Characterization of large array of plasmonic nanoparticles on layered substrate: dipole mode...

Characterization of large array of plasmonic nanoparticles on layered substrate: dipole mode analysis integrated with complex image method

Mohammad Mahdi Tajdini1,2

and Hossein Mosallaei1,3

1Applied EM and Optics Laboratory, Northeastern University,

360 Huntington Ave, Boston, Massachusetts 02115, USA [email protected]

[email protected]

Abstract: In this paper, an efficient analytical method for characterizing large array of plasmonic nanoparticles located over planarly layered substrate is introduced. The model is called dipole mode complex image (DMCI) method since the main idea lies in modeling a subwavelength spherical nanoparticle at its electric scattering resonance with an induced electric dipole and representing the electromagnetic (EM) fields of this electric dipole over the layered substrate in terms of finite complex images. The major advantages of the proposed method are its accuracy and rapid calculation in characterizing various kinds of large periodic and aperiodic arrays of nanoparticles on layered substrates. The computational time can be reduced significantly in compared to the traditional methods. The accuracy of the theoretical model is validated through comparison with numerical integration of Sommerfeld integrals. Moreover, the analytical results are compared well with those determined by full-wave finite difference time domain (FDTD) method. To demonstrate the capability of our technique, the performances of large arrays of nanoparticles on layered silicon substrates for efficient sunlight energy incoupling are studied.

©2011 Optical Society of America

OCIS codes: (260.2110) Electromagnetic optics, (250.5403) Plasmonics, (230.4170) Multilayers, (350.6050) Solar energy, (040.5350) Photovoltaic.

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1. Introduction

Recently, plasmonic materials have been studied extensively in various research communities owing to the fascinating capability for manipulating electromagnetic waves in subwavelength dimensions by means of surface plasmon polaritons (SPP) [1–6]. These surface modes are created by interaction of electromagnetic (EM) fields with plasmons at the interfaces of noble metals and dielectrics. Another interesting aspect is associated with the scattering properties of plasmonic nanoparticles [7]. While the scattering from a small dielectric particle can be disregarded when the size is much smaller than a wavelength, for a subwavelength particle made of plasmonic material, the EM scattering can be very large at certain resonant wavelength specified by the material parameters and the particle geometry. Utilization of scattering resonance of the plasmonic nanoparticle is currently enjoying a substantial growth in research activities both theoretically and experimentally owing to the advancement in nanofabrication technologies. Examples can be plasmonic nanoantennas functioning in transmitting mode as interconnect [8–13], or in receiving mode with application for solar-cells sunlight engineering [14–17]. Integration of plasmonic particles with quantum-dots can also manipulate other interesting applications in molecular level [18–20]. Array of gold nanoparticles can also be used for bio application to create new biomaterials and sensing features [21–23]. These are just a few examples in the multidisciplinary area of plasmonics.

In most of these applications, one needs to realize a large array of plasmonic or core-shell dielectric-plasmonic nanoparticles on a layered substrate, where their EM characterization can be very complicated and time-consuming. This is because of the particle configuration and the Green’s functions of layered substrate. The particle is made of a dielectric core coated with a thin shell frequency dispersive plasmonic material, and hence a fine meshing is required to accurately model the element. The generation of SPP on the circular boundary of sphere adds another difficulty in numerical modeling. In addition, the effect of the substrate appears as Sommerfeld integrals that are very low-convergent. Above all these, a large array of particles must be analyzed (which can be quasiperiodic or aperiodic creating novel physics [24–27]). Thus, modeling such structures with a conventional numerical method requires significant memory and computational time.

In this paper, we present a capable and powerful model with the use of dipole mode analysis integrated with complex image method which can successfully surpass the aforementioned issues and solve the problem very efficiently. The particles are modeled with array of dipoles (their modes profiles), and the substrate contribution is taken into account with only few sources’ images applying complex image method. This can enable a significant reduction in memory and time, by orders of magnitude.

The Mie analysis for the scattering behavior of a spherical particle is used to represent the subwavelength nanoparticle with its equivalent dipole mode profile. This technique has been used successfully in [11–13,28] for modeling array of plasmonic nanoantennas. Then, the EM fields for the array of dipoles on multilayered substrate are obtained using the dyadic Green’s function approach leading to Sommerfeld-type integrals [29–31]. Special attention must be given to efficient calculation of the Sommerfeld integrals as they have low-convergent behaviors. There have been various techniques to accomplish this. Steepest descent path (SDP) method [32] is useful but only applicable for far-field calculation. Another approach is based on Gaussian Green’s function (GGF) method where the integrand is expanded into a series of Gaussian functions [33,34]; however, this method cannot model the singularities of the Green’s functions. The complex images (CI) method which is used in this paper approximates the integrand by series of complex exponential functions. The Sommerfeld integral will then be reduced to a closed form summation of few terms of images of an infinitesimal source radiating in a homogenous unbounded area, when both the amplitudes and locations of these images are complex values. The CI method has been formulated and utilized in open literature [35–40] for various applications like the analysis of waveguides,


patch antennas, and photonic crystals. Here, we investigate the structures of very large arrays of core-shell dielectric-plasmonic nanoparticles (periodic or nonperiodic) on multilayered substrates, enabling efficient energy incouplings.

