Saddle Point Method Of Integration Research Articles

On the Complete Radiation Pattern of a Vertical Hertzian Dipole Above a Low-Loss Ground

The complete radiation field pattern of a vertical Hertzian dipole antenna on or above a lossless or low-loss dielectric half-space is studied using a rigorous Sommerfeld formalism. The reflected fields in the air above the interface and the subsurface fields transmitted into the dielectric are computed by numerical integration of the Sommerfeld integrals. Furthermore, to facilitate the physical interpretation of these results, a detailed asymptotic saddle-point integration method analysis is presented, which includes terms that vary in magnitude with the second power of the inverse distance from the dipole. It is shown that the second-order field constituents are dominant at the interface, where the first-order geometrical fields vanish. These second-order terms comprise an evanescent wave propagating along the interface in the upper half-space and a lateral wave, also known as the head wave, which propagates in the subsurface along the direction of the critical angle. The two waves only exist between two cones whose half-angles are equal to the critical angle, and their interference with the geometrical-optics fields determines the radiation pattern for elevation angles near the horizon. The far zone surface fields on either side of the interface comprise two second order waves that propagate along the interface, one with the phase velocity in the air, and the other with the phase velocity in the dielectric. Away from the interface, the leading field components vary with the first power of the inverse distance, which explains the sharp dip in the field pattern at the interface-a phenomenon known as the interface pattern extinction. Another distinctive phenomenon, observed in the subsurface field pattern, is the rippling that occurs in the angular range between the critical angle cone and the interface. The asymptotic analysis has shown that this pattern scalloping results from the interference of the lateral wave with the geometrical-optics spherical wave.

Scattering from a PEMC strip deeply buried beneath a dielectric slab using Kobayashi potential method

Kobayashi potential method is applied to get the scattering behavior by a perfect electromagnetic conducting strip deeply buried underneath a flat dielectric slab. Using the two-dimensional solution of homogeneous Helmholtz’s equation, Fourier transformed representation of scattered field is written in terms of the unknown characteristic functions. Application of the boundary conditions on tangential components of electromagnetic fields gives dual integral equations (DIEs) which through Weber–Schafheitlin’s discontinuous integrals and Jacobi polynomials are used to satisfy the edge condition. The resulting DIEs are written in matrix equations, and those that involve infinite integrals are computed numerically. The saddle point method of integration is used to obtain an expression for the far-zone scattered field. To validate the results, obtained results are firstly compared with corresponding obtained through the physical optics method. The width of the strip, the thickness of the slab, and the angle of incidence of plane wave are taken as parameters to study their influence on the scattered field.

On the Sommerfeld half-space problem: appraisal of approximate solutions with extensions to plasmonics

The classical Sommerfeld problem of a vertical electric dipole radiating over an imperfectly conducting half-space is revisited. Two variants of the modified saddle-point integration method are employed to derive new, closed-form, second-order approximations of the far-zone electromagnetic fields. The new formulas are applied to compute radiation patterns and surface fields for sample problems including both ordinary and plasmonic media, and their accuracy is assessed by comparison with the results of rigorous numerical evaluation of the Sommerfeld integrals.

Strong-field ionization via a high-order Coulomb-corrected strong-field approximation

Signatures of the Coulomb corrections in the photoelectron momentum distribution during laser-induced ionization of atoms or ions in tunneling and multiphoton regimes are investigated analytically in the case of an one-dimensional problem. High-order Coulomb corrected strong-field approximation is applied, where the exact continuum state in the S-matrix is approximated by the eikonal Coulomb-Volkov state including the second-order corrections to the eikonal. Although, without high-order corrections our theory coincides with the known analytical R-matrix (ARM) theory, we propose a simplified procedure for the matrix element derivation. Rather than matching the eikonal Coulomb-Volkov wave function with the bound state as in the ARM-theory to remove the Coulomb singularity, we calculate the matrix element via the saddle-point integration method as by time as well as by coordinate, and in this way avoiding the Coulomb singularity. The momentum shift in the photoelectron momentum distribution with respect to the ARM-theory due to high-order corrections is analyzed for tunneling and multiphoton regimes. The relation of the quantum corrections to the tunneling delay time is discussed

Modified saddle-point integral near a singularity for the large deviation function

Long-time-integrated quantities in stochastic processes, in or out of equilibrium, usually exhibit rare but huge fluctuations. Work or heat production is such a quantity, for which the probability distribution function displays an exponential decay characterized by the large deviation function (LDF). The LDF is often deduced from the cumulant generating function through the inverse Fourier transformation. The saddle-point integration method is a powerful technique to obtain the asymptotic results in the Fourier integral, but special care should be taken when the saddle point is located near a singularity of the integrand. In this paper, we present a modified saddle-point method to handle such a difficulty efficiently. We investigate the dissipated and injected heat production in equilibration processes with various initial conditions, for example, where the generating functions contain branch-cut singularities as well as power-law ones. Exploiting the new modified saddle-point integrations, we obtain the leading finite-time corrections for the LDFs, which are confirmed by numerical results.

