Download - Three-dimensional extended nonlocal photopolymerization driven diffusion model. Part I. Absorption

Three-dimensional extended nonlocalphotopolymerization driven diffusion

model. Part I. Absorption

Haoyu Li, Yue Qi, and John T. Sheridan*

School of Electrical, Electronic and Communications Engineering, UCD Communications and Optoelectronic ResearchCentre, SFI-Strategic Research Cluster in Solar Energy Conversion, College of Engineering and Architecture,

University College Dublin, Belfield, Dublin 4, Ireland*Corresponding author: [email protected]

Received July 24, 2014; accepted September 3, 2014;posted September 8, 2014 (Doc. ID 217691); published October 13, 2014

When holographic gratings are recorded in a photopolymer material layer, the spatial distributions of the photo-reactions taking place lead to the formation of nonuniform gratings in depth. In an effort to study such effects, inthis series of papers a three-dimensional (3D) nonlocal photopolymerization driven diffusion (NPDD) model isdeveloped. In Part I, we focused on describing the photoinitiationmechanisms by introducing a 3D dye absorptionmodel that more accurately and physically describes the processes taking place. Then, the values of physicalparameters are extracted by numerically fitting experimentally obtained normalized transmittance growth curvesfor a range of layer thicknesses in an acrylamide/polyvinyl alcohol (AA/PVA) photopolymer material sensitizedby Erythrosine B (EB). In Part II [J. Opt. Soc. Am. B 31, 2648 (2014)], applying the results in Part I, the full 3Dphotophysical and photochemical evolutions are modeled. Then the resulting 3D NPDD model is validatedexperimentally. © 2014 Optical Society of America

OCIS codes: (050.7330) Volume gratings; (090.0090) Holography; (090.2900) Optical storage materials;(160.5335) Photosensitive materials; (010.1030) Absorption.http://dx.doi.org/10.1364/JOSAB.31.002638

1. INTRODUCTIONPhotopolymers have been actively studied for many reasonsincluding their potential as self-processing holographic datastorage media [1–5]. One way to optimize their holographicmemory capacity is to more fully use the medium thicknessfor storage. For example, thicker volume gratings, with corre-spondingly narrower Bragg angular selectivity widths, allowmore closely angularly multiplexed recording [6–8]. To makethe storage materials more sensitive, i.e., requiring lowerexposure intensities and shorter exposure times, higher con-centrations of photosensitizers having higher absorptivitymust be chosen [9–11]. However, while both thicker layersand higher sensitivities can increase the medium’s data stor-age capability, they also lead to increasing nonuniformity ofrecording with material depth. In order to further develop andattain the full potential of these photopolymer materials, a de-tailed understanding of the evolutions of grating formationthroughout the physical depth of the layer during and afterexposure is crucial [12–14].

The recording illumination intensity becomes attenuated asit passes through the layer as a result of absorption by the dyemolecules. This complicates the photophysical and photo-chemical processes taking place [15–17]. In order to developmore accurate theoretical predictions and evaluate the utilityand true storage capacity of such volume holographic media,a complete three-dimensional (3D) model is required that notonly includes the temporal �t� and in-plane spatial variations(in x and y) but also those in the volume thickness �z�.Assume a material layer illuminated as shown in Fig. 1. Note

that the grating does not vary in y; therefore in this paper spa-tial variations only exist in the x and z directions.

Previously, work has been presented discussing the photo-physical and photochemical evolutions in the photopolymermaterials in both the one-dimensional (1D) and 3D cases.For the 1D case detailed examinations of the photoabsorptivebehavior of the dye and of the resulting photochemical evo-lutions have been presented during grating formation in pho-topolymer materials. Gleeson et al. [18–22] have presented a1D nonlocal photopolymerization driven diffusion (NPDD)model developed to describe polymer chain growth and diffu-sion effects both during and after exposure. Gallego et al. [23–25] have discussed thick grating formation in the 3D case.Based on their results they have demonstrated that there ex-ists an attenuation coefficient that governs the nonuniform re-cording in depth. It has been shown how this value can beextracted by fitting the model to experimental measurements.One of the results of their analysis is that there is an effectiveoptical thickness, which for thicker layers is less than thephysical thickness that limits the grating strength recordableand thus limits the data storage capacity of the material [23].

Starting from these previous studies, we first explore thelight absorption and the dye consumption at all points inthe material layer as a function of time during the photoinitia-tion process. The aim is to better understand what takes placeinside the material during exposure and in particular to ex-plain the variations of the absorbance of the photosensitivedye and the attenuation of the light intensity distribution.In this paper the spatial and temporal variations of the

2638 J. Opt. Soc. Am. B / Vol. 31, No. 11 / November 2014 Li et al.

0740-3224/14/112638-10$15.00/0 © 2014 Optical Society of America

http://dx.doi.org/10.1364/JOSAB.31.002638

absorbed light intensity are evaluated using the Beer–Lambertlaw [9]. It is shown that the resulting nonuniform exposingintensity is most obvious when the dye is highly sensitive(absorptive) or the material layer is very thick. To do thiswe solve both the physical photoreaction equations and thephotochemical rate equations. This is done using the finite-difference method (FDM) in both time and space to numeri-cally predict when and where photons are either absorbed ortransmitted. Depending on the resulting nonuniform record-ing light energy absorption, the time-dependent 3D dye con-centration distribution can be estimated. These results arethen used to determine the photochemical reactions initiatedduring recording. In this way, by understanding the relation-ship between the photophysical evolutions, i.e., the light inten-sity absorption, and the varying concentrations of monomer,polymer, dye, initiator, and inhibitor in the layer, we ultimatelyaim in this series of papers to improve the validity of the 3DNPDD model. Given the model results, i.e., the predictedmaterial components’ concentrations, it is then possible, usingthe Lorentz–Lorenz relation [26–30], to calculate the corre-sponding refractive index profile in the volume. In this waythe formation of the 3D volume grating can be accuratelydescribed, as discussed in Part II [31].

