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In this publication, the Luneberg integrals are revisited and the conditions of the using of such integrals have been recalled. Additivity law of Luneberg's integrals and the link with the Frenel kernel for the propagation are discussed. By means of the definition of the Luneberg's integrals,the propagation of a vectorial electromagnetic field (Hertz potentials) is developped and a new approach of the computation have been proposed based on Zernike polynomials. With this new approch simulations of holograms is illustrated in the case of the digital in-line holography with an opaque disk.

We sharpen the bound n 2 k on the maximum modulus of the k th radial derivative of the Zernike circle polynomials (disk polynomials) of degree n to n 2 ( n 2 − 1 2 ) ⋅ . . . ⋅ ( n 2 − ( k − 1 ) 2 ) ∕ 2 k ( 1 ∕ 2 ) k . This bound is obtained from a result of Koornwinder on the non-negativity of connection coefficients of the radial parts of the circle polynomials when expanded into a series of Chebyshev polynomials of the first kind. The new bound is shown to be attained within a factor O ( 1 ∕ n ) for Zernike circle polynomials of degree n and azimuthal order m when m = O ( n ) by using an explicit expression for the connection coefficients in terms of squares of Jacobi polynomials evaluated at 0.

In this paper we highlight that extreme events such as freak waves are a transient phenomenon in keeping with the old fisherman tale that these extreme events seem to appear out of nowhere. Janssen (J. Phys. Oceanogr., vol. 33, 2003, pp. 863–884) obtained an evolution equation for the ensemble average of the excess kurtosis, which is a measure for the deviation from normality and an indicator for nonlinear focusing resulting in extreme events. In the limit of a narrow-band wave train, whose dynamics is governed by the two-dimensional nonlinear Schrödinger (NLS) equation, the excess kurtosis is under certain conditions seen to grow to a maximum after which it decays to zero for large times. This follows from a numerical solution of the problem and also from an analytical solution presented by Fedele (J. Fluid Mech., vol. 782, 2015, pp. 25–36). The analytical solution is not explicit because it involves an integral from initial time to actual time. We therefore study a number of properties of the integral expression in order to better understand some interesting features of the time-dependent excess kurtosis and the generation of extreme events.

In this paper, we study a single-server polling model with two queues. Customers arrive at the queues according to two independent Poisson processes. The server spends random amounts of time in each queue, regardless of the amounts of work present at the queues. The service speed is not constant; it is assumed that the server works at speed (Formula presented.) at queue i when its current workload equals xi, i = 1, 2. We first focus on the case that all visit times are constant. In the two-queue case, we then compute the Laplace–Stieltjes transform (LST) of the steady-state joint workload distribution. Using a different method in which we exploit the independence of the workloads at visit endings, we compute the joint LST of workloads in the case of an arbitrary number of queues with constant visit times. Next, we consider a two-queue polling model with constant visit times at the first queue and general visit times at the second, and we derive the marginal workload distribution at the first queue. We also investigate the case of a two-queue polling model with exponentially distributed visit times. We determine the steady-state marginal workload distributions, and we formulate a two-dimensional Volterra integral equation for the LST of the steady-state joint workload distribution. We finally show that this equation can be solved by a fixed-point iteration.

The Struve functions Hn(z), n=0, 1, ... are approximated in a simple, accurate form that is valid for all z≥0. The authors previously treated the case n = 1 that arises in impedance calculations for the rigid-piston circular radiator mounted in an infinite planar baffle [Aarts and Janssen, J. Acoust. Soc. Am. 113, 2635–2637 (2003)]. The more general Struve functions occur when other acoustical quantities and/or non-rigid pistons are considered. The key step in the paper just cited is to express H1(z) as (2/π)−J0(z)+(2/π) I(z), where J0 is the Bessel function of order zero and the first kind and I(z) is the Fourier cosine transform of [(1−t)/(1+t)]1/2, 0≤t≤1. The square-root function is optimally approximated by a linear function ct+d, 0≤t≤1, and the resulting approximated Fourier integral is readily computed explicitly in terms of sin z/z and (1−cos z)/z2. The same approach has been used by Maurel, Pagneux, Barra, and Lund [Phys. Rev. B 75, 224112 (2007)] to approximate H0(z) for all z≥0. In the presen...

It is a well-known problem in Gabor analysis how to construct explicitly given dual frames associated with a given frame. In this paper we will consider a class of window functions for which approximately dual windows can be calculated explicitly. The method makes it possible to get arbitrarily close to perfect reconstruction by allowing the modulation parameter to vary. Explicit estimates for the deviation from perfect reconstruction are provided for some of the standard functions in Gabor analysis, e.g., the Gaussian and the two-sided exponential function.

