Addendum to Research on Problems of Wave Processes: Diffractive Edge and Singularity, Facts and Ideas Derived from Analysis Part II

NOTE 1: You have arrived at the INVESTIGATION: ANALYSIS SECTION of this website.

This section is Under-Edit or Analysis Addition if necessary: Construction began on December 7, 2010 and was finished on April 28, 2011.

Update: February 16, 2012.

NOTE 2: This section contains the relevant facts and associated ideas that have been derived from each reference that is used in this investigation. They are presented in Summary form using the following SECTION GUIDE:

Item Number. Reference Description: Derived Fact (s) from the Reference Document [Reference No.] Derived Ideas.

NOTE 3: GTD: Geometrical Theory of Diffraction; PTD: Physical Theory of Diffraction

A8. L.A. Vainshtein's 1963 paper on Open Resonators for Lasers:

Institute of Physical Problems, Academy of Sciences of the U.S.S.R.

"Introduction":

Laser (quantum light generator): oscillating system, an open resonator (simplest case, two opposing plane parallel mirrors spaced apart)

A.G. Fox & T. Li (1961): first quantitative paper on theory of plane mirror resonators (theory required computations with fast computing machines

Important feature of open resonators:

Wavelength is much smaller than all of dimensions;
Spectrum of natural frequencies is sparse (as referenced to the volume of closed 
resonator); &
Wide applications (millimeter & submillimeter waves technology).

New theory of Open Resonators formed by plane mirrors & Resonators based on sections of waveguide with open ends

Key basis of theory (G.L. Suchkin, 1960):

"smallest of the radiation loss of an open resonator is due to the fact that the wave reaching the edge or end of the resonator is a guided wave of frequency only slightly higher than the critical frequency";

"such a wave scarcely emerges as radiation at all, but is reflected back with a reflection coefficient of absolute value close to unity."; &

but missing development in calculating the main characteristics of these resonators' main characteristics (frequency, radiative damping, field, & current distributions).

Vainshtein's approach is based on the Rigorous theory of diffraction at the open end of a waveguide

"Diffraction at an Open End of A Waveguide":

Plane Waveguide with an open end in the zx-plane (wave is independent of x coordinate)

The wave's wave number (k) is coupled with the distance (2a) separating the waveguide's plates:

ka = pi (q/2 + p), where q is a large integer & |p| < 1/2.

A wave H_0q or E_0q arrives at the open end of the guide:

Frequency is close to the critical frequency (for p = 0, these frequencies are equal).

Diffraction field F(w) of this wave is connected with surface density of current f(z) via a contour integral

F(w) is determined from functional equations for case z < 0 & z > 0 with the introduction of the L(w) function into a contour integral for z > 0

The wave number w_j is for the wave H_0q+2j or E_0q+2j in the plane waveguide:

Waves (propagating, j = 1, 2,…; damped, j = -1,-2, …; & arrived, j = 0).

L(w) = L₁(w)L₂(w), L₂(w) = L₁(-w)

An approximation solution for |w| << k & 2ak^-3|w|^4 << 1) is arrived at where:

F(w) is a complex function of w,w₀,A, L₁(w₀), L₂(w) parameters; &

f(z) is complex function of w,w₀, A, and a series of R_0,j component scaled by a complex exponential, where A is the current amplitude of the arriving wave & R_0,j reflection coefficient (R_0,0 is coefficient of the incident wave with respect to current).

Reflection & transformation coefficients for current: Figure 2, Absolute values & Figure 3, Phase:

for small values of p the wave with the index j = 0 undergoes strong reflection, with almost no transformation into the waves with the indices j = +/- 1; &

R_0,0 = - exp{iβ(1 + i) s₀},

where β = 0.824 (a function of the zeta function of Riemann) & s₀ = (4pi * p)^1/2.

Physical meaning of these results:

Propagating wave in a waveguide can be composed as the sum of two plane waves (two parallel rays (beams)

"If the frequency of the wave is close to the critical frequency, these rays make a small angle ε with the normal to the walls (Fig. 4) and therefore can easily be turned, owing to diffraction, through the angle twice ε, which results in the formation of a reflected wave. There is practically no transformation into waves with other indices, since in such a transformation the rays have to be turned through much large angles."

