Supplementary Material
Surface plasmon sensors on ZnO: Ga layer surfaces
Electric field distributions and absorption-sensitivity enhancements
Hiroaki Matsui 1, 2, Akifumi Ikehata3 and Hitoshi Tabata 1, 2
1
Department of Electrical Engineering and Information Systems,
University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan.
2
Department of Bioengineering, University of Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan.
3
Analytical Science Division Nondestructive Evaluation Laboratory,
National Food Research Institute, Tsukuba, Ibaraki 305-8642, Japan.
1. Fresnel Reflection calculations
We employed a Fresnel reflection for the purpose of calculating an absorption spectrum of SPR. The
intensity of reflectance can be expressed as a function of various parameters, such as the incident
angle (), the thickness of a ZnO: Ga layer (t), and amplitude reflectance rij at boundaries between
media i and j for light incident from i to j. Reflectance (R) is expressed by the following equation 1, 2:
R( , t ) 
rpm  rmw exp( 2ik mz t )
1  rpm rmw exp( 2ik mz t )
(1)
The amplitude reflectance of each boundary is described as follows:
rpm 
rmw 
k pz m  k mz p
(2)
k pz m  k mz p
k mz w  k sz m
k mz w  k sz m
(3)
where p, m and w indicate prism, ZnO: Ga and water, respectively. kwz, kmz and kpz denote the z component
(parallel to the surface) of wave vectors of incident light inside water, ZnO: Ga and the prism, respectively.
Three factors are calculated as follows:
 2
k jz   j  2
 c
kx   p

c


  k x 2 


1/ 2
for j = p, m, s
sin 
(4)
(5)
j is the dielectric constant of medium j. kx is the component of the incident wave vector parallel to the
interface.  is the angular frequency of the incident light; and c is the velocity of light. p is the
dielectric constant of the prism layer (BK-7); p = np2 (np is the refractive index of the prism).
The rij is expressed for each polarization direction, for example, rij-p for p-polarized light and rij-s for
s-polarized light. Therefore, we can obtain the reflectance for p-polarized light (Rp) and s-polarized
light (Rs) using Eq. (1) with rij-p and rij-s, respectively. Additionally, Rp and Rs can be written by the
intensity of the input light (Iin) and output light (Iout) as follows:
RP  I out / I in
(6)
Rs  I //out / I //in
(7)
The s-polarized light is fully reflected. Accordingly, Rs must be 1. A p-polarized reflection spectrum
is obtained by the Rp / Rs ratio. Moreover, the observable absorbance (A) is denoted by as follow:
A   log( R p / Rs )
(8)
2. Calculations for evanescent field distributions based on a Fresnel model
Depth analyzes of evanescent fields can be calculated by a plane-boundary multilayer system 3. A
dielectric constant and layer thickness of the jth layer are defined to ˆ j and dj (j = 1, 2, …., N). A
dielectric constant of the prism is p. The semi-infinitely thick dielectric substrate has a dielectric
constant ˆD . In addition, the complex dielectric constant of each layer is related to its complex
refractive index n̂ j by ˆ j  n̂ j ( nˆ j  n j  ik j ; nj is the refractive index. and kj is the absorption index).
Radiation impinges at the prism/multilayer interface with an incident angle of .
The following matrix M is given concerning the experimental
E//i
parameters of multilayer.
i

sin( k zj d j )
qj

cos( k zj d j ) 

N
cos( k zj d j )
M  

J 1
 iq j sin( k zj d j )
H//i

(9)
where q j  k zj / ˆ j and kzj are the z component of the wave vector in
p
d1
ˆ1  Re[ˆ1 ]  i Im[ ˆ1 ]
d2
ˆ2  Re[ˆ2 ]  i Im[ ˆ2 ]
2 1/ 2
vector in the prism kzP by k zj  [(2 /  ) 2 ˆ j  k zP
] with kzP =
dj
ˆ j  Re[ˆ j ]  i Im[ ˆ j ]
(2/)[Psin2]1/2.
dN
ˆN  Re[ˆN ]  i Im[ ˆN ]
the jth layer. kzj is given in terms of the x component of the wave
Figure S1. Schematic
illustration
H//t
shows the interaction between a
plane wave and stratified medium.
Parallel polarized (//) radiation are
only illustrated.
within the lth layer is described by the following equations:
E zz2 
E xz2 
2
k xP
(2 /  ) ˆ j
2
2
P
2
V// z , and
2
(2 /  )
U // z
2
(10)
ˆ j


ˆ


i
sin( k zj d j )  t // 
cos( k zj z )
i l sin( k zl z ) N  cos( k zj d j )

