1

Prof. David R. JacksonECE Dept.

Spring 2014

Notes 37

ECE 6341

2

Line Source on a Half Space

1/2 1/22 2 2 20 0 1 1y x y xk k k k k k

There are branch points at 0 1,x xk k k k

r

y

x

00 0

0

11

4y x

z z

jk y jk xTEz x x

y

E j A

IA k e e dk

j k

01

1

TE

y

Zk

00

0

TE

y

Zk

1 0

1 0

TE TEx xTE

x TE TEx x

Z k Z kk

Z k Z k

3

Steepest-Descent Transformation

0 0 0sin cosx yk k k k

There are no branch points in the plane from ky0 (cos is analytic).

There are still branch points in the plane from ky1:

Steepest-descent transformation:

1/22 2 21 1 0 sinyk k k

1sin n

4

Line Source on a Grounded Substrate

1/2 1/22 2 2 20 0 1 1y x y xk k k k k k

There are branch points only at 0xk k

00 0

0

11

4y x

z z

jk y jk xTEz x x

y

E j A

IA k e e dk

j k

01

1

TE

y

Zk

00

0

TE

y

Zk

0

0

TE TEin x xTE

x TE TEin x x

Z k Z kk

Z k Z k

r

y

x

1 1tanTE TEin x yZ k jZ k h

(even function of ky1)

5

Steepest-Descent Path Physics

0 0 0sin cosx yk k k k

There are no branch points in the plane (cos is analytic).

Both sheets of the kx plane get mapped into a single sheet of the plane.

Steepest-descent transformation:

We focus on the grounded substrate problem for the remaining discussion.

6

Steepest-Descent Path Physics

Examine ky0 to see where the plane is proper and improper:

0 0

0

cos

cos cosh sin sinh

y r i

r i r i

k k j

k j

0 0Im sin sinhy r ik k

0

0

: Im 0

: Im 0

y

y

k

k

Proper

Improper

7

SDP Physics (cont.)

0 0Im sin sinhy r ik k

P: properI: improper i

I

2

r

2

I

IIP P

P P

C

0

0

: Im 0

: Im 0

y

y

k

k

Proper

Improper

8

SDP Physics (cont.)

0 0sin sin cosh cos sinhx r i r ik k k j Mapping of quadrants

in kx plane

Non-physical “growing” LW pole

(conjugate solution)

xrk

xik

12

3 4

0k0k

/ 2 / 2conjr r r

conji i

i

I

2

r

2

LWP

SWP

I

II

PP

PP

4

1

1

4

3

2

2

3

C

(symmetric about /2 line)

*LWxp xk k

9

SDP Physics (cont.)

SDP: cos cosh 1 r i

A leaky-wave pole is considered to be physical if it is captured when

deforming to the SDP (otherwise, there is no direct residue contribution).

i

2

r

2

SDP

LWP

SWP

C

0

10

SDP Physics (cont.)

LWP:

0

3/2

jk x

z

eE A

x

SDP:

2 ResLWxjk x

zE j e (exists if pole is captured)

The leaky-wave field is important if:

1) The pole is captured (the pole is said to be “physical”).2) The residue is strong enough.3) The attenuation constant is small.

Comparison of Fields:

LWxk j

11

SDP Physics (cont.)

LWP captured: b

b rp

Note:

The angle b represents the

boundary for which the leaky-wave field exists.

i

r

SDP

b rp

p rp ipj LWP

12

SDP Physics (cont.)

0

cos0 cos

y xjk y jk x

z x x

j k

C

E F k e e dk

F e k d

cos

02 Res cos pj kLWz p pE j F k e

Behavior of LW field:

0xp y pjk x jk yLWzE Ae e

In rectangular coordinates:

LWxp xk k j where

(It is an inhomogeneous plane-wave field.)

13

SDP Physics (cont.)

Examine the exponential term:

cos cos

cos cosh sin sinh

p rp ip

rp ip rp ip

j

j

Hence

0

0

sin sinh

sinh sin

rp ip

ip rp

k

k

e

e

cos pj ke

0ip since

14

Radially decaying:

rp

SDP Physics (cont.) 0 sinh sinip rpk

e

LW exists:

b

b rp

Also, recall that LW exists

r

rp

b

y

x

LW decays radially

Line source

15

Power Flow

0

0 0

0

ˆ ˆRe Re( )

ˆ ˆRe sin cos

ˆ ˆsin cosh cos cosh

x y

rp ip rp ip

rp ip rp ip

k x k yk

x k j y k j

k x y

0

x

y

Power flows in the direction of the vector.

16

Power Flow (cont.)

0 rp Hence

0tan tanxrp

y

Note that

0 ˆ ˆsin cosh cos coshrp ip rp ipk x y

Also, 0 0ˆ ˆsin cosx y

rp

r

y

x

b

0

Note: There is no

amplitude change along the rays ( is perpendicular to in a lossless region).