Part I of this paper is structured as follows. In Section 2, wereview the primary photochemical mechanisms involved inthe photoinitiation process. In Section 3, applying the finite-difference time-domain (FDTD) method, both the physicalphotoreaction equations and the photochemical rate equa-tions are solved. The aim is to accurately describe the lightdistribution inside the material and the photosensitizer ab-sorptive behavior. In Section 4, the numerical simulation re-sults are presented and discussed. In Section 5, the validityand generality of the resulting 3D dye model presented hereare examined by applying the numerical predictions to fit a setof experimental results. It is shown that compared to theprevious 1D model case, more stable values of physicalparameters, i.e., the molar absorptivity, εD, and the quantumefficiency, ϕ, can be obtained using the 3D model proposed.Finally, a brief conclusion is provided in Section 6.

In Part II [31], a set of 3D differential rate equations gov-erning the polymerization process is derived. The resultingequations are solved numerically using the explicit-Runge–Kutta method [18], iteratively performing calculations usingthe same time and space steps used for the FDTD calculationspresented in Part I. In this way, a 3D NPDD model isdeveloped. The material parameters, i.e., the nonlocal re-sponse parameter, σ, the diffusion rate of monomer, Dm,

the chain initiation kinetic constant, ki, and the terminationrate, kt, are extracted by numerically fitting the predictionsof the 3D model to experimentally measured refractive indexmodulation growth curves.

2. PHOTOINITIATION MECHANISMSWe begin by describing the photoinitiation mechanisms takingplace in the material layer. Under suitable illumination, thephotosensitizers absorb photons, and the ground state dyemolecule, Dye, is promoted to the singlet excited state,1Dye� [9–11,20,32–34],

Dye� hν→kaS1Dye�; (1)

where kaS �s−1� is the rate of photon absorption by which theground state is excited to the singlet state. The singlet statecan undergo intersystem crossing into the triplet state,3Dye� [9,10,20]. For simplicity we reduce this process to

Dye� hν→kaT3Dye�; (2a)

where kaT �s−1� is the effective rate in going from the ground totriplet state as discussed in several previous studies [32–34].The singlet state dye can also recover to the ground state,

1Dye�→kr Dye; (2b)

where kr �s−1� is the rate constant of recovery of the excitedstate dye back to ground state. The primary radicals (R•) areproduced when the triplet state dye reacts with the coinitiator(electron donor, ED) [18–22]. The major reactions takingplace between these molecules are listed as follows:

3Dye� � ED→kd R• �HDye•; (3)

ED�HDye•→kb H2Dye� EDint; (4)

R• �M→kiM•

1: (5)

In these equations, HDye• represents a radicalized dye andkd �cm3 mol−1 s−1� is the rate constant of dissociation of theinitiator; see Eq. (3). H2Dye is the bleached di-hydro transpar-ent form of the dye, EDint is an intermediate form of the co-initiator, and kb �cm3 mol−1 s−1� is the rate constant of thebleaching process; see Eq. (4). M is the monomer, and ki�cm3 mol−1 s−1� is the chain initiation kinetic constant. Theseprimary radicals (R•) can react with the monomer to producechain initiators, M•

1; see Eq. (5).All of these photoreactions are summarized in the flow

chart in Fig. 2. We draw attention to the use of kaS and kaTto designate the rates of photon absorption and of transitionfrom the ground to triplet state following the expression inRefs. [32–34]. Note that the recovery of the triplet state dyeis neglected for the sake of simplicity. This dye model allowsus to treat the ground state dye consumption process sepa-rately from the rest of the photoinitiation process. As a result,two independent rate equations governing the ground statedye excitation and the singlet state dye production can be

Fig. 1. Volumeunslanted transmission holographic grating geometry.

Li et al. Vol. 31, No. 11 / November 2014 / J. Opt. Soc. Am. B 2639

solved by applying the finite-difference approximations in thespace and time domains.

Applying the Beer–Lambert law [9–11,31–34], the time-varying light intensity in depth (in z), I 0in�z; t� (Einsteincm−3 s−1), during the dye excitation can be described by

I 0in�z; t� � Tsf I 00

�exp

�−εD

Zz

0�Dye�z0; t��dz0

��∕z; (6)

where I 00 (Einstein cm−2 s−1) is the incident intensity, εD�cm2 mol−1� is the molar absorption coefficient of the photo-sensitizer, and �Dye�z; t�� mol cm−3� is the time-varyingground state dye concentration distribution that varies withdepth.

