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

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.

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

A1. H.M. MacDonald's 1915 paper on Diffraction Problems:

Problems of the diffraction of electric waves:

Perfectly conducting right circular cone, solution- constants of the essential series (series expressed by appropriate harmonic functions);

These constants are identical to constants in a series for a corresponding potential problem.

Solution of the problem of diffraction of electric waves by perfectly conducting wedge: use same approach

Purpose of this paper: demonstrate the analytical form of this solution method and apply this method to the problem of diffraction of sound waves by a rigid wedge

Linear partial differential equation: Ew + (k^2) w = 0

Potential equation: Ew = 0

where E is some operator, w satisfies certain boundary conditions.

Green's Function (Wedge): complex derivation

For case of straight edge or half plane: integrand singularity, deform path of integration

No diagrams of geometric problem or experimental results presented.

[A1]

Approximate analytical result based on the commonality of two series (diffraction & potential):

In 1915, did the technology exist to measure a wedge's acoustic diffraction field????

A2. P.A.M. Dirac's famous 1931 paper on Singularities and Electromagnetic Field:

"Introduction" (1):

Advancement of Theoretical Physics and Abstraction of Mathematics:

"…to perfect and generalize the mathematical formalism that forms the existing basis of theoretical physics, and after each success in this direction, to try to interpret the new mathematical features in terms of physical entities…"

Prediction of negative Kinetic energy values for an electron

Negative-energy states - hole --> Anti-electron (& anti-proton)

This paper's object: reason for the existence of a smallest electric charge.

Theory of this paper: connection between the smallest electric charge & the smallest magnetic pole

"Non-integrable Phases for Wave functions" (2):

Particle's motion is represented by a Wave function;

Uncertainty of phase (B);

"wave function always satisfying the same wave equation, whether there is a field or not, and the whole effect of the field when there is one is in making the phase non-integrable";

definite derivatives of B; and

Principle of Gauge Invariance and reference to V. Fock (1929).

"Nodal Singularities" (3):

Nodal line (points where the wave function disappears)

"that the end points of nodal lines must be the same for all wave functions. These end points are then points of singularity in the electromagnetic field."

"Electron in Field of One-Quantum Pole" (4):

Radiating singular line from one-quantum pole (magnetic);

"No stable states for which the electron is bound to the magnetic pole"

"Conclusion" (5):

"quantum mechanics does not preclude the existence of isolated magnetic poles";

"wave equation whose only physical interpretation is the motion of an electron in the field of a single pole"

quantum of magnetic pole and the electronic charge

Unobservability of isolated magnetic poles:

Attractive force between 2 one-quantum poles of opposite sign is very large 4692 (1/4) times between electron and proton.

[A2]

V. Fock was studying the connection of non-integrability of phase and the electromagnetic field.

A3. A. Einstein and N. Rosen's 1935 paper on "The Particle Problem in the General Theory of Relativity":

Mathematical representation of physical space: concept of "bridge"

Singularities of the field

"For these reasons writers have occasionally noted the possibility that material particles might be considered as singularities of the field." (page 73)

"Every field theory, in our opinion, must therefore adhere to the fundamental principle that singularities of the field are to be excluded." (page 73)

Removal of Singularities: Equations modified so that they are free of denominators

Electron or proton as a two-bridge problem

[A3]

Einstein worked on problems of field representations containing singularities.

A4. C.J. Bouwkamp's 1946 paper on Singularities and Sharp Edge:

Lord Rayleigh: certain singularities will occur at sharp edge of diffracting screen

Sommerfeld's well-known solution of electromagnetic diffraction by a semi-plane

Uniqueness theorem

Wave functions with finite derivatives up to a certain order in the complete space

Sommerfeld's explicit solution for a perfectly conducting semi-plane: incoming plane wave propagating in a direction normal to the edge

Edge singularity

Hankel function of the first kind

Page 473:

"'edge' of a physical non-infinitely thin screen - field does not become infinitely large.";

"No energy radiation from edge domain"; and

Maxwell's field equations - if solution is not free of any singularity, the solution must allow that no energy is actually generated in the apparent sources at those places ("Consequently, we should admit the occurrence of singularities, in such a way, however, that no energy is actually generated in the apparent sources at those places.").