Also, R_0,0 reflection coefficient depends on the parameter s₀ that characterizes the phenomenon of "diffraction creep" of the reflected beam of rays in the waveguide:

where s₀ = ε(2ka)^1/2;

The transverse diffusion of this beam of rays occurs as they are reflected from each semi-infinite wall to the other wall located at 2a distance which forms the reflected wave from the turned rays.

Case for circular waveguide with an open end is presented & similar results are derived:

"We note that the vector character of electromagnetic waves does not manifest itself near the critical frequency (the same results are obtained)."

The derived asymptotic laws derives for plane or circular waveguide with an open end describes the reflection of waves of high index near their critical frequencies (ka >> 1) "gives satisfactory accuracy, which improves rapidly as ka increases, i.e., as the number of the wave increases."

"The asymptotic laws are not affected by the behavior of the current near a sharp edge, which is different for waves of different polarizations"

"Open Cylindrical Resonator (Open Tube)"

Applications of Open Tube using the Theory of Diffraction for open end of a tube:

ka << 1, an acoustical resonator with high figure of merit; &

ka >> 1, "good resonance properties for both electromagnetic and acoustical oscillations.

"Two-Dimensional Resonator Formed by Plane Mirrors"

"Open Resonator Formed by Rectangular Mirrors"

"Open Resonator Formed by Circular Mirrors"

"The Spectrum of Natural Frequencies":

The spectrum of the natural frequencies is sparser in Open Resonators than for closed resonators.

In the excitation of open resonator a considerable fraction of the power supplied is taken up by oscillations with large radiative damping (unavoidable radiative losses).

"Summary":

The theory of a natural oscillating system (Resonator) is developed so that its quantitative characteristics of oscillation (frequency, radiative damping, field, and current distributions) can be easily calculated & "admit an intuitive physical interpretation."

V.P. Bykov was involved in the development of this paper on open resonators.

Note that this paper references a number (6) of L.A. Vainshtein's works

[A8]

This 1963 paper by Professor Vainshtein is very important since it presents a simplified mathematical theory for modeling of Open Resonators for the purpose of generating a laser beam or millimeter or submillimeter waves.

This theory is based upon an extremely important idea that certain waves in the waveguide (whose frequency is near the critical frequency of the waveguide) are turned (reflected) at the open end (edge) by diffraction so that they go back into the waveguide as a reflected wave.

This paper's derivation is based upon the relationship between the induced surface current density on the waveguide's plate and the diffraction field. A key result for this modal reflection (diffraction) in the open-ended waveguide is the formulation of the reflection and transformation coefficients for current.

It is important to realize that this Diffraction of the incident modal wave's field is related to the induced surface current density of the plates of the Open-ended waveguide.

Figure 4 (a waveguide with angled parallel rays against the wall) and its accompanying text does not provide any physical explanation into why this reflection or turning (diffraction) of a "modal" wave occurs.

From an engineering point-of-view, Vainshtein's theory can be used to model on realistic computing machines and to build potentially powerful open resonators that could be used in a range of applications (military & commercial).

A9. L.A. Vainshtein's 1964 paper on Diffraction in Open Resonators & Open Waveguides with Plane Mirrors:

Institute of Physical Problems, Academy of Sciences of the U.S.S.R.

"Introduction":

Theory of resonators with plane mirrors was presented in a Vainshtein's 1963 paper from a "rigorous discussion of the diffraction at the open end of a plane waveguide."

This paper's purpose is to present the Open resonators' theory using M.A. Leontovich & V.A. Fok's work of the parabolic equation (1944 & 1946)

Open resonator: an oscillatory system whose dimensions are large as compared to the wavelength & has good resonance properties ("spare" natural modes has larger radiation losses)

Open waveguide: a transmission line whose dimensions are large as compared to the wavelength & only one wave (or a small waves) is transmitted efficiently, the rest filtered out by radiation losses.

Asymptotic Methods of Diffraction is applied: simple solution

"A Two-Dimensional Resonator":

Two dimensional problem: xz-plane, independent of y coordinate, plane mirrors -a < x < a, separated by 2l (z = +/- l), where the wave propagates in the x direction

Wave equation [Φ, a scalar function of W(x,z) & W(x,-z) where W is the wave amplitude & for wave number k] reduces to the Parabolic Equation [only transverse diffusion (x direction) of W]:

∂²W/∂x² + i2k∂W/∂z = 0;

A Green function G(x - x',z - z') is contained with this Parabolic Equation;

The function W(x,z) is the amplitude of a wave propagating in the +z direction & excited by the currents in the lower mirror z = -l;

This W(x,z) is expressed as an definite integral (|x'| < a) of the G() & W() functions;

The function f(ξ) is a definite integral (ranges from -M/2 to M/2) of f(ξ') scaled by a complex exponential, where the function f(ξ), is proportional to W(x,-l) is the current density on the lower mirror.

