Nonlinear recording effects on holographic reflection gratings recorded on BB640 emulsions have been studied. Using an analyticalapproach to the characteristic curve of these emulsions, we obtained the recorded density profiles corresponding to the original light pattern generated by the holographic setup. The final spatial profiles of the refraction index resulting from different exposure levels and different bleaching processes were evaluated studying the experimental diffraction efficiency and comparing it with Kogelnik’s theory. The quality of the fittings of this model with the experimental spectral responses of the final gratings was used as a measure of concordance between the original sinusoidal, periodical light distribution and the resulting modulation profile of the processed hologram. The range of applicability of the theoretical model to the experimental results was evaluated using different bleaching processes.
© 2003 Optical Society of America
Of all the existing photosensitive materials, holographic recording materials have been chosen by their ability to store, without distortions, the original interference pattern generated by the holographic setup. One of the most frequently used materials of this type are the fine and ultra-fine grain silver halide emulsions. The most relevant difference between these and other recording materials the mechanism of image formation in each case. In conventional holographic silver halide emulsions, the image is primarily stored as an absorption pattern, with developed silver, that is later converted into a phase pattern after the bleaching process. The first step consists in the exposure and development of the emulsion, to form an absorption grating with poor holographic properties. At this stage, both the emulsion and the corresponding processing are characterised by the D-Log E curve . This response curve plays an important role with regard to the exposure levels, since it relates the exposure energy to the optical density of the recording, which is proportional to the mass of developed silver . The characteristic curve has a zone that is close to linear, with exposure energy proportional to the corresponding density. The way to ensure a distortion free storage of the holographic pattern is to use exposure energies corresponding to the range of this linear zone.
The best way to check that the recorded pattern is similar to that originally generated by the holographic setup consists in comparing the response of the recorded hologram with the results predicted by a theoretical model that simulates ideal holograms with perfect periodical and sinusoidal modulations. Several models have been proposed, for both reflection and transmission gratings, with approximated or rigorous solutions using couple wave or modal theories [3, 4, 5] as well as for non periodical and non sinusoidal gratings [6, 7]. Of all these, the most widely used is the well-known coupled wave theory formulated by Kogelnik in 1969 , which applies to sinusoidal, periodic holographic structures.
In this study we recorded holographic reflection gratings. These are one of the most demanding holographic structures in terms of the recording material used to store them, since the interference patterns have the highest spatial frequencies, requiring both good volume properties and low distortion of the recorded light profile. These gratings were recorded on ultra fine grain silver halide emulsions, with a layer thicknesses of less than 10 µm, so no important distortion of the interference fringes due to absorption of the recording beams along their propagation direction was to be expected. The development and bleaching processes generate oxidation products that tan the emulsion, so that the losses due to absorption are important and must be considered when analysing the recorded gratings. This absorption is different at the exposed and unexposed zones of the interference pattern, so that some absorption modulation remains in the final recording, and mixed amplitude-phase responses must be taking into account . Additionally, in both the grainy structure of the final hologram and the gelatin of silver halide emulsions the absorption of light is greater at shorter wavelengths than at longer wavelengths, so the absorption coefficient depends on the wavelength.
The resulting reflection holograms were evaluated by means of the spectral transmittance of the recordings, a method widely used to characterise reflection holograms [10, 6]. The typical shape of a transmission curve, as opposed to a typical bandpass optical filter, together with the grating characteristics, is shown in Fig. 1. Diffraction efficiency (η), replay wavelength (λ 0) and bandwidth (Δλ) can be obtained from the numerical analysis of the experimental data.
Based on these premises, we have applied Kogelnik’s expressions to check the suitability of this model for analysing these spectral responses. From the fittings of the spectral responses we can obtain the main gratings parameters of each recording: refractive index modulation n 1, absorption modulation α 1 and effective thickness d, as well as a fitting quality factor that will be a measure of the degree of applicability of the Kogelnik’s sinusoidal model to the experimental results.
Holographic reflection gratings were recorded on red sensitive ultra-fine grain BB640 emulsions manufactured by Colourholographics Ltd. They were pre-sensitised by soaking for 15 minutes in a 3% Triethanolamine water solution, dried with a photographic roll and warm air and left in the exposure room to cool down for half an hour in normal laboratory conditions (200 and 60% RH) .
