Exponential Response Function Research Articles

Stronger compensatory thermal adaptation of soil microbial respiration with higher substrate availability.

Ongoing global warming is expected to augment soil respiration by increasing the microbial activity, driving self-reinforcing feedback to climate change. However, the compensatory thermal adaptation of soil microorganisms and substrate depletion may weaken the effects of rising temperature on soil respiration. To test this hypothesis, we collected soils along a large-scale forest transect in eastern China spanning a natural temperature gradient, and we incubated the soils at different temperatures with or without substrate addition. We combined the exponential thermal response function and a data-driven model to study the interaction effect of thermal adaptation and substrate availability on microbial respiration and compared our results to those from two additional continental and global independent datasets. Modeled results suggested that the effect of thermal adaptation on microbial respiration was greater in areas with higher mean annual temperatures, which is consistent with the compensatory response to warming. In addition, the effect of thermal adaptation on microbial respiration was greater under substrate addition than under substrate depletion, which was also true for the independent datasets reanalyzed using our approach. Our results indicate that thermal adaptation in warmer regions could exert a more pronounced negative impact on microbial respiration when the substrate availability is abundant. These findings improve the body of knowledge on how substrate availability influences the soil microbial community-temperature interactions, which could improve estimates of projected soil carbon losses to the atmosphere through respiration.

Using the softplus function to construct alternative link functions in generalized linear models and beyond

AbstractResponse functions that link regression predictors to properties of the response distribution are fundamental components in many statistical models. However, the choice of these functions is typically based on the domain of the modeled quantities and is usually not further scrutinized. For example, the exponential response function is often assumed for parameters restricted to be positive, although it implies a multiplicative model, which is not necessarily desirable or adequate. Consequently, applied researchers might face misleading results when relying on such defaults. For parameters restricted to be positive, we propose to construct alternative response functions based on the softplus function. These response functions are differentiable and correspond closely to the identity function for positive values of the regression predictor implying a quasi-additive model. Consequently, the proposed response functions allow for an additive interpretation of the estimated effects by practitioners and can be a better fit in certain data situations. We study the properties of the newly constructed response functions and demonstrate the applicability in the context of count data regression and Bayesian distributional regression. We contrast our approach to the commonly used exponential response function.

Stochastic Model of Emotional Stress Development in the Educational Process

The aim of the study is the development of mathematical models that describe the reaction of the individual to the occurrence of stresses of various nature, including those that appear during the implementation of the educational process. The complexity of developing such a model is confirmed by the lack of theoretical results to substantiate the classic experiments of Holmes and Rae on the effect of stress on personality. In this regard, the task is to develop such a mathematical model that would allow not only to give a theoretical explanation of the experimental results used in the Holmes and Rae stress calculator, but also to become a tool for studying the effect of stress on a person in other conditions, including in the process of educational activities of both the teacher and the learner.The research method consists in a mathematical description of the process of occurrence of stresses that develop over time, and it is believed that stresses occur at random times and are characterized by relative stress values indicated in the classical table of Holmes and Rae. The need to involve these results is that they allow us to confirm the correspondence of the developed theoretical mathematical models to already known practical results. The following main assumptions are accepted in the paper. It is believed that a person is exposed to stresses that can occur at random localized points in time and are interpreted as a sequence of points on the time axis, the number and location of which is random. The response of a person to a particular stress is described by a decreasing exponential function of three arguments – current time, random time of stress occurrence, and stress magnitude. The reaction of the individual to a sequence of stresses is the sum of the responses of the person to individual stresses, i.e. it is assumed that the personality exhibits the properties of linearity. In the process of developing a mathematical model, the distribution of the number of random stresses is substantiated according to the Poisson law, which is used to describe the occurrence of random events with a distinct discreteness. The paper introduces one of the key indexes - the coefficient of emotional load, equal to the ratio of the mathematical expectation of everyday stressful background and the mathematical expectation of the sum of this background and additional stress. The response of a person to a particular stress is described by an exponential response function widely used in natural science applications. The total relative value of processed, experienced, random relative values of stress, as well as their non-random mathematical expectations, is introduced into consideration.The new results of the study are: – development of a stochastic mathematical model for the development of stresses over time, depending on the parameters included in the model. It is shown that the behavior of a person’s reaction to stress, predicted by a mathematical model, corresponds to the previously mentioned experimental results. – a study of the behavior in time of a person’s reaction to stress for a situation that was not previously considered and in which the person is affected by a single stress of great intensity, as well as regular emotional imposition.In conclusion, it is noted that the developed model not only makes it possible to theoretically explain the experimental data, but also significantly expand the scope of the studied effect of stress on personality. Thus, it turned out to be possible to predict the impact of a single impact, as well as to indicate a way to account for periodic intentional exposure (emotional suppression). In addition, the results can be used in the study of emotional stresses in the educational process in order to predict and consider them in practical activities.