Wireless networks equipped with the CSMA protocol are subject to collisions due to interference. For a given interference range, we investigate the tradeoff between collisions (hidden nodes) and unused capacity (exposed nodes). We show that the sensing range that maximizes throughput critically depends on the activation rate of nodes. For infinite line networks, we prove the existence of a threshold: When the activation rate is below this threshold, the optimal sensing range is small (to maximize spatial reuse). When the activation rate is above the threshold, the optimal sensing range is just large enough to preclude all collisions. Simulations suggest that this threshold policy extends to more complex linear and nonlinear topologies. Keywords: Carrier-sensing range; Markov processes; collisions; exposed nodes; hidden nodes; random-access; throughput; wireless networks

In this paper, a new methodology is presented to derive the aberration state of a lithographic projection system from wafer metrology data. For this purpose, new types of phase-shift gratings (PSGs) are introduced, with special features that give rise to a simple linear relation between the PSG image displacement and the phase aberration function of the imaging system. By using the PSGs as the top grating in a diffraction-based overlay stack, their displacement can be measured as an overlay error using a standard wafer metrology tool. In this way, the overlay error can be used as a measurand based on which the phase aberration function in the exit pupil of the lithographic system can be reconstructed. In practice, the overlay error is measured for a set of different PSG targets, after which this information serves as input to a least-squares optimization problem that, upon solving, provides estimates for the Zernike coefficients describing the aberration state of the lithographic system. In addition to a detailed method description, this paper also deals with the additional complications that arise when the method is implemented experimentally and this leads to a number of model refinements and a required calibration step. Finally, the overall performance of the method is assessed through a number of experiments in which the aberration state of the lithographic system is intentionally detuned and subsequently estimated by the new method. These experiments show a remarkably good agreement, with an error smaller than 5¿¿m¿ , among the requested aberrations, the aberrations measured by the on-tool aberration sensor, and the results of the new wafer-based method.

Consider the all-time maximum of a Brownian motion with negative drift. Assume that this process is sampled at certain points in time, where the time between two consecutive points is rendered by an Erlang distribution with mean 1 / ω . The family of Erlang distributions covers the range between deterministic and exponential distributions. We show that the average convergence rate as ω → ∞ for all such Erlangian sampled Brownian motions is O ( ω − 1 / 2 ) , and that the constant involved in O ranges from − ζ ( 1 / 2 ) / 2 π for deterministic sampling to 1 / 2 for exponential sampling. The basic ingredients of our analysis are a finite-series expression for the expected maximum, an asymptotic expansion of ∑ j = 1 k − 1 ( 1 − exp ( 2 π i j / k ) ) − s , s ∈ R , as k → ∞ using Euler–Maclaurin summation, and Fourier sampling of functions analytic in an open set containing the closed unit disk.

We develop many-server asymptotics in the QED regime for models with admission control. The admission control, designed to reduce the incoming traffic in periods of congestion, scales with the size of the system. For a class of Markovian models with this scaled control, we identify the QED limits for two stationary performance measures. We also derive corrected QED approximations, generalizing earlier results for the Erlang B, C and A models. These results are useful for the dimensioning of large systems equipped with an active control policy. In particular, the corrected approximations can be leveraged to establish the optimality gaps related to square-root staffing and asymptotic dimensioning with admission control., Comment: IFIP WG 7.3 Performance 2013, 31st International Symposium on Computer Performance, Modeling, Measurements and Evaluation 2013, September 24-26, Vienna, Austria

The dominant pole approximation (DPA) is a classical analytic method to obtain from a generating function asymptotic estimates for its underlying coefficients. We apply DPA to a discrete queue in a critical many-sources regime, in order to obtain tail asymptotics for the stationary queue length. As it turns out, this regime leads to a clustering of the poles of the generating function, which renders the classical DPA useless, since the dominant pole is not sufficiently dominant. To resolve this, we design a new DPA method, which might also find application in other areas of mathematics, like combinatorics, particularly when Gaussian scalings related to the central limit theorem are involved.

Ultrashort X-ray pulses from free-electron laser X-ray sources make it feasible to conduct small- and wide-angle scattering experiments on biomolecular samples in solution at sub-picosecond timescales. During these so-called fluctuation scattering experiments, the absence of rotational averaging, typically induced by Brownian motion in classic solution-scattering experiments, increases the information content of the data. In order to perform shape reconstruction or structure refinement from such data, it is essential to compute the theoretical profiles from three-dimensional models. Based on the three-dimensional Zernike polynomial expansion models, a fast method to compute the theoretical fluctuation scattering profiles has been derived. The theoretical profiles have been validated against simulated results obtained from 300 000 scattering patterns for several representative biomolecular species. Keywords: Zernike polynomials; fluctuation X-ray scattering; structure

Random-access networks may exhibit severe unfairness in throughput, in the sense that some nodes receive consistently higher throughput than others. Recent studies show that this unfairness is due to local differences in the neighborhood structure: nodes with fewer neighbors receive better access. We study the unfairness in saturated linear networks, and adapt the random-access CSMA protocol to remove the unfairness completely, by choosing the activation rates of nodes as a specific function of the number of neighbors. We then investigate the consequences of this choice of activation rates on the network-average saturated throughput, and we show that these rates perform well in non-saturated settings.