Solving Eq. (6) involves calculating a variable upper limitintegral function,

Rz0 �Dye�z0; t��dz0. We define the fraction of

light lost by Fresnel boundary reflections and material scatter-ing to be Tsf [31–34]. In Eq. (6) the unit of light intensity hasbeen converted from mW cm−2 into Einstein cm−2 s−1 usingI 00 � λBI0∕�Nmhc� in order to be consistent with the rate equa-tions in the following section [20,21]. I0 �mWcm−2� is the in-cident intensity, λ (nm) is the wavelength of the exposing light,Nm � 6.02 × 1023 is Avogadro’s number, c �ms−1� is the speedof light in free space, and h (J s) is Plank’s constant. We alsodefine the parameter B � 1 − exp�−εDAoΔzD� to be the initialabsorptive fraction [20], where Ao �mol cm−3� is the initial con-centration of dye, and ΔzD is the spatial step in depth, i.e., z,used for the FDTD method calculations, which will be dis-cussed in detail in Section 3. The light intensity I 0in�z; t� inEq. (6) can be used to evaluate the absorbed intensity Ia�z; t�(Einstein cm−2 s−1). The light energy absorbed over onespatial step in depth is

Ia�z; t� � −ΔzD�I 0in�z� ΔzD; t� − I 0in�z; t��: (7)

Recalling Fig. 2, kaS and kaT [32–34] present the photon ab-sorptive rates for the ground state to singlet state and thetriplet state conversions, respectively. As described [10,20]these photon absorptive rates can be written as

kaS�x; z; t� � εDϕSzI�x; z; t�; (8a)

kaT�x; z; t� � εDϕTzI�x; z; t�; (8b)

where ϕS and ϕT (mol Einstein−1) denote the quantum effi-ciencies of the reactions in going from the ground state tothe singlet and the triplet states, and I�x; z; t� (Einsteincm−2 s−1) is the time-varying incident intensity distribution.Assume illumination by a sinusoidal exposure distributedalong x. In practice two finite coherent exposing beams areused and an elliptic area on the material surface is illuminated.From Eq. (6) the exposing intensity can be written:

I�x; z; t� � I 0in�z; t��1� V cos Kx� cos θ; (9)

whereK � 2π∕Λ is the grating vector magnitude, with the gra-ting period Λ. The incidence angle of the two symmetric re-cording beams is θ, and cos θ is included to allow for variationof the exposing illumination intensity with illumination angle.The visibility of the fringes is given by V � 2�I1I2�1∕2∕�I1 � I2�, where I1 and I2 are the intensities of the two inci-dent beams. Substituting from Eq. (9) into Eqs. (8a) and (8b),the rates, kaS�x; z; t� and kaT�x; z; t�, can be rewritten as

kaS�x; z; t� � kaS�z; t��1� V cos Kx�; (10a)

kaT�x; z; t� � kaT�z; t��1� V cos Kx�: (10b)

The time-varying photon absorptive, kaS�z; t� and kaT�z; t�,are given by substituting from Eq. (6) into Eqs. (8a) and (8b),respectively,

kaS�z; t� � Tsf εDϕSI 00 cos θ exp�−εD

Zz


�;

(11a)

kaT�z; t� � Tsf εDϕTI 00 cos θ exp�−εD

Zz


�:

(11b)

3. CALCULATION USING THE FDTDMETHODThis section contains two subsections. The first provides a setof governing differential equations describing the time-varyingdye concentration and the light intensity distribution in depth.The second describes how to use a specially modified versionof the FDTD method to solve these rate equations.

A. Photoinitiation EquationsRecalling Eqs. (1) and (2) in Section 2, the processes ofground state dye consumption and singlet state dye produc-tion are treated separately. Consistent with Fig. 2, the follow-ing set of partial differential equations, governing theevolutions of the material concentrations, is derived:

Fig. 2. Flow chart illustrating the photoinitiation mechanisms,following the expression in Refs. [32–34].


∂�Dye�x; z; t��∂t

� −kaS�x; z; t��Dye�x; z; t��− kaT�x; z; t��Dye�x; z; t�� kr �SDye��x; z; t��; (12)

∂�SDye��x; z; t��∂t

� kaS�x; z; t��Dye�x; z; t�� − kr �SDye��x; z; t��:(13)

In these equations �Dye�x; z; t�� and �SDye��x; z; t��(mol cm−3) represent the ground and singlet state exciteddye concentrations, respectively. Note that the ground statedye concentration can be obtained by solving Eqs. (12) and(13) independently. Once we obtain the ground state dyeconcentration, the behavior of the light intensity and photo-physical conditions can be derived using Eqs. (6)–(11). Forthis reason we solve Eqs. (12) and (13) using the FDTDmethod.

B. Finite-Difference Time-Domain MethodPreviously, Wu and Glytsis [35] applied a FDTD method tosolve the 1DNPDDmodel in its dimensionless form forDuPontpolymer material. Kelly et al. [36] extended the 1D NPDDmodel using the FDM to include nonlocal temporal effectsfor use in modeling Acrylamide/polyvinyl alcohol (AA/PVA)material. In those papers [35,36] the space step was in x andthe time step was in t. Gallego et al. [37] used the FDM to solvethe NPDD in 3D. In their calculations, the space steps are inboth x and z, and the time step is in t. While the FDTD methodis very flexible when solving the two-dimensional space andtime cases numerically, it has the disadvantages of slow speedor large errors. These problems are in some cases unavoidabledue to the nature of the FDTD method itself [38].

