Comparison of holographic photopolymer materials by use of analytic nonlocal diffusion models

The one-dimensional diffusion equation governing holographic grating formation in photopolymers, which includes both nonlocal material response and generalized dependence of the rate of polymerization on the illuminating intensity, has been previously solved under the two-harmonic expansion assumption. The resulting analytic expressions for the monomer and polymer concentrations have been derived and their ranges of validity tested in comparison with the more accurate numerical four-harmonic case. We used these analytic expressions to carry out a study of experimental results presented in the literature over a 30-year period. Automatic fitting of the data with these formulas allows material parameters, including the nonlocal chain-length variance sigma, to be estimated. In this way, (i) a quantitative comparison of different materials can be made, and (ii) a standard form of experimental result presentation is proposed to facilitate such a procedure.     

周期性材料非经典热传导时空间多尺度分析方法

研究了一种空间-时间多尺度的方法,来分析周期性材料中非傅立叶热传导问题。计算模型是根据空间-时间尺度的高阶均匀化理论建立的,通过引入放大空间尺度和缩小时间尺度,研究了由空间非均匀性引起的非傅立叶热传导的波动效应和非局部效应。合并不同阶的均匀化非傅立叶热传导方程,消去缩小时间尺度参数,得到四阶微分方程。并进一步用C0连续修正了高阶非局部热传导方程的有限元近似解,使问题的求解避免了对有限元离散的C1连续性要求。给出的数值算例讨论了各种情况下方法的正确性与有效性。      

A Spatially Nonlocal Model for Polymer Desorption

In order to describe diffusion of a penetrant in a polymer entanglement network, one must incorporate nonlocal effects. Most previous models have included nonlocality in time only; however, by exploiting the disparate length scales in such systems, one can model these effects by a partial integrodifferential equation which is nonlocal in space. When considering the case of diffusion near the glass-rubber transition, a moving boundary separates the polymer into two regions, each governed by a different set of PDEs. The desorption of a semi-infinite polymer is studied using singular perturbation methods. Layers arise at the exposed surface, at the moving boundary, and initially. Analytical and phase-plane solutions are obtained for the solution, which exhibits physically realistic forms of desorption overshoot. Thus, spatially nonlocal models have the potential to replicate experimental systems, and should be considered in concert with other viscoelastic models of polymer-penetrant systems.     

Propagation of optical pulses with spatial and temporal dependence

A convenient approximate formula is proposed for the study of free-space propagation of spatial and temporal pulses with an identifiable carrier frequency. It does not contain a time derivative operation on the pulse's temporal envelope explicitly. It is shown that once a short (for example, picosecond or subpicosecond) pulse is created with a spatial and a temporal structure, it does not last forever. The approximation discussed is valid over a certain distance as dictated by the wave equation. Beyond this distance, the spatial and temporal characteristics begin to influence each other significantly. Two examples are presented. The first example is that of a pulse with a factored form of a spatial envelope times a temporal envelope. The second example is that of a clear aperture with a grating, by which pulse stretching or temporal distortion is examined and the result is in agreement with that found in the literature.     

Analysis of dual-wavelength oscillation in a broad-area diode laser operated with an external cavity

Apodizing holographic gratings used in an external cavity have shown to be effective to control the modal content of multimode broad-area diode lasers, providing single longitudinal-mode and single lateral-mode emission. They can also be designed to provide Littrow reflection at two wavelengths. We observed stable oscillation at two wavelengths in a diode laser with an external cavity ended with such a grating. This is not a common behavior for homogeneously broadened gain media. We present simulations of the behavior of this laser based on a rate equation analysis. The effects of spatial hole burning and spontaneous emission are examined.     

Size-dependent nonlinear vibration of an electrostatic nanobeam actuator considering surface effects and inter-molecular interactions

In this paper, the size-dependent nonlinear vibration of an electrostatic nanobeam actuator is investigated based on the nonlocal strain gradient theory, incorporating surface effects. A comprehensive model regarding the von Karman geometrical nonlinearity, inter-molecular forces and both components of the electrostatic excitation (AC and DC) is proposed to explore the system behavior near the primary resonance. Utilizing Hamilton’s principle, the nonlinear equation of motion of the system is derived. The natural frequency and dynamic response of the system, comprising frequency and force response diagrams, are obtained analytically via multiple scales technique in conjunction with the differential quadrature method and validated through a numerical approach. The roles of the nonlocal and strain gradient parameters, surface elasticity, inter-molecular forces and quality factor on the system oscillations are examined. The acquired results unveiled that the size-dependent parameters can significantly displace the multi-valued portions and instability thresholds of the dynamical response. Furthermore, it is deduced that the surface effects induce the stiffness hardening of the nanobeam, whereas the inter-molecular forces impose the stiffness softening effect.

