Green's Function

The Green's function method has applications in several fields in Physics, from classical differential equations to quantum many-body problems. In the quantum context, Green's functions are correlation functions, from which it is possible to extract information from the system under study, such as the density of states, relaxation times and response functions. Despite its power and versatility, it is known as a laborious and sometimes cumbersome method. Here we introduce the equilibrium Green's functions and the equation-of-motion technique, exemplifying the method in discrete lattices of non-interacting electrons. We start with simple models, such as the two-site molecule, the infinite and semi-infinite one-dimensional chains, and the two-dimensional ladder. Numerical implementations are developed via the recursive Green's function, implemented in Julia, an open-source, efficient and easy-to-learn scientific language. We also present a new variation of the surface recursive Green's function method, which can be of interest when simulating simultaneously the properties of surface and bulk.

In this paper, Green's function and surface current density for planar structure has been calculated. The approach makes use of the popular and rigorously used spectral domain full wave analysis method in conjunction with method of moment as numerical analysis tool. In present approach, boundary conditions are applied at patch metallization, which leads to integral equation with the involvement of green's function in spectral domain, which includes the effect of dielectric, conductor loss, surface wave modes and space wave radiation. By applying Galerkin's moment method integral equation are transformed to linear set of equations. Entire domain basis function is used to improve the efficiency of the solution.

Research in solid-state nanophotonics and quantum optics has been recently pushing the limits in semiconductor microcavity design. High quality microcavities that confine light into small volumes are now able to drastically alter the local density of states (LDOS). Plasmonic systems can provide smaller effective confinements, however it is unclear if the benefits of confinement are good enough to balance material losses due to non-radiative
processes. This thesis presents a compendium of techniques for calculating photonic Green functions in various lossy, inhomogeneous magneto-dielectric systems. Subsequently we derive a rigorous theory of quantum light-matter interactions, valid in both weak and strong coupling limits, and show how the classical photonic Green function is developed to calculate Purcell factors, Lamb shifts, and the near and far field spectra from a single photon emitter. Using these techniques, this work investigates the classical and quantum optical properties of a variety of inhomogeneous structures, including their coupling to single photon emitters. This includes examining Purcell factors above negative index slabs and showing the convergence of many slow-light modes leads to a drastic increase in the LDOS along with large Lamb shifts. The optical trapping of metallic nanoparticles is examined
above a negative index slab and a silver half-space, showing the importance of interparticle coupling on the optical forces. Then the interaction between a quantum dot and a metallic nanoparticle is studied where far-field strong coupling effects are observed only when the metallic nanoparticle is considered beyond the dipole approximation. Finally, this work addresses the issue of the LDOS diverging in lossy materials, which necessitates a description of spontaneous emission beyond the dipole approximation; the “local field problem” in quantum optics is revisited and generalized to include local field corrections for use in any photonic medium. The strength of finite-difference time-domain techniques is demonstrated in a number of cases for the calculation of regularized Green functions in lossy inhomogeneous media. This thesis presents a comprehensive study of Green function approaches to model classical and quantum light-matter interactions in arbitrary nanophotonic structures, including quantum dots, semiconductor microcavities, negative index waveguides, metallic half-spaces and metallic nanoparticles.

We show explicitly how the commonly adopted prescription for calculating effective mode volumes is wrong and leads to uncontrolled errors. Instead, we introduce a generalized mode volume that can be easily evaluated based on the mode calculation methods typically applied in the literature, and which allows one to compute the Purcell effect and other interesting optical phenomena in a rigorous and unambiguous way.

Seismic survey is a program for mapping geological structure by observation of seismic wave. Creating seismic wave with artificial sources and observing the arrival time of the waves reflected from acoustic impedance contrasts or refracted through high velocity members. Real-time vertical seismic profile (VSP) imaging provides information ahead of the drill bit that confirmed the well trajectory. A key objective was to determine the best location for the seismic source while drilling. Geological bodies like salt causes ray bending which leads to higher amplitude reflections between sediment layers and causes multiple rays scattering in the data collected. Conventional source location and interpolation methods heavily rely on the accuracy of the velocity models. In this paper interferometric techniques will be use to extract multiple scatter information in ocean bottom seismic data which will provide reliable results without knowledge of the velocity models. The sparse ocean bottom seismic data are interferomatically correlated with the model -based Green's functions for the sea floor model. This generates densely recorded ocean bottom seismic traces. Local matching filter will be applied to reduce the artifacts in the interpolated data. To mitigate the artifacts in the interferometry correlation results, an anti -aliasing theory developed will establish a fundamental criterion for numerical application of seismic interferometry to any seismic data. Mathematical Time Revisal Mirror models will be developed in MATLAB to determine the position of seismic source by comparing the phases of signals at the output terminals. Simulation will also be run using the data generated to plot a two dimensional ocean bottom seismic profile (OBS). Hence seismic interferometry can be applied during exploration for hydrocarbon and during down hole drilling to provide reliable information about complex geology.

