Babeva et al. Vol. 27, No. 2 /February 2010 /J. Opt. Soc. Am. B 197
Two-way diffusion model for short-exposureholographic grating formation in acrylamide-
based photopolymer
Tsvetanka Babeva,1,2,* Izabela Naydenova,1,3 Dana Mackey,1,4 Suzanne Martin,1,3 and Vincent Toal1,3
1Centre for Industrial and Engineering Optics, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland2Central Laboratory of Photoprocesses, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., 1113 Sofia, Bulgaria
3School of Physics, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland4School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland
. INTRODUCTIONnterest in photopolymer systems has increased markedlyn the past few years. Due to their high sensitivity, self-rocessing and low cost, they find applications in variousreas such as holography [1], manufacturing of optical el-ments [2,3], holographic data storage [4–6], etc. Theain disadvantage of many photopolymer systems is
heir poor response at high-spatial-frequency recording.Photopolymer systems usually consist of one or twoonomers, an electron donor, and sensitizing dye, all dis-
ersed in a binder matrix [7]. Upon uniform illuminationhe monomer polymerizes, and the refractive index of theystem changes. When the photopolymer is exposed to annterference pattern, more monomers are polymerized inhe bright regions than in the dark ones. This sets up aoncentration gradient of the monomer, which then startso diffuse from a dark to a bright area where it is poly-erized. The formation of surface-relief grating in an
crylamide-based photopolymer with peaks coincidingith the areas where illumination intensity is maximum
8] as well the swelling of material in an illuminated spotbserved experimentally [9] or predicted by calculations10] can be regarded as an experimental evidence foronomer diffusion from a dark to a bright area. It isorth noting that if the monomer mass transfer is not in-olved in the relief formation process and the polymerhrinkage is the only mechanism involved, then the peakshould appear in the nonilluminated areas. Due to poly-erization and monomer diffusion, a polymer density
patial distribution is formed, which results in aefractive-index modulation of a similar form. Therefore,he recorded phase holographic grating is due to a spatial
ariation of the refractive index resulting from changes ofhe density of the photopolymer components.
Grating evolution in photopolymer systems has beentudied theoretically and experimentally by several au-hors [7,11–16]. The common feature of the proposed mod-ls is that they fail to describe the high-spatial-frequencyesponse of photopolymers. The low diffraction efficiencyt a high-spatial-frequency can be explained using twopproaches, both referring to the nonlocal response of theaterial. This means that the response of the material at
ne point and time depends on what happens at otheroints and times in the medium. The first model, the non-ocal photopolymerization-driven diffusion (NPDD) model17,18] assumes that the chains grow away from their ini-iation point, resulting in “spreading” of the polymer. Theodel predicts that improvement at high spatial frequen-
ies can be achieved if shorter polymer chains are createduring the holographic recording [17]. Despite the suc-essful theoretical modeling, no experimental evidence formprovement of an acrylamide-based photopolymer re-ponse at a spatial frequency higher than 3000 lines/mmas so far been achieved adopting this approach [19]. Al-ernatively, the two-way diffusion model [20,21], which islso based on the nonlocal response of the materials, as-umes that short polymer chains diffuse away from theright fringes, thus reducing the refractive-index modula-ion. Such processes could be responsible for the decreasef diffraction efficiency at high spatial frequencies athich the fringe spacing is small, and there is enough
ime for some of the polymer chains to escape from theright fringes before the medium becomes less permeableue to complete polymerization.
010 Optical Society of America
mppsmrcpsos
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198 J. Opt. Soc. Am. B/Vol. 27, No. 2 /February 2010 Babeva et al.
To verify this assumption we propose a theoreticalodel for the formation of a weak grating after a short ex-
osure time. This model accounts for both monomer andolymer diffusion, and moreover distinguishes betweenhort polymer chains capable of diffusing and long poly-er chains that are immobile. The time evolution of
efractive-index modulation after a short exposure is cal-ulated and compared with experimental results. The im-act of diffusion coefficients, polymerization rates, inten-ity, and spatial frequency of recording on the propertiesf weak diffraction gratings are investigated by numericalimulations.
