c) can be avoided only by improvements of the recording medium, which is a task of material research and shall not be considered here. Factors a) and b) may be suitably discussed in mathematical terms. In a holographic recording process, using a constant-amplitude reference wave, the intensity distribution I(X) in the hologram plane may generally be written
Z(x) = K2+.2(x)+2Ku(x)
COS[~(X)+~~YX]
,
(1)
where K and a(x) are the amplitudes and 2nc~x and $(h.) are the phases of the reference and object wave, respectively *. By suitable processing, e.g. development and bleaching, a phase distribution is produced, which, assuming constant modulation transfer functions, may be written as a function of intensity. Rather than the phase distribution itself the most interesting part is the phase variation P(X) due to the variation of intensity. This intensity variation, denoted by s(x) , is given by s(x) = [a2(x) - a2(x)] + 2Ku(x) cos [4(x) + 211~~x1 = s,(X) + Si(X) ,
(2)
where a2(x) is the mean value of a2(x). The response of the recording medium phase variation V(X) = g[a(x)]
is the (3)
,
where g denotes the response function. The ultimate object of concern, namely the amplitude transparency t(x) of the hologram, is thus given by t(x) = e-iq(X)
= 1 - icp(x) -+~2(x)+~icp3(x)+.
.. . (4)
Image degradation
is caused by the term sn(x) in
* In all cases,
2-D spatial variables will be replaced by 1-D variables for simplicity of notation.
263
Volume 8. number (i
OP1’ICS
COMMUNICA’rIOKS
(2)) by non-linearities possibly involved in the response function &S(X)] in (3) and by the intrinsic non-linear terms in (4). It is the “noise” term S,(x) in (2) which causes the random phase variation mentioned in b). However, S,(X) can be neglected for lar e reference-object irradiance ratios E2__K 2,+ ,~n (x), since ; sn(x)i is of the order +a= K2 j262 and T;-i(,Y) j is of the order K(~r=)l/~ = K~,/E. Upatnieks and Leonard [4] have recently investigated image degradation due to the random phase variations caused by the “noise” term sn(x) in the range 1 -‘r c 2 -r: 1000. In their theoretical considerations, they have assumed the linear phase response o?(x) =cs(x) (c = real constant). i.e. g[s(x)] =cs(,Y) in (3). Their results will be taken into consideration in the discussion of our results (section 3). Regarding the response function g[s(?c)], it will be shown in the detailed discussions below that the linear form ,L~[s(x)] =CS(X) may be a satisfying approximation in the low exposure region corresponding to optical densities below about 3. For higher exposures, non-linearities of g will no longer be insignificant. The intrinsic non-linear terms in (4) will be of main importance in the present paper. It will be shown that the reconstruction properties of the bleached holograms of diffuse objects obtained in our experiments can essentially be derived from these intrinsic non-linearities. In particular, a characteristic minimum of the image contrast, plotted as a function of exposure (or, equivalently of mean density before bleaching), is conclusively explained by a corresponding maximum mean phase modulation of the hologram. A “curious” minimum in this contrast function reported by Upatnieks and Leonard [4] and also observed by McMahon and Franklin [2], which heretofore remained unexplained, is believed to be identical with the described minimum.