The scattering of guided waves by a conductive post placed inside a rectangular waveguide with a step discontinuity on its sidewalls is analyzed through an integral-equation formulation. The interaction between post and discontinuity is examined with use of the Green's function. Numerical computations are carried out based on the mode-matching technique, providing a stable and accurate solution. An equivalent-circuit model is developed from the electromagnetic analysis and the computed scattering -parameters. The developed solution accuracy is validated with simulation software. became an Associate Professor with the Department of Electrical Engineering, NTUA, and in 1987, he became a Professor. He has authored or coauthored over 120 papers in refereed international journals, and has authored three books in Greek on microwaves, fiber-optics telecommunications, and radar systems. His research interests include electromagnetic scattering, propagation of electromagnetic waves, fiber-optic telecommunications, and high-speed circuits operating at gigabit/second rates. Since 1988, he has been the national representative of Greece to the Technical Telecommunication Committee, European Cooperation in the Field of Scientific and Technical Research (COST), and has actively participating in several COST telecommunications projects.

PACS 71.15.Mb, 78.20.Bh Theoretical approaches for ab initio studies of the electronic and optical properties of matter are here reviewed. Examples within Density Functional Theory, Many-Body perturbation Theory and Time Dependent Density Functional Theory are presented and discussed, pointing out advantages and drawbacks of the different schemes.

The coexistence of BCS superconductivity and itinerant ferromagnetism in uranium based intermetallic systems is analyzed using a Hubbard Hamiltonian. To obtain the superconducting transition temperature C T and Curie temperature FM T , we used the Green's function method. The order parameter of superconductivity (∆) and ferromagnetism () I or m are obtained in the mean field approximation. It is found that there generally exist coexistent solutions to coupled equations of the order parameter in the temperature range () FM C T T T , min 0 < <. In our model, ferromagnetism is itinerant and therefore carried by the conduction electrons. This arises from a splitting of the spin-up and spin-down band. A consequence is that the ferromagnetism and superconductivity is carried by same electrons. Expressions for specific heat, energy spectra and density of states are derived. The specific heat has linear temperature dependence as opposed to that of the exponential decrease in the BCS theory. The density of states for a finite magnetic order parameter increases as opposed to that of a ferromagnetic metal. The theory is applied to explain the observations in uranium based intermetallic compoundUCoGe. The agreement between theory and experiments is quite encouraging.

The heat transfer analysis of concrete-filled steel circular hollow sections (CHS) subjected to fire is essentially classified as the process of deriving a solution for transient heat conduction in a composite medium in a cylindrical coordinate system. In this paper, an efficient numerical approach based on an analytical Green's function (GF) solution is developed, where the spatial variable is analytically tractable without recourse to spatial discretization. The thin steel layer is conveniently. treated as a lumped heat capacitance with uniform temperature distribution to avoid tedious analytical solutions for hollow cylinders. A computational model is developed based on Duhamel's principle, incorporating time-varying fire conditions. The choice of sampling time is not constrained by numerical stability and can be as large as 30 min in one time step. The numerical approach can be used to predict the temperature field inside the entire composite domain and the heat flux at the fire and the steel-concrete interfaces, as well as to verify more sophisticated numerical codes, e.g. finite element codes, of heat transfer.

We investigate the quantum optical properties of a quantum-dot dipole emitter coupled to a finite-size metal nanoparticle using a photon Green-function technique that rigorously quantizes the electromagnetic fields. We first obtain pronounced Purcell factors and photonic Lamb shifts for both a 7- and 20-nm-radius metal nanoparticle,without adopting a dipole approximation.We then consider a quantum-dot photon emitter positioned sufficiently near the metal nanoparticle so that the strong-coupling regime is possible. Accounting for nondipole interactions, quenching, and photon transport from the dot to the detector,we demonstrate that the strong-coupling regime should be observable in the far-field spontaneous emission spectrum, even at room temperature. The vacuum-induced emission spectra show that the usual vacuum Rabi doublet becomes a rich spectral triplet or quartet with two of the four peaks anticrossing, which survives in spite of significant nonradiative decays. We discuss the emitted light spectrum and the effects of quenching for two different dipole polarizations.

