KEYWORDS: Diffraction, Diffraction gratings, Holography, Data storage, Polymers, Data modeling, Diffusion, Polymerization, Photopolymers, Refractive index
The development of an optimized scheduling technique based on an accurate model is necessary for the continued
development of holographic data storage technology. In this paper we examine an algorithm based on the non-local
polymerization driven diffusion model (NPDD), which determines an appropriate recording schedule for use in data
storage. The NPDD model accounts for nonlocal spatial and temporal material effects present in material involving free
radical chain polymerization. The model is solved using a finite difference technique and an optimized schedule
determined. Results are compared to experimental work. The inverse-square scaling law of holographic diffraction is
also examined and is shown to hold for low diffraction efficiency gratings but breaks down for a low number of high
efficiency gratings.
Development of a theoretical model of the processes present during the formation of a holographic grating in photopolymer material is crucial in enabling further development of holographic applications. To achieve this, it is necessary to understand the photochemical and photo-physical processes involved and to isolate their effects enabling them to be modelled accurately. Photopolymer materials are practical materials for use as holographic recording media, as they are inexpensive and self-processing. Understanding the recording mechanisms will allow their limitations for certain processes to be improved and a more efficient, environmentally stable material to be produced. In this paper we further develop our Non-local Polymer Driven Diffusion (NPDD) model to include the effects of absorption and inhibition effects. Thus we attempt to increase the accuracy of our existing model.
Incorporating the effects of volume changes into the Nonlocal Polymerization Driven Diffusion Model we examine these effects on grating evolution. The concept of free volume or hole formation is explored and the subsequent decay and diffusion of holes examined. The inclusion of a nonlocal temporal response function is also shown to be critical to the modeling of grating formation for short recording times. The model is solved using a finite difference technique and results converted initially to refractive index modulation and then to diffraction efficiency using rigorous electromagnetic theory. Fits are carried out to experimental data and model parameters determined.
Holographic data storage systems, utilising various photopolymer materials as the recording medium, are currently being developed. The photopolymer recording material used in this study is an Acrylamide/PVA based material. In this paper, having determined values for some basic properties of the material such as diffusion of Polyacrylamide and diffusion of water, we now look at chemically modifying the material and experimentally determine the impact. An important material characteristic, which determines the performance of any photopolymer medium, is the spatial frequency response of that material. Previously, applying our Non-local Photo-Polymerisation Driven Diffusion Model, (NPDD), we have discussed the effects on material behaviour of the length of the polymer chains and the rates of diffusion within the material. These parameters have been shown to be important in determining the response of the material. If the average length of the Polyacrylamide chains is shortened, an increase in the diffusion coefficient might be observed. Shorter Polyacrylamide chains should then result in an increase in the materials spatial frequency response, and ultimately in an increase in holographic data storage capacity. One possible method of doing this is to modify the chemical composition of the material to control chain length. The rates of diffusion of the material, both before and after modification of the chemical composition, are compared to determine the impact. Shortening the chain lengths should result in the possibility of creating smaller structures in the photopolymer material.
The Nonlocal Polymerization Driven Diffusion model, NPDD, is can be used to describe holographic grating formation in Acrylamide-based photopolymer. The free radical chain polymerization process results in polymer being generated nonlocal both in space and time to the point of chain initiation. Temporal nonlocality can be used to describepost exposure dark effects. Nonlinear response and the effects of dye bleaching have been examined. Both primary and bimolecular chain termination mechanisms have been included and examined. Recently 3-D, and inhibition effects have also been included. In this paper we review of our recent work. It is shown that temporal effects become most notable for short exposres and the inclusion of the nonlocal temporal response function is shown to be necessary to accurately describe the process. In particular, brief post exposure self-amplification 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. 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). 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). Inhibition is typically observed at the start of grating growth during which the formation of polymer chains is suppressed. In this paper experiments are reported, carried out with the specific aim of understanding of these processes. The results support our description of the inhibition process in an PVA/Acrylamide based photopolymer and can be used to predict behaviour under certain conditions.