     

Hygromorphic response dynamics of oak: towards accelerated material characterization

When a wood board is exposed to a change in relative humidity on only one of its surfaces, e.g. in case of flooring or a panel painting, the resulting asymmetric moisture content profile induces differential expansion over the thickness. Consequently a bending moment causes the board to curve. A theory is presented to describe the bending of a wood board due to a step change in relative humidity. The board is assumed to be homogeneous, isotropic, and linearly elastic. Moisture transport is presumed to obey the diffusion equation with constant coefficients, such that moisture transport can be directly related to the bending of the board. It is shown that the transient deflective behavior provides the diffusion coefficient and the final length change yields the linear hygroscopic expansion coefficient. Derived diffusion coefficients are in good agreement with values in literature. Furthermore, a scaling law for the deflection of the board is proposed, which is seen to be followed qualitatively but not quantitatively by experiments. Finally, by assuming the deflection of the board to be the response of a linear system, the deflective frequency response of the board can be predicted from its step response. The results allow upscaling of deflection and expansion, such that behavior of thick boards can be determined from an experiment using a thin board.     

Global stability of the rotating-disk boundary layer

The global stability of the von Kármán boundary layer on the rotating disk is reviewed. For the genuine, radially inhomogeneous base flow, linearized numerical simulations indicate that convectively propagating forms of disturbance are predominant at all radii. The presence of absolute instability does not lead to the formation of any unstable linear global mode, even though the temporal growth rate of the absolute instability increases along the radial direction. Analogous behaviour can be found in the impulse solutions of a model amplitude equation, namely the linearized complex Ginzburg–Landau equation. These solutions show that, depending on the precise balance between spatial variations in the temporal growth rate and the corresponding shifts in the temporal frequency, globally stable behaviour can be obtained even in the presence of a strengthening absolute instability. The radial dependency of the absolute temporal frequency is sufficient to detune the disturbance oscillations at different radial positions, thus overcoming the radially increasing absolute growth, thereby giving rise to a stable global response. The origin of this form of behaviour can be traced to the fact that the cylindrical geometry of the rotating-disk flow dictates a choice of a globally valid time non-dimensionalization that, when properly employed, leads to a significant radial variation in the frequency for the absolute instability.     

Femtosecond single-shot correlation system: a time-domain approach

We introduce, analyze, and experimentally demonstrate what to the best of our knowledge is a new pulse correlation technique that is capable of real-time conversion of a femtosecond pulse sequence into its spatial image. Our technique uses a grating at the entrance of the system, thus introducing a transverse time delay (TTD) into the transform-limited reference pulse. The shaped signal pulses and the TTD reference pulse are mixed in a nonlinear optical crystal (LiB(3)O(5)), thus producing a second-harmonic field that carries the spatial image of the temporal shaped signal pulse. We show that the time scaling of the system is set by the magnification of the anamorphic imaging system as well as by the grating frequency and that the time window of the system is set by the size of the grating aperture. Our experimental results show a time window of ~20 ps. We also show that the chirp information of the shaped pulse can be recovered by measurement of the spectrum of the resulting second-harmonic field.     

