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1 Provided by the author(s) and University College Dublin Library in aordane with publisher poliies., Please ite the published version when available. Title Reording beam modulation during grating formation Authors(s) Gleeson, M. R.; Kelly, John V.; O'Neill, Feidhlim T.; Sheridan, John T. Publiation date Publiation information Applied Optis, 44 (26): Publisher Optial Soiety of Ameria Link to online version Item reord/more information Publisher's statement Publisher's version (DOI) This paper was published in Applied Optis and is made available as an eletroni reprint with the permission of OSA. The paper an be found at the following URL on the OSA website: Systemati or multiple /AO reprodution or distribution to multiple loations via eletroni or other means is prohibited and is subjet to penalties under law. Downaloaded T03:43:24Z The UCD ommunity has made this artile openly available. Please share how this aess benefits you. Your story matters! Some rights reserved. For more information, please see the item reord link above.

2 Reording beam modulation during grating formation Mihael R. Gleeson, John V. Kelly, Feidhlim T. O Neill, and John T. Sheridan Holography has been of inreasing interest in reent years, with developments in many areas suh as data storage and metrology. Photopolymer materials provide potentially good materials for holographi reording, as they are inexpensive and self-proessing. Many experiments have been reported in the literature that desribe the diffration effiieny and angular seletivity of suh materials. The majority of these reports disuss the performane of the holographi optial element after the reording stage. It has been observed, however, that sometimes, during exposure, the transmitted reording beam intensities vary with time. A simple phenomenologial model is proposed to explain the beam modulation, whih inorporates the growth of the phase grating, time-varying absorption effets, the mehanial motion of the plate, the growth of a lossy absorption grating during the reording proess, and the effets of nonideal beam ratios Optial Soiety of Ameria OCIS odes: , , , , The authors are with the Department of Eletroni and Eletrial Engineering, Faulty of Engineering and Arhiteture, University College Dublin, Belfield, Dublin 4, Republi of Ireland. J. T. Sheridan s address is john.sheridan@ud.ie. Reeived 10 September 2004; revised manusript reeived 25 April 2005; aepted 27 April /05/ $15.00/ Optial Soiety of Ameria 1. Introdution In reording a sinusoidal unslanted holographi grating in a photopolymer material 1 we have observed, 2,3 as have others, 4 that the intensities of the two transmitted beams oasionally flutuate in an approximately sinusoidal fashion. Signifiantly, this variation in the intensities does not always our. In this paper we provide one possible explanation for suh behavior. Analysis of the reording proess is arried out, and an approximate model of the behavior is proposed. The resultant preditions are ompared with experimental data. In the proposed model we attempt to take into aount four effets, namely, (i) any in-plane motion of the grating with respet to the two exposing beams, (ii) the time-varying absorption of the photopolymer material that is due to bleahing of the photosensitive dye, (iii) the growth of the phase grating owing to the inrease in the refrative-index modulation, and, finally, (iv) the existene of a lossy absorption grating during exposure. As the lossless grating is being formed, an absorption grating will be formed simultaneously beause of the redution in dye onentration in the exposed regions of the grating. Therefore a lossy grating is formed exatly out of phase with the refrative-index grating. The effets of this absorption or lossy grating are examined in an attempt to inrease the auray of the approximate analysis. We propose that the variations in the transmitted reording intensities during exposure arise primarily as a result of a ombination of these effets. We have arried out a series of experiments to examine the validity of our proposed model. The first experiment involves measuring the transmitted intensities of the two exposing beams to examine their variation as a funtion of time. The seond examines the effet that the absorption of the material has on a single beam. The third experiment examines the hange in the refrative index that takes plae during the growth of the phase grating. The fourth and final experiment arried out involves the estimation of the in-plane mehanial motion of the grating with respet to the two exposing beams. We present these experiments and, following analysis, show that our model approximately predits the intensities of the beams during the reording proess. Thus our first-order model is shown to be in general agreement with our experimental results. 