Note that ξ = (k/2l)^1/2 & M = (2ka^2/l)^1/2;

A key equation is kl = pi (q/2 + p) where q is a large integer (kl >> 1) & p is a complex parameter determining the frequency and radiation damping of the free mode;

A solution is arrived for the case of infinite limits (a = infinity) at that "corresponds approximately to waveguide modes propagating or being attenuated between the parallel planes z = +/- l & which are expanded into plane waves whose wave vectors (k_x, k_y, k_z) satisfy the conditions |k_x| << k, k_y = 0, k_z ≈ +/- k";

Also, a solution is arrived for the case of diffraction of a wave with index j = 0 at the open end of a plane waveguide consisting of parallel semi-infinite mirrors, z = +/- l, x > 0;

An indefinite integral equation (eqn 17) involving f(ξ') is developed for f(ξ) with the result that f(ξ) is a function of A, j = 0 traveling wave exponential and a series of R_0,j component scaled by a complex exponential;

Next, equations are derived for the unknowns R_0,j & R_0,0 that involve the Fresnel's integral;

Using one approach, R_0,0 = - F(-s₀) / F(s₀) indicates that the strong reflection of waveguide modes at a frequency near cutoff is caused by Fresnel diffraction and if p & s₀ are small,

R_0,0 = -exp[iβ(1 + i)s₀] with β = 2/(pi)^1/2 = 1.13;

Using the Wiener-Hopf-Fok method, eqn 17 is solved & one derives a slightly different R_0,0 value for β = 0.824; &

The parameter s₀ = (4pi * p)^1/2 "characterizes the field arising from the diffraction of the traveling wave at one of the semi-infinite mirrors forming the waveguide near the edge of the other mirror".

"The application of the Parabolic Equation means that the interaction of mirrors is taken into account within the framework of physical optics (Huygens-Fresnel principle).

Diffraction at small angles produces a equivalent results of physical optics & rigorous theory.

Physical consideration:

A wave (mode j = 0) of frequency close to cutoff (s₀ from it (reflection coefficient close to -1), this is repeated at the open end x = -a, and this repeats forever; &

"Waves of other numbers (j = +/- 1, +/- 2, …) arising at each reflection can be neglected to a first approximation since they have small amplitudes &, on reaching the opposite edge, are almost entirely radiated into space."

Next, consider the case where Φ is equal to one of the components of the vector potential (Φ = Ax or Φ = Ay)

A solution is obtained where W(x,z) and f(ξ) are derived (dependent on mode m, odd or even).

The modes for an open cavity are determined:

The radiation damping of free modes is determined by the imaginary part of p; &

"The surface current density drops at the edges to small values of order 1/M." As all components of the electromagnetic field are small at the edges of the resonator, it follows that the radiation loss is also small."

"Rectangular Mirrors"

"Circular Mirrors"

"An Open Waveguide with Plane Mirrors":

An open waveguide formed by periodically located parallel mirrors

Proposed by B.Z. Katsenelenbaum (1963) & N.G. Bondarenko & V.I.Talanov (1952)

Parabolic equation solution

Waves in this type of waveguide obey the same laws as the modes in an open resonator.

"Open Resonators and Open Waveguides":

The theories of open resonator & the open waveguide hold for mirrors of arbitrary (but identical) shapes.

"Conclusion":

Serious barrier to possible application of cavity resonators for millimeters & submillimeters: asymptotic Rayleigh-Jean law, crowding together of the natural frequencies

The modes in the open resonators are determined by three indices (m, n, & q) for E in x & y coordinates.

Modes in open waveguides are determined by two indices (m & n).

"The coincidence of the various modes is removed to a high degree in open resonators since radiation damping increases rapidly as the indices m and n increase. The dominant mode may have a sufficiently small radiation loss."

Note that this paper references papers by M.A. Leontovich (1944, 1946 & 1958), V.A. Fok (with Leontovich, 1946 & 1958), and G.D. Malyuzhinets.

