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 November 4, 2009 and was finished on January 11, 2010.

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

Beginning of RESULTS PAPER derived from RESEARCH on PROBLEMS of DIFFRACTION, SCATTERING, and PROPAGATION of WAVES.

**2. P.Y. Ufimtsev’s 2007 book on Physical Theory of Diffraction (PTD) [2]**:

In the

**Foreword** by Kenneth M. Mitzer, once a Northrop employee, he extols Ufimtsev’s work of PTD.

Dr. Mitzer worked on the Norhrop’s B-2 that cost the American taxpayer billion of dollars.

K.M. Mitzer received a Ph.D. in 1964 from California Institute of Technology. His Ph.D. dissertation (“Theory of the Scattering of Electromagnetic Waves by Irregular Interfaces”) does not cite Ufimtsev but does cite other Russians’ work.

Few scientists or engineers in the world can critique this complex work for correct derivations or formulas.

**3. Ufimtsev’s 2003 book on PTD:**In the

**Foreword** by K.M Mitzner, once a Northrop employee, he talks about the Lockheed F-117 Stealth Fighter & Northrop B-2 Stealth Bomber: extols PTD & Ufimtsev.

Using the B-2 as a basis, Dr. Mitzner claims experimental proof of principles of PTD.

PTD, an improvement of Physical Optics (PO)

**Book Editor:** Andrew J. Terzuoli, U.S. Air Force Institute of Technology (AFIT)

**Translator’s note** by Richard D. Moore, former employee of USAF's Foreign Techhnology Division's (FTD) Low Observables Office:

Soviet Scientist responsible for the theoretical developments of the F-117

Lockheed's Bill Schroeder & Denys Overholser: acquired 1971 FTD's translated PTD monograph

"In Intelligence Community Ufimtsev's name became almost mythic.": "mysterious man"

**Acknowledgements**: V.A. Fock, L.A. Vainshtein, E.N. Maizel’s, & L.S. Chugunova.

Ufimtsev's Publicly Disclosed Affiliation (

**About Author**):

Central Research Radio Engineering Institute, Moscow, Soviet Union

Engineer, Senior Engineer, Senior Scientist; 1954 - 1973

**[3]**
Excellent presentation of Edge Diffraction works
Very complicated presentation
Concept of “uniform” & “non-uniform” currents & radiation (fields)

149 References: Ufimtsev, V.A. Fock, L.A. Vainshtein, & other Russians’ works, W. Braunbek

Important contributors are Soviets/Russians;

Few scientists or engineers in the world can critique this work for correct derivations or formulas.

Ufimtsev is knowledgeable about Radar.

In 1956 & 1957, Ufimtsev worked in Moscow at the Central Research Radio Engineering Institute.

**4. Institute for Defense Analyses 2003 DARPA Assessment Paper:**Offset Strategy & Stealth Combat Aircraft:

U.S. Early Air Superiority Goal by a Stealth Tactical aircraft: "New types of deep air attacks"

Provides insight to players in government involved in Stealth aircraft program development;

Major Roles: William J. Perry & Paul Kaminski;

Top-level Government decision-making about stealth in weapon platform:

**Fund aircraft companies to propose Stealth conceptual designs that would defeat Soviet Vietnam War & Yom Kippur War era integrated anti-aircraft systems (radar subsystem)**1974: First Low observable characteristics of Aircraft Weapon Platform concept
1975 Summer: first phase results, Lockheed & Northrop
1976: Second phase, Aircraft Prototype development began (HAVE BLUE) Lockheed

1977: Successful flights; Accelerate development of real weapon system “Black” program F-117A aircraft (SENIOR TREND)

1981: first F-117A delivered
1990: 59 F-117A deployed;

Future Stealth Applications: B-2 Stealth Bomber (Northrop) & advanced cruise missiles where both benefited from the TACIT BLUE program; F-22 and Joint Strike fighter aircraft; and

Stealth Countermeasures: ultrawideband & multistatic radar & radar data fusion.

**[4]****5. Jim Cunningham's 1991 Paper on Secrecy and F-117A:**USA stealth technology development: Background & Problems (Secrecy); and

1950s stealth development for U-2 reconnaissance airplane; and

Ability of Soviet Intelligent organizations and community to penetrate classified programs.

**[5]**Soviet radar enhancement and Soviet low observable research (scattering and diffraction) physics and applications would have been extremely important work in the USSR in 1950s.

Soviet Intelligent Community would have targeted the West (U.S., UK, etc.) and Soviet Open-literature Controls would have been implemented.

1950 and 1960s U.S. DOD, Air Force, & CIA would have funded studies on Soviet/Russian theoretical & applied electromagnetics (radar) and its associated countermeasures (stealth & invisibility).

The 1956 and subsequent Soviet physics Russian physics papers translated into English papers; Ufimtsev’s 1962 monograph translated in 1971.

**6. Aronstein and Piccirillo's 1997 book on the development of the U.S. Have Blue and the Stealth Fighter (F-117A) aircraft: **HAVE BLUE

1974 - 1975 Have Blue DARPA (Defense Advanced Research Projects Agency) Study

Very Low Observable (VLO) Military Aircraft: Reduce signature levels to avoid detection & how to do it

Physics of Signature reduction: reduce detection range by a factor of 10, requires the radar cross section (RCS) to be reduced by a factor of 10,000

In the Spring of 1975 the DARPA goals of RCS signature levels are established (McDonnell Douglas & Hughes).