In wavefront characterization, often the combination of a Shack-Hartmann sensor and a reconstruction method utilizing the Cartesian derivatives of Zernike circle polynomials (the least-squares method, to be called here Method A) is used, which is known to introduce crosstalk. In [J. Opt. Soc. Am. A31, 1604 (2014)10.1364/JOSAA.31.001604], a crosstalk-free analytic expression of the LMS estimator of the wavefront Zersectnike coefficients is given in terms of wavefront partial derivatives (leading to what we call Method B). Here, we show an implementation of this analytic result where the derivative data are obtained using the Shack-Hartmann sensor and compare it with the conventional least-squares method.

Wireless networks equipped with the CSMA protocol are subject to collisions due to interference. For a given interference range we investigate the tradeoff between collisions (hidden nodes) and unused capacity (exposed nodes). We show that the sensing range that maximizes throughput critically depends on the activation rate of nodes. For infinite line networks, we prove the existence of a threshold: When the activation rate is below this threshold the optimal sensing range is small (to maximize spatial reuse). When the activation rate is above the threshold the optimal sensing range is just large enough to preclude all collisions. Simulations suggest that this threshold policy extends to more complex linear and non-linear topologies.

On-axis and far-field series expansions are developed for the sound pressure due to an arbitrary, circular symmetric velocity distribution on a flat radiator in an infinite baffle. These expansions are obtained by expanding the velocity distributions in terms of orthogonal polynomials R(2n) (0)(sigma/a)=P(n)(2(sigma/a)(2)-1) with P(n) the Legendre polynomials. The terms R(2n) (0) give rise to a closed-form expression for the pressure on-axis as well as for the far-field pressure. Furthermore, for a large number of velocity profiles, including those associated with the rigid piston, the simply supported radiator, and the clamped radiators as well as Gaussian radiators, there are closed-form expressions for the required expansion coefficients. In particular, for the rigid, simply supported, and clamped radiators, this results in explicit finite-series expressions for both the on-axis and far-field pressures. In the reverse direction, a method of estimating velocity distributions from (measured) on-axis pressures by matching in terms of expansion coefficients is proposed. Together with the forward far-field computation scheme, this yields a method for far-field loudspeaker assessment from on-axis data (generalized Keele scheme). The forward computation scheme is extended to dome-shaped radiators with arbitrary velocity distributions.

Power series expansions in ka are derived for the pressure at the edge of a radiator, the reaction force on the radiator, and the total radiated power arising from a harmonically excited, resilient, flat, circular radiator of radius a in an infinite baffle. The velocity profiles on the radiator are either Stenzel functions (1-(sigma/a)2)n, with sigma the radial coordinate on the radiator, or linear combinations of Zernike functions Pn(2(sigma/a)2-1), with Pn the Legendre polynomial of degree n. Both sets of functions give rise, via King's integral for the pressure, to integrals for the quantities of interest involving the product of two Bessel functions. These integrals have a power series expansion and allow an expression in terms of Bessel functions of the first kind and Struve functions. Consequently, many of the results in [M. Greenspan, J. Acoust. Soc. Am. 65, 608-621 (1979)] are generalized and treated in a unified manner. A foreseen application is for loudspeakers. The relation between the radiated power in the near-field on one hand and in the far field on the other is highlighted.

This paper presents new Gaussian approximations for the cumulative distribution function P(A λ ≤ s) of a Poisson random variable A λ with mean λ. Using an integral transformation, we first bring the Poisson distribution into quasi-Gaussian form, which permits evaluation in terms of the normal distribution function Φ. The quasi-Gaussian form contains an implicitly defined function y, which is closely related to the Lambert W-function. A detailed analysis of y leads to a powerful asymptotic expansion and sharp bounds on P(A λ ≤ s). The results for P(A λ ≤ s) differ from most classical results related to the central limit theorem in that the leading term Φ(β), with is replaced by Φ(α), where α is a simple function of s that converges to β as s tends to ∞. Changing β into α turns out to increase precision for small and moderately large values of s. The results for P(A λ ≤ s) lead to similar results related to the Erlang B formula. The asymptotic expansion for Erlang's B is shown to give rise to accurate approximations; the obtained bounds seem to be the sharpest in the literature thus far.

