AD-A278 788

RL-TR-93-265Interim ReportDecember 1993

GREEN'S FUNCTIONS FOR ANANISOTROPIC MEDIUM: PART II. TWO-LAYER CASE

ARCON Corporation

Saba MudalarTIC

I ELECTEMAY 0 2 19941

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Rome Laboratory

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4. TITLE AND SUBTrTLE 5. FUNDING NUMBERS

GREEN'S FUNCTIONS FOR AN ANISOTROPIC MEDIUM: PART II. C - F19628-89-C-0186TWO-LAYER CASE PE - 62702F

8. AUTHOR(S) PR - 4600TA - 14

Saba Mudaliar WU - 26

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13. ABSTRACT(Ma*-2 ad!nI

The Dyadic Green's Functions (DGF) of a two-layer biaxially anisotropic medium arederived. The principal coordinate system of the anisotropic medium is allowed tohave arbitrary orientation with respect to the layer geometry. The formulation isbased on the unbounded Dyadic Green's Function derived in Part I of the sequel.Using the matrix method the coefficients of the two-layer DGF are expressed in termsof half-space Fresnel reflection and transmission coefficients. To complete thisprocedure the various relevant half-space Fresnel coefficients are derived. Theform in which the results are presented has a physically meaningful and compactstructure. A numerical example is provided where we have computed the reflectivities.

14. SUBJECT TEFMS 11 UMBER OF PAGES

Green's Functions, Electromagnetic Waves, Anisotropic Medium, 36

Layered Medium Ia PRCE COoE

17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19, SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACTOF REPORT OF THIS PAGE OF ABSTRACT

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TABLE OF CONTENTS

1. Introduction ............................................ 2

2. Geometry of the problem ................................. 2

3. Formulation ............................................. 3

4. Matrix method ............................................. 6

5. Half-space Fresnel coefficients .......................... 10

6. Numerical example ........................................ 16

7. Conclusion ............................................... 17

References .............................................. 19

Appendix A ............................................... 20

Appendix B ............................................... 21

Appendix C ............................................... 23

ILLUSTRATIONS

Figure 1. Geometry of the problem .......................... 25

Figure 2. Amplitude vectors of

waves in two layer medium ................. 26

Figure 3. Reflectivities versus incident angle ............. 27Accesion ForNTIS CRA&I

DTIC TABUnannounced 0Justification

By .......Dist, ibution /

Availability Codes

Avail and I orDist Special

1I

1. INTRODUCTION

In Part I [Mudaliar, 1993]1 we obtained the dyadic Green's

function (DGF) of a biaxially anisotropic medium for the unbounded case.

It may be noted that the results obtained there are in a form very

convenient for deriving the DGF's for anisotropic layered structures.

Indeed that was one of the underlying motives of Part I. We consider in

this report the two-layer case.

The plan of the report is as follows. In Section 2 we briefly

describe the geometry of the problem. We formulate the solutions of the

problem in Section 3. In order to facilitate the evaluation of the

coefficients of the DGF's we introduce amplitude vectors and adopt a

matrix method in Section 4. The procedure is completed in Section 5 by

deriving various half-space Fresnel coefficients. Finally the report

concludes in Secion 6.

2. GEOMMETRY OF THE PROBLEM

The geometry of the problem is shown in Figure 1. It consists of three

regions: Region 0 ( z > 0 ) is an isotropic medium with permittivity co,

Region 1 ( 0 > z > - d ) is the biaxially anisotropic medium with

permittivity 7 and Region 2 ( z < - d ) is an isotropic medium of

permittivity c 2" All the three regions have the same permeability A. In

this coordinate system the permittivity 7 is represented by the following

symmetric matrix.

- 2

[£11 12 13]

1 - Vl2 £22 £23 (1)

L 13 £23 33J

where the elements c J are explained in (3) of Part I.

3. FORMULATION

We have a unit impulse electric current source in Region 0 away from

the boundary. Our interest is to find the DCF's: U 00, (r,) and

U2 0 (rr). Here as usual the second subscript denotes the region where

the source is located while the first subscript denotes the region

containing the observation point. In order to obtain the above DGF's we

need to solve the system of equations given in Appendix A. Based on the

knowledge about the structure of the DGF for the unbounded case obtained

in Part I and that of the DGF's for the layered isotropic media2 we can

now construct solutions for our two-layer problem as follows.

3

G00 (r~r2j A- iR fF~~I {~e A+ 1k 0of

-hv vJh 0 e ]h

•- i•0• A+ ko.?

