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A PERIODIC LAYERED MEDIUM GREEN’S FUNCTION

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CHRISTOPHER P. TRAMPEL

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A PERIODIC LAYERED MEDIUM GREEN’S FUNCTION
By

Christopher P. Trampel

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Electrical and Computer Engineering

2004

ABSTRACT
A PERIODIC LAYERED MEDIUM GREEN’S FUNCTION
By

Christopher P. Trampel

In this thesis, we derive a periodic Green’s function for dipoles radiating inside a lay-
ered medium. In order to do so, we proceed as follows: first the spatial Green’s function
for a dipole inside a layer is derived in terms of Hertz potentials. Next, it is shown that
this periodic Green’s function can be calculated in the spectral domain provided that
the Fourier integrals do not have poles on the real axis. The derived expressions indicate
that this spectral sum is rapidly converging for most source-observation pairs. However,
they are not so for the source and observation pair lying on either the top or bottom
interfaces. To overcome this, a Kummer’s transformation is proposed. We validate our
Green’s function via reduction to a canonical half-space problem. The periodic layered
medium Green’s function is validated numerically by comparison with analytical data

for reflection and transmission from a single layer.

Copyright by
CHRISTOPHER PAUL TRAMPEL
2004

ACKNOWLEDGEMENTS

I would like to thank first and foremost my wife Mary for supporting me through
the rigors of graduate school with her customary grace and empathy. Thanks also to
my parents Christine and Darrell for their love and encouragement. I am also indebted
to Dr. Shanker Balasubramaniam, Dr. Gregory Kobidze, Dr. Edward Rothwell, Dr.

Dennis Nyquist, and Dr. Leo Kempel for their guidance during the writing of this thesis.

iv

TABLE OF CONTENTS

LIST OF FIGURES .............................. vii
NOMENCLATURE .............................. viii
CHAPTER 1 INTRODUCTION ...................... 1
1.1 Plasmons ................................... 1
1.2 Numerical modelling of plasmons ...................... 1
1.3 Periodic layered medium Green’s function ................. 2
1.4 Organization ................................. 2

CHAPTER 2 PERIODIC LAYERED MEDIUM GREEN’S FUNC-

TION ..................................... 4

2.1 Motivation ................................... 4
2.2 Problem Definition .............................. 5
2.2.1 Tangential components ........................ 8

2.2.2 Normal component .......................... 10

2.2.3 Dyadic Green’s function ....................... 11

2.3 Derivation of spectral series for general periodic Green’s function ..... 13
2.4 Kummer’s Transformation .......................... 16
2.5 Surface Waves ................................. 19
CHAPTER 3 RESULTS ........................... 21
3.1 Analytical validation of dyadic Green’s function .............. 21
3.1.1 The Gxx and ny components ................... 21

3.1.2 The sz and Gzy components .................... 22

3.1.3 The Gzz component ......................... 24

3.2 Numerical validation of dyadic Green’s function .............. 24
CHAPTER 4 CONCLUSION ........................ 28
APPENDIX A Components of dyadic Green’s function ........ 29

BIBLIOGRAPHY

vi

2.1
2.2

3.1

3.2

LIST OF FIGURES

Layered medium formulation ....................

A current element embedded in a single layer ...........

(a) Transmission from 20 nm slab of c, = 4 as a function of
wavelength; (b) Transmission from 40 nm slab of c, = 4 as a
function of wavelength .......................

Transmission from 20 nm slab of 6,. = 4 as a function of angle . .

vii

4
5

27

Lower case

6n

60

Upper case

Acronyms

EM

DE

IE

FDTD

F MM

AIM

NOMENCLATURE

permittivity in layer n (s2 - C2 - kg"l - III—3)
permittivity in free space (s2 - C2 . kg“l - III—3)
permeability of free space (H - m—1 )

temporal frequency (radians - s“)

unknowns
electric field intensity (V - m“1)
electric flux density (C - m_2)

magnetic field intensity (A - m’l)

electromagnetic

Differential Equation

Integral Equation

Finite Difference Time Domain
Fast lNIultipole Method

Adaptive Integral Method

viii

CHAPTER 1 INTRODUCTION

1 .1 Plasmons

Periodically structured dielectrics support waves bound to the surface of the object
upon plane wave illumination. Specifically, dielectrics with negative real permittivity
support charge density oscillations known as surface plasmons. The coupling between a
surface wave and charge density oscillation is the so-called surface plasmon polariton.

Research into nano—plasmon optics is growing into a rich research field, with far
reaching implications. Surface plasmon resonance interferometry techniques have yielded
sensors for biological media [1]. Specifically, plasmonic sensors have been applied to the
study of DNA [2]. Plasmons may also have application in quasi-planar lightwave circuits.

Plasmons have been studied extensively from a theoretical perspective. The dis—
persion relation for SPPs has been derived for a periodic array of scatterers at the
interface between two infinite media [3], and the resulting dispersion curves show strong
band gaps. Even more interesting optical properties of SPPs have been observed ex-
perimentally. Metal films perforated by periodically distributed holes exhibit stronger
transmission than that predicted by geometrical optics [4]. These transmission peaks
occur at the same frequencies as SPP modes. The mechanism of this enhanced trans-
mission has been well studied and is related to the excitation of surface plasmons on
both interfaces [5]. Coupling in this so—called surface plasmon polariton (SPP) molecule

funnels energy from one interface to the other.

1.2 Numerical modelling of plasmons

Full wave analysis is necessary for study of plasmons supported by arbitrarily shaped
geometries. However, the large negative real permittivity of metals at optical frequencies
necessitates dense discretization in order to capture the wave physics. Since the holes

occupy only 5% of the volume of a typical thin film, a sizeable region must be meshed.

We seek an integral equation scheme that will allow us to mesh only the hole region.