2. EXPERIMENTS RESULTS
AND EXPERIMENTAL
In a. commonly used holographic arrangement, we have produced interference patterns according to (l), with a He-Ne-laser. The object consisted of part of a usual, binary resolution card and was placed directly behind a ground glass diffusor. Except for the speckle patterns, uniform intensity of the object wave was achieved in the hologram plane. We have used a collimated reference wave, the spatial frequency N 264
November 1970
in the hologram plane being 1000 and 200 lp,‘mm, respectively. The low carrier frequency 200 lp ‘mm was chosen to facilitate direct inspection of the hologram phase distribution by an interference microscope. Moreover, in this case the holograms can be considered thin, so that eq. (4) is valid. We use the well known [7] discriminator Q = 2rXL ?zor12in order to decide whether the hologram is thin or thick. X is the wavelength of light, L the thickness and ?zo the refractive index of the emulsion, and d the fringe spacing of the holographic interference pattern. A hologram is thick if Q ) 1. In our case, L - 7 p , N 1.5, and X =0.63 p, one obtains Q = 15 for 120 1000 lp :‘mm and 6 N 0.5 for 200 lp ‘mm. Thus [7] the hologram is thin in the latter case and thick for 1000 lp ‘mm. The reference-object irradiance ratio was 4 : 1 in both cases and additionally 10 : 1 for 200 lp ‘mm. We have used Agfa-Gevaert 8375 plates, which were processed according to the following directions for use * : 1. Stress-relieve the emulsion in a water vapor atmosphere. 2. Expose the hologram. 3. Preharden the emulsion for 10 minutes in SH-5 bath**. 4. Wash 2 minutes. 5. Develop in Agfa-Gevaert G3P 5 minutes. 6. Stop bath. 7. Fix. 8. Wash. 9. Bleach. 10. Wash 5 minutes. 11. Successive baths 50%, ‘75?&, 90% alcohol, each for 2 minutes. 12. Dry in air. The bleach bath consists of two components: Part A: 120 g CuSO4, 7.5 g KBr and 150 g citric acid in 1 1 water. Part B: 1 part H202 (30% solution) in 7 parts water. For total exposure below 100 PWsec cmm2, the plates are bleached for 12 minutes in a mixture 1 : 1 for A and B. For exposure beyond 100 nWsec cm-2 the plates are bleached for 6 minutes in solution A and for 6 minutes in the 1 : 1 mixture A and B. This precaution is necessary because otherwise, for high exposure levels, H202 removes the emulsion or parts of it from the glass base. The net effect of the bleach is that the metallic silver image, which is obtained after fixing, is converted into an AgBr image. The dielectric constant and hence the refractive index of AgBr is considerably larger than that of the emulsion, * The essentials of this procedure are proposed by Pennington and Harper [a]. ** SH-5 bath: 10 g NaZC03, 50 g NazS04, 40 cm3 benzotriazole (0.5% alcoholic solution), Hz0 to make 1 1. Add formaline at 5 cm3/1 before use.
Volume
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OPTICS
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so that a phase variation p(x) is generated in the hologram. The processing steps l., 3. and 11. are inserted in order to remove inherent scattering effects of the emulsion [c) in section
F, is the total area and F the actually illuminated area of the object frame. The estimate (7) is sufficiently accurate for our purposes. (In fact, it is rather good: R, of course, increases to the edge of the image frame, but this effect is approximately compensated by the distortion flux outside the frame.) From (6) and (7) we thus have:
We have measured the efficiency and the contrast of the reconstructed images. These data do not change remarkably if the holograms, in reconstruction, are inserted into a liquid gate. The contrast R in the binary reconstructed image is defined as the ratio (5)
,
where I is the mean intensity in the “luminous” part of the image and In is that in the “dark” part (fig. 1). This contrast was measured in the centre of the image.
Fig.
1. Ideal
binary
image
(hatched
area)
and image
It was observed that the “noise”-intensity I, has an almost gaussian distribution across the area of the image and somewhat outside of it. This is very plausible if the noise is due to nonlinearities [a) in section 11, because in this case it is the higher correlations of the image which cause the image degradation [5,6]. These higher correlations approach very soon a gaussian form for the aperiodic object used. Regarding the reconstruction efficiency, we have to distinguish between total diffraction flux and ideal image flux [5,6]. It is: total diffraction flux= ideal image flux+ distortion flux or equivalently nt=n+nn
9
1970
where
11.
R = (I-In)/In
November
(‘3
where qt, n and vn denote total efficiency, ideal image efficiency, and distortion efficiency, respectively. The distortion flux is defined as that radiant flux of the image, which is supplied by the noise intensity In. The distortion efficiency can be derived from the image contrast R [eq. (5), fig. 11:
We have measured nt, F,/F and R, and then calculated 77. We have varied the mean total exposure
distortion.
The mean
E =
height
7[K2+a2(x)]
of the “noise”
is In.
(7 = exposure
time)
in the range 10 nWsec
cmm2 G E s 2600 PWsec
cm -2 .