With the present study we demonstrate that the two-ay diffusion model can satisfactorily predict the poorigh-spatial-frequency response in highly permeable pho-opolymers. It also predicts that the improvement of theigh-spatial-frequency response of such systems must beirected towards decreasing the permeability of the pho-opolymer matrix and avoiding the creation of diffusinghort polymer chains. In a more realistic picture, prob-bly both nonlocal processes—nonlocal polymerizationnd short polymer diffusion—take place and should beaken into account in order to achieve the ultimate high-patial-frequency response.
. TWO-WAY DIFFUSION MODELs stated above the variation of monomer concentration
in time and space) during illumination is due to theonomer polymerization and monomer diffusion. Gener-
lly, these two processes are expressed mathematicallysing a standard one-dimensional diffusion equation [7]:
�m�x,t�
�t=
�
�x�Dm�x,t��m�x,t�
�x � − F�x,t�m�x,t�, �1�
here m�x , t� is the monomer concentration, F�x , t� is theolymerization rate, and Dm�x , t� is the monomer diffu-ion coefficient. The polymerization rate depends on theree-radical concentration that is a function of radicaleneration and termination rates. For constant intensitynd for the short-exposure time of 0.1–0.3 s used in ourtudies, we can assume that the rate of free-radical gen-ration is constant because there are plenty of unbleachedye molecules available to absorb and generate radicals.urthermore, due to the insignificant changes in the ma-erial’s viscosity for such a short exposure time, we canssume that the termination rate is also constant (i.e, therommsdorff effect can be ruled out). Therefore, in thease of short-exposure time we can assume the polymer-zation rate not to change in time. Further, we supposehat the polymerization rate is proportional to the inten-ity of illumination,
F�x� = kpIa�x� = kpI0a�1 + V cos�Kx��a = F0�1 + V cos�Kx��a
� F0f�x�, �2�
here I�x�=I0�1+V cos�Kx�� is the illumination patternntensity, I0 is the average intensity, V=2�I1I2 / �I1+I2� ishe fringe visibility, I1 and I2 the intensities of the writingeams,K=2� /� the grating vector, � the grating period,nd F =k Ia, where k =0.1 s−1�mW/cm2�−a is a fixed con-
0 p 0 p
tant [22]. For recording intensities fromto 100 mW/cm2 and a=0.3–0.5, the polymerization
ime is between 10 and 1 s. Thus, for short-exposureimes �0.1–0.2 s� the changes in the permeability of theedium are insignificant, and one can assume that Dm is
onstant in time. Concerning the spatial variation of Dm,t was shown that even the first-order term in Fourier se-ies expansion of Dm has a rather small effect [7], so wean assume that Dm is constant. With these assumptions,q. (1) describing the rate of change of monomer concen-
ration takes the simpler form,
�m�x,t�
�t= Dm
�2m�x,t�
�x2 − ��t�F�x�m�x,t�, �3�
here we introduce the step function ��t� to account forhe short-exposure regime with te being the exposureime,
��t� = 1, if t � te
0 if t � te . �4�
s could be seen from Eqs. (3) and (4), the proposed modelssumes that the polymerization stops as soon as the il-umination is turned off. This is a simplification that cane justified by the fact that experimentally we observeittle or no change in diffraction efficiency following ter-
ination of longer exposures. Further, in our model weistinguish two types of polymer chains: short chains, p1,apable of diffusing and long chains, p2, that are immo-ile. We assume that short chains are converted to longhains at a rate proportional to monomer and short poly-er concentrations and introduce a parameter �, which is
he conversion rate constant. Then the equations for tem-oral and spatial evolution of p1 and p2 take the form:
�p1�x,t�
�t=
�
�x�Dp�x��p1�x,t�
�x � + ��t��F�x�m�x,t�
− �m�x,t�p1�x,t��,