A new approximation to the Stokes drift velocity profile based on the exact solution for the Phillips spectrum is explored. The profile is compared with the monochromatic profile and the recently proposed exponential integral profile. ERA-Interim spectra and spectra from a wave buoy in the central North Sea are used to investigate the behaviour of the profile. It is found that the new profile has a much stronger gradient near the surface and lower normalized deviation from the profile computed from the spectra. Based on estimates from two open-ocean locations, an average value has been estimated for a key parameter of the profile. Given this parameter, the profile can be computed from the same two parameters as the monochromatic profile, namely the transport and the surface Stokes drift velocity., 27 pp, 7 figs, one table in Ocean Modelling, 2016

Freak wave generation and decay is discussed in the context of the Hamiltonian description of ocean waves. In linear theory the probability distribution function (p.d.f.) of the surface elevation is given by the Normal distribution, while for finite amplitude waves there are, for extreme events, always deviations from the Normal, giving finite skewness and excess kurtosis. These deviations are caused by the presence of the bound waves and, in case of almost one-dimensional propagation, by non-resonant four-wave interactions.

The impact of wave-dependent surface stress on the ocean circulation has been studied using surface stresses calculated from a numerical wave model. The main questions to be investigated were what the effect would be on the Ekman currents in the upper ocean and what the impact would be on storm surge predictions. To answer the first question, the response of wave-dependent forcing on an Ekman type of model was studied. Here, the wave forcing was provided by a one-gridpoint version of the wave model. Second, the impact of the waves was studied with a three-dimensional ocean circulation model for the North Sea. Three different experiments were performed for a period of 1 yr. To test the effect on the storm surge signal, the results have been compared with sea level observations from 22 stations along the Norwegian and Dutch coasts. One of the main findings is that calculating stresses in the wave model, thereby introducing sea-state-dependent momentum fluxes, has a strong positive impact on the storm surge modeling compared with applying a traditional parameterization of surface stresses from the 10-m wind speed. When all cases with sea level deviation from the mean of less than 0.5 m were removed, the root-mean-square error for 1 yr averaged over all stations was reduced by approximately 6 cm. Splitting the momentum budget into an Eulerian and a wave part (Stokes drift) has only a negligible effect on the modeling of the sea surface elevation but increases the angular turning of the Eulerian surface drift to the right of the wind direction with an angle of about 4°.

Finite-amplitude deep-water waves are subject to modulational instability, which eventually can lead to the formation of extreme waves. In shallow water, finite-amplitude surface gravity waves generate a current and deviations from the mean surface elevation. This stabilizes the modulational instability, and as a consequence the process of nonlinear focusing ceases to exist when kh < 1.363. This is a well-known property of surface gravity waves. Here it is shown for the first time that the usual starting point, namely the Zakharov equation, for deriving the nonlinear source term in the energy balance equation in wave forecasting models, shares this property as well. Consequences for wave prediction are pointed out.

In this article we investigate the computational aspects of some recently proposed iterative methods for approximating the canonical tight and canonical dual window of a Gabor frame (g, a, b). The iterations start with the window g while the iteration steps comprise the window g, the k-th iterand ¿k, the frame operators S and Sk corresponding to (g, a, b) and (¿k, a, b), respectively, and a number of scalars. The structure of the iteration step of the method is determined by the envisaged convergence order m of the method. We consider two strategies for scaling the terms in the iteration step: Norm scaling, where in each step the windows are normalized, and initial scaling where we only scale in the very beginning. Norm scaling leads to fast, but conditionally convergent methods, while initial scaling leads to unconditionally convergent methods, but with possibly suboptimal convergence constants. The iterations, initially formulated for time-continuous Gabor systems, are considered and tested in a discrete setting in which one passes to the appropriately sampled-and-periodized windows and frame operators. Furthermore, they are compared with respect to accuracy and efficiency with other methods to approximate canonical windows associated with Gabor frames.

Let $X_1,X_2,...$ be independent variables, each having a normal distribution with negative mean $-\beta, Comment: Published at http://dx.doi.org/10.1214/105051606000000781 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

A solution of Rayleigh’s instability equation, which circumvents the apparent critical-layer singularity, is provided. The temporal and spatial growth rates of water waves exposed to a logarithmic wind profile are calculated and discussed. The findings are similar to previously published results, except for shear velocity–to–wave celerity ratios larger than 2, where the newly calculated growth rates start to decrease after having reached a distinct maximum. The ratio of the spatial to temporal growth rates is examined. It is shown to deviate by up to 20% from the leading-order value of 2. The implications of the growth rate to the modal distributions of energy input from wind to waves, for young and mature seas, and in temporal/spatial growth scenarios, are analyzed.