In this subsection a modified FDTDmethod is introduced tosolve Eqs. (12) and (13). Instead of solving these equations infull 3D, the dimensionality of the FDTD calculation is reducedby increasing the number of equations to be solved. First,Eqs. (12) and (13) are rewritten by substituting Fourier seriesexpansions in x for the various concentration distributions.Then when performing the calculations, only the space step in-crementΔzD, inmaterial depth (z), and the time step incrementΔtD are necessary. Such a substitution is entirely consistentwith other approaches used in this area [18,21] since in the caseof a cosinusoidal illumination, it is typically assumed that thedistributions of components are periodic functions in x, andcan be written as a Fourier series,

�Q�x; z; t�� X∞l�0

Ql�z; t� cos lKx; (14)

where �Q�x; z; t�� represents a material component concentra-tion, andQl�z; t� represents the lth harmonic amplitude, whichonly varies in depth and time. Using Eq. (14), �Dye�x; z; t�� and�SDye��x; z; t�� (mol cm−3) in Eqs. (12) and (13) are rewritten as

�Dye�x; z; t�� X∞l�0

Dyel�z; t� cos lKx; (15a)

�SDye��x; z; t�� X∞l�0

SDyel�z; t� cos lKx: (15b)

Substituting Eqs. (10a), (10b), (15a), and (15b) into Eqs. (12)and (13), and retaining the first four concentration harmonicamplitudes, the resulting coupled differential equations are

∂Dye0�z; t�∂t

� −kaS�z; t�Dye0�z; t� − kaT�z; t�Dye0�z; t�

−

V2kaS�z; t�Dye1�z; t� −

V2kaT�z; t�Dye1�z; t�

� krSDye0�z; t�; (16a)


� −VkaS�z; t�Dye0�z; t� − VkaT�z; t�Dye0�z; t�− kaS�z; t�Dye1�z; t� − kaT�z; t�Dye1�z; t�

−



� krSDye1�z; t�; (16b)


� −



− kaS�z; t�Dye2�z; t� − kaT�z; t�Dye2�z; t�

−



� krSDye2�z; t�; (16c)


� −



− kaS�z; t�Dye3�z; t� − kaT�z; t�Dye3�z; t�� krSDye3�z; t�; (16d)

∂SDye0�z; t�∂t

� kaS�z; t�Dye0�z; t� �V2kaS�z; t�Dye1�z; t�

− krSDye0�z; t�; (17a)


� VkaS�z; t�Dye0�z; t� � kaS�z; t�Dye1�z; t�

� V2kaS�z; t�Dye2�z; t� − krSDye1�z; t�; (17b)


� V2kaS�z; t�Dye1�z; t� � kaS�z; t�Dye2�z; t�

� V2kaS�z; t�Dye3�z; t� − krSDye2�z; t�; (17c)


� V2kaS�z; t�Dye2�z; t� � kaS�z; t�Dye3�z; t�

− krSDye3�z; t�: (17d)

In these equations kaS�z; t� and kaT�z; t� can be obtained us-ing Eqs. (11a) and (11b). Comparing Eqs. (16) and (17) to pre-vious FDTD implementations [35], we note that our numerical


algorithm, using the modified FDTD, is greatly simplified. Inthe previous algorithm, the stability criterion required thatΔtD ≤ �ΔxD�2∕2, where ΔxD was the space step in x [35]. Inx, the length scale is determined by the grating period Λ,which is of the order of hundreds of nanometers [19,22].Meanwhile, in z, it is related to the layer thickness d, whichis of the order of hundreds of micrometers [6–8]. This resultsin values of ΔtD and ΔxD that must be chosen to be very smallto satisfy the stability criterion [35]. Returning to our modifiedFDTD method, Eqs. (16) and (17), the time scale t and thespace scale z are independent, and therefore no explicit sta-bility criterion is required. The ith and jth steps correspond toz � i × ΔzD and t � j × ΔtD, respectively, where i and j areintegers. Then kaS�z; t� and kaT�z; t� are discretized as

kaS�i�1; j�� kaS�i; j�exp�−ΔzDεD

�Dye0�i; j��

12Dye1�i; j�

��;

(18a)

kaT�i� 1; j� � kaT�i; j�exp�−ΔzDεD

�Dye0�i; j��

12Dye1�i; j�

��:

(18b)

The main calculation loop of the modified FDTD algorithm(in z − t) is as shown in Fig. 3.

In Eqs. (18a) and (18b), the values of kaS�i� 1; j� andkaT�i� 1; j� can be calculated using the values at the previousdepth step, kaS�i; j� and kaT�i; j�. The initial conditions for theiterative calculation loop of Eqs. (18a) and (18b) are

kaS�1; j� �I0λBεDϕS cos θ

Nmhc; (19a)

kaT�1; j� �I0λBεDϕT cos θ

Nmhc: (19b)

Rewriting Eqs. (16) and (17) at the ith space values and atthe jth time step gives that

Dye0�i; j � 1� � ΔtD�−kaS�i; j�Dye0�i; j� − kaT�i; j�Dye0�i; j�

−

V2kaS�i; j�Dye1�i; j� −

V2kaT�i; j�Dye1�i; j�

� krSDye0�i; j�� Dye0�i; j�; (20a)

Dye1�i; j � 1� � ΔtD�−VkaS�i; j�Dye0�i; j� − VkaT�i; j�Dye0�i; j�− kaS�i; j�Dye1�i; j� − kaT�i; j�Dye1�i; j�

−

V2kaS�i; j�Dye2�i; j� −

V2kaT�i; j�Dye2�i; j�

� krSDye1�i; j�� Dye1�i; j�; (20b)

Dye2�i; j � 1� � ΔtD�−

V2kaS�i; j�Dye1�i; j�

−

V2kaT�i; j�Dye1�i; j� − kaS�i; j�Dye2�i; j�

− kaT�i; j�Dye2�i; j� −V2kaS�i; j�Dye3�i; j�

−

V2kaT�i; j�Dye3�i; j� � krSDye2�i; j�

�

� Dye2�i; j�; (20c)

Dye3�i; j � 1� � ΔtD�−


−

V2kaT�i; j�Dye2�i; j� − kaS�i; j�Dye3�i; j�

− kaT�i; j�Dye3�i; j� � krSDye3�i; j��

� Dye3�i; j�; (20d)