[A9]

The derivation in Professor Vainshtein's 1964 paper is based upon the application of the wave equation, appropriate boundary conditions, and the Parabolic Equation to the problem of Diffraction in an Open Resonator [parallel plane mirrors or waveguide]. Then, a relationship between the induced surface current density on the mirror(s) and the diffraction field is developed.

A key result for this modal reflection (diffraction) in the open-ended waveguide is the formulation of the reflection and transformation coefficients for current.

Also, the Wiener-Hopf-Fok method is applied to the diffraction problem to arrive at a similar solution (slightly different value for β, 0.824 vs 1.13).

Unfortunately, this paper does not provide any physical explanation of why the Open Resonator or Open-ended waveguide reflection or turning (edge diffraction) of a certain "modal" wave (whose frequency is near the critical frequency) occurs.

But this reflection of a certain modal wave (that occurs at a small angle with the mirror's normal) does not follow the normal law of reflection. Thus, it is called Diffraction.

It is important to realize that this Diffraction of the incident modal wave's field is related to the induced surface current density of the mirrors.

A10. D.S. Jones 1964 book on the Theory of Electromagnetism (EM):

"Boundary conditions and Uniqueness" (1.27):

Sommerfeld radiation conditions

"Reciprocity theorems" (1.32):

Theory of electrostatics, harmonic EM field

"Reciprocity theorem states the field produced at the 2nd source by the 1st source is the same as that produced at the 1st source by the second."

"Diffraction by Obstacles with Edges" (Chapter 9):

Arbitrary shape bodies with the presence of an edge (where the direction of the surface normal changes discontinuously)

Radius of curvature is zero at an edge so that no matter how high the frequency, it does not diffract waves in the some way as the region of glancing incidence on a spherical object.

"Uniqueness" (9.1):

Uniqueness Theorem (S surface, E electric field vector, H magnetic field vector)

If the scattered field satisfies the radiation condition and Surface integral of Cross Product of E & H* dot dS -> 0 as S contracts to edge, and if the obstacles are finite in size, there is one and only one solution to the diffraction problem

"Edge Condition" (9.2):

Physical interpretation of the Surface integral of Cross Product of E & H* dot dS -> 0 as S contracts to edge is that no energy is radiated from the edge.

Thus, a derived approximate solution that describes the field behavior must produce the result of the absence of edge radiation.

No charge is induced on the edge

"The Radiation from a Semi-infinite circular pipe" (9.10):

Reference to L.A. Vajhshtejn's 1948, 1949, and 1950 Russian papers

Wiener-Hopf technique

New feature is the solution of simultaneous Wiener-Hopf equations: no general method of solution, approximate solution

Reciprocity theorem is applied: "relation between the field produced inside the guide by an external source and that produced externally by an internal source."

Waveguide's Edge conditions

Incident modes of the guide & modes produced in the pipe by the plane wave

TE Mode Analysis: Singularities of integrand, contour of integration is deformed

"The reflection coefficients are significant at large distances from the open end only for the propagated modes."

TM Mode Analysis: "behavior of the reflected coefficients is qualitatively similar but they are very much larger than those for TE waves."

"TE waves radiate more energy outside the guide than TM waves."

Dr. Vajhshtejn has compared approximation results to the results derived from the Huygens principle:

Forward direction ("reasonable accurate") and backward direction ("hopelessly inadequate").

"The Perfectly Conducting" Strip (9.12)

"The Kontorowich-Lebedev Transform" (9.13)

"Application to the Wedge" (9.14):

Problem of the wedge with approximate solution:

"when a plane wave strikes a wedge a ray incident on the edge is diffracted onto a right-circular cone with axis the edge and of which one generator is the continuation of the incident ray. The existence of this cone has been confirmed experimentally at optical wavelengths by Savornin."

"Keller's Method" (9.22):

"at a straight edge the diffracted wave spreads on a right circular cone with axis the edge and the continuation of the incident ray as generator."

"It is now assumed that even if the edge is curved the same is true locally (Fig. 9.10). Now neighbouring rays of the same cone intersect each other on the edge so that the edge is a caustic of the diffracted rays."

[A10]

A11. P.Ya. Ufimtsev's 1968 paper on Current Waves in dipoles and parabolic equation method:

Published in 1968 (Moscow, Soviet Union/Russia)

Parabolic equation method:

Approximation method to find asymptotic solutions to wave equation problems; &

Used to find this paper's solutions to the Problem of diffraction by a thin cylindrical wire (dipole) and by a ribbon.

Properties of current waves in thin vibrators have become a recent study in the Theory of dipoles.