2. Transmitted Beam Intensity Variations In this setion we briefly outline the first experiment used to monitor the variations of the two reording beams during the reording of a grating and the theoretial notation used. 10 September 2005 Vol. 44, No. 26 APPLIED OPTICS 5475

3 always the ase), whih orresponds to exposure with a nonunity or nonideal beam ratio. B. Theoretial Notation Using Kogelnik s notation 7 yields the following expressions for permitivity ε and ondutivity : 0 1 os(kx), (1) 0 1 os(kx), (2) Fig. 1. beam. Shemati of the reording geometry used and the probe A. Experimental Setup Shown in Fig. 1 is the experimental setup, whih enables both transmitted beams to be observed in real time during reording. The power of both beams was measured before they entered the material. The effet of the material on the two beams was then monitored during and immediately after exposure. The unslanted sinusoidal transmission volume hologram shown in Fig. 1 was reorded by use of a HeNe laser of wavelength 633 nm. Our photopolymer material onsists of a monomer (arylamide), a binder [poly(vinyl alohol)], a ross linker (bisarylamide), a photosensitive dye (Methylene Blue) for reording with the HeNe laser, and an eletron donor (triethanolamine). 5 A probe beam of wavelength 488 nm, whih does not interat with the dye, is also used. The transmitted light intensity is measured with an appropriate number of samples per seond, and the data are proessed with LabView software. 6 One of the experimental data sets obtained is shown in Fig. 2. We note that a less than ideally stable plate holder was used during this experiment. As an be seen, the intensity of I R is greater than that of I S (this is not Fig. 2. Transmitted exposing beam intensities. Variation with time of the intensities an be seen for the nonunity beam ratio. where 1 and 1 are the amplitudes of the spatial modulations, 0 is the average dieletri onstant, and 0 is the average ondutivity. Grating vetor magnitude K is given by 2, where is the period of the grating. In Kogelnik s analysis, 7 ε and are assumed to be modulated in phase. In Eq. (2) we introdue a phase shift of rad, resulting in a hange of sign. The expression for oupling onstant is then ( 0 ) j ( 0 ) 1 2, (3) where is the replay wavelength in free spae, 0 is the permeability of the medium, whih is assumed to be equal to that of free spae, and is the speed of light in vauo. This oupling onstant desribes the oupling between the two beams that is due to both the phase and lossy gratings, and it an be rewritten as n 1 j 1 2 r j i, (4) where n 1 is the amplitude of the spatial modulation of the refrative index as in Ref. 7, is the amplitude of the absorption modulation assoiated with the lossy dye grating, and r and i are the real and imaginary parts, respetively, of the oupling onstant. 3. Absorption Parameters ( 0, 1 ) The major effets involved in the reording of a grating must be separated (deoupled) to enable a simple phenomenologial model of the modulation of the transmitted reording beams to be onstruted. The seond experiment measured the variation of the absorption of the material with time 0 t, whih was ahieved by passage of a single beam through a standard (hemial omposition and thikness) material layer. The variation of the resultant transmitted intensity was measured many times, and an average plot was obtained. Time-varying absorption effets aused by the material take plae primarily beause of the photosensitive dye present in the material. As the light is passed through the material, the dye undergoes an oxidation reation. The dye absorbs a photon in the presene of the eletron donor, whih then produes free radials. These ause the polymerization of the arylamide, whih auses a density vari APPLIED OPTICS Vol. 44, No September 2005

