high-frequency incremental techniques for scattering and

ALBERTO TOCCAFONDIDept. of Information EngineeringUniversity of Siena - Italy

HighHigh--frequency Incremental Techniques frequency Incremental Techniques for Scattering and Diffractionfor Scattering and Diffraction

Siena, Siena, FebruaryFebruary 23, 200523, 2005

SUMMARYSUMMARY

•• BackgroundBackgroundBasic terminologyGTD versus PTDBasic concepts of Incremental Theories

•• IncrementalIncremental TheoryTheory of of DiffractionDiffraction (ITD)(ITD)Localization processWedge-shaped configurationsCylindrical configurationDouble edge diffraction

•• Overwiev Overwiev of of Incremental TheoriesIncremental TheoriesCurrent based methodsField based methods

HF SCATTERING FROM LARGE OBJECTSHF SCATTERING FROM LARGE OBJECTS

design and analysis of antennas (e.g., reflectors)

An efficient and accurate description of scattering phenomena at high frequency is of interests in many applications.

Rx

prediction of Radar Cross Section

propagation in complex environments (e.g., urban propagation)

antenna installation and inter-antenna coupling (e.g., on ships, aircrafts, space platforms…)

HF EDGE DIFFRACTION TECHNIQUESHF EDGE DIFFRACTION TECHNIQUES The most important theory that has dominated this scenario in past

few decades is:

Geometrical Theory of Diffraction (GTD) in its Geometrical Theory of Diffraction (GTD) in its Uniform version (UTD, UAT)Uniform version (UTD, UAT)

Both UTD and PTD are ray-optical theories: at high frequency, the scattering from a complex object may be obtained as a superposition of rays emanating from “isolated” flash points.

* P. Ya Ufimtsev “Method of edge waves in the physical theory of diffraction” Soviet Radio Publication House, 1962. English Version: Air Force Foreign Technology Division, FTD-HC-23-259-71, Sept. 1971 (normal incidence).

Physical Theory of Diffraction (PTD*) in its original Physical Theory of Diffraction (PTD*) in its original versionversion

Other important theory is:

GEOMETRICAL THEORY OF DIFFRACTIONGEOMETRICAL THEORY OF DIFFRACTION The exact field is the sum of a geometrical optics field (GO) and a

diffracted field.

The space surrounding the object is divided into regions separated by the reflection and shadow boundaries.

go diff= +E E E The GO field accounts for

direct, and reflected ray-fields.

The diffracted field is the contribution which must be added to the GO to obtain the exact field.

The GTD field provides an asymptotic approximation of the diffracted field.

diff gtdE E∼

RSB

ISB

PHYSICAL THEORY OF DIFFRACTIONPHYSICAL THEORY OF DIFFRACTION

The scattered field is the sum of a physical optics field (PO) and a fringe wave field.

i po fw= + +E E E E The PO field arises from the

induced surface currents on the object . poJ

The FW field arises from the non uniform surface currents on the objects .nuJ

poJ nuJ

Ufimtsev’s basic approach:

“First, we will investigate the rigorous solution of this problem(canonical). Then we will find its solution in the physical optics approach. The difference of these solutions determines the field created by the non uniform part of the current (fringe wave current).”

PHYSICAL THEORY OF DIFFRACTIONPHYSICAL THEORY OF DIFFRACTION

The NU currents must be added to the PO currents to obtain the total currents, that radiates the scattered field.

po nu= +J J J

poJ fwJ

The PTD field provides an asymptotic approximation of the fringe wave field.

fw ptdE E∼

One major advantage of the PO approach is that both the incident and the PO fields are continuous across the RSB and ISB. As a consequence, also the FW field and its asymptotic approximation (PTD field) are continuous across the same SB.

GTD VERSUS PTDGTD VERSUS PTD

The PO surface contribution is the leading term of the asymptotic evaluation of the PO surface radiation integral for k → ∞.

As a consequence, the sum of the fringe wave and the PO diffracted field is equal to the diffracted field.

The PO field is divided into a PO surface contribution and a PO diffracted field .

posE

podE

po po pos d= +E E E

poJ

po rs =E E

0pos =E

po is = −E E

po fw diffd + =E E E fw diff po

d= −E E E

GTD VERSUS PTDGTD VERSUS PTD

The PO edge-diffracted contribution is the leading term of the asymptotic evaluation of the PO radiation integral for k → ∞.

The difference between the GTD field and the PO edge-diffracted field is the PTD field.

The PO diffracted field is asymptotically approximated by the PO edge-diffracted contribution .po

edEpo pod edE E∼

ptd gtd poed= −E E Efw diff po

d= −E E E

At high frequency, GO and PO predictions may be augmented by GTD and PTD field respectively.

go gtd

i po ptd

⎧ +⎨

+ +⎩

E EE

E E E∼

RAY METHODS AND CANONICAL PROBLEMSRAY METHODS AND CANONICAL PROBLEMS

The locality principle is essential: when a radiating object is large in terms of a wavelength the scattering and diffraction is found to be a local phenomenon,i.e., it depends only on the nature of the boundary surface and the incident field in the immediate neighborhood of the “flash” points.