The sum of waves of different types (External electromotive force source waves; Primary boundary waves; & Multiple diffraction waves) determine the field along the surface of the conductor.

Current waves on a ribbon:

"…the parabolic equation method gives Sommerfeld's exact expression for primary boundary current wave excited by the plane wave on the ribbon."

In Conclusion, Ufimtsev states "The author thanks L.A. Vainshtein for formulating the problem and for his interest."

This Russian paper was translated by J. Berry.

[A11]

This Ufimtsev's paper is important because it indicates the following ideas:

i) Ufimtsev's study of another diffraction approximation method: Parabolic equation method;

ii) his work with Current Waves; and

iii) his dependency on L.A. Vainshtein's technical expertises and work on the theory of diffraction as indicated by the Conclusion section of this paper.

A12. T.B.A. Senior and P.L.E. Uslenghi's March 1971 paper on High-frequency backscattering:

High-frequency backscattering from a finite cone

Perfectly conducting, right circular cone with a flat base

High-frequency incident plane wave (field), E polarization

Geometrical theory of diffraction is applied to obtain the first two terms of the asymptotic expansion

Axial caustic

References 1958 work of Ufimtsev on Secondary diffraction of Electromagnetic wave by a disk

Appendix:

Diffraction matrix for a wedge (local edge geometry) for the problem of diffraction at the base of the cone;

Edge coincides with z axis, plane wave incident in plane perpendicular to z axis with electric (or magnetic vector parallel to the edge; &

but Keller (1957) set of base vectors applied here.

[A12]

A13. J. Meixner's 1972 paper on Edges & Electromagnetic (EM) field behavior:

Solution of diffraction problems at sharp edges of diffracting obstacle, EM field vector may become infinite

Order of this singularity is subject to the so-called edge condition

Edge condition: "states that the electromagnetic energy density must be integrable over any finite domain even if this domain contains singularities of the electromagnetic field. In other words, the electromagnetic energy in any finite domain must be finite."

Perfectly conducting surface with edge:

Near the edge, singular components of EM field are of the Order of p^-1/2 where p is the distance from the edge

Points: well-defined tangents along the edge (locally straight)

Space filled with wedges of homogeneous material with a common straight edge

Periodic fields

Complex mathematical derivation

Components of E and H parallel to the edge are finite

Figures 3 and 4 graphs ???

No experimental results

Electrostatic & magnetostatic cases: same behavior as in the electromagnetic case

Results valid if surfaces separated different materials do not carry surface charges.

[A13]

German scientist

Work funded by US Air Force.

A14. T.B.A. Senior and P.L.E. Uslenghi's November 1972 paper on Experimental Evidence to support the Geometrical Theory of Diffraction (GTD):

GTD (Keller, 1958 & 1962): diffracted rays, new kind of optical rays

Authors state that:

"The GTD is especially important in studying the scattering by edges because it cannot, at the present time, be replaced by other techniques in a satisfactory manner."

Experimental Setup (Figure 1):

He-Ne gas laser (wavelength = 0.6328 micrometer) produces a visible beam;
Laser beam is projected on edge of a razor blade;
Screen perpendicular to edge;

Model: Optical ray impinge edge of a wedge (sharp narrow-angle)

Observed intersection of cone of diffracted rays with screen (Figure 2):

Incident electric field linearly polarized in vertical direction(parallel to faces & 
perpendicular to edge); &
Photo of "intersection of the edge-diffraction cone with the screen."

Doctors Senior and Uslenghi assert that these observations are consistent with the predictions of GTD (cone of diffracted rays)

Note that the research for this paper was funded by the U.S. Air Force Office of Scientific Research.

[A14]

As of 1972, Doctors Senior and Uslenghi asserts that GTD is only the viable theory of scattering (diffraction) by edges.

=> one can interpret this paper in that they are informing the U.S. Air Force that GTD, not PTD, is a viable method for the design of stealth features of an aircraft.

Scientifically, a Question can be asked if this Optical Experiment's setup actually models & simulates an optical ray interacting with an edge:

Does a Gaussian irradiance distributed (finite width) laser beam model accurately an optical ray?

Does this laser beam's interaction with a razor's edge model a plane, cylindrical, or spherical transverse wave's interaction with a metallic edge?