The dipole mode analysis integrated with complex image technique as the DMCI method can provide a state-of-the-art approach for performance characterization of large arrays of plasmonic nanoparticles on layered substrates. The main advantage of the DMCI method lies in its precision as well as rapid convergence. The speed is by several orders of magnitude better than the traditional full-wave methods.

This paper is organized as follows. In Section 2, a brief review of a core-shell dielectric-plasmonic nanoparticle and its representation with a dipole mode is reviewed. In Section 3, the EM fields of an arbitrary array of nanoparticles in a planarly multilayered medium are derived and the effect of the layered medium with the Sommerfeld-type integrals is determined. The complex image method is applied to solve the Sommerfeld integrals. Numerical results along with some potential applications of periodic and aperiodic (Fibonacci) arrays of plasmonic nanoparticles on layered substrates are demonstrated in Section 4. Full-wave finite difference time domain (FDTD) numerical technique is applied to verify the analytical results.

2. Performance of a core-shell dielectric-plasmonic nanoparticle: A review

Scattering characteristics associated with subwavelength plasmonic particles have extensively been studied in literature [7,11,41]. This scattering phenomenon originates from the local surface plasmon resonance at the plasmonic frequency, i.e., the oscillation of free electrons at the boundary of a metal. The plasmonic frequency varies from ultraviolet for most metals to visible range for some noble metals such as silver or gold. A plasmonic particle manufactured as core-shell with dielectric core and noble metal coating enables greater control over the scattering features of plasmonic nanoparticles. For a subwavelength plasmonic spherical particle made of a uniform material embedded in the air, the local plasmonic resonance occurs

when the real part of the particle’s relative permittivity is nearly 2. In contrast, the resonance of a concentric nano core-shell spherical particle can be tailored at a desired wavelength by tuning the ratio of inner to outer radii. Considering the focus of this paper in modeling arrays of core-shell dielectric-plasmonic nanoparticles, a review of the field analysis of a core-shell spherical structure is provided first.

Fig. 1. Geometry of a concentric core-shell particle.

For a core-shell sphere as depicted in Fig. 1 with the core of c and radius a and the

coating of s and radius b, if the incident field

iE is expanded in terms of spherical wave

vectors of first kind 1

pqM and 1

pqN , one can obtain scattered fieldsE as [42]:

1 1 3 3( ) ( )

0 0

[ [Γ Γ ]] .e h

i pq pq pq pq s p pq pq p

p p

p q p p q

pq pq

p

a b a b

E N M E N M (1)


It is noticed that 1

pqM ( 3

pqM ) and 1

pqN ( 3

pqN ) are correspond to the solenoidal and radial fields

representing the incoming (outgoing) waves. Coefficients ( )Γ e

p and ( )Γ h

p can be obtained

applying the boundary conditions. For instance, for ( )Γ e

p one has [11]:

( )

( )

( ) ( )

UΓ

U V

e

pe

p e e

p pj

(2a)

( )

( ) ( ) ( ) 0

( ) / ( ) / ( ) / 0U .

0 ( ) ( ) ( )

0 ( ) / ( ) / ( ) /

p c p s p s

d d d

p c c p s s p s se

p

p s p s p m

d d d

p s s p s s p m m

j k a j k a y k a

j k a j k a y k a

j k b y k b j k b

j k b y k b j k b

(2b)

( )V e

p is similar to ( )U e

p except that in the last column, the functionspj and d

pj are substituted

by py andd

py , respectively. Here,ck ,

sk , andmk are the wave numbers of dielectric core,

coating shell, and surrounding medium. pj and py stand for the pth order spherical Bessel

functions of the first and second kind, respectively.d

pj is defined as

[ ( )] /d

p pj x xj x x andd

py is by the same token determined.( )Γ h

p can be determined in the

same matter using the duality principle, i.e., in Eq. (2) superscript e and permeability must

be replaced with h and permeability , respectively. When the size of a plasmonic

nanoparticle is considerably smaller than the wavelength of the external EM field’s

excitement, it can be approximated with an induced Hertzian dipole as 0 ( ') J J r r

linearly related to the total electric field upon that particle by a polarizability factor as [17]:

( )

0 13

6Γ .em

total

mk

J E (3)

The plasmonic particles with other shapes can also be utilized and modeled with induced electric dipoles provided the size of the particles is small respect to wavelength. A T matrix approach can be applied to solve the problem [43,44].

The EM field of a Hertzian dipole in an arbitrary direction in a homogeneous medium can be derived through the dyadic Green’s function approach as [29]:

0 02

exp( ) exp( ).

4 4

jkr jkrj

r rk

E r I J H r J (4)

when

0 .x y zJ J J J x y z I xx yy zz (5)

3. Array of nanoparticles on layered substrate

To investigate the values and polarizations of induced dipoles for an array of N plasmonic nanoparticles over a planarly layered medium (as shown in Fig. 2), the induced dipole moment for each nanoparticle is derived analogous to Eq. (3) by solving the following linear

system of equations for , {1,2,..., }p q N as:

. .

0 . . .