Both GTD and PTD are ray-optical methods: upon an identification of stationary surface and edge points (ray tracing), the proper ray-field is associated to each ray.

The actual scatterer is substituted by local canonical configurations at the points of reflection and diffraction.

P

Q' L

S

z

LIMITATIONSLIMITATIONS

Observation point is located at a caustic: all points along the edge, or on a finite portion, are stationary.

Ray-Optical methods are not applicable when isolated stationary edgepoints cannot be identifies.

Observation points is located outside the cones of diffraction: no edge stationary points exists.

near field caustic

near field caustic

caustic at infinity

caustic at infinity

RCS of poligonal flat plate

INCREMENTAL THEORIES FOR EDGED BODIESINCREMENTAL THEORIES FOR EDGED BODIES

The original idea is that an illuminating field undergoes a diffraction process at each incremental element of the edge (Young, Maggi, Sommerfeld).

The observed “field pattern” (ex: diffracted field) is thus caused by the interaction between the “superposition” of elementary contributions distributed along the actual edge, and the GO field.

The mathematical formulation of this “idea” may be asymptotically expressed as a line integral along the edge contour.

( ) ( )diff diffe

l

dl∫E r F r∼ scatterer

edge contour l

eβ

'eβ

Q’e

( , )i iE Hˆ 's

ˆes P

r

rx

y

z

er

er

e e= −R r r

INCREMENTAL THEORIES FOR EDGED BODIESINCREMENTAL THEORIES FOR EDGED BODIES

At high frequency, each incremental field contribution is estimated from a local uniform, infinite, canonical configuration that more appropriately models the actual discontinuity.

Q' L

S

z

The two prevailing techniques to derive incremental contributions are the Current based and the Field based methods.

( )diffc eF r

( )diffeF r

( ) ( )diff diffe

l

dl∫E r F r∼

Incremental fields contributions are deduce from the currents of local canonical problems.

Incremental Length Diffraction Coefficients (ILDC):

K. M. Mitzner, Northrop Corp., Tech. Rep., Apr. 1974R. A Shore and A. D Yaghjian, IEEE AP, Jan. 1988.A. D. Yaghjian......

Equivalent Edge Currents (EEC):

A. Michaeli, IEEE AP, Mar. 1984A. Michaeli, IEEE AP, July,1986.

Equivalent Edge Wave (EEW):

D. I. Bouturin and P Ya. Ufimtsev, Sov. Phys. Acoust. Jul.-Aug. 1986.P. Ya. Ufimtsev, Electromagnetics, Apr.-June 1991.

CURRENT BASED METHODSCURRENT BASED METHODS

Incremental fields contributions are deduce from the field of local canonical problems.

Incremental Diffracted Field:

A. Rubinowicz, Acta Phisica Polonica, 1965.

Equivalent Current Method (ECM):

R. F. Millar, Proc. IEE , Mar. 1956.C. E. Ryan and L. Peters, Jr., IEEE AP, May 1969.E.F. Knott and T. B. A. Senior, IEEE AP, Sept.1973.

Incremental Theory of Diffraction (ITD):

R. Tiberio, S. Maci, IEEE AP, May 1994. R. Tiberio, S. Maci, and A. Toccafondi IEEE AP, 1996-2004.

FIELD BASED METHODSFIELD BASED METHODS

EQUIVALENT CURRENT METHODEQUIVALENT CURRENT METHOD The incremental field attributed to the edge element at re may be

thought as generated by traveling-wave electric Ie and magnetic Me line current, located at re and directed along the edge tangent .

( )diffc eF r

e

Q’e

ˆ 's

ˆes

er

eMeI

e

e

ˆ ˆ ˆ ˆ ˆ( ) [ ( ) ( ') ( ) ( ')]4

jkdiffc e e e e e e

jk es s e I Q s e M Qζπ

−

= × × + ×R

F rR

e e= −R r r

The constant propagation factor is .'cos ek β

It assumed that the value of Ie and Me depend linearly on the incident electric and magnetic field.

EQUIVALENT CURRENT METHODEQUIVALENT CURRENT METHOD Each traveling-wave current radiates a conical wave

e

e

/ 4'

e

/ 4'

e

ˆ ( ) ( ')sin8

ˆ ( ) ( ')sin8

jkjI

e e

jkjM

e e

k e ee P I Qk

ke ee P M Qk

π

π

ζ βπ

βζ π

−

−

⋅ −

⋅ −

R

R

ER

HR

∼

∼

If and , the components of the GTD edge-diffracted field at P by a local canonical configuration at Q’, can be approximated as:

e λR e eρR

e

e

' '

e

' '

e

ˆ ˆ( ) [ ( ')] ( , , )