Plates were exposed to a single collimated He-Ne (wavelength 632.8 nm) beam in a Denisyuk configuration . The setup consists of an optical sandwich composed by a first surface mirror that reflects the collimated beam back into the emulsion. The emulsion side of the plate is in contact with the mirror via an index matching fluid, and the glass side is in contact with an anti-reflection coated glass via another thin layer of index matching fluid to prevent reflections. With this configuration a spatial frequency of 4992 l/mm is recorded on the plate (considering a refractive index of 1.579 for the unexposed emulsion). Transmittance of the unexposed emulsion at 632.8 nm is 71.3 %, resulting in an average beam ratio is 1:0.713, that corresponds to a fringe visibility of 0.98. The sandwich was mounted on a computer controlled motorised holder which enables us to record 9 gratings with different exposure energies on a 2.5″×2.5″ plate.
Exposed plates were developed with AAC developer (Ascorbic Acid 18 g/l+Sodium Carbonate 60 g/l) . After washing they were bleached with fixation-free rehalogenating bleaches R-10 (Potassium Dichromate 2 g/l+Sulphuric Acid 10 cc/l+Potassium Bromide 35 g/l) and reversal bleach R-9 (Potassium Dichromate 2 g/l+Sulphuric Acid 10 cc/l). Plates used for density measurements were developed with AAC and fixed with non-hardening fixer F-24 . After bleaching or fixing, plates were washed and soaked in deionized water with a few drops of Photoplo and Acetic Acid to prevent printout.
After drying, the plates were analysed using a fibre fed spectroradiometer. This device enable us to measure the zero order of the grating with a replay angle of 0°, matching the recording geometry. A short arc Xenon lamp was used as the light source, collimated and polarised perpendicular to the plane of incidence to match the recording conditions. Light is collected by an optical fibre that feeds the spectrophotometer and data were transferred to a computer for storage and analysis. Reflection losses were experimentally evaluated an found to have a value of 6.7%.
3. Kogelnik’s theory
Kogelnik’s coupled wave theory is a very powerful tool to study off-Bragg replay conditions for reflection holograms recorded in silver halide materials, as has been previously described. In this study we started with the expression for the amplitude of the direct transmitted waves for lossy mixed unslanted reflection gratings of Kogelnik’s theory, given by Eq. (1).
For a mixed lossy dielectric unslanted reflection grating with normal incident recording beams, mean refractive index n, refractive index modulation n 1, absorption coefficient α, grating period Λ and absorption modulation α 1, the grating parameters γ 1 and γ 2 are given by Eqs. (2) and (3).
This model was applied to the spectral response of mixed unslanted reflection gratings recorded in BB640 holographic emulsions. As mentioned in a previous section, the absorption coefficient α is a function of the wavelength. This dependence was calculated with the envelope curve of the transmitted spectrum of the grating without considering the diffracted band. As an example, the envelope shown in Fig. 1 was obtained by fitting the transmission curve in the intervals of [450–550] nm and [650–700] nm. We used a semi-empirical saturation curve corrected by a sinusoidal expression given by Eq. (6) as the envelope of the transmittance curve.
where A, B, C, and D are fitting parameters obtained using a non linear fit technique . In most of the calculations, regression coefficients are better than 0.99.
From this expression we obtain the wavelength dependence of the absorption coefficient given by Eq. (7), taking into account that Te=100×e -2αd.
where d is the thickness of the emulsion.
The absorption modulation coefficient α 1 was considered, to a first approximation, proportional to the average absorption coefficient α, as expressed in Eq. (8).
The replay wavelength λ 0 is obtained from the experimental transmittance curve as the wavelength corresponding to minimum transmission (and maximum η). The corresponding value of the spatial frequency Λ for an unslanted reflection grating is calculated as λ 0=2nΛ.
In order to obtain the main grating characteristics, n 1 and α 1, we fitted the experimental data to Eqs. (1), (6) and (8) using the merit function given by Eq. (9) with nonlinear fit calculations .
where N is the number of experimental points, Ti are the experimental transmission data at wavelength λi and T(λi) are the theoretical transmittances evaluated at wavelength λi.