1, 1

{ }.N N

ext q ext q

p p p p dip p ref p ref p

q q p q

J r E r E r E r E r (6)


The total electric field includes four terms. The first term represents the external excitation, the second term provides direct interactions between the particles, and the third and forth terms consider the reflection from the layered substrate. Equation (6) can be used to obtain the induced dipole moments for all N plasmonic nanoparticles. As illustrated in Eq. (4), the EM fields of a plasmonic nanoparticle can be attained by applying specific vector operators to a spherical wave. The spherical wave can be expressed as an integral summation of plane waves

in the z direction multiplied by conical waves in the direction over all wave numbers k via

Sommerfeld identity as [37]:

Fig. 2. Array of plasmonic nanoparticles on a planarly multilayered medium.

2

0

exp( )( )exp( ' )

2z

z

kjkrdk H k jk z z

r j k

(7)

where 2

0 (.)H is the second kind Hankel function of zero order. It is well-known that any

conical wave is a superposition of plane waves as [29]:

2 ' '

0

1 1exp .x x y

y

H k dk jk x x jk y yk

(8)

Obviously, the EM fields of a plasmonic nanoparticle are superpositions of plane waves and thus its scattering performance above a layered substrate can be treated with plane wave reflection-transmission formulations. To solve the problem one can decompose Maxwell’s equations into two decoupled scalar equations characterized by transverse magnetic and

electric waves ( TMzand TE z

) as discussed in [29]. Writing Eq. (4) for each layer and using

Sommerfeld identity, the EM fields generated by the qth plasmonic nanoparticle

( {1,2,..., }q N ) above a planarly stratified medium can be derived for each layer as [29]:

2 2 2

2

2

2

0

1

8

[exp exp 2 ]

q q q q

ix x i i

i

TM TM

i iz i iz i

z

iz

y

q q q

E J J Jx y x zx

kdk H k T jk R jk z z d

kz z

r

(9a)

2 2 22

2

2

0

1

8

[exp exp 2 ]

x y z

q q q q

iy i i

i

TE TE

q i iz q i iz q i

iz

E J J Jy x y zy

kdk H k T jk z z R jk z z d

k

r

(9b)


2 2 22

2

2

0

1

8

[exp exp 2 ]

q q q q

iz i i

i

TM TM

q i iz q i iz q i

i

x y z

z

z x z y zE


k

J J J

r

(9c)

2

0

8

[exp exp 2 ]

q q q

yix

TE TE

q i iz q i iz q i

iz

z

jH J J

z y


k

r

(9d)

2

0

8

[exp exp 2 ]

q q q

iy

TM TM

q i iz q i i

x z

z q i

iz

jH J J

z x


k

r

(9e)

2

0

8

exp exp 2[ ]

q q q

xiz

TE TE

q i iz q i iz q i

iz

y

jH J J

y x


k

r

(9f)

where 2 2( ) ( ) .q q qx x y y The TM

iT , the total transmission coefficient, is the ratio of

the amplitude of the total transmitted TMz wave in the ith layer to the amplitude of the

incident TMz plane wave and TM

iR , the total reflection coefficient, is the ratio of the

amplitude of the upgoing TMz wave to the amplitude of the downgoing TMz

wave in the ith

layer. TM

iT includes the effect of all layers above the ith layer and TM

iR takes the effect of all

layers below the ith layer into account. The total transmission and reflection coefficients TM

iT

and TM

iR are indicated as [29]:

1 1 1 1

1, 1

1

1 , 1 1 1 1

exp exp

{exp( ( )) }1 exp( 2 )

TM TM

i z iz i

TMil l

lz l l TM TMl l l l l z l l

T T jk d jk d

tjk d d

r R j k d d

(10a)

, 1 1 1 1

, 1 1 1 1

exp( 2 )

1 exp( 2 )

TM TM

i i i i z i iTM

i TM TM

i i i i z i i

r R j k d dR

r R j k d d

(10b)

when 1

TMT is the amplitude of the incident TMz plane wave and , 1

TM

i it and , 1

TM

i ir are the Fresnel

transmission and reflection coefficients for TMz case given by

1 1 1

, 1 , 1

1 1 1 1

2.TM TMi iz i iz i i z

i i i i

i iz i i z i iz i i z

k k kt r

k k k k

(11)

For the case of TE z mode, all the relations are the same except that all are defined and

prescribed now for TE z and:


1 1 1

, 1 , 1

1 1 1 1

2.TE TEi iz i iz i i z

i i i i

i iz i i z i iz i i z

k k kt r

k k k k

(12)

In general, the integral representation of the transverse EM field can also be acquired in each of the homogeneous layers provided the integral representation of the z component of the EM fields and the boundary conditions are given. Here, we achieve all the EM field components

directly from Eq. (4) because it allows applying the CI method elegantly on the problem. .

q

dipE

in Eq. (6) is the total electric field produced by the qth plasmonic nanoparticle above the

substrate when in Eq. (9) all the reflection coefficients equal to zero. .

q

refE for the qth

plasmonic nanoparticle is accomplished as ..q q

dipE E

The major challenge in solving Eq. (9) is the Sommerfeld integrals whose integrands are composed of total transmission and reflection coefficients and the Hankel functions (kernels). They usually are slow-convergent and require extensive time for calculation. Here the complex images (CI) technique is introduced. The basic idea behind it is to approximate the integrand by a series of complex exponential functions [35–40]. Then, the Sommerfeld integral will be reduced to a closed form summation of a series utilizing Sommerfeld identity.