__Lockheed's Involvement:__In late 1974 or early 1975 Lockheed discovers of the DARPA's VLO project.

"Lockheed did not have any rigorous theory or predictive techniques that could be used to design an aircraft from the beginning for low RCS."

Signature reduction & flat panels: D-21 drone vs. faceting approaches

Bill Schroeder, Denys Overholser and Ed Lovick: RCS prediction computer program:

Spring 1975 start, 6 weeks development of this program

ECHO 1, a simple RCS prediction program, provide 95% solution

ECHO 1 Concept:

RCS calculation: body is large relative to incident radiation wavelength, electrical currents, surface integral, and scattered radiation;

Physical optics (PO) and Perfectly electrically conducting (PEC) object: no neighboring surface coupling.

Schroeder's accomplishment: explicit expression for calculating of integral for a triangular flat panel;

Schroeder's solution & nonspecular angles: slant (slope) of each surface;

Past efforts of determination of surface slant for RCS reduction: intuition or empirical rules;

ECHO 1: inaccurate edge returns calculations (did not incorporate Diffraction effect due to edges)

Diffraction: radiation produced by these currents along the edges.

Overholser came across the Ufimtsev's translated 1962 Russian monograph in Physical Theory of Diffraction (PTD) "sometime after the initial development of ECHO."

During the early design stages of Experimental Survivable Testbed (XST)/Have Blue, Ufimtev's Theory of Diffraction (PTD) had not been implemented in ECHO 1.

Pure luck is being attributed as the cause of the accuracy of ECHO 1.

By the time of F-117A design, ECHO was upgraded for edge diffraction and special materials and coatings.

RCS model test: RCS "spikes" matched with ECHO predictions

RCS test agreed well with ECHO predictions: provided confident in ECHO & faceted design concept

ECHO's facets number capability limitation: Increase of the number of facets saturated this computer program.

Experimental Survivable Testbed (XST) Proposals are submitted in late Summer of 1975.

DARPA & USAF developed an algorithm for reduction of all signatures to a single number.

Lockheed Proposal: Hopeless Diamond predicted and measured signature data & configuration data

__Northrop's Involvement:__Northrop's RCS prediction work since 1966

RCS computer program developed by 1970 (incorporated Ufimtsev's PTD idea)

GENSCAT, second generation RCS prediction computer program (1974):

Contained K.M. Mitzner's explicit mathematical expression for PO scattering from a polygonal flat panel

Implemented the Incremental length diffraction coefficients (ILDC), an extension of Ufimtsev PTD work

**"Thus, at the time the pre-XST studies began, Northrop actually had a much more comprehensive RCS prediction capability than Lockheed."**GENSCAT as well as ECHO did not design aircraft.

Northrop proposed a faceted aircraft during the Spring of 1975.

__XST Program: __Lockheed and Northrop proposals contained a revolutionary breakthrough in RCS reduction technology.

Dr. Malcolm Currie, Director of Defense Research and Engineering (DDR&E) assistance

Phase 2: XST/Have Blue program became Top Secret

XST Phase 1: On November 1, 1975, Lockheed and Northrop were selected for Phase 2 (two demonstrator aircraft to be built and tested)

Lockheed's RCS prediction ECHO computer program used to design changes for XST:

Lockheed and Northrop XST models' Signatures are lower by orders of magnitude than any other previous aircraft

Lockheed is winner of XST Phase 1 (April 1976)

DARPA wanted the Northrop XST team to stay together:

BSAX (Battlefield Surveillance Aircraft, Experimental), Tacit Blue (data for B-2 bomber program)

General Dynamics, one of participants of the earlier study believed that a minimum RCS level exists where no further substantial RCS reduction could be achieved through airframe shaping or treatment.

DDR&E William J Perry (starting in 1977) informed about progress of the Have Blue program

Aircraft Signature & Detection Reduction dependent on Basic Physics: detection range varies only as the fourth root of RCS

Two Have Blue were constructed

Have Blue No. 1 building began in July 1976, completed in November 1977 and its first flight was in December 1, 1977

**Have Blue aircraft's Limited performance and unusual stability characteristics**Have Blue No. 1 was lost on May 4, 1978

XST/Have Blue program ended in December 1979:

Outcome: 20-30% reduction of aerodynamic efficiency in exchange for orders-of-magnitude reduction in RCS

Technology Breakthrough in U.S. RCS reduction:

'A considerable effort in the field of RCS prediction was made during the latter half of the 1960s. Although Dr. P.I. Ufimtsev is widely credited with providing the key to Lockheed's breakthrough, the work that took place both before and after the "discovery" of Ufimtsev's theory was critically important to developing useful RCS prediction capability at both Northrop and Lockheed.'

DARPA chose the Northrop, the loser of the XST program to work on another program and then on the Tacit Blue program. And, Lockheed went on to designing and constructing the F-117A, the Stealth Fighter aircraft.

