The electromagnetic fields of a single optic fibre mode are quantized based on the observationthat these fields can be derived from a scalar harmonic oscillator function depending on onlytime and the axial wavenumber. Asymptotic results for both the one-photon probabilitydensity and two-photon correlation density functions within the forward light cone arepresented, showing an algebraic decay for large times or distances. This algebraic decay,increasing the uncertainty in the arrival time of the photons, also remains in the presence ofdispersion shift, in qualitative agreement with experimental results. Also presented are explicitformulae to be used in parameter studies to optimize quantum optic fibre communications.
We obtain Born's rule from the classical theory of random waves in combination with the use of thresholdtype detectors. We consider a model of classical random waves interacting with classical detectors and reproducing Born's rule. We do not discuss complicated interpretational problems of quantum foundations. The reader can select between the "weak interpretation," the classical mathematical simulation of the quantum measurement process, and the "strong interpretation," the classical wave model of the real quantum (in fact, subquantum) phenomena.
The distance dependence of the probability of observing two photons in a waveguide is investigated and the Glauber correlation functions of the entangled photons are considered. First the case of a hollow waveguide with modal dispersion is treated in detail: the spatial and temporal dependence of the correlation functions is evaluated and the distance dependence of the probability of observing two photons upper bounds and asymptotic expressions valid for large distances are derived. Second the generalization to a real fibre with both material and modal dispersion, allowing dispersion shift, is discussed.
This paper gives a detailed derivation of the classical electromagnetic modes of a layered circularly symmetrical dielectric waveguide. The corresponding Hamilton function is derived by using suitable canonical observables and the standard analogy to the classical harmonic oscillator. The derivation is generic in the sense that it can be used as an "algorithm" to compute the electromagnetic field of the waveguide. The associated Hamilton operator can then be obtained by using the standard quantization procedure where the canonical observables are replaced by the corresponding operators i.e., the creation and the annihilation operators of the photon (or equivalently, the position and the momentum operators of the harmonic oscillator) and by taking the appropriate commutation relations into account.
The paper explores the fundamental physical principles of quantum mechanics (in fact, quantum field theory) that limit the bit rate for long distances and examines the assumption used in this exploration that losses can be ignored. Propagation of photons in optical fibers is modelled using methods of quantum electrodynamics. We define the "photon duration" as the standard deviation of the photon arrival time; we find its asymptotics for long distances and then obtain the main result of the paper: the linear dependence of photon duration on the distance when losses can be ignored. This effect puts the limit to joint increasing of the photon flux and the distance from the source and it has consequences for quantum communication. Once quantum communication develops into a real technology (including essential decrease of losses in optical fibres), it would be appealing to engineers to increase both the photon flux and the distance. And here our "photon flux/distance effect" has to be taken into account. This effect also may set an additional constraint to the performance of a loophole free test of Bell's type-to close jointly the detection and locality loopholes.
This paper presents a digital directional coupler (DDC) that separates forward- and backward-traveling waves on a transmission line. Based on two independent wideband measurements of voltage and current and on frequency-domain digital wave splitting using a fast Fourier transform (FFT), the DDC is a versatile device for direction separation. A practical procedure is described for the calibration of the digital processor with respect to the particular transmission line and the voltage and current sensors that are being used. As an experiment, a DDC was designed and implemented using low-cost wideband sensors and was installed with medium-voltage equipment in a power distribution station. Partial discharge (PD) measurements were conducted on cross-linked polyethylene (XLPE)-insulated power cables to illustrate the directional separation capabilities of the DDC.
A mathematical framework is introduced for optimization of antenna near-field imaging problems, based on the multipole expansion of the electromagnetic field, the Fisher information to quantify the quality of data and use of modern interior point convex optimization techniques. We consider the general problem of optimizing the measurement sensor allocation for parameter estimation in distributed systems, and in particular the problem of optimizing the measurement set-up for antenna near-field estimation. As an application example for antenna near-field imaging, we consider a relevant measurement set-up using cylindrical probing coordinates. The convex optimization problem is examined using duality theory, and it is shown that several structural properties of the optimal measurement problem can be exploited in developing an efficient interior point optimization method. In particular, we show that the cylindrical measurement set-up yields a Fisher information matrix with block diagonal structure, a feature which can be directly exploited in the optimization algorithm by reducing the number of dual decision variables.