4 ation in the material and a hange in the refrative index. 5 The absorbane of light by the dye is neessary to produe this photohemial reation. During the early stages of the exposure, more light is absorbed as more ative dye is available, but, as the dye is used up (bleahed), more and more of the light passes through the material This explains the observed inrease in transmitted intensity with time shown in Fig. 3. To estimate the value of the absorption 0 t, we proessed the data obtained from these experiments. The intensity values measured at the start were nonzero, as the material is partially transparent; the material begins to absorb light as soon as light is inident upon it. A best fit to the data of Fig. 3 was obtained. The resultant analyti fit to the output intensity in terms of parameters a0, a1, and a2 was then substituted into Eq. (5). The result is an empirial expression for the time evolution of absorption 0 t as defined for use in Kogelnik s oupled-wave equations (7): 0 (t) log e{1 exp[ a0 a1(t 2) a2(t 2) 2 ]}, 2 d (5) where d is the measured thikness of the material layer. (The shift in time is inluded to allow for the inhibition period; see Ref. 11.) The oeffiient values obtained are a , a , and a2 0. A plot of the resultant absorption 0 t is shown in Fig. 4. As was mentioned above, a lossy grating that is rad out of phase is generated simultaneously as the lossless phase grating is being formed, with 1 t the absorption modulation assoiated with it. 8,12 1 t might reasonably be expeted to have a form similar to that shown in Fig. 5, whih is governed by an equation of the form 1 (t) exp( pt)[1 exp( qt)]. (6) When the light is inident upon the material the dye Fig. 4. Change in absorption with time 0 t. absorbs some of the light in the bright regions. As shown in Fig. 5, we propose that 1 t initially inrease in region t1, as there is an abundane of dye in the bright region and this dye is being bleahed by the exposure of the light. Eventually, however, the lossy amplitude reahes a maximum, region t2, and then starts to derease as the dye available in the bright region is used up and the dye is bleahed throughout the volume. The lossy grating amplitude would then be expeted to derease to zero, as shown in region t3. We use this heuristi desription of the lossy grating amplitude s temporal behavior in Setion 6 below. Next we examine the effet of the growth of the lossless phase grating on the modulation of the two reording beams. 4. Grating Strength ( ) As a grating is formed by the reation of a modulation in refrative index n 1, absorption grating 1 t, or both, oupling takes plae between the exposing beams used to reord the grating. 13,14 When oupling onstant 0, there is no oupling and therefore there is no diffration. To enable a value for r to be obtained from the measured experimental data, we aptured a growth urve 15,16 by using a probe beam with a wavelength that does not interat greatly with the dye or affet the grating being reorded; see Fig. Fig. 3. time. Single beam (standard layer) fit to the loss data with Fig. 5. Example of the form that 1 t might be expeted to take with time, where p 0.05 and q September 2005 Vol. 44, No. 26 APPLIED OPTICS 5477

5 1. In this ase the lossy grating has no effet on the probe beam. The wavelength of the probe beam used is 488 nm. In the third experiment we examine the hange in the refrative index aused by the growth of the phase grating. Using the growth urve data, we find an expression for the grating strength,. To inrease the auray of our results, we orret the data to allow for Fresnel refletions at the boundaries of the material layer As the refrative index of air is different from that of the photopolymer material and the glass substrate, and while we assume that the refrative index of the material layer is lose to that of glass, there are hanges in the angles of refration aross the interfaes. Furthermore, refletions from the surfaes redue the intensity of the transmitted beams. This redution in intensity will hange the measured diffration effiienies, and therefore the data must be orreted for these effets. We use Snell s law to alulate the angle at whih the light travels within the photopolymer material, n r sin r n i sin i, where n r is the refrative index of the material