An Incremental Theory of Diffraction formulation for the scattering by a thin elliptical cylinder, a strip, or a

slit in a conducting surface

* Dept. of Information Engineering, University of Siena, Siena, Italy

M. M. AlbaniAlbani*, A. Toccafondi*, *, A. Toccafondi*, C. Della C. Della Giovampaola*Giovampaola*, D. , D. Erricolo°Erricolo°

2007 USNC / URSI North American Radio Science Meeting

July 22 - 26, 2007, Ottawa, Canada

° Dept. of ECE, University of Illinois at Chicago, Chicago, IL, USA

augments Physical Optics (fringe formulation)

MOTIVATIONSMOTIVATIONS

The Incremental Theory of Diffraction (ITD) formulation has been introduced to evaluate the field scattered by moderately sized, local circular cylinder-shaped structures.

overcomes difficulties in applying in ray-field descriptions close and at causticscaustics

smoothly blends into the field predicted by UTD (when applicable)

improves upon the field estimate whenever a stationary phase condition has not yet been established

Advantages

Incremental field contributions may be deduced from either the currents or the field of local canonical problems

Elementary Edge Waves (PTD)

Incremental Theoryof Diffraction (ITD)

BACKGROUNDBACKGROUND

ELLIPTIC CYLINDER SHAPED CONFIGURATION

To extend the applicability of the ITD approach

( ) ( ) ( )t i sP P PΨ = Ψ + Ψ

Scattered field representation

The incremental field ψ(Ql) may be deduced from an appropriate local canonical problem tangent at Ql.

For this purpose, we need to find a convenient field representation for the exact solution of the local canonical problem

Then, at high-frequency, it is assumed that

ITD LOCALIZATION PROCESSITD LOCALIZATION PROCESS

( ) ( )s ll

P Q dlΨ = ψ∫

( ) ( )cs cP z dz∞

−∞

′′ ′′Ψ = ψ∫

( ) ( )0l c z

Q z′′=

′′=ψ ψ

P

P’

Ql

l

z

l

INCREMENTAL FIELD CONTRIBUTIONINCREMENTAL FIELD CONTRIBUTION

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

( )3 , ' ( , , ) ( ' , ', ')D q q qG r r u z z u v u z z u v dz∞

−∞

= − ⋅ −∫

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

zjk z zDzG r r U k u v U k u v e dkρ ρπ

∞− −

−∞

= ⋅∫

2 ( , , , ', ')DG k u v u vρ

Spectral synthesis

P

P’

Ql zzq z

z'

cosh cos2

sinh sin2

2

dx u v

dy u v

e d a

=

=

= ( ,0)A a≡

(0, )B b≡

,02d

F ≡

x

y

au uv

INCREMENTAL FIELD CONTRIBUTIONINCREMENTAL FIELD CONTRIBUTION

1 1 '

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

q

- -q z

z u z u v u z u v F U k u v F U k u vρ ρψ = = − ⋅ = ⋅

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

Inverse Fourier transform

P

P’

Ql zzq z

z'

( )3 , ' ( , , ) ( ' , ', ')D q q qG r r u z z u v u z z u v dz∞

−∞

= − ⋅ −∫

Canonical solution (spectral synthesis) for an infinite elliptic cylinder with both source and observation point at finite distance

( )2'' 2 ''zk k kρ = −

LOCAL CANONICAL PROBLEMLOCAL CANONICAL PROBLEM

( ) ( ) ( ), , ,e oG G G′ ′ ′= +r r r r r r

( ) ( ) ( )

( ) ( )( )

, , (4), ,

(4), ,

1, , , ,

2 2 2 2

, ' , ' 2 2

n n

z

n n

e o e on n a e o e o

jk z zze o e o

d d dG C k u S k v R k u

d dS k v R k u e dk

ρ ρ ρ

ρ ρ

π

∞

−∞

′′ ′− −

′ ′′ ′′ ′′= ⋅

′′ ′′ ′′

∫r r

( ) ( ), ,( , )0

1, ,e o e one o

n n

G j G∞

=

′ ′= −Ω∑r r r r

( ,0)A a≡

(0, )B b≡

,02d

F ≡

x

y

au

Canonical solution (spectral synthesis) for an infinite elliptic cylinder with both source and observation point at finite distance