1.3 Periodic layered medium Green’s function

To that end, we introduce the periodic Green’s function for planarly layered media.
The layered medium Green’s function for a single arbitrarily oriented current element
is well known. The electric dyadic Green’s function for planarly layered media can
be found in Chew [6]. The layered medium Green’s function was derived in terms of
potentials for planar media by Michalski et al. [7]. Both Chew and Michalski express
the Green’s function in the spatial domain in terms of inverse Fourier transforms. The
spatial periodic Green’s function is an infinite sum over a two-dimensional lattice of
the Green’s function for a single current element. However, any lattice sum may be
computed in the spectral domain [8]. Pozar et al. derived a periodic Green’s function
for a current element above a grounded dielectric layer via spectral techniques [9].

In this thesis, we derive a spectral periodic Green’s function for current elements
radiating inside a layered medium. Our Green’s function differs from Pozar’s in that
our current elements reside inside a layer backed by a dielectric, while his lie on top of a
grounded dielectric slab. A new derivation of the Green’s function for a current element
inside a single layer is presented. We show how a spectral periodic Green’s function
may be obtained for an arbitrary spatial lattice sum. The resulting spectral layered
medium periodic Green’s exhibits exponential convergence for most source-observation
pairs. However, the convergence is extremely slow when both the source and observation
point lie on the same interface. We propose a technique to improve convergence for this

case based on a Kummer’s transformation [10].

1.4 Organization

The remainder of the thesis is organized as follows. Chapter 2 begins with the volume
integral formulation for EM scattering from homogeneous dielectric bodies. This chapter

includes a new derivation of the Green’s function for a current element inside a single

layer, a derivation of the spectral series periodic Green’s function for an arbitrary lattice
sum, and the Kummer’s transformation acceleration technique. A derivation of the
dispersion relationship for surface waves supported by a single dielectric layer concludes
the chapter. Chapter 3 details the analytical and numerical validation of the layered

medium Green’s function. Chapter 4 summarizes our conclusions and future work.

CHAPTER 2 PERIODIC LAYERED MEDIUM GREEN’S
FUNCTION

2.1 Motivation

We begin by discussing the geometry in Figure 2.1. Consider a volume (I bounded

 

 

 

 

Figure 2.1 Layered medium formulation

by a surface 80 that is embedded in a slab of permittivity en; either side of this slab
are dielectric regions whose permittivity is denoted by 5,,_1 and an“. The permittivity
of the volume Q is 50. A plane wave described by {Ei(r), H‘(r)} is incident upon the
dielectric. At this point, one can use either a surface or a volume equivalence theorems;
while we have both working codes, we have chosen the latter presentation. Volume
equivalence theorem permits us to replace free space with background permittivity 6,,
and introduce an equivalent current density J (r) = —jw(en — 50)E(r) such that the same
fields are produced everywhere. Then using the fact that the the total field E(r) is a

superposition of the incident field and the scattered field results in the desired integral

equation :

E<r> = E'lr) —- mm» (2.1)

where

 

£{X(r)} i {k3, + VV-}/ndv'Gp(r, r’) - KW) (2.2)

jwen ’

Gp(r, r’) is the periodic dyadic Green’s function for a current element radiating in layered

media, and kn is the wavenumber in medium 77..

2.2 Problem Definition

The starting point for the derivation of the Green’s function for periodically arranged
unknowns is the layered medium Green’s function that is very well known. However, we
have derived our own based on Hertz potentials which reduces to that given in [7] with
appropriate scaling [11].

Consider a current element inside a layered medium (Figure 2.2). The radiated

E‘
81’ “'1

83, u, E3 %

Figure 2.2 A current element embedded in a single layer

electric field is given by:

Ef(r), z 2 h
E(r) = E§(r) + E§(r) + E2‘(r), 0 g 2 g h (2-3)
E§(r), z 5 0

The field in region 1 consists of a single up going wave. In region 2, the field has three
constituents: a principal wave which acts as if it is in an unbounded medium (E3), an
up going wave (Ea,L ), and a down going wave (E; ). A single down going wave represents
the field in region 3.

We prefer to work with Hertz potentials as intermediate quantities. The electric field

may be recovered from the Hertzian potential thus:

Eflr) = kirflr) + V(V - 76%)) (24)
EU) = kiwar) + WV - 760)) (25)
Ear) = kivrar) + WV - «3(0) (26)

where n is the layer number, and kn is the wavenumber in layer 11. Each component of

the Hertzian potential is expressed as an inverse transform:

1 00 7 ' ~ ' 2
”mm = (2753 [[00 wna(k)eikpeiw<k> d2k (2.7)

where Wna(k) is an unknown amplitude spectrum, p r: .7337: + yy, k 2 kg“: + kyy,

 

p(k) = \/k,2, — (k3 + kg), and k;c and fry are the transform variables with respect to :1:
and y. The sign before p(k) determines the direction of propagation in the z direction:
a positive sign indicates a down going wave while a negative sign indicates an up going

wave. The up going and down going waves are homogeneous solutions to the Helmholtz

 

equation, while the principal wave is an inhomogeneous solution to the following :

J(r)

1'an

 

V27rfl(r) + k,2,7r£(r) = — (2.8)

where J (r) is the electric current. The principal wave W5 (r) may be regarded as a
particular solution to the above, while 7r; (r) and 7r; (r) are homogeneous solutions of
the same.