ADVANCED TACTICAL AIRCRAFT GENESIS

Have Blue phase 1 model test results: "breakthrough of tremendous military value" (Spring 1976)

U.S. Air Force weapons system preliminary requirement studies (Summer 1976)

Secretary of Defense Harold Brown and DDR&E William J. Perry supported the establishment of a special project office for a Low Observables program (June 1977).

Advanced Technology Aircraft (ATA)

Lockheed Advanced Development Projects received $11.1 million contract on October 10, 1977

ATA "A" becomes the F-117A (the eventually deployed Stealth Fighter)

FULL-SCALE DEVELOPMENT

F-117A was a new design but used the Have Blue aircraft shaping concepts.

Lockheed's ECHO computer program was enhanced for the F-117A design:

The number of facets was increased;

Edge diffraction calculation was added (based on Ufimtsev's work of PTD); and "limited ability to calculate the effects of non-metallic materials"

Denys Overholser, developer of ECHO, learned of Ufimtsev's PTD (USAF's 1971 translated version of Ufimtsev's 1962 PTD Russian monograph published in the USSR) "shortly after the initial design of Have Blue."

For the design of the F-117A, the "best" case calculation for edges was based on the basic ECHO 1 results and the "worst case" was based on the full diffraction calculation. "A prediction was chosen between the two case results.

In the mid-1980s, Lockheed started to apply the Method of Moments to the problem of predicting analytically the performance of special edges.

**[6]**It is amazing that Lockheed, the designer of the original U2, did not possess the analytical capability to reduce the RCS of CIA and military U.S. government aircraft prior to the Have Blue project in the 1970s.

Overholser's discovery of the USAF translated Ufimtsev 1962 monograph (1971) probably occurred in late Spring or early Summer 1975.

It is most likely that Lockheed used Ufimtsev's work of Physical Theory of Diffraction to correct the design of the F-117A given the parent prototype aircraft's aerodynamic flight problems while attempting to achieve the necessary Low Observability requirements. An Enhanced ECHO or similiar computer program would have been used in the aircraft/airframe design for the tradeoff of the flight characteristic and RCS.

**7. Everest E. Riccioni reveals major issues and problems with the F-22, the second generation U.S. Air Force stealth fighter aircraft:**The F-22 jet aircraft, 'Raptor' per unit cost has increased three times over its original cost.

The design goal of Raptor was to "fly stealthily at supersonic speeds deep into the Soviet Union to take out bombers headed toward the United States or its allies."

"The Raptor's stealth capabilities have been largely misrepresented."

It has limited direction radar evasion.

**[7]**Another example of failed U.S. expensive technology based supposedly on Ufimtsev’work.

**8. Panter’s 1982 book on Communication Systems:**Path Loss Prediction: Tropo-scatter radio-relay system;

Tropospheric Propagation & Path Loss;

Single diffraction knife-edge and Smooth-earth diffraction phenomenon; and

Geometric Optics Methods applied to model these forms of diffraction.

**[8]**U.S. Communications Engineering Approach in 1982. It indicates degree of knowledge about applied Diffraction Theory in U.S. Communications systems;

Both forms of diffraction may be actually very complicated. And, Geometric Optics method may be not appropriate. It is dependent on the transmission wavelength and characteristic length of diffraction element.

Soviet scientists M.A. Leontovich and V.A. Fock were pioneers in this applied diffraction theory research. This research could be used for

**Radar Transmit and Receive Site Selection**. Knowledge of Diffraction theory and how to apply it practically would become very important in the Soviet Union.

**9. Solimeno, Crosignani, and Di Porto's 1986 book on Optical Radiation:****"Ray Optics":**"Scalar Ray Equations in Curvilinear Coordinates: the Principle of Fermat":

Optical length [AB], a metric, a line integral with its integrand being the index of refraction n(

**r**);

"This principle asserts that the optical length of a ray between two points A and B is smaller than the optical length of any other curve joining A and B and lying in the neighborhood of the ray."; &

Laws of reflection and refraction.

**Diffraction theory and its associated Asymptotic approximation theory:** many key references;

Fundamentals of Diffraction Theory

Reduction of Diffraction Integrals to Line Integrals

Thomas Young (1801,1802, & 1803), Arnold Sommerfeld (1896), Rubinowicz (1917), & Miyamoto & Wolf

Two waves superposition: edge & interior field
Boundary-diffracted wave (BDW)

Asymptotic Evaluation of Diffraction Integrals:

Reference to Ufimtsev’s two works (monograph & paper)

Aperture Diffraction and Scattering from Metallic and Dielectric Obstacles

Diffraction problems in most practical cases do not allow closed analytical solutions.

Canonical examples of the wedge with finite surface impedance, the circular cylinder, and the sphere

Wedge, Diffraction Matrix (coefficients)

Sensitivity to wavelength is a very important characteristic of all diffraction fields.

G.D. Maliuzhinets asymptotic solution for Wedge Surface Impedances

No analytical solution exists for Diffraction from a Dielectric Wedge.

J.B. Keller: Generalized Fermat Principle and Geometric Theory of Diffraction (GTD): Procedural approach to the description of rays diffracted by any kind of obstacle.