SDye0�i; j � 1� � ΔtD�kaS�i; j�Dye0�i; j� �


− krSDye0�i; j�� SDye0�i; j�; (21a)

SDye1�i; j � 1� � ΔtD�VkaS�i; j�Dye0�i; j� � kaS�i; j�Dye1�i; j�

� V2kaS�i; j�Dye2�i; j� − krSDye1�i; j�

�

� SDye1�i; j�; (21b)

SDye2�i; j � 1� � ΔtD�V2kaS�i; j�Dye1�i; j� � kaS�i; j�Dye2�i; j�

�V2kaS�i; j�Dye3�i; j� − krSDye2�i; j�

�

� SDye2�i; j�; (21c)Fig. 3. Flow chart illustrating the calculation loop of light absorptionand dye consumption processes using the FDTD method.


SDye3�i; j � 1� � ΔtD�V2kaS�i; j�Dye2�i; j� � kaS�i; j�Dye3�i; j�

− krSDye3�i; j�� SDye3�i; j�: �21d�

The initial concentrations j � 1 of the ground and singletstate dyes are defined as follows:

Dye0�i; 1� � Ao; (22a)

Dye1�i; 1� � Dye2�i; 1� � Dye3�i; 1� � 0; (22b)

SDye0�i; 1� � SDye1�i; 1� � SDye2�i; 1� � SDye3�i; 1� � 0:

(23)

As summarized in Fig. 3 the resulting concentrations at the(j − 1)th time step are used to update those at the next jth timestep. When the material rate equations, Eqs. (20) and (21), aresolved in time steps of ΔtD, we assumed the variations ofincident light intensity transmission and absorption change ef-fectively instantaneously. Therefore over the entire materialdepth the kaS and kaT values are updated at each time stepusing the values of those calculated in the previous time step.Then the FDTD time space calculations associated with thedye model are treated as two independent parts and dealt withseparately. This process is reasonable because the time takenfor light to propagate through the material layer is very shortcompared to the time constants associated with the materialphotochemical effects. Typically the time taken for the light topropagate through the material is less than 10−11 s [19,20],while the photochemical reactions and mass transport effectstake place over milliseconds or seconds [39,40]. The algorithmtherefore works as follows: during the first time step the lightpropagates through the material and the dye concentration isobtained by solving Eqs. (19), (20), (21), (22), and (23). Then,the next values of kaS and kaT can be calculated using the pre-vious values and using Eqs. (18a) and (18b). After this thematerial equations governing the concentrations at the nexttime step can be found.

4. SIMULATIONS AND DISCUSSIONBefore presenting the simulation results, note that it isassumed that the dye diffusion effects are negligible[34,41,42]. To begin a set of initial physical reasonable condi-tions must be chosen as follows. The spatial frequency ofthe exposing illumination is assumed to be SF � 1∕Λ �1428 lines∕mm. The exposing fringe visibility is unity, i.e.,V � 1. The dry material layer is assumed to have a reasonablylarge thickness, d � 500 μm [23–25]. The two recordingbeams are assumed coherent and uniform, of wavelengthλ � 532 nm, delivering a total exposing intensity ofI1 � I2 � 10 mWcm−2. The initial concentration of the photo-sensitizer is �Ao� � 1.22 × 10−6 mol cm−3. The physical param-eters include εD � 2.0 × 108 cm2 mol−1, ϕS � 6.0 × 10−3 molEinstein−1, and ϕT � 12.0 × 10−3 molEinstein−1, and the re-covery rate is kr � 0.33 × 10−3 s−1 [43–42]. The Fresnel andscattering loss fraction used is Tsf � 0.70, which is atypical value for this thickness [23–25]. All the values used

are similar to those previously estimated from fits to standardAA/PVA experimental data [9,10,31,33].

We assumed that the FDTD spatial iteration calculationloop starts at z � 0 and ends at z � d, and the total exposuretime is texp � 240 s. The numerical parameters areΔzD � 10 μm, the number of sampling steps in z isNzmax � 50. ΔtD � 0.1 s, and the number of sampling stepsin time is Ntmax � 2400.

Applying the initial conditions above and using Eqs. (19),(22), and (23), we solve Eqs. (18), (20), and (21), with the firstfour harmonic amplitudes being retained in the calculations.The spatial distributions of the ground state dye concentra-tions, �Dye�x; z; t��, are predicted at various times as shownin Fig. 4. The distributions are shown over one grating period−Λ∕2 ≤ x≤� Λ∕2.

As can be observed in Fig. 4, the ground state dye is con-sumed most rapidly at the center of the bright regions wherex � 0 [9,10,31,33,44]. In Fig. 4(a) when texp � 30 s,�Dye�x; z; t�� increases with z (material depth). The initial layerabsorbance primarily depends on the illumination intensity asthere are dye molecules available at every depth. In agreementwith previous analyses [23–25], if the position in z is closer tothe input layer’s surface, i.e., z � 0, it receives a higher inten-sity exposure, leading to a more rapid consumption of theground state dye. Then, as the exposure time increases atlower depths, in the bright regions the ground state dyemolecules are used up more rapidly, and more ground statedye molecules are consumed in the darker regions. Sucheffects have previously been discussed in the 1D NPDD case[9,10,44].

