|
The nonlocal polymerization driven diffusion model is used to describe holographic grating formation in acrylamidebased
photopolymer. The free radical chain polymerization process results in polymer being generated nonlocal both in
space and time to the point of chain initiation. A Gaussian spatial material response function and an exponential
temporal material response function are used to account for these effects.
In this paper we firstly examine the nature of the temporal evolution of grating formation for short recording
periods. It is shown that in this case, temporal effects become most notable and the inclusion of the nonlocal temporal
response function is shown to be necessary to accurately describe the process. In particular, brief post exposure selfamplification
of the refractive index modulation is noted. This is attributed to continued chain growth for a brief period
after exposure. Following this a slight decay in the grating amplitude also occurs. This we believe is due to the
continued diffusion of monomer after exposure. Since the sinusoidal recording pattern generates a monomer
concentration gradient during the recording process monomer diffusion occurs both during and after exposure. The
evolution of the refractive index modulation is determined by the respective refractive index values of the recording
material components. From independent measurements it is noted that the refractive index value of the monomer is
slightly less than that of the background material. Therefore as monomer diffuses back into the dark regions, a
reduction in overall refractive index modulation occurs.
Volume changes occurring within the material also affect the nature of grating evolution. To model these
effects we employ a free volume concept. Due to the fact that the covalent single carbon bond in the polymer is up to
50% shorter than the van der Waals bond in the liquid monomer state, free volume is created when monomer is
converted to polymer. For each bond conversion we assume a hole is generated which then collapses at some
characteristic rate constant.
Incorporating each of these effects into our model, the model is then solved using a Finite-Difference Time-
Domain method (FDTD). The Lorentz-Lorenz relation is used to determine the overall evolution refractive index
modulation and the corresponding diffraction efficiency of the resulting grating is calculated using Rigorous Coupled
Wave Analysis (RCWA). Fits are then carried out to experimental data for 1 second exposures. Good quality fits are
achieved and material parameters extracted. Monomer diffusion rates are determined to be of the order of D ~ 10-10
cm2/s and the time constant of the nonlocal material temporal response function being of the order of τn ~ 10-2s.
Material shrinkage occurring over these recording periods is also determined.
|
