Improvement of the diffraction efficiency and kinetics ofholographic gratings in photochromic media by auxiliary light
Norbert Hampp
Department of Chemistry, Philipps-University of Marburg, Hans-Meerwein-Strasse Building H, D-35032 Marburg, Germany, andMaterial Science Center, D-35032 Marburg, Germany
Thorsten Juchem
Department of Chemistry, Philipps-University of Marburg, Hans-Meerwein-Strasse Geb. H, D-35032 Marburg, Germany
Films made from the biological photochrome bacterio-rhodopsin (BR) are recording materials of highsensitivity to light. Since the earliest days withholographic experiments with BR films hologramgrowth curves that show an overshooting initial peakhave been reported.2 – 6 This observation was qualita-tively explained to be due to the optical nonlinearity ofthe BR material. However, we found that the experi-mental conditions themselves might be responsible forthis effect. Here we introduce a method that is usefulfor photochromic materials in general and makes itpossible to prevent these disturbances.
To model the photoresponse of BR a simple two-statemodel that comprises an initial B state and a long-living M state is suff icient. Photochemical rate k1 cor-responds to the hologram-forming intensity, k2 is thereadout wave, k3 is an auxiliary blue wave, and k4 rep-resents the thermal decay rate of the M state. I1, I2,and I3 are assumed to be plane waves:
B568
k1,k2!
√k3, k4
M412 . (1)
I1�x� represents the intensity that results fromthe superimposition of both interfering planewaves, object I11, and reference beam I12. A si-nusoidal intensity modulation results: I1�x� ��I11 1 I12� �1 1 V cos�2px�G�� with grating constantG � l1��2 sin uW �, where 2uW is the angle betweenthe recording beams and l1 is the recording wave-length. The contrast V � 2�I11I12�1�2��I11 1 I12� ofthe intensity pattern equals 1 for I11 � I12 and isless than 1 for I11 fi I12. The rate constants aregiven by k1�x� � eB�l1�FBl1CI1�x� := a1I1�x�, k2 �eB�l2�FBl2CI2 := a2I2, k3 � eM �l3�FMl3CI3 := a3I3,and k4 � 1�tM with C := 2.303��NAch�. NA repre-sents the Avogadro constant, c is the speed of light,
0146-9592/04/242911-03$15.00/0
and h is Planck’s constant. In the experiments thewavelength for recording is l1 � 532 nm, for readoutit is l2 � 647 nm, and for the auxiliary light it isl3 � 413 nm. The M lifetime of the particular BRfilm was tM � 212 s. The quantum eff icienciesFB � FM � 0.64 and the decadic extinction coef-ficients eB�532� � 45,000 l mol21 cm21, eB �647� �7465 l mol21 cm21, and eM �413� � 45,000 l mol21 cm21
were used. The following a values resulted:a532 � 0.275, a647 � 0.0595, and a413 � 0.229.Intensities I1 to I3 used with the a values need tobe given in milliwatts per centimeters squared. Thelight-induced changes of the two-state model intro-duced above are described by the following equation:dB�t��dt � 2�k1 1 k2�B�t� 1 �k3 1 k4�M �t�. Through-out this paper k values with more than one subscriptrepresent the sum of the k values in the subscript,e.g., k1234 := k1 1 k2 1 k3 1 k4. The time-dependentlight-induced changes in the local B-state popula-tion during hologram rise BR �t� and decay BD �t�are derived from boundary condition BR �0� � B0,which represents the total concentration of BR, andMR�0� � 0.7 The diffraction eff iciency h in a f irstapproximation is proportional to the square of theM-state population, i.e., h ~ �M �t��2 � �B0 2 B�t��2:
MR�x, t� �k12
k1234B0�1 2 exp�2k1234t�� . (2)
For V fi 1 the spatially varying k1 values range fromthe minimum k1D in the dark areas to the maximalvalue k1B in the bright areas. Using the definitionI1 := I11 1 I12 the following k1 values are derived:for I11 � I12 � V � 1, k1B � 2a1I1 and k1D � 0,and for I11 fi I12 � V fi 1, k1B � �1 1 V �a1I1 andk1D � �1 2 V �a1I1. Instead of treating the cases V � 1and V fi 1 separately, for V fi 1 the following should be