100 e + R v 0 e 0

+ Rvh h0v+ e ]0- h v 0

(2)

1i0-,) f k • { a e + B ha e

A- ixb.orhA+ ib -f h+ Ahbb e Bhbe 0

._a .-a

A_ A r .r A+ ik.y[ a e + B a eva va

-b . -

+ Avb b e + B vb e

(3)

"4

"- I "-•20(,)- 8 2 k Oz 2+Xv2

+[ A- A- A- ];ei2"r -ei0"r'+ [ vv V 2 + Xvh h 2 ]V0

(4)

where for n - 0,2

Ah 1. A (Sa)hn k nX z

p

v xi- k h (bn k n n

n

1 - (5c)V n i P- n xhnni

A (6a)-=k +zkn p nz

-k A-n - z k (6b)n p nz

k - (k 2 - k 2 )1/2 (7a)n p

k2 2 (7b)n n

A+ A+ - a -b b A

and a ,b, v ,a ,k, X are defined in Appendix B. Here h and v denote

5

the horizontal and vertical polarizations, which are two characteristicA A

linear polarizations in an isotropic medium. a and b denote the

polarizations of the a-wave and the b-wave, which are two characteristic

(extraordinary) waves in a biaxially anisotropic medium. Note that the

superscripts + and - indicate the upward and downward propagating waves,

respectively. The task here is to determine the sixteen reflection and

transmission coefficients: R's, A's, B's and X's.

4. MATRIX METHOD

The unknown coefficients appearing in (2), (3) and (4) can be

evaluated by using (A4) and (AS). But this is at best a tedius procedure;

moreover, the results thus obtained are in a very complicated form. We

therefore follow an alternative method 3 in which we first express the

unknowns in terms of half-space Fresnel coefficients. Later in the next

section we evaluate the various half-space Fresnel coefficients.

In this procedure we denote the amplitude vectors of various waves as

p, q, P, Q, s (see Figure 2). These are two-element vectors whose

elements are the characteristic components of the corresponding wave. For

example

S" Pv

where ph and pv are the amplitudes of horizontally and vertically

polarized components of the p-wave in an isotropic medium. Similarly

" Pb

- - • i • m m m m l m m m m m I ,6

where P and Pb are the a- and b- components of the P-wave in a biaxially

anisotropic medium and so on. Accordingly these amplitude vectors satisfy

the following equations

R 10121

where F01' 501' ý10' 710' ý12 and 712 are the various half-space

reflection and transmission matrices defined as follows

0• R01Rvh

Svhi (10a)01 0 RJ0

Xhb Xvb

910 _ F::: ((la)

R ab b7

10 X ah Xbh(1bXI0 " (llb)

X av Xbv

R 1 2 exp i(kad+kau)d ] 2 exp [ i +kau)daa z zaz (1a12 R, [ exp [i(-k.d+,ku)d 12 , x, ,bd+b)u, d] (12

[ 12 exp [(ka -k )d ] 2 exp [ (kb z2)d I12 u 12 exp ad 2z ] bd 2z bud

Lav 'kz k2ý Ozdkz

Note that the exponential terms are included in (12) to take into account

the phase shift at the boundary, z - -d. For a given incident wave

corresponding to f, Equations (8) and (9) represent four equations for

four unknowns -, F, Q and '9. The solution is readily obtained and is given

as follows.

. •(13a)

8

P - (13b)

Q -B~(13c)

I -(13d)

where

] R1 + o i 1 2 ( T - RlO '12 )I R0 1 (14a)

Rha A va 1 0 F12 ) 01 (14b)

- Ahb Awh X(1b

B12 ( T - 110 112 R01 (14c)B b B vb

- x12 ( T - R1 10 12 ) 0 1 (14d)Xhv Xv

We note that through (14) we have represented all the unknown coefficients

in the two-layer DGF's in terms of half-space Fresnel coefficients. In the

uniaxial limit, i.e. when c - c and 2- 0, our results agree with those

9

3of Lee and Kong . We shall take up the task of deriving explicit

expressions for these half-space Fresnel coefficients in the next section.

5. HALF-SPACE FRESNEL COEFFICIENTS

We notice that for our two-layer problem there are three situations

to consider.

SITUATION 1

Here we have free space in the region above the interface z - 0 and

the biaxially anisotropic medium in the region below (Figure 2). A plane

wave from above is incident on the interface. Depending on the

polarization of the incident wave there are two cases to consider.

(i) h-wave incidence

For this case the electric fields in the two regions may be

formulated as follows.

E0(Y) - e + R hoe + v eO z > 0 (15a)

A_ iiia.Y A- iK b.-Er(Y) - Xha a e + Xhb b e 9 z < 0 (15b)

The associated boundary conditions are given as follows.

10

U•I ][ n•l h l ]I lIig

z x E0(7) - zA x E( at z - 0 (16a)

A A- AzxV E0 () - zx Vx E() at z - 0 (16b)

On substituting (15) in (16) we obtain the following solutions for the

Fresnel coefficients.