�p2�x,t�
�t= ��t��m�x,t�p1�x,t� �5�
here Dp is the polymer diffusion coefficient. Further, wessume that Dp is proportional to the interference pat-ern, which means that the maximum values of Dp coin-ide with the peaks in intensity. On the other hand, it isnown that at higher intensity more short-chain polymerolecules are formed [23]. Considering that the center of
he bright fringes will be rich of short polymers and thedges will be poor, the assumption in Eq. (6) means thathe diffusion coefficient for shorter chains will be higherhan the diffusion coefficient for longer chains:
Dp�x� = Dp�1 + V cos�Kx��a � Dpf�x�. �6�
n this simplified picture we also assume, as it is seenrom Eq. (5), that the conversion from short to long poly-er chains also stops when the exposure is stopped. For
he purpose of the subsequent analysis and numericalimulations, we introduce dimensionless variables
wasdaffrpvtetsfstwl
ws
Wettt(bm
3MAs
wliiw�au
wnptttws5tac
I�emmtddstct
cEctEaif
Babeva et al. Vol. 27, No. 2 /February 2010 /J. Opt. Soc. Am. B 199
x̄ =x
�, t̄ =
t
t0, m̄ =
m
m0, p̄i =
pi
m0�i, = 1,2�, �7�
here m0=m�x ,0� is the initial monomer concentration,nd t0=1 s. The value of t0 is chosen to be 1 s for two rea-ons. The first one is for convenience. When t0 is 1 s, theimensionless time used in computations will be the sames the real time (that is, in seconds). For values of t0 dif-erent from 1 s, a correction factor will be needed to trans-orm the dimensionless time to the real time. The secondeason is that when nondimensionalizing a system ofhysical equations, it is customary to scale variables byalues with a similar order of magnitude. As the exposureime is 0.2–0.3 s and the total simulation times did notxceed 10 s, we considered t0=1 s as a good referenceime. It is also common practice to choose scales that havepecial significance in the physical problem, such as dif-usion or polymerization time. However, we avoided suchcales here as we varied these parameters, which affecthe diffusion and polymerization rates and consequentlyould distort the time and dynamics of the whole prob-
em.The model equations become
�m̄
�t̄= �t0
�2m̄
�x̄2− ��t̄�F0t0f�x̄�m̄,
�p̄1
�t̄= ��t0
�
�x̄�f�x̄��p̄1
�x̄ � + ��t̄��F0t0f�x̄�m̄
− m̄p̄1�,
�p̄2
�t̄= ��t̄�m̄p̄1, �8�
here �=Dm /�2, �=Dp /Dm and =m0t0�. The nondimen-ional initial and boundary conditions are
m̄�x̄,0� = 1, p̄i�x̄,0� = 0,
�m̄
�x̄�x̄, t̄� =
�p̄1
�x̄�x̄, t̄� =
�p̄2
�x̄�x̄, t̄� = 0, for x̄ = 0,1. �9�
e have imposed a zero-flux boundary conditions as wexpect the final monomer and polymer concentration pat-erns to exhibit minima or maxima at the ends of the in-erval �0,��, which are maximum points for the illumina-ion intensity. It should be noted that by integrating Eqs.8) one can obtain the conservation law that is expectedecause the total concentration of different phases (mono-er, short, and long polymers) remains constant:
s explained in the previous paragraph, a polymer den-ity spatial modulation is formed upon illumination,
hich results in refractive-index modulation with a simi-ar pattern to that of the illumination. The refractive-ndex modulation is the difference between the refractivendices in the illuminated and nonilluminated areas. Ife consider both areas as effective mixtures of monomer
m�, short �p1�, and long �p2� chain polymer molecules andbinder �b�, their refractive indices could be expressed
sing the Lorentz–Lorenz equation in the form [24]
ne2 − 1
ne2 + 2
= m
nm2 − 1
nm2 + 2
+ p1
np12 − 1
np12 + 2
+ p2
np22 − 1
np22 + 2
+ b
nb2 − 1
nb2 + 2
,
b + m + p1 + p2 = 1, �11�
here ne is the effective refractive index of the mixture;m, p1, p2, and b are the volume fractions of compo-ents i=Vi /Vtot where Vi and Vtot are the volume occu-ied by the ith component and the total volume, respec-ively, and nm, np1, np2, and nb are the refractive indices ofhe components. In our numerical simulations we usedhe values of the refractive index of each component thatas determined previously from spectrophotometric mea-