Our derivation of the Green’s function relies upon boundary conditions on the Hertz
potential [11]. These boundary conditions are a direct consequence of continuity of
tangential electric and magnetic field across an interface. The electric and magnetic
fields are written in terms of Hertz potentials and the continuity of electric and magnetic
fields across a boundary is imposed. A system of four equations results, one equation
for each of the four tangential components of the electric and magnetic fields. When the
components of the excitatory current J are considered independently, it is found that not
all components of the Hertz potential are necessary to represent electric and magnetic
fields that satisfy the boundary conditions. When the proper components of the Hertz
potential are conjectured for the associated component of the current, the system of four
equations imply boundary conditions on the components of the Hertz potential. The

boundary conditions for the components of the Hertzian potential are summarized as

follows [12] :

 

 

7rm,(r) = N2+1,n7r(n+1)a(r), a = :c, y, z (2.9)
87am, 1‘ , an n a r
8:: ) : “3+”: ( 312) ( )a a = xay (2'10)
871120.) a”(fl+1)=(r) 2 a7T(rl+l)ac(r) 677(n+1)y(r)
__ _ _— z .— N — 2.11
( az 82 ) ( n+1,n 1)( ax + 83/ ) ( )
6..
N3+l,n = (+1 (2.12)

n

where a is the component of the Hertzian potential (:13, y, z).

2.2.1 Tangential components

The unknowns W10,(k), W2+a(k), W2_a(k), W30(k) for the tangential components (a =
:r, y) were found by applying boundary conditions 1 and 2 at interface 1-2 and 2-3. The

first boundary condition at interface 1-2 leads to:

n10,(:r, y, h) = N2217T20(;L', y, h) (2.13)

«Mas/.12) = N31{W§a(x. (J. h) + ”£2202, 31, h) + ”21.00, 11. h)} (2-14)

Next, we substitute for the potentials in terms of inverse transforms :

 

1 fm d2kW10(k)ejk'pe‘jpl(k)” =

 

(27f)2 —oo
2 1 +°° 2 63k") —jp2(k)h 2+ jk-p -jp2(k)h - 'k-p much
(2.15)

+00 2 —jp1(k)h_ Ema” + —jp2(k)h
d k[w..,(k)e N21{——: W ) + W2a(k)e
-00 (2.16)
+ W2;(k)e’m(k)h}]ejk‘9 = 0

Invoking the Fourier transform theorem:

‘ kh
l/Vla(k)e—jp1(k)h _ N2 {e___ JP“ )

2, 252 192(k) + Wimp—Wk)" + W2;(k)eip2“‘)"} = 0 (2.17)

The above represents the first of four equations in the four unknowns Wla(k), W2:(k),

Wz‘a(k), W3a(k). The second boundary condition enforced at interface 1-2 yields:

afllaa, ya h) _ N2 BWQQCB’ 93h) 87710:”,an h) _
_ 21 _

882 ('92 Oz
N22182: ai7I2a(~T Blah) + "20(3 311(1) + 7r§a(:v.y,h)}

 

 

(2.18)

The first boundary condition applied at interface 2-3 yields :

«20(1), y, 0) = N§2w3a(x, y, 0) (2.19)

nae, y, 0) + «as, y, 0) + use. 9.0) = N3227rsa(:r. y, 0) (2.20)

The second boundary condition enforced at interface 2-3 yields :

8W20($9 ya 0) _ 2 871-3001;: y: 0)

83 _ N32 82 (2.21)
B _ 8n 0, 1:, ,0
Evan, 11.0) + was, y. 0) + Nae, 31.0)} = Nat—(7D (2.22)

The final equations (2.23) are derived in a manner completely analogous to (2.17) by
substituting for the potentials as transforms and using the Fourier transform theorem.
For brevity, we summarize the equations resulting from imposing boundary conditions

at both interfaces in matrix form:

 

 

 

 

 

 

 

“Wm male-WM macaw“: 0‘ 'wlkkf ' Nanak) '
6-.....» —N§.5§§—:§%e-m<k>h N221gfi,[%€jm(k’h 0 Wm) _ N§.%V2¥<k)
0 —1 —1 N32 W2;(k) — 143(k)

_ 0 222:3 {.fii N3. _W3.<k>_ _ gait/213(k) _

(2.23)

For notational simplicity we introduce:

Wm. 5M” (2 24)
2a ) — 26219200 I

1
V23 k = —— 2.25

2.2.2 Normal component

The unknowns W1z(k), W2:(k), W2;(k), W3z(k) for the normal component were found
by applying boundary conditions 1 and 3 at interface 1-2 and 2-3. The first boundary

condition enforced at interface 1-2 leads to:

”141(13)an h) : N2217T22($7 ya h)’/T1z($, y: h) :
(2.26)

N221{7r52(x. 31, h) + ”$06.11. h) + ”54:11.31. 11)}
The third boundary condition imposed at interface 1-2 gives :

8w z(x,y,h) 87v Jazz/ah) 37f a:(:r_.y,h 8W (cc.y,h
‘l—a—‘JTLma—m 2 )+ 21 l

81; 8;, ) (2.27)

 

8W1z(17, y, I?) a p + — —
82 32] 22k“ 31. h) + 772206.11, h) + 2243:, y, 1.)] _

8
_(N221-1){5;l7r§x($9y9h)+ 7r2::(13 y! h”) + ”23(xvya h)l (228)

a _
+ a—ylflgyh') 31: h) + ”gr/(1.131311) + ”231(13’ y’ h)”

The first boundary condition enforced at interface 2-3 leads to:

7r2203,31, 0) = News 31,0)”;(33, y, 0) + 76206, y, 0) + 7r2211?, 31.0) = N§2W3z($, y, 0)
(2.29)

The third boundary condition imposed at interface 2—3 gives :

8W22(:v,y,0) _ 023435.20) _

am By ) (2.30)

 

10

a __ 6W :(x,y,0
Elam, y. 0) + rate 1) 0) +m<r0>1 — —3;,;——’

8
— (N322 — waste, 1). 0) + use. 11.0) + age, 3). 0)) (2.31)

a _
+ Riki/(13,3130) + ”2:10;, 3110) + 7T2y($’y’0)]}

The final equations (2.32) are derived by substituting for the potentials as transforms
and using the Fourier transform theorem. For brevity, we summarize the equations

resulting from imposing boundary conditions at both interfaces in matrix form:

r- -I - q r- -I

 

 

 

 

 

 

 

 

e-jp1(k)h _N2216—Jp2(k)h —N2218j”2(k)h 0 le(k) A
_ -'1(k)h BEER -’ (kih _MEZ ' 00’! +
6 JP P1006 JP2 19100671)2 0 W2z(k) ___ B (232)
0 —1 —1 N32 W2';(k) C
3 k
. 0 1 —1 37% _W3z(k)_ _D_
where
- - ' Nanak) ’
A '2 _ . .
[— gj—gfigvmk) — megkpvmw + k,W,§(k)e-m<k>
C = +kxw,;(k)eiP2<k>+kyw2;(k)em<k>+k,v2;2(k)+kyvgg2(k)}] (2.33)
[D V223(k)
- . v233(k) + %§fi”{ka3.(k) + kyW3y(k)} _

 

 

2.2.3 Dyadic Green’s function

After solving these two systems, the total Hertzian potential for region two may be

expressed as a superposition of three waves.