We present a study of light-induced forces between two coupled plasmonic nanoparticles above various slab geometries including a metallic half-space and a negative index material (NIM) slab waveguide. We investigate optical forces by nonperturbatively calculating the scattered electric field via a Green function technique which includes the particle interactions to all orders. For excitation frequencies near the surface plasmon polariton and slow-light waveguide modes of the metal and NIM, respectively, we find rich light-induced forces and significant dynamical back-action effects. Optical quenching is found to be important in both metal and NIM planar geometries, which reduces the spatial range of the achievable interparticle forces. However, reducing the loss in the NIM allows radiation to propagate through the slow-light modes more efficiently, thus causing the light-induced forces to be more pronounced between the two plasmonic particles. To highlight the underlying mechanisms by which the particles couple, we connect our Green function calculations to various familiar quantities in quantum optics.

1] The Green's function of a parallel plate waveguide with cascaded step discontinuities is computed by using analytical methods. The expression is written as a sum of two terms, the first of which corresponds to the effect of the horizontal plates on the primary source's field taking into account the symmetric images with respect to the metallic boundaries. The second term is related to the reflections from the vertical discontinuity edges, which are treated by applying successively a mode-matching technique. The solution of the posed boundary value problem is obtained through a well-conditioned linear system. The behavior of the guided waves far from the discontinuity region is determined with the help of far field approximations. Numerical computations for specific cases are presented and examined and many general purpose procedures of educational and tutorial interest are extensively described.

A genuinely two-dimensional discretization of general drift-diffusion (including incompressible
Navier-Stokes) equations is proposed. Its numerical fluxes are derived by computing
the radial derivatives of “bubbles” which are deduced from available discrete data by exploiting the
stationary Dirichlet-Green function of the convection-diffusion operator. These fluxes are reminiscent
of Scharfetter-Gummel’s in the sense that they contain modified Bessel functions which allow to
pass smoothly from diffusive to drift-dominating regimes. For certain flows, monotonicity properties
are established in the vanishing viscosity limit (“asymptotic monotony”) along with second-order
accuracy when the grid is refined. Practical benchmarks are displayed to assess the feasibility of the
scheme, including the “western currents” with a Navier-Stokes-Coriolis model of ocean circulation.

A new 3-D analytical model of subthreshold current based on the three dimensional analytical solution of Poisson's equation with suitable boundary conditions of enhancement mode fully depleted SOI-MOSFET is presented in this paper. The effect of channel length, width and doping concentration for uniformly doped SOI MOSFET has been investigated. The results obtained from the proposed model have been compared for validation with ATLAS device simulator.

The application of Fourier series to the heat conduction on a circular ring is considered. The temperature distribution function u(θ;t)and so u(θ;t)=u(θ+2πn;t), where n is an integer, positive or negative.  It is periodic as after integer number of revolutions, we return to the same position on the circle.  u(θ;t) satisfies diffusion equation. This is given by the convolution of the initial distribution at time t=0,  viz.,u(x;0) and heat kernel  G(x-x^';t). This kernel is also called propagator or Green’s function in different context. This heat kernel is identified as Jacobi ϑ_3 (πx:2πt),  which is periodic in x and t. The physical significance of the kernel is explained clearly.  The extraction of heat kernel as the impulse response for an initial ‘delta heating’ on a very small part of the ring and how it can be measured are clearly explained.

Green's function approach provides a new possibility of analytical procedure for heat transfer analysis of structures in fire. In many cases, it has been applied successfully to predict the temperature field. Heat transfer analysis of structure members protected by intumescent paint is complicated by active expansion of the base coat at elevated temperature. Because of this complex thermal behavior of intumescence, as well as commercial concern about the release of information, very little research has been reported in the literature. In this article, a Green's function method based on a one-dimensional reduced heat transfer model is presented. The highly nonlinear thermal conductivity of the intumescent paint is simplified to be temperature-independent to yield analytically tractable solutions of heat equations. The proposed approach is verified against the experimental investigation as well as finite-element analysis.