To be more explicit, if F r is a spatial function defined as the integral of a spectral function

zf k as below:

2 '

0 exp ,2

z z

z

kF dk f k H k jk z z

j k

r (13)

if the function zf k is expressed as a series of exponential terms over zk as:

exp( ),z n z n

n

f k a k b (14)

and since the plasmonic nanoparticles are above the substrate and obtaining the EM fields

inside the layers is interested, we can assume 'z z here and then, by substituting the exponential series of Eq. (14) into Eq. (13) we determine:

2 '

0

2 '

0

( ) exp( ) ( )exp( ( ))2

exp( )( )exp( ( ))

2

n z n z

n z

n

n z n n

n nz n

kF dk a k b H k jk z z

j k

k jkRa dk H k jk z z jb a

j k R

r

(15)

where

2 2 ' 2( ') ( ') ( ) .n nR x x y y z z jb (16)

In other words, the integral is represented as a series of images of an infinitesimal source radiating in a homogenous unbounded area, where both the amplitudes and locations of these images can be complex values.

As it was demonstrated, the EM fields of an arbitrary array of plasmonic nanoparticles in each homogeneous layer is derived by taking integrals over total transmission and reflection

coefficients for both TMz and TE z

modes. The total transmission coefficient of each layer

depends on that layer and all the layers above. On the other hand, the total reflection coefficient of each layer depends on that layer and all the layers below. More clearly, for the

ith layer of a planarly N-layered medium for either TMz or TE z

mode:


1 2 1, , , , , , .i z z iz i iz i z NzT f k k k R g k k k (17)

because

2 2 2 ,jz jk k k (18)

for two different homogeneous layers we can write

2 2 2 2 .jz j i izk k k k (19)

By applying Eq. (19) to all homogeneous layers, all four total coefficients (reflection and

transmission coefficients for TMz and TE z

modes) for each layer can be written merely as

functions of zk of that layer which makes the utilization of the CI method possible. Therefore,

if the four coefficients for each layer can be expanded as series of exponential terms successfully to require only a few terms, the EM fields in each layer can be derived as series with few terms. In other word, if

' 'exp ( ) exp( )TM TE

i iz il iz il i iz in iz in

l n

T k k T k k (20a)

' 'exp ( ) exp( )TM TM TE TE

i i iz im iz im i i iz io iz io

m o

T R k k T R k k (20b)

when il , '

in , il , '

in , im , '

io , im , and '

io are coefficients determined by fitting the

exponential series to the four total coefficients, substituting Eq. (20) in Eq. (9) the EM fields of a plasmonic nanoparticle on top of a planarly stratified medium for the ith layer can be designated as

2 2 22

24

exp exp[ ]

q

ix x y

q q q

i i

i

i ilq i imq

il im

l milq imq

zE J J Jj

x y x zx

jk P jk Q

P Q

r

(21a)

2 2 22

2

' '

' '

' '

4

exp exp[ ]

q

iy

q q q

i i

i

i inq i ioq

in io

n oinq ioq

x y zEj

y x y zy

jk P

J

jk

P

J

Q

Q

J

r

(21b)

2 2 22

24

exp exp[ ]

q

iz

q q q

i i

i

i i

x y

lq i imq

il im

l milq imq

zEj

z x z y z

jk P jk

P

J

Q

J

Q

J

r

(21c)

' '

' '

' '

exp exp1[ ]

4

q

ix

i inq i ioqq q

in io

n oinq ioq

y zHjk P jk Q

z yJ

P QJ

r (21d)

exp exp1

[ ]4

q

iy

i ilq i imqq q

il im

l milq imq

x zHjk P jk Q

z x PJ

QJ

r (21e)


' '

' '

' '

exp exp1[ ]

4

q

iz

i inq i ioqq q

in io

n oinq ioq

x yHjk P jk Q

y xJ

P QJ

r (21f)

where

2 2 ' 2 ' 2( ) ( )ilq q q il inq q q inP z z j P z z j (22a)

2 2 ' 2 ' 2( 2 ) ( 2 ) .imq q q i im ioq q q i ioQ z z d j Q z z d j

(22b)

Thus, all the components of the EM fields can be calculated by applying the above differential operators on four spherical waves radiated from the Hertzian dipoles located at homogeneous complex space. In other words, an arbitrary array of plasmonic nanoparticles on top of a planarly multilayered structure can successfully be modeled by an array of induced dipole particles and their images, weighted and positioned optimally in a homogeneous complex space to require only a few terms in the series. Equation (21) provides the EM fields through a very efficient analytical method without any integration appropriate particularly for characterizing large arrays of plasmonic nanoparticles deposited on multilayered media. This is a comprehensive analytical solution for the total EM fields in a closed form and includes all the phenomena of interactions between the nanoparticles and inhomogeneity as series with coefficients which can be independent of the number or location of nanoparticles or the observation point.

To expand the spectral integrand in each layer the Prony technique is used [45]. The

inverse Hankel transform in Sommerfeld-like integrals is carried out along the real axis rC in

the complex k -plane as shown in Fig. 3(a). The equivalent contour of rC in the complex zk -

plane ( a aj k k k j ) is demonstrated in Fig. 3(b) when

2 2(Re{ } )ak k k . Basically the Prony method is a method for expanding a function of a

real variable into an exponential series. The spectral coefficients we have here are functions of

the complex variable zk . However, the Prony method can be utilized by defining a proper and

robust path in the complex zk -plane and a simple transformation of the complex variable zk

in that path to a real modified contour parameter t as it has been done in [35].