A partition of unity FEM for time‐dependent diffusion problems using multiple enrichment functions

An enriched partition of unity FEM is developed to solve time‐dependent diffusion problems. In the present formulation, multiple exponential functions describing the spatial and temporal diffusion decay are embedded in the finite element approximation space. The resulting enrichment is in the form of a local asymptotic expansion. Unlike previous works in this area where the enrichment must be updated at each time step, here, the temporal decay in the solution is embedded in the asymptotic expansion. Thus, the system matrix that is evaluated at the first time step may be decomposed and retained for every next time step by just updating the right‐hand side of the linear system of equations. The advantage is a significant saving in the computational effort where, previously, the linear system must be reevaluated and resolved at every time step. In comparison with the traditional finite element analysis with p‐version refinements, the present approach is much simpler, more efficient, and yields more accurate solutions for a prescribed number of DoFs. Numerical results are presented for a transient diffusion equation with known analytical solution. The performance of the method is analyzed on two applications: the transient heat equation with a single source and the transient heat equation with multiple sources. The aim of such a method compared with the classical FEM is to solve time‐dependent diffusion applications efficiently and with an appropriate level of accuracy. Copyright © 2012 John Wiley & Sons, Ltd.     

Investigation of the diffusion processes in a self-processing acrylamide-based photopolymer system

Results from the investigation of the diffusion processes in a dry acrylamide-based photopolymer system are presented. The investigation is carried out in the context of experimental research on optimization of the high-spatial-frequency response of the photopolymer. Tracing the transmission holographic grating dynamics at short times of exposure is utilized to measure diffusion coefficients. The results reveal that two different diffusion processes contribute with opposite sign to the refractive-index modulation responsible for the diffraction grating buildup. Monomer diffusion from dark to bright fringe areas increases the refractive-index modulation. It is characterized with diffusion constant D0 = 1.6 x 10(-7) cm2/s. A second diffusion process takes place during the recording. It decreases the refractive-index modulation and we ascribe it to diffusion of short-chain polymer molecules or radicals from bright to dark fringe areas. The estimated diffusion coefficient for this process is D0 = 6.35 x 10(-10) cm2/s. The presence of the second process could be responsible for the poor high-spatial-frequency response of the investigated photopolymer system. Comparison with the diffusion in photopolymer systems known for their good response at high spatial frequencies shows that both investigated diffusion processes occur in a much faster time scale.     

Reinforced concrete bridge deck reliability model incorporating temporal and spatial variations of probabilistic corrosion rate sensor data

The reliability of reinforced concrete (RC) bridge decks depends significantly on the rate of corrosion of the reinforcing steel. Structural health monitoring (SHM) techniques, including embedded corrosion rate sensors, can greatly improve the quantification of the steel corrosion rate, which can lead to improved estimates of structural safety and serviceability. Due to uncertainties in concrete properties, environmental conditions, and other factors, the rate of corrosion of reinforcing steel can be highly variable, both within a given structural component and over time. By placing multiple corrosion rate sensors throughout a structural component, such as a bridge deck, these spatial and temporal variabilities can be monitored and as such better predicted, for use in a reliability model. The objective of this investigation is to present a reliability model for a RC bridge deck incorporating both spatial and temporal variations of probabilistic corrosion rate sensor data. This objective is accomplished using a computational reliability model and Monte Carlo simulation. Corrosion rate sensor data is assumed for multiple critical sections throughout a RC bridge deck over time by applying empirical spatial and temporal relationships. This data is then used to improve an existing spatially invariant reliability model. The improved reliability model incorporates several sub-models to determine the changes in load effects on and resistance of a RC bridge deck slab over time, as well as spatial correlation of corrosion and a system approach to account for spatial variability. The improved reliability model incorporating both spatial and temporal variations in corrosion rate data provides a better estimate of the service life of a RC bridge deck slab.     

Concentration-dependent diffusion in a finite slab with and without chemical reaction

Concentration-dependent diffusion in symmetrical and asymmetrical geometrical systems is investigated in this work. Also included is the chemical reaction of general order. Different types of concentration dependence of diffusion coefficient are considered. The governing nonlinear diffusion equation is solved by the orthogonal collocation method which involves the approximation of the spatial derivative term by an orthogonal polynomial of the Legendre type. The main advantage of the orthogonal collocation method is that the resulting ordinary differential equations can be integrated with precision and stability. Due to these advantages, a wide range of nonlinear diffusion problems can be tackled without much difficulty.     