01 + + b(1a

01 I xh v.+ h b (17a)

01 + - +

" -2 k h- +b- h- + 'I (17c)

Xhb-- 2 koz hb x+ - h 1 (17d)ya- xYa- }/(1d

where all the unknown quantities are defined in Appendix C.

(ii) v-wave incidence

Here the electric fields in the two regions are formulated as follows.

- AFA+ iko"FE() - v e + v e + vh(18a)

11

-- b-A- ir.- A-- ic YX1(Y)- XV&a e +Xb b e , z<0 (18b)

By substituting (18) in (16), we obtain the following solutions.

Rl I- + X h+ + Xvb (19a)vv va a- v

01 + +Rv Xa Va_- + Xb vb_ (19b)

v va - vb ) + b-+yb- T +Xva -b [ (T x b IK (19c)

Xb [ Tv+x VT )y a-(T y -. )x ]a- (19d)

SITUATION 2

Here we have the biaxially anisotropic medium above the boundary z - 0

and the isotropic medium 2 below it. A plane wave from above impinges on

the interface. There are two cases to consider depending upon the type of

the incident wave

(i) a-wave incidence

12

The electric fields E1 and E2 in the two regions are formulated as

follows.

-a -bE1(•)"a ei a + R 2a a e +R ab z > 0 (20a)

12 A- X1 .ir y 12 A-_ ic2yE2(y) - Xah h 2 + av v2 , z < 0 (20b)

Substituting (20) in (16) we obtain the following solutions.

R,12 - - $- ( , K..++ (21a)

R12 " - - (21b)

12 12 - 12- -Xah - Raava+ + RabVb+ + Va- (21c)

av _ R hi +R 1 2 hb + -(21d)X1 Ra abb + a

where

i- T + k h- T (22)

for t-(x,y], C-(a,b],(i,j]- (+,-]

All other quantities are given in Appendix C; but we have to replace

13

kOz by k2z.

(i) b-wave incidence

The electric fields E and E2 in the two regions are formulated as

follows.

A-.. i•b, 12 A+ ik b-r . 1 2 A+ -iVa- + %bb e + %aa z > 0 (23a)

- 1 2 A- i,~R '-A iR-Y(7)- Xhh 2 e 2 12v 2 e 2 z < 0 (23b)

From (23) and (16) we obtain the following solutions.

12 R1%bb aa

12 12R ab Replacements

a "'b (24)

12 x12bXah baa

12 12v av

SITUATION 3

In this situation the geometry is the same as in Situation 1; the

only difference is that the wave is incident from below. Once again

14

there are two cases to consider.

(M) a-wave incidence

The electric fields in the two regions are formulated as

-A+ ik -F A+ ik -YE0 (7) - X ah 0 e + Xav v 0 e , z > 0 (25a)

A i-af /ý i-aYA r -r- e + Raa e + Rab e , z < 0 (25b)

Applying the boundary conditions (16), we have the following solutions.

R -R12aa aa

Rab ab Replacement of

Superscripts (26)

Xh X 12+"ah ah

-4+

x - x12av av

Besides the above replacements, also note the following changes: is

here defined by (A2) and in other equations k2z is replaced by k 0 z.

(ii) b-wave incidence

15

The electric fields in the two regions are formulated as

- ei 0"YA ik Y-Eo(Y) - Xbh + Xbv Vo e , z > 0 (27a)

.b -b -a,1 bA.+ ik-Y' A im-c*r A- ixY"e + Rbb b e + Raa e , z < 0 (27b)

From (27) and (16) we have the following solutions.

Rbb - Raa

Ra - Rab Replacements

a -1 b (28)

Xbh - Xah b -a

Xbv - Xav

It is to be noted that in the uniaxial limit, our results (17), (19),

(21), (24), (26) and (28) agree with corresponding results of Lee and

3Kong

6. NUMERICAL EXAMPLE

One quantity of interest in our results is the reflection coefficient

of the two layer medium. To illustrate its characteristics we have taken

the following example. Region 0 is free space with permittivity c0 while

16

Region 2 has permittivity (6.0+i.006)cO. The anisotropic medium is

characterised by the following permittivities: cx - ( 2 . 8+i.001)0o , -Y

( 3 .O+i. 0 0 2 )c0, Cz - (3.2+i.003)c 0. The tilt angles are 01 - 20* and #2 -

40". The azimuthal angle of incident wave is 45". The thickness of the

layer is 1.0 m and the frequency is chosen as 10 GHz. In Figure 3a we have

plotted the reflectivities, rh - IRhhI 2 + 1Rhvj2 and rv - IRvv12 + IRvh12.