Near-source response of a resistive layer to a vertical or horizontal electric dipole excitation

Frequency and time-domain analytical expressions are derived for the electromagnetic field response of a resistive layer inserted in an otherwise homogeneous whole-space as observed in the vicinity of an electric dipole source. The analysis of closed-form solutions demonstrated that at near-source-receiver offsets, the spectral and spatial distribution of the fields is better described by a superposition of images, associated with the dipolar character of the charges distributed in the boundaries, rather than the guided mode behavior dominating the response in the far offset regime. Approximate solutions of the fields in the frequency domain were derived using the saddle point method of integration. The formulas describing the fields were in good agreement with semianalytical calculations. However, a lower frequency bound was found, below which the expressions are inaccurate, and thereby they cannot be used to obtain time-domain solutions. A kernel modulation scheme was used instead, which yields an infinite series representation for the fields. The expressions thus derived produce accurate fields to very low frequencies, and thereby they were also used to obtain time-domain formulas. The analysis indicated that for a vertical electric dipole (VED) excitation, the late time response of the image field associated with the charge density induced on the upper boundary appears to cancel the direct field, thus providing the response of the layer. For a horizontal electric dipole (HED) source, the superimposed contributions of the transverse electric (TE) and transverse magnetic (TM) modes appeared to oppose the image field, resulting in the direct field dominating over the response of the layer, and thereby masking any sensitivity to the properties of the layer in this configuration.

Diffraction by an imperfect half plane in a bianisotropic medium

A general theory to study the electromagnetic diffraction by imperfect half planes immersed in linear homogeneous bianisotropic media is presented. The problem is formulated in terms of Wiener‐Hopf equations by deriving explicit spectral domain expressions for the characteristic impedances of bianisotropic media, which allow one to exploit their analytical properties. In the simpler case of perfect electric conducting and perfect magnetic conducting half planes, the Wiener‐Hopf equations involve matrices of order 2, which can be factorized in closed form if the constitutive tensors of the bianisotropic material are of special form. Four of these special cases are discussed in detail. In order to deal with the more general problem, a technique to numerically factorize the Wiener‐Hopf matrix kernels is presented. Our numerical approach is discussed on one example, by considering the previously unsolved problem of a perfect electric conducting half plane in a gyrotropic medium. The reported numerical results show that the diffracted field contribution is obtained by use of the saddle point integration method.

On the Probability Density Function of an Optical DPSK Signal in the Presence of Intrachannel Crosstalk and ASE Noise

The application of the saddle point integration method for evaluating the probability density function of a DPSK signal in the presence of intrachannel crosstalk and ASE noise is investigated. Numerical results for a 10 Gbit/s optically pre-amplified direct detection receiver show that the shape of the density has an asymmetric two-pronged structure. Moreover, the reliability of a saddle point approximation is also studied, and shown that this technique estimates very well the tails of the density, but lacks accuracy in the central region.

Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre- and postdetection filtering

A novel approach to analytically evaluate the bit error probability in optically preamplified direct-detection systems is presented, which can take into account the effects of pulse shaping, chirping, filtering at the transmitter and the receiver, both pre- and postdetection, chromatic dispersion, and ASE noise. The method is computationally very fast in that the saddle point integration method for solving the resulting line integral of a particular moment generating function is adopted. A closed-form approximation for the bit error probability is also provided, which is within 0.01 dB from the exact numerical results.

Performance analysis of DS/CDMA systems with shadowing and flat fading

A new method is developed for evaluating the error probability ( P e) for direct sequence, code division multiple access (DS/CDMA) wireless systems that includes the effects of shadowing and fading. The method is based on saddle point integration (SPI) of the test statistic's moment generating function (MGF) in the complex plane. The SPI method is applicable to both ideal and wireless channels. For wireless channels, a Padé approximation (PA) of the MGF, which is derived from the moments of the channel's shadowing and fading distributions, allows efficient evaluation of the P e. The SPI method can be used to model independent channels using separate shadowing and fading moments for each individual channel. The relative error between the probability density function (PDF) of the composite variate representing log-normal shadowing and Rayleigh fading and the PDF found from the inverse Laplace transform of the PA is negligible. Results show that log-normal shadowing increases the P e by 100–1000% compared to channels exhibiting fading only.

Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre- and postdetection filtering

A novel approach to analytically evaluate the bit error probability in optically preamplified direct-detection systems is presented, which can take into account the effects of pulse shaping, chirping, filtering at the transmitter and the receiver, both pre- and postdetection, chromatic dispersion, and ASE noise. The method is computationally very fast in that the saddle point integration method for solving the resulting line integral of a particular moment generating function is adopted. A closed-form approximation for the bit error probability is also provided, which is within 0.01 dB from the exact numerical results.

Computing the distribution of sums of random sine waves and of Rayleigh-distributed random variables by saddle-point integration

The method of saddle-point integration is applied to calculating the cumulative distribution of sums of independent random variables that either have a Rayleigh distribution or are sine waves with independently random phases in the presence of Gaussian random noise.