A15. B.L. Moiseiwitsch's 1977 book on Integral Equations:

"Preface":

Mathematical Physics

Differential equations, boundary conditions imposed externally Integral equations, boundary conditions incorporated within formulation

Integral equation: solution of problem as an infinite series (Neumann expansion, successive terms arise from an iterative procedure, convergence)

Integral equations of the first kind - Laplace and Fourier transform

Hilbert transform - singular kernel

Integral transform - integral equations having Kernels of difference or convolution type

Theory of linear integral equations of 2nd kind - Volterra & Fredholm

Lebesgue square integrable functions - Hilbert space

Riemann & Lebesgue integration

"Classification of integral equations" (Chapter 1):

Fredholm equation of the first and second kind

Special type of kernel- Separable kernels

Square integrable functions and kernels

Singular integral equations

"Connection with differential equation" (Chapter 2):

Green's function

"Integral equation of the convolution type" (Chapter 3)

"Method of successive approximation" (Chapter 4):

Neumann series - solves integral equation of the 2nd kind; iterative procedure- sequence of approximations; infinite series solution

"Integral equations with singular kernels" (Chapter 5):

Generalization to higher dimensions

Green's function in 2 & 3 dimensions

Dirichlet's problem

"The resolvent" (Chapter 8):

Resolvent kernel

"Fredholm theory" (Chapter 9)

"Bibliography":

V.I. Smirnov: 1964 book on Integral Equations

[A15]

A16. A.D. Rawlins' 1977 paper on Diffraction by a Dielectric Wedge:

Electromagnetic (EM) Diffraction by Dielectric Wedge:

Physical Situation: "Radio propagation through warm air front over the sea".

Approximate expressions for field solution of boundary value problem

E-polarized (H-polarized) EM line source & arbitrary angled dielectric wedge

Solution is in the form of Fredholm integral equation: solution to this equation is made by the use of the standard perturbation technique (Rayleigh-Gans-Born approximation)

Perturbation parameter is dependent on the refractive index of the dielectric wedge

"General Integral Equation Formulation for the Dielectric Wedge":

Vertex field: "Edge condition" by D.S. Jones, 1964 & Radiation condition

Green's function singularity exists, "a small region around the singularity of the Green function must be cut out of Sd and an appropriate limiting process taken"

"Diffraction of an E-polarized Plane Wave by a Right Angle Dielectric Wedge":

E-polarized plane wave incident & wedge's index of refractive (n), 1 <= n <= √2

Figure 4 & 5, graphical plots of the far field: displays the focusing effect (constructive/destructive interferences) of transmitted waves from the two faces of the wedge

[A16]

Scottish scientist

A17. D.R. Wilton and S. Govind's November 1977 paper on Edge Condition:

Use of an Edge Condition in moment method solution is investigated.

Failure to incorporate can result in erroneous currents and anomalous behavior of the solution near edges.

TM plane wave scattering by a conducting strip

The following statement is based on the D. S. Jones 1964 Electromagnetism book reference:

"In electromagnetic scattering and radiation problems, it is well known that currents parallel to a conducting edge may be singular and must obey an edge condition."

[A17]

A18. P. Parhami, Y. Rahmat-Samii and R. Mittra's November 1977 paper on Antennas and Scattering:

Referenced from Ufimtsev's 2000 paper

Antennas mounted on a finite conducting body (rectangular plate)

Antenna and plate currents

Combination of the finite-difference approach and the method of moments to E-integral equation

Efficient evaluation of the current on the structure

Numerical Green's function

"Generation of [Ypp] and numerical results for rectangular plates" (3):

"Incorporation of edge condition" (3.2):

Near fields determination: Importance of edge condition, accuracy

"Numerical results" (3.3):

Gaussian quadrature routines

Current distribution and radiation patterns

"As expected, the incorporation of the edge condition has a negligible effect on the shape of the far-field scattering patterns."

Plate's RCS (radar cross section): more accurate with edge condition incorporation

"Antenna-plate system" (4):

Thin-wire model for antenna (monopole)

Base of Antenna rests at exactly the junction of four adjacent plate patches

"Numerical results" (4.2):

Current distribution and radiation patterns

"Conclusions" (5):

"The procedure developed in this paper is capable of handling the case of a thin, vertical, wire antennas arbitrarily located on a rectangular ground plate."