LOCAL CANONICAL PROBLEMLOCAL CANONICAL PROBLEM

( ) ( ) ( )

( ) ( )( )

, , (4), ,

(4), ,

1, , , ,

2 2 2 2

, ' , ' 2 2

n n

z

n n

e o e on n a e o e o

jk z zze o e o

d d dG C k u S k v R k u

d dS k v R k u e dk

ρ ρ ρ

ρ ρ

π

∞

−∞

′′ ′− −

′ ′′ ′′ ′′= ⋅

′′ ′′ ′′

∫r r

soft b.c.

(1)( , )

,

(4)( , )

R ,2,

2 R ,2

n

n

e o ae on a

e o a

dk u

dC k u

dk u

ρ

ρ

ρ

′′ ′′ = ′′

modal series representationmodal series representation, rapidly converges for small small (moderately sized)elliptic cylinderselliptic cylinders

(1)'( , )

,

(4) '( , )

R ,2,

2 R ,2

n

n

e o ae on a

e o a

dk u

dC k u

dk u

ρ

ρ

ρ

′′ ′′ = ′′

hard b.c.

first derivative

( ) ( ) ( ) ( ) ( )( ), , (4) , (4)

, , , ,

1r,r , , , , , ' , ' 2 2 2 2 2 2 2

z

n n n n

jk z ze o e o e on n a n a ze o e o e o e o

d d d d d dG k u S k v R k u k u S k v R k u e dkρ ρ ρ ρ ρ ρσ σπ

∞′′ ′− −

−∞

′ ′′ ′′ ′′ ′′ ′′ ′′ ′′= ∫

(1)( , )

, , ,

(4)( , )

R ,2, , ,

2 2 2R ,2

n

n

e o ae o e o e on a n a n a

e o a

dk u

d d dC k u k u k u

dk u

ρ

ρ ρ ρ

ρ

σ σ

′′ ′′ ′′ ′′= = ′′

( )'' , ,nU k u vρ ( )'' , ', 'nU k u vρ

THE APPROPRIATE SPECTRUM FUNCTIONSTHE APPROPRIATE SPECTRUM FUNCTIONS

The GF is cast in the form of the product of 2 spectrum functions, depending only on either the incidence or the observation aspects (soft case)

Use of the ITD FT-convolution process yields

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

z zjk z jk zn n n z n zu z u v u z u v U k u v e dk U k u v e dkρ ρπ π

∞ ∞− −

−∞ −∞

− ⋅ = ⋅∫ ∫

( ) ( ), (4)

, ,

1( , , ) , , ,

2 2 2 2z

n n

jk ze on n a ze o e o

d d du z u v k u S k v R k u e dkρ ρ ρσπ

∞−

−∞

′′− = ∫

Asymptotic expression of ( ) ( )(4), ,ne oR uχ

( )

cosh (2 1) / 4(4) 2

,

1, cosh

2 2cosh

2

n

dj k u n

e o

d dR k u e k u

dk u

ρ π

ρ ρ

ρ

− − + → ∞

;

HIGHHIGH--FREQUENCY EXPRESSIONSFREQUENCY EXPRESSIONS

cosh2d

u ρ→Since for large u

( )(2 1) / 4(4)

,

1,

2nj k l

e o

dR k u e u

kρ ρ π

ρρ ρ

− − + → ∞

;

( )

(2 1) / 4, cos( )

,

1( , , ) , , sin

2 2 2 sinn

j le o jkr

n n a e oC

e d du z u v k u S k v e k d

rθ

πθ β

ρ ρσ θ θπ β

+− − ′′−

∫;