This conference illuminates and benefits from relations between different types of waves, i.e., quantum physics methods in classical wave modeling. The proceedings are intended for researchers and graduate students in fundamental as well as applied sciences. The preface classifies and summarizes the conference giving relations between the papers.
This paper discusses criteria for establishing uniqueness of wave propagation problems. Causality, or passivity that implies causality, is adopted as the fundamental principle. It is stressed that radiation conditions are not applicable for waveguide modes that carry no active power. The Jones' criteria for causality in the frequency domain, which covers the convectively unstable case, are presented and analysed, the vanishing absorption principle, VAP, in particular. It is proposed to use L^{2} for a lossy medium but a weighted L^{2} for the lossless case.
With the purpose of source localization in high voltage direct current submarine power cables general layered waveguide models are investigated. It is concludes that the non-discrete modes can usually be ignored in comparison with the least attenuated discrete mode. Only a finite number of discrete modes are of importance for a band limited signal. Accurate analytical expressions are derived in the time domain for both discrete and non-discrete electromagnetic waveguide modes for large distances together with an error analysis.
This paper presents accurate analytical expressions in the time domain for discrete electromagnetic waveguide modes for large distances together with a discussion of the errors. The main application is source localization for high voltage direct current submarine power cables. The research behind this paper is part of the project “Fundamental wave modelling for signal estimation on lossy transmission lines” funded by the Swedish Research Council and ABB High Voltage Cables AB in Sweden.
There are two optic fibre properties in particular that obstruct the transfer over long distances in quantum communications. One of them is dispersion, which reduces the maximum bit rate. In classical communication with modern highly purified fibres, dispersion is the major limiting factor. The other property is the material loss that causes fluctuations in addition to a general annihilation with distance of photons. It is believed that losses are the major limiting factor in quantum communication over long distances if the bit rate is not an issue. Of major interest in quantum communication is the photon source. It is therefore a high priority to develop methods for the determination of properties like initial temporal modes, repeatability, independence of sequentially emitted photons, etc., for a source emitting single photons in a given spatial mode. In the current paper we suggest that the source properties can be estimated using statistics of the run times of the photons. This requires that the fibre is modelled with sufficient accuracy. To simplify the analysis, it is assumed that the fibre losses can be neglected and that the photons are independent and identically generated. Energy detection in one spin state is employed, making the modelling scalar. The one photon initial temporal mode is found by maximizing a maximum-likelihood function based on running time statistics. Unfortunately, this optimization problem is, in general, not convex. However, for photon detection in the so-called asymptotic radiation zone, where the probability density can be determined to a sufficient degree with asymptotic methods, the optimization problem is convex. In the current paper, quantum tomography in fibres based on this convex optimization method is presented, and its generalization to more complicated situations like the introduction of losses in the modelling is discussed.
For the purpose of determining the twist of a homogeneous, locally reacting, uniaxial cylinder, an inverse microwave scattering theory is presented. Remote measurements of the spiral grain of trees and logs are the prime application. Based on practical considerations, it is assumed that the transmitting and receiving antennas are collocated, requiring a three-dimensional modelling. A general theory is first developed, followed by an asymptotic analysis assuming that the distance from the antennas to the cylinder is many wavelengths and many cylinder radii. In this way, a substantial reduction of the numerical complexity, to the level of the two-dimensional case, is achieved. The error of the determined twist angle as function of inherent parameters of the problem using a CramEacuter-Rao analysis is given. The results from numerical simulations show that this error is low enough for determining the grain angle. Presented parameter studies of the error can be used for minimizing the errors in a measurement set up, of particular interest for non-sophisticated instruments and non-ideal laboratory conditions, by selecting optimum parameters such as frequency and antenna gain. Finally, it is stated that the model has a great potential for developing efficient algorithms for measuring the twist angle.
An inverse microwave scattering theory is presented for the determination of twist of a homogeneous locally reacting uniaxial cylinder. The prime application is remote measurements of spiral grain of trees and logs. From practical considerations, the transmitting and receiving antennas are near each other
requiring a three-dimensional modelling. First a general theory is developed. Then asymptotic formulae are derived assuming that the distance from the antennas to the cylinder is many wavelengths and many cylinder radii granting a substantial reduction of the numerical complexity compared to the twodimensional case. Finally, the error of the determined twist angle as function of inherent parameters of the problem is given using a Cramer Rao analysis.