Optical Resonators and Fabry-Perot Interferometers

"Vaynshteyn Theory of Concentric and Plane-Parallel Resonators":

Open-ended waveguide and Diffraction: elegant procedure for solution of an integral equation.

Figure VII.34: Diffraction at the waveguide mode:

A diagram illustrating the mode(s) that are strongly reflected by the edges.

**[9]**This excellent book is very technical and presents the mathematical physical complexity of diffraction theory as of 1986. From this book, we learn the importance of the wedge canonical example: available diffraction pattern (field) solution for practical applications.

It has no discussion of Ufimtsev's Physical Theory of Diffraction but it refers just to two of his works ([24] & [33]).

**10. Bowman, Senior, and Uslenghi's 1969 book on Scattering Cases:**PREFACE:

University of Michigan Radiation Laboratory: ~20 years of work in Radar Scattering Behavior Prediction

Scattering Theory:

Perfect Conducting body to Electromagnetic Waves
15 Geometrically Simple Scattering Shapes

Handbook for Radar & Antenna Specialist

U.S. Air Force (USAF) Cambridge Research Laboratory

The USAF provided financial support for this book.

INTRODUCTION:

The sphere, circular cylinder, wire, cone, wedge, half-plane, disc, and paraboloid geometric shaped bodies have important applications in radar and antenna theories.

Additional Boundary condition for an Edged body:

1) Uniqueness of Electric and Magnetic Fields Solution

2) "the electromagnetic energy contained in any finite volume about the surface singularity must be finite" (Meixner integrability condition, 1949)

Reciprocity Theorem: Transmitting and Receiving properties of an Antenna

Integral Equations: Reference to Mitzner's 1967 paper

Keller's Theory

Physical Optics

Perfectly conducting body - postulated current distribution
Quadratures
High frequency approximations
Estimate of specular return becomes more accurate with increasing frequency

Postulated current in vicinity of edges is seriously in error but diminishes with increasing frequency

Physical optics in general fails to satisfy the Reciprocity theorem everywhere except in the direction of a specular return.

Physical optics method extensions: No mention of Ufimtsev

Fock's Theory

Principle of the local field in penumbra region is basic to the analysis of the high-frequency diffraction by a convex perfectly conducting object with continuously varying curvature.

Fock's 1946 two papers: 1) distribution of induced currents and 2) field near surface

"Universal" function for the current near the shadow boundary ("Fock functions")

Cullen 1958 paper: direct solution to "Universal" current distribution

Function-theoretic Methods

Maliuzhinets: developed a method of solving diffraction problems in angular regions, representing the field in a wedge-shaped region by a Sommerfeld integral for which an inversion formula exists.

Numerical methods

BODY OF BOOK:

Ufimtsev's results appear in the Strip, Wire, and Disc chapters, Chapter 4, 12, and 14, respectively.

**[10]**These three Scientists and Professors: Bowman, Senior, and Uslenghi are clearly U.S. and World experts in the fields of Scattering, Diffraction, Radar: depth of knowledge and their own contributions.

No mention of PTD (Ufimtsev) in Introduction

Fock in 1946 formulated a key idea that may be the precusor to Ufimtsev's extension of Physical Optics theory (1957) for High-frequency approximation model of Diffraction by a convex, perfectly conducting body:

Fock : Ufimtsev
Local curvature : small element
Paraboloid of revolution : wedge

**11. Mitzner's 1967 Paper on Scattering:**In 1967, Dr. Mitzner worked at Northrop Norair, Hawthorne, California.

Paper represents work paid for by U.S. Air Force.

References:

M.A. Leontovich's 1948 work/paper, Translation from 1957 V.A. Fock's work: Appendix from Air Force Cambridge Research Center; and

1940 Russian paper of S.M. Rytov (Mitzner possesses a translation.).

**[11]**No mention of Ufimtsev

Very important paper:

Mitzner is studying Scattering from practical metallic scatterers (finite conductivity) in 1967. The Soviets were studying this scattering physics back in 1948.

Leontovich boundary condition Improvement: curvature-dependent boundary conditions

U.S. Air Force is translating Soviet (Russian) papers.

Someone in the U.S. Air Force is following Soviet/Russian Diffraction/Scattering work in the 1950s.

Obviously, Dr. Mitzner had access to Russian papers (translated via the U.S. Air Force).

**12. Jackson’s 1975 book on Classical Electrodynamics:**"Fields and Charge Densities in Two-dimensional Corners and Along Edges":

A Model of Conducting Surfaces joined together: the intersection of Two Conducting Planes that form a corner or edge;

Neighborhood of sharp "corners" or edges (assumed to be infinitely sharp);

The fields, and the surface charge densities: vary with distance near ρ = 0 as ρ^(pi/β)-1 where β is the opening angle & ρ is the distance from the origin (intersection);

For small β, no charge essentially accumulates in this deep corner;

For β > pi, the 2-D corner becomes an edge and the field & the surface charge density become singular as ρ → 0;

For edge of a thin sheet (β = 2pi), the singularity is as ρ^-1/2.