that the light is passing into (holographi layer air), n i is the refrative index of the material that the light is passing from (air holographi layer), i is the angle of inidene, and r is the refrated angle inside the material. For the results reported here the refrative index of free spae used is n i 1, while the refrative indies of the glass plates and the polymer layer are assumed to be idential, n r 1.5. The angle of inidene of the exposing beam in air is , giving a grating period of 1 m for 633 nm and r Applying the Fresnel transmission and refletion oeffiients yields the resultant values for the orret diffration effiienies. The probe-beam wavelength used was 488 nm, and the wave was inident on-bragg in air at i The expression derived by Kogelnik 7 for the diffration effiieny of a lossless phase transmission grating replayed on-bragg, B, an be used to give an expression for the grating strength: (t) sin 1 [ FC (t)], (7) where FC t is the time-dependent Fresnel-orreted diffration effiieny. The variation of the grating strength, t, is then found by appliation of Eq. (7) to the Fresnel-orreted data. The following funtion was then used to provide a best fit: (t) F 0 [1 exp( f 1 t f 2 t 2 )]. (8) Typial best-fit oeffiient values are F , f , and f From Eq. (8), values for the real part of oupling onstant r t an be obtained by use of the following equation: (t) r(t)d os B. (9) Beause the probe beam used has a different wavelength 488 nm we assume that the lossy grating has no effet and that B is the on-bragg replay angle for that wavelength. In Eq. (9) the thikness of the material has the measured value d 100 m. When no monomer remains in the layer (when it has all been polymerized), grating formation ends. By this time in the exposure proess the refrative index of modulation, n 1, has reahed its maximum (saturation) value and it,, and all remain onstant. In Fig. 2, saturation ours at an exposure time of approximately 84 s. After this point the absorption effets are also observed to be negligible; i.e., we assume that Our fourth and final experiment involves examining the effets that mehanial vibrations have on the intensities of the two reording beams during saturation. 5. Effets of Mehanial Vibration Muh emphasis is plaed on reduing mehanial vibration in holographi reording setups, for example, through the use of floating optial tables. The reason for trying to minimize these effets is that vibrations derease the visibility of the reorded grating fringes owing to overwriting, produing smearing effets. The gratings reorded during this researh were fabriated with a period of 1 m. In this ase a highfrequeny relative in-plane motion of the plate during exposure of amplitude 0.5 m ould potentially eliminate the grating ompletely. We arried out a series of experiments, using plate holders with different levels of mehanial stability (rigidity). We note that the experimental results shown in Fig. 2 were produed by use of a relatively poor holder, whih was not rigid. Therefore vibrational motion ould our. The mehanial shutter (Uniblitz shutter, ontrolled by a VMM-D1 shutter driver ontroller), used at the start of the exposure, has been found to be a signifiant potential soure of mehanial noise in our setups. Suh system noise an be minimized by use of a rubber damping material, i.e., Sorbothane, 20 whih is a visoelasti polymer. This material exhibits speial visoelasti properties of both liquids (visous solutions) and solids (elasti materials). They ombine shok absorption, vibration isolation, and vibration damping harateristis. For the experimental results shown in Fig. 2 a large-intensity osillation is observed during reording. We propose that, following a long period of exposure (by whih time the grating is fully formed and absorption effets do not vary with time), the main ause of the intensity variations is in-plane mehanial vibrations. We onstruted a model desribing the transmitted intensity from a lossless grating replayed simultaneously by the two reording beams by applying appropriate replay boundary onditions. 7 We then applied this model to determine the form of the relative in-plane motion of the grating from both of the simultaneously measured sets of intensity data orresponding to I R and I S APPLIED OPTICS Vol. 44, No September 2005

