Prof. David R. Jackson ECE Dept. Spring 2014 Notes 35 ECE 6341 1

1

Prof. David R. JacksonECE Dept.

Spring 2014

Notes 35

ECE 6341

2

Higher-order Steepest-Descent Method

Assume

g z

C

I f z e dz

0 0f z

This important special case arises when asymptotically evaluating the field along an interface (discussed later in the notes).

r

3

Higher-order SDM (cont.)

Along SDP:

g z

C

I f z e dz

00 g z g zg z

SDP

I e f z e dz

20 ( )s g z g z real

20 ( )g z sdz

I e f z s e dsds

Hence

Note: The variable s is taken as positive after we leave the SP, and negative before we enter the SP.

4


Define

Assume

20 ( )g z sdz

I e f z s e dsds

( )dz

h s f z sds

20g z sI e h s e ds

210 0 0

2h s h h s h s

0 0f z

Then

5


Note: (s is odd)2

0ss e ds

Hence 01

~ 02

g zsI h e I

2 22 2

02s s

sI s e ds s e ds

where

Use2

2

t s

dt s ds

6


0

1

23 02

3 3

2 2

3

2

12

2

1

1 3 1 1 1

2 2 2

1

2

ts

t

tI e dt

t

t e dt

1

0

x tx t e dt

1/ 2

1 ! 1 2 !

1 1

x x x x

x x

Recall:

7


Hence

0

3

2

1 1~ 0

4g zI h e

We now we need to calculate 0h

Note: The leading term of the expansion now behaves as 1 / 3/2.

8


2

3( ) 2

( )

h s f z s z s

h s f z s z s f z s z s

h s f z s z s f z s z s z s

f z s z s z s f z s z s

3

0 0 00 0 3 0 0 0h f z z f z z z f z z

9


Hence 3

0 00 0 3 0 0h f z z f z z z

We next take two derivatives with respect to s in order to calculate 0 , 0z z

2

2

2

s g z z s

g z z s g z z s

(1st)

(2nd)

20s g z g z

10


From (2nd):

or

1

2

0

20z

g z

0

20 SDPjz e

g z

2

0 02 0 0g z z g z z

We then have

0

2 2

arg

SDP

g z

11


3

3

0 2

3

g z z s g z z s z s g z z s z s g z z s

g z z s g z z s z s g z z s

(3rd)

22 g z z s g z z s (2nd)

3

0 0 00 0 3 0 0 0g z z g z z z g z z

2

0 00 0 3 0g z z g z z

or

12


Hence,

02

0

20

3

g zz

g z

2

0

0

010

3

g z zz

g z

1

2

0

20z

g z

Therefore,

where

13


We then have

3

23

00

00 2

0 0

20

2 23

3

SDP

SDP

j

j

h f z eg z

g zf z e

g z g z

14

Higher-order SDM (cont.)Summary

3

23

00

00 2

0 0

20

2 23

3

SDP

SDP

j

j

h f z eg z

g zf z e

g z g z

g z

C

I f z e dz 0 0f z

0

3

2

1 1~ 0

4g zI h e

0

2 2

arg

SDP

g z

15

Example

A line source is located on the interface.

y

0Ix

0

1Semi-infinite lossy earth

16

Example

zA

The field is TMz and also TEy. ˆ zE zE

y

0Ix

0


h

Consider the line source at a height h above the interface.

E j A

TE

17

Example

y

0Ix

0


h

TE

0 0 00 0

0 0

1 1

4y y y xjk y h jk h jk y jk xTE

xy y

Ie e e e dk

j k k

1 0

1 0

TE TETE

TE TE

Z Z

Z Z

00

0

01

1

TE

y

TE

y

Zk

Zk

TEy:

18

Example

00 0 1 0

0 1 0

11

4y x

TE TEjk y jk x

xTE TEy

I Z Ze e dk

j k Z Z

1/22 20 0y xk k k

y

0Ix

0


The line source is now at the interface (h = 0).

zA

19

Example (cont.)

1 0 1

1 0 0 1

21

TE TE TE

TE TE TE TE

Z Z Z

Z Z Z Z

00

0

01

1

TE

y

TE

y

Zk

Zk

1/22 20 0y xk k k

1/22 21 1y xk k k

Simplifying,

20

Example (cont.)

1 01

0 1 0 1

0 1

12

221 1

TEy y

TE TEy y

y y

k kZ

Z Z k kk k

Hence

Therefore,

00 0

0 1

1

2y x

jk y jk xx

y y

Ie e dk

j k k

Now use

0

0

0 0

sin

cos

cos

x

x

y

k k

dk k d

k k

sin

cos

x

y

21

Example (cont.)

or

0 cos0 0 0

12 2 2 2

0 1 0

cos

2cos sin

j k

C

I ke d

jk k k

0 cos0 0

12 2 21

cos

2cos sin

j k

C

Ie d

jn

1 1 0/n k k

Note: 2 2 21

0 1 0 0 12 2 2 21 1 0

2 2 20 0 1 0 1

sin , sinsin

sin , siny

k k k k kk k k

jk k k k k

22

Example (cont.)