Role of a Metallic structure in the Generation and Transmission of Electromagnetic Radiation:

Its dimensions are comparable to the wavelengths involved
Metallic boundaries and Electromagnetic fields
Surface and Interior of Conductor & Fields

Conductor's neighborhood & fields:

Surface of a perfect conductor: only the normal component of Electric field and the tangential component of the Magnetic field can exist;

Inside of a perfect conductor: fields' components are zero;

Surface of a Good conductor: All fields' components are non-zero; and

Inside of a Good conductor: All fields' components penetrate to some depth (skin depth).

Electromagnetic Wave Generation:

Source as a Localized Oscillating System of Charge and Current Density
Simple Current Configurations
Vector Potential A(**x**,t) where **x** is a position vector.
Derived Magnetic Induction (B) and Electric Field (E)
Near Zone, Intermediate Zone, & Far Zone: relative to source dimensions and wavelength.

Long Wavelength Scattering

Scattering & diffraction: separate treatment but are not logically separate

Diffraction, departures from geometric optics (GO), caused by the waves’ finite wavelength

Kirchhoff approximation: major deficiency, missing vector character

Vectorial Diffraction Theory

Surface current density for conducting surfaces

Scattering in the Short-Wavelength Limit:

Absence of knowledge about the correct surface E & B fields;
On the surface, we make some approximations;
Wavelength is short compared to the dimensions of obstacle;

‘In the shadow region the scattered fields on the surface must be very nearly equal & opposite to the incident fields, regardless of the nature of the scatterer, provided it is “opaque.”’;

Shape of illuminated portion of the scatterer, as well as its electromagnetic properties; and

“Opaque” scatterer.

**[12]**No mention of Ufimtsev; Cites A. Rubinowicz, L.A. Weinstein (Vaynshteyn), V.A. Fock, Senior, and Uslenghi.

Very Complex integrals are involved in diffraction theory.

**13. Rojansky’s 1979 book on Electromagnetics:**"Plane Electromagnetic Waves"

"Radiation from a sheet of alternating current"
"Reflection from Nonconductors; Normal Incidence"
"Reflection from Perfect Conductors; Normal Incidence": Standing Wave in axis of
Electric Field & Surface Current

"Reflection from Perfect Conductors; Oblique Incidence"

**[13]**Simple plane interface as reflector & Perfect Conductor Surface Current

**14. Ramo, Whinnery, & Van Duzer’s 1984 & 1994 book on Electromagnetics:**"PLANE-WAVE PROPAGATION and REFLECTION"

A complex Electromagnetic Wave Patterns may be represented as a Superposition of Plane Waves.

Uniform Plane Wave Concept

"RADIATION"

"FIELD and POWER CALCULATIONS WITH CURRENT ASSUMED ON THE ANTENNA"

"RADIATION FROM FIELDS OVER AN APERTURE"

"Fields as Source of Radiation"

Huygens' principle
Equivalent current sheets in Aperture

"FIELD ANALYSIS OF ANTENNAS"

"Numerical Methods: The Method of Moments" (New section found in the 1994 edition)

Sum of Incident and Scattered Electric fields is equal to zero at the surface of a Perfectly conducting conductor.

Induced Surface currents are represented as sums of weighted current basis functions.

"RECEIVING ANTENNAS AND RECIPROCITY"

"A Transmitting-Receiving System"
"Reciprocity Relations"
"Equivalent Circuit of the Receiving Antenna"

"OPTICS"

"RAY OR GEOMETRICAL OPTICS"

"Geometrical Optics through Applications of Laws of Reflection and Refraction"

Uniform plane wave model: the wave is Uniform over an Infinite Wavefront.

Uniform plane wave & boundary interaction: the wave interacts with an infinite plane boundary between two media.

“the wavefront extends & is uniform over many wavelengths, and when the boundaries are large in comparison with wavelength. Both boundaries and wavefront may be non-planar so long as radii of curvature are also large in comparison with wavelength.”

"Geometrical Optics as Limiting Case of Wave Optics"

Eikonal equation of geometrical optics; Uniform plane wave.

Rays in Inhomogeneous Media

Curvature of ray

**[14]**Current vs. field as Sources of Fields

**15. Arfken‘s 1985 book on mathematical methods:**Solution of a Nonhomogeneous Partial Differential Equation:

Green's function method G(r,x)

Green's Function Symmetry: basis of various Reciprocity theorems

G(r,x) = G(x,r)

Reciprocity principle: a cause at x produces the same effect at r as a cause at r creates at x.

Infinite Series & Asymptotic or Semiconvergent Series

Bessel Functions & Asymptotic Expansions.

**[15]**Mathematical Basis for the Reciprocity Principle (via the Influence function, Green's functions)

Introduction to complexity of Asymptotic theory

**16. Hecht and Zajac's 1979 book on Optics:**Diffraction theory; and Concept of Boundary Diffraction Waves (BDW) (edge waves)

Eugen Maey (doctoral thesis, 1893): demonstrated that an edge wave could be extracted from a "modified Kirchhoff formulation for a semi-infinite half-plane"

Arnold Sommerfeld's rigorous solution of the half-plane problem (1896): cylindrical wave proceeds from the screen's edge; propagates in the geometrical shadow region and the illuminated region; combines with geometric waves in accord with Young's theory

Reference to A. Rubinowicz's 1965 work.