Stabilized nonlocal model for dispersive wave propagation in heterogeneous media

In this paper, a general-purpose computational model for dispersive wave propagation in heterogeneous media is developed. The model is based on the higher-order homogenization with multiple spatial and temporal scales and the C⁰-continuous mixed finite element approximation of the resulting nonlocal equations of motion. The proposed nonlocal Hamilton principle leads to the stable discrete system of equations independent of the mesh size, unit cell domain and the excitation frequency. The method has been validated for plane harmonic analysis and for transient wave motion insemi-infinite domain with various microstructures.This work was supported by the Sandia National Laboratories under Contract DE-AL04–94AL8500, the Office of Naval Research through grant number N00014–97–1-0687, and the Japan Society for the Promotion of Science under contract number Heisei11-nendo 06542.     

Dynamic hologram recording characteristics in DuPont photopolymers

Hologram formation and diffusion reactions in photopolymer films are examined at different stages of exposure and at different spatial frequencies. Different properties of the grating formation process are evaluated from efficiency data, including the relative rates of diffusion and photoinitiated polymerization, dye absorption, and residual efficiency enhancement after UV curing. It was also found that gratings with larger periods (~1.4 mum) are susceptible to erasure effects with postexposure laser illumination. In addition, crossed gratings were found to have an extended dynamic range. This effect can have a significant impact on the number of holograms formed with rotational or peristrophic multiplexing.     

The Boussinesq-Flamant problem for an elastic nonlocal half-infinite space

Summary Transmission of a concentrated force into a half-infinite elastic medium is examined assuming that the medium has nonlocal properties. The values of the nonlocal moduli are adopted from the known studies on the wave propagation in nonlocal media. Field equations are solved by using the Fourier transform technique. Inversion of the equations obtained for the stresses and displacements shows that the equations for stresses coincide with those predicted by the conventional theory. The equations for the displacements differ, however, from their classical counterparts, and can only be evaluated approximately. The first approximation leads to the classical equations. The fourth approximation derived for the Poisson material close to the line of loading displays deviations from the classical values amounting to 35%.     

Hyperbolic heat conduction due to a mode locked laser pulse train

In situations which involve repetitive pulsing of a material with a mode locked Nd:YAG laser, the pulse duration can be sufficiently small (i.e. in the picosecond range) that the classical parabolic heat conduction equation fails to adequately predict the resulting temperature distribution in the material. In such cases, the hyperbolic heat conduction equation, which accounts for the finite time to the commencement of heat flow, is appropriate. In the present work, the hyperbolic heat conduction equation is used to predict the temperature distributions in both semi-infinite and finite isotropic media due to a train of temporally rectangular pulses which approximate the Gaussian temporal profile of mode locked laser pulses. The energy carried in the pulses is assumed to be absorbed in the surface plane of the material. The spatial profile of the pulses can be either Gaussian, doughnut or a combination of the two. The parabolic and hyperbolic models are examined for selected pulse frequencies.     

多维非经典热传导问题的时间2空间多尺度高阶均匀化分析

采用一种时间-空间多尺度高阶渐近均匀化分析方法,模拟了热冲击载荷条件下多维微尺度多相周期性结构中的非经典热传导问题。通过引入放大空间尺度和缩小时间尺度,在不同时间尺度上获得由空间非均匀性引起的波动效应和非局部效应。根据高阶均匀化理论在空间和时间上进行均匀化,消去缩小时间尺度,确定各阶等效均匀化热传导系数的关系并对该系数进行数值求解,获得了多维非傅里叶热传导高阶非局部温度场控制方程。进而对二维周期性多相材料中的非傅里叶热传导问题进行分析,结果证明了本文中所提出的多维非傅里叶热传导高阶非局部模型的正确性与有效性。     

Diffraction characteristics of spatial and temporal Gaussian-shaped femtosecond laser pulse by rectangle reflection grating

The exact intensity distribution expression for the spatial and temporal Gaussian-shaped femtosecond laser pulse diffracted by a rectangle reflection grating is derived. The spatial and temporal diffraction characteristics are theoretically investigated in detail, and a criterion for judging whether or not the diffraction pulse is just split into two independent pulses in the temporal domain is obtained. The results show that the diffraction intensity in the temporal domain consists of three parts: the intensity diffracted by the upper reflection surface of the grating, the intensity diffracted by the nether reflection surface, and their temporal coherent intensity. The temporal coherent intensity becomes weaker, even is zero, for the higher height from the nether surface to the upper surface of the grating. The principal maximum becomes more sharply bright for the bigger waist width of the femtosecond laser pulse in the spatial domain.