Research dealing with models to predict and understand the behaviour of photopolymers have generated many interesting studies considering a 2-dimensional geometry. These models suppose that the photopolymer layer is homogeneous in depth. Using this approximation good results can be obtained if the thickness of photopolymers is less than 200 μm. However, it is well known that Lambert-Beer's law predicts an exponential decay of the light inside the material. In recent years intensive efforts have been made to develop new holographic memories based on photopolymers. For this application the thickness of the layer is increased, usually to more than 500 μm, and Lambert-Beer's law plays a significant role in the recording step. The attenuation of the index profile inside these materials has been measured, showing that it is an important phenomenon. This attenuation limits the maximum effective optical thickness of the grating and shows that the 2-D models can not be applied in these cases. For this reason in this work a 3-dimensional model is presented to analyze the real behaviour of the photopolymers and study the variations in the
index profile in depth. In this work we examine the predictions of the model in the case of a general dependence of the polymerisation rate with respect to the intensity pattern, and the effects of varying the exposure intensity are also compared in 3-D cases. Finally, the limitation of the data storage capacity of the materials due to the Lambert-Beer law is evaluated.
The Nonlocal Polymer Driven Diffusion (NPDD) model successfully predicts high spatial frequency cut-off and higher
harmonic generation, experimentally evident in holographic gratings recorded in free radical chain photopolymer
materials. In this paper the NPDD model is extended to include a nonlocal material temporal response. Previously it
was assumed that following a brief transient period, the spatial effect of chain growth was instantaneous. However,
where the use of short exposures is necessary, as in optical data storage, temporal effects become more significant.
Assuming that the effect of past chain initiations will have less effect on monomer concentration at a later point in time
than current initiations, a normalized exponential function is proposed to describe the process. The extended diffusion model is then solved using a Finite-Difference Time-Domain technique to predict the evolution of the monomer and
polymer concentrations during and after grating recording. The Lorentz-Lorenz relation is used to determine the
corresponding refractive index modulation and The Rigorous Coupled Wave Method applied to determine and/or
process diffraction efficiencies. A fitting technique is then used which first solves the diffusion model as described and
determines a set of parameters which give best fits to the experimental data. Results show that the inclusion of the
nonlocal temporal response is necessary to accurately describe grating evolution for short exposures i.e. continued
polymer chain growth for some period after recording resulting in an increase in the refractive index modulation.
Monomer diffusion is also shown to influence refractive index modulation post-exposure. Monomer diffusion rates
determined to be of the order of D ~ 10-11 cm2/s and the time constant of the nonlocal material temporal response
function being of the order of τn ~ 10-2s.
Holographic data storage systems, utilising photopolymer material as the recording medium, have recently been
presented. Because of their relatively low cost and ease of use, due to their self-processing nature, photopolymers
provide many potential advantages as the holographic recording material and data storage medium of choice.
Photopolymers show promise, for example, for Write-Once Read-Many (WORM) storage systems. The photopolymer
recording medium used in this study is an Acrylamide/Polyvinylalcohol (A/PVA) based dry layer. An important
material characteristic, which determines the performance of any photopolymer medium, is the spatial frequency
response of that material. Previously, applying our Non-local Photo-Polymerisation Driven Diffusion Model, (NPDD),
we have discussed the effects on material behaviour of the length of the polymer chains and the rates of diffusion within
the material. These parameters have been shown to be critically important in determining the response of the material.
If the average length of the Polyacrylamide (photopolymerised Acrylamide monomer, PA), chains is shortened, an
increase in the diffusion coefficient of these molecules might be observed. According to the NPDD shorter PA chains
should then result in an increase in the materials spatial frequency response, and ultimately in an increase in holographic
data storage capacity. In this paper we report on several experiments carried out (a) to determine the diffusion constant
of the PA and (b) to also determine the diffusion constants of both water and Propanol in our material layers.
In recent years there has been an increasing interest in holography and its applications. One such application is data
storage. Optimising the holographic recording materials is therefore of critical importance in the capacity and clarity
of the information stored. Photopolymer materials are practical materials for use as holographic recording media, as
they are inexpensive and self-processing. Understanding the photochemical mechanisms present during recording in
these materials is crucial in enabling further developments. Obtaining critical material parameters allows
improvements of the performance of these materials, such as its spatial frequency response or its environmental
stability. This also allows a better understanding of the photochemical processes that occur during the formation of
the holographic grating. Our current work, which is presented in this paper deals with two of the processes that
occur during holographic grating formation. The first of these is the photochemistry involved in the absorption of
the light by the photosensitive dye. We monitor the power of the transmitted beams, which are used for recording
the gratings. The second process we concentrate on is the inhibition effect present during grating growth. It has
been noted in the literature that there is a slight delay at the start of grating growth. The reason for this delay is due
to an inhibition process, which is present to some extent in all photopolymer recordings. The work presented here
explains why it occurs. A theoretical model is developed to predict the behaviour of the temporal evolution of the
grating. This model has been improved to account for the absorption effects of the material due to the photosensitive
dye and the inhibition period, which results in a reduction in the rate of polymerisation.