The electromagnetic field excited by an electric line source in the presence of a printed structure periodic along two directions with arbitrary geometry of the metallization is calculated. The Array Scanning Method (ASM) is adopted to express the field excited by the aperiodic source in terms of an integral superposition of suitable auxiliary Floquet-periodic fields. The latter are solutions of electric-field integral equations that are cast in a mixed-potential formulation in the unit cell and are numerically discretized with an efficient implementation of the method of moments in the spatial domain. The accuracy of the proposed approach is validated through an independent ASM code developed for metal-strip gratings and also against consolidated analytical models available for specific strip-and patch-based periodic structures.

A ring source of arbitrary current backed by a perfectly conducting sphere is analyzed through Green's function formulation. The infinite double sum of the Green's function is written in terms of a single series by performing a transformation of the coordinate system. The resulting form is used for the numerical evaluation of the scattering integral. The operation of the coupled loop-sphere structure is understood via the discussion of several numerical results.

The problem we study cane be defined as follows: In three-dimensional space, we consider two homogeneous media - "air" and "water" - separated by a plane interface. There is a point source of sound in "air" at height h above the interface. Its emission is given by a function F(t) of time. We are interested in the behavior of pressure in water, at time t, depth z, and distance r from a vertical line going through the source.
This problem has been discussed innumerable times, both for monochromatic and transient sources [1-6]. The feature of interest to us is the existence, first recognized in the monochromatic regime, of contributions to the total solution called lateral waves. They tend to be concentrated near the interface, and have properties of propagation and attenuation different from waves in homogeneous media. They decrease exponentially with depth, and their attenuation length is frequency-dependent. Consequently, their penetration depth decreases with frequency. For a recent discussion [7], it should be mentioned that this attenuation does not correspond to absorption of energy, a lateral wave is merely a contribution to an elastic propagation.
The classical methods of resolution, while well adapted to the case of monochromatic sources, are less suited to the description of transients. This is due to the fact that waves of different frequencies both follow different paths and undergo different attenuations. In such time-and-frequency dependent situations, it is natural to apply wavelet transform techniques.
We shall first, briefly describe the time behavior of the WT  of the acoustic field at a fixed point under water.  (This is not the same as attempting to use wavelet techniques to solve partial differential equations of the propagation problem). It is straightforward to write down an expression for the propagator for our problem. This expression is much smoother than the propagator itself. and allows selective reconstitutions with arbitrary precision. A natural decomposition into three contributions appears corresponding to branch points of the integrand. In a second part,  we use the different contributions to obtain a formula for reconstruction of the time-dependent source.

Bulk modes (BM) are basis solutions to the Schro ̈dinger equation and they are useful in a number of physical problems. In the present work, we establish a complete set of BMs for graphene ribbons at arbitrary energy. We derive analytical expressions for these modes and systematically classify them into propagating or evanescent mode. We also demonstrate their uses in efficient electronic transport simulations of graphene-based electronic devices within both the mode-matching method and the Green’s function framework. Explicit constructions of Green’s functions for infinite and semi-infinite graphene ribbons are presented.

In this study, we present the exact solution of certain fractional partial differential equations (FPDE) by using a modified hom otopy perturbation method (MHPM).The exact soluti ons are constructed by choosing an appropriate initial approximation and only one term of the series obtai ned by MHPM. The exact solutions for initial value problems of F PDE are analy tically derived. The methods i ntroduced an efficient tool for solving a wide class of time-fractional partial differential equations.

An accurate numerical approach is developed in this paper for thermal analysis of contour-insulated concrete-filled circular hollow sections subjected to fire, based on analytical solutions of transient diffusion equations in radial coordinate system obtained using the Green's function approach. The steel layer is conveniently treated as a thin film with lumped heat capacitance, i.e. the temperature distribution inside the steel section is assumed to be uniform. Perfect contact conditions are assumed at both of the insulation-steel and the steel-concrete interfaces. Solutions of the heat equations are represented by large time series expansions obtained from homogeneous boundary conditions using the eigenfunction approach. The time-varying boundary conditions, i.e. the fire conditions are incorporated in terms of Stieltjes integral by means of Duhamel's theorem. Numerical models are developed using temporal discretization, while no spatial discretization is required since spatial variables are analytically tractable. The proposed numerical scheme is not restricted to any specific fire conditions, and can be used in conjunction with parametric fire as proposed in the Eurocode.