Vector character of diffraction - singularities generated - derivatives in expressions for field components

Wave equation: 2nd Order partial derivatives, finiteness / singularity

[A4]

Singularity behavior in mathematical description versus energy finiteness

A5. J. Elliott's 1951 paper on Singular Integral Equations of Cauchy Type:

Solutions of Cauchy-type Integral Equations involving a complex plane contour and is expressed as a Cauchy principal value (normalized by pi)

[A5]

A6. A. Sommerfeld's famous 1954 book on "Optics" Lectures on Theoretical Physics, Vol. IV:

Arnold Sommerfeld was a Professor at the University of Munich, Germany.

This book is an English translation of a record of Dr. Sommerfeld's lectures on optics as recorded by L. Waldmann in 1934 (last few subjects were added to the contents of his lectures).

"Reflection & Refraction of Light" (Chapter 1):

"Fresnel's Formulae. Transitions from Rarer to Denser Media" (3):

"Artificial Suppression of Reflection for Perpendicular Incident" (C)

"Total Reflection" (5):

Transfer the incident wave into the denser medium with its reflection into the same medium and its refraction in the rarer medium:

sin (alpha) /sin(beta) = 1/n for angles alpha & beta (measured against the interface's normal) & index of refraction n

"Light Penetrating into the Rarer Medium" (B):

"Inhomogenous" wave is created whose structure differs from a "homogenous" plane wave:

Its strength attenuates perpendicularly to the boundary surface while it propagates without weaking along the surface.

"The Tunnel Effect of Wave Mechanics" (C):

Existence of the inhomogeneous wave, a difficult problem to prove experimentally:

Using Hertz waves & two asphalt prisms (separated by air space) set-up, this wave type (inhomogeneous) was shown to exist (Bose-Institute, Calcutta, India, 1897).

This phenomenon is analogous to the process of "tunnel effect" of wave mechanics.

"Metallic Reflection" (6):

"Some Remarks on the Color of Metals, Glasses and Pigments" (C):

Colored glass: color from the transmitted light; "a very weak selective absorptivity suffices to color the glass intensively"

Very beautiful colors are created by diffraction of light by colloidal particles

Green of leaves: transparent green grains

The light green color appearance of foliage: created by its interior's light reflections by many inhomogeneities.

Nature's most beautiful colors displays via colors of interference:

Need for synthetic interference color technique exists.

"Theory of Dispersion (Chapter 3):

Nature of refractive media (matter's optical property)

"Electron fluid": analogous to the continuity of fluids of the theory of Hydrodynamics

"Phase Velocity, Signal Velocity, Group Velocity" (22):

"Phase velocity (u = c/n) refers exclusively to purely periodic states of Light & Matter" (steady state) where n is the index of refraction & c is the speed of light in a vacuum

"Fourier Representation of a Bounded Wave Train" (A):

Fourier Integral: replaced by a converging contour integral in the complex plane where singularities exist

"Propagating of the Wave Front in a Dispersive Medium" (B)

"The Precursors" (C)

"The Signal in its Final Steady State" (D)

"Group Velocity and Energy Transport" (E):

Group Velocity: measure of propagation of energy (amplitude), g = dw/dk where w is the wave frequency & k is the wave number (2pi/wavelength)

Contour integrals, saddle-point method (precise mathematically) & stationary phase method

"The Theory of Diffraction" (Chapter 5):

Key definition of Diffraction: "Any deviation of light rays from rectilinear paths which cannot be interpreted as reflection or refraction"

Reflection & refraction: deviation of ray from a straight line caused be a body's surface whose radius of curvature is large compared to the wavelength of the light

Geometrical optics: no diffraction as wavelength as approaches 0 (limiting case of wave optics)

Diffraction phenomena: "generally not noticeable to the naked eye" (low intensities & small dimensions)

Regular grating: increase intensity

Huygens' principle: classical theory of diffraction, scalar spherical wave

"Theory of Gratings" (32):

"Line Gratings" (A):

Diffraction or a more general description, Scattering:

Reflection gratings: A cylindrical wave is propagated from each grating line after its interaction with incident light wave;

The interference of waves emerging from the different grating lines occurs (same incident wave)

"Space Gratings" (C):

Max von Laue, X-rays & crystal, Laue's Fundamental Equations, crystal: transmission grating

P.P. Ewald's "dynamic theory" of X-ray interference

"Huygens' Principle" (34):

Kirchhoff's work: "... the Huygens' principle is an exact consequence of the differential equations of optics."