The three-dimensional frequency transfer function for optical imaging systems was introduced by Frieden in the 1960s. The analysis of this function and its partly back-transformed functions (two-dimensional and onedimensional optical transfer functions) in the case of an ideal or aberrated imaging system has received relatively little attention in the literature. Regarding ideal imaging systems with an incoherently illuminated object volume, we present analytic expressions for the classical two-dimensional x-y-transfer function in a defocused plane, for the axial z-transfer function in the presence of defocusing and for the x-z-transfer function in the presence of a lateral shift δy with respect to the imaged pattern in the x-z-plane. For an aberrated imaging system we use the common expansion of the aberrated pupil function with the aid of Zernike polynomials. It is shown that the line integral appearing in Frieden's three-dimensional transfer function can be evaluated for aberrated systems using a relationship established first by Cormack between the line integral of a Zernike polynomial over a full chord of the unit disk and a Chebyshev polynomial of the second kind. Some new developments in the theory of Zernike polynomials from the last decade allow us to present explicit expressions for the line integral in the case of a weakly aberrated imaging system. We outline a similar, but more complicated, analytic scheme for the case of severely aberrated systems.

We present a characterization of the modulation space S0 in terms of the Zak transform of its elements. We illustrate our result by considering S−λh, where h is the standard Gaussian, S is the “frame” operator corresponding to the critical-density Gabor system (h, a = 1, b = 1), and % % MathType!MTEF!2!1!+- % feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWMaey % icI48aaKGeaeaacaaIWaGaaiilamaalaaabaGaaG4maaqaaiaaikda % aaaacaGLBbGaayzkaaaaaa!3DF1! $$\lambda \in \left[ {0,\frac{3}{2}} \right)$$ . Both the proof of the main result and the example require basics from Gabor frame theory; these are developed in a separate section. We further use a result from recent work by Grochenig and Leinert on Wiener-type theorems in a non-commutative setting. We also present an extension of our main result to more general modulation spaces.

This study presents an experimental method to determine the resist parameters at the origin of a general blurring of a projected aerial image. The resist model includes the effects of diffusion in the horizontal plane and image blur that originates from a stochastic variation of the focus parameter. We restrict ourselves to the important case of linear models, where the effects of resist processing and focus noise are de- scribed by a convolution operation. These types of models are also known as diffused aerial image models. The used mathematical frame- work is the so-called extended Nijboer-Zernike ENZ theory, which al- lows us to obtain analytical results. The experimental procedure to ex- tract the model parameters is demonstrated for several 193-nm resists under various conditions of postexposure baking temperatures and bak- ing times. The advantage of our approach is a clear separation between the optical parameters, such as feature size, projection lens aberrations, and the illuminator setting on one hand, and process parameters intro- ducing blur on the other. © 2006 Society of Photo-Optical Instrumentation Engineers.

The judgment of the imaging quality of an optical system can be carried out by examining its through-focus intensity distribution. It has been shown in a previous paper that a scalar-wave analysis of the imaging process according to the extended Nijboer-Zernike theory allows the retrieval of the complex pupil function of the imaging system, including aberrations as well as transmission variations. However, the applicability of the scalar analysis is limited to systems with a numerical aperture (NA) value of the order of 0.60 or less; beyond these values polarization effects become significant. In this scalar retrieval method, the complex pupil function is represented by means of the coefficients of its expansion in a series involving the Zernike polynomials. This representation is highly efficient, in terms of number and magnitude of the required coefficients, and lends itself quite well to matching procedures in the focal region. This distinguishes the method from the retrieval schemes in the literature, which are normally not based on Zernike-type expansions, and rather rely on point-by-point matching procedures. In a previous paper [J. Opt. Soc. Am. A 20, 2281 (2003)] we have incorporated the extended Nijboer-Zernike approach into the Ignatowsky-Richards/Wolf formalism for the vectorial treatment of optical systems with high NA. In the present paper we further develop this approach by defining an appropriate set of functions that describe the energy density distribution in the focal region. Using this more refined analysis, we establish the set of equations that allow the retrieval of aberrations and birefringence from the intensity point-spread function in the focal volume for high-NA systems. It is shown that one needs four analyses of the intensity distribution in the image volume with different states of polarization in the entrance pupil. Only in this way will it be possible to retrieve the "vectorial" pupil function that includes the effects of birefringence induced by the imaging system. A first numerical test example is presented that illustrates the importance of using the vectorial approach and the correct NA value in the aberration retrieval scheme.

We show that $(g_2,a,b)$ is a Gabor frame when $a>0, b>0, ab0$. This is accomplished by expressing the Zak transform of $g_2$ in terms of the Zak transform of the Gaussian $g_1(t)=(2\gamma)^{{1/4}} \exp (-\pi \gamma t^2)$, together with an appropriate use of the Ron-Shen criterion for being a Gabor frame. As a side result it follows that the windows, generating tight Gabor frames, that are canonically associated to $g_2$ and $g_1$ are the same at critical density $a=b=1$. Also, we display the ``singular'' dual function corresponding to the hyperbolic secant at critical density.