17

ESDP (Extreme SDP)

/ 2

ESDP

i

r

2

Fast

Slow

Set

We can show that the ESDP divides the LW region into slow-wave and fast-wave regions.

The ESDP is important for evaluating the fields on the interface (which determines the far-field pattern).

18

ESDP (cont.)

(SDP)

(ESDP)

cos cosh 1

sin cosh 1

r i

r i

Recall that

0

0

sin

sin

xp p

rp ip

k k

k j

0

Re

sin cosh

xp

rp ip

k

k

To see this:

Hence

19

ESDP (cont.)

Fast-wave region:

Slow-wave region:

0

sin cosh rp ipk

0

1

k

0

1

k

Hence

sin cosh 1r i

sin cosh 1rp ip

sin cosh 1rp ip

Compare with ESDP:

20

ESDP (cont.)

/ 2

i

r

ESDP

2

Fast

Slow

The ESDP thus establishes that for fields on the interface, a leaky-wave pole is physical (captured) if it is a fast wave.

LWP captured

LWP not captured

SWP

21

SDP in kx Plane

0

0

sin

sin

x

r i

k

k j

k

0

0

sin cosh

cos sinh

xr r i

xi r i

k k

k k

cos cosh 1r i SDP:

We now examine the shape of the SDP in the kx plane.

The above equations allow us to numerically plot the shape of

the SDP in the kx plane.

so that

22

SDP in kx Plane (cont.)

2

(Please see the appendix for a proof.)

LW

SDP

xik

0kxrk

0 sinxk k

C

SW

23

Fields on Interface

2

The leaky-wave pole is captured if it is in the fast-wave region.

LW

ESDP

xik

0kxrk

SW

fast-wave region

The SDP is now a lot simpler!

24

Fields on Interface (cont.)

SW CS

z z z

SW LW RWz z z

E E E

E E E

The contribution from the ESDP is called the “space-wave” field or the “residual-wave” (RW) field.

(It is similar to the lateral wave in the half-space problem.)

2

LW

ESDP

xik

0kxrk

SW

25

Asymptotic Evaluation of “Residual-Wave” Field

( 0)xjk xRWz x x

EDSP

E F k e dk y

Use

0x

x

k k js

dk j ds

xik

0k

xrk

- + s

26

Define

0

0

00

00

jk xRW sxz

jk x sx

E je F k js e ds

je F k js e ds

H s F s F s

0

0

00

0

0

jk xRW sxz

jk x sx

E je F k js e ds

je F k js e ds

Asymptotic Evaluation of “Residual-Wave” Field (cont.)

27

Then

0

0

jk xRW sxzE je H s e ds

x x for

Assume ~ 0H s As s as

0

1

1~ jk xRW

z

AE je

x

Watson’s lemma (alternative form):

We then have

Asymptotic Evaluation of “Residual-Wave” Field (cont.)

28

It turns out that for the line-source problem at an interface,

1

2

Hence0

3/2

3~

2

jk xRWz

eE j A

x

Note: For a dipole source we have0

1 2

jkRWz

eE A

Asymptotic Evaluation of “Residual-Wave” Field (cont.)

29

Discussion of Asymptotic Methods

We have now seen two ways to asymptotically evaluate the fields on

an interface as x for a line source on a grounded substrate:

1) Steepest-descent ( ) plane

2) Wavenumber (kx) plane

There are no branch points in the steepest-descent plane. The function f ( ) is analytic at the saddle point 0 = = /2, but is zero there. The fields on the interface correspond to a higher-order saddle-point evaluation.

The SDP becomes an integration along a vertical path that descends from the branch point at kx = k0. The integrand is not

analytic at the endpoint of integration (branch point) since there is a square-root behavior at the branch point. Watson’s lemma is used to asymptotically evaluate the integral.

30

Summary of Waves

0

3/2~

jk xRWz RW

eE A

x

LWxjk xLW

z LWE A eSWxjk xSW

z SWE A e

x

y

LW

SWRW

Continuous spectrum

LW LW LWxk j SW SW

xk

31

Interpretation of RW Field

The residual-wave (RW) field is actually a sum of lateral-wave fields.

x

y

c

32

Proof of angle property:

0tan

tan

xr xr

xi xi

r i

k k k

k k

Hence ~ r

Appendix: Proof of Angle Property

0

0

sin cosh

cos sinh

xr r i

xi r i

k k

k k

The last identity follows from

or ~ r

2

tanxrr

xi

k

k

33

As

Hence

2

i

r

~2

2

Proof (cont.)

On SDP:

~2

or

(the asymptote)

2

0u

r

2

0u

0u

0u

0u

i

SDP

SAP

2

2

34

2

ESDP:2

0

Hence

To see which choice is correct:

In the kx plane, this corresponds to a vertical line for which

Proof (cont.)