urements (nm=1.55, np1=np2=1.64 and nb=1.496 at32 nm). Further, using the normalized concentrations ofhe components calculated from the model (m̄, p̄1 and p̄2)nd considering the densities for all components, we cal-ulated the volume fraction of each component:
m =m̄/�m
�b/m0���b/�m� + �m̄/�m� + �p̄1 + p̄2�/�p, �12a�
p1 =p̄1/�p
�b/m0���b/�m� + �m̄/�m� + �p̄1 + p̄2�/�p, �12b�
p2 =p̄2/�p
�b/m0���m/�b� + �m̄/�m� + �p̄1 + p̄2�/�p. �12c�
n Eq. (12) m0 is the initial monomer concentration and �ii=m ,b ,p� are the densities of the components. They arequal to 1.3 g/cm3 for the polymer [19], 1.15 g/cm3 for theonomer, and 1.19 g/cm3 for the binder. The values foronomer and binder densities are obtained considering
he masses of the components (see Section 5.A) and theirensities [19]. To make the picture more realistic, in theenominator of Eq. (12) which is the total volume of theample, we introduce the parameter b /m0 as the ratio ofhe masses of the binder and monomer. In this way we ac-ount both for the presence of the binder and for the facthat the monomer occupies about 17% of the total volume.
The temporal changes of the volume fractions of allomponents (m, p1, p2, and b) can be estimated fromq. (12) where the variations of m̄, p̄1, and p̄2 in time arealculated by the model equations [Eqs. (8) and (9)]. Fur-her, m, p1, p2, and b �=1−m+p1+p2� are used inq. (11) for calculation of the effective refractive index asfunction of time. The temporal growth of refractive-
ndex modulation that gives rise to the first order of dif-raction was then calculated as
wdt
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cacdmtei9
ichs edttqibcam
smt5tow2msapf
ratcsiC
Fps=
Fps=
200 J. Opt. Soc. Am. B/Vol. 27, No. 2 /February 2010 Babeva et al.
�n = nemax�t� − nemin�t�, �13�
here nemax�t� and nemin�t� are the effective refractive in-ices in the centers of the bright and dark fringes, respec-ively.
. NUMERICAL SIMULATIONShe nondimensional model equations were integrated nu-erically using a standard Crank–Nicolson finite-
ifference method [25]. The numerical value for the mono-er diffusion coefficient �Dm=1.3 10−8 cm2/s� was taken
rom the experimental data previously published in [16]nd the ratio � between the polymer and monomer diffu-ion coefficient was varied between 0.001–0.1. The influ-nce of the polymerization rate F0 was studied for valuesf 0.1, 0.3, and 1 s−1. The spatial frequencies of recordingere varied from 200 to 5000 l /mm, which covered grat-
ng periods from 5 to 0.2 �m, respectively. The propor-ionality constant values between the recording intensitynd the polymerization rate are a=1 [7,13], 0.5 [12], or.3 [22]. It may not be straightforward to determine ex-erimentally the rate of conversion of short to long poly-er chains ; that is why it was varied between 0 and
00. The exposure time was 0.2 s unless otherwise speci-ed.Some concentration profiles of monomers and polymers
alculated from Eqs. (8) are shown in Figs. 1 and 2 for lownd high spatial frequencies, respectively. Allomponents—monomer, short, and long polymers—eveloped spatial modulation with a concentration mini-um for the monomer and a maximum for the polymer in
he center of the bright fringes �x=0,1,2�. Because thexposure time �0.2 s� is small compared to the polymer-zation time �3.3 s�, most of the monomer (more than5%) remains unpolymerized after such a short period of
ig. 1. (Color online) Numerical results for the concentrationrofiles of (a) monomer, (b), short and (c) long chain polymers forpatial frequency of 500 lines/mm. Exposure time is 0.2 s, F00.3 s−1, a=0.5, =1, D /D =0.01.