Fig. 3. Integration contours rC and dC (a) in complex k -plane and (b) in complex zk -

plane.

Different paths lead to different levels of the CI approximations which have extensively been pointed out in [46], i.e., it is possible to approximate the integration path by a robust path so that the exponential expansion is accurate respect to the exact results. Choosing different levels means selecting different approximation paths. Here, we exploit two-level CI


approximation for the exponential expansion of the spectral coefficients by means of transformation as follows

(1 ) 0 .iz i

tk k jt t T

T

(23)

In other words, the contour rC in the complex zk -plane is approximated via the two-level CI

approach by line dC , illustrated in Fig. 3(b). This is based on the fact that the integration path

can be modified accordingly as long as no integrand’s pole is encountered in the integration

path. The corresponding contour of dC in the complex k -plane is demonstrated in Fig. 3(a)

which is along a deformed integration contour passing through the origin and lying in the first and third quadrants of coordinates. Any deformed integration contour can be used as far as no more singularity is encountered in the deformation.

The mathematical details of the Prony method used in this paper are given in Appendix A. In this method, the only parameters which should be set priori are the number of exponential terms N and the truncation point T [36]. It is noticed that the truncation point of the approximation path should be sufficiently selected near to or far from the origin of the

complex zk -plane to afford adequate information from the spectral functions for the far field

or near field approximation, respectively [34]. In the numerical results of this paper, T is taken usually around 18. The coefficients can then be modified to obtain the best results.

The dipole mode analysis integrated with the complex image technique as the DMCI method, presented for the first time in this paper, can provide a state-of-the-art approach for performance characterization of large arrays of plasmonic nanoparticles on planarly multilayered structures. The main advantages of the DMCI method lie in its precision as well as its extremely fast computational procedure.

4. Numerical results

In this section, we apply the developed DMCI approach to the modeling of array of core-shell dielectric-plasmonic nanoparticles deposited on layered structures offering potential photovoltaic applications [14–17].

First, the scattering performance of a single particle made of dielectric core of SiO2

material (rc = 2.2-j0.01) and radius a with silver coating of plasmonic material with Drude

parameter rs (demonstrated in Table 1 [47]) and radius b = 30 nm is reviewed. Figure 4

demonstrates the magnitude and phase of the polarizability factor . It is noted that by tuning

the radii ratio a/b the plasmonic resonance of the nanoparticle can be controlled. For instance,

the polarizability factor of the concentric nanoparticle reaches its maximum at 0 = 500 nm

provided a = 0.785b. Another interesting feature is that at the resonance the phase of the polarizability factor is close to zero, which means at the resonance the induced dipole is nearly in phase with the incident field. For the sake of comparison, the magnitudes of the

electric dipole moment ( )

1Γ e , magnetic dipole moment ( )

1Γ h , and electric quadrupole moment

)

2

(Γ e are plotted in Fig. 5. As it can be seen, at the resonant wavelength of plasmonic

nanoparticles, the influence of other electric and magnetic multipoles can be neglected compared to the electric dipole moment which justifies that a plasmonic subwavelength particle can be considered as an induced electric dipole.


Fig. 4. (a) Magnitude and (b) phase of the plarizability factor of a concentric nano core-

shell dielectric-plasmonic particle of Fig. 1 vs. a/b for different wavelengths with SiO2 core and silver coating when b = 30 nm.

Let us assume our substrate is made of three layers as shown in Fig. 6. The first layer is

SiO2 ( r = 2.2-j0.01) with thickness of 0.160 , the second layer is a composition of GaAs-

AlGaAs (r = 8-j0.1) with thickness of 0.3

0 , and the third layer is gold (r = 2.28-j3.81)

with thickness of 0.40 when Drude model [47] is used for its material representation. This

configuration can have significant photovoltaic application where one is interested to confine the light in the middle layer with the use of plasmonic particles on the top layer [16]. Thus, it will be of remarkable importance to obtain the field profile within the GaAs-AlGaAs layer.

Since the total transmission and reflection coefficients are determined solely by the topology of the structure, the mechanism of EM wave propagation in each layer depends on

the treatment of these coefficients (for both TMzand TE z

modes). The spectral coefficients

for TMzand TE z

modes varies with changing 0/k k , i.e., the shift of the angle of the

incident EM wave. The peaks in the magnitudes of the coefficients are associated with the poles of the total transmission and reflection coefficients. In other words, the spectral

coefficients may have poles in the complex k -plane. These poles influence the definition of

the integration path if the Sommerfeld integrals must be calculated numerically as well as in the designation of the approximation path if the CI method is going to be obtained. According to Eq. (10), it can be seen that a pole for a spectral coefficient (whether total transmission or reflection one) takes place when a wave, after reflecting from the top and bottom interfaces with a phase shift through the layer, becomes in phase with the wave itself which is precisely the condition of positive feedback. Physically, it can specify two important circumstances in the structure. The first situation is when the wave number of the layer is larger than the wave numbers of its both adjacent layers which is called the guidance condition, i.e., the circumstance in which guided modes are excited through the slab. The second condition happens when the layer is contiguous to an outermost layer and the wave number of the layer is lower than the wave number of that adjacent layer. In such condition surface and leaky waves are created.