The behaviour here appears to be a familiar one. For small angles on

incidence we notice that rh -rv and there is a null (Brewster angle!) at

around 60° for rv. The anisotropic nature of the medium is not very

evident by looking at this reflectivity plot. This is perhaps the axis of

symmetry is very close to the incident plane. Next we consider the case

when #2 - 80°. Since the axis of symmetry is well away from the incident

plane the anisotropic behaviour is very apparent in Figure 3b. Just to

illustrate the contributions of like-polarised and cross-polarised

reflectivities to rv we have plotted IRvv1 2 and JRvhJ2in Figure 3c.

Perhaps in this example the cross-polarised reflectivity is rather small

compared to the like polarised reflectilvity. But the fact that there

exists cross-polarization is significant.

7. CONCLUSION

We have derived in this report the dyadic Green's function of a two

layer biaxially anisotropic medium. The formulation is based on the

solution for the unbounded case obtained in Part I. But the evaluation of

the various coefficients required the use of a matrix method which

expresses these two-layer coefficients in terms of half-space reflection

and transmission coefficients. This procedure is completed by deriving

17

explicit expressions for the various half-space Fresnel reflection and

transmission coefficients. To illustrate the computational simplicity we

have provided a numerical example where we have plotted the reflectivities

versus the incident angle.

18

REFERENCES

1. Mudaliar, S., "Green's Functions for an Anisotropic Medium:

Part I. Unbounded Case", Rome Laboratory Technical Report, 1993.

2. Tsang, L, Njoku, E. and Kong, J.A., "Microwave thermal emission from a

stratified medium with nonuniform temperature distribution",

J. Appl. Phys., vol.46, 5127-5133, 1975.

3. Lee, J.K. and Kong, J.A., "Dyadic Green's functions for layered

anisotropic medium", Electromagnetics, vol.3, 111-130, 1983.

19

APPENDIX A

The DGF's satisfy the following equations

V o-,f) WpC0 00 r' T 6(y YE) (AI)

2 •

x V x G(0 R ') - 2 ; a O(Y,') - 0 (A2)

V x V x t2 0 (or') - 42 G 2 0 (Yr) - 0 (A3)

where w is the angular frequency. The boundary conditions associated

with these DGF's are given as follows:

Sx 0 0 (i,?') - x G1 0 (Y,V) at z - 0 (A4a)

A -Aat 0z x V x G0 0 (Y,i') - z x V x G10 (Y,r) at z - 0 (A4b)

A ~ -A

z x G1 0 (rr') - z x a2 0 (Y,I') at z - -d (A5a)

A -A-z x V x G1 0 (Y,Y') - z x V x G20 (Y,Y?) at z - -d (A5b)

20

APPENDIX B

A+ 12 A+a - (u) -1/2 * A (Bla)

- [ -1/2 - (ia - •ya') - -A (Bib)

--- (¶ vl-1/2 -A (B2b)

tu A~-2 A+' - x • , t-aorL, (B3a)

Xd A- -2 A-X - x • " , t- a or b (B3b)

where

A a A Aa A

A+ 1 2 xa -- +A (B4a)hau a A

^Ab Ab A Ab Ab A

1 kx(kx 1) + kx(kx 2) (B4b)

+hbu ibx 1 1 Ikx 2 .21

k - k +aU , -aorb (B5)p z

h( J + -x -).-- • .... / (B6)A A.

1 k x O0 1k x o 2 1

21

k k• (B7)

A " (± siin2 + 2 sin#1 eosi 2 A

0z cos 02 (B8)

- L (L- ) (B9a)•y (cz- c x)

[ x(c z- cy) ]1/21/2" (B9b)

2 cy (cz- c x)

For definiteness we have assumed in the above equations that c < c < tx y

Aa A Aa A

A- 1 [ c X 01 Ic x IC (BlOa)h Ic x olI lIc x 0 21

Ab Ab A Ab Ab A17 [K x(ic x 01) +~ x(X x 02) (BlOb)

-hd I Ab11 AbI A

A -pk +A id, -a orb (Bll)p zh d -h u -- d} (B12)

22

APPENDIX C

jjii ii 6j17 17 - 7 C" ) / 61(cl)

x y y x

wheref f

Ij f I

Jf -h\, v II f

I' a, b I

f h - +"1 -

1V "- --

h T + vTi kOz h- T (C2)

f xwhere I-\X, Yf

6 hi v- _ i (C3)x y y x

T k k i - ki (C4)

23

T k kvj jC5Tv z ~ 0 Vf CS

A -x 8 ~~ (C6)

~ ~A { -~- 1.(C7)

24

z

1E0 REGION 0

z=O

REGION I

z= -d

REGION 2

Figure 1. Geometry of the Problem

25

43-C

Figure 2. Amplitude Vectors of Waves in Two-Layer Medium

26

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