«200) = «3.0) + nan) + vac) (2.34)

11

”20(1‘) =

 

1 +°° ejk'p . . . . .
(2765/ W e‘m‘“: + Wheelie-11’2"": + W.:.(k)e“‘"’em<k>:)

26213200

 

 

772(1'): 1 /+ood2k{ ejk'P 6—37’20‘)z
0‘ (2702 -oo 262p2(k)

321(k)R‘t23(k)e—jp2(k)‘/2102(k)+R23(k)v22(3(k)ejk-pe—jp2(k)z
(1+R‘12(k)R,f,3(k)e ‘32““0’ )
+1111(k)e-m<*>v:3<k)+R;.(k)Ra3(k)e-j2m<k>a3<k) awake}
(1 -Rzi(k)R$3(k )8 "2"? k”’)

+

 

 

. __ pmac) — wk)
Emu" ‘ Mk) +pn<k)

Substituting for V2102(k) and Vfiflk):

 

7m (1‘): “—1 /+OO d2k{ ejk’P e”jp2(k)z
a (27il2 —00 2621920‘)

+ { RiifklR/t23(k)e—jm(k)h

262102000 - 331(k)Réa(k)e‘12m(klh)

313(k) . ., _,,,, .

+ 2€2P2(k)(1 — RitzllklRi3(k)€_j2p2(k)h) }eJ" e (k)
+ { R‘tzlfkle-flmk)

262m(k)(1 — R1.<k)R13<k)e-j2m<k>h)

RillklR/tz (Me-1210200 J p m

+ 2.,p,(1.)(1_ H1.ik)a3<k1<e)-flm We k ‘3’ “‘2 }

 

 

 

 

12

(2.35)

(2.36)

(2.37)

(2.33)

 

———1 +00 (12k jk-p €“”’""”
6200‘) _ (2702 /—00 8 {2521?20‘)

Rt”(k)333(k)e—j2p2(k)he—jp2(k)z + R23(k)e_jp2(k)z

 

 

 

. 2.39

i 262m(k)(1- R1.<k)R33<k)eJk~p) ( ’

+ szl(k)e-j2p2(k)ejp2(k)z + R51(k)R,t23(k)e—J2m(k)ejm(k)z

2€2P2(k)(1 - 341(k)3}23(k)6‘j2p2‘k)")
The above implies that the Green’s function is :
G R 1 +00 (121: e-jm(k)lz-Z’|
2‘“ ’ ‘ (24)? f... { 26243)
t t ~j2p2(k)h —J'102(k)(z-2’) t —' (k)(z+z’)
+ R21(k)R’23(k)e 6 + R223(k)€ JP? (240)

262M100 - Rél(k)Réa(k)eJ'"'P)
Rf”(k)e-j2m(k)ejm(k)(z+2’) + R31(k)R§3(k)e-J'2p2(k)ejm(k)(z-z’)
262p2(k)(1 - Rx31(kW/3:)(k)€""i’2“‘)")

 

+ }ejk'(P-P’)

The normal components of the dyadic Green’s function are derived in a manner analo-
gous to that used above. The total dyadic Green’s function can be written in the spectral

domain as :

G(k, R) = Gp(k, R)I + (31%)., R) (2.41)

 

where G”(k, R) = e;;k}:R, R = \/(a: — 2")2 + (y — y’)2 + (z — z’)2, and

 

G’”(k, R) = Rn(k, min“: + (k, R)yy
R” (2.42)
+ R..(k, Rm + 12.,(k, way + R2,,(k, Rm

(See Appendix for coefficients Rag(k, R), a = :L‘, y, z, B = x, y, z.)

2.3 Derivation of spectral series for general periodic Green’s

function

Electromagnetic (EM) scattering from a periodic structure can be captured via anal-
ysis of a single mother cell by employing a periodic Green’s function. Linearity of our in-

tegral equations allows us to represent the field radiated by current elements distributed

13

uniformly along the lattice as a superposition of isolated current elements distributed

over the entire lattice. Thus the periodic Green’s function is given by:

= 2:00 W: G( (:r—rr —uD,, y— y —vDy ,”z—z)e ”510‘ e’jvfiVD" (2.43)

where G(R) is the dyadic layered medium Green’s function, D: and 0,, is the lattice
spacing in the x and y directions, and fix and fly are the :1: and y components of the wave
vector of the incident field. In most situations, the above converges slowly in the spatial
domain. We seek an alternative form of the periodic Green’s function with exponential

convergence. To that end, the previous equation may be rewritten as follows:

Gp(R) '2 *2: Z 6(1—23’ —qu,y— y —vDy)e J"fi’D’efif’fii’Di’ (2.44)

u=—oo v=-—oo

where :1: denotes convolution. Next we Fourier transform with respect to both x and y

and invoke the multiplication property of the Fourier transform:

2 , Zoo Zoom< : I 7 u 481 1) g
u 00!): x y

where G(kz, kg, 2 — z’) is the Fourier transform of the Green’s function. Next, we take

the inverse Fourier Transform to restore the spatial representation Gp(R) :