ˆ ˆ( ) [ ( ')] ( , , )

jkd i

s e e e

jkd i

h e e e

ee P e Q D

ee P e Q D

φ φ β

φ φ β

−

−

⋅ ⋅

⋅ ⋅

R

R

E ER

H HR

∼

∼

Q’e

ˆ 's

es

er

eMeI

e e= −R r r

EQUIVALENT CURRENT METHODEQUIVALENT CURRENT METHOD Comparing the above expressions outside the shadow boundary

transition region:/ 4

' ''

/ 4' '

'

ˆ8 [ ( ')]( ') ( , , )sin

ˆ8 [ ( ')]( ') ( , , )sin

j i

e s e e ee

j i

e h e e ee

e e QI Q Dk

e e QM Q Dk

π

π

π φ φ ββζ

ζ π φ φ ββ

−

−

⋅= −

⋅= −

E

HQ’e

ˆ 's

es

er

eM

eI

e e= −R r r

The current are “equivalent” since it depends on the angles of incidence and observation at the point of diffraction.

Unlike GTD, the fields radiated by EC remain valid even within the caustic regions of edge-diffracted rays.

These currents are defined only for observation points lying on the Keller cone of half-angle . '

eβ

An heuristic extension to include points outside the Keller cone have been suggested (Knott and Senior).

/ 4' '

'

/ 4' '

'

ˆ8 [ ( ')]( ') ( , , , )sin sin

ˆ8 [ ( ')]( ') ( , , , )sin sin

j i

e s e e e e

e e

j i

e h e e e e

e e

e e QI Q Dk

e e QM Q Dk

π

π

π φ φ β βζ β β

ζ π φ φ β ββ β

−

−

⋅= −

⋅= −

E

H

EQUIVALENT EDGE CURRENTEQUIVALENT EDGE CURRENT The diffracted field by a local canonical configuration (pec) locally

tangent at Q’ is (half-plane):

sr

s= −R r r

( )sJ r

z

x

y rˆ ˆ( ) [ ( )]

4c

jkdiffc s

S

eP jk dSζπ

−

× ×∫∫R

E R R J rR

∼

total induced electric current density

The surface radiation integral is approximated in far field as:

ˆ ˆ( )

0

ˆ ˆ ˆ ˆ( ) ( ) ( )4

jkrdiff jk r u uc s

eP jk u e r r e du dzr

ζπ

∞ ∞−⋅

−∞

⎡ ⎤× × ×⎢ ⎥

⎣ ⎦∫ ∫E J r∼

e

u( )sJ r

z

x

y r

incremental strip

Q’ e

Q’

The incremental diffracted field associated to Q’ is the end-point contribution to the field generated by the surface current density on an infinite extended incremental strip, starting at that edge point.

cS

EQUIVALENT EDGE CURRENTEQUIVALENT EDGE CURRENT

ˆ ˆ( )2

0

ˆ ˆ( )2

0

ˆ ˆ ˆ ˆ ˆ( ') ( ) ( )ˆ ˆ

ˆ ˆ ˆ ˆ( ') ( )ˆ ˆ

jk r u us

jk r u us

u eI Q r e r e dur e

u eM Q e r e dur e

ζ

∞⋅

∞⋅

×= ⋅ × ××

×= ⋅ ××

∫

∫

J r

J r

It is found that the equivalent edge current are

When the total current is applied, the above expressions provide an asymptotic approximation to the diffracted field.

( ') ( )

( ') ( )

gtd Is

gtd Ms

I Q LM Q L

=

=

J rJ r

( ') ( )

( ') ( )

po I poed s

po M poed s

I Q LM Q L

=

=

J rJ r

( ') ( )

( ') ( )

ptd I nus

ptd M nus

I Q LM Q L

=

=

J rJ r

When the FW (non uniform) surface current density is applied, the above expressions provide an asymptotic approximation to the PTD field.

When the PO surface current density is applied, the above expressions provide an asymptotic approximation to the PO edge-diffracted field.

Originally the strip was chosen normal to the edge

Michaeli (1984) derived a set of EEC for the wedge diffracted field.

The crosspolar component vanish at stationary point . 'β β=

The incremental contributions are singular on a half-cone with tip at the edge, axis along the incremental strip and interior tip angle . 'pθ β=

ˆ ˆu x=

zQ’

'β

EQUIVALENT EDGE CURRENTEQUIVALENT EDGE CURRENT

1 1ˆ ˆ( ') ( ')

ˆ( ')

i I i IE H

i MH

I Q e D Q e Djk jk

M Q e Djk

ζζ

⎡ ⎤ ⎡ ⎤= ⋅ + ⋅⎣ ⎦ ⎣ ⎦

⎡ ⎤= ⋅⎣ ⎦

E H

H

Valid for arbitrary direction of observation.

INCREMENTAL LEGTH DIFFRACTION COEFFICIENTSINCREMENTAL LEGTH DIFFRACTION COEFFICIENTS

Mitzner (1984) first considered the fringe wave surface current densityalong the strips normal to the edge.

He provided expressions in terms of Incremental LenghtDiffraction Coefficients (ILDC).