Parameters n 1 and α 1 are obtained by finding the minimum value of Q(n 1, α 1), with the residual values Qmin (minimum value of merit function Q(n 1, α 1)) being a measurement of the quality fit. As discussed in the introduction, Qmin can be considered a parameter that gives an idea of the deviation of the real profile of the index and absorption modulation from the original holographic profile.
4. Density profiles
As has been stated, the final phase reflection hologram is primarily stored as a volume absorption pattern, since we are using an ultra fine grain photographic emulsion as the recording material. The sinusoidal light pattern generated by the holographic set up is stored in the photographic emulsion. The exposure energy, commonly considered the recording energy, is actually the average value of all the local light energies of the real sinusoidal light energy pattern. In a system with recording beam energies E 1 and E 2, the interference pattern range from values through . Taking into account that the beam ratio used in the experimental setup was 0.71, we are using a resulting fringe visibility ( ) of 0.98. Fringe spacing is 0.20 nm, corresponding to the recorded reflection holograms. With this data, the light energy distribution as a function of the spatial coordinate x taken perpendicular to the plate substrate is given by Eq. (10).
where E 0=E 1+E 2 is the average exposure.
This light pattern is stored in the developed photographic emulsion as an optical density distribution. As mentioned above, the relation between the exposure energy and the corresponding density is characterised by the D-Log E curve of the developed material. The D-Log E curve of BB640 plates for a reflection setup processed with developer AAC and non hardening fixer F-24, was measured and data fitted with the analytical expression given by Eq. (11) [16, 17]. The regression coefficient of the non linear fit was 0.999. Results are presented in Fig. 3.
The real spatial density pattern obtained after developing, stored inside the volume of the emulsion, which is proportional to the spatial silver concentration pattern, can be obtained by introducing the spatial light energy distribution of the sinusoidal pattern expressed by Eq. (10) into the analytical expression for the D-Log E curve, Eq. (11). The resulting spatial density profiles for four different average exposure energies can be seen in Figs. 4(a), (b), (c) and (d), each compared with their original spatial light energy recording profiles. As the total exposure energy increases, the deviations from the sinusoidal profile become important. In the case of the two higher energies, the profiles are closer to a step function than to a sinusoidal function. In these figures, it must be understood that the area under the solid line of the profile represents the concentration of the developed silver grains (proportional to density), and the area above this line represents the concentration of the remaining undeveloped silver halide grains.
5. Modulation profiles
Absorption reflection gratings obtained with chemical developers on silver halide holographic emulsions show very low or null diffraction efficiency values. They need to be transformed into phase gratings, and this is done by the bleaching process[18, 19]. There are several bleaching procedures but with all of them, at the end of the process, only zones with either gelatin or gelatin with silver halide grains remain in the emulsion, rendering a spatial phase structure instead of the original spatial density profile. The relation between these two structures depends on the method of bleaching used . In order to understand this relation, we must recall the different ways in which a reversal bleach  and a fixation free rehalogenating bleach [22, 23, 24] act.
A reversal bleach works by dissolving and eliminating all the developed silver from the emulsion, leaving only the original non developed silver halide grains. Therefore the regions with high levels of exposure end up with a low concentration of silver halide crystals, whereas in the regions with low levels of exposure the original concentration of the unexposed plate is maintained. No diffusion processes have been reported with this bleaching technique. The index modulation is produced as the result of the different silver halide concentrations in the exposed (low concentration) and unexposed (high concentration) regions. This bleach leads to a slightly distorted recorded profile since material is removed from the zones that contained the developed silver, and this is something that also causes a modification in the grating period. Therefore, the spatial distribution of the concentration of silver halide crystals, directly related to the final index modulation, can be obtained by turning Figs. 4 upside down and would be represented by the periodic profiles shown in Figs. 5(a) and (b). The first was obtained with an average recording energy of 300 µJ/cm2 and the second with 900 µJ/cm2, and both are compared with a sinusoidal profile included for reference. The first profile maintains a shape close to sinusoidal, while the second results in a periodic non sinusoidal structure.