As exposure continues, dye at deeper locations begins to beexcited in Fig. 4(b). When texp � 60 s, the dye concentrationprofiles at z � 100 and 200 μm have decreased significantly.Moreover, the �Dye�x; z; t�� distribution distorts from the ex-posing sinusoidal pattern and exhibits narrowing in the darkregions, i.e., when z � 10 μm. In results for texp � 240 s,shown in Fig. 4(c), this distortion becomes visibly stronger,and more ground state dye is consumed in the dark regions.It is also noticeable that at deeper material depths, i.e., z �100 and 200 μm, the distortions are also apparent. Thereforenot enough ground state dye concentration remains at thesedepths by these exposure times to produce an undistorteddistribution.

When texp � 240 s in Fig. 4(d) the �Dye�x; z; t�� distributionat z � 500 μm begins to be consumed in the bright regions.Comparing the results at z � 10 and 500 μm, we note thatat material depths closer to the input surface, the distortionbecomes visibly stronger earlier. Thus there is a nonlinear re-sponse of the material to the exposure intensity, and thethicker in the initial layer, the stronger the nonlinear responseeffects taking place within the material depth. Furthermore,comparing the results for z � 10, 100, and 200 μm, the profilesin this figure are much more similar to each other, in terms ofboth the profile distortions and the concentration distribu-tions. It is clear that if the consumption of the ground statedye continues for a long time, i.e., until texp � 240 s,�Dye�x; z; t��will be almost totally used up in the bright regionsat almost all depths. As a result, the light intensity is no longersimply attenuated with depth. This would seem to indicatethat the resulting refractive index profile inside the materialwill no longer exhibit a simple exponential attenuation in


depth for such longer exposure times. Such effects will bediscussed in detail in Part II [31].

The nonlinear material response can also be examined in x,z, and t by examining the predicted light absorption usingEq. (7) in Section 2. In Fig. 5, Ia�x; z� is plotted for the sameexposure times and depths examined in Fig. 4, i.e., texp =(a) 30, (b) 60, (c) 120, and (d) 240 s, and z � 10, 100, 200,and 500 μm, over two grating periods –Λ ≤ x ≤ �Λ.

For the shortest exposure time, texp � 30 s in Fig. 5(a), andat lower depth, i.e., z � 10 μm, in the layer a higher percent-age of the total absorptive dye concentration is consumedmuch more rapidly, which leads to the production of manyexcited dye molecules. As the concentration of the ground

state dye decreases, the amplitude of Ia�x; z� increases. Atthe location where the dye consumption rate is largest,Ia�x; z� will have a peak value. As shown in Fig. 5(a) theinitially absorbed light maximum value is located close tothe input surface of the material, z � 0, and in the brightlyilluminated regions.

As the exposure continues, at lower depths relatively fewerabsorptive ground state dye molecules are available com-pared to at the deeper positions. For this reason the effectof light absorption decreases close to the input surface, i.e.,at lower depths, and at the same time the total transmittedlight intensity will increase leading to the absorption peakmoving deeper into the layer, as shown in Figs. 5(b)–5(d).For example, when most of the dye molecules are excitedat z � i × ΔzD, the dye concentrations available to be excitedat z � �i − 1� × ΔzD and �i� 1� × ΔzD are smaller. This meansthat the light intensity transmitted by the �i − 1�th space stepis high, while the light intensity transmitted from the ith intothe �i� 1�th space step is lower. As the exposure continuesinto the next space step, the maximum dye consumption rateposition moves through the material, and therefore the peaklocation also moves through the layer from z � 0 μm to z �500 μm with time. Furthermore, it should also be noted thatthe peak value of Ia�x; z� decreases significantly with time asindicated in Fig. 5; i.e., in Fig. 5(a), the maximum value ofIa�x; z� is ∼20 × 10−7 Einstein cm−2 s−1 at z ≈ 0, and inFig. 5(d), the maximum value is ∼5 × 10−7 Einstein cm−2 s−1

at z ≈ 500 μm. This is because some light always reachesthe deepest parts of the layer (even when texp � 0) and excitessome of the dye. Throughout the entire exposure time theground state dye concentration is always being consumedat all depths of the material layer. When the absorbed intensitypeak location moves down into the deeper regions, theavailable ground state dye concentration at that time is lessthan it was initially when at t � 0 and at z � 0.

5. EXPERIMENTAL RESULTSIn this section, the proposed 3D dye model is examined andvalidated by a range of experimental results. Applying themodel predictions and a set of experiments for different layerthicknesses, the material absorptive characteristics can beinvestigated. This is done by measuring the normalized trans-mittance growth curves for a typical dye, Erythrosine B (EB),involved in a standard free-radical photopolymer system, i.e.,acrylamide/polyvinyl alcohol (AA/PVA) [41,42]. Then variousmaterial parameter values are extracted by fitting the modelpredictions to the experimentally obtained results.

First, the resulting normalized transmission curves fora range of material layer thicknesses are measured. Theexperimental setup is shown in Fig. 6. It involves a stablegreen laser source of intensity I0 � 10 mWcm−2 at a wave-length of λ � 532 nm. The beam passes normally throughthe material layer, and the resulting transmitted intensity ismeasured. The area of illumination is 0.25 cm2. The spatialfilter is composed of a microscope objective EFL 8.00 anda pinhole 2.5 μm. The lens placed in front of the spatial filteris of focal length 8 cm and diameter 4 cm.