ˆ ˆu x=

zQ’

'β

' ' '

' '

( ')( ')

0 ( ') 4

ptd i jkrptd

ptd i

D DF E Q eQDF E Q r

ββ βφβ β

φφφ φ π

−⎡ ⎤ ⎡ ⎤⎡ ⎤= =⎢ ⎥ ⎢ ⎥⎢ ⎥

⎣ ⎦⎣ ⎦ ⎣ ⎦F

The ILDC’s are singular on a half-cone with tip at the edge, axis along the incremental strip and interior tip angle . 'pθ β=

It is found (Knott, 1985) that if this coefficients is cast in the form of EEC, the two methods are equivalents (they differ by terms due to PO).

REDUCTION OF SINGULARITIESREDUCTION OF SINGULARITIES In general, for arbitrary strip orientation, the singularities lie on a half-

cone along the strip with interior angle:

[ ]ˆ ˆˆ2arccos (cos ' sin ' )p x z uθ β β= + ⋅

To reduce the number of singular directions, Michaeli (1986) and Ufimtsev (1991) considered the elementary strip in the direction of the “grazing diffracted ray”.

ˆ ˆˆ(cos ' sin ' ) 0px z uβ β θ+ = → =

The resulting incremental fields are singular only along the strip direction, when the incident field is grazing.

x

z

Q’

'β

'β'β

PTDPTD--EECEEC

Michaeli derived a set EEC by asymptotic evaluation of the fringe wave current radiation integral for the canonical wedge solution.

x

z

Q’

'β

'β'β

It exhibits a nonnon--reciprocalreciprocal behavior w.r.t. the aspects of incidence and observation.

(Half-Plane)

2

sin sin 'cos cos (cos ' cos )sin '

β β φ β β βµβ

+ −=

PTDPTD--EEWEEW Ufimtsev (1991) derived a set PTD incremental diffraction coefficients

based on the concept of Elementary Edge Wave.

EEWs are the waves scattered “…by the vicinity of an infinitesimal element of a curved edge.”

They are calculated as the spherical edge waves radiated by the nonuniform currents flowing along the elementary strips

It exhibits a nonnon--reciprocalreciprocal behavior w.r.t. the aspects of incidence and observation.

High-frequency asymptotics are provided in the form:

x

z

'β

'β'βQ’

1 1ˆ ˆ( ) ( ', ) ; 2

jkrptd ptd ptd

E e H EeP D Q r r

rπ ζ

−

= = ×F F F

ˆ ˆ ˆ ˆ ˆ( ', ) ( ') ( ', ) ( ') ( ', )i ieD Q r Q e Q r Q e Q rζ⎡ ⎤ ⎡ ⎤= ⋅ + ⋅⎣ ⎦ ⎣ ⎦E M H N

OTHER CONTRIBUTIONSOTHER CONTRIBUTIONS

Ando (1991) introduced a set of GTD equivalent edge currents in order to completely eliminates the false singularities, in which the orientation of the strip depends on the observation aspects.

Shore and Yaghjian (1989) developed a simple substitution method for obtaining ILDC’s directly from the 2-D scattered far fields of PEC scatterer.

It allows to obtain the ILDCs without any integration, or differentiation or specific knowledge of the currents.

ILDC’s for aperture (Michaeli 1995), strips and slits (Yaghjian 1997) as well as for cylindrical scatterer (Yaghjian 2001) have also been proposed.

Explicit ITD formulations have been obtained for:

INCREMENTAL THEORY OF DIFFRACTIONINCREMENTAL THEORY OF DIFFRACTION

Pec local wedges. Pec local thin circular cylinders.Double local edges

Heuristic ITD formulations have been obtained for:

Local edges in impedance planar surfaces. Local edges in thin coated and dielectric panels.

ITD LOCALIZATION PROCESSITD LOCALIZATION PROCESS

Q' L

S

( ) ( ')diff diff

L

P Q dl= ∫E F Diffracted field

the incremental field can be deduced from an appropriate local canonical problem

Q' L

S

z

( ) ( '') ''diff dc cP z dz

+∞

−∞

= ∫E F

Find a convenient representation for the diffracted field of the canonical problem.

Assume that at high-frequency'' 0

( ') ( '')diff dc z

Q z=

=F F

Reciprocity suggests

Q'

S

z

'' ( ')3 2 '' ''1( , ') ( , ', )2

zjk z zD DzG r r G k e dkρρ ρ

π

∞− −

−∞

= ∫

'' ( ')3 '' '' ''1( , ') ( , ) ( , ')2

zjk z zDzG r r L U k U k e dkρ ρρ ρ

π

∞− −

−∞

⎡ ⎤= ⋅⎢ ⎥

⎣ ⎦∫

For an object, with a uniform cylindrical canonical configuration along the z-axis, by spectral synthesis:

A USEFUL REPRESENTATIONA USEFUL REPRESENTATION

It is found that G2D may be represented as the result of the application of a linear operator L[].