[A18]

A19. P. Parhami, Y. Rahmat-Samii, and R. Mittra's 1980 paper on numerical evaluation of the Sommerfeld integrals:

Raj Mittra, University of Illinois, Urbana

P. Parhami, University of Illinois, Urbana and then at TRW Defense and Space Systems Group

Efficient numerical evaluation of Sommerfeld Integrals

Antenna problems

Radiative Fields Determination/Derivation

Fields due to current elements over a lossy ground (imperfect ground)

Sommerfeld Integrals: highly oscillatory, based on Hankel functions

Numerical Integration Technique approach: steepest descent path

Asymptotic approximation comparisons

Vector potential

Gaussian quadrature integration applied

Asymptotic and exact expressions are valid only when No Singularities are intercepted during the Steepest descent path deformation

Reference [5] indicates that University of Illinois, Urbana has an Electromagnetic Laboratory and Mittra and Parhami are working with applications of EMP simulators.

[A19]

P. Parhami is associated with Professor Raj Mittra

Working with Current element of Radiating antenna

Mirror Image problem involves Sommerfeld integrals.

A20. C. Joo, J. Ra, and S. Shin's 1980 paper on Edge Diffraction and Right-Angled Dielectric Wedge:

Wedge applications:

Radar reflections from dielectric objects; &
Tip diffraction for GTD.

Rigorous solution for dielectric wedge problems is not available.

Right-angled dielectric wedge, a two wave numbers & regions electromagnetic problem:

kd wave number inside wedge (Sd);
kv wave number outside wedge (Sv); &
ε relative dielectric constant.

Dual integral equation is first formulated

Physical Optics (PO) approximation is made & then a correction is performed where the PO field consists of:

reflected plane waves;
refracted plane waves; & 
edge-diffracted field (v1 in Sv, v2 in Sd), cylindrical waves.

Corrected edge-diffracted field:

Sommerfeld integral along the steepest descent path in complex w-plane.

A new Dual integral equation is arrived at:

Virtual source in free space and in unbounded dielectric;

An Arbitrary multipole line source is now assumed to exist at the origin; &

A Sommerfeld integral is formed & a dual series equation is created.

The total field is the sum of the reflected & refracted plane waves and the corrected edge-diffracted waves

"Results and Conclusion":

The dual series is solved numerically by multipole expansion coefficients truncation.

Figure 3: Diffraction pattern:

"Fig. 3 assures the validity of this theory by showing that the diffraction pattern approaches to that of a perfect conducting wedge as the wedge dielectric constant increases from 2 to 10.".

"The four angles where the pattern blows up are exactly the transition angles of refracted and reflected geometric optical fields".

[A20]

South Korean investigators

The statement describing Figure 3 does not make sense: for a perfectly conducting wedge the relative dielectric constant ε should be exactly one and not approaching 10. Very strange???

Also, this paper is referenced by S. Solimeno et al's 1986 work (Chapter VI Section 2.3 & Figure VI.5, pages 383-384).

A21. Max Born and E. Wolf's 1980 book (6th edition) on Optics:

"Foundations of Geometrical Optics" (Chapter 3):

"Approximation for Very Short Wavelengths" (3.1):

The Light Ray is modelled as an infinitely thin pencil that is associated with an electromagnetic field where its corresponding wavelength approaches Zero (0) in the limit.

This field has the same character as a plane wave.

Laws of refraction and reflection on a plane boundary

Light Rays (Paths & Intensity) and Geometry of Physical Boundary Interaction

Eikonal (Image) Equation - can be derived from the scalar wave equation.

Geometrical optics - eikonal equation is not valid at the boundaries of shadows or at a focus.

Geometrical optics model, the light ray, exists in essence because a "field behaves locally as a plane wave."

"General Properties of Rays" (3.2):

Light ray trajectory in homogeneous medium (constant index of refraction): straight line

Light ray trajectory in heterogeneous medium (variable index of refraction): curved line

"The laws of refraction and reflection" (3.2.2):

Unit Normal to boundary surface: fundamental geometric parameter

"Other Basic Theorems of Geometrical Optics" (3.3):

"The principle of Fermat" (3.3.2):

"Principle of the shortest optical path"

Optical length

Regular neighbourhood

If a ray's regular neighbourhood condition is not met, its optical length may not be a minimum.

An important example of regularity violation is a field of rays and reflection off a plane mirror.

"The theorem of Malus and Dupin and some related theorems" (3.3.3):

Huygens' theorem (construction):

"that each element of a wave-front may be reported as the centre of a secondary disturbance which give rise to spherical wavelets; and moreover that the position of the wave-front at any later time is the envelope of all wavelets."