Useful representation of

HIGHHIGH--FREQUENCY EXPRESSIONSFREQUENCY EXPRESSIONS

( , , )nu z u v− for large u

Asymptotic evaluation for large r

( )

( 1) / 2,

,( , , ) sin , sin ,2 22 n

j n jkre o

n n a e o

e d d eu z u v k u S k v

r

π

σ β βπ

+ − −

;

By analogy

( )

( 1) / 2 ',

,( ', ', ') sin ' , sin ' , '2 2 '2 n

j n jkre o

n n a e o

e d d eu z u v k u S k v

r

π

σ β βπ

+ −

;

HIGHHIGH--FREQUENCY EXPRESSIONSFREQUENCY EXPRESSIONS

Scattered scalar field by the actual elliptic cylinder

( )

( )

12

(1)( 1) ( , )

,,0

(4)( , )

12

(1)( , )

,(4)( , )

R sin ,2( ) sin ,

2 2R sin ,2

R sin ' ,2 sin ' , '

2R sin ' ,2

n

nq

n

n

n

n

j n e o ae on q e oz

e o a

e o a

e o

e o a

dk u

e dz S k v

dk u

dk u

dS k v

dk u

π βψ β

π β

ββ

β

+

=

= ⋅

⋅

'

'

jkr jkre er r

− −

Incremental contribution for a soft elliptic cylinder

( ), ,( , )0

1, ( )e o e on le o

n n l

j Q dlψ∞

=

′Ψ = −Ω∑ ∫r r

( ) ( ) ( ), , ,e o′ ′ ′Ψ = Ψ + Ψr r r r r r

P

P’

Ql

l

z

l

In the electromagnetic case for directedẑ

is obtained after replacing in( ),s sz z nd E d Hζ

( )'' (') ('), ,nU k u vρ

,e onψ

by ( )( ) '' (') ('), ,nk U k u vρ ρ′

,,

e os hΨ EM Vector Potentials

( )2

,,2,

j m e oz z s h

kE H j

kρζ ω= − Ψ

Thus,

dipole illumination and observation

ELECTROMAGNETIC CASEELECTROMAGNETIC CASE

,e osΨ,e o

hΨ

jzA

mzF

soft hard

THE DYADIC SCATTERING COEFFICIENTTHE DYADIC SCATTERING COEFFICIENT

Dyadic closedDyadic closed--form highform high--frequencyfrequency expression for the ITD incremental scattered field contribution

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

1 1, , , ( ) ( )

n n

e os h s h l s h le o

n nn n

D j Q j Qβ β φ φ ψ ψ∞ ∞

= =

′ ′ = − −Ω Ω∑ ∑

( )( )

( )( )

( )( )

, , , 00 , , , 4

i jkrls

ih l

dE P E QD eD rdE P E Q

β β

φ φ

β β φ φβ β φ φ π

−′

′

′ ′ = ′ ′

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. 'P

'iEφ

'iEβ

'βlQ

β

sdEβ

sdEφ

P

( 1), , ,

( , ) ( , ) ( , ) ( , ) ( , )( ) sin , sin ' , sin , sin ' , '2 2 2 2 2n n n n nj n

e o e o e os h l s h a s h a e o e o

e d d d dQ k u k u S k v S k v

π

ψ σ β σ β β βπ

+ =

INCREMENTAL SCATTERING BY A STRIPINCREMENTAL SCATTERING BY A STRIP

x y

z

ITD coefficient for a strip

the odd part vanishes (soft case)

( )

( )

( )0

( )0

1, , , ( )

1, , , ( )

n

n

es s le

n n

oh h lo

n n

D j Q

D j Q

β β φ φ ψ

β β φ φ ψ

∞

=

∞

=

′ ′ = −Ω

′ ′ = −Ω

∑

∑

( 1)

( ) sin ,0 sin ' , 0 sin , sin ' , '2 2 2 2 2n n n n n

j ne e es l s s e e

e d d d dQ k k S k v S k v

π

ψ σ β σ β β βπ

+ =

12

(1) (')

(')