The purpose of this paper is to present a procedure for the estimation of the smallest eigenvalues and their associated eigenfunctions of nth order linear boundary value problems with homogeneous boundary conditions defined in terms of quasi-derivatives. The procedure is based on the iterative application of the equivalent integral operator to functions of a cone and the calculation of the Collatz-Wielandt numbers of such functions. Some results on the sign of the Green functions of the boundary value problems are also provided.

— We give well-posed statements of the main initial–boundary value problems in a rectangular domain and in a half-strip for a second-order parabolic equation that contains partial Riemann–Liouville fractional derivatives with respect to one of the two independent variables. We construct Green functions and representations of solutions of these problems. We prove existence and uniqueness theorems for the first boundary value problem and the problem in the half-strip with the boundary condition of the first kind.

We have calculated the optical absorption for InGaNAs and GaNSb using the band anticrossing (BAC) model and a
self-consistent Green’s function (SCGF) method. In the BAC model, we include the interaction of isolated and pair N
levels with the host matrix conduction and valence bands. In the SCGF approach, we include a full distribution of N
states, with non-parabolic conduction and light-hole bands, and parabolic heavy-hole and spin-split-off bands. The
comparison with experiments shows that the first model accounts for many features of the absorption spectrum in
InGaNAs; including the full distribution of N states improves this agreement. Our calculated absorption spectra for
GaNSb alloys predict the band edges correctly but show more features than are seen experimentally. This suggests
the presence of more disorder in GaNSb alloys in comparison with InGaNAs.

The calculation of the local density of states (LDOS) in lossy materials has long been disputed due to the divergence of the homogeneous Green function with equal space arguments. For arbitrary-shaped lossy structures, such as those of interest in nanoplasmonics, this problem is particularly challenging. A nondivergent LDOS obtained in numerical methods such as the finite-difference time-domain (FDTD) technique, at first sight appears to be wrong. Here we show that FDTD is not only an ideal choice for obtaining the regularized LDOS, but it can address the local-field problem for any lossy inhomogeneous material. We exemplify the case of a finite-size photon emitter (e.g., a single quantum dot) embedded within and outside a lossy metal nanoparticle and show excellent agreement with analytical results.

The Green function of the diffraction-radiation theory of time-harmonic waves under a moderate water depth is considered. The paper presents a newly developed enhanced algorithm based on Endo's approach, which employs a direct Gauss-Laguerre quadrature in the calculation of the principal value integrations. This methodology was firstly originated from Endo (1983), and had been subsequently improved by Liu et al. (2008) and Shan et al. (2019). Unfortunately, the latest version of Endo's approach still has two fatal problems with (1) incredibly large or small values at some "weird frequencies", and (2) nonsense values in the high-frequency region under a large depth. The present algorithm proposes special techniques for the removal of the singularities at these "weird frequencies", and provides special treatments to avoid exceeding hardware's limitation when the input parameter k 0 h is overlarge. The developed algorithm is thereafter tested against a variety of sea conditions, via a comparison with Newman's polynomial algorithm (Newman, 1985), certifying its good accuracy in various circumstances. By carrying out a benchmark test on the DeepCwind semisubmersible, through a comparison with Shan's algorithm (Shan et al., 2019) and the commericial software Hydrostar®, the cause of the occurrence of the "weird frequencies" is found, in association with other important findings. The computations demonstrate that the present enhanced algorithm is free from the preceding problems and is sufficiently accurate to compute wave loads in practice.

The 2-D time-domain Green's function of a graphene sheet is here derived, by assuming a local Drude-like model for the graphene conductivity valid in the absence of biasing magnetic fields and when both spatial-dispersion effects and interband terms are negligible (i.e., up to the low terahertz range). The sought Green's function is derived in a semi-analytical form through a modified Cagniard-De Hoop approach. This allows for deriving simple semi-analytical expressions for the fields radiated by a pulsed line source in the presence of a graphene sheet, which can be computed in a fast and straightforward way. Theoretical and numerical validations are presented by obtaining the known results for nondispersive metallic sheets as limiting cases and through comparisons with results obtained numerically through an exact canonical double inverse Fourier transform.