Then applying Eq. (6) in Section 2 and the 3D dye modeldiscussed in Section 3, the absorptive parameter values canbe estimated by fitting the simulations to these experimentallymeasured curves. It is assumed that the rate of excitation of

50 100 150 200

2. 10 7

4. 10 7

6. 10 7

8. 10 7

1. 10 6

1.2 10 6

[Dye(x, z)] ×10-6

(mol cm-3)

Λ/2 -Λ/2 0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

Bright Dark Dark

texp = 30 (s)

x

(a)

z = 10 µm z = 100 µm

z = 200 µmz = 500 µm

50 100 150 200

2. 10 7

4. 10 7

6. 10 7

8. 10 7

1. 10 6

1.2 10 6

[Dye(x, z)] ×10-6

(mol cm-3)

Λ/2-Λ/2 0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

Bright Dark Dark

texp = 60 (s)

x

(b)

z = 10 µm

z = 100 µm

z = 200 µm

z = 500 µm

50 100 150

2. 10 7

4. 10 7

6. 10 7

8. 10 7

1. 10 6

1.2 10 6

[Dye(x, z)] ×10-6

(mol cm-3)

Λ/2-Λ/2 0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

Bright Dark Dark

texp = 120 (s)

x

(c)

z = 10 µmz = 100 µm

z = 200 µmz = 500 µm

2. 10 7

4. 10 7

6. 10 7

8. 10 7

1. 10 6

1.2 10 6

[Dye(x, z)] ×10-6

(mol cm-3)

Λ/2-Λ/2 0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

Bright Dark Dark

texp = 240 (s)

x

(d)

z = 10 µm

z = 100 µm

z = 200 µm z = 500 µm

Fig. 4. Simulations of the ground state dye concentration in x − z,�Dye�x; z�� mol cm−3, for four different exposure times: texp =(a) 30, (b) 60, (c) 120, and (d) 240 s, at four different material depths:10 (green full curve), 100 (red short dashed curve), 200 (purple dashedcurve), and 500 μm (blue long dashed curve).

Fig. 5. Comparison of numerical simulations of the absorbed inten-sity Ia�x; z� Einstein cm−2 s−1 in depth, of the spatial frequency expo-sure of SF � 1428 linesmm−1, for a material thickness of d � 500 μmand incident intensity I0 � 10 mWcm−2; results for four different ex-posure times are presented: texp = (a) 30, (b) 60, (c) 120, and (d) 240 s.


the photosensitizer is much faster than the regeneration orrecovery rates, as has been verified experimentally[9,10,32–34]. Therefore, during exposure, the diffusion, recov-ery, and bleaching of the photosensitizer and its excited statesare neglected. In this way the complexity of the model isreduced, and it is no longer necessary to distinguish thetwo excited dye states; i.e., we assume that the singlet andtriplet state photosensitizer concentrations can be treatedas a single lumped parameter [41,42]. We refer to a new quan-tum efficiency, ϕ � ϕS � ϕT. The experimental and theoreti-cal normalized transmittance curves for different materiallayer thicknesses are given in Fig. 7.

The parameter values estimated by fitting the experimentaldata using both the 1D and 3D dye models are extracted andlisted in Table 1. A mean square error (MSE) value is also

shown to quantify the quality of the fitting. The numericalparameters for the 3D model calculations are as follows:ΔzD � 50 μm (the space step length) and ΔtD � 0.1 s(the time step length). Meanwhile a 1D dye model previouslyreported in [9,10,41,42] is also used to estimate the corre-sponding parameter values as a comparison. All these resultsare shown in Table 1.

In Table 1 we first note that the estimated values of molarabsorptivity εD using the 1D and 3D models are consistent(but not identical) for the different material layer thicknesses.Before exposure when t � 0, both Eq. (12) in Section 2 andthe 1D model [41,42] predict that the transmittance equationsreduce to

T0 � Tsf exp�−εDAod�; (24)

where T0 is the initial transmittance value, the initial dye con-centration is Ao � 1.22 × 10−6 mol cm−3, and d is the materiallayer thickness. As a result, the value of εD can be independ-ently estimated using the value of initial transmittance T0

in Eq. (24).Based on the demonstrations in Table 1, it is worth noting

that the values of quantum efficiency ϕ estimated aresignificantly different between the 1D and 3D model cases.In the 1D model case, the values of ϕ decrease greatly asthe material layer thicknesses d increase from 50 to150 μm. When the value of d is small, i.e., 50 μm, the valuesof ϕ are consistent between the 1D and 3D models,i.e., ϕ � 4.90 molEinstein−1.

The reason for these results is mainly due to the limitationof the 1D model. In the previously developed 1D models[9,10,32,33,41,42], the time-varying dye concentration is nottaken into account along the material depth, and the incidentlight intensity is therefore assumed to be a constant, i.e.,I0 � 10 mWcm−2. However, taking into account the complexphotoinitiation processes taking place in the material volume,there is a significant variation of the light intensity inside thematerial, and the light absorption is not uniform in depth[23–25]. When applying the 1D model to fit the experimentaldata, the theoretically estimated absorbed light is more than isactually the case; thus the values of ϕ extracted are lower thanin the 3D case. Such errors become worse the larger the layerthickness.

To overcome such errors, the proposed 3Dmodel is appliedto simulate the light intensity variations inside the material.We observe that the extracted parameter values, i.e., εDand ϕ, are consistent when the 3D model is used. In Table 1we found that the maximum deviation of ϕ is 2.60 × 10−2 when

Fig. 6. Setup for the transmission experiments.

Fig. 7. Normalized transmission in AA/PVA photopolymer materialsensitized by Erythrosine B (EB), for three typical material layerthicknesses: (a) d � 50 μm, (b) d � 100 μm, and (c) d � 150 μm. Boththe experimental data points (circles, squares, and triangles) and the3D dye model fits (red, green, and purple solid lines) are presented.The total illuminating intensity is I0 � 10 mWcm−2.