**[16]**BDW concept was developed a long time before Ufimtsev (Young, etc.). Generalized concept to EM and elastic waves.

**17. A. Rubinowicz’s 1965 paper on Diffraction Wave:**"The Miyamoto-Wolf Diffraction Wave"

"The History of the Diffraction Wave":

"Edges of illuminated objects shine when observed from their shadow" (based on empirical evidence for the case of light)

“The first wave theory of diffraction was suggested by Thomas Young [1807, 1855].” “The diffraction phenomenon arises then from the interference of the contributions of the different line elements of the diffracting edge with each other and with the geometric-optical wave. A characteristic feature of Young’s theory is the fact that the diffracted wave is formed by a scattering of the incident wave on the line elements of the diffracting edge at the line elements in question."

Gian Antonio Maggi (1888)
Eugen Maey (1893)
A. Sommerfeld (1896)
A. Rubinowicz (1917, 1924)
R.S. Ingarden (1955)

Miyamoto and Wolf (1962): Miyamoto-Wolf diffraction wave differs from a Young's wave, i.e., "wave motion diverging from a line element of the diffracting edge does not depend only upon the local behavior of the incident wave at the particular element."

"The Scalar Diffraction Theory":

"The Helmholtz-Huygens Principle" (1):

Each regular solution of the Helmholtz equation can be expressed by the Helmholtz-Huygens principle for a region R of space bounded by a closed surface S

u(P), an surface integral of Vector field

**V**(P,Q) over S where P, an observation point in R & Q, an integration point on S

**V**(P,Q) is equal to the Vector Potential

**W**(P,Q) through a curl operation.

"Modified Helmholtz-Huygens principle (Miyamoto-Wolf 1962):

"

**W**(P,Q) must have, as a function of Q, at least one singularity on any surface surrounding the point P if u(P) is regular within S."; &

"

**W**(P,Q) must have singularities also in space points which are regular points of u(P)."

"…the singularities of a vector potential for a given solution of the Helmholtz equation are not uniquely determined…":

thus, probably these singularities have no certain physical meaning.

"Diffraction Wave for an Arbitrary Incident Wave" (2):

Kirchoff representation (1882) of wave motion in the shadow half-space:

u

_{K}(P) = ∫

_{A} curl

_{Q} **W**(P,Q) ∙

**n**dS

_{Q}.

Using Stokes' theorem & no singularities of

**W**(P,Q) at the Q-points on the surface A, we obtain the diffraction wave as:

u

_{d}(P) = ∫

_{Γ}**W**(P,Q) ∙ d

**s**_{Q}where Γ is the diffracting edge.

Interpretation: wave motion is determined by the contributions of the different line elements d

**s**_{Q} of the diffracting edge Γ.

If singularities exist at some Q points on A, the integral expression for u

_{d}(P) is obtained plus integrals representing contributions from the singularities:

Characteristics of these singularity cases are discussed in this section.

"The Miyamoto-Wolf Derivation of the Vector Potential

**W**(P,Q)" (3)

"A Geometric Derivation of the Vector Potential

**W**(P,Q)" (6):

Figure 1, Conical frustum bounded by the covering surface σ and the lateral area λ:

Infinite conical frustum with the vertex at point P_{0} where the point light source is
located;
λ is shadow boundary of σ, open surface;
Rim Γ, a bounding curve of σ & λ;
Q is a point on the rim Γ & on rim's line element ds_{Q}; &
A & A' are generators of surface λ that pass through the endpoints of d**s**_{Q} where A & Q
intersect.

"Physical Properties of the Vector Potential

**W**(P,Q) and of the Corresponding Diffraction Wave" (10):

"The whole diffracted wave u

_{d}(P) is determined in point P by the retarded values of the incident wave in the shadow boundary of a point source of light in this point."

No general transformation has been found that would convert the Miyamoto-Wolf diffraction wave into a Young wave.

"Reflection Cone" (12):

Rubinowicz (1924, 1938) applied the method of stationary phase to show for the spherical wave case [1st approximation, optical waves (large k)] that:

The diffracted wave is generated by a reflection of the incident light from the diffracting edge;

Similar to the case of a reflecting wire, the incident light is reflected into
reflection half-cones;
For the diffracting edge, the amplitude of the reflected light is modulated by a
directional factor;
Significant contributions will stem only from the neighbourhoods of 'active'
points Q_{j} on the diffracting edge at which the phase k(r + S) has a stationary value
(from other points, interference occurs);

At Q

_{j}, cos(

**i**_{r},

**i**_{s}) = -cos(

**i**S,

**i**_{s})|grad S|

where

**i**_{r} is an unit vector in the direction of

**r** = PQ,

**i**_{s} is an unit vector in the direction of the line element dS

_{Q},

**i**_{S} is unit normal to the phase surface S(Q) (the direction of incidence of the incident wave) ;

**Reflection cone:**
Its axis of symmetry is defined by ds_{Q};
Angle of incidence, angle between **i**_{S} & **i**_{s};
Angle of reflection, angle between a generator of the reflection cone and its
axis of symmetry ds_{Q} (supplementary to the angle between **i**_{s} & **i**_{r}); &
"these two angles are equal only in the geometric-optical case |grad S| = 1."