Let $(g_{nm})_{n,m\in Z}$ be a Gabor frame for $L_2(R)$ for given window $g$. We show that the window $h^0=S^{-1/2} g$ that generates the canonically associated tight Gabor frame minimizes $\|g-h\|$ among all windows $h$ generating a normalized tight Gabor frame. We present and prove versions of this result in the time domain, the frequency domain, the time-frequency domain, and the Zak transform domain, where in each domain the canonical $h^0$ is expressed using functional calculus for Gabor frame operators. Furthermore, we derive a Wiener-Levy type theorem for rationally oversampled Gabor frames. Finally, a Newton-type method for a fast numerical calculation of $\ho$ is presented. We analyze the convergence behavior of this method and demonstrate the efficiency of the proposed algorithm by some numerical examples.

A deep-water approximation to the Stokes drift velocity profile is explored as an alternative to the monochromatic profile. The alternative profile investigated relies on the same two quantities required for the monochromatic profile, viz the Stokes transport and the surface Stokes drift velocity. Comparisons with parametric spectra and profiles under wave spectra from the ERA-Interim reanalysis and buoy observations reveal much better agreement than the monochromatic profile even for complex sea states. That the profile gives a closer match and a more correct shear has implications for ocean circulation models since the Coriolis-Stokes force depends on the magnitude and direction of the Stokes drift profile and Langmuir turbulence parameterizations depend sensitively on the shear of the profile. The alternative profile comes at no added numerical cost compared to the monochromatic profile., Accepted for publication in J Phys Oceanogr. 24 pages, 10 figures

Recently, in the context of orthogonal frequency division multiplexing (OFDM), a new method (FAB-method) for constructing tight Gabor frames (with redundancy 2) from a (non-tight) Gaussian g was proposed by LeFloch et al. (see Proc. IEEE, vol.83, no.6, p.982-96, 1995). In this letter, we prove that the FAB-method yields the tight window function canonically associated to the Gaussian. We furthermore provide a necessary and sufficient condition on the initial window function g in the Zak transform domain for the FAB-method to yield a tight Gabor frame. This yields a characterization of all initial window functions g for which the FAB-method works.

We introduce and analyze a method for stabilizing test parameters and estimating the non-linearity of mildly non-linear quantizers by Fourier analysis of the quantizer's output to sinusoidal input signals. Stabilization for the combined effects of the rounding operation of the quantizer and the variations in offset and amplitude of the applied sinusoid is achieved by a deterministic form of subtractive dithering, called wobbling. It thus appears that one can retrieve low-to-medium degree non-linearities whose contribution to the quantizer's distortion in the relevant frequency range is of the same order as or well below the distortion due to rounding before wobbling. For the analysis of the method a thorough investigation of the spectra of sampled, quantized sinusoids (wobbled or not) has to be made. As such we complete and extend existing results in the literature of spectra of this type. The method is potentially useful for estimating the integral non-linearity of AD-converters.

The partial derivatives and Laplacians of the Zernike circle polynomials occur in various places in the literature on computational optics. In a number of cases, the expansion of these derivatives and Laplacians in the circle polynomials are required. For the first-order partial derivatives, analytic results are scattered in the literature. Results start as early as 1942 in Nijboer’s thesis and continue until present day, with some emphasis on recursive computation schemes. A brief historic account of these results is given in the present paper. By choosing the unnormalized version of the circle polynomials, with exponential rather than trigonometric azimuthal dependence, and by a proper combination of the two partial derivatives, a concise form of the expressions emerges. This form is appropriate for the formulation and solution of a model wavefront sensing problem of reconstructing a wavefront on the level of its expansion coefficients from (measurements of the expansion coefficients of) the partial derivatives. It turns out that the least-squares estimation problem arising here decouples per azimuthal order m, and per m the generalized inverse solution assumes a concise analytic form so that singular value decompositions are avoided. The preferred version of the circle polynomials, with proper combination of the partial derivatives, also leads to a concise analytic result for the Zernike expansion of the Laplacian of the circle polynomials. From these expansions, the properties of the Laplacian as a mapping from the space of circle polynomials of maximal degree N, as required in the study of the Neumann problem associated with the transport-of-intensity equation, can be read off within a single glance. Furthermore, the inverse of the Laplacian on this space is shown to have a concise analytic form.