Twist in wood, being closely related to spiral grain, may cause
serious problems in building structures, furniture, and joinery. It is therefore of great interest to sort out, at an early stage in the manufacturing process, trees, logs and boards that have an access of spiral grain. The spiral grain pattern is described by a helical deviation of the fibre direction in relation to the longitudinal direction of a living tree or a log and seems to be an indicator for other defects such as compression wood. Remote microwave sensing of spiral grain has received a large interest during the latest two decades. Its development has been impeded by the large variation with moisture content of the microwave properties of wood and by the complexity in modelling the electromagnetic field in a log with spiral grain. A review is presented of a direct method with no requirement of information on moisture content for boards. This procedure has recently been generalized to cylindrical logs and trees having a constant slope of the grain. A further generalization is presented here to allow for the normal spiral grain pattern with radially changing slope of grain in wood under bark. Based on this theory, a measurement procedure is proposed for the detection of wood grain angle with radial dependence, requiring no information on moisture content in the sapwood, also applicable for completely or partially frozen wood. A suitable application would be an instrument to use in the forest for measurements on living trees or logs.
This paper provides a general framework for electromagnetic modeling, computation and measurements regardingthe wave propagation characteristics of High-Voltage Direct Current (HVDC) power cables.The modeling is focused on very long (10 km or more) HVDC power cables andthe relevant frequency range is therefore in the low-frequency regime of about 0-100 kHz.An exact dispersion relation is formulated together with a discussion on practical aspectsregarding the computation of the propagation constant and the related characteristic impedance.Experimental time-domain measurement data from an 80 km long HVDC power cable is used to validate the model.It is concluded that a single-mode transmission line model is not adequate to account for the mismatch between the power cableand the instrumentation.A mismatch calibration procedure is therefore devised to account for the connection between the measurement equipmentand the cable. A dispersion model is thus obtained that is accurate for early times of pulse arrival.To highlight the potential of accurate electromagnetic modeling, an example of high-resolution length-estimation is discussedand analyzed using statistical methods based on the Cram\'{e}r-Rao lower bound.The analysis reveals that the estimation accuracy based on the present model (and its related model error)is in the order of 100 m for an 80 km long power cable, and that the potential accuracy using a ``perfect'' model based on the givenmeasurement data is in the order of centimeters.
An adjoint field approach is used to formulate a general numerical framework for Fisher information based sensitivity analysis in electrical impedance tomography. General expressions are given for the gradients used in standard least squares optimization, i.e., the Jacobian related to the forward problem, and it is shown that these gradient expressions are consistent with commonly used electrode models such as the shunt model and the complete electrode model. By using the adjoint field formulations together with a variational analysis, it is also shown how the computation of the Fisher information can be integrated with the gradient calculations used for optimization. It is furthermore described how the Fisher information analysis and the related sensitivity map can be used in a preconditioning strategy to obtain a well balanced parameter sensitivity and improved performance for gradient based quasi-Newton optimization algorithms in electrical impedance tomography. Numerical simulations as well as reconstructions based on experimental data are used to illustrate the sensitivity analysis and the performance of the improved inversion algorithm in a four-electrode measurement set-up.
An adjoint field approach is used to formulate a general numerical framework for Fisher information-based sensitivity analysis in electrical impedance tomography. General expressions are given for the gradients used in standard least-squares optimization, i.e. the Jacobian related to the forward problem, and it is shown that these gradient expressions are compatible with commonly used electrode models such as the shunt model and the complete electrode model. By using the adjoint field formulations together with a variational analysis, it is also shown how the computation of the Fisher information can be integrated with the gradient calculations used for optimization. It is furthermore described how the Fisher information analysis and the related sensitivity map can be used in a preconditioning strategy to obtain a well-balanced parameter sensitivity and improved performance for gradient-based quasi-Newton optimization algorithms in electrical impedance tomography. Numerical simulations as well as reconstructions based on experimental data are used to illustrate the sensitivity analysis and the performance of the improved inversion algorithm in a four-electrode measurement set-up.