"Rigorous Diffraction Theory" (Chapter 11):

"Introduction" (11.1):

First rigorous solution: Sommerfeld (1896), 2-D plane wave, infinitely thin perfectly conducting half plane - expressed exactly & simply in terms of the Fresnel integrals.

Next, treatment of Wedge

Solution approaches:

-Separation of variables, infinite series & series converges

-Integral equation, Wiener & Hopf

"Boundary conditions & Surface currents" (11.2):

Infinite conductivity & No penetration of electric field in this conductor

Electric current exists purely on surface of the conductor

Screen is infinitely thin.

"Two-Dimensional Diffraction by a Plane Screen" (11.4):

Two-dimensional diffraction problem: completely independent of one Cartesian coordinate

An angular spectrum of plane waves

Dual integral equations

"Two-Dimensional Diffraction of a Plane Wave by a Half-Plane" (11.5):

Direction of propagation of an E-polarized (parallel to z-axis) plane wave is normal to the diffracting edge (aligned with z-axis);

Direction of propagation was normal to the diffracting edge;

Sommerfeld's famous result (complete field solution) derived;

Evaluation of Sommerfeld's solution at any field point by using the Fresnel integrals tables;

2 Cases for k wave number and r distance (point of observation): kr >> 1 (Optical experiments) and kr << 1 (Behavior of field in vicinity of sharp edge - cm radio wavelengths):

kr >> 1: experimental fact, diffracting edge when viewed from the shadow section appears illuminated and fringes are created by the Interference between the geometrical optics field & the diffraction in regions where they are comparable;

kr << 1: idealized concept of infinitely sharp edge, existence of singularities, uniqueness of solution.

Numerical calculations for perfectly conducting half-plane for normally incident E-polarized plane wave are shown in Figure 11.11 (E_z amplitude at distance of 3 wavelengths behind screen).

Numerical calculations for perfectly conducting half-plane for normally incident H-polarized plane wave are shown in Figures 11.12 (H_z amplitude contours), 11.13 (H_z phase contours), and 11.14 (Average Energy flow lines).

Experimental study of diffraction of electromagnetic waves can be advanced by the application of Microwave radio techniques:

Diffracting screen (edge) & 3 cm wavelength waves, a more optimal measurement test bed as compared to an optical test bed (approaches the idealization of a perfectly conducting half-plane).

"Three-Dimensional Diffraction of a Plane Wave by a Half-Plane" (11.6)

"Diffraction of a Localized Source by a Half-Plane" (11.7):

Line-source: Hankel function; and

Dipole: 3-D field spectrum of plane waves.

"Other Problems" (11.8):

Two parallel half-planes:

References L.A. Vainshtein's two 1948 Soviet (Russian) papers that pertain to Open Resonators & guided wave diffraction;

Mathematical derivation is devoid of any parameters of guided wave diffraction

A strip - perfectly conducting, infinitely long with parallel edges

"Uniqueness of Solution" (11.9):

General diffraction problem, Surface S, Perfectly Conducting, tangential component;

Infinite Surface;

Obstacle with infinitely sharp edge;

Singularity at the edge;

Permitted infinite sequence of solution;

Physical solution: singularity of the lowest possible order;

Unique solution -> induced current over surface, vanishing at the edge; and

Integrals convergent -> preclude them having singularities of too high an order.

[A21]

Max Born et al is working with current and radiated waves.

Doctors Born and Wolf appear to be concern with and knowledgeable about the work of experimental verification of the derived formulas of Rigorous Diffraction Theory and Edged Obstacles.

A22. A.E. Heins' 1982 paper on Sommerfeld's Half-plane diffraction Problem:

First problem in diffraction theory: single type boundary condition

1896 Solution: his (Sommerfeld) mathematical methods, not generalizable

Wider class of geometries - problems as integral equations of Wiener-Hopf type: Mixed type boundary conditions

Sommerfeld radiation condition: boundary conditions at infinity

System of Wiener-Hopf integral equations

Carleman's ideas on singular integral equations (pages 75 & 79)

Application of Fourier integral theorem

Mathematical solution but no physical insight provided

Application of J. Elliott's 1951 work on Singular Integral Equations: Reduction of Equations (7.6 & 7.7), page 86

[A22]

University of Michigan Professor

Problems involve increasing complexity of boundary conditions.

No diagrams to visualize problem or solution are presented.

No physical insight into problem is provided.

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