Approximate integral formulations for the diffracted wave u

_{d}(P) & u

_{d\Qf}(P) are derived but are only valid some distance from the shadow boundary

"Diffraction Theory of Electromagnetic and Dirac-Electron Waves"
"Conclusion"

**[17]**Incident wave – diffracting edge – diffracted wave

Very important idea: a Miyamoto-Wolf diffraction wave differs from a Young's wave

Reflection cone: an derived approximate result of a model of diffraction of an spherical incident light wave (the case of the reflection of a spherical incident light wave by a diffracting edge)

Is the shining of a diffracting edge as viewed from the object's shadow real?

**18. Klein and Furtak's 1986 Optics book:****Fermat's Principle** (light & index of refraction):

Optical path length (OPL), mathematically, a line integral

"Any deviation of the path from that taken by the true ray, that is, that is, of first order in small quantitites, will produce a deviation in optical path length that is at least second order in small quantities."

Deviant path - Virtual path

"The true ray is that for which the optical path is an extremum with respect to small deviation in OPL.":

the actual path is stationary with respect to small changes in OPL.

**Diffraction Theory**.

**[18]**Fermat's principle is presented relative to the Ray Model of Light energy propagation, an approximate representation of the Propagation of Electromagnetic waves of Light.

**19. Incremental Length Diffraction Coefficients (ILDC) 1974 Report to U.S. Air Force:**by K.M. Mitzer, Northrop employee

"Calculation of Radar Cross Section"

Dec 1 1970 – June 1 1973 Work

Goal: "develop a semi-automated system for computing the radar cross section (RCS) of aerospace vehicles over the frequency range of 500-20,000 MHz."

Cites USAF translated Ufimtsev 1962 monograph, reference No. 16

3-D bodies with edges. Perfect conductors.
Scattering associated with incremental length of edge (ILEDC)
2-D Diffraction coefficients
Well-known edge diffraction results of Ufimtsev (PTD)
Geometric wave & fringe wave diffraction (Surface current based diffraction)

Page 3: “Solutions calculated by this approach are compared with experimental results in Section 4.2 and it is confirmed that the approach is accurate over a wide range of conditions."

Focus: ILDC & solution of 2-D canonical problems

Physical optics

Page 4: “We note, however, that efficient techniques for the evaluation of physical optics integrals are crucial to the development of PTD as a practical tool for the treatment of realistic scattering problems and, indeed, much of the advantage of PTD over GTD lies in the ability to treat physical optics diffraction phenomena which have no two-dimensional counterpart.”

**[19]****20. Ufimtsev's August 1957 paper on Diffraction (by a Wedge & by a Ribbon):**“New method for the approximate computation of the scattering of plane electromagnetic waves at certain ideally-conducting convex bodies whose surface has bends.”

Received July 30, 1956.

**[20]**Concept expression (Requires Small Wavelength) for computation of diffraction field: Sum of fields of Uniform & Non-uniform currents

The concept is essentially an expression of the

**Electromagnetic Finite Element** approach.

Complex (Complicated) body composed of many small elements of irradiated surface

A Body element is modeled as a flat conducting plane at the point of tangency and a small wedge if the element is on a curved surface (vicinity of point of maximum curvature).

Diffraction (Total) field: Linear sum of a large number of small elements’ individual fields

Describes fundamental drawback of the expressed method: Lack of solution of Problem of Secondary diffraction: interaction of fringe currents (Non-uniform currents)

First translated paper of Ufimtsev

Complex mathematics and No experimental results are presented

**21. Ufimtsev‘s second Diffraction (by a Strip) paper (March 1958): **Interaction of edge currents

Introduces concept of Scattering object as a series of light sources

Key problem is to find functions that will determine the gradual changes in the field of each source (light-dark boundary).

Secondary and Tertiary Diffraction

Angle parameter (s) specification for Radar diffraction

“In conclusion, I should like to express thanks to L.A. Vainshtein for his guidance.”

Received March 25, 1957.

**[21]**Complex mathematics and No Experimental results are presented.

From this paper, we learned of L.A. Vainshtein involvement in this work.

The UCI SL volume that possesses this paper contains “ASTIA Library” & “US Air Force” stamps.

ASTIA: Armed Services Technical Information Agency

**22. Ufimtsev‘s third Diffraction (by a Disk) paper (March 1958): **Interaction of edge currents

Two Point Sources

Small luminous region at the edge of the disk

Secondary and Tertiary Diffraction

“I take this opportunity to express thanks to L.A. Vainshtein for his guidance.”

Received March 25, 1957.

**[22]**Complex mathematics; Experimental results (Figure 3) are presented but where did the results come from???

From this paper, we learned of L.A. Vainshtein involvement in this work.

The UCI SL volume that possesses this paper contains “ASTIA Library” & “US Air Force” stamps.

**23. Ufimtsev‘s fourth Diffraction (by a Disk & by a Finite Cylinder) paper (November 1958):**Two "luminous" points on the rim of the disk -> "luminous" segment

On page 2391, Ufimtsev references W. Braunbek’s 1950 work.