This principle serves as the basis of the classical theory of diffraction:

"This theory is only an approximation which is valid only for sufficiently small wavelengths": boundary conditions are not known precisely; &

The vectorial nature of electromagnetic field (optical field) is not taken into account.

"The Spherical Wave" (A):

Acoustics: Scalar spherical wave u, solution to scalar wave equation

Electrodynamics: Vectorial spherical wave, complex, e.g., Hertz vector & physical light source

"Green's Theorem and Kirchhoff's Formulation of Huygens' Principle" (B):

Spherical wave u, mathematical auxiliary function, "probe" & v, desired solution of scalar wave equation

Violation of a theorem in Riemann's theory of functions

"Green's Function, Simplified Formulation of Huygens' Principle" (C):

"Fraunhofer and Fresnel Diffraction" (D)

"Black or Reflecting Screen" (F):

Opaque screen (possesses infinite conductivity per Maxwell theory): perfectly reflecting

Black screen: completely non-reflecting (not specified in Maxwell theory)

"The Problem of the Shadow in Geometrical and in Wave Optics" (35):

"The Eikonal" (A)

S, eikonal, wave surface; the normal to surface represents the ray direction

Variation of index of refraction: curved rays

Diffraction at edge of screen: appears as a luminous line, eyes do a false extrapolation of rays of cylindrical wave (field in edge's vicinity is actually continuous)

The differential equation of the eikonal is a very simple case of Hamilton's partial differential equations of dynamics.

"The Origin of the Shadow according to Wave Optics" (B)

"Diffraction behind a Circular Disc" (C):

Diffraction at the elliptically curved portion of the edge of the screen

"There is no darkness anywhere along the central perpendicular behind an opaque disc (except immediately behind the disc)":

This result is inconsistent with the shadow of a disc based on geometrical optics (rectilinear ray paths).

Spherical waves vs. plane wave light intensity effect

"The Circular Opening and Fresnel Zones" (D):

Infinite number of dark places along the central axis

"The Similarity Law of Diffraction" (E):

"the similarity law says: the same diffraction phenomena observed with a small object are also observed with an object magnified by a similarity transformation, provided only that the distances of source and the point of observation from the object are correspondingly magnified."

W. Arkadiew (Moscow, Russia, 1913) observed experimental this law in his laboratory

"Fraunhofer Diffraction by Rectangles and Circles" (36):

"Supplement to Section 35 B. Light Fans arising from Polygonally Rounded Apertures" (E):

In certain direction, a Light fan of finite intensity is radiated & no shadow appears

Independent of wavelength

"Fresnel Diffraction by a Slit" (37):

"Fresnel's Integrals" (A):

F(w); F = C + iS; entire transcendental function of w, can be represented by an infinite series

"Discussion of the Diffraction Pattern" (B):

Cornu's spiral

"Diffraction by a Straight Edge" (C):

Slit, infinitely wide (left-hand edge at infinity & right-hand edge fixed) & incident plane wave

Microscopic view of razor edge: parabolic cylinder vs. sharp edge of theoretical half-plane

"It is very remarkable that patterns on precise diffraction photographs exhibit almost no dependence on the material and shape of the diffraction edge." (refer to Arkadiew, 1913, pg. 221)

"Rigorous Solutions of Certain Diffraction Problems" (38):

"Call a solution of a diffraction problem exact only if it satisfies Maxwell's equations both outside and inside the diffracting object and if it satisfies the proper boundary conditions on the surface of that object."