The corresponding mean densities before bleaching according to these mean exposures are shown in fig. 2. The resulting ideal image efficiencies and the corresponding central image contrasts are shown in fig. 3. All values are mean values of several experiments (about 5 - 10). Considerable variations of the measurements are observed for high exposures (beyond about 200 PWsec cm-2, i.e. densities of 3, see fig. 2), and the best results in image contrast in this region are better by a factor of 2 and more. Therefore, distinct conclusions can be based on the form of the curves in this region, rather than on their actual height.
265
1:olume
b’
i:
2. number
2 hlenn density
OPTICS COMMUiYICATIONS
6
of the emulsions
before
3. DISCUSSION OF EXPERIMENTAL
hicaching versus exposure. usual D =Lqlog (E)] curve.
RESULTS
Let us first discuss the curves for the carrier frequency 200 lp,‘mm, in which case the hologram can be treated as thin and thus can be described by (4). The main feature of the two efficiency curves (200 lp mm, fig. 3a) are the two maxima, looking like the two humps of a camel. In each curve, the first maximum value is urnax = (18 i l)‘[. This is in very good agreement with theory 161, which predicts a maximum ideal image efficiency of 18.4”; for thin phase holograms of diffuse obIrcts. the phase response (3) of which is ,,l(S) (,a( \-) ((, real constant). In theory this maximum is reached for a mean phase variation ‘:;:ic;~&,r _ 1. As is well known also for phase gratings, the first order diffraction efficiency will decrease beyond this optimum phase variation. Accordingly, the decrease of the efficiency curves in fig. 3a is caused by an increase of the mean phase modulation (,91beyond this optimum value. This last statement we have proven experimentally (fig. 4). In fig. 4, some interferograms are shown, taken with an interference microscope from bleached phase holograms. The corresponding exposure rates E of the holograms 266
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are indicated in fig. 3 by the corresponding characters at the bottom. The characters a -e correspond to the 10: 1 curves while f corresponds to the 4 : 1 curves. The interference lines in the photographs of fig. 4 are produced in the microscope and are perpendicular to the mean direction of the holographic interference lines. A variation of the optical path length across the holographic interference lines causes a corresponding fringe shift of the interference lines produced in the microscope. In fig. 4a, for instance, these phase variations across the holographic interference lines are very small, so that the interference lines produced by the microscope are essentially straight lines. The fringe shifts are proportional to the phase variations ,,v(x) of the hologram. A fringe shift of one line width (e.g. from dark to dark) corresponds to a phase variation / U(X / = 2n. G (x))“’ The mean phase modulations ~1 = (@ can be obtained from the photographs of fig. 4. One can clearly recognize that this mean phase modulation increases from a to d and decreases from cl to e. The mean phase modulation thus passes a maximum in the region around d. The first maximum efficiency, however, is obtained closely before r. and the corresponding ~71 is the
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Fig. 3. Ideal image efficiency17 (fig. 3a) and image -referencecontrast R (fig. 3b) versus exposure. object irradiance ratio c2 =lO : 1, spatial frequency of
carrier wave 200 lp/mm; - .- E2= 4 : 1, spatial frequency 200 Ip/mm; ---- E2= 4 : 1, spatial frequency 1000 lp/mm. The characters a-f at the bottom correspond to fig. 4.
optimum value for maximum efficiency. This optimum value (cpl = 1 in theory, see above) is again reached on the opposite side of the ‘plmaximum, thus causing the second maximum in the efficiency curves. The actual height of this second maximum for the 10 : 1 curve is somewhat lower than that of the other maxima. This is probably caused by a non-linear phase response and, additionally, by the considerable variations of the measured values in this region mentioned above.