“The results obtained allow us to hope that this method will make it possible to solve problems for more complicated bodies.”

“I take this occasion to express my gratitude to L.A. Vainshtein, who directed this work.”

Received July 30, 1956

**[23]**Complex mathematics; Experimental results (Figure 7) are presented but where did these results come from???

The UCI SL volume that possesses this paper contains “US Air Force” & “ASTIA Library” stamps.

It appears that L.A. Vainshtein played a huge role in Ufimtsev’s first four papers (work leading up to 1956 & 1957 received dates of papers).

These four papers do not list where the work of Ufimtsev’s paper was done.

**Note that the US Air Force stamps imply that the US Air Force personnel were aware of these Ufimtsev four papers in 1957 and 1958.****24. Ufimtsev's 1962 monograph on Diffraction (PTD) (Translated in 1971 by USAF):**FOREWORD by L.A. Vaynshteyn:

Elegant description of "physical theory of diffraction"

Currents become excited on surface of a body: Geometric Optics (uniform part)

Additional currents arising in vicinity of edges or borders = edge waves & rapidly attenuates with increasing distance from edge or border (nonuniform part of the current)

One may find the radiation field created by the additional currents by comparing the edge or border with the edge of an infinite wedge or the border of a half-plane.

Secondary Diffraction

**Physical theory of diffraction allows for an advance into longer wave region:**

Obtain results "of interest for radar"Applicable to resonating structures & diffraction (pages vi & vii)

Additional currents (# of cases) give main contribution to radiation field

Final equations are not asymptotic in the strict sense of the word. Complicated slowly varying function describe the decay of fields & currents

INTRODUCTION

Description of physical theory of diffraction approach:

An Electromagnetic Finite Element approach (requires small wavelength of incident radiation)

Physical Optics approach: surface current; irradiated part of body's surface = element

j0 uniform current density

j1 additional current density result from curve of surface:

i) additional current concentrated mainly in vicinity of the boundary between illuminated & shadowed parts of body's surface but

ii) additional current also arises near edge, bend or point of a body.

j1 is comparable to j0

Key statements about non-uniform current distribution in regards to Physical representation of nonuniform part of current:

No explicit mathematical expression cited.

"This part of current is generally not expressed in terms of well-known functions."

Fringing field created by non-uniform part of current.
Direct integration: "very complicated & immense equations"
Basic indirect consideration without direct integration of it

Issues of multiple order diffraction

Key idea: "Approximate solutions of diffraction problems would be impossible without the use of results obtained in the Mathematical Theory of Diffraction."

The rigorous solution to the problem of diffraction by a wedge is widely used in this book:

Attributed to Sommerfeld via Reference 16, a Russian source published in 1937 (appears to be related to the works of P. Frank and R. von Mises, indirect source to the work of Sommerfeld).

Contributions of Fok (1945 & 1946 papers) & Vainshtein (1953 paper): Vainshtein's key work, diffraction, open end of waveguide & shadowing by the opposite end of waveguide

BODY of MONOGRAPH

"Diffraction by a Wedge" (Chapter 1):

The rigorous solution to the problem of diffraction of a wedge (first derived by Sommerfeld) is based on the method of branching wave functions;

Graphic derivation by Sommerfeld:

Contour integral is transformed into a series;

Ufimtsev proposes a reverse direction: the solution is first in the form of series & then in its integral representation, "such a path seems to us more graphic";

Ufimtsev's claims: "Necessity for a detailed derivation needed" because the Sommerfeld's results "are not represented in a sufficiently clear form, which hinders their use."

"Brief Review of the Literature" (Section 25):

W. Braunk (1950 & 1954 papers):

Diffraction of a scalar wave by a circular hole in a flat screen;

The approximate solution is in the form of a surface integral, where its boundary values of the integrand were taken from the Sommerfeld's rigorous solution to the problem of diffraction by a half-plane (pages 155-156).

Diffraction works for 3-D bodies having edges are comparatively scarce (page 159)

"Diffraction problems arising in antenna theory are usually distinguished by their great complexity since their corresponding metal bodies (mirror, horn, etc.) have a complicated shape." (page 160)

These bodies' dimensions are larger than transmission wavelength.

Application of physical theory of diffraction to antenna problems: "very promising"

Diffraction of short wave by smooth bodies: Fok & Leontovich fundamental works (pages 160-161)

CONCLUSION

Solution of number of diffraction problems

Radar reflection application

Reciprocity principle

Various kinds of interpolation & simplified equations

Admission of a limited number of diffraction problems that have a solutions due to theoretical studies

L.A Vaynshteyn's contribution to this work.

92 References (some in Russian, German, & English)

Ufimtsev & Vainshtein clearly has knowledge about radar principles.

**[24]**No disclosure of Ufimtsev's affiliation for this document.

Edited Translation

Why does Ufimtsev refer to an indirect source for A. Sommerfeld' landmark works on wedge diffraction and not to A. Sommerfeld original publications/works?

Beginning of RESULTS PAPER derived from RESEARCH on PROBLEMS of DIFFRACTION, SCATTERING, and PROPAGATION of WAVES.