6 To start, we measured the average intensity values of the two beams after saturation, shown in Fig. 2. The values found are I R 0.56 mw and I S 0.4 mw. The field amplitudes used as part of the replay boundary onditions are thus given by r I R and s I S. For an unslanted transmission grating the obliquity fators R and S are equal to os, where is the inident angle of the exposing beams (Fig. 1). They are equal beause of the symmetri illumination used when an unslanted grating was reorded. The boundary onditions must take aount of the motion of the grating relative to the two beams. In-plane motion simply gives rise to a phase hange between the two beams and the grating. It was assumed in the researh reported in Ref. 7 that the referene wave, R, at the input boundary z 0 has unit amplitude R 0 1 and that as it propagates through the grating it ouples to the diffrated wave, S. It was also assumed that S 0 0. Relative in-plane motion has no effet in this ase. However, in our model the initial onditions used are S(0) s exp[ ikdx(t)], (10) R(0) r exp[ ikdx(t)], (11) where dx t is the relative in-plane movement of the plate. 21 The oupled-wave differential equations used to model this situation, in whih n 1 is onstant, the replay is on-bragg, and we assume that losses are negligible, i.e., that 1 0 i 0, are given by R (z) i r S(z) 0, (12) S (z) i r R(z) 0, (13) z is the distane into the thikness of the material layer. These oupled equations are solved under the boundary onditions in Eqs. (10) and (11), and expressions for the transmitted intensities at z d are found: I R (t) 2 r 1 2 s 2 (r 2 s )os 2 rd 2 2rs sin 2 rd sin[2kdx(t)], (14) I S (t) 1 2 r 2 s 2 ( r 2 s 2 )os 2 rd 2rs sin 2 rd sin[2kdx(t)]. (15) Given r, s, d, r,, K, and the Fresnel-orreted data for both I R and I S, estimates of the motion of the plate, dx t, an be obtained. A plot of the estimated motion of the plate extrated from Fig. 2 is shown in Fig. 6. This motion, estimated from the measured data for both of the beams, I R and I S, should be equal for the two beams but does not oinide exatly beause of (i) Fig. 6. Motion of the plate extrated from experimental data for both beams with a fit to the motion inluding extrapolation to estimate the motion before saturation. experimental errors, (ii) the simpliity of the oupledwave model, and (iii) the simple form of in-plane motion assumed. However, good general agreement is found, whih indiates that our in-plane motion assumption has some validity. Returning to the intensity data, we note that the frequeny of the osillations varies slightly as a funtion of time. Obtaining a best fit to the frequeny variations of the osillations, we should then be able to extrapolate our predition for dx t bak into the growth region of the experimental data. We aomplish this by first onverting the frequeny variation in the saturation region into the orresponding phase variations. We then identify the points where the intensity data ross the intensity average as zero phase values. A pieewise linear representation of the phase variation is shown in Fig. 7. A ubi polynomial best fit to the phase of the form given in Eq. (16) below is used: (t) a bt t 2 dt 3. (16) Typial oeffiients obtained for the best-fit polynomial were a 0.284, b 0.274, , and d Then, using this expression for the phase, we proeed, using the analyti funtion shown Fig. 7. Phase variations with time for the R beam (unwrapped). A similar plot was obtained for the S beam. 10 September 2005 Vol. 44, No. 26 APPLIED OPTICS 5479

7 S (z) i[ r (t) i i (t)]r(z) 0 (t)s(z) 0. (19) Fig. 8. Experimental data for the intensity of the R beam (filled irles), theoretially fitted to the data with i 0 (thin dashed line), and inluding the lossy grating, where i 0 (thik dashed urve). in the following equation, to find a least-squares fit to desribe the in-plane motion of the plate: dx(t) A B exp( Ct)sin[ (t)]. (17) The exponential is introdued into the fitting funtion beause the amplitude of the osillation in Fig. 6 is observed to derease with time. Typial values of A , B , and C were found. Having defined Eq. (17), we an make preditions about the motion during the growth of the phase grating, i.e., during the first 84 s of the reorded data. Bak interpolation provides a rude approximation to the motion of the plate from the moment the shutter