12 2 21

cos

cos sinf

n

0 Saddle point:

We then identify

Note: There are branch points in the complex plane arising from ky1, but we are ignoring these for a lossy earth (n1 is complex). (There are no branch points in the plane arising from ky0, since the steepest-descent transformation has removed them.)

cosg j

k

1/22 20 0 cosy xk k k k

23

Example (cont.)

As

090 , 90

0 0f (unless 1 = 2 )

12 2 21

cos

cos sinf

n

Hence

24

Example (cont.)

0 12 2 21

cos

cos sinf

n

Far-field (antenna) radiation pattern:

0g j

k

0 / 2

0g j

0 /40 0

12 2 021

cos 2

2cos sin

j k jIe e

j k jn

0

00

2~ SDPg z jI f z e e

g z

25

Example (cont.)

Final far-field radiation pattern:

0 /40 0

12 2 021

cos 2

2cos sin

j k jz

IE j e e

j kn

y

0Ix

0


0 / 2

Note: The pattern has a null at the horizon.

26

Example (cont.)

Interface field: / 2

0 cos0 0

12 2 21

cos

2cos sin

j k

C

Ie d

jn

0

3

2

1 1~ 0

4g zI h e

0 0

2

II

j

From the higher-order steepest-descent method, we have:

27

Example (cont.)

00 0

3/2

0

1 10

2 4j k

z

IE j h e

j k

We then have

28

Example (cont.)We have

cosg j

with

3

23

00

00 2

0 0

20

2 23

3

SDP

SDP

j

j

h f eg

gf e

g g

0 / 2

12 2 21

cos

cos sinf

n

4SDP

29

Example (cont.)We have

cosg j

0g j

0 0g

0g j

0 0g

0 / 2

30

Example (cont.)

We therefore have

with

3

3 /4200 2 jh f e

12 2 21

cos

cos sinf

n

31

Example (cont.)We then have

12 2 21

cos

cos sinf

n

1 1

2 2 2 22 21 1sin cos sin sin cos 1 sin cosN n n

2

D N N Df

D

1 12 2 2 22 21 1

212 2 21

sin cos sin sin cos 1 sin cos

cos sin

n n

f

n

21

2 2 21cos sinD n

32


2

D N N Df

D

1 12 2 2 22 2

1 1

1 1 32 2 2 2 2 2 22 2 2

1 1 1

cos cos sin sin sin 1 sin cos

cos sin 1 sin cos sin cos sin sin cos sin sin cos

N n n

n n n

1 1

2 2 2 22 21 12 cos sin sin sin sin cosD n n

where

33

Example (cont.)At the saddle point we have

0 0N

20 12 1D n

0 21

1

1f

n

20 1 1D n

0 21

2

1f

n

20 1 1N n

34


3

3 /4221

20 2

1jh e

n

0

33 /40 0 2

3/221 0

1 2 12

2 4 1j kj

z

IE j e e

j n k

The field along the interface is then

35

Example (cont.)

The field in space ( < / 2) is

0

33 /40 0 2

3/221 0

1 2 12

2 4 1j kj

z

IE j e e

j n k

The field along the interface ( = / 2) is

0 /40 0

12 2 021

cos 2

2cos sin

j k jz

IE j e e

j kn

Summary

36

Example (cont.)

Space wave Line Source

Lateral wave

01 jke

0

3/2

1 jke

Lossy earth

37

Extension to Dipole

Dipole SourceSpace wave

0

2

1 jk rer

01 jk rer

Lateral wave

Lossy earth

38

Important Geophysical Problem

Lateral wave

TX line source

RX line sourceSpace wave

c c

R

1R 2RrR

Lossy earth

The field is asymptotically evaluated for R

Two types of wave fields are important for large distances:

Space wave Lateral wave

Note: More will be said about these waves in the next chapter on “Radiation Physics of Layered Media.”

39

Important Geophysical Problem (cont.)

1 11 1

rjk R jk RTE

r

e eR R

1 1 1 2

0

3/ 21 2

1 jk R jk Rjk e e

eR R

Space wave:

Lateral wave:This will be the dominant

field for a lossy earth (k1 is complex).

Note: Amplitude constants have been suppressed here.

Lateral wave

TX line source

RX line sourceSpace wave

c c

R

1R 2RrR

Lossy earth

h1h2

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Prof. David R. Jackson ECE Dept. Spring 2014 Notes 35 ECE 6341 1