Stochastic Dynamics of Emotional Characteristics of Cognitive Systems

The aim of the research is to analyze the emotional state of a person using stochastic methods and to study the dynamic characteristics of this state. The study uses statistical characteristics that are suitable for creating artificial intelligent systems, as well as for further development and design of organizational and technical systems. It is assumed that the emotional state of a person is determined by the amount of stress he experiences over a certain period of time. To account for the emotional responses of the cognitive system corresponding to the reactions of the human personality, the known results are used. In particular, to characterize the relative values of stress, the Holmes and Rahe scale was used, which contains 43 typical life situations with corresponding relative values of stress. It should be emphasized that all known results relate to time-independent stress models, and only the results of statistical processing of measurements of the impact of stress on the individual at certain time intervals are published.Materials and methods of research are the application of the Poisson model of stress occurrence in the process of system functioning. It is assumed that stresses are those impacts that are then processed, perceived, processed, and used by the individual and the organizational and technical system that models it. The occurrence of stresses over time is modeled by point Poisson processes, and the stress of each type according to Holmes and Rahe is described by a process with the corresponding individual function of the intensity of the occurrence of points (stresses). Dynamic responses of the system are described by well-known response functions in the theory of control systems. The key parameter of the response function is the time constant, which characterizes the typical response time of the system to individual stress.New results of the study are estimates of the average frequency of occurrence of various types of stress, which allowed us to determine the mentioned functions of the intensity of occurrence of stress in the Poisson model. This, in turn, made it possible to develop analytical relations for the relative amount of stress processed by the system for the current time. Thus, a model of the emotional state of the cognitive system (the development of the process of experiencing stress over time) is proposed in the form of a decreasing function of time with a certain time constant that characterizes the inertial properties of experiencing stress. Specific results and corresponding curves are obtained for the exponential response function of the system, which depends on the current time, times of stress occurrence, and relative stress values. In general, they correspond to the predictions of similar reactions for the individual obtained in published studies. In this regard, the model can be applied in the engineering design of organizational and technical systems that require accounting or modeling of emotional reactions.In conclusion, the directions of further development of the theory are indicated. In particular, the possibility of studying the reaction of the system to events, the occurrence of which is described by functions of the intensity of their occurrence, depending on time, which, of course, will bring the research results closer to real situations. Another important area may be the introduction of indicators and criteria for stress tolerance of systems to the consideration and study of behavior.

Effects of Changing Temperature on Gross N Transformation Rates in Acidic Subtropical Forest Soils

Soil temperature change caused by global warming could affect microbial-mediated soil nitrogen (N) transformations. Gross N transformation rates can provide process-based information about abiotic–biotic relationships, but most previous studies have focused on net rates. This study aimed to investigate the responses of gross rates of soil N transformation to temperature change in a subtropical acidic coniferous forest soil. A 15N tracing experiment with a temperature gradient was carried out. The results showed that gross mineralization rate of the labile organic N pool significantly increased with increasing temperature from 5 °C to 45 °C, yet the mineralization rate of the recalcitrant organic N pool showed a smaller response. An exponential response function described well the relationship between the gross rates of total N mineralization and temperature. Compared with N mineralization, the functional relationship between gross NH4+ immobilization and temperature was not so distinct, resulting in an overall significant increase in net N mineralization at higher temperatures. Heterotrophic nitrification rates increased from 5 °C to 25 °C but declined at higher temperatures. By contrast, the rate of autotrophic nitrification was very low, responding only slightly to the range of temperature change in the most temperature treatments, except for that at 35 °C to 45 °C, when autotrophic nitrification rates were found to be significantly increased. Higher rates of NO3− immobilization than gross nitrification rates resulted in negative net nitrification rates that decreased with increasing temperature. Our results suggested that, with higher temperature, the availability of soil N produced from N mineralization would significantly increase, potentially promoting plant growth and stimulating microbial activity, and that the increased NO3− retention capacity may reduce the risk of leaching and denitrification losses in this studied subtropical acidic forest.