(3,,(R) = (flip/f: dk dky é( k ,k,.z — z')°
u=-<;v=-oo 3 (2.46)
u 'U )( . _ , . __ I
(5 ,x____-Tk _____yeka(-’I: I)Jky(y y)
( Dz 27r’ y Dy 277) e

 

u=-oov=-oo (2.47)
exmeix + 5—er — 4: ))exp()<-g; + 57”.“?! — y'))

14

This spectral series periodic Green’s function is a superposition of the Fourier transform
of the isolated Green’s function sampled at appropriate intervals. Observe that the
periodic Green’s function is computed in the spectral domain. One cautionary remark
must be made. Spectral domain summation is valid only if the Green’s function has
no poles along the real axis. Fortunately, our models for the metal contain a small loss
which raises these poles off the real axis. With these caveats the Green’s function may

be computed thus:

“Zoo :2; GP(:r—:r'—qu,y—y’— vDy,2— W)exp(—juBD —jvflyD y)

+ Z lime—I’— quy— y'—vDy,z— z’)exp( 446.0 422440 )

II
I
8
C
II
8

(2.48)

where D; and Du is the lattice spacing in the :1: and y directions, ,8; and fly are the
a: and 3] components of the wave vector of the incident field, and GP and GR are the
same as in (2.41). The sum over the principal wave is computed using an Ewald series
for on—plane sources and observations and a spectral series for the off-plane cases. The

phase accumulated as a result of reflections facilitates spectral domain summation of

mm).

 

 

 

em = 2: 2 23,26: 83+ 4— +4, - )-
uzmgy 27”) (2.49)
eXP(J'( + 34W: - I'))exp(j(—- + Hwy - y'))

Dan 4 Dy

where GR(kI, kg, 2 — 2’) is the Fourier transform of GR(R).

15

2.4 Kummer’s Transformation

Next we consider the efficiency of the spectral domain periodic Green’s function.

Our goal is exponential convergence for all source-observation pairs. Let us consider

Rzz(k, R) term by term:

 

1 , , ,
22 k R = P k n, k ~J2p2(k)h —Jp2(k)(z-—z)
R ( ’ ) 2(27T)2€2p2(k)[R21( )R25( )5 6
+ R33(k)e-jp2(k)(z+z’) + R311(k)€-j2m(k)hejpa(k)(z+z') (2.50)

ejk'(p-—p’)

+ Ran(k)R33<k)e-I2II“‘>"eII‘*>‘I':"I1 _ R511(k)R33(k)e—j2p2(k)h

 

Nfinpm(k) - 191(k)
NziipMk) + Mk)

 

" (k)=

ml

(2.51)

The phase represented by the e‘j”?(")(z+z') terms provide exponential convergence for
source-observation pairs that are well separated in the z-direction. However, the sum-
mation is inefficient for source and and observation points which lie near the same

interface. The two problem terms are:

 

 

 

 

1 - . , ejk-(p—p’)
z k R = n k "1217200,! JP2(k)(z+z) ‘
R 21( , ) 2(27T)2€2I)2(k) 1( )6 6 1 — R31(k)R33(k)e‘J2P2(k)h
(2.52)
1 . , ejk-(p-p’)
z k = ,n —JP2(k)(2+z) . 2.
R 23( ’R) 2(27r)262p2(k) 23(k)e 1 — R51,(k)}233(k)e—22p2<k)h ( 53)

Consider the spectral summation of R212(u, v) :

 

+00 +00 +00 +00 1 n
2 Z R;21(U,v)= Z Z 2(27Tl2621)2(uaUlRmm’v).

_ _ - _ eXp(J'k- (p - p’))
1 - 331(u, v)Ré’3(u, v)exp(-J'2m(u, WI)
(2.54)

 

exp(—j2p2(u, v)h. + jp2(u, v)(2 + 2'))

 

where pn(u, v) = \/kf, — (kw + 14%;?)2 - (kyo + 11127:)2, DI and Dy are the lattice spacing

in the :r and y directions, respectively. It is easy to see that when (kxo + iii—D2 +

16

(kyo + 11—201")2 becomes larger than k3,, pn(u, v) becomes a complex number. Choosing
the negative root results in a quantity of the form pn(u, v) = - j L where L is a positive
real number. Substituting into the phase term yields e‘j”?(“"’)(z+z') = e‘L(‘+z') which
is a decaying exponential. Thus source-observation pairs that lie above the bottom
interface yield exponential convergence. While all components of the Green’s function
are adversely affected in this case, most still converge as C(55) as u —+ 00. Consider the

asymptotic behavior of a typical term from the G” component evaluated at 2 = 2’ = 0:

1 N2 —1
t 32 t
R I —_ I 2.
”(u, L) 2(27r)2(;2p2(u, v) p3(u, v) + Nggpfiu, v) 23(u, v) ( 55)

 

We analyze the behavior of each term sequentially. Consider :

 

2 . , I 2
T23(u,v) = p2(u 1’ = __ (2.56)
MM. U) + p3(u, ’U) 1 + p3 u, *0)
p2 ”’7 U
NOW 11:13,: —’ 1 as u,v —> 00. Thus T§3(u, v) -+ 1. Without loss of generality we set

u = I). Since p2(u) —+ u as u —-> 00:

24/2(27r)222 1 + N322 u?

 

122.01.) —+ (2-57)

Clearly this sum converges as 0(fi).
The problem term is Rzz(u, v) which converges as (96) for the on plane case. In the

interest of brevity we support this claim for Rz21(u, v) only :

1
2(27T)2€2])2(U, v)

exp(—j2p2(u, v)h + jp2(u, v)(2 + 2’))

 

R221 (u, v) = R31 (u, v):

exr>(jk- (p - p’))
1 " Ram ”1233“" v)exp(—j2p2(u, W")
(2.58)

 

We begin with the asymptotic behavior of the reflection coefficients. To that end we

17

rewrite

u, v
Nag—H, ,, -
1

R31(u, v) = (2.59)

priH‘” +1
17.1.?)

