Sensors and Actuators B 86 (2002) 49±57

Asymmetric SPR sensor response curve-®tting equation for theaccurate determination of SPR resonance angle
Kazuyoshi Kuriharaa, Kaori Nakamurab, Koji Suzukia,b,*aKanagawa Academy of Science and Technology, 3-2-1 Sakado, Takatsu-ku, Kawasaki 213-0012, Japan
Department of Applied Chemistry, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japanb

Received 27 September 2001; received in revised form 4 March 2002; accepted 19 March 2002

Abstract
An asymmetric surface plasmon resonance sensor response curve (SPR curve) equation derived from the three-layer Fresnel equation
regarding p-polarization was proposed that can be used as a more general and appropriate SPR curve equation than Kretschmann's equation
given by R X  1 a=f X b2  g2 g, where R is the re¯ectance of the incident light, a, b and g are real parameters, and X the incident
angle. The proposed equation is expressed as R X  A1 fB  C X Dg=f X D2  E2 g with a minimum re¯ectance at a conp
ventional resonance angle of X  D  B  B2  C2 E2 =C, where A, B, C, D and E are ®ve real parameters that are very useful as ®tting
functions for a gold-based SPR curve in real-time measurement. In this equation, a modi®ed resonance angle de®ned as parameter D is more
appropriate than the conventional resonance angle as a variable that gives a linear response to changes in the refractive index of the sensing layer.
# 2002 Elsevier Science B.V. All rights reserved.
Keywords: Surface plasmon resonance; Resonance angle; SPR curve; Real-time measurement

1. Introduction
The surface plasmon