(4) (')

R sin ,02sin , 0

2 R sin ,02

n

n

n

ees

e

dk

dk

dk

βσ β

β

=

, 02d

0

21

au

a de

=

==

x

y

the even part vanishes (hard case)

( 1)

( ) sin ,0 sin ' ,0 sin , sin ' , '2 2 2 2 2n n n n n

j no o oh l h h o o

e d d d dQ k k S k v S k v

π

ψ σ β σ β β βπ

+ =

12

(1)' (')

(')

(4)' (')

R sin ,02sin ,0

2 R sin ,02

n

n

n

ooh

o

dk

dk

dk

βσ β

β

=

INCREMENTAL SCATTERING BY A SLITINCREMENTAL SCATTERING BY A SLIT

x y

z, 0

2d

0

21

au

a de

=

==

x

y

ITD coefficient for a slit (dual of the strip)

the even part vanishes (soft case)

( )

( )

( )0

( )0

1, , , ( )

1, , , ( )

n

n

os s lo

n n

eh h le

n n

D j Q

D j Q

β β φ φ ψ

β β φ φ ψ

∞

=

∞

=

′ ′ = −Ω

′ ′ = −Ω

∑

∑

( 1)

( ) sin ,0 sin ' , 0 sin , sin ' , '2 2 2 2 2n n n n n

j no o os l s s o o

e d d d dQ k k S k v S k v

π

ψ σ β σ β β βπ

+ =

12

(1)' (')

(')

(4)' (')

R sin , 02sin , 0

2 R sin , 02

n

n

n

oos

o

dk

dk

dk

βσ β

β

=

the odd part vanishes (hard case)

( 1)

( ) sin ,0 sin ' , 0 sin , sin ' , '2 2 2 2 2n n n n n

j ne e eh l h h e e

e d d d dQ k k S k v S k v

π

ψ σ β σ β β βπ

+ =

12

(1) (')

(')

(4) (')

R sin ,02sin , 0

2 R sin ,02

n

n

n

eeh

e

dk

dk

dk

βσ β

β

=

excitation: electric electric hertzianhertzian dipoledipoleResults from simulations are compared with a MoMMoM solution (Feko™)

#1 #1 straight uniform elliptic cylinder – azimuthal scan

NUMERICAL RESULTSNUMERICAL RESULTS

x

y

z

r'r

0.15a λ= 0.075b λ= 0.26d λ=

0.8662dea

= = 5r λ= ' 7r λ=

-6

-4

-2

0

2

4

6

8

10

0 20 40 60 80 100 120 140 160 180

ITDMoM

| |

(dBV/m)

Eθ

ϕ

' 0φ =

#2 #2 strip – azimuthal scan

NUMERICAL RESULTSNUMERICAL RESULTS

x

y

z

r 'r

d

-2

0

2

4

6

8

-80 -60 -40 -20 0 20 40 60 80

| |

(dBV/m)

Eθ

ϕ

ITDMoM

0.4d λ= 5r λ= ' 7r λ=

'2π

φ =

#3 #3 strip – azimuthal scan

NUMERICAL RESULTSNUMERICAL RESULTS

0.4d λ= 5r λ= ' 7r λ=

x

y

z

r'r

-8

-6

-4

-2

0

2

4

6

8

0 20 40 60 80 100 120 140 160 180

| |

(dBV/m)

Eθ

ϕ

' 0φ =

ITDMoM

ITD provides a quite general procedure for defining incremental field contributions from the exact solution of canonical problems with a uniform cylindrical configuration.

Here, this procedure has been applied to elliptic cylinders, and to strips and slits as particular cases (EM case).

Comparisons with results obtained by MoM simulations have shown that the proposed incremental representation may be used to estimate the scattered field from moderately sized elliptic structures.

Explicit closed form, high-frequency expressions of a dyadic incremental scattering coefficient have been obtained for moderately sized elliptic cylinders, strips and slits.

CONCLUDING REMARKSCONCLUDING REMARKS