Exponential Growth Regime Research Articles

Evidence of convective constraint release during hole growth in freely standing polystyrene films at low temperatures

Hole growth measurements were performed using optical microscopy on freely standing polystyrene films at temperatures that were slightly larger than the bulk value of the glass transition temperature T(bulk)g. For the measured range of temperatures, we have observed a transition from linear growth of the hole radius R during the early stages to exponential growth of R at later times. We have characterized this transition as a function of molecular weight 120 x 10(3) < Mw <2240 x 10(3) , film thickness 61 nm<h<125 nm , and temperature 101 degrees C<T<117 degrees C . The viscosity at the edge of the hole inferred from the long time exponential growth regime exhibits shear thinning due to the large shear strain rates present at the edge of the hole. The R (t) data for all times can be fit very well using an expression that describes exponential hole growth with a time-dependent viscosity that allows for an initial, transient response due to the decay of elastic entanglements. The time scale for the decay of the transient behavior is interpreted in terms of the decay of entanglements by the convective constraint release mechanism of the tube theory of entangled polymer dynamics.

The simulation of non-linear stellar pulsations

Details of a method for the numerical simulation of stellar pulsations and instabilities in the non-linear regime are presented. Important properties of the method are intrinsic mass and energy conservation. These are indispensable if the connection between stellar pulsations and mass loss is to be investigated. A prescription for the construction of initial models is given, which allows for the study of the linear regime of exponential growth of instabilities and a comparison with independent linear stability analyses. The dependence on numerical parameters of the simulations is investigated in detail. As a result, the acoustic luminosity generated by strange mode instabilities in a massive star is found to depend sensitively on the numerical viscosity necessary to handle shock waves. The values derived have to be regarded as lower limits and reach the order of magnitude of the kinetic luminosities observed in the winds of corresponding objects. This result is taken as further indication for a connection between strange mode instabilities and stellar mass loss.

Free electron lasers with slowly varying beam and undulator parameters

A self-consistent theory of a free electron laser (FEL) with slowly varying beam and undulator parameters is developed using the WKB approximation. The theory is applied to study the performance of a self-amplified spontaneous emission (SASE) FEL when the electron beam energy varies along the undulator as would be caused by vacuum pipe wakefields and/or when the undulator strength parameter is tapered in the small signal regime before FEL saturation. We find that a small energy gain or an equivalent undulator taper slightly reduces the power gain length in the exponential growth regime and can increase the saturated SASE power by about a factor of 2. Power degradation away from the optimal performance can be estimated based upon knowledge of the SASE bandwidth. The analytical results, which agree with numerical simulations, are used to optimize the undulator taper and to evaluate wakefield effects.

Molecular basis for the explanation of the exponential growth of polyelectrolyte multilayers.

The structure of poly(l-lysine) (PLL)/hyaluronan (HA) polyelectrolyte multilayers formed by electrostatic self-assembly is studied by using confocal laser scanning microscopy, quartz crystal microbalance, and optical waveguide lightmode spectroscopy. These films exhibit an exponential growth regime where the thickness increases exponentially with the number of deposited layers, leading to micrometer thick films. Previously such a growth regime was suggested to result from an "in" and "out" diffusion of the PLL chains through the film during buildup, but direct evidence was lacking. The use of dye-conjugated polyelectrolytes now allows a direct three-dimensional visualization of the film construction by introducing fluorescent polyelectrolytes at different steps during the film buildup. We find that, as postulated, PLL diffuses throughout the film down into the substrate after each new PLL injection and out of the film after each PLL rinsing and further after each HA injection. As PLL reaches the outer layer of the film it interacts with the incoming HA, forming the new HA/PLL layer. The thickness of this new layer is thus proportional to the amount of PLL that diffuses out of the film during the buildup step, which explains the exponential growth regime. HA layers are also visualized but no diffusion is observed, leading to a stratified film structure. We believe that such a diffusion-based buildup mechanism explains most of the exponential-like growth processes of polyelectrolyte multilayers reported in the literature.

Comparison of the Structure of Polyelectrolyte Multilayer Films Exhibiting a Linear and an Exponential Growth Regime: An in Situ Atomic Force Microscopy Study

We report here on the structural characterization of polyelectrolytes multilayer films formed by poly(l-glutamic acid) and poly(l-lysine) (PGA/PLL). The growth of this system is compared to that of poly(styrenesulfonate)/poly(allylamine hydrochloride) (PSS/PAH) multilayers by means of in situ atomic force microscopy (AFM) and by optical waveguide lightmode spectroscopy (OWLS). In contrary to the (PSS/PAH)i films that are growing linearly with the number of deposited layer pairs i, optical data evidenced that the (PGA/PLL)i films are characterized by an exponential growth. The analysis of the structure of the (PSS/PAH)i films reveals a smooth featureless surface covered by small globules. On the other hand, (PGA/PLL)i films form extended structures that appear with a vermiculate pattern. We propose a new growth mechanism based on polyelectrolyte diffusion in and out of the film coupled to the formation of polyanion/polycation complexes at the surface of the film in order to explain the whole results.

Self-amplified spontaneous emission in Smith–Purcell free-electron lasers

We present an analysis of a Smith–Purcell system in which a thin current sheet of electrons moves above a grating surface in the direction perpendicular to the grating grooves. We develop a consistent theory for evolution of the electromagnetic field and electron distribution in the exponential growth regime starting from the initial electron noise and the incoming amplitude. The dispersion relation for the complex growth rate for this system is a quadratic equation.