p m
nitial illumination. From Fig. 2 it can be seen that theoncentration profile of the monomer is almost flat forigher spatial frequencies. Because of the small fringepacing �200 nm�, the monomer needs less time (about 810−4 s) to diffuse from dark to bright fringes and to re-
stablish the concentration equilibrium disturbed by theecreased number of monomer molecules due to their par-icipation in the photopolymerization process. As a result,he spatial modulation of the monomer disappears veryuickly. Similarly equalizing the monomer concentrationn space takes place for low-spatial-frequency recording,ut this process is slower compared to the high-frequencyase. The monomer diffusion time at 500 l /mm �2 �m� isbout 0.08 s and can be observed with the present experi-ental arrangement.A comparison of Figs. 1(b) and 2(b) shows that the
preading of the polymer out from the bright fringes isore pronounced for high spatial frequencies at which
he distances and diffusion times are smaller. For000 l /mm the widening of the illuminated area is fasterhan at 500 l /mm where more time is needed for diffusionf short polymer chains away from bright fringes. It isorth noting the different time scales for Figs. 1(b) and(b). The concentration profiles for long-chain polymerolecules [Figs. 1(c) and 2(c)] do not change after expo-
ure, and neither is further widening of the illuminatedrea observed with time. The reason for that is that longolymer chains are assumed to be immobile, and onceormed at a particular location, cannot move to another.
Figure 3 presents the evolution of the calculatedefractive-index modulation with time at low �500 l /mm�nd high �5000 l /mm� spatial frequency of recording forhree different ratios of polymer and monomer diffusionoefficients. It is seen that, after the illumination istopped, refractive-index modulation decreases more rap-dly at both higher spatial frequencies and higher Dp.onsidering that this decrease is due to diffusion of short
ig. 2. (Color online) Numerical results for the concentrationrofiles of (a) monomer, (b), short and (c) long chain polymers forpatial frequency of 5000 lines/mm. Exposure time is 0.2 s, F00.3 s−1, a=0.5, =1, Dp /Dm=0.01.
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Babeva et al. Vol. 27, No. 2 /February 2010 /J. Opt. Soc. Am. B 201
olymer chains away from the bright fringes, it can be ex-ected that the decrease will be more rapid for higher val-es of Dp as well as for higher spatial frequencies wherehe fringe spacing is smaller.
The influence of the polymerization rate on the post-xposure dynamics of refractive-index modulation at low500 l /mm� and high �5000 l /mm� spatial frequencies cane seen from Fig. 4. Considering that F0 is proportional tontensity of illumination [see Eq. (2)] the dependences inig. 4 can be also regarded as intensity dependences ofefractive-index modulation. It is seen that as F0 de-reases the refractive-index modulation also decreases.he reason for this is that at low F0 less monomer is con-erted to polymer during the illumination. It is seen that,fter illumination ceases, �n decreases more rapidly forigher values of F0 (i.e, higher intensity). This can be ex-lained by the fact that at higher intensity more shorthain polymers are formed [23]. They are mobile and canasily escape from bright fringe regions, resulting in a de-rease of �n.
The results presented in Figs. 3 and 4 show that theodel predicts the drop in refractive-index modulation at
igher spatial frequency, which is experimentally ob-erved. This will be discussed in more detail in the nextection.