� � 0, and k2 � �1 2 V �a1I1.The difference DMR �t� � MRB �t� 2 MRD �t� between themaxima and the minima is given by
DMR �t�B0
��k1B 2 k1D �k34
k1B234k1D 2342
k1B2
k1B234exp�2k1B234t�
2k1D2
k1D234exp�2k1D234t� ,
k1 � k1B , k1D � 0 ,
DMR �t�B0
�k1k34
k1234k2342
k12
k1234exp�2k1234t�
2k2
k234exp�2k234t� . (3)
The DM steady-state value for I11 � I12 andk2 � k3 � 0 is DM0 � k1�k14. For I11 � I12 but with areadout wave k2 fi 0 (which in the case of V fi 1 is con-sidered to also receive a contribution from the record-ing wavelength) DM1 and DM2, with and withoutauxiliary blue light k3, are obtained:
DM2 := DMk2fi0,k3�0 �k1k4
k124k24
,
DM1 := DMk2fi0,k3fi0 �k1k34
k1234k234
. (4)
Of course the maximal modulation DM for both caseswith k2 fi 0 is less than in the undisturbed case DM0with k2 � 0, i.e., DM2�DM0 # 1 and DM1�DM0 # 1.The question is whether the use of auxiliary blue lightincreases the contrast, i.e., whether DM1 . DM2.This is true as long as the following equation isfulfilled:
DM1
DM2
�k124k34k24
k1234k234k4. 1 $
k2k12
k4k34. 1 . (5)
The diffraction eff iciency is proportional to DM2.To test the auxiliary light improvements the ratioDM1
2�DM22 has to be considered. There is an op-
timal k3 value k3opt that for a given set of k values
may be calculated from d�dk3�DM12�DM2
2� � 0 to be
k3opt �
pk2k12 2 k4 �
pk2k12 . (6)
In most cases k4 ,, �k2k12�1�2 holds, and in a firstapproximation k4 � 0 may be assumed. Using k3
opt
yields the highest population contrast enhancementthat is possible by use of auxiliary blue light.
In Fig. 1A the experimentally determined depen-dence of the improvement of steady-state diffractionefficiency on the auxiliary light’s intensity is shown.The related numerical simulation (Fig. 1B) fullysupports the model introduced above. The experi-mentally determined value I413 � 1.9 mW�cm2
and the numerically predicted value for k3opt to be
I413 � 2.1 mW�cm2 match well. With increasingI413 (�k3 value) the diffraction eff iciency for a givenset of parameters goes through a well-defined maxi-
mum. Of course at higher intensities deviations ofthe numerical model from the experimental valuesmust be accepted because in the numerical model theapproximation of pure phase gratings is assumed,which is crucial for photochromic media such as BRfilms. The use of auxiliary blue light is the key to asignificant improvement of the diffraction efficiencyby compensating for distortions that arise from photo-chemically active and (or) comparably strong readoutlight.
There are two different ways in which hologramgrowth curves are monitored. In the f irst case thereadout beam k2 is switched on together with thewriting beams k1 (all-on-simultaneously, AOS). Inthe second case the readout beam k2 is on all thetime, and only the writing beams k1 are turned onand off (write-switched-only, WSO). As far as thephotochemistry in the BR film is considered, theinitial population distribution for WSO is significantlydifferent from that for AOS. In experiments allsituations between AOS and WSO may occur. Thetiming of the readout beams is almost never described
Fig. 1. Improvement of the steady-state diffraction eff i-ciency of holograms in a BR film exposed to readout lightby blue auxiliary light (write, 532 nm, 5 mW�cm2; readout,647 nm, 1 mW�cm2; auxiliary, 413 nm). A, Dependenceof steady-state diffraction eff iciencies in a BR film on theauxiliary light’s intensity. B, Numerical simulation(DM1
2 ~ h) of the experimental data from A. Symbolsrepresent experimental or related numerical results. Thecurves are a guide to the eye.