This paper presents an asymptotic analysis of non-discrete radiating modes with applications in waveguide theory. As a main application, the radiating modes of an open waveguide structure with circular geometry is considered. A generalized Jordan's lemma is used to justify that field components can be calculated as the sum of discrete and non-discrete modes, that is, as the sum of residues of poles and an integral along the branch-cut defined by the transversal wavenumber of the exterior domain. An asymptotic expression is derived for field components at large distance along the waveguide and supplemented with rigorous upper and lower error bounds. A numerical example regarding the axial symmetric 0th order transverse magnetic modes of a thin copper wire in water is included to demonstrate that there may be a non-trivial balance between the contributions from discrete and non-discrete modes.
This paper presents a detailed modeling and analysis regarding the dispersion characteristics ofmultilayered open coaxial waveguides or cables. The electromagnetic model is based on a layer recursive computation of axial-symmetric fields in connection with a magnetic frill generator excitation that can be calibrated to the current measured at the input of the cable. The layer recursive formulation enables a stable and efficient numerical computation of the related dispersion functions, as well as a detailed analysis regarding the analytic and asymptotic properties of the associated determinants. Modal contributions as well as the contribution from the associated branch-cut (nondiscrete radiating modes) are defined and analyzed. Measurements and modeling of pulse propagation on an 82-km-long HVDC power cable are presented as a concrete example. In this example, it is concluded that the contribution from the dominating axial-symmetric transverse magnetic mode is sufficient, and that the contribution from the branch-cut is negligible for all practical purposes, and in particular if the exterior domain is lossy. The main contribution of this paper is to provide the necessary modeling and analysistools for a quantitative study of these phenomena.
An optimal plasmonic resonance is derived for small homogeneous and isotropic inclusions in a lossy surrounding medium. The optimal resonance is given in terms of any particular eigenmode (electrostatic resonance) associated with the double-layer potential for a smooth, but otherwise arbitrary surface.
This paper presents a study of the physical limitations for radio frequency absorption in gold nanoparticle (GNP) suspensions. A spherical geometry is considered consisting of a spherical suspension of colloidal GNPs characterized as an arbitrary passive dielectric material which is immersed in an arbitrary lossy medium. A relative heating coefficient and a corresponding optimal near field excitation are defined, taking the skin effect of the surrounding medium into account. The classical Mie theory for lossy media is also revisited, and it is shown that the optimal permittivity function yielding a maximal absorption inside the spherical suspension is a conjugate match with respect to the surrounding lossy material. A convex optimization approach is used to investigate the broadband realizability of an arbitrary passive material to approximate the desired conjugate match over a finite bandwidth, similar to the approximation of a metamaterial. A narrowband realizability study shows that for a surrounding medium consisting of a weak electrolyte solution, the electromagnetic heating, due to the electrophoretic (plasmonic) resonance phenomena inside the spherical GNP suspension, can be significant in the microwave regime, provided that the related Drude parameters can be tuned into (or near to) resonance. As a demonstration, some realistic Drude parameters are investigated concerning the volume fraction, mass, and friction constant of the GNPs. The amount of charge that can be accommodated by the GNPs is identified as one of the most important design parameters. However, the problem of reliably modelling, measuring and controlling the charge number of coated GNPs is not yet fully understood, and is still an open research issue in this field. The presented theory and related physical limitations provide a useful framework for further research in this direction. Future research is also aimed at an expansion towards arbitrary suspension geometries and the inclusion of thermodynamical analysis.
Classical homogenization theory based on the Hashin–Shtrikman coated ellipsoids is used to model the changes in the complex valued conductivity (or admittivity) of a lung during tidal breathing. Here, the lung is modeled as a two-phase composite material where the alveolar air-filling corresponds to the inclusion phase. The theory predicts a linear relationship between the real and the imaginary parts of the change in the complex valued conductivity of a lung during tidal breathing, and where the loss cotangent of the change is approximately the same as of the effective background conductivity and hence easy to estimate. The theory is illustrated with numerical examples based on realistic parameter values and frequency ranges used with electrical impedance tomography (EIT). The theory may be potentially useful for imaging and clinical evaluations in connection with lung EIT for respiratory management and control.