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The mean phase modulation in f (fig. 4) is similar to that in d and hence larger than the optimum modulation. Accordingly, the exposure rate corresponding to f lies beyond the first maximum efficiency for the 4 : 1 curve, see fig. 3a. This first maximum is reached at lower exposure rates than for the 10 : 1 curve, because for constant mean exposure E, the signal is larger for the 4 : 1 curve. Turning now to the contrast curves in fig. 3b, the most striking characteristics are the distinct minima for moderate exposures. The exposure rates for these minima coincide with those for the corresponding efficiency curves. These contrast minima are clearly explained by the corresponding maximum phase modulations ~1, which cause maximum image degradation due to the intrinsic non-linearities in (4). In theory [6], the image contrast is proportional to @I4 for the linear-phase response P(X) = CS(X) in (3). Let us now briefly discuss the curves for the carrier frequency 1000 lp,/mm. The overall appearance of these curves is the same as that for the thin holograms, and the dependence of the phase modulation on exposure is quite similar. For low exposures, the 1000 lp,‘mm efficiency curve nearly coincides with the corresponding 200 lp/mm curve. This corresponds to the well known fact, that for low phase modulations the efficiency of a thick phase grating is essentially the same as that for a thin one. In the exposure region 20 - 100 PWsec cmm2, the thick holograms are more advantageous in both efficiency and contrast. However, the increase in maximum efficiency * is not so dramatic, as perhaps has been expected, and the increase in image contrast is much more pronounced in increasing the reference-object irradiance ratio from 4 : 1 to 10: 1 rather than increasing the spatial frequency from 200 to 1000 lpi/mm. In the above discussions, image degradation due to factors b) and c) in section 1 and nonlinearities involved in the phase response function g[s(x)] in (3) have not been taken into ac count. However, these effects will also be more or less present, and will affect, in particular, the actual height of the contrast curves. In our opinion, factor C) of section 1 can be largely neglected in our investigations because * We have made very few holograms with 10 : 1 reference-object irradiance ratio and 1000 tp/mm spatial frequency of the carrier wave. The efficiency was in no case greater than about 25%. Pennington and Harper [8] have reported 28% efficiency, and Burckhardt and Doherty [3] have obtained 25% for bleached thick holograms of diffuse objects. 267
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b
e Fig.
taken with an mterference microscope from phase holograms with different exposures. The interference lines are perpendicular to those in the interferofframs. The corresponding exposure values are indicated by the corresponding characters in fig. 3.
4. lnterferograms
holographic
of the precautions in the plate processings mentioned above. We have confirmed this experimentally in a wide exposure region. However, the decrease of the ~2 = 10 : 1 contrast curve for low exposures may be caused by this effect, which comes out here because of the very low efficiencies. The same effect is probably present also at extremely high exposures for all curves. Factor b) has been investigated by Upatnieks and Leonard [4]. Although their results cannot be compared directly, it can be derived from their results that image degradation due to b) is relatively negligible in the minimum region of our contrast curves. For high contrasts, on the other hand, this factor will affect the actual height of the curves, in particular for the lower reference-object irradiance ratio. The non-linearities involved in the phase response function q(x) =g[s(X)] will cause similar effects. From the bleaching process, which converts Ag into AgBr, it may be reasoned that the 268
phase response q(x) is roughly proportional to density before bleaching, if we assume q(x) to be proportional to the AgBr concentration. In fig. 2, density before bleaching is plotted versus exposure. This somewhat unusual density curve is approximately linear in the low exposure region (say below 100 yWsec cm-2), so that the linear-phase characteristic Q(X) = es(x) may be a sufficient approximation. In fact, as shown above, the assumption q(x) = es(x) leads to a satisfying agreement of theoretical and experimental results in this region. For higher exposure rates, a deeper understanding of the properties of bleached holograms will be possible only if the phase response function g[s(x)] is better known. A contribution of Chang and George in their recent paper [9] may be a fruitful step in this direction.
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REFERENCES [l] J. N. Latta, Appl. Opt. 7 (1968) 2409; J. Upatnieks and C. Leonard, Appl. Opt. 8 (1969) 85; and references therein; L.H.Lin, Appl. Opt. 8 (1969) 963. [2] D.H.McMahon and A.R.Franklin: Appl. Opt. 8 (1969) 1927. [3] C. B. Burckhardt and E. T. Doherty, Appl. Opt. 8 (1969) 2479.
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[4] J. Upatnieks and C.Leonard, J. Opt. Sot. Am. 60 (1970) 297. [5] J. W.Goodman and G.R. Knight, J. Opt. Sot. Am. 58 (1968) 1276. [6] H.Dammann, to be published. [7] W.R.Klein, Proc. IEEE 54 (1966) 803. [S] K. S. Pennington and J. S. Harper, J. Opt. Sot. Am. 69 (1969) 481A and private communication. [9] M.Chang and N.George, Appl. Opt. 9 (1970) 713.
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