september 2003©2003 by h.l. bertoni1 vii. diffraction by an absorbing half-screen kirchhoff-huygens...

$: September 2003©2003 by H.L. Bertoni1 VII. Diffraction by an Absorbing Half-Screen Kirchhoff-Huygens Approximation for Plane Wave Diffraction by an Edge$
September 2003 ©2003 by H.L. Bertoni 1

VII. Diffraction by an Absorbing Half-Screen

• Kirchhoff-Huygens Approximation for Plane Wave Diffraction by an Edge

• Geometrical Theory of Diffraction• Uniform Theory of Diffraction


Plane Wave Illumination of an Absorbing Half-Plane

E x,y,0( ) =jk4π

E0 1+cosα( )e−jkr

rd ′ y d ′ z

−∞

∞

∫0

∞

∫

where

r = x2 + y− ′ y ( )2+ ′ z ( )

2

zx

y

r

dy

dz

€

E in =E0e−jkx

( x, y, 0 )


Evaluation of the Integration Over z

€

Let r = x2 +(y−y')2 +(z')2 = ρR2 +(z')2 where ρR = x2 +(y− ′ y )2

In the exponent r ≈ρR +( ′ z )2 2ρR

In the denominator r ≈ρR

Then

1+cosα( )e−jkr

rd ′ z

−∞

∞

∫ ≈1+cosα

ρR

e−jkρR exp−jk

2ρR

( ′ z )2⎡

⎣ ⎢ ⎤

⎦ ⎥ d′ z

−∞

∞

∫

Let u=z'e j π 4 k 2ρR so that dz'= e−j π 4 2ρR k( )du and then

1+cosα( )e−jkr

rd ′ z

−∞

∞

∫ =1+cosα

ρR

e−jkρR e−j π 4 2ρR

ke−u2

du−∞

∞

∫

=1+cosα

ρR

e−jkρR e−j π 4 2πk

π

• The Contribution to the integral come from a small region about z = 0


Evaluation of the Integration Over y

E x,y,0( ) =E0

jk4π

1+cosαρR

e−jkρR e−j π 4 2πk

d ′ y 0

∞

∫

=E0

ejπ 4

2k

2π1+cosα

ρR

e−jkρRd ′ y 0

∞

∫

where ρR = x2 +(y− ′ y )2 , cosα =x ρR

We distinguish two cases that are most easily solved

1) Well inside the region y > 0 illuminated by the plane wave.

2) Well inside the shadow region y < 0.R

Xy’

y

R

X y’

y


Inside the Illuminated Region y > 0

€

E x,y,0( ) =E0

e j π 4

2k

2π1+cosα

ρR

e−jkρR d ′ y −1+cosα

ρR

e−jkρR d ′ y −∞

0

∫−∞

∞

∫⎧ ⎨ ⎩

⎫ ⎬ ⎭

Contribution to the first integral comes from ′ y near y

Contribution to the second integral comes from ′ y near 0

y

y

Re1+cosα

ρR

e−jkρR⎡

⎣ ⎢ ⎤

⎦ ⎥

Interruptedcancellation

Cancellation ofalternate half cycles


Evaluating y Integral for y > 0

€

First Integral: use ρR = x2 +(y−y')2 and expand about y'=y

ρR ≈x+(y− ′ y )2

2x in exponent

ρR ≈x and cosα ≈1 in amplitude

Second Integral: ρR = ρ2 −2y ′ y +( ′ y )2 where ρ≡ x2 +y2 and expand

about y'=0

ρR ≈ρ−y ′ y ρ

in exponent

ρR ≈ρ and cosα ≈x/ρ in amplitude

E x,y,0( ) =E0e j π 4

2k

2π2x

e−jkx e−j k

2x(y− ′ y )2

d ′ y −1+x/ρ

ρe−jkρ e

jkyρ

′ y

d ′ y −∞

0

∫−∞

∞

∫⎧ ⎨ ⎩

⎫ ⎬ ⎭


Evaluating y Integral for y > 0 - cont.