By Fourier analysis, this spectral integral representation is interpreted as the spatial convolution product of two functions:

( )3 , ' ( '' , ) ( ' '', ') ''DG r r L u z z u z z dzρ ρ∞

−∞

⎡ ⎤= − ⋅ −⎢ ⎥

⎣ ⎦∫

d 1 1 'c '' 0

F ( '') ( , ) ( ', ') ( , ) ( , ')- -

zz u z u z F U k F U kρ ρρ ρ ρ ρ

=⎡ ⎤⎡ ⎤= − ⋅ = ⋅⎣ ⎦ ⎣ ⎦

This above spatial integral representation allows to directly define the local incremental field contribution

Inverse Fourier transform

INCREMENTAL FIELD CONTRIBUTIONINCREMENTAL FIELD CONTRIBUTION'' ( ')3 '' '' ''1( , ') ( , ) ( , ')

2zjk z zD

zG r r L U k U k e dkρ ρρ ρπ

∞− −

−∞

⎡ ⎤= ⋅⎢ ⎥

⎣ ⎦∫

P

Q'

S

z

φφ'

ββ'

nπ

This "FT-Convolution" process is directly applicable to the exact solution for the wedge

Plane wave spectral representation

'' ''

'

cos 'cos '2 '' 1( , '; ) ( , '; , ') '2

jk jkD e

c c

G k G e e d dρ ρ

α α

ρ α ρ αρρ ρ α α φ φ α α

π− −= ∫ ∫

WEDGEWEDGE--SHAPED CONFIGURATIONSSHAPED CONFIGURATIONS

'' ( ')3 2 '' ''1( , ') ( , ', )2

zjk z zD DzG r r G k e dkρρ ρ

π

∞− −

−∞

= ∫

ITD diffracted field contribution

'

2

2( ') ( , '; , ') ( , ) ( ', ') '2( 2 )

d ec

c c

kdF Q G u z u z d dj

α α

α α φ φ ρ ρ α απ

= −∫ ∫' 'cos '' '1( ', ')

2zjk jk z

zu z k e e dkρ ρ αρρ

π

∞−

−∞

= ∫

' '

[ ( , , ) ' ( ', ', ')]( ) [( '), , '] sin sin ' ' 'd e jk rf r fc l

C C C C

dF Q G e d d d dα α θ θ

α θ β α θ βα α φ φ θ θ θ θ α α− += −∫ ∫ ∫ ∫

Spectral representation (EM case)

1 cos coscossin sin

θ βαθ β

−= 1 cos 'cos 'cos 'sin 'sin '

θ βαθ β

−=

(') (') (') (') (') (') (') (') (')( , , ) sin sin cos cos cos cosf α θ β β θ α β θ γ= + =

The desired stationary phase condition is achieved at(')cos 1γ =

ASYMPTOTIC ANALYSIS (I)ASYMPTOTIC ANALYSIS (I)

Two more condition are required

(Rubinowicz)

2sin sin sin 'θ β β=

Also

ASYMPTOTIC ANALYSIS (II)ASYMPTOTIC ANALYSIS (II) In the asymptotic analysis, the structure of the spectrum (spectral synthesis)

requires.

independent of , 'θ θ( ')υ α α= −

( ) 1 cos cos 'cos 'sin sin '

β βα αβ β

−− =

cos cos 'sin( ')sin sin '

j β βα αβ β−− =

'θ θ= This is achieve d by ( analytic continuation of the diffraction cone).

The value of at the stationary point is:( ')υ α α= −

Q'

S

z

φφ'

ββ'

nπ

1 cos cos 'cossin sin '

β βυβ β

−=

explicitly satisfies reciprocity

well-behaved at any aspect, including observation points lying on theaxis of the local canonical wedge.

'φφ −=Φ±

ITD SOLUTION FOR WEDGE PROBLEMITD SOLUTION FOR WEDGE PROBLEM

ITD stationary phase point

'

'

( ')( , , ') 0( ')

( ')0 ( , , ') 2

d jkred

c dh

E QF eQE QF r

ββ

φφ

υ φ φυ φ φ π

−⎡ ⎤ ⎡ ⎤⎡ ⎤= =⎢ ⎥ ⎢ ⎥⎢ ⎥

⎣ ⎦ ⎣ ⎦⎣ ⎦F

DD

( ), ( , , ') [ , ] , [ , ]e h n nD Dυ φ φ υ υ− += Φ − + ΦD

sin1[ , ]2 cos cos

nnd

nn n

π χ

υ χπ χυ

−⎛ ⎞⎜ ⎟⎝ ⎠=

−⎛ ⎞⎛ ⎞−⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

[ , ] [ , ] [ , ]n n nD d dυ υ υ± ± ±Φ = Φ + −Φ

ITD vs. UTDITD vs. UTD When a stationary phase condition has been well established, the

integration of ITD incremental fields smoothly blends into the field predicted by the UTD.