opens until saturation is reahed. Figure 6 shows the preditions of the analyti best fit to the motion in both regions. Combining this result and the three previous experimental results, we an now provide a fit to the omplete experimental data set originally presented in Fig Combined Model We wish to fit the data presented in Fig. 2 that desribe the modulation of the transmitted beams. From these data we were able to analyze the different proesses involved in reording a holographi grating. From the seond experiment we were able to express the absorption of the material, 0 t, asa funtion of time. The third experiment enabled us to obtain an expression for lossless grating strength t, as the grating was being grown. From the fourth experiment we derived an expression for the motion of the plate relative to the exposing beams as a funtion of time, dx t. All these expressions are now substituted into the two first-order oupled-wave equations, whih are thus modified to allow for the time evolution of all the major (deoupled) grating parameters during exposure: R (z) i[ r (t) i i (t)]s(z) 0 (t)r(z) 0, (18) In partiular, we note that, although we have no analyti expression for 1 t, the effets of the timevarying lossy grating amplitude is retained in this analysis. We solve Eqs. (18) and (19), using the boundary onditions presented in Eqs. (10) and (11) at z d, obtaining I R (t) 1 4 exp 2d[ 0(t) i (t)] ( 1 exp 4d i(t) (r 2 s 2 ) 2 1 exp 4d i(t) {rs os[2kdx(t)]} 2 exp 2d i(t) (r s)(r s)os 2d r(t) 2rs sin[2kdx(t)]sin 2d r(t) ), (20) I S (t) 1 4 exp 2d[ 0(t) i (t)] ( 1 exp 4d i(t) (r 2 s 2 ) 2 1 exp 4d i(t) {rs os[2kdx(t)]} 2 exp 2d i(t) (r s)(r s)os 2d r(t) 2rs sin[2kdx(t)]sin 2d r(t) ). (21) Equations (20) and (21) are then used to fit the full data urves with the hope of aurately desribing the modulation of the reording beams data during grating formation and saturation. Plots of I R t [Eq. (20)] and the orresponding experimental data, assuming that i is negligible, are shown in Fig. 8. Also, a plot of Eq. (20) inluding i is shown in Fig. 8, where 1 t has the form that was presented in Eq. (6) and Fig. 5. The form hosen met ertain onditions: all time-varying absorption effets happened within the first 84 s [therefore, when t 84 s, 1 t 0] and the size of 1 t at all times was less than 0 t. These onditions plae onstraints only on the possible form of the behavior of the lossy grating. The values for the oeffiients in Eq. (6) that govern the form hosen are p 0.05 and q An aurate model for its behavior must be found to improve the overall fit to the experimental data. But, as an be seen from Fig. 8, the inlusion of the values used for the lossy grating in the analysis (even though it is not an aurate model of its behavior) an be used to improve the fit to the experimental data. Similarly aeptable fits an also be found to the I S t data APPLIED OPTICS Vol. 44, No September 2005

8 Fig. 9. S beam intensity data, with an almost unity beam ratio (filled irles), with a theoretial fit to the data where i 0 (dashed line) and a fit to the data inluding the lossy grating, i.e., where i 0 (solid urve). The above analysis was repeated several times for data sets produed with beam ratios loser to unity. For the experimental data shown in Fig. 9, the R beam intensity is I R 0.45 mw and the S beam intensity is I S 0.42 mw. The resultant theoretial fit to those data, where i was assumed to be negligible for the S beam is shown in Fig. 9. An improved fit ould be ahieved when the lossy grating was inluded in the analysis, as shown by the solid urve in Fig. 9. A form for the lossy grating similar to that used for Fig. 8 was used, with the oeffiients in Eq. (6) given by p 0.05 and q Conlusions In this paper one possible explanation for the modulation of the reording beams that is sometimes observed when sinusoidal unslanted gratings are reorded has been proposed. A first-order analysis was presented, whih assumes that different parts of the physial reording proess an be deoupled. The simple phenomenologial model proposed inorporates the growth of the phase grating, time-varying absorption effets, in-plane motion of the plate owing to vibration, growth of a lossy absorption grating during reording, and the effets of beam ratio to explain the beam modulation. A series of experiments was arried