Observe that M —> 1 as u, v —I 00. Also, the e’flmu'vw term in the denominator

p1(u,v)

rapidly decays to 0. Thus the asymptotic behavior of R221(u, v) is :

 

Rz . -> .
21(u, v) 2(27r)262p2(u, v) N122 + 1

e-flm(u.v)hejp2(u.v)(z+z’)8jk-(p-p’) (2.60)

 

Setting 2 + 2’ = 2h (both source and observation points he on the first interface) and

U21):

N1:2_ 1 ejk-(p-p’) N122 _ 1 1 ejkIp-p’)
22+ 112\/2(27r)2 P201”) N122 +12\/§(27I)2 u

 

 

Rz2l(u)_’ (261)

Therefore, the sum to converges as 0%). We accelerate the convergence by first observ-

ing that R212(u, v) is simply a constant times the principal part of the Green’s function.

Rz21(u,v) ————G'P(u,v) (2.62)

1V3:+
We can exploit this term by term convergence towards the principal Green’s function. In
a Kummer’s transformation, a series that can be summed quickly is added and subtracted
to a slowly converging series. Here we simply add and subtract a scaled version of the

principal Green’s fucntion. The summation over R212 is broken into two parts, 51 and

S12.

‘+00 +00 +00
ZiR221(u,v)‘—-ZZSIUU+ZZSZU'U (2.63)

u=-00 U=—(X3 uz—oo U=—CXJ UZ—W U=—(X3

18

l

 

 

 

S , = Rn —j2p2(u,v)hejp2(u,v)(z+z’) _
1(u v) 2(27r)2€2p2(U,v) 21(u,v)e
ejk-(p-p') N2 _1 (2'64)
. — ’2 GP(u v)
1 - R3196”ligawwle—flp’w’mh N122 + 1 ’
S (u v) - Na - le(u v) (2 65)
2 a _ N122+1 a ‘

Consider the summation of Sl(u, 2)). As u, v —-> 00, the first term approaches the second
and thus Sl(u, v) -—> 0. The asymptotic behavior of the first term causes 51 to converge
rapidly. The summation of 82 is evaluated by the usual spectral or Ewald series and

therefore displays exponential convergence.

2.5 Surface Waves

Coupling between plasmons and surface waves is responsible for many of the inter-
esting optical properties of thin films. Investigating the dispersion relationship for these
bound waves will allow us to excite specific surface wave modes and provide further
insight into the enhanced transmission phenomenon. For dielectrics with negative real
epsilon, only the TM modes yield proper surface waves. The dispersion relationship for

TM waves may be found by setting the denominator of the Rzz(k, R) term to zero :

1 — Hzl<k>R33<k>eI2II<k>h = 0 (2.66)

As a check on the above, we derive an equivalent expression from first principles. We

begin our analysis be writing expressions for the field in each region.

Cexp(—q2 — jwt +jI3y), 2 2 h
Ha:(yv 75) = {Acos(u2) + Bsin(u2)}exp(—~jwt + jfiy), O S 2 S h (2-67)
Dexp(v2 — jwt + jfiy), 2 S O

19

Here, q, u, v are the spatial frequencies in region 1, 2, and 3 respectively and A, B, C and

D are unknown amplitudes. Maxwell’s equations provide an expression for By :

1 8HI ,2
Ey(y.2)=————(y——)

jwc 62 (268)

Next, the we enforce continuity of the tangential electric and magnetic fields at both

interfaces. The resulting system of equations may be written in matrix form thus :

cos(uh) sin(uh.) —exp(—qlz) 0 A 0
—%sin(uh) %COS(Uh) %exp(—qh.) 0 B 0 (2.69)
1 0 0 —1 C 0
u v
. 0 a ‘a 0- _D- .01

 

 

 

 

 

 

Setting the determinant of this matrix to zero leads to the following implicit relationship

between u, v, and q :

,/qh .1
P"— : fgsinum) _ —1--cos(uh) (2.70)
6263 62 6162

We seek a relationship between the spatial frequency, L3, and temporal frequency, w. To

that end, we recognize that phase continuity requires that :

 

 

 

’82 __ 2 ___ LIP/1.061 (2.71)
52 + “2 = “12/1062 (2.72)
,62 _ U2 ___ w2m€3 (2.73)
Eliminating u, q and v :
2 ”:3 \/—3§_—'§— (2'74)
V “’ “0:2 ‘ 3 sin( w2floé2 - 4'24) — 6‘: “mash/ml.)
2 1

20

CHAPTER 3 RESULTS

Validation of the dyadic Green’s function was accomplished via both analytical and
numerical techniques. First, we show analytically that our Green’s function reduces to
the Green’s function for a current element radiating above a dielectric half space when
the permittivity of medium 1 is made equal to the permittivity of region 2. We analyze

each component of the dyadic Green’s function in turn.

3.1 Analytical validation of dyadic Green’s function

3.1.1 The Gxx and ny components

Consider the Rm,(k, R) component of the Green’s function :

ROAR, R) = {Ra}(k,R53(k){finite—.2424» + Raa<k)e-III"><I+I’)

+ Rjzl(k)e—J'2p2(k)hejp2(k)(z+z’)

 

 

t t 6-12P2(k)h 'p2(k)(z—z’) ejk.(p_pr)

+ R21(k)R’23(k) 6’ }2(27r)262p2(k)(1 _ R21(k)Ré3(k)e-12p2(k)h)
(3.1)
R5..(k> — W") - wk) (32)

_ pm(k) + p..(k)

Let the permittivity of region 1 be made equal to the permittivity of region 2. Then

121(k) becomes equal to 192(k) and :

123,0.) = W") - 101(k) _ 201(k) — 131(k)

 

 

_ = o 3.3
121(k) + 242(k) 101(k) + 221(k) ( ’
Substituting for R§1(k) into (3.1) :
}, —jm(k)(2+2’) jk-(p-p’)
Raa(k, R) = R2302" e (3.4)

 

2(27r)262p2(k)

21

Adding the principal portion to the reflected part we arrive at the total tangential

component:

e—JP2(k)|z-z'l + 353(k)e-J'pz(k)(z+z’)ejk-(p—p’)
2(27r)262p2(k)

 

000(k, R) =

This expression agrees with that given in the Nyquist class notes, pp. 6-21 [12].