UTDUTD

ITDITD

ITD vs. MOM, UTDITD vs. MOM, UTD Far-zone pattern of a circular disc

PTD vs. MOM, UTDPTD vs. MOM, UTD Far-zone pattern of a circular disc

The GO ray field is well-behaved within its domain of existence, the total field is

( ')ggo d

lQ dl= + ∫E E F

( ') ( ')i po d podl

Q Q dl⎡ ⎤= + + −⎣ ⎦∫E E E F F

The GO ray field regime has not been well established or when anaugmentation of the PO field is desired

TWO ALTERNATIVE APPROACHESTWO ALTERNATIVE APPROACHES

ggo i po po god= + −E Ε E E E∼ diff utdE E∼

It is more convenient from numerical point of view

Note: the fringe contributions are different from those of PTD, only the terminology is the same

fw ptd gtd poed= −E E E E∼

Well-behaved at any aspect

Tends to remedy the typical non-reciprocity of PO.

ITD FRINGE FORMULATION (I)ITD FRINGE FORMULATION (I)

( ')s po f

lQ dl= + ∫E E F

P

Q' l

S

z

( ') ( ') ( ') ( ')f f d poc c edQ Q Q Q= −F F F F∼

The incremental PO edge-diffracted field is obtained by applying the ITD localization process to a local canonical problem with PO currents.

( ')poed QF

A localized incremental, fringe field contribution is obtained as

( ')po poed ed

l

Q dl= ∫E F

The PO edge-diffracted field is obtained by integration of the incremental PO end-point contributions

They are found by applying the ITD localization process to the local canonical problem of a plane half-lit and half-shadowed

( )posQF

( ')poed QF

PO EDGEPO EDGE--DIFFRACTED FIELDDIFFRACTED FIELD

'φφ −=Φ±

'11 12

'22

( )( , , ') ( , , ')( ')

( ) 20 ( , , ')

po po po jkrlpo

ed po pol

E QF eQE QF r

ββ

φφ

υ φ φ υ φ φπυ φ φ

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤= = ⋅⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦⎣ ⎦⎣ ⎦F

D DD

11 sin[ , ]2 cos cos

d υυ

ΩΩ =+ Ω

cos ( ') 1 sin sin 'sgn z zθ β β= + −

1 111

1 122

12

( ; , ') [ , ] [ , ]

( ; , ') [ , ] [ , ]

( ; , ') cos

po

po

po

d d

d d

υ φ φ υ υ

υ φ φ υ υ

υ φ φ θ

− +

− +

= Φ − Φ

= Φ + Φ

=

D

D

D

INCREMENTAL PO EDGEINCREMENTAL PO EDGE--DIFFRACTED FIELDDIFFRACTED FIELD

1 cos cos 'cossin sin '

β βυβ β

−= ITD stationary phase point

P

Q'

S

z

φφ'

ββ'

x

y

'φφ −=Φ±

'11 12

'22

( ')( , , ') ( , , ')( ')

( ') 20 ( , , ')

f f f jkrf

c f f

E QF eQE QF r

ββ

φφ

υ φ φ υ φ φπυ φ φ

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤= = ⋅⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦⎣ ⎦⎣ ⎦F

D DD

( )( ) ( )2

cos 21[ , ]2 cos 2 sin 2

fd υυ

ΩΩ =

+ Ω

cos ( ') 1 sin sin 'sgn z zθ β β= + −

11 2 2

22 2 2

12

( ; , ') [ , ] [ , ]

( ; , ') [ , ] [ , ]

( ; , ') cos

f f f

f f f

f

d d

d d

υ φ φ υ υ

υ φ φ υ υ

υ φ φ θ

− +

− +

= Φ − Φ

= Φ + Φ

= −

D

D

D

ITD FRINGE FORMULATION FOR A LOCAL HALFITD FRINGE FORMULATION FOR A LOCAL HALF--PLANEPLANE

FRINGE ITD vs. PTDFRINGE ITD vs. PTD

ITD fringe augmentation tends to remedy the typical non-reciprocity of current based methods


Circular CylindersCircular Cylinders

INCREMENTAL SCATTERED FIELD CONTRIBUTIONINCREMENTAL SCATTERED FIELD CONTRIBUTION

( ) ( ) ( )t i sP P PΨ = Ψ + Ψ

Scattered field

( ) ( )sl

l

P Q dlψΨ = ∫

The incremental field may be deduced from an appropriate local canonical problem tangent at Ql. To this purpose, we need to find the convenient field representation for the local canonical problem

( ) ( )cs

cP z dzψ∞

−∞

Ψ = ∫ Then, at high-frequency, it is assumed that

0( ) ( )l c zQ zψ ψ

==

Total field (scalar formulation)

LOCAL CANONICAL PROBLEMLOCAL CANONICAL PROBLEM

where

( )1( , ) ( , )8

s jn sc n

nr r e r r

jφ φ

π

∞′− −

=−∞

′ ′Ψ = − Ψ∑

, ''( , , )e hnG kρρ ρ′

Canonical solution (spectral synthesis)

''

(2) ''( , ) ''

''

(2) ''

( )soft b.c. ( )

( )( )

' ( )hard b.c. ( )