We present a self-consistent Green’s function (SCGF) approach for the Anderson
many-impurity model to calculate the band dispersion and density of states near the
conduction band edge in GaNx As1−x dilute nitride alloys. Two different models of the N states
have been studied to investigate the band structure of these materials: (1) the two-band model,
which assumes all N states have the same energy, EN ; (2) a model which includes a full
distribution of N states obtained by allowing for direct interaction between N sites. The
density of states, projected onto extended and localised states, calculated by the SCGF
two-band model, are in excellent agreement with those previously obtained in supercell
calculations and reveal a gap in the density of states just above EN , in contrast with the results
of previous non-self-consistent Green’s function calculations. However, including the full
distribution of N states in a SCGF calculation removes this gap, in agreement with
experiment.

We present a mean-field theory for charged polymer chains in an external electrostatic field in the weak and strong coupling limits. We apply the theory to describe the statistical mechanics of flexible polyelectrolyte chains in a hexagonal columnar lattice of stiff cylindrical macroions, such as DNA, in a bathing solution of a uni-univalent salt (e.g. NaCl). The salt effects are first described in the Debye-Hückel framework. This yields the macroion electrostatic field in the screened Coulomb form, which we take to represent the mean field into which the chains are immersed. We introduce the Green's function for the polyelectrolyte chains and derive the corresponding Edwards equation which we solve numerically in the Wigner-Seitz cylindrical cell using the ground state dominance ansatz. The solutions indicate the presence of polyelectrolyte bridging, which results in a like-charge attraction between stiff macroions. Then we reformulate the Edwards theory for the strong coupling case and use the standard Poisson-Boltzmann picture to describe the salt solution. We begin with the free energy which we minimize to obtain the Euler-Lagrange equations. The solutions yield self-consistently determined monomer density and electrostatic fields. We furthermore calculate the free energy density as well as the total osmotic pressure in the system. We again show that bridging implicates like-charge attractions of entropic origin between stiff cylindrical macroions. By analyzing the osmotic pressure we demonstrate that, in certain parts of the parameter space, a phase transition occurs between two phases of the same hexagonal symmetry.

In the present paper, we consider the problem on a transverse impact of a viscoelastic sphere upon an elastic Bernoulli-Euler beam in a viscoelastic medium, the viscoelastic features of which are described by certain viscoelastic operators. The Young's modulus of the impactor is the time-dependent operator, which is defined via the standard linear solid fractional derivative model, while the bulk modulus is considered to be constant. Within the contact domain the contact force is defined by the modified Hertzian contact law with the time-dependent rigidity function. For decoding the viscoelastic operators involving in the problem under consideration, the algebra of Rabotnov's fractional operators is employed. The integral equation is obtained for the contact force.

A tiny metallic cylinder placed into a planar dielectric waveguide scatters the field developed by a current-carrying skew strip centralized at the middle of the slab. Due to the small size of the scatterer, the induced surface current is taken independent of the azimuthal angle. The Green's function of the problem is expressed in closed form and it is inserted to the scattering integral after the polar equation of the strip has been determined. The behavior of nearfield quantities in the slab, with respect to geometrical and material parameters, is observed and examined.

The Ewald method is efficiently applied for the computation of the Green's function for a 3-D array of phased point sources. A best splitting parameter , which avoids the loss of accuracy occurring at high frequencies while maintaining rapidly converging properties, is derived. The loss of significant digits in the calculation of the modified spatial and spectral series involved in the Ewald summation technique can be set by the user, thus limiting the total error in the computation to a controllable level. Particular care is also given to the best way of summing the series that appear in the Ewald method and a new efficient algorithm for the reduction of the involved three-index series to one-index series is proposed, which allows for a further acceleration of the computation. Numerical results are presented to show the validity and efficiency of the proposed formulation.

We introduce the complex band structure and a medium-dependent (Green’s function) quantum-optics formalism to study the enhanced spontaneous emission factors and Lamb shifts from a quantum dot or atom near the surface of a slow-light metamaterial waveguide. Using a realistic loss factor of γ/2π=2 THz, Purcell factors of approximately 250 and 100 are found at optical frequencies for p-polarized and s-polarized dipoles, respectively, placed 28 nm (0.02λ0) above the slab surface. For smaller loss values, we demonstrate that the slow-light regime of odd metamaterial waveguide propagation modes can be observed and related to distinct resonances in the Purcell factors. Correspondingly, we predict unusually large and rich Lamb shifts of approximately −1 to −6 GHz for a dipole moment of 50 Debye. We also make a direct calculation of the far-field-emission spectra which contains direct measurable access to these enhanced Purcell factors and Lamb shifts.