Figure 5 presents the influence of the rate of conversionfrom short to long polymer chains on the refractive-
ndex modulation for weak gratings with spatial frequen-ies of 2000 l /mm. Small values of mean that the con-ersion from short to long chains is slow, so polymerolecules are mobile for longer times and can diffuse
way from bright fringes, reducing the refractive-indexodulation. On the other hand higher values of mean
hat short chains are converted to long chains faster, lead-
ig. 3. (Color online) Time evolution of refractive index modu-ation for weak gratings with spatial frequency of (a) 500 l /mmnd (b) 5000 l /mm at different ratios Dp /Dm (F0=0.3 s−1, a=0.5,=1). The dotted vertical line shows the time when light isurned off.
ig. 4. (Color online) Time evolution of refractive index modu-ation for weak gratings with spatial frequency of (a) 500 l /mmnd (b) 5000 l /mm at different polymerization rates (Dp /Dm0.01, a=0.5, =1). The dotted vertical line shows the time when
ight is turned off.
ng to a slow decrease of refractive-index modulation dueo the fact that long polymer chains are incapable of dif-using away from bright fringes. As expected, it is seenrom Fig. 5 that the refractive-index modulation de-reases very rapidly when is small. On the other hand,or high values of , the decrease in �n is slower.
From the numerical simulations presented in Fig. 3(b)nd Fig. 5, it can be seen that the high-spatial-frequencyesponse could be improved by suppressing the diffusionf short polymer chains and by choosing the recording pa-ameters so as to favor the rapid conversion of short toong chain polymers. Following this strategy and choosingbinder with low permeability, we have already been suc-
essful in recording reflection holograms in acrylamide-ased photopolymer [26,27].
. EXPERIMENTAL DATA. Materialshe photosensitive layers were prepared by adding 2 mlf triethanolamine, 0.6 g acrylamide, 0.2 g N,N-ethylene bisacrylamide, and 4 ml Erythrosin B dye of
.1 mM dye stock solution to 17.5 ml stock solution ofolyvinilalcohol (10 w/w) [28]. Amounts of 2 ml of theell-mixed solution were gravity settled on leveled glass
ubstrates so that the upper sides of the layers were openo the air. The thickness of the layers after drying for 24 hn darkness under normal laboratory conditionst° �21–23� °C and RH= �40–60�%) was 150±3 �m.
. Recording of Gratingsransmission gratings with spatial frequency in theange 200–3000 l /mm and diffraction efficiency of a fewercent were recorded using an NdYVO4 laser (Verdi 05)�532 nm� using short exposure times �0.2–2 s�. A He–Neaser, ��633 nm� was used for monitoring the real-timevolution of diffraction efficiency. The refractive indexodulation was calculated from the measured diffraction
fficiency using Kogelnik’s coupled-wave theory [29].
. RESULTS AND DISCUSSIONSn this section we illustrate with two examples the goodgreement that has been obtained between the results ob-
ig. 5. (Color online) Time evolution of refractive index modu-ation for weak gratings with spatial frequency of 2000 l /mm atifferent rates of conversion from short to long polymer chainsDp /Dm=0.01, a=0.5, F0=0.3 s−1). The dotted vertical line showshe time when light is turned off.
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202 J. Opt. Soc. Am. B/Vol. 27, No. 2 /February 2010 Babeva et al.
ained from gratings and those obtained from the modelresented here for refractive-index modulation of weakratings.