Growth of transverse coherence in SASE FELs

We introduce the correlation function between the electric field at two different points in the transverse plane as a parameter to quantify the degree of transverse coherence. We also propose a more realistic model for the initialization of the radiation in computer codes used to study SASE FELs. We make these modifications in the code tda and use it to study the growth of transverse coherence as a function of electron beam size, beam current and transverse emittance. Our results show explicitly that the onset of full transverse coherence in SASE takes place much before the power saturates. With the more realistic model the onset of the exponential growth regime is delayed, and to get a given power from the FEL one needs a longer undulator than would be predicted by the original tda code.

Power spectrum calculation for the Cornell Wiggler A SASE experiment at BNL

Recently, we showed (Yu, Phys. Rev. E 58 (1998) 4991) that the widely used simulation code TDA3D, even though a single-frequency code, can be used to determine the power spectrum in the SASE process with excellent approximation in the exponential growth regime. In this paper, we apply this method to the BNL Cornell Wiggler A SASE experiment as an example. When the gain is not very high, there are many modes in the radiation, which seems to make the analytical calculation very difficult. However, we show that the increment of the radiation due to SASE over the spontaneous radiation can be expanded in terms of guided modes with rapid convergence. Thus when the spontaneous radiation is subtracted from the SASE power during the calculation, there is a good agreement between the analytical theory and the numerical simulation.

Calculation of the output power in self-amplified spontaneous radiation using scaling of power with number of simulation particles

Recent advances in self-amplified spontaneous emission (SASE) experiments stimulate interest in quantitative comparison of measurements with theory. In this paper we show that the widely used simulation code TDA3D, developed by Tran and Wurtele [Comput. Phys. Commun. 54, 263 (1989)] even though a single frequency code, can be used to determine the output power in the SASE process with excellent approximation in the exponential growth regime. The method applies when the gain is not very high, which is a special advantage, because when the gain is not very high, the analytical calculation is particularly difficult since the exponential growing term does not dominate. The analysis utilizes a scaling relation between the output power and the number of simulation particles in the code TDA3D: $〈P〉{=N}_{\ensuremath{\lambda}}^{\ensuremath{'}}{/N}_{\ensuremath{\lambda}}〈{P}^{\ensuremath{'}}〉,$ where $〈P〉$ is the output power and ${N}_{\ensuremath{\lambda}}$ is the line density of the electrons, while $〈{P}^{\ensuremath{'}}〉$ is the calculated output power using a line density ${N}_{\ensuremath{\lambda}}^{\ensuremath{'}}$ of the number of simulation particles in the code TDA3D. Because of the scaling property, the number of simulation particles can be taken to be many orders of magnitude less than the actual experiment. Comparison of our results with experiment yields new insight into the SASE process.

Enhanced phase stability for a Raman free-electron laser amplifier in the exponential growth regime

The wave propagation number for the growing wave of a free-electron laser in the exponential growth regime is a complicated function of electron beam current and voltage. As a result if the electron beam voltage drifts the imaginary part of the propagation number in general will change too, leading to a change in the radio-frequency phase of the output power. For typical gains, the two largest contributions to the shift in the imaginary component of the propagation constant are due to the change in the beam’s reduced plasma wave number and the change in the detuning from resonance. This paper demonstrates that these changes can be made to cancel each other with the correct selection of plasma reduction factor and initial detuning from resonance if the interaction strength is weakly dependent on beam velocity, leading to a device that is first-order phase and gain stable under voltage fluctuations. The analysis is performed for the axial free-electron laser interaction and the results also apply for transverse-coupling free-electron lasers.

Computer simulations of cometary‐ion/ion instabilities and wave growth

The growth of electromagnetic fluctuations driven by newborn cometary ions is studied by means of hybrid computer simulations of homogeneous plasmas. The simulations are one‐dimensional with the direction of spatial variation in the direction of the ambient magnetic field B0 Newborn ions are injected into the simulations at a constant rate, and are given a velocity relative to the solar wind which makes a nonzero angle α with respect to B0. For relatively strong free energy injection rates corresponding to oxygen ion injection at relatively small values of α, the fluctuating field energy density exhibits exponential temporal growth of the ion/ion right‐hand resonant instability. At the end of this growth regime there is a peak value of the fluctuating magnetic field energy density; at 0°≤α≲45° this quantity is approximately proportional to cos8α. For relatively weak free energy injection rates corresponding to oxygen ion injection at relatively large α and proton injection at all α values, there are two distinct regimes. At early times the fluctuating field energy density exhibits linear temporal growth, whereas at later times the fluctuating field energy approaches a constant value in an asymptotic regime. The exponential growth regime corresponds to fluctuating magnetic field energy density spectra with clear peaks at the oxygen ion cyclotron resonance wave number in agreement with observations. In contrast, spectra of the linear temporal growth regime exhibit less distinct peaks at shorter wavelengths which then show a transfer of wave energy to longer wavelengths (an apparent “inverse cascade”) as the asymptotic regime is approached. This absence of a clear spectral peak at the proton cyclotron resonance frequency is similar to cometary observations. The simulations and observations share two additional properties: a linear correlation between cometary proton energy density and proton resonant magnetic fluctuations, as well as a shift of magnetic fluctuation energy of the oxygen‐ion/proton modes to shorter wavelengths and weaker amplitudes as α increases toward 90°.

Variational solution of the wave equation for a high gain FEL and a finite wiggling radius

The driving current for an FEL amplifier operating in the exponential growth regime is derived for a frequency close to the reasonance one and for a finite value of the wiggling radius of the electrons. The wave equation is solved by applying the generalized Rayleigh-Ritz variational principle. In particular, it is shown that the dispersion equation for the propagation constant can be put into the same form as that derived in absence of wiggling.