The characterization of the behavior of photopolymers is an important fact in order to control the holographic memories
based on photopolymers. In recent years many 2-dimensional models have been proposed for the analysis of
photopolymers. These models suppose that the photopolymer layer is homogeneous in depth and good agreement
between theoretical simulations and experimental results has been obtained for layers thinner than 200 μm. The
attenuation of the light inside the material by Beer's law is an important factor when higher thickness are considered. In
this work we use a Finite-Difference method to solve the 3 dimensional problem. Now diffusion in depth direction and
the attenuation of the light inside the material by Beer's law are also considered, the influence of the diffusivity of
material in the attenuation of the refractive index profile in depth is analyzed.
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.
In recent years developments in holographic materials have lead to an increasing interest in many areas such as data storage and metrology. Materials such as Acrylamide-based photopolymers are good holographic recording materials, as they are inexpensive and self-processing. The diffusion rate of monomer and the molecular weight of polymerised monomer determine many material characteristics. The length (size) of the polymer chains has a direct effect on the diffusion rate of the polymer and shortening the chain length leads to an increase in the diffusion rate. Shorter chains also decrease the non-local material response parameter and, in consequence, lead to an increase in the spatial frequency response of the material. Thus it is expected that by controlling the polymer chain length (molecular weight) one might control the material spatial frequency response. In this paper we look at the effect of varying the quantity of crosslinking agents on the material and its impact on the rate of diffusion. We then look at determining the rate of diffusion of water within the material to provide a lower limit to the maximum rate of monomer diffusion.
In this paper we analyze the evolution of the refractive index modulation when recording gratings in an acrylamide based photopolymer. A nonlocal diffusion model is used to predict theoretically the grating evolution. The model has been developed to account for both nonlocal spatial and temporal effects in the medium which can be attributed to polymer chain growth. Previously it was assumed that the temporal effect of chain growth could be neglected. However temporal effects both due to chain growth and monomer diffusion are shown to be significant, particularly of short recording periods. The diffusion model is solved using a Finite-Difference Time-Domain technique to predict the evolution of the monomer and polymer concentrations throughout grating recording. Using independently measured refractive index values for each component of the recording medium, the Lorentz-Lorenz relation is used to determine the corresponding refractive index modulation. Gratings recorded for short exposure times with the diffraction efficiency growth monitored in real time both during and after recording are presented. The effect of volume shrinkage of polymer on grating evolution is also examined. The temporal response of the material and monomer diffusion is shown to influence refractive index modulation post-exposure. The inclusion of the nonlocal temporal response and the use of the Lorentz-Lorenz relation are shown to be necessary to accurately describe this polymerization process.
Photopolymer materials are practical materials for use as holographic recording media, as they are inexpensive and self-processing. Understanding the mechanisms present during the fabrication of gratings in these materials is crucial in enabling further development. One such mechanism is the presence of an inhibition period at the start of grating growth during which the formation of polymer chains is suppressed. Some previous studies have indicated possible explanations for this effect and mathematical models have been proposed to approximate the observed behaviour. We examine the kinetic behaviour involved within the photopolymer material during recording to enable a clear picture of the photochemical processes present. Sets of experiments were carried out with the specific aim of developing an improved understanding of these processes. Here we discuss these experimental results and provide a theoretical model, which attempts to describe the inhibition process in our Acrylamide based photopolymer and predicts this behaviour under certain conditions.
Micro-optical devices are very important in current high-tech consumer items. The development of future products depends on both the evolution of fabrication techniques and on the development of new low cost mass production methods. Polymers offer ease of fabrication and low cost and are therefore excellent materials for the development of micro-optical devices. Polymer optical devices include passive optical elements, such as microlens arrays and waveguides, as well as active devices such as polymer based lasers. One of the most important areas of micro-optics is that of microlens design, manufacture and testing. The wide diversity of fabrication methods used for the production of these elements indicates their importance. One of these fabrication techniques is photo-embossing. The use of the photo-embossing technique and a photopolymer holographic recording material will be examined in this paper. A discussion of current attempts to model the fabrication process and a review of the experimental method will be given.