Figure 6 presents the comparison between the modelednd measured curves of refractive-index modulation inhe case of weak gratings at different spatial frequenciesetween 200 and 3000 l /mm and different intensity val-es (see the captions of Fig. 6). It is seen that the modelredicts qualitatively very well the behavior of theefractive-index modulation. For the same intensity (i.e,0) the initial slope of the graph of �n versus exposure
ime is the same for gratings with a spatial frequency of00, 500, and 1000 l /mm. With decreasing intensity, thelope also decreases. Further, with increasing spatial fre-uency the amplitude of refractive-index modulation de-reases. Additionally, the model predicts very well theost-exposure increase of �n for low spatial frequency200 l /mm� [20]. It can be seen from Fig. 6 that the valuesf �n predicted by the model are higher than the mea-ured ones. This difference may be due to the discrepan-ies between the real values of refractive indices of thehotopolymer components and the values assumed in theodel. From Eqs. (11) and (13) it is seen that the calcu-
ated �n is a function of the monomer, polymer, andinder refractive index as well as their volume fractions.t is relatively easy to determine the monomer and theinder refractive indices. (For example, transmittancend reflectance measurements on the respective layers30]). However, the determination of the polymer refrac-ive index is not so straightforward because it depends onhe degree of polymerization. In our simulations we usedhe np value that is determined from the spectrophoto-etric measurements of a bulk polymerized layer, i.e, one
hat is uniformly illuminated. However, under conditionsf spatially nonuniform polymerization it may happenhat the degree of polymerization is different. Moreover,t a short illumination time it is possible that the mono-ers are not fully polymerized as it is in the case of bulk
ig. 6. (Color online) (a) Numerically simulated and (b) experi-entally measured refractive index modulation for weak grat-
ngs at different spatial frequencies (texp=0.3 s, a=0.5, F00.15 s−1 for 200, 500, and 1000 l /mm; F0=0.10 and 0.05 s−1 for000 and 3000 l /mm; Dp /Dm=0.01, =10). The dashed verticaline shows the time when light is turned off.
olymerization, and the refractive index of the polymerraction is different than 1.64 (the refractive index of uni-ormly polymerized material). Our additional simulationsave shown that if we decrease the value of np from 1.64o 15.8 keeping all parameters the same, the refractive in-ex modulation decreases three times.Figure 7 presents the comparison between simulated
nd measured refractive-index modulation in the case of aonstant exposure of 7 mJ/cm2 for a short-exposure grat-ng of the spatial frequency of 500 l /mm. We obtained aonstant recording exposure of 7 mJ/cm2 using exposureimes of 2, 0.5, and 0.1 s and recording intensities of 3.5,4, and 70 mW/cm2, respectively. The good agreement re-arding the shapes and slopes of the curves can be easilyeen, but the calculated values of �n are again higherhan the measured ones.
ONCLUSIONtwo-way diffusion model for short-exposure holographic
rating formation in an acrylamide-based photopolymers presented. Accounting for both monomer and polymeriffusion, the model predicts the experimentally observedrop in refractive-index modulation at high spatial fre-uency. Moreover, the model distinguishes between shortolymer chains capable of diffusing and long polymerhains that are immobile.
The numerical simulations show that the suppressionf short polymer diffusion improves the high-spatial-requency response and that fast conversion of short toong polymer chains has a positive effect on the finalefractive-index modulation. Further, higher recording in-ensities generate larger numbers of short polymerhains, leading to higher post-exposure reduction inefractive-index modulation. Therefore, low-intensity re-ording is more appropriate for high-spatial-frequency re-ording. Following this strategy and choosing a binderith low permeability improved the spatial-frequency re-
ponse, and we have already been successful in recordingeflection holograms in acrylamide-based photopolymer26,27].
It was demonstrated that a good agreement betweenhe theoretically predicted and the experimentally mea-ured refractive-index modulation curves can be obtainedsing the two-way diffusion model.
CKNOWLEDGMENThis publication has resulted from research conductedith the financial support of the Science Foundation Ire-
ig. 7. (Color online) (a) Numerically simulated and (b) experi-entally measured refractive index modulation for weak grat-
ngs 500 l /mm at exposure of 7 mJ/cm2 (te, recording time; I, in-ensity, a=0.5, Dp /Dm=0.01, =1, F0=0.84, 0.37, and 0.19 s−1 fore=0.1, 0.5, and 2 s) The dashed vertical lines show the timehen light is turned off.
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Babeva et al. Vol. 27, No. 2 /February 2010 /J. Opt. Soc. Am. B 203
and grant 065/RFP/PHY085 and European Cooperationn Science and Technology (COST) Action MP0604. Theuthors acknowledge the School of Physics at DIT and Fa-ility for optical characterization and spectroscopy andIT for technical support. T. Babeva thanks the Arnold Fraves Postdoctoral Programme at DIT.
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