In the case of gratings processed with fixation free rehalogenating bleach, it should be remembered that this process involves material diffusion . The bleaching bath includes a halide compound that reacts with the metallic silver atoms, and the silver halide crystals that result from this rehalogenating process precipitate in the zones were the original non exposed silver halide crystals of the emulsion are located, and not in the exposed regions where the developer has been acting. These diffusion processes imply that there is no material removal and no variation in the grating period. In addition, it helps in recovering a final sinusoidal profile when the distortion is not too severe, as represented in Fig. 6.
6. Results and discussion
The experimentally recorded reflection gratings were studied using the model based on Kogelnik theory for mixed volume reflection gratings with the transmittance curve correction. The model was applied to the analysis of the experimental data obtained from the spectral responses of a set of holographic mirrors recorded with different exposure energies and processed with AAC developer and R-10 and R-9 bleaches in order to obtain their main holographic parameters. In Figs. 7(a) and (b) the experimental transmittance curves of mirrors recorded with exposure energies of 300 and 900 µJ/cm2 and processed with AAC and fixation free rehalogenating bleach R-10 are compared with the resulting transmittance curves calculated using the model. Both figures show good agreement between experimental and theoretical data. Figures 7(c) and (d) show the same graphs for mirrors recorded with the same exposure energies and processed with AAC and reversal bleach R-9. In this last figure the deviation between the experimental results and the theoretical simulation is remarkable.
The residuals of the fittings Qmin, calculated from the merit function given by Eq. (9), are a measure of the applicability of Kogelnik’s theory to these recordings. They are also an evaluation of the linear response of the recording material, in that low residuals correspond to a good reproduction of the original sinusoidal light profile, while high values are understood to be important deviations of the recorded profile from the holographic light pattern. Figure 8 presents the residuals for both processings studied. In the case of fixation free rehalogenating bleach R-10, there are two well separated zones. The first, up to a exposure energy of 900 µJ/cm2, presents a trend close to a saturation curve, with low values of parameter Qmin because, although the profile was distorted during the development process, shown in Fig. 4(d), the diffusion of rehalogenated silver associated with the bleaching helps in recovering the sinusoidal profile, as shown in Fig. 6. However, for energies higher than 1200 µJ/cm2 this trend is broken and residuals grow in a random and irregular way. We understand that this is the exposure energy at which the sinusoidal profile cannot be recovered by the diffusion processes of rehalogenating bleach. Considering the solvent bleach, residuals grow steadily up to a maximum, since in this case the distortion of the spatial profiles, with no diffusion processes involved in the bleaching step, are caused only by the development step as shown in the sequence of Figs. 4(a) to (d). Except for very low exposure energies, residuals are much higher for this processing than for the case of the rehalogenating bleach.
In this study we have discussed the deviations from the original sinusoidal profile generated by the holographic setup caused by the processing of the emulsions. These distortions do not imply a poor holographic performance of the material, but a non linear response and a different index modulation spatial profile, and therefore, a limitation to the use Kogelnik’s theory with these recordings. Diffraction efficiencies in both bleaching techniques are very high in the case of R-10 bleach (close to 72%), and high in the case of R-9 bleach (reaching 60%), as can be seen in Fig. 9.
Nonlinear recording effects on holographic reflection gratings recorded on BB640 emulsions have been analysed. Using an analytical approach to the characteristic curve of these emulsions, the density profiles corresponding to the original light pattern generated by the holographic setup were obtained. The resulting refractive index modulation profiles were evaluated for fixation free rehalogenating and solvent bleaching processes. Index modulation profiles were evaluated using Kogelnik’s theory as a tool to study the applicability of the model to the experimental results obtained after processing gratings with AAC developer and post-processing them with a fixation free rehalogenating bleach and a reversal bleach.
The quality of the fittings of this theoretical model with the experimental spectral responses of the recorded holograms was used as a measure of the concordance between the original sinusoidal light distribution and the modulation profile of the processed emulsion. It was found that only for low energies was a sinusoidal modulation index obtained with the bleaching techniques studied.
This work was supported by the Comision Interministerial de Ciencia y Tecnologia (CICYT) of Spain (Project No. MAT2000-1361-C04-03).
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