Fig. 5. Magnitudes of ( )

1 Γ e,

( )

1 Γ h, and

( )

2 Γ e of a concentric nano core-shell dielectric-

plasmonic particle of Fig. 1 vs. a/b with SiO2 core and silver coating when b = 30 nm and 0

= 500 nm.

Fig. 6. A three layer substrate with SiO2 ( r = 2.2-j0.01) slab of thickness 00.16 , GaAs-

AlGaAs ( r = 8-j0.1) slab of thickness 00.3 , and gold ( r = 2.28-j3.81) slab of

thickness 00.4 (at operating wavelength of

0 = 500 nm).

The main key to calculate the EM fields in each layer is to expand the total transmission and reflection coefficients. The spectral coefficients are expanded here in terms of only few exponential terms. For the GaAs-AlGaAs slab, the total transmission and reflection coefficients are expanded with 8 and 9 exponential terms, respectively, while T is opted 18. The reflection coefficients from the entire substrate are expanded into 10 term series when T

is taken 20 for the TMz mode and 14.5 for the TE z

mode. The results are demonstrated in

Figs. 7–9 vs. modified contour parameter t. Excellent agreement can be observed between the exact calculation and CI method. Tables 1–3 list the complex terms for the six spectral coefficients to show the typical values in the complex image series.

Fig. 7. Total spectral reflection coefficients vs. modified contour parameter t for (a) TE z and

(b) TMzmodes from the entire substrate via the exact formulation and CI method.


Fig. 8. Total spectral transmission coefficients vs. modified contour parameter t for

(a) TE zand (b) TMz

modes in the GaAs-AlGaAs layer via the exact formulation and CI

method.

Fig. 9. Total spectral reflection coefficients vs. modified contour parameter t for (a) TE zand

(b) TMzmodes in the GaAs-AlGaAs layer via the exact formulation and CI method.

Table 1. Complex Terms for the Total Reflection Coefficients of the Entire Substrate

TMz TE z

n na 6( 10 ) nb na

6( 10 ) nb

1 2 3 4 5 6 7 8 9 10

0.000-j0.000

0.009-j0.005 0.054 + j0.036

0.158-j0.077 0.470 + j0.184

0.903-j0.547 0.619-j3.019

0.818 + j1.643 0.290-j0.319

0.752 + j0.969

0.000 + j0.000

0.000-j0.008

0.001-j0.021

0.003-j0.039

0.006-j0.066

0.013-j0.115

0.048-j0.162

0.124-j0.208 0.181-j0.216

0.213-j0.248

0.000 + j0.000 0.000 + j0.000 0.002 + j0.003 0.013 + j0.015 0.067 + j0.026 0.273-j0.028

0.195-j1.009 0.157 + j0.141 0.126-j1.870

0.863 + j4.959

0.000-j0.003

0.000-j0.010

0.001-j0.022

0.003-j0.039

0.005-j0.064

0.011-j0.100

0.032-j0.153 0.183-j0.215

0.174-j0.251

0.086-j0.225


Table 2. Complex Terms for the Total Transmission Coefficients of the GaAs-AlGaAs Layer

TMz TE z

n na 6( 10 ) nb na

6( 10 ) nb

1 2 3 4 5 6 7 8

0.000-j0.000

0.005-j0.025

0.031-j0.097

0.150-j0.207

0.331-j0.264

0.608-j0.053 0.241-j0.346

1.741-j1.432

0.000 + j0.000

0.000-j0.002

0.000-j0.007

0.000-j0.014

0.000-j0.025

0.000-j0.041 0.019-j0.085 0.020-j0.131

0.000-j0.000

0.002-j0.011

0.013-j0.042

0.060-j0.086

0.126-j0.108

0.212-j0.032 0.013-j0.234

0.248-j1.034

0.000 + j0.000

0.000-j0.002

0.000-j0.007

0.000-j0.014

0.000-j0.024

0.000-j0.039 0.006-j0.085 0.033-j0.158

Table 3. Complex Terms for the Total Reflection Coefficients of the GaAs-AlGaAs Layer

TMz TE z

n na 6( 10 ) nb na

6( 10 ) nb

1 2 3 4 5 6 7 8 9

0.000-j0.000 0.000 + j0.000

0.029-j0.012

0.013 + j0.010

0.350 + j0.076

0.018 + j0.350 0.227 + j1.292

0.635-j0.876

0.083 + j0.425

0.000 + j0.000

0.000-j0.003

0.000-j0.007

0.001-j0.014

0.003-j0.023

0.005-j0.036

0.011-j0.053

0.028-j0.085 0.034-j0.095

0.000-j0.000

0.005 + j0.003 0.021-j0.025

0.069 + j0.048 0.053-j0.201

0.265 + j0.246

0.280-j0.323

0.174 + j0.917

0.228 + j0.487

0.000-j0.001

0.000-j0.003

0.000-j0.008

0.001-j0.014

0.002-j0.022

0.004-j0.033

0.006-j0.048

0.010-j0.067

0.008-j0.137

It is valuable to compare the numerical results obtained via the complete integration of the spectral coefficients with the expressions accomplished by the CI method. Figures 10–12 demonstrate the magnitude of the total transmission and reflection coefficients of the structure

of Fig. 6 for both TMzand TE z

modes in the spatial domain. Excellent agreement is observed

over a wide range of distances, i.e., from 0k = 0.01 up to

0k = 50 which validates the

accuracy of our method in finding the EM fields from the near to far region. Now let us consider the characteristic of 3 plasmonic particles over the slabs of Fig. 6.