Fig. 2. Hologram rise kinetics: A, Transition in the ki-netics of the hologram growth from AOS (Dt647 � 0) to WSO(Dt647 . 50 s). B, Dependence of hologram growth curvesin the AOS mode on intensity I413 of the blue auxiliarylight.
in the literature, as in general it is assumed that theintensity has no or little inf luence on the results.
First we treat the AOS case. The kinetics in thedark and bright regions of the hologram rise (�MR)are different, but the initial value for M is identicalfor both MRD �0� � MRB �0� � 0. This causes modula-tion amplitude DMR �t� to go through a maximum andthen returns to a lower value [Eq. (3)]. The same be-havior is observed in the diffraction eff iciency, whichis proportional to DMR
2�t�. To identify time tmax atwhich the maximal diffraction efficiency is obtained,the f irst derivative of the time-dependent modulationamplitude d�dt�DMR�t��B0�2 � 0 has to be solved:
tmax �1k1
lnk12
k2
. (7)
It was found that tmax is characteristic for the inten-sity of the readout light (k2) but not dependent on theauxiliary blue light (k3). For k2 ! 0, no maximum isobserved at all.
Second the WSO case shall be treated. It is as-sumed that all waves except k1 were on for a longtime and that the steady-state was reached. With the
start values MRD �0� � MRB �0� � k2�k234B0, the time-dependent modulation amplitude DMR �t� � MRB 2MRD � k1k34�k1234k234B0�1 2 exp�2k1234t�� results.Obviously now only one time-dependent e functionis involved, and the maximal modulation is reachedwithout the modulation passing through a maximum.This simple difference between AOS and WSO causesquite different hologram rise curves. In Fig. 2A thetransition from AOS to WSO is shown in several stepsas time Dt647 between turn on of k2 (l � 647 nm) andk1 (writing beams) was changed in constant stepsof Dt647 � 0 70 s. In the AOS case (Dt647 � 0) anovershooting peak in diffraction eff iciency is observed,whereas for Dt647 . 50 s, which is close to the WSOcase, the f inal diffraction efficiency is reached in asingle-exponential time dependence without any initialpeak at all.
As derived in the simulation [Eq. (7)], time tmax atwhich the maximal diffraction efficiency is reacheddoes not depend on k3, i.e., the intensity of theauxiliary light. In Fig. 2B a set of experimentalgrowth curves with a wide range of blue auxiliarylight intensities conf irms this. At k3 , k3
opt an over-shooting peak is observed; k3
opt causes the hologramnot to show any overshooting in the rise phase and toreach the maximally achievable diffraction efficiencyin the shortest possible time. The use of k3 .. k3
opt
diminishes the diffraction eff iciency. The decay ofholograms is identical and singly exponential in AOSas well as in WSO, as DMR�`� reached in steady stateis identical for both schemes.
In photochromic media nondestructive reconstruc-tion is always a problem because of the readout beam’sintensity. It should be as high as possible to produce ahigh intensity of diffracted light, but as low as possiblein order not to disturb the holographic grating. It hasbeen demonstrated that the use of properly adjustedauxiliary light allows one to compensate for the dis-turbances caused by the readout light. The hologramkinetics become more readily reproducible, and higherdiffraction intensities can be obtained. The results re-ported here were obtained with BR films but they arevalid for photochromic media in general.
This work was supported through Bundesmin-isterium für Bildung und Forschung grant FKZ13N8196. N. Hampp’s e-mail address is [email protected].
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