out to provide an empirial basis for our model. The first experiment involved measuring the transmitted intensities of the two exposing beams to permit us to examine their variation as a funtion of time. As modulation in the beams does not always our, this experiment was rerun a number of times with different plate holders of different rigidity. Several beam ratios were also used when we examined the modulation effets. Two partiular sets of data were presented here for different beam ratios; the resultant intensity osillations differed signifiantly. In both ases the intensity osillations measured are in agreement with the preditions of our model and as predited beome more signifiant for nonunity reording beam ratios. The seond set of experiments examined the effet the absorption of the material has on a single beam as a funtion of time. 22,23 The same standard layer thikness and omposition were used in all the experiments disussed, and an empirial expression for the absorption oeffiient was presented. The third set of experiments examined the growth of the lossless phase grating. By apturing the growth urves of gratings, using a probe beam of a different wavelength that is unaffeted by the dye, we found the grating s refrative-index modulation and in partiular the maximum or saturation value. The fourth and final experimental results presented involve the determination of the motion of the grating with respet to the two exposing beams. The magnitude of this motion was found to derease from maximum amplitudes of approximately 30 nm in the ase of Fig. 8 and of approximately 20 nm in the ase of Fig. 9. We note that the amplitude and frequeny of the osillations varied among setups, indiating some dependene on the reording setup. As mentioned, one plausible soure of the vibrations is the Uniblitz mehanial shutter, and damping the shutter, inreasing the rigidity of the plate holder, or both was found to redue these effets. We note that the motion present is large enough to ause some smearing of the grating during reording. We also expeted some variation of the exposing beam s pattern amplitude with depth. All suh effets are ompletely negleted in our analysis. When one is reording, it is desirable to have unity beam ratio, beause this produes maximum fringe visibility. We note that both our model and data indiate that if the beams are unequal, the effets of modulation in intensity are more signifiant. To inrease the validity of our model we inluded the effets of the formation of an absorption grating that arise as results of the dye. As shown in Fig. 9, the inlusion of a physially reasonable lossy grating an improve the fit to the experimental data. However, the form and size of the lossy grating must be further studied. 9 Changes in the exposing intensity as a funtion of both time and depth within the material are important. Reently attempts to predit threedimensional behavior in photopolymers were reported. 10 Inlusion of these effets and also the variation of average intensity I 0 within the material with any time variation of the absorbane of the dye 24 are urrently being investigated. It is assumed that only in-plane motion of the plate ours; the other five degrees of freedom available are negleted. However, as noted, assuming simple motion, the vibrations estimated by use of the R and S beams are not exatly equal. Therefore other types of motion are taking plae and should be inluded in the model. Furthermore, as the largest amplitude of motion ours at the start of the exposure, it is possible that nonlinear mehanial oupling takes plae at that time. Experiments are urrently being arried out to measure the vibrations present in our system on the interferene pattern produed by the two plane waves used for reording. The movement of the fringes an be observed and qualitatively mathes the behavior 10 September 2005 Vol. 44, No. 26 APPLIED OPTICS 5481