We consider Markovian many-server systems with admission control operating in a QED regime, where the relative utilization approaches unity while the number of servers grows large, providing natural Economies-of-Scale. In order to determine the optimal admission control policy, we adopt a revenue maximization framework, and suppose that the revenue rate attains a maximum when no customers are waiting and no servers are idling. When the revenue function scales properly with the system size, we show that a nondegenerate optimization problem arises in the limit. Detailed analysis demonstrates that the revenue is maximized by nontrivial policies that bar customers from entering when the queue length exceeds a certain threshold of the order of the typical square-root level variation in the system occupancy. We identify a fundamental equation characterizing the optimal threshold, which we extensively leverage to provide broadly applicable upper/lower bounds for the optimal threshold, establish its monotonicity, and examine its asymptotic behavior, all for general revenue structures. For linear and exponential revenue structures, we present explicit expressions for the optimal threshold., Comment: 36 pages, 7 figures

Item does not contain fulltext Various authors have presented the aberration function of an optical system as a power series expansion with respect to the ray coordinates in the exit pupil and the coordinates of the intersection point with the image field of the optical system. In practical applications, for reasons of efficiency and accuracy, an expansion with the aid of orthogonal polynomials is preferred for which, since the 1980s, orthogonal Zernike polynomials have become the reference. In the literature, some conversion schemes of power series coefficients to coefficients for the corresponding Zernike polynomial expansion have been given. In this paper we present an analytic solution for the conversion problem from a power series expansion in three or four dimensions to a double Zernike polynomial expansion. The solution pertains to a general optical system with four independent pupil and field coordinates and to a system with rotational symmetry in which case three independent coordinate combinations have to be considered. The conversion of the coefficients is analytically in closed form and the result is independent of a specific sampling scheme or sampling density as this is the case for the commonly used least squares fitting techniques. Computation schemes are given that allow the evaluation of coefficients of arbitrarily high order in pupil and field coordinates.

Loudspeakers are often modelled as a rigid piston in an infinite baffle. As a model for real loudspeakers, this approach is limited in two ways. One issue is that a loudspeaker cone is not rigid, and a second issue is that a loudspeaker is mostly used in a cabinet. Both issues are addressed here by developing the velocity of the radiator in terms of orthogonal polynomials known from optical diffraction theory as Zernike circle polynomials. Using these polynomials we develop semi-analytic expressions for the sound pressure from the radiator in two different cases: as a flexible flat radiator mounted in an infinite baffle, and as the cap of a rigid sphere. In the latter case the comparison is done not only for the pressure but also for other quantities viz. the baffle-step response, sound power and directivity, and the acoustic centre of the radiator. These quantities are compared with those from a real loudspeaker. Finally, in the case of the baffled-piston radiation the spatial impulse response is presented.

Let $a>0, b>0, ab

In this paper we give a further investigation of the method introduced by the author in [1, Frequency-domain bounds for nonnegative unsharply band-limited functions] for proving bounds for functions with nonnegative Fourier transforms. We also dealt with the question of how large the supremum KS of all numbers |f(u)| is with f the Fourier transform of a nonnegative integrable function F and f(0) = 1, |f(ku)| ≤ e for k ∈ S. Here u > 0 and S ⊂ {2, 3, . . .}. This problem was related in [1] to finding the infimum MS of all numbers Mh = maxϑ [(1−h(ϑ))/(1− cos ϑ)] over all 2π-periodic even, smooth functions h whose Fourier cosine coefficients ak vanish for k ∉ S, and it was proved and announced for several cases that MS (1−KS ) = 1. In this paper we prove the results announced in [1]. To that end we generalize the method given in [1] to include Fourier transforms f of probability measures on R and a certain generalized function h, and we show that the numbers KS, MS are assumed as |f(u)|, Mh for certain allowed f,h. Moreover, we establish a fundamental relation between finding the numbers KS, MS and the numbers KT, MT where T = {2, 3, . . .}\S. In particular, we show that MT = 2KS (2KS − 1)−1,KT = 1/2 MS(MS − 1)−1 and that MT (1 − KT) = 1,KSKT = 1/2 , whenever MS (1 − KS) = 1.

We present a Fourier Transform Infrared spectrometer for use with a frequency comb laser as source. The spectrometer can completely resolve the modes of the frequency comb at 100 MHz.

The theory of orthogonal polynomial (Zernike) expansions of functions on a disk, as used in the diffraction theory of optical aberrations, is applied to obtain (semi-) analytical expressions for the spatial impulse responses arising from a non-uniformly moving, baffled, circular piston. These expressions are in terms of the expansion coefficients of the non-uniformity and the responses of the orthogonal expansion functions. The latter impulse responses have a closed formas finite series involving Legendre functions and the sinc function.The method is compared with a similar method, proposed in P.R. Stepanishen, J. Acoust. Soc. Am. 70, 1176-1181, 1981 where zero-th orderorthogonal Bessel functions, rather than Zernike polynomials, are used as expansion functions. The method is also considered for retrieval of a non-uniform velocity profile from measured spatial impulseresponses on the level of expansion coefficients, which can be applied to acoustic holography for loudspeakers.