Method of Series can be applied to Diffraction problem but it can have the issue of slow series convergence rate for a given mathematical diffraction model

"The Problem of the Straight Edge" (A) (page 249):

Problem is physically not rigorous (screening half-plane, infinitely thin but opaque)

But a mathematically rigorous solution in closed form & applicable to all wavelength domains

Fresnel diffraction as shown by this problem: a well-defined mathematical boundary value problem (A. Sommerfeld, Mathem. Ann., Vol. 47, page 317, 1896)

Edge of Screen: z-axis of cylindrical coordinate system

Monochromatic plane wave is incident on screen's front surface at an angle

Linearly polarized wave where electric field is directed parallel to the z-axis

2-Dimensional problem (diffracted electric field is parallel to the z-axis)

Figure 76, problem diagram, "illuminated" & "shaded" region (shadow)

Scalar field u and its condition on it for a unique solution:

scalar wave equation with u = 0 at the surfaces of screen; 
u is finite & continuous everywhere (no singularity) including screen's edge; 
radiative field in illuminated & shadow region

Edge of screen neither radiates or absorbs energy

Importance of Energy conditions: physically uniqueness of problem

J. Meixner: "that the energy density at the edge of the screen shall be integrable with respect to space"

Scalar field u & function U are a solution of the wave equation on a Two-sheeted Riemann surface

Mathematical construction of the branched solutions: applicably for only the simplest case (half-plane)

"Construction of Branched Solutions" (B):

"Wave Bundle", u, an integral formulation for the solution of the diffraction problem:

Integral equation (5), solution of the 2-D wave equation (differential equation) for 
any arbitrary, possible complex, path of integration;
Field u represents an "inhomogeneous" wave if integration path is complex;
Deform the path around a pole in an arbitrary way as long as it does not cross any 
other singularities of the integrand;
Create U by variable replacement; &
U is now single-valued on the Riemann surface.

"Upper sheet" is illuminated by the incident plane wave; "Lower sheet" lies in the shadow, & sheets are connected along the "shadow boundary"

'The contradistinction of "light and shadow" here finds its simplest mathematical formulation.'

Function U, a unique solution of the diffraction problem only if all conditions are satisfied (radiation condition, wave at large distance from incident)

Solution of a diffraction problem must be Unique (required by Nature).

Alternative integral evaluation method is applied instead of the general Saddle-point Method

"Representation of U by a Fresnel Integral" (C)

Function U is converted into analytical function that is valid in the shadowed & illuminated sheets of the problem's two-sheeted Riemann surface

"The Diffraction Field of the Straight-Edge" (D)

"The light is diffracted far into the region of the geometrical shadow. The infinitely large value which this expression assumes on the shadow boundary is, of course, illusory and is due to the fact that our asymptotic approximation is not valid there."

Cylindrical wave emitted by the edge of the screen:

A. Kalaschnikow (1912): photographs

The appearance of a screen's edge as a "thin luminous line" is caused by eye extrapolation:

"Hence the 'luminous edge' is not real."

Phase jump is not real

Huygens' principle's boundary values differ from exact boundary values near the edge of the screen and at large numerical distances from the edge

"Generalization" (E):

Two-sheeted Riemann surface can be generalized to an n-sheeted Riemann surface

Rectangular wedge (Fresnel's integral representation is not possible)

"Black screen":

Infinitely valued integral function U (infinitely number of sheets): no returned energy to the physical space; &

"Black" cannot be defined by boundary conditions condition for the application of Maxwell's theory.

"Basic Remarks on Branched Solutions" (F)

Huygens' principle: solution of diffraction problem by summation method

Solution of diffraction problem by boundary values approach: correctness & uniqueness doubted because boundary values not generally known

Method of images leads to the problem of constructing branched solutions of the wave equation:

Edge of screen (perfectly reflecting plane screen of arbitrary shape): branch line

Mathematical construction of the branched solutions is possible only for the case of the half-plane

Branches of algebraic function (nth degree equation: defined on an n-sheeted Riemann surface

Connection between the edge's energy condition & the theory of functions of a complex variable z

Edge's energy condition is a consequence that the u (field) be everywhere continuous (also at the edge of the screen)

"Addenda, Chiefly to the Theory of Diffraction" (Chapter 6):

"Diffraction by a Very Narrow Slit" (39):

Huygens' principle is meaningless when the diffracting aperture's dimensions become small relative to wavelength

"On Young's Interpretation of Diffraction" (44):

Thomas Young's concept of creation of Diffraction Fringes (1802):

"A kind of reflection" by incident light at the diffraction opening's Edges; &

the interaction of the Edge rays & Incident light rays via the Principle of Interference.