' ( )

n

ne hn

n

n

J k ae

H k ac k a

J k ah

H k a

ρ

ρρ

ρ

ρ

⎧⎪⎪= ⎨⎪⎪⎩

and '' 2 '' 2( )zk k kρ = −

( ) ( ) ( ) ( ) ( ), , (2) (2)1r, r 2

zjk z ze h e hn n n n zc k a H k H k e dkρ ρ ρψ ρ ρ

π

∞′′ ′− −

−∞

′ ′′ ′′ ′′ ′ ′′= ∫

THE APPROPRIATE SPECTRUM FUNCTIONS (I)THE APPROPRIATE SPECTRUM FUNCTIONS (I)

Soft boundary conditions

'' (1) '''' '' ''

(2) '' (2) ''

( ) ( )2 ( ) 2 1 1 ( ) ( )

( ) ( )n

n ne e en n

n n

J k a H k ac k a k a k a

H k a H k aρ ρ

ρ ρ ρρ ρ

σ σ ∗= = + = +

(1) ''''

(2) ''

( )( )

( )ne

nn

H k ak a

H k aρ

ρρ

σ =

(1)2 ( )

(2)

( )( )

nj k an

n

H k ae

H k aρϑρ

ρ

=for kρ real ( )( ) nj k aen k a e ρϑ

ρσ =

Thus, the typical n-th term in the integrand of the exact solution is

, '' '' '' (2) '' (2) ''1( , , ) 1 ( ) ( ) ( ) ( )2

e h e en n n n nG k k a k a H k H kρ ρ ρ ρ ρρ ρ σ σ ρ ρ∗′ ′⎡ ⎤= +⎣ ⎦

THE APPROPRIATE SPECTRUM FUNCTIONS (II)THE APPROPRIATE SPECTRUM FUNCTIONS (II)

This leads to the useful representation

(1) (2)1( , ', ) ( , , ) ( , , ) 2n n nG k G k G kρ ρ ρρ ρ ρ ρ ρ ρ′ ′⎡ ⎤= +⎣ ⎦

where

(1) (2) (2)( , , ) ( ) ( )n n nG k H k H kρ ρ ρρ ρ ρ ρ′ ′=

( )( ) ( real)nj k aen k a e kρϑ

ρ ρσ =

(2) (2) (2)( , , ) ( ) ( ) ( ) ( )e en n n n nG k k a H k k a H kρ ρ ρ ρ ρρ ρ σ ρ σ ρ∗′ ′⎡ ⎤ ⎡ ⎤= ⎣ ⎦ ⎣ ⎦

INCREMENTAL CONTRIBUTIONINCREMENTAL CONTRIBUTION

Using it in the ITD-FT convolution process leads to

(Reciprocity)

by inverse FT

(1) (2)( ) ( ) ( )n l n l n lQ Q Qψ ψ ψ= +

( )( ) ( ) ( )2

1 ( ) ( )(2 )

z zj k z k zi i in n n z zU k U k e dk dkρ ρψ ρ ρ

π

∞ ∞′ ′− −

−∞ −∞

′ ′ ′= ∫ ∫

with( ) ( ) ( ) (2) ( ) ( )

( )

1 1( ) ( ) ; ( )

2( )i

n nen

U k H k ik aρ ρ

ρ

ρ ρσ

⎧ ⎫ ⎧ ⎫⎪ ⎪′ ′ ′ ′= =⎨ ⎬ ⎨ ⎬′ ⎩ ⎭⎪ ⎪⎩ ⎭

ELECTROMAGNETIC CASEELECTROMAGNETIC CASE

In the electromagnetic case for z-directed dipole illumination and observation

is obtained after replacing in( ),s sz z n

d E d Hζ

( ) ( ) ( )( )inU kρ ρ′ ′

( )inψ

by ( ) ( ) ( ) ( )( )ink U kρ ρ ρ′ ′ ′

( , )e hΨ EM Vector Potentials

( ) ( )2 2 ( , ), e hz z zE H k kζ = − Ψ

Thus,

HIGHHIGH--FREQUENCY EXPRESSIONS (I)FREQUENCY EXPRESSIONS (I)

In order to find tractable expressions for first we use

Next, collecting terms of the product of the two spectrum functions yields the four fold spectral integral representation for each as( ),s s

z z nd E d Hζ

( )inψ

cos(2) 21( ) jk jn jnn

C

H k e e e dρ

α

ρ α α πρ ρ α

π− − −= ∫

[ ]( , ) ( , ), , ( ) 2 2( ) ( ) sin sinjk r re h e h jnn n

C C C C

k a k a e e d d d dθ θ α α

θ α θ αα αρ ρσ σ θ θ α α θ θ

′ ′

′ ′ ′− +′∗ − + ′ ′ ′∫ ∫ ∫ ∫

In the spherical spectral domain ( ) ( )cos , ( , )zk k C j jηθ π′ ′= ≡ − ∞ + ∞

( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( , ) sin sin cos ( / ) cos cosr rθ α β θ α β θ′ ′ ′ ′ ′ ′ ′ ′ ′= + −

Standard ITD form.