E x,y,0( ) =E0ejπ 4

2k

2π2x

e−jkx e−j

k2x

(y− ′ y )2

d ′ y −1+x/ ρ

ρe−jkρ e

jkyρ

′ y d ′ y

−∞

0

∫−∞

∞

∫⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

Let −u =ejπ 4 k (2x)(y−y' ) =1

jky ρe

jkyρ

′ y ⎤

⎦ ⎥ ⎥ −∞

0

du=ejπ 4 k (2x)dy' =ρ jky( )

e−j

k2x

(y− ′ y )2

d ′ y =-∞

∞

∫ e−jπ 4 2xk

e−u2du=e−jπ 4 2x

k−∞

∞

∫ π


Evaluating y Integral for y > 0 - cont.

E x,y,0( ) =E0ejπ 4

2k

2π2x

e−jkx e−jπ 4 2πxk

⎛

⎝ ⎜ ⎞

⎠ ⎟ −

1+x/ ρρ

e−jkρ ρjky

⎛

⎝ ⎜ ⎞

⎠ ⎟

⎧ ⎨ ⎩

⎫ ⎬ ⎭

=E0e−jkx+E0e

−jπ 4 e−jkρ

ρD θ( )

where y ρ =sinθ, x ρ =cosθ

and

D θ( ) =−12πk

1+cosθ2sinθ

The first term is the incident plane wave.

The second term is a cylindrical wave

propagating away from the edge

with pattern function D θ( ).

Y

x

Incident plane wave


Inside the Shadow Region y < 0

€

E x,y,0( ) =E0

e j π 4

2k

2π1+cosα

ρR

e−jkρR d ′ y 0

∞

∫

Contribution to the integral comes from ′ y near 0

yy

Re1+cosα

ρR

e−jkρR⎡

⎣ ⎢ ⎤

⎦ ⎥

Interruptedcancellation

Cancellation ofalternate half cycles


Evaluating y Integral for y < 0

€

ρR ≈ x2 +(y− ′ y )2 = ρ2 −2yy'+(y')2 ≈ ρ2 −2yy' where ρ≡ x2 +y2

In the exponent use ρR ≈ρ−y ′ y ρ

in exponent

In amplitude use ρR ≈ρ and cosα ≈x/ρ=cosθ

E x,y,0( )=E0

ej π 4

2k

2π1+x/ρ

ρe−jkρ e

jkyρ

′ y

d ′ y 0

∞

∫

Since ejk

yρ

′ y d ′ y

0

∞

∫ =ρjky

exp jkyρ

y'⎛

⎝ ⎜ ⎞

⎠

⎤

⎦ ⎥ 0

∞

=−1

jkyρ=

e−j π 2

−ky ρ( )

Then E x,y,0( )=E0e−jπ /4 e−jkρ

ρD θ( )

Where D θ( )=12

k2π

1+x/ρ−ky ρ( )

=−1

2πk

1+cosθ2sinθ

This is a cylindrical wave propagating away from the edge with pattern function D(θ)


Geometrical Theory of Diffraction (GTD)Total field

EGTD =EG0U θ( )+ED

Incident plane wave

EG0 =E0e−jkx

Diffracted wave

ED =E0e−jπ / 4 e−jkρ

ρD θ( )

D θ( ) =−12πk

1+cosθ2sinθ

GTD is not valid near the shadow boundary θ =0 where D θ( ) =∞.

()

y

x

Shadowboundary

DiffractedCylindrical wave

IncidentPlanewave


GTD Valid Outside Transition Region

Fresnelzone

Shadowboundary

Incidentplanewave

y

x

€

Example: If x =10 m and f =1GHz, λx = 3=1.7 m.