9 estimated. Quantitative orrelation remains to be ahieved. Any signifiant improvements to our approximate model will neessitate inluding rigorous eletromagneti analysis, inluding time variation effets, with the material models desribing the grating formation proess. An example of one suh material model is the nonloal polymer driven diffusion model. 25,26 We onlude by noting that, despite all the assumptions made, our first-order phenomenologial model is plausible and physially appealing and provides insights into and general agreement with the experimental data measured. The authors aknowledge the support of Enterprise Ireland and Siene Foundation Ireland through the Researh Innovation Fund and the Basi Researh Program. F. T. O Neill urrently holds an Embark (Irish Researh Counil for Siene, Engineering and Tehnology) postdotoral researh fellowship. Referenes 1. F. T. O Neill, J. R. Lawrene, and J. T. Sheridan, Automised testing and reording of holographi optial element arrays, Optik (Stuttgart) 111, (2000). 2. M. R. Gleeson, F. T. O Neill, and J. T. Sheridan, Modulation of reording beams during grating formation, in Photon Management, F. Wyrowski, ed., Pro. SPIE 5456, (2004). 3. M. R. Gleeson, F. T. O Neill, and J. T. Sheridan, Reording beam modulation during grating formation, in Organi Holographi Materials and Appliations II, K. Meerholtz, ed., Pro. SPIE 5521, (2004). 4. W. L. Wilson, InPhase Tehnologies ( personal ommuniations, 2003). 5. J. R. Lawrene, F. T. O Neill, and J. T. Sheridan, Photopolymer holographi reording material, Optik (Stuttgart) 112, (2001). 6. LabView User s Manual (National Instruments Corporation, Austin, Tex., January 1998), pp H. Kogelnik, Coupled wave theory for thik holographi gratings, Bell Syst. Teh. J. 48, (1969). 8. S. Caron, J. J. A. Couture, and R. A. Lessard, Real time holographi reinforement demonstrated by thionine PVA photoreduible thin layers, Appl. Opt. 29, (1990). 9. S. Caron, R. A. Lessard, and P. C. Roberge, Photodarkening and partial photobleahing: appliation to dihromated gelatin, Appl. Opt. 40, (2001). 10. S. Gallego, M. Ortuño, C. Neipp, A. Márquez, A. Beléndez, I. Pasual, J. V. Kelly, and J. T. Sheridan, Physial and effetive optial thikness of holographi diffration gratings reorded in photopolymers, Opt. Express 13, (2005). 11. A. V. Galstyan, R. S. Hakobyan, S. Harbour, and T. Galstian, Study of the inhibition period prior to the holographi grating formation in liquid rystal photopolymerizable materials, Eletroni-Liq. Cryst. Commun. ( 05_05_11_13_17/). 12. Y. L. Lee, C. H. Kwak, J. H. Kwon, Y. S. Im, and O. S. Choe, Observation of a fast formed absorbtion grating and a slowly formed phase grating in undeveloped dihromated gelatin, Appl. Opt. 40, (2001). 13. J. V. Kelly, M. R. Gleeson, C. E. Close, F. T. O Neill, J. T. Sheridan, S. Gallego, and C. Neipp, Temporal and non-ideal behaviour in photopolymers, in Opto-Ireland 2005: Photoni Engineering, B. W. Bowe, G. Byrne, A. J. Flanigan, T. J. Glynn, J. Magee, G. M. O Connor, R. O Dowd, G. D. O Sullivan, and J. T. Sheridan, eds., Pro. SPIE 5827, (2005). 14. C. Carre and D. J. Lougnot, Photopolymerizable material for holographi reording in the nm domain, J. Opt. (Paris) 21, (1990). 15. F. T. O Neill, J. R. Lawrene, and J. T. Sheridan, Improvement of holographi reording material using aerosol sealant, J. Opt. A Pure Appl. Opt. 3, (2001). 16. F. T. O Neill, J. R. Lawrene, and J. T. Sheridan, Thikness variation of self-proessing arylamide based photopolymer and refletion holography, Opt. Eng. 40, (2001). 17. R. R. A. Syms, Pratial Volume Holography (Clarendon, 1990). 18. E. Heht, Optis, 2nd ed. (Addison-Wesley, 1987). 19. M. Born and E. Wolf, Priniples of Optis, 6th ed. (Pergamon, 1980) J. T. Sheridan, Staked volume holographi gratings: Part I, Transmission gratings in series, Optik (Stuttgart) 95, (1993). 22. N. Capolla and R. A. Lessard, Proessing of holograms reorded in Methylene Blue sensitized gelatin, Appl. Opt. 27, (1998). 23. C. Solano, R. A. Lessard, and P. C. Roberge, Methylene Blue sensitized gelatin as a photosensitive medium for onventional and polarizing holography, Appl. Opt. 26, 1989 (1987). 24. S. Blaya, L. Carretero, R. F. Madrigal, M. Ulibarrena, P. Aebal, and A. Fimia, Photopolymerization model for holographi gratings formation in photopolymers, Appl. Phys. B 77, (2003). 25. J. T. Sheridan and J. R. Lawrene, Non-loal response diffusion model of holographi reording in photopolymer, J. Opt. So. Am. A 17, (2000). 26. J. V. Kelly, F. T. O Neill, J. T. Sheridan, C. Neipp, S. Gallego, and M. Ortuno, Holographi photopolymer materials: nonloal polymerization driven diffusion under nonideal kineti onditions, J. Opt. So. B 22, (2005) APPLIED OPTICS Vol. 44, No September 2005