Spatial diversity equalization for MIMO ocean acoustic communication channels

The available bandwidth of ocean acoustic channels is inherently narrow that impedes high-transmission rate and makes it difficult for multiple users to communicate simultaneously. To alleviate this problem, spatial diversity antennas are used to increase the date rate. In this paper, we employ the spatial diversity equalizers (SDE) to increase the effective channel bandwidth by minimizing the mean-square error (mse). Although joint equalizers have been used in digital telephone subscriber lines to suppress crosstalk, we apply the concept to ocean acoustic channels and show that multiuser communication is possible despite the narrow-channel bandwidth. In addition, we will show that the advantage of SDE is not because we use more taps, but because we collect the data carried through various ocean paths. By applying the saddle point integration method to multiinput multioutput (MIMO) channels, we compute the probability of error (PE) to show that a factor of 3-4 of channel reuse is possible.

Triple Plasmon Decay of Electromagnetic Pump Wave

A fluid description of the triple plasmon decay process has been provided. The linear and nonlinear analyses suggest that this parameter process will occur very likely at the local area where the plasma density is about nc/9. The analyses with saddle point method of integration indicate also that the triple plasmon decay process is absolutely unstable.

Performance evaluation of QAM and QPRS systems with interfering channels in frequency selective fading

An analytical technique is developed to evaluate the error rate performance of M-ary quadrature amplitude modulation (QAM) and quadrature partial response signalling (QPRS) in the presence of adjacent channel and cochannel interference with frequency selective fading. Error probability expressions are derived and evaluated numerically using the method of saddle-point integration. In addition to the above impairments, the system model also incorporates other disturbances, namely stationary white gaussian noise, quadrature channel interference, random timing misalignments and carrier phases of interfering signals. An extensive set of results and comparisons are presented.

Binary nucleation kinetics: A matrix method

The kinetics of binary nucleation is investigated and an exact method to evaluate the nucleation rate is developed. The exact method accounts for all flows to the stable clusters, and thus the nucleation rate is obtained independently of determining the location of the saddle point in the Gibbs free energy surface. The exact method is based on determination of cluster concentrations, which can be obtained after solving a set of linear equations. The values of the nucleation rate and the direction of flow at the saddle point are calculated for water–sulphuric acid and water–ammonia systems. The results are compared to the results obtained by Stauffer’s saddle point integration method for binary nucleation kinetics. The saddle point integration method is found to give a reasonably accurate estimation of nucleation rate in the water–sulphuric acid system. However, in the water–ammonia system the difference is found to be as high as three orders of magnitude.

The dyadic Green's function for the cylindrical chirostrip antenna

Using the eigenfunction expansion of dyadic Green's function in unbounded chiral medium, and the method of scattering superposition, the expressions of dyadic Green's function for the one and two—layer cylindrical chirostrip antennas are derived, when the field sources are placed in any layer of cylindrical chirostrip antenna. Also, using the method of saddle point integration, the asymptotic expression of dyadic Green's function for the wraparound chirostrip antenna is obtained. The results can be directly used to analyse the radiation characteristics of cylindrical chirostrip dipolc antenna, et al.

Computing the generalized Marcum Q-function

A modified version of the method of saddlepoint integration is shown to compute the generalized Marcum Q-function accurately over a broad range of parameters. Bounds on the truncation error incurred by stopping the numerical quadrature at a certain point are presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Plane Wave Diffraction by an Open-Ended Parallel Plate Waveguide Cavity

The diffraction of a plane electromagnetic wave by an open-ended parallel plate waveguide cavity with a planar termination is rigorously analyzed for the E -polarized case using the Wiener-Hopf technique. Introducing the Fourier transform for the unknown scattered field and applying boundary conditions in the transform domain, the problem is formulated in terms of the simultaneous Wiener-Hopf equations, which are solved via a factorization and decomposition procedure. The solution is exact but formal since branch-cut integrals with unknown integrands and an infinite number of unknowns are involved. Approximation procedures based on rigorous asymptotics are further presented and the approximate solution to the Wiener-Hopf equations is derived. The scattered field inside and outside the cavity is evaluated by taking the inverse Fourier transform and applying the saddle point method of integration. Based on these results, numerical computations of the bistatic scattered pattern as well as the radar cross sec...

Detection probabilities for correlated Rayleigh fading signals

Both the method of saddlepoint integration and its associated saddlepoint approximation are applied to calculating the probability of detecting correlated Rayleigh-fading signals in Gaussian noise by means of a detector that integrates M samples of the output of a quadratic rectifier. The quadrature components of the signal samples are modeled as an autoregressive moving-average process, and specific results are exhibited for a first-order Markov process. By these methods the fluctuation loss can be computed for much larger values of M and for larger values of the detection probability than previously. Values calculated by the saddlepoint approximation prove to be close enough to the exact values to be useful over a broad range of signal parameters. >