λx

€

GTD is not valid in

the transition region


Example of Shadowing at Building Corners

Building

y

x

FromTransmitter

2 m

For f =1 GHz at y =−2 m

in shadow region

ρ = 102 +22 =10.2

θ =arctanyx

⎛ ⎝

⎞ ⎠

=−11.3°

k =2πf / c=20.9

€

ED =E0

1

10.2

1

2π ×20.9

1+cos11.32sin11.3

=0.138E0

−20log(0.138)=17 dB

Signal at 2 m into the shdow region is 17 dB below incident signal.


Uniform Theory of Diffraction for Small y

€

ρR = x2 +y2 −2y ′ y +( ′ y )2

In exponent use Taylor series expansion in ′ y up to second order

ρR ≈ρ−yρ

( ′ y )+x2

2ρ3 ( ′ y )2 where ρ = x2 +y2

In amplitude ρR =ρ, cosα ≈x/ρ=cosθ

End point integrals for ED :

(y>0) ED =E0

e jπ /4

2k

2π−

1+cosθ

ρe−jkρ⎡

⎣ ⎢ ⎤

⎦ ⎥ exp−jk

x2

2ρ3 ( ′ y )2 −yρ

( ′ y )⎡

⎣ ⎢ ⎤

⎦ ⎥ ⎧ ⎨ ⎩

⎫ ⎬ ⎭ d ′ y

−∞

0

∫

=E0ejπ /4

2k

2π−

1+cosθρ

e−jkρ⎡

⎣ ⎢ ⎤

⎦ ⎥ exp−jk

x2

2ρ3 ′ y 2+

y

ρ′ y

⎡

⎣ ⎢ ⎤

⎦ ⎥ ⎧ ⎨ ⎩

⎫ ⎬ ⎭ d ′ y

0

∞

∫

(y<0) ED =E0

e jπ /4

2k

2π1+cosθ

ρe−jkρ⎡

⎣ ⎢ ⎤

⎦ ⎥ exp −jk

x2

2ρ3 ( ′ y )2 +yρ

′ y ⎡

⎣ ⎢ ⎤

⎦ ⎥ ⎧ ⎨ ⎩

⎫ ⎬ ⎭ d ′ y

0

∞

∫


Evaluating End Point Integrals

€

For both y>0 and y <0

ED =E0

e jπ /4

2k

2π−sgn(y)

1+cosθρ

e−jkρ⎡

⎣ ⎢ ⎤

⎦ ⎥ exp0

∞

∫ −jkx2

2ρ3 y'2+

y

ρy'

⎡

⎣ ⎢ ⎤

⎦ ⎥ ⎧ ⎨ ⎩

⎫ ⎬ ⎭ dy'

Complete the square of the exponent

kx2

2ρ3 y'2+

yρ

y'⎡

⎣ ⎢ ⎤

⎦ ⎥ =

xρ

k2ρ

y'+kρ2

yx

⎡

⎣ ⎢ ⎤

⎦ ⎥

2

−kρ2

yx

⎛

⎝ ⎜ ⎞

⎠ ⎟

2

Define s=kρ2

y

x

⎛

⎝ ⎜ ⎞

⎠ ⎟

2

so that - jkx2

2ρ3 y'2+

yρ

y'⎡

⎣ ⎢ ⎤

⎦ ⎥ =−j

xρ

k2ρ

y'+ s⎡

⎣ ⎢ ⎤

⎦ ⎥

2

+js


Evaluating End Point Integrals - cont.