Templates for Modeling Temperature Response of Passive Greenhouses

A previous publication [1] Stiles, M. 2018. “A Simple Model for Temperature Response of Single Zone Passive Buildings,” Journal of Building Physics, April 2018. Available at http://journals.sagepub.com/doi/full/10.1177/1744259118763843 (last accessed November 27, 2018). [Google Scholar] introduced a model of interior air temperature response to exterior temperature for a single-zone structure with no active heating, ventilation, or cooling. The model was based on a simple exponential step response function relating the two temperatures. Results were sufficiently encouraging to apply the model to other simple passive structures. This article applies the model to two cases of passive greenhouses. One of the greenhouses is a rudimentary plastic-covered hoop house. The other is a super-insulated net-zero structure that grows summer crops in cold climates using sunlight as the source of heating. A model for the interior air temperature of both of these structures was developed using two time-series templates. The first template is an exponential step response function of outdoor air temperature. This template synchronizes the model to the actual interior temperature but does not match its amplitude. A second template was developed by subtracting the model using the first template from the actual interior air temperature data. The second template scaled according to available sunlight as measured by an inexpensive photometric sensor. The superposition of the two templates tracks the actual interior temperature reasonably well given the limitations described in this article. Average difference between actual and modeled data for both structures was within 2°F, although maximum deviations ranged up to 30°F. This article concludes with a discussion of refinements to the model and of its commercial deployment should it prove generally applicable to passive greenhouses.

Stabilization of two-dimensional dark and anti-dark solitons in a nonlocal media

The existence and elastic collision of ring dark and anti-dark solitons in a nonlocal media with an exponential nonlocal response function was first investigated by Horikis and Frantzeskakis (2016 Opt. Lett. 41 583), who found that the initial rings expand outward, keeping their shape during evolution for relatively short propagation distances. In this paper, we study the stability of two-dimensional dark and anti-dark soliton in a nonlocal media by direct numerical simulations. We found that the shapes of the nonlocal response function have a great effect on the propagation of these solitons. In particular, the ring dark solitons can be completely stabilized in a nonlinear media with a cylindrical nonlocal response function due to the boundary force. We also investigate the dynamics of gray ring dark solitons and elliptic dark solitons in detail.

Analysis of dynamic characteristics of stochastic influences in cognitive systems

The aim of the study is to provide an analytical description of the dynamics of the processes to form images in the cognitive system and their subsequent processing by the consciousness, as well as the study of the simplest characteristics of the quality of the cognitive system functioning in the form of the signal/noise ratio. In accordance with the ideas of the cognitive theory, it is believed that images (schemes, categories, Gestalt, systems, archetypes, etc.) are firstly generated in the human brain and then processed by the consciousness. These images are formed at random in time and are characterized by a random force of effects and subsequently processed by the consciousness. The images are characterized by random numbers, the common interpretation of which is the amount of information corresponding to the appearance of a certain image. The times of appearance are points on the time axis; their number and position are random as well. The work consists of a logically completed model including the following components: • Justification of a statistical model of the appearance of effects during the operation of the cognitive system in the form of the Poisson point process, characterized by the intensity of occurrence of effects and the random values of those effects. • Development of a mathematical model in the consciousness processing of the random effects in the form of reducing response function, which depends on the current time, the time of occurrence of effects and the magnitudes of these effects. To obtain applied results, exponential response function was applied and the analytical results for the mathematical expectations of the processed and not processed information by the consciousness were received. • Introduction for consideration of the signal/noise ratio, characterizing the performance of cognitive systems in the presence of interference and study of its behavior in the situations with the presence of random background noise, regular and single impact of a large information capacity. The use of mathematical apparatus of random point processes allows us to formulate a dynamic model of random effects and use it to investigate the cognitive systems. The results can be used, in particular, for planning a sequence of control actions in cognitive systems as well as for the formation and formalization of requirements for cognitive and intellectual organizational-technical systems.

Effect of competing cubic-quintic nonlinearities on the modulational instability in nonlocal Kerr-type media

We investigate analytically and numerically the modulational instability (MI) of plane waves under competing nonlocal cubic-local quintic nonlinearities. The generic properties of the MI gain spectra are then demonstrated for the Gaussian response function, exponential response function, and rectangular response function. Special attention is paid to competing nonlocal cubic-local quintic nonlinearities on the MI. We observe that the focusing local quintic nonlinearity increases the growth rate and bandwidth of instability contrary to the small values of defocusing local quintic nonlinearity which decrease the growth rate and bandwidth of instability. Numerical simulations of the full model equation describing the dynamics of the waves are been carried out and leads to the development of pulse trains, depending upon the sign the quintic nonlinearity.