2 1 1 cos cossin sin sin ; cossin sins

β βθ β β γβ β

− ′−′= =′

HIGHHIGH--FREQUENCY EXPRESSIONS (II)FREQUENCY EXPRESSIONS (II)

ITD stationary point

1 cos cos; cos( )sin sin

β βθ θ α αβ β

′−′ ′= + =′

The incremental electromagnetic scattered field is,

( )( )

( )( , )

( )(2)

sin sin( , , )

sin sin

ne hn

n

J kac a

H ka

β ββ β

β β

′ ′′ =

′ ′

well-behaved at any incidence and observation aspects

( ) ( , ), ( , , ) sin sinsjkr jkr

s s e hz z nn

jnjn e ed E d H e e c ar r

γπζ β β β β′− −

−⎡ ⎤′ ′= ⎣ ⎦ ′

THE DYADIC SCATTERING COEFFICIENTTHE DYADIC SCATTERING COEFFICIENT

( ) ( , )( , ) ( , , ) 4 ( , , )sjns jn jn e he h n

na j e e e c aγφ φ πβ β β β

∞′− − −

=−∞

′ ′= ∑D

( ) ( , , , ' ) 040 ( , , , ' )( )

s is jkre

ss ih

dE P Ea eradE P E

β β

φ φ

β β φ φπβ β φ φ

−′

′

⎛ ⎞ ⎛ ⎞′⎛ ⎞⎜ ⎟ ⎜ ⎟= ⎜ ⎟⎜ ⎟ ⎜ ⎟′⎝ ⎠⎝ ⎠ ⎝ ⎠

DD

( )( )

( , )

( , )

(2)( )

sin sin( , , ) , sin sin

ne hn

n

J kac a

H ka

β ββ β

β β

′ ′′ =

′ ′

well-behaved at any incidence and observation aspect, including 0β = 0β′ =

The expected transitional behavior of the field is reconstructed by numerical integration of the incremental contributions along the curved axis of the actual cylindrical configuration.

NUMERICAL RESULTSNUMERICAL RESULTS

Results from simulations are compared with a MoM solution (Feko™) Straight uniform cylinder – azimuthal scan

x’ = 6 λr = 8 λ

L = 20 λ

ka = π /5

ka = 2π /5

ka = 3π /5

CPU timesCPU times

ITDITD

MoMMoM

ka = π /5

1.1 s (nav = 3.6)

15.3 s

ka = 2π /5

2.3 s (nav = 3.8)

148.4 s

ka = 3π /5

4 s (nav = 4)

279.8 s


Results from simulations are compared with a MoM solution (Feko™)

x’ = 8 λr = 10 λ

L = 20 λ

Straight tapered cylinder – azimuthal scan

amin = 0.05 λ

amax = 0.4 λ

CPU timesCPU times

ITDITD

MoMMoM

1.2 s (nav = 5.5)

17 min 6 s


Results from simulations are compared with a MoM solution (Feko™)

Circular torus – elevation scan

z’ = 5 λr = 15 λ

a = 0.1 λRT = 8 λ

causticscaustics

CPU timesCPU times

ITDITD

MoMMoM

2.2 s (nav = 3.7)

2 min 25 s


Double Edge DiffractionDouble Edge Diffraction

ITD FORMULATION FOR DOUBLE EDGE DIFFRACTIONITD FORMULATION FOR DOUBLE EDGE DIFFRACTION

( )21dd PE

( )21dd PE

( ) ( ) ( )1

2 2 1 2 1 2 1 1,i d d

lQ Q Q Q dl= = ∫E E F


( )21dd PE

( ) ( )2

2 2 2 2,d d

lP P Q dl= ∫E F


( ) ( ) ( )1

2 2 1 2 1 2 1 1,i d d

lQ Q Q Q dl= = ∫E E F

( ) ( )2 1

21 21 2 1 1 2, ,dd dd

l lP P Q Q dl dl= ∫ ∫E F

( )21dd PE

( ) ( )

( ) ( )1

2

1 1 1 1

2 2 2 2

,

,

d d

l

d d

l

P P Q dl

P P Q dl

=

=

∫∫

E F

E F


( )( )

( ) ( ) ( )( )( )

12 22 1

2 1

2 1 12 2 2 2 12 1 1 1 1

12 22 1 1

, ,, , , ,

2 2, ,

dd i jks jks

dd i

F P Q Q E Q e es sF P Q Q E Q

β β

φ φ

υ φ φ γ υ φ φπ π

− −′

′

⎛ ⎞ ⎛ ⎞′ ′⎜ ⎟ ⎜ ⎟= ⋅ ⋅ ⋅

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

D M D

( ) ( )2 1

21 21 2 1 1 2E , ,dd dd

l lP P Q Q dl dl= ∫ ∫ F

( )2 1 1

2 1 1

12 1212

12 12

cos sinsin cos

i d d

i d d

E E E

E E Eβ β β

φ φ φ

γ γγ

γ γ′

′

⎛ ⎞ ⎛ ⎞ ⎛ ⎞−⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟= ⋅ = ⋅⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠

M


high-frequency incremental techniques for scattering and

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