€

Then

ED =E0

ejπ /4

2k

2π−sgn(y)

1+cosθ

ρe−jkρ⎡

⎣ ⎢ ⎤

⎦ ⎥ e js exp−j

xρ

k2ρ

′ y + s⎛

⎝ ⎜ ⎞

⎠ ⎟

2⎡

⎣ ⎢

⎤

⎦ ⎥ d ′ y

0

∞

∫

Using variable u=xρ

k2ρ

′ y + s so that d ′ y =ρx

2ρk

du=s

y ρ( )

2k

du

ED =E0

ejπ /4

j2kk

2π−sgn(y)

1+cosθy ρ( ) ρ

e−jkρ⎡

⎣ ⎢

⎤

⎦ ⎥ 2j se js exp−ju2

( )dus

∞

∫⎡

⎣ ⎢ ⎤

⎦ ⎥

or

ED =E0e−jπ /4

2πk−

1+cosθ2sinθ

e−jkρ

ρ

⎡

⎣ ⎢ ⎤

⎦ ⎥ 2j sejs exp−ju2

( )dus

∞

∫⎡

⎣ ⎢ ⎤

⎦ ⎥


Evaluating End Point Integrals - cont.

€

Thus

ED =E0e−jπ /4

2πk−1+cosθ2sinθ

⎡ ⎣

⎤ ⎦ e−jkρ

ρF s( )=E0e

−jπ /4 e−jkρ

ρD θ( )F s( )[ ]

where F s( )=2j se js e−ju2

dus

∞

∫

is a transition function that eliminates the singularity at θ=0.

Since s=kρ2

y

x

⎛

⎝ ⎜ ⎞

⎠ ⎟

2

=kρ2

tanθ( )2 so that s>>1 when θ is large enough.


Approximation for F(s)

€

Rational Approximation F s( ) = 2πs f 2s/π( )+jg 2s/π( ){ }

Here f ξ( ) ≈1+0.926ξ

2+1.792ξ +3.104ξ2

gξ( ) ≈1

2+4.142ξ +3.492ξ2 +6.670ξ3

where ξ = 2sπ

For s→ 0, F s( ) ≈ 2πse jπ /4

2=

πkρ2

y

xe jπ /4


Field at the Shadow Boundary

€

For y→ 0, ρ ≈x, cosθ ≈1, sinθ ≈y x so that D(θ) ≈-1

2πk y x( )

Then ED ≈E0e−jπ /4 −1

2πk1

y x⎡

⎣ ⎢ ⎤

⎦ ⎥ ⎧ ⎨ ⎩

⎫ ⎬ ⎭

e−jkx

xπ kx

2yx

ejπ /4⎧ ⎨ ⎩

⎫ ⎬ ⎭

≈12

E0e−jkx −y

y

⎛

⎝ ⎜ ⎞

⎠ ⎟

For y=0+

E =E0e−jkx+ED =E0e

−jkx 1−12

⎛ ⎝

⎞ ⎠ =

12

E0e−jkx

For y=0−

E =ED =12

E0e−jkx

Total field is continuous at y=0 and is 12

incident field, or 6 dB lower

than the fiel in the absence of the edge.


Value of F(s) for Large s

€

Rational Approximation F s( ) = 2πs f 2s/π( )+jg 2s/π( ){ }

When s→ ∞, f ξ( ) ≈1+0.926ξ

2+1.792ξ +3.104ξ2 →0.3ξ

=0.3π2s

gξ( ) ≈1

2+4.142ξ +3.492ξ2 +6.670ξ3 → 0

F(s)→ 2πs 0.3π2s

⎛

⎝ ⎜ ⎞

⎠ ⎟ =0.3π ≈1

On the edge of the transition region y = λx, s=kρ2

yx

⎛ ⎝

⎞ ⎠

2

=kλ2

ρx

=π

and ξ = 2. Since f ( 2)=0.215, g( 2)=0.0297 and

F(π)= 2π(π) 0.215+j0.0297[ ] ≈0.937

Outside the transition region F(s) ≈1


Variation of Field Near Shadow Boundary

f =900 MHz

12

Width of Fresnel zone

λx =303

=3.16

Y

x

x= 30 m

-10 -5 0 5 10-30

-25

-20

-15

-10

-5

0

5

Rec

eive

d S

igna

l(dB

)

y(meters) WF-WF

september 2003©2003 by h.l. bertoni1 vii. diffraction by an absorbing half-screen kirchhoff-huygens...

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