Modulational instability in nonlocal media with competing non-Kerr nonlinearities

We investigate analytically and numerically the modulational instability (MI) and propagation properties of light in nonlocal media with competing cubic–quintic nonlinearities where the response functions are assumed to be equal. By using the linear stability analysis, the generic properties of the MI gain spectra are demonstrated for the exponential and rectangular response functions. Special attention is paid to investigate the competition between the spatial scale of the cubic and quintic nonlinearities. For media with exponential response function, we have obtained the range of the wave numbers where instability occurs. It is found that the increase in the absolute value of the quintic nonlinearity suppresses the instability in the regime where the cubic nonlinearity prevails over the quintic one and promotes its development in the opposite case. For media with negative response function, additional MI bands are excited at higher wave numbers when the width of the nonlocal response function exceeds a certain threshold. In the regime where the quintic nonlinearity is dominant, the increase in the absolute value of the quintic coefficient leads to the enhancement of the gain value and the movement of the maximum gain to higher wave numbers. On the other hand, in the case of the predominance of the cubic nonlinearity, the position of the maximum gain bands move to lower wave numbers and MI domain becomes increasingly narrows when the quintic term increases. The numerical simulations fully confirm our analytical results.

The Physiology of Chilling Temperature Requirements for Dormancy Release and Bud-break in Temperate Fruit Trees Grown at Mild Winter Tropical Climate

It is studied that inadequate winter chilling may interfere with the normal processes of plant growth, reproductive development and subsequent yield. As much of the evidences behind these studies are subjective and region based, the available information was collated and evaluated to further investigate the impacts of winter chill that is currently an issue of fruit growers in mild winter areas. Though, the period of adequate low temperatures is insufficient in warmer regions to satisfy the chilling requirements of temperate fruit trees, this call up on many option for chill compensation; like planting low chill cultivars, use of dormancy breaking chemicals, forced defoliation, pruning and some other techniques as an alternative strategies. However, the diverse agro-climatic conditions due to major differences in altitude, rainfall as well as in slope characteristics showed the existence of different chilling temperature requirements across locations that favor many temperate fruit trees to grow: including apple, pear, grape, peach, nectarine, plum, cherry, walnut, almond and other fruit tree species. Temperate fruit production in mild winter areas now days increased rapidly and even more new industries are being developed in regions where none previously existed. To date, in tropical highlands, fruit productivity and quality have been gradually improved through introduction and selection of better adaptable varieties based on their chilling requirements. To supplement the present existing knowledge gap in relation to the cultivar-environment interactions, the use of chill models to quantify chill accumulation during winter months would help in classifying the environment (potential growing areas) according to the amount of chill hours existed in that location. Up to date, a number of valid methods were applied for quantification of chill accumulation in tropical and sub-tropical conditions showed varying results depending on the types of species, existing warm temperature and other climatic variables. The limitation in their predictive performance from region to region is due to their designing approaches that were primarily for temperate climate. This requires a model comparison for specific location, i.e. by using more than one model to avoid the potential mis­leading in calculation and chill estimation. Of the chill models tested to quantify the chill accumulation in the mild winter areas, the use of dynamic model gave good estimation that it nullifies the chilling reversal by high temperature. Also, the model of Positive Chill Units (PCU, or Positive Utah) is a competent under warm climate, next to the dynamic model. The Positive Utah model, an iteration of the original Utah, excludes the negation influence of high temperatures. The procedure for PCU is the same as for the original Utah model except that, when negative, the chill unit value is set equal to zero. Therefore, the accumulated chill units are equal to zero until the temperatures drop into the effective zone and positive chill units begin to accumulate. Though, for these areas with warm climate, using the 0 - 7.2 °C model is not recommended, because of its sensitivity to changes in temperatures that represent different weighing factors recorded for other models. Other important alternatives to these classical models include, the Growing Degree Hour Model (GDH), the Mean Temperature Model, Exponential temperature response functions and others are applied as independently, or in combination with classical chill models for a better chill estimation. Winter chill should be studied like other weather dependent processes because the present trends in chill decline across locations significantly affect fruit culture in areas with mild winter. Therefore, identifying the problems related with lack of insufficient winter chilling would help in designing possible strategies for the changing scenarios and understanding the current physiological responses of the plant against these changes.