Several quantities related to the Zernike circle polynomials admit an expression, via the basic identity in the diffraction theory of Nijboer and Zernike, as an infinite integral involving the product of two or three Bessel functions. In this paper these integrals are identified and evaluated explicitly for the cases of (a) the expansion coefficients of scaled-and-shifted circle polynomials, (b) the expansion coefficients of the correlation of two circle polynomials, (c) the Fourier coefficients occurring in the cosine representation of the radial part of the circle polynomials.

We have investigated correlation patterns generated by a frequency-comb laser in a dispersive unbalanced Michelson interferometer and apply the developed formalism to the case of distance metrology. Due to group velocity dispersion, the position of the brightest fringe of the correlation pattern, which is used for distance determination, cannot be derived by simply using the definition of group refractive index of the dispersive medium. It is shown that the discrete spectrum of the optical frequency comb gives rise to correlation functions which can be represented by a series, namely a discrete correlation series. We have developed a general formalism, valid for any pulse train, extending the discrete model to a continuous model of cross-correlation functions using the Poisson summation. Our model is relevant for any offset and repetition frequency of the frequency comb. From the continuous cross-correlation model we show that, even for a homogeneous dispersive medium the position of the brightest fringe varies nonlinearly for small delay distances and stabilizes for longer ones. We have compared the theoretical results to measurements of pulse propagation in air for path-lengths up to 200 m.

The author discusses the asymptotic behaviour of the Pearcey-type integralI'alpha(X, Y)=2 (sup) approximately=0 ualpha +1 exp(i(u4+Xu2))Jalpha(Yu) du for -1( alpha (5/2, where Jalpha is a Bessel function, as X to +or- infinity , Y fixed, as Y to infinity , X fixed, and as Y= rho (2/3 mod X mod )32/, X to - infinity , rho fixed. The case alpha =- 1/2 gives the classical Pearcey integral whose asymptotics has been investigated recently by Kaminski (1989) and Paris (1991). In the case alpha =0, I'alpha(X, Y) as a function of Y>or=0 represents the radial part of the impulse-response function describing the image formation in high resolution electron microscopes at normalized defocus X. He uses the approach of Paris by representing I'alpha(X, Y) in terms of Weber parabolic cylinder functions, and he augments this approach by invoking the Chester-Friedman-Ursell method (1957) to obtain the leading asymptotics of I'alpha(X, Y) around the caustic Y2=(2/3 mod X mod )3, X to - infinity.

We study the image formation by a high-numerical-aperture optical imaging system in the presence of a multilayer structure in the region around the image plane. Earlier references to this subject in the literature use numerical solutions of the diffraction integrals. In this paper, we use a numerical approach based on the semi-analytic Extended Nijboer-Zernike (ENZ) theory to solve the diffraction integrals in the presence of a multilayer structure. The specific ENZ calculation scheme uses the complex Zernike expansion of the complex amplitudes of forward and backward propagating plane wave components in a certain layer of the multilayer stack. By its nature, the ENZ approach enables an accurate and fast calculation of the vector field in the stratified image region. Examples of multilayer imaging that are encountered in high-numerical-aperture optical systems and in optical lithography for semiconductor manufacturing are presented and the accuracy of the ENZ approach is examined.

A Brownian motion observed at equidistant sampling points renders a random walk with normally distributed increments. For the difference between the expected maximum of the Brownian mo- tion and its sampled version, an expansion is derived with coefficients in terms of the drift, the Riemann zeta function and the normal distribution function.

Multi-access networks exhibit severe unfairness in throughput. Recent studies show that this unfairness is due to local differences in the neighborhood structure: Nodes with less neighbors receive better access. We study the unfairness in tandem networks, and adapt the multi-access protocol to remove the unfairness completely, by choosing the activity rates of nodes appropriately as a function of the number of neighbors.

We discuss two independent, large scale experiments performed in two wave basins of different dimensions in which the statistics of the surface wave elevation are addressed. Both facilities are equipped with a wave maker capable of generating waves with prescribed frequency and directional properties. The experimental results show that the probability of the formation of large amplitude waves strongly depends on the directional properties of the waves. Sea states characterized by long-crested and steep waves are more likely to be populated by freak waves with respect to those characterized by a large directional spreading.

Results are presented of mask imaging using the Extended Nijboer-Zernike (ENZ) theory of diffraction. We show that the efficiency of a mask imaging algorithm, derived from this theory, can be increased. By adjusting the basic Finite Difference Time Domain (FDTD) algorithm, we can calculate the near field of isolated mask structures efficiently, without resorting to periodic domains. In addition, the calculations for the points on the entrance sphere of the imaging system can be done separately with a Fourier transformed Stratton-Chu near-to-far-field transformation. By clever sampling in the radial direction of the entrance pupil, the computational effort is already reduced by at least a factor of 4.