The Fisher Information Integral Operator (FIO) and related sensitivity analysis is formulated in a variational framework that is suitable for analytical Green's function and gradient-based approaches in microwave tomography. The main application considered here is for parameter sensitivity analysis and related preconditioning for gradient-based quasi-Newton inverse scattering algorithms. In particular, the Fisher information analysis can be used as a basic principle yielding a systematic approach to robust preconditioning, where the diagonal elements of the FIO kernel are used as targets for sensitivity equalization. The infinite-dimensional formulation has several practical advantages over the finite-dimensional Fisher Information Matrix (FIM) analysis approach. In particular, the FIO approach avoids the need of making a priori assumptions about the underlying discretization of the material such as the shape, orientation and positions of the assumed image pixels. Furthermore, the integral operator and its spectrum can be efficiently approximated by using suitable quadrature methods for numerical integration. The eigenfunctions of the integral operator, corresponding to the identifiable parameters via the significant eigenvalues and the corresponding Cramr-Rao bounds, constitute a suitable global basis for sensitivity and resolution analysis. As a generic numerical example, a two-dimensional inverse electromagnetic scattering problem is analysed and illustrates the spectral decomposition and the related resolution analysis. As an application example in microwave tomography, a simulation study has been performed to illustrate the parameter sensitivity analysis and to demonstrate the effect of the related preconditioning for gradient-based quasi-Newton inverse scattering algorithms.
This paper presents a systematic approach to robust preconditioning for gradient-based nonlinear inverse scattering algorithms. In particular, one- and two-dimensional inverse problems are considered where the permittivity and conductivity profiles are unknown and the input data consist of the scattered field over a certain bandwidth. A time-domain least-squares formulation is employed and the inversion algorithm is based on a conjugate gradient or quasi-Newton algorithm together with an FDTD-electromagnetic solver. A Fisher information analysis is used to estimate the Hessian of the error functional. A robust preconditioner is then obtained by incorporating a parameter scaling such that the scaled Fisher information has a unit diagonal. By improving the conditioning of the Hessian, the convergence rate of the conjugate gradient or quasi-Newton methods are improved. The preconditioner is robust in the sense that the scaling, i.e. the diagonal Fisher information, is virtually invariant to the numerical resolution and the discretization model that is employed. Numerical examples of image reconstruction are included to illustrate the efficiency of the proposed technique.
An electromagnetic analysis is presented for experiments with strong permanent disc magnets. The analysis is based on the well known experiment that demonstrates the effect of circulating eddy currents by dropping a strong magnet through a vertically placed metal cylinder and observing how the magnet is slowly falling through the cylinder with a constant velocity. This experiment is quite spectacular with a super strong neodymium magnet and a thick metal cylinder made of copper or aluminum. A rigorous theory for this experiment is provided based on the quasi-static approximation of the Maxwell equations, an infinitely long cylinder (no edge effects) and a homogeneous magnetization of the disc magnet. The results are useful for teachers and students in electromagnetics who wish to obtain a deeper insight into the analysis and experiments regarding this phenomenon, or with industrial applications such as the grading and calibration of strong permanent magnets or with measurements of the conductivity of various metals, etc.. Several experiments and numerical computations are included to validate and to illustrate the theory.
This paper presents a Fisher information based Bayesian approach to analysis and design of the regularization and preconditioning parameters used with gradient based inverse scattering algorithms. In particular, a one-dimensional inverse problem is considered where the permittivity and conductivity profiles are unknown and the input data consist of the scattered field over a certain bandwidth. A priori parameter modeling is considered with linear, exponential and arctangential parameter scalings and robust preconditioners are obtained by choosing the related scaling parameters based on a Fisher information analysis of the known background. The Bayesian approach and a principal parameter (singular value) analysis of the stochastic Cramer-Rao bound provide a natural interpretation of the regularization that is necessary to achieve stable inversion, as well as an indicator to predict the feasibility of achieving successful reconstruction in a given problem set-up. In particular, the Tikhonov regularization scheme is put into a Bayesian estimation framework. A time-domain least-squares inversion algorithm is employed which is based on a quasi-Newton algorithm together with an FDTD-electromagnetic solver. Numerical examples are included to illustrate and verify the analysis.