Table 1. Parameter Values from Fits to Experimental Transmittance Curves for a Range of Material Layer

Thicknesses in AA/PVA Photopolymer Material Sensitized by EB, using 1D and 3D Models, Respectivelya

Models d (μm) texp (s) εD �×108� �cm2 mol−1� ϕ �×10−2� �molEinstein−1� Tsf (–) T0 (–) MSE �10−3�50 30 1.36 4.90 0.81 0.320 2.82

1D model 100 40 1.41 3.00 0.78 0.165 5.75150 60 1.40 2.30 0.73 0.066 3.94

Average – – 1.39 3.40 – – –50 30 1.36 4.90 0.81 0.320 2.51

3D model 100 40 1.40 5.00 0.78 0.165 3.47150 60 1.38 5.00 0.73 0.066 3.26

Average – – 1.38 4.97 – – –aIn all cases I0 � 10 mWcm−2.


using the 1D model. Simultaneously it is only 0.10 × 10−2 whenusing the 3D model. This result supports the validity andgenerality of the 3D model and the procedure used.

In order to obtain greater accuracy, more experiments forthicker material layers should be used. In the model the Fres-nel and scattering losses are considered and estimated usingthe empirical parameter Tsf . Experimentally, care is alwaystaken to try to produce uniform layers. As the thickness in-creases, however, it becomes more difficult to produce suchlayers, and this will lead to deterioration in the quantitativeagreement between the experimental data and the predictionsof the model. In such cases validating the proposed modelbecomes increasingly difficult. As the material layer thicknessincreases, the loss due to scattering becomes much morepronounced. The exposure time for large thicknesses mustalso be longer. Therefore the laser source used must be ex-treme stable. All these conditions critically affect the accuracyof further experimental measurements.

6. CONCLUSIONSIn order to further develop a 3D time-dependent NPDDmodel,it is necessary to more fully and accurately model the effectsof light absorption and dye consumption inside the materialvolume. Following an analysis of the major photochemical re-actions taking place during photoinitiation, a more completephysical representation of the material behavior is presented,and a 3D dye model is developed.

In this article, a modified FDTD method is introduced tonumerically solve the photoreaction equations governing boththe ground state dye concentration and the light intensitytransmission and absorption effects. As is commonly donewhen solving the 1D NPDD model, Fourier series expansionsin x of the component material concentrations are used torewrite the two governing partial differential photoreactionequations into eight first-order coupled differential equations.Then suitable FDTD grids in depth (z) and time (t) areidentified, and a discrete FDTD iterative algorithm isobtained, i.e., Eqs. (18), (20), and (21). Using the initialconditions, i.e., Eqs. (19), (22), and (23), the dye concentrationvariations with time and space are numerically predicted. Thelight intensity values are then iteratively updated in both z andt. This numerical method reduces the total FDTD calculationtime significantly, and permits the calculation accuracy to beimproved. Using this algorithm several new predictions havebeen made:

(i) The previously identified light intensity attenuation ef-fects with depth [23–25] are only observed during the earlystages of the exposure. Our results clearly indicate that asthe exposure continues, and the ground state dye concentra-tion is used up, the variation produced does not remain a sim-ple exponential attenuation with depth.(ii) Similarly, as the dye is used up a nonlinear response is

produced, resulting in a spatial distribution that is no longersinusoidal (like the exposing pattern), but exhibits significantdistortion, i.e., loss of fidelity. This distortion is visiblystronger closer to the input layer surface at z � 0 but becomesprogressively worse at all depths for longer exposures.

The model is then validated using a set of experiment data.Transmittance measurements are undertaken for a range ofdifferent material layer thicknesses. The materials’ absorption

parameters are then estimated from the results using the new3D dye model, and the different data sets were compared tothose previously predicted using the corresponding 1D model[9,10,32,33,41,42]. No significant differences in the parametervalues extracted by the 3D model fittings for different thick-nesses were observed. Meanwhile, in the 1D model case, thevalues of quantum efficiency ϕ decrease significantly as thematerial layer thicknesses increase. These results indicatethat the 3D model can well describe the light absorptionand the dye consumption during the photoinitiation processin the material.

Much work remains to be done in relation to related dyeeffects. A more complete and detailed quantum photophysicaland photochemical analysis of photon absorption needs to beused to describe the photoinitiation process [9–11]. Compar-isons of the predictions of this model to reproducibleexperimental data for various types of materials, e.g., epoxy[11], PQ/PMMA [45–47], and DuPont OmniDex613 [43], andvarieties of dyes, e.g., Eosin Y, Phloxine B, Rose Bengal[41], and 2-(4-(N,N-dimethylamino)benzylidene)-1H-indene-1,3(2H)-dione (D_1) [42], are also necessary.

In Part II of this paper [31], the results above are used incombination with the 3D NPDD to describe the photopolyme-rization process. The refractive index modulation is calcu-lated using the Lorentz–Lorenz relation [26–30]. Using thisextended 3D NPDD model, the nonuniform volume gratingformation evolution taking place in the material is more fullyexamined, and the predictions of the model are compared toexperimental results.

ACKNOWLEDGMENTSH. Li is supported by a University College Dublin–China Schol-arship Council joint scholarship. Y. Qi is supported by the EUERASMUS Mundus fund. The authors also acknowledge thesupport of the Irish Research Council for Science, Engineer-ing and Technology (IRCSET), Enterprise Ireland and ScienceFoundation Ireland (SFI), under the National DevelopmentPlan (NDP).

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