Recent improvements in holographic materials have led to advances in a variety of applications, including data storage and interferometry. To further increase the possibility of commercial applications in these areas it is necessary to have available an inexpensive, self-processing, environmentally stable material that has a good spatial frequency response. One promising type of material is Acrylamide-based photopolymer recording materials. The material can be made self-processing and can be sensitised to different recording wavelengths using different photosensitive dyes. The self-processing capability of this material simplifies the recording and testing processes and enables holographic interferometry to be carried out without the need for complex realignment procedures. Although this material has a lot of advantages over others it has significant disadvantages such as its spatial frequency response range (500-2500 lines/mm). Therefore, it is of ever-increasing importance to resolve uncertainties regarding optical and material properties, i.e. the refractive index and the diffusion constants. Using experimental diffraction efficiency measurements, a value for the refractive index modulation can be obtained. Then carrying out analysis using the Polymerisation Driven Diffusion model (PDD) values for the diffusion coefficients of various materials in the grating layer can be found. Applying the Lorentz-Lorenz relation, refractive index variations within the material can be more fully understood. With the resulting improved understanding it will be possible to improve the characteristics of photopolymer materials by altering the chemical composition, for example by controlling the crosslinker concentration or through the careful use of inhibitor and/or retarders to control the polymer chain growth.
The inclusion of a nonlocal spatial response function in the Nonlocal Polymer Driven Diffusion model (NPDD) has been shown to predict high spatial frequency cut-off in photopolymers and more recently it has been shown that use of the nonlocal model is necessary to accurately predict higher order grating components. Here the nature of the temporal response of photopolymer is discussed and a nonlocal temporal response function proposed. The extended model is then solved using a finite element technique and the results discussed. Based on this model we examine the nature of grating evolution when illumination is stopped during the grating recording process. Refractive indices of the components of the photopolymer material used are determined and predictions of the temporal evolution of the refractive index modulation described. Material parameters are then extracted based on fits to experimental data for non-linear and both ideal and non-ideal kinetic models.
The well known scaling law of holographic diffraction states that the replay diffraction efficiency η = Γ/M2, where M is the number of gratings (pages) stored, and G is a constant system parameter. This is an important metric used to quantify HDS material performance, and a great deal of experimental work to validate this rule for a wide variety of materials, (photorefractives, polymers etc.) have been presented over the years in the literature. No paper detailing the theoretical basis of this law, (i.e. including specific material characteristics, the recording geometry and/or the electromagnetic replay conditions), for photopolymers has previously been presented. In a recent paper [1] we have described in a clear way the optimization of the recording schedule in a photopolymer material governed by the Nonlocal Polymerization Driven Diffusion model (NPDD). One of the main assumptions made in [1] is that a long material relaxation time can be permitted between exposures. Another was that no phase shifts of the exposing pattern took place between exposures. In this paper we discuss these assumption and develop an intermediate first-order model. In a second paper [2], based on the results presented in [1] we have shown that our optimized predictions are in agreement with the scaling law of holographic diffraction. Thus the law is shown to hold for photopolymer recording media governed by the predictions of the NPDD. Based on our analysis: (i) A media inverse scaling law is proposed; (ii) G is for the first time related to photopolymer material parameters and the hologram geometry and replay conditions; and (iii) The form and validity of the diffraction efficiency inverse square scaling law for higher diffraction efficiency gratings is commented upon. In this paper we also review this result.
Many of the next generation of high-tech consumer products will take advantage of the manufacturing advances in micro-optics that we are currently taking place. One of the most important areas of micro-optic research is that of microlens design and fabrication. The importance of this area is perhaps highlighted by the range of competing fabrication technologies. Each having important advantages/disadvantages for a given application. It is therefore important to pursue other possible fabrication methods. In this paper we examine two of these novel fabrication techniques: (1) Photo-embossing in holographic recording materials, and (2) Microfluidic lens fabrication. The first of these techniques offers the possibility of combining the advantages of diffractive optical elements with those of conventional refractive optical elements. The second technique in combination with inkjet deposition technology can be used to produce a wide range of optical elements (lenses) and offers the possibility of controlling the lens profile in real time during formation using electric fields.