Modelling in situ activities of enzymes as a tool to explain seasonal variation of soil respiration from agro-ecosystems

Understanding in situ enzyme activities could help clarify the fate of soil organic carbon (SOC), one of the largest uncertainties in predicting future climate. Here, we explored the role of soil temperature and moisture on SOM decomposition by using, for the first time, modelled in situ enzyme activities as a proxy to explain seasonal variation in soil respiration. We measured temperature sensitivities (Q10) of three enzymes (β-glucosidase, xylanase and phenoloxidase) and moisture sensitivity of β-glucosidase from agricultural soils in southwest Germany. Significant seasonal variation was found in potential activities of β-glucosidase, xylanase and phenoloxidase and in Q10 for β-glucosidase and phenoloxidase activities but not for xylanase. We measured moisture sensitivity of β-glucosidase activity at four moisture levels (12%–32%), and fitted a saturation function reflecting increasing substrate limitation due to limited substrate diffusion at low water contents. The moisture response function of β-glucosidase activity remained stable throughout the year. Sensitivity of enzymes to temperature and moisture remains one of the greatest uncertainties in C models. We therefore used the response functions to model temperature-based and temperature and moisture-based in situ enzyme activities to characterize seasonal variation in SOC decomposition. We found temperature to be the main factor controlling in situ enzyme activities. To prove the relevance of our modelling approach, we compared the modelled in situ enzyme activities with soil respiration data measured weekly. Temperature-based in situ enzyme activities explained seasonal variability in soil respiration well, with model efficiencies between 0.35 and 0.78. Fitting an exponential response function to in situ soil temperature explained soil respiration to a lesser extent than our enzyme-based approach. Adding soil moisture as a co-factor improved model efficiencies only partly. Our results demonstrate the potential of this new approach to explain seasonal variation of enzyme related processes.

Characterization of calibration curves and energy dependence GafChromic™ XR-QA2 model based radiochromic film dosimetry system.

The authors investigated the energy response of XR-QA2 GafChromic™ film over a broad energy range used in diagnostic radiology examinations. The authors also made an assessment of the most suitable functions for both reference and relative dose measurements. Pieces of XR-QA2 film were irradiated to nine different values of air kerma in air, following reference calibration of a number of beam qualities ranging in HVLs from 0.16 to 8.25 mm Al, which corresponds to effective energy range from 12.7 keV to 56.3 keV. For each beam quality, the authors tested three functional forms (rational, linear exponential, and power) to assess the most suitable function by fitting the delivered air kerma in air as a function of film response in terms of reflectance change. The authors also introduced and tested a new parameter χ = netΔR·e(m netΔR) that linearizes the inherently nonlinear response of the film. The authors have found that in the energy range investigated, the response of the XR-QA2 based radiochromic film dosimetry system ranges from 0.222 to 0.420 in terms of netΔR at K(air)(air) = 8 cGy. For beam qualities commonly used in CT scanners (4.03-8.25 mm Al), the variation in film response (netΔR at K(air)(air) = 8 cGy) amounts to ± 5%, while variation in K(air)(air) amounts to ± 14%. Results of our investigation revealed that the use of XR-QA2 GafChromic™ film is accompanied by a rather pronounced energy dependent response for beam qualities used for x-ray based diagnostic imaging purposes. The authors also found that the most appropriate function for the reference radiochromic film dosimetry would be the power function, while for the relative dosimetry one may use the exponential response function that can be easily linearized.

Acclimation and soil moisture constrain sugar maple root respiration in experimentally warmed soil

The response of root respiration to warmer soil can affect ecosystem carbon (C) allocation and the strength of positive feedbacks between climatic warming and soil CO2 efflux. This study sought to determine whether fine-root (<1 mm) respiration in a sugar maple (Acer saccharum Marsh.)-dominated northern hardwood forest would adjust to experimentally warmed soil, reducing C return to the atmosphere at the ecosystem scale to levels lower than that would be expected using an exponential temperature response function. Infrared heating lamps were used to warm the soil (+4 to +5 °C) in a mature sugar maple forest in a fully factorial design, including water additions used to offset the effects of warming-induced dry soil. Fine-root-specific respiration rates, root biomass, root nitrogen (N) concentration, soil temperature and soil moisture were measured from 2009 to 2011, with experimental treatments conducted from late 2010 to 2011. Partial acclimation of fine-root respiration to soil warming occurred, with soil moisture deficit further constraining specific respiration rates in heated plots. Fine-root biomass and N concentration remained unchanged. Over the 2011 growing season, ecosystem root respiration was not significantly greater in warmed soil. This result would not be predicted by models that allow respiration to increase exponentially with temperature and do not directly reduce root respiration in drier soil.