A statistical signal analysis for the inverse source problem of electromagnetics is given. We consider the problem of estimating either the near field or the radiating current distribution from a measurement of the far field. The solution is derived via a linear operator formalism, and the ill-posedness of the reconstruction is quantified by using the Cramer-Rao lower bound which is explicitly given in terms of the multipole expansion of the electromagnetic field. A numerical study is included to illustrate the theoretical results.
This paper presents a c ylindrical multipole expansion for periodic sources with applications for three-phase power cables.It is the aim of the contribution to provide some analytical solutions and techniques that can be useful in the calculation ofcable losses. Explicit analytical results are given for the ﬁelds generated by a three-phase helical current distribution andwhich can be computed efﬁciently as an input to other numerical methods such as, for example , the Method of Moments.It is shown that the ﬁeld computations are numerically stable at low frequencies (such as 50 Hz) as well as in the quasi-magnetostatic limit provided that sources are divergence-free. The cylindrical multipole expansion is fur thermore usedto derive an efﬁcient analytical model of a measurement coil to measure and estimate the complex valued permeability ofmagnetic steel armour in the presence of a strong skin-effect.
This paper provides a mathematical framework for Fisher information analysis forinverse problems based on Gaussian noise on infinite-dimensional Hilbert space. The covariance operator for the Gaussian noise is assumed to be trace class, andthe Jacobian of the forward operator Hilbert-Schmidt. We show that the appropriatespace for defining the Fisher information is given by the Cameron-Martin space. This is mainly because the range space of the covariance operator always is strictlysmaller than the Hilbert space. For the Fisher information to be well-defined, it is furthermore required that the range space of the Jacobian is contained in the Cameron-Martin space. In order for this condition to hold and for the Fisher information tobe trace class, a sufficient condition is formulated based on the singular values ofthe Jacobian as well as of the eigenvalues of the covariance operator, together withsome regularity assumptions regarding their relative rate of convergence. An explicit example is given regarding an electromagnetic inverse source problem with “external”spherically isotropic noise, as well as “internal” additive uncorrelated noise.
This paper presents a convex optimization approach to study optimal realizations of passive electromagnetic structures. The optimization approach complements recently developed theory and techniques to derive sum rules and physical limitations for passive systems operating over a given bandwidth. The sum rules are based solely on the analytical properties of the corresponding Herglotz functions. However, the application of sum rules is limited by certain assumptions regarding the low- and high-frequency asymptotic behavior of the system, and the sum rules typically do not give much information towards an optimal realization of the passive system at hand. In contrast, the corresponding convex optimization problem is formulated to explicitly generate a Herglotz function as an optimal realization of the passive structure. The procedure does not require any additional assumptions on the low- and high frequency asymptotic behavior, but additional convex constraints can straightforwardly be incorporated in the formulation. Typical application areas are concerned with antennas, periodic structures, material responses, scattering, absorption, reflection, and extinction. In this paper, we consider three concrete examples regarding dispersion compensation for waveguides, passive metamaterials and passive radar absorbers.
This paper provides a quantitative analysis of the optimal accuracy and resolution in electrical impedance tomography (EIT) based on the Cramér–Rao lower bound. The imaging problem is characterized by the forward operator and its Jacobian. The Fisher information operator is defined for a deterministic parameter in a real Hilbert space and a stochastic measurement in a finite-dimensional complex Hilbert space with a Gaussian measure. The connection between the Fisher information and the singular value decomposition (SVD) based on the maximum likelihood (ML) criterion (the ML-based SVD) is established. It is shown that the eigenspaces of the Fisher information provide a suitable basis to quantify the trade-off between the accuracy and the resolution of the (nonlinear) inverse problem. It is also shown that the truncated ML-based pseudo-inverse is a suitable regularization strategy for a linearized problem, which exploits sufficient statistics for estimation within these subspaces. The statistical-based Cramér–Rao lower bound provides a complement to the deterministic upper bounds and the L-curve techniques that are employed with linearized inversion. To this end, electrical impedance tomography provides an interesting example where the eigenvalues of the SVD usually do not exhibit a very sharp cut-off, and a trade-off between the accuracy and the resolution may be of practical importance. A numerical study of a hypothetical EIT problem is described, including a statistical analysis of the model errors due to the linearization.