Figure 13(a) shows the configuration of the concentric core-shell dielectric-plasmonic nanoparticles in an isosceles triangle. The sides of the triangle are assumed to be

1 2 0 / 2l l and 03 / 4l . The radii of the particles and their cores are

00.06

and00.047 , respectively. The nanoparticles are assumed to be excited by an incident plane

wave of ( )exp( )exp / 2x zjk x jk z E x z with the wavelength of 500 nm where

0 / 2.x zk k k The magnitude of the normalized x-component of the electric field via the

array when y = 40 nm and z = 70 nm is obtained by means of the exact formulation and CI method as illustrated in Fig. 13(b). The results are in good agreement.


Fig. 10. Magnitude of total spatial reflection coefficients for (a) TE zand (b) TMz

modes

from the entire substrate via the exact integration and CI method.

Fig. 11. Magnitude of total spatial transmission coefficients for (a) TE zand (b) TMz

modes

for the GaAs-AlGaAs layer via the exact integration and CI method.

Fig. 12. Magnitude of total spatial reflection coefficients for (a) TE zand (b) TMz

modes for

the GaAs-AlGaAs layer via the exact integration and CI method.

Fig. 13. (a) Configuration of three plasmonic nanoparticles. (b) Normalized Exvia the array

(deposited on the layered substrate) using exact integration and CI method.

To validate the accuracy of the analytical approach, a full-wave numerical technique based on an in-house developed FDTD method is also utilized to characterize the performance of


array of particles on the layered structure. Figures 14(a)–16(a) show the configurations of two z-directed and one y-directed unit electric dipoles, nine z-directed unit electric dipoles and 8 core-shell nanoparticles with one z-directed unit electric dipole at the center, respectively. The wavelength is 500 nm and all the plasmonic nanoparticles are same as in Fig. 13 and in their resonance frequency. Figures 14(b)–16(b) demonstrate the analytical results for these structures and Figs. 14(c)–16(c) illustrate the FDTD results. All the electric field distributions are normalized to their maximum values in the observation plane. The results are in good agreement. If the spectral coefficients of the layer are expanded into exponential terms accurately, the calculation of the EM fields is extremely fast and takes only few minutes in compared to hours for FDTD method. It is noticed that owing to the small size and spherical complexity of the nanoparticles and the cubical meshing of our FDTD method, a fine discretization with stair-casing around the boundaries is required for FDTD modeling (making FDTD computation very complex). The resonant frequency achieved by the FDTD technique for the core-shell plasmonic nanoparticles in Fig. 16 has almost 2.8% error compared to the theoretical prediction. Utilizing smaller cell-size may reduce the error.

Fig. 14. (a) Configuration of three unit electric dipoles (with different polarizations) above the layered substrate shown in Fig. 6, (b) DMCI result, and (c) FDTD result for the

normalized 2 zE (dB) at the middle of the GaAs-AlGaAs layer.

Fig. 15. (a) Configuration of nine unit electric dipoles above the layered substrate shown in

Fig. 6, (b) DMCI result, and (c) FDTD result for the normalized 2 zE (dB) at the middle of the

GaAs-AlGaAs layer.


Fig. 16. (a) Configuration of eight plasmonic nanoparticles and a z-directed unit electric dipole at the center above the layered substrate shown in Fig. 6, (b) DMCI result, and (c) FDTD result

for the normalized 2 zE (dB) at the middle of the GaAs-AlGaAs layer.

To demonstrate the capability of the analytical method for efficient characterization of large arrays of plasmonic nanoparticles, two examples are studied here. The first one is an array of 100 plasmonic nanoparticles in a periodic configuration as illustrated in Fig. 17(a) deposited on the layered substrate of Fig. 6. The distance between the centers of the adjacent nanoparticles is 125 nm. The nanoparticles are assumed as concentric core-shell particles with SiO2 cores and silver coatings. The external electric field is a plane wave as in Fig. 13. The diameter of all particles is 60 nm. All plasmonic nanoparticles are at their maximum scattering and therefore, the diameter of the cores for all particles is 47 nm. The z-component of the electric field intensity is calculated analytically inside the GaAs-AlGaAs medium. The result is demonstrated in Fig. 17(b). All results are normalized to the maximum of the z-component of the electric field intensity in case of no nanoparticle to show the effect of the plasmonic nanoparticles.

Fig. 17. (a) A periodic array of 100 plasmonic nanoparticles above the substrate of Fig. 6 and

(b) 2 zE (dB) at the middle of the GaAs-AlGaAs layer. It is normalized to the case that no

nanoparticle exists.


Fig. 18. (a) An aperiodic array of 100 plasmonic nanoparticles above the substrate of Fig. 6 and

(b) 2 zE (dB) at the middle of the GaAs-AlGaAs layer. It is normalized to the case that no

nanoparticle exists.