RPID — A new digital particle identification algorithm for CsI(Tl) scintillators

A new digital algorithm for online particle identification in CsI(Tl), called Reconstructive Particle IDentification (RPID) is reported. The concept is based on a model for the signal generation with two exponential components for the scintillation light output and an exponential response function of the preamplifier. To provide a fast algorithm which could be used online in modern FPGA-based electronics a simplified method for signal deconvolution was developed. Within a few simple processing steps the original pulse shape is modified to recover the amplitudes of the two scintillation components that determine the type of particle. Data from an experiment 12C $(p,p')$ 12C* at 21 MeV allowed to separate $\gamma$ -rays and protons even of very low energies. We present the performance of this new algorithm concerning the $\gamma$ -proton separation as well as the identification of protons not fully stopped within the CsI(Tl).

Compressed Exponential Response Function Arising From a Continuous Distribution of Gaussian Decays – Distribution Characteristics

AbstractA compressed exponential function (CEF) is shown to be represented by a distribution of Gaussian functions. The properties and characteristics of the distribution are detailed and discussed, and are illustrated with reference to the change in the NMR spectroscopy proton Hahn echo relaxation response that takes place in a composite material (prepared by melt compounding a mixture of graphite nanoparticles and pyromellitic anhydride modified polypropylene carbonate (PPC)) during aging at 90 °C. The results show that both the width and the average value of the spin‐spin relaxation rate distribution decrease during aging, suggesting that the molecular motion becomes less constrained and less heterogeneous.magnified image

Solitons in nonlocal nonlinear kerr media with exponential response function

In this paper, we find some exact analytical solutions including bright soliton solution, dipole-mode soliton solution, double soliton solution and periodic solution when a slit laser beam propagates in Kerr-type nonlinear, nonlocal media with exponential response function. Furthermore, we address the energy flow is a monotonically growing function of d₂ and the Hamiltonian decreases while the energy flow increases. And we also obtain an Airy-like soliton by numerical method.

Unsteady turbulent round jets and vortex motion

A new model to predict the velocity distribution in round jets with time-varying injection profiles has been formulated as an extension of steady jet theory. The approach introduces an effective injection velocity within the jet based on a representative response time. It is assumed that the instantaneous injection velocity affects the velocity within the jet with an exponential response function and that the response time is related to the fluid particle’s residence time within the jet, consistent with the theory of translation of jet vortex rings from Helmholtz’s vortex motion analysis [P. G. Tait, London Edinburgh Dublin Philos. Mag. J. Sci. 33, 485 (1867)]. The Helmholtz theory is also shown to reduce to the well-known velocity decay rate in the case of steady turbulent gas jets. A Duhamel superposition integral is used to determine the effective injection velocity for time-varying injection rates. The model is tested with different injection profiles and different ambient densities. The results are also compared with numerical results from a computational fluid dynamics code. The comparisons agree very well and the new model is shown to offer an efficient method to predict jet tip penetrations for unsteady jets.

Effect of dispersion on the relaxation-retardation time scale ratio

A simple relation between relaxation (modulus) and retardation (permittivity and compliance) times is determined for cases of dispersive instead of purely exponential response functions. An analytical solution for the Cole-Cole case and numerical treatments of the Cole-Davidson function reveal that the ratio of the relaxation- to retardation time constant is given by the ratio of the high- to low-frequency dielectric constant raised to the power of 1x, where x is the exponent that gauges the deviation from exponentiality.

Temporal analysis of grating formation in photopolymer using the nonlocal polymerization-driven diffusion model

The nonlocal polymerization-driven diffusion model (NPDD) has been shown to predict high spatial frequency cut-off in photopolymers and to accurately predict higher order grating components. We propose an extension to the NPDD model to account for the temporal response associated with polymer chain growth. An exponential response function is proposed to describe transient effects during the polymerization process. The extended model is then solved using a finite element technique and the nature of grating evolution examined in the case when illumination is stopped prior to the saturation of the grating recording process. Based on independently determined refractive index measurements we determine the temporal evolution of the refractive index modulation and the resulting diffraction efficiency using rigorous coupled wave theory. Material parameters are then extracted based on fits to experimental data for nonlinear and both ideal and non-ideal kinetic models.