U
k zj
 // z 
k zl
   k zD  H i

V    k
t //  //
k

 
zl
zj


J 1
 // z  i
sin( k zl z )
cos( k zl z )
ˆ D 

i
sin(
k
d
)
cos(
k
d
)

zj
j
zj
j


 ˆi

 ˆ j

where d is the distance between z and the surface of the (l-1) th layer, H yz2
magnetic field in the y direction, E xz2
z
ˆD
E//t
The electromagnetic field at a distance z from the prism surface
2
H yz2  U // z ,
z
(11)
is the mean-square
is the mean square electric field in the x direction, E zz2
is
the mean square electric field in the z direction, and t// = 2qD /[(M11+M12qD)qD+(M21+M22qD)] is the
complex Fresnel transmission coefficient. H //i represents the magnetic field of the incident radiation,
which is expressed in terms of the corresponding electric field by H //i   1P/ 2 E //i . The plane of incident
radiation is defined by the x-z plane with the x direction and the propagating direction of the radiation.
3. Penetration depth and propagation length of a SP wave
3.1 Penetration depth of a SP wave
The electric field of a SP wave normal to the ZnO: Ga layer surface exponentially decreases. The
decay depth is evaluated when the field falls to 1/e. The penetration depths of SPs in dielectric media
(d: water and methanol) and metallic ZnO: Ga (m) layers can be expressed by the following
equations 4:
d 
0
2
  m'   w'
m 
0
2
  m'   w'
(12)
 w'2
(13)
 m'2
where m and d are the complex dielectric permittivities of ZnO: Ga and water, respectively. ’m and
’d indicate real part of m and d, respectively. Also, ”m and ”d indicate real and imaginary part of m
and d, respectively. 0 is wavelength.
3.2 Propagation length of a SP wave
The penetration length (LSP) along the metal-dielectric interface, after the field amplitude decreases to
e-1, can be given as follows 5:

1
L "  0
2k x 4
  m'   d'
 ' '
  m d



3/ 2
  d"  m"
 '2  '2
d m



1
4. A penetration depth and electric field distribution at a water/Au interface in the visible range
10
<E⊥2 2
<Ezz
<Hyz2
<Exz2>
5
0
-100
0
100 200 300 400 500
Distance, d (nm)
(b)
200
d = 139 nm
( = 652 nm)
150
600
650
700
Wavelength, (nm)
750
Propagation length, Lsp (nm)
15
Penetration depth, d (nm)
Au
(a)
Water
20
10
BK-7
Electric field <E2>
25
8
6
(c)
d = 2.64 m
( = 652 nm)
4
2
0
600
650
700
Wavelength, (nm)
750
Figure S1. Depth-dependent mean square electromagnetic fields at a water/Au interface. <Hy2> is the mean
square magnetic field. <Exz2> is the mean square electric field in the z-direction, <Ezz2> is the mean square
electric field in the z-direction, and <E⊥2> is the mean square electric field of p-polarized radiation; and <E
2
2
2
⊥ > = <Exz > + <Ezz >. The field amplitudes are normalized to the electric field amplitude of the incident
radiation. (b) Penetration depth (d) and (c) propagation length (Lsp) of a SP wave in water on Au metal
layer surface as a function of wavelength ().
Figure S1(a) shows depth-dependent mean square electromagnetic fields at a resonant angle ( ) of
75o for SPR at a water/Au interface. The field calculations were carried out at a wavelength at 652 nm.
A thickness of 50 nm was chosen to calculate the field distributions into water from an Au layer
surface. The field distribution of <E⊥2> at an water/Au interface was similar to that at an water/ZnO:
Ga interface. In addition, the penetration depth (d), at which the field falls to 1/e, was within 0.15 to
0.25 m in the visible range using Eq. (12) [Fig. S1(b)], which was close to the penetration depth of
ZnO-SPR in the NIR range. A propagation length (Lsp) of Au in the visible range was similar to that of
ZnO in the NIR range [Fig. S1(c)], as calculated using Eq. (13). These results supported that the
detection sensitivity of ZnO-SPR in the NIR range was comparable to that of Au-SPR in the visible
range. The dielectric functions of Au and water in these theoretical estimations were taken from Refs.
6 and 7.
5. References
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M. Golosovsky, V. Lirtsman, V. Yashunsky, D. Davidov, and B. Aroeti, J. Appl. Phys. 105, 102036
(2009).
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K. Johansen, H. Arwin, I. Lundstron, and B. Liedberg, Rev. Sci. Instrum. 71, 3530 (2000).
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I. Thormahlen, J. Straub, and U.J. Grigull, J. Phys. Chem. Ref. Dat. 14, 933 (1985).
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