3.1.2 The sz and Gzy components
Consider the Rzo(k, R) component of the Green’s function :

1
2(27T)2€2p2(k)

k) R513(k)T2ti(k)€—j2p2(k)h

Rm(k, R) 2 k0

 

l
9200 + N221P1(
(N32 - 1)
1130‘) + N322P2(k)
N2 —

+ { 21 21
192(k) + N21P1(k)
N32 _ 1 Tt k —jp2(k)(z+2’)

23( )}e

P3(k) + N2 132(k)
N313: 1 (3.6)

 

 

321 (k)T2’3(k)e-j2p2(k’h }e-J'p2(k)(z—z’)

 

333(k)T2‘1(k)R,f,3(k)e-j2p2(k)h

+

 

 

 

 

 

+ Tt (k 6—12P2(k)h
{192(k) + N221p1(k) 21 )
N322 - 1 t t _ .4 ‘ ’
n T k J p2(k)h p2(k)(z+z)
+ 193(k) + N§2p2(k)R21(k) 210‘) 23( )e }eJ
+ { N22’ _ 1 R§3(k)T2‘,(k)e-12m(k)h
192(k) + Nglplaq
+ T ’ k " k 6 12mm»: €Jm(k)(z .)
133(k) + N322P2(k) 23( )R21( ) } ]

ejk-(p-p’)

(1 — R31(k)R2’3(k)€—j2p2(k’h)(1 " R21(k)RIt23(k)€—flp2(k’h)

 

 

ann(k) = pm(k) - Mk)

 

p.00 + p.(k> (3‘7)
t _ 2pm(k)
W") “ pmac) +p..(k> (3'8)

n _ szpm(k) -pz(k)
0” _ Niinpm(k) +pz(k)

22

Let the permittivity of region 1 be made equal to the permittivity of region 2. The

contrast ratio between layer 1 and 2 is :

N122 = 51 = 6—1 :1 (3.10)

62 61
Again, the reflection coefficient between layer 1 and 2 vanishes thus:

N122P2(k) — 191(k) _ p1(k) —p1(k) _
Nf2P2(k) +p1(k) _ 121(k) +p,( ) _ 0 (3.11)

 

 

331(k) =

r

Also, the transmission coefficient between region 1 and 2 becomes unity :

 

 

t _ 222(k) _2192(k)_
TM") ‘ mm +p2<k> ‘ 214(k) ‘ ’ (3'12)

Making the appropriate substitutions into (3.6) :

1 N322 —- 1 , .
2(27r)262p2(k) 103(k) + N§2p2(k)T23(k) (3,13)

exp(-J'P2(k)(I + I') + jk- (p - p’))

 

Rm(k, R) 2 k0

Adding the principal portion to the reflected part we arrive at the total Green’s function

component:

1
2(27rl262p20‘)

T;3(k)e-jpz(k)(z+z’)}eik-(p-p’)

e—ijknz—z I

Gm(k, R) 2 k0

 

3.14

+ 223(k) + Nam)

 

This expression agrees with that given in the Nyquist class notes, pp. 6-22 [12].

23

3.1.3 The Gzz component

Consider the R2,,(k, R) component of the Green’s function :

 

1 . . I
= n n —J2p2(k)h —Jp2(k)(z—z)
R22(k1 R) 2(27T)2€2p2(k) [I121 (k)R’23(k)e 8
+ 333(k)e-jp2(k)(z+z’) + 33211(k)e-j2p2(k)hejp2(k)(z+z’) (3'15)

ejk-(p-p’)

n n, —12p2(k)hejm(k)(z—z')
+ R21(k)R23(k)6 ] 1 _ R32’1(k)@3(k)8—j2p2(k)h

 

 

n (k) _ lempm(k) — pl(k)

"’1 _ Ninpm(k) +pz(k) (3'16)

Let the permittivity of region 1 be made equal to the permittivity of region 2. Then

131(k) becomes equal to p2(k) and :

_ Nf2pz(k) - 121(k) p1(k) - 101(k)

 

 

 

R2“ ’ N122P2(k)+191(k) p1(k)+p1(k) ( ’
Substituting for R’l‘2(k) into (3.1) :
n k —jp2(k)(z+z’) jk-(p—p’)
R..<k, R) = R2“ ’6 e (3.18)

2(27T)2€2P2(k)

Adding the principal portion to the reflected part we arrive at the total tangential

component:

-J'p2(k)|z-2’| 71, -J‘m(k)(2+z’) jk°(p-p’)
G..(k, R) = e + 323“? 6 (3.19)
2(270 6219200

 

This expression agrees with that given in the Nyquist class notes, pp. 6—21 [12].

3.2 Numerical validation of dyadic Green’s function

We validate our dyadic Green’s function numerically using the integral equation
formulation described previously. We first generate a brick shaped mesh of height 12,,

and width wg. At the same time, we specify the thickness of the background slab to

24

be hl and the period D1. = D, = mg. The volume equivalence principle allows us to
replace the meshed region of the background slab with free space. Thus, we are able to
analyze reflection and transmission from a slab of height h = h] — hg. We compare the
numerical solution to the analytical solution for a dielectric slab of height h. Specifically,
we consider reflection and transmission from a dielectric slab as a function of thickness,
incident angle, and wavelength. Figure 3.2 shows the transmission from two slabs of
different height as a function of wavelength. Figure 3.2 displays transmitted intensity
from a dielectric slab as a function of angle of incidence. We observe excellent agreement

between analytical and numerical values.