The performance of a deterministic aperiodic array of plasmonic particles on layered substrate for photovoltaic application [24] is also investigated here. A Fibonacci array is considered [25,26]. Fibonacci sequence is generated by adding two most preceding members in a recurrence fashion. Although it is not periodic, it is long-range correlated. Moreover, it has some mesmerizing fractal properties such as being self-similar with a scaling factor equal to the Golden ratio [26]. Owing to these unique structural features, arrays of plasmonic nanoparticles in Fibonacci configuration demonstrate distinguishing diffraction patterns from the ones by either periodic or random arrays. Spectral band-gaps and localized light have been investigated recently in Fibonacci arrays of plasmonic nanoparticles [25,27]. Figure 18(a) illustrates the Fibonacci array of 100 plasmonic nanoparticles. Two different particles are used to implement the array. One has core diameter of 42 nm and the other of 51 nm. The result is shown in Fig. 18(b) illustrating a more uniform field distribution than the periodic case. Although the aperiodic array enhances the energy coupling through the layer (in compared to the case that no particle exists), its intensity is not as good as the periodic case. One needs special optimization for Fibonacci elements or implementing other aperiodic or quasiperiodic configurations to feature unique physics [24,27]. The goal here has been to just highlight the power and efficiency of our technique in modeling aperiodic cases.

The DMCI approach enables a fast and efficient technique to comprehensively manipulate the performance of large array of plasmonic nanoparticles on layered substrate and can significantly reduce the computational time and memory (in compared to conventional numerical methods such as FDTD). For FDTD method, even if the structure is periodic and one wants to model one unit-cell, since the particle is small (can be λ/100) and it is made of thin frequency dispersive plasmonic material, and because of the generation of SPP, the calculation requires considerable memory and time. One can obviously imagine the challenge of FDTD in characterizing finite and non-periodic large array of particles.

5. Conclusions

In this paper, we demonstrate a theoretical modeling framework for performance analysis of large arrays of periodic/aperiodic plasmonic particles located above layered substrates. The plasmonic nanoparticles are represented by their induced electric dipole modes and the interactions between them and the substrate are taken into account considering the relevant spectral coefficients. Complex image method is applied to calculate the Sommerfeld integrals in closed forms, replacing the effects of the layers with the images of the dipole sources in the complex space. This can offer a state-of-the-art method with excellent accuracy, versatility, and rapid convergence over all distances from the complex structure. The speed of DMCI method is by several orders of magnitude faster than the traditional full-wave techniques. The accuracy of the analytical results is compared well with rigorous full-wave numerical results obtained through FDTD. Unique applications such as periodic and Fibonacci arrays of


plasmonic particles (10 x 10) on layered substrates to enhance coupling and provide uniform field distribution for GaAs-AlGaAs thin layer (sandwiched between a SiO2 layer on the top and a gold substrate in the back) are demonstrated. The DMCI technique is a very capable and powerful method that can be used for characterizing various plasmonic applications involving large arrays of nanoparticles on layered substrates, leading to energy-efficient photovoltaic applications.

Appendix A: The Prony method

The Prony method [45] is an efficient approach to interpolate a function of exponential terms on a set of available data. Basically, one can expand an arbitrary function F(t) in terms of N exponential terms as

1

exp( )N

n n

n

F t A B t

(A1)

where the coefficients nA and

nB should be obtained and t belongs to the interval 0 .t T

To determine the series of Eq. (A1) uniquely, 2N samples of data are required. If the 2N equally spaced data are sampled from the function, i.e.,

1 2 2( ) ( ), ( ), , ( )m NF t F t F t F t (A2)

when

1 1,2, ,22 1

m

Tt m m N

N

(A3)

then, according to the Prony method, by solving the N N matrix equation as follows

1 11

1 22 1

2 1 2 2 21

( ) ( ) ( )( )

( ) ( ) ( )( ),

( ) ( ) ( ) ( )

N N NN

N N NN

N N N N

F t F t F tF t C

F t F t F tF t C

F t F t F t F tC

(A4)

the exponents can be accomplished as ln )(n nB where1 2 , , , N are the roots of the

Nth order polynomial equation of

1 2

1 1 0.N N N

N NC C C

(A5)

Ultimately, the weighting coefficients can be easily achieved via solving the N N matrix

equation as follows

11 1

22 2

1 11 2

2 21 2

1 2

( )

( ).

( )NN N

tt t

N

tt t

N

tt tN NN

A F t

A F t

A F t

(A6)

The total transmission and reflection coefficients obtained in this paper are functions of the

complex variable z

k . However, the Prony method is to expand a function of a real variable

into an exponential series. To surpass this, the integration contour is deformed in complex z

k -

plane to obtain a robust path to operate over real parameter t. Two-level CI approximation is exploited in this paper for the exponential expansion of the spectral coefficients through transformation relation of Eq. (23). By changing the variable of the total transmission and


reflection coefficients from t to z

k according to Eq. (23), one can obtain the series in Eq. (A1)

as a function of z

k given by

1

expN

z n n z

n

F k k

(A7)

where the coefficients in Eq. (A7) can be related to the coefficients in Eq. (A1) by Eq. (23) as:

exp .1 (1 )

n n

n n n

TB TBA

jT k jT

(A8)

Acknowledgments

This work is supported in part by U.S. Air Force Office of Scientific Research (AFOSR) Grant No. FA9550-10-1-0438. The authors would like to acknowledge Z. Hao at Applied EM and Optics Laboratory, Northeastern University, for his precious help in acquiring the FDTD results. The authors would like also to thank Prof. Adibi’s research group at Georgia Institute of Technology for providing useful comments about the material parameters.


Characterization of large array of plasmonic nanoparticles on layered substrate: dipole mode...

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