A Diffusion Model for Spatially Dependent Photopolymerization
ABSTRACT Photopolymers represent an attractive class of optical recording materials due to properties such as high refractive index
modulation, dry film processing, low cost, etc. Applications include holographic data storage disks, optical interconnections,
memories and filters. This paper addresses the dynamics of short-exposure holographic grating formation; a new model is proposed
to explain the experimental observations of low diffraction efficiency in high spatial frequency gratings.
Dublin Institute of Technology
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A Diffusion Model for Spatially Dependent
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Mackey, D., Babeva, T., Naydenova, I., Toal, V.: A diffusion model for spatially dependent photopolymerisation. Progress in Industrial
Mathematics at ECMI 2008. Mathematics in Industry, 2010, Volume 15, Part 2, 253-258, DOI: 10.1007/978-3-642-12110-4_36
A diffusion model for spatially dependent
D. Mackey1, T. Babeva2, I. Naydenova2, and V. Toal2
1School of Mathematical Sciences, Dublin Institute of Technology, Ireland
2Centre for Industrial and Engineering Optics, Dublin Institute of Technology,
Summary. Photopolymers represent an attractive class of optical recording mate-
rials due to properties such as high refractive index modulation, dry film processing,
long shelf life, etc. Applications include holographically based devices for optical
storage disks, optical interconnections, optical memories and filters. This paper will
address the dynamics of short-exposure holographic grating formation; a new math-
ematical model is proposed with the aim of understanding the experimental obser-
vations of low diffraction efficiency in high spatial frequency gratings.
The basic formulation of a dry photopolymer system usually consists of
monomer and photoinitiator, all dispersed in a binder matrix. Upon uniform
illumination, a monomer polymerizes and the refractive index of the system
changes. When material is exposed to an interference pattern more monomers
are being polymerized in the bright regions than in the dark ones. This sets
up a concentration gradient and the monomer starts to diffuse from dark to
bright areas where it has been polymerized. The recorded holographic grating
(refractive index space and time profile) is a result of changes in the relative
density of components.
The grating evolution in photopolymer systems has been studied theo-
retically and experimentally by several authors (, , etc.). However, the
common feature of all models proposed to date is that they cannot describe
the experimental observation of poor diffraction efficiency at higher spatial
frequencies. One of the theories which attempts to explain this poor high fre-
quency response focuses on the counter diffusion of short chain (terminated
or non-terminated) polymer molecules from bright to dark fringes, which de-
creases the overall refractive index modulation (see, for example, ). To verify
this theory, we propose a mathematical model for hologram formation which
includes both monomer and polymer diffusion, and moreover differentiates
between short polymer chains capable of diffusing and long polymer chains
2D. Mackey, T. Babeva, I. Naydenova, and V. Toal
which are immobile. The time evolution of the refractive index modulation
after short exposures is then compared with experimental results.
2 Problem formulation/Refractive index analysis
The photopolymer is exposed to two coherent beams of intensities I1and I2
which create the following illumination pattern
I(x) = I0(1 + V cos(kx))
where k is the grating wavenumber, I0= I1+I2and V = 2√I1I2/(I1+I2) is
known as the beam intensity modulation. Upon illumination, the sensitizing
dye absorbs a photon and reacts with the electron donor to produce free rad-
icals. In the presence of monomer these free radicals initiate polymerization.
During the propagation step, free radicals and monomer molecules interact
and produce growing polymer chains. At the termination step, two free radi-
cals or two polymer chains interact and the polymer chains stop growing.
The faster consumption of monomer near the illumination peaks sets up
a concentration gradient and the free monomer starts to diffuse from dark
to bright areas. The combination of spatially nonuniform polymerization and
free monomer diffusion increases the spatial modulation of the refractive in-
dex and yields a phase grating. It was conjectured in ,  that secondary
diffusion processes (involving short-chain polymer molecules or radicals) take
place during recording that decrease refractive index modulation and that this
process is responsible for the observed poor high-spatial-frequency response.
The refractive index of a material consisting of a mixture of components
can be calculated with the well-known Lorentz-Lorenz equation:
where n is the effective refractive index of the mixture, niare the refractive
indices of the components (monomer, polymer and binder), determined sep-
arately from spectrophotometric measurements, and Φi are the normalized
concentrations of the components (e.g. Φm= m/(b + m + p), where m,p and
b denote concentrations of monomer, polymer and binder, respectively). The
details of this calculation are not important so will not be included here.
The refractive index modulation measures grating strength and is usually
calculated as the difference between the values in the bright and dark fringes,
∆n(t) = nmax(t) − nmin(t)
Ideally, ∆n(t) should exhibit fast growth followed by convergence to an equi-
A diffusion model for spatially dependent photopolymerization3
3 Proposed model
We now propose a generalization to existing models which takes into account
monomer and polymer diffusion, creation of short polymer chains and in-
troduces a simple “immobilization” mechanism which mimics the growth of
polymer chains to the extent where they cannot diffuse any longer. The short
exposure r´ egime is also reflected in the model, whereby all polymerization and
immobilization processes stop once the light beam is terminated.
In what follows, the spatial domain is assumed to be x ∈ [0,Λ], where
kis the grating period or fringe spacing. The classical model (see, for
example, ) consists of a polymerization-diffusion equation for the monomer
molecules but assumes diffusion stops once monomer is polymerized,
∂x2− F(x)m. (1)
Here m(x,t) denotes monomer concentration, Dm is the monomer diffusion
constant and the polymerization rate is proportional to the illumination
F(x) = F0(1 + V cos(kx))a≡ F0f(x)
where F0is a constant. The concentration of free monomer in the material is
initially considered spatially uniform,
m(x,0) = m0.
In addition, we now assume that short polymer chains are capable of dif-
fusing and the diffusion coefficient is also proportional to the illumination,
D(x) = Dpf(x), meaning that at a higher intensity, more short-chain poly-
mer molecules are formed, which are more mobile. We also assume that short
chains are converted to long chains at a rate proportional to monomer and
polymer concentrations and that the long chains are immobile once formed.
The resulting equations are
+ Φ(t) [F(x)m − Γmp1](3)
where, p1(x,t) is the concentration of short polymer chains, p2(x,t) is the
concentration of long polymer chains, and the new initial conditions are
p1(x,0) = 0,p2(x,0) = 0. We assume these equations are supplemented
by zero-flux boundary conditions. To account for a short exposure r´ egime in
equations (2)-(4) we have introduced the step function
4D. Mackey, T. Babeva, I. Naydenova, and V. Toal
, if t ≤ te
, if t > te
where teis the beam exposure time. Note that this model is not valid for long
exposure times when the diffusion coefficients for both monomer and polymer
molecules are known to be time dependent.
With the choice of non-dimensional variables
¯ x =x
¯t = tF0,¯ m =
(i = 1, 2),
the model becomes
+ Φ(t) [f(x)m − γmp1] (6)
≪ 1;γ =Γm0
We also have the initial and boundary conditions
m(x,0) = 1, pi(x,0) = 0;
∂x(x,t) = 0,for x = 0, 1.(10)
On adding and integrating equations (5)–(7) we get the conservation law
[m(x,t) + p1(x,t) + p2(t)] dx = 1,
which is to be expected, since monomer is converted into polymer while the
total concentration of particles remains constant.
4 Results and conclusions
The non-dimensional model (5)–(10) was integrated numerically using a stan-
dard finite difference method. The numerical values used for the diffusion con-
stants are Dm= 10−7cm2/s, Dp= 10−9cm2/s (close to the values determined
in ), so ε = 0.01. The polymerization rate is assumed to be F0= 0.3s−1,
and Λ is varied between 2·10−7m and 1·10−5m (corresponding to a range of
100–5000 lines/mm). The exposure time is te= 0.2s.
A diffusion model for spatially dependent photopolymerization5
Figures 1 and 2 show the evolution of the monomer and long polymer
concentrations for various values of the system parameters. The spatial mod-
ulation of these species is represented here over two grating periods. Note
there is a departure from the expected sinusoidal pattern occurring for a com-
bination of low spatial frequency (Λ = 10−5m) and longer exposure times
(te = 4s). A detailed analysis of how the ratio between diffusion and poly-
merization rates, as well as the exposure time, affect the holographic grating
formation will form the subject of further study.
Fig. 1. Relative concentrations of monomer (m/m0) and long polymer (p2/m0).
Here Λ =
Fig. 2. Relative concentrations of monomer (m/m0) and long polymer (p2/m0).
Here Λ =
Figure 3 shows a remarkable resemblance of the qualitative behaviour of
the refractive index modulation between experiment and model simulation,
for three values of the spatial frequency (200, 350 and 500 lines/mm). Fig-
ure 4 shows the time evolution of the refractive index modulation for a wide
range of spatial frequencies, between 100 and 5000 lines/mm. Note that the
model simulations reflect the experimental observations of refractive index
modulation drop at high spatial frequencies.
The model validates the theory that the high spatial frequency response
can be improved by suppressing the diffusion of short-chain polymer molecules.
This requires that the holographic recording conditions be chosen so as to
6D. Mackey, T. Babeva, I. Naydenova, and V. Toal
Fig. 3. Comparison of experimental (left) and numerical (right) results for the
refractive index modulation. The three curves correspond to Λ = 5 · 10−6m,Λ =
2.9 · 10−6m and Λ = 2 · 10−6m.
Fig. 4. Refractive index modulation for several values of Λ. Note the deterioration
of the grating strength for high spatial frequencies.
favour the production of longer polymer chains which will not be able to
diffuse readily from the bright to the dark fringe regions. A successful experi-
mental strategy for achieving a high diffraction efficiency reflection hologram
in an acrylamide-based photopolymer was recently presented in .
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