25

Continuous slab, TE, 60 degree incidence, h = 20 nm

0.9 Tr f f i

 

 

 

 

 

— nunfen'cal
- - theoretical
0.85» .
0.8
8
To 0.75
.9
E
2
g 0.7
l—

0.65 .

0.6 ~

 

 

 

l

0.5 1 1 4 1 1 1
200 450 500 550 600 650 700 750 800
Wavelength (nm)

 

(3)

Continuous slab. TE, 60 degree incidence, h = 40 nm
0.65 . . . . r

 

 

— numerical
- - theoretical

 

 

 

0.55 -

.0
on

Transmission
.0
A.
01

.O
a.

0.35 .

 

 

 

l

0. 1 1 1 1 1 1
200 450 500 550 600 650 700 750 800
Wavelength (nm)

 

(b)

Figure 3.1 (a) Transmission from 20 nm slab of 6, = 4 as a function of
wavelength; (b) Transmission from 40 nm slab of e, = 4 as a
function of wavelength

26

 

Continuous slab, TE, I. = 600 nm. h = 20 nm

1 r r r r v

 

— numerical

___x\ - — theoretical

 

 

 

Transmission
0 P 9 .0
O) V m (O

.0
or

0.3- ~
\

 

1 1 1

10 20 30 40 50 60 70 80
Angle 6

 

 

Figure 3.2 Transmission from 20 nm slab of 6, = 4 as a function of angle

27

CHAPTER 4 CONCLUSION

In this thesis, we derived a spectral periodic Green’s function for current elements
radiating inside a layered medium. We introduced a volume integral equation formu-
lation for electromagnetic (EM) scattering from arbitrarily shaped dielectric bodies in
terms of unknown current J. A new derivation of the Green’s function for a current
element inside a single layer was presented. It was shown that this periodic Green’s
function could be calculated in the spectral domain provided that the Fourier integrals
do not have poles on the real axis. A Kummer’s transformation technique was used to
improve convergence for source-observation pairs lying on either interface was discussed.
We validated our Green’s function via reduction to a canonical half-space problem. The
periodic layered medium Green’s function was validated numerically by comparison with
analytical data for reflection and transmission from a single layer.

The periodic layered medium Green’s function approach has two major computa-
tional advantages. As mentioned previously, the holes in a typical perforated metal only
occupy roughly 5% of the area of the film. The layered medium approach allows us
to mesh the hole region rather than the metal. Secondly, the periodic layered medium
Green’s function is evaluated inside the metal region. The large negative real permit-
tivity causes the periodic layered medium Green’s function to converge more quickly in
the metal than in free space.

Future work will doubtless focus on analysis of larger structures. While our algorithm
is much more eflicient than conventional surface or volume integral equation formula-
tions, the cost still scales as 0(N2). Fast methods such as the fast multipole method
(FMM) or the adaptive integral method (AIM) need to be developed for layered media.
Other interesting physics can be observed when one introduces multiple layers into the
perforated thin film geometry. Our algorithm can be easily modified to accommodate

substrate and superstrate layers.

28

APPENDIX A Components of dyadic Green’s function

The tangential components (a = 2:, y) of the dyadic Green’s function are given by:

Raa(k, R) = {R51(k)Ré3(k)e—flpflklhe—J'Pflsz—z’) + 353(k)e—jpz(k)(z+z')

+ Ra](k)e-flpz(k)hejp2(k)(z+z’)
ejk-(p—p’)
2(27rl2627)2(k)(1 — 321(klR23(kle_j2m(k’h)
(A.1)

+ RSI(k)R33(k)e_j2p2(k)hejm(k)(z"z’)}

 

The normal components of the dyadic Green’s function are given by:

1
2(27T)2€2p2(k)[

R213(k)T21(k)6—j2p2(k)h

 

12mm, R) = kc,
N3, — 1
102(k) + N§1p1(k)
(N32 - 1)
111(k) + N32222(k)
+ { N22, — 1
192(k) + N221P1(k)

N2 — 1 _- Z ,I
+ 143(k) :QNsmIk)T53("”e W“ M

N31 ‘1 —'2
+ T’ k e .1 172(k)h
{192(k) + N221pl(k) 2” )

N2 —1 n -‘ . z z,
1330‘) :2N2 p2(k)1?}21(k)R21(k)T2tg(k)€ J4P2(k)h}63p2(k)( + )

32

N2 —1

+ { 21 2
Wk) + N..p.<k)

N322 " 1
193(k) + N§,p2(k)

 

{

 

Rf),(k)T2’3(k)e—j2p2(k)h}e’jm(k)(z_z')

 

333(k)Té1(k)R§3(k)e"2p’("’h

 

 

 

+

 

HI..<k)T;.<k)e-I2II‘*>"

 

+ T§3(k)sz’1(k)e—j2m(k)h}ejm(k)(z‘z')]
ejk-(p-p’)

(1 ‘ @1(k)R33(k)e-j2m(k’hl(1 _ R’zi(k)R23(k)e—j2p2(k)h)

 

29

where a = 2:, y.

1 n
20702414200 [R2103

+ use-jp2(k)(z+z’) + R511(k)e-j2p2(k)hejp2(k)(z+2’)

 

Rzz(k, R) =

R33e—j2p2(klhe-JP2(k)(z—Z')

'k. _ I
+R310‘) n3e’j2p2(k)hejp2(k)(z-z')] 8'7 (p p)

 

1 — R2’1(k)R33(kle—j2m(k)h

_ pm(k) - Mk)

 

 

RM“) ‘ mek) + p.(k)
t _ 2pm(k)
”“1"" ‘ mek) + Mk)
".(k) _ N.I...pm<k) _p,(1.)

 

— N12 m(k) +p1(k)

30

(A.3)

(A4)
(A.5)

(A.6)

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31

)I]IIIII,II]I]]I]I][311])