Photopolymer materials are good materials for the recording of holographic optical elements (H.O.E's), as they are inexpensive and self-processing. Understanding the mechanisms present during the fabrication of gratings in these materials is crucial in enabling further developments. One such mechanism is the presence of an inhibition period at the start of grating growth during which the formation of polymer chains is suppressed. Some previous studies have indicated possible explanations for this effect and mathematical models have been proposed to approximate the observed behaviour. We have carried out a set of experiments with the specific aim of developing an improved understanding of this process. In this paper we discuss these experimental results and provide a theoretical model, which describes the inhibition process in our Acrylamide based photopolymer and predicts this behaviour under certain conditions.
The inclusion of a nonlocal spatial response function in the Nonlocal Polymer Driven Diffusion model (NPDD) has been shown to predict high spatial frequency cut-off in photopolymers. Here the nature of the temporal response of photopolymer is discussed and a nonlocal temporal response function proposed. The extended model is then solved using a finite element technique and the results discussed. Based on this model we examine the nature of grating evolution when illumination is stopped during the grating recording process. Refractive indices of the components of the photopolymer material used are determined and predictions of the temporal evolution of the refractive index modulation described.
Many new high-tech consumer products that are now under development require micro-optical elements. The development of these micro-optical devices has been carried out by many different researchers working in a variety of areas. This has lead to a large number of different fabrication techniques. We examine a novel fabrication technique that may allow the development of large arrays of elements quickly and cheaply. It is known that the exposure of dye sensitised Acrylamide layers to light can lead to material refractive index and volume changes. It is therefore proposed that a patterned exposure can be used to form a mixture of volume and surface relief patterning, enabling the production of optical elements. The examination of this fabrication technique, in particular the study of the processes that result in this volume change, may also lead to improvements in the photopolymer material so as to control shrinkage of these materials. The development of low shrinkage holographic recording materials is an active area in holography as most current photopolymer materials exhibit some volume change during the recording process. This has implications for the fidelity of the replayed image. This is of crucial importance in areas such as data storage systems. The further study of this process also has implications for the wider holographic research community. It is important to understand the surface relief profile of the holographic element prior to extracting grating parameters as surface relief effects may influence the experimental data. In this paper we describe initial experimental attempts to produce micro-optical elements for use in the visible spectrum using patterned exposure of an Acrylamide based photopolymer material.
Non-local and non-linear models of photopolymer materials, which include diffusion effects, have recently received much attention in the literature. The material response is non-local as it is assumed that monomers are polymerised to form polymer chains and that these chains grow away from a point of initiation. The non-locality is defined in terms of a spatial non-local material response function. The numerical method of solution typically involves retaining either two or four harmonics of the Fourier series of monomer concentration in the calculation. In this paper a general set of equations is derived which allows inclusion of higher number of harmonics for any response function. The numerical convergence for varying number of harmonics retained is investigated with special care being taken to note the effect of the; non-local material variance σ, the power law degree k, and the rates of diffusion, D, and polymerisation F0. General non-linear material responses are also included.
Two extensions of the Nonlocal Polymer Driven Diffusion model (NPDD) describing grating formation in photopolymers are proposed. The inclusion of a nonlocal spatial response function has been shown to predict high spatial frequency cutoff in photopolymers. Here the nature of the temporal response of photopolymer is discussed and a nonlocal temporal response function proposed. A number of different possible spatial response functions are also examined. The extended model is then solved using a finite element technique and the results compared to the purely spatial response case. The kinetics of photosensitive polymer holographic recording are then examined. Assuming primary termination dominates the chain formation process a model is proposed based on a quadratic relationship which is shown to exist between the monomer concentration and polymerisation rate. The model is solved using a four-harmonic expansion. Results are then fitted numerically to growth curves of experimentally monitored diffraction efficiencies of gratings recorded in an acrylamide-based photopolymer system. Physical parameters such as diffusion constant and polymerisation rate are extracted and compared to the literature.
The kinetics of photosensitive polymer holographic recording materials is examined. We discuss why a linear relationship between monomer concentration and polymerization rate does not satisfactorily explain current experimental results and propose a possible solution. Then, using the Rigorous Coupled Wave Model(RCWM) we examine the higher order grating components so as to more clearly understand the non-linear relationship between exposing intensity and polymerization.
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