Half-plane & cylindrical wave appears to be emanated by the edge

"The cylindrical wave does not, of course, radiate uniformly in all directions; rather its intensity depends in a definite way on the angle of diffraction."

"Furthermore, the edge of the screen is not an actual light source with infinite amplitude but only appears as such to a sufficiently distant observer because of a representation of the light field which is valid only asymptotically at large distances."

A. Rubinowicz (1917 & 1924): Young's work extended to arbitrary diffraction screens

"Reformulation of Kirchhoff's Solution of the Problem of Diffraction" (A):

Scalar light field, diffraction problems of acoustics (not a vectorial problem of optics)

Problem Geometry:

P' point of luminous source at finite distance to screen S (non-plane wave emission);
σ surface spanning diffraction opening;
P observation point;
r' & r distances;
ƒ surface of cone whose apex is at P';
Curve s forms the boundary of screen opening; &
Differential of curve s, ds & Differential of cone surface ƒ, dƒ.

Transformation of surface integral ∫dσ to line integral ∫ds for curve s by the method of Rubinowicz:

Cone of surface ƒ is formed by rays emitted by source at P' and passing through the boundary of diffraction aperture (Figure 90)

"Reduction of the Surface Integral over the Cone to a Line Integral over the Boundary of the Diffraction Opening. Sharpening of Young's Theory" (B):

A complicated contour integral is derived that describes the following structure of diffraction:

Spherical wave reflected by the edge;

(p_s, ds) is the angle of incidence at the curve element of the edge; &

(r_s, p_s) is the angle of reflection at the edge.

Diffraction fringes in the illuminated region result from interference of the incident light with a wave reflected by the edge (edge wave).

The shadow region contains only this Edge wave.

The derived results are only valid within the Kirchhoff theory (the diffraction opening is large as compared to wavelength & a non-vectorial electromagnetic problem).

"Discussion of the Contour Integral" (C):

Rubinowicz approximates the contour integral by the method of stationary phase (simplified saddle-point method & adapted to a real domain):

"Only those points on the boundary curve yield a substantial contribution to the integral at which the phase is stationary with respect to translation along he curve;"

the phase of the integrand on the boundary curve: ik(p_s + r_s);

this phase remains constant under translation along the contour when dp_s/ds = -dr_s/ds or

when the "reflection condition" is satisfied, cos(p_s,ds) = -cos(r_s,ds);

"In general, a finite number of points s = s₁, s₂, … on the curve will satisfy this condition. Each of these points radiates a substantial intensity to the point of observation P, and the line integral may be evaluated with sufficient accuracy as the sum of these radiations."; &

"The locus of points P which receive radiation from any one point s_ν on the edge is a circular half-cone with the apex at s_ν and with the axis ds."

E. Maey (1893) has proven experimentally this phenomenon for a half-plane for which only one point s₁ exists.

The discontinuity of the diffraction field at the shadow boundary does not in fact exist.

"Light fans" in Section 36E (explained from the point of view of Fresnel's zones):

Young-Rubinowicz reflections perspective:

"Each point s_ν on the diffraction edge radiates a conical light fan; when there are not merely discrete points sv but continuous sequences of such points, the light fans become particularly strong.";

Polygonally bounded diffraction apertures (rectangles, etc): "the specific intensity in these light fans is of the same order of magnitude as that of the incident light; the phenomenon of shadow formation thus disappears."

"Diffraction Near Focal Points" (45):

Caustics of variously shapes (focal lines)

Geometrical optics: a focal point is an infinite concentration of rays

"Wave optics resolves this (physically obviously inadmissible) singularity into a strong light concentration of finite amplitude and finite extent."

Focal point's phase jump is pi (cylindrical wave, pi/2 jump)

"The Diffraction Field in the Neighborhood of the Focal Point" (B):

Singularity of the light field at the focus (Geometric optics) does not really exist; wave optics: the field is entirely regular (no singularity)

"Amplitude and Phase along and near the Axis of the Light Cone" (C)

"The Cylindrical Wave and its Phase Jump" (D):

Geometrical focal line: No singularity appears from an optical wave perspective.

|U|^2 calculation for intensity distribution for finite value U

The phase as well as the amplitude change continuously in the vicinity of the focus.

"The Huygens' Principle of the Electromagnetic Vector Problem" (46):

Scalar acoustic & vectorial optical problems

Vector (simple polarization) problem may be reduced to two scalar problems

Hugyens' principle & its application (Fresnel-Kirchhoff): scalar formulation

Vectorial formulation: "an exact solution of the diffraction could be obtained only if the exact boundary values E₀ or H₀ were known. This is not the case. Rather, these boundary values can only be found simultaneously with the solution of the diffraction problem."

"The vectorial Huygens' principle is no magic wand for the solution of boundary value problems"

"Cerenkov Radiation" (47):

"The Field of the Cerenkov Electron" (A):

Mach cone: electromagnetic field is zero outside the Mach zone; exists in the interior of the Mach cone

"The Radiation of the Cerenkov Electron" (B):

"Cerenkov Radiation with Dispersion taken into Account" (C):

The electrons are made visible by the Cerenkov effect

Point-electron: singularity of the field on the Mach cone, creates difficulties with the Fourier analysis

Electron of finite extent: "singularity is smoothed out into a transition zone with finite field strength"

"On the Nature of White Light, Photon Theory and Complementarity" (49):

'A line grating diffracts a single "plane" pulse in the same manner as a plane wave' (Gouy, 1886):

A slanted incident primary encounters the various grating lines at equal time intervals; and, then a spread out sequence of secondary pulses are emitted by the grating lines.

"Fermat's principle of the shortest time of arrival" is the same as "the principle of the shortest (geodetic) line in mechanics of a force-free mass point"

Hamilton: "geometrical optics is identical with the ordinary mechanics of mass points not only for homogeneous & inhomogeneous media"

[A6]

Dr. Arnold Sommerfeld (1868 - 1951) is one of the founding fathers of modern theoretical physics; thus, he is one of the most famous physicists of the 20th century.

Optics, the 1954 book by Professor Sommerfeld, is a foundational (thus, extremely important) scientific work in the fields of Electrodynamics of Light and the Theory of Diffraction. It contains many unique theoretical ideas, mathematical formulations, and experimental results including important works of Russian scientists.

This book was published three years after he died in 1951; the preface of his book was written by him about two years before his death.

Scientific investigators in the field of the Theory of Diffraction reference through their papers and books directly or indirectly the works of Professor Sommerfeld (e.g., Optics).

For example, consider two extremely important works such as the 1957 and 1962 papers on the Geometrical Theory of Diffraction by J.B. Keller (New York University) where the concepts of the Law of Edge Diffraction, the "Cone of Diffracted Rays," and the Diffraction Coefficient are discussed.

It is important to realize that Optics does not provide a complete mathematical derivation that supports the existence (structure) of the "Cone of Diffracted Rays" nor does it address the issue of the energy distribution in such a light cone. One could consult Sections 44 C (Rubinowicz & E. Maey) and 36 E for a very incomplete discussion on the "circular half-cone" of radiation and "light fans," respectively.

Sommerfeld does not specify in detail (physical & mathematical description) the phenomenon of the Edge Wave (simply stated as "a kind of reflection"). Note that there is no such thing as a "sharp" Edge.

Also, no solution to the Problem of Diffraction by a Wedge is presented in Optics.

The "luminous edge" of a diffracting structure is not real according to Sommerfeld.

Further, one must understand the key concept of the Rigorous Definition of a Problem of Diffraction from a physical perspective, a mathematical perspective, and their mutual limitations so that one can derive the correct diffraction solution, if possible.

Singularities in Diffraction fields (i.e., infinities of field values) do not exist physically. They are problems of constructs of mathematical models of Diffraction.

This very important fact about field singularities implies that the mathematical or physical models of the Diffraction phenomena do not actually describe the true phenomenon of Diffraction in Nature. Further, the generally accepted Theories of Diffraction may not be correct, too.

Finally, there are common ideas between Optics (Optical Electrodynamics) and Hamiltonian dynamics (mechanics).

Thus, what new model (s) of Diffraction can be found using these ideas?



A7. J.B. Keller's 1957 paper on Diffraction by an Aperture:

Geometrical theory of diffraction, a new theory of wave propagation: extension of Geometrical optics ("diffracted rays")

Diffraction by an Aperture of a screen or by flat thin obstacles

Results agree with the leading terms in the asymptotic expansion of the exact solutions of diffraction problems where the propagation constant approaches infinity (wavelength approaches zero)

"Ray Tracing":

Edge of screen produces diffracted rays that satisfy the Law of Edge Diffraction:

A diffracted ray & the corresponding incident ray make equal angles with the edge & lie on opposite sides of the plane normal to the edge at the point of diffraction;

Figure 1. Cone of diffracted rays;

It follows that infinitely many diffracted rays are produced by one incident ray;

The diffracted rays lie on the surface of a cone having as axis the tangent to the edge at the point of diffraction; &

These rays are a consequence of a special form of an extended Fermat's principle.

Law of Vertex diffraction

"Field on a Ray":

Amplitude variation satisfies the Principle of Energy Conservation;

Present theory produces an infinite value for the field on a Caustic, not a valid value

"Reflection and Diffraction Coefficients":

Neighboring diffracted rays of the same cone intersect each other at the diffracting edge;

Edge is a caustic of the set of diffracted rays: this results in a infinite value for the field at the edge;

"We cannot determine the field on a diffracted ray simply by equating the diffracted field at the edge to a multiple of the incident field.";

Alternative derivation for Edge field & diffraction coefficient for cone of diffracted rays;

Diffraction coefficient D depends on:

Type of field;
Nature of the screen at the point of diffraction;
The directions of the incident & diffracted rays; &
Propagation constant (Wave number) k.

Two examples of diffraction coefficients for scalar field, reduced wave equation, & u = 0 (field) or ∂u/∂n = 0 (normal derivative):

Diffraction of a plane wave by a screen (modeled as a half-plane) whose boundary-value solution was derived by Sommerfeld (A.J.W. Sommerfeld, Optics, pp 249, 1954)

Refers to Appendix II (Diffraction Coefficient D) for half-plane and wedge derivation comparison: claims agreement

Corner-diffracted rays: emanates in all directions from a corner

Claims that GTD derivation applies to diffraction by any edge

"Total Field":

The total field u(P) is the sum of the fields on all the rays through point P including:

the sum of all diffracted fields on all diffracted rays through P; &
the geometric optics field that is the sum of the fields on all incident & 
reflected rays through P.

"Each time a ray is diffracted by an edge or corner its field is multiplied by a diffraction coefficient D or C which contains the factor k^-1/2 or k^-1."

"Conclusion":

"Although the fields which we have constructed are not the exact solutions of the field equations, they are presumably the leading terms of asymptotic expansions of such solutions, for k large.";

"The infinities in our results then correspond to changes in the asymptotic behavior of the solution.";

"Our method utilizes certain special or canonical solutions of simplified problems for the determination of diffraction coefficients, and thus provide a new motivation for their solutions. Canonical solutions can also be used to supplement our results at caustics and shadow boundaries."; &

Thus, at an edge, the edge field which is infinite, and the geometric optics field, which is discontinuous, may be replaced by the field which occurs in diffraction by a half-plane.

"Appendix I. Caustics of Diffracted Rays":

"The caustic contains the curve of centers of curvature of the edge, i.e., the evolute of the edge."; &

"If the edge is the edge of a thin flat obstacle, then the caustic will appear as a bright line in any cross section of its shadow. Bright lines of exactly this shape have been observed by J. Coulson and G.G. Becknell (Phys. Rev. 20, 594, 1922)."

[A7]

Diffraction investigator at the Institute of Mathematical Sciences, New York University

This paper/research was funded by U.S. Air Force, Air Force Cambridge Research Center.

This paper does not prove physically or mathematically why the "Cone of Diffracted Rays" exists as an edge diffraction phenomenon nor does it provide a clear mathematical description of the "Cone of Diffracted Rays":

An Edge is a caustic with an infinite number of diffracted rays being emitted at a given point of diffraction (principle of energy conservation????).

Keller claims that Sommerfeld's 1954 book Optics provides key proof to support his GTD work and the necessary associated Diffraction Coefficients. Thus, Sommerfeld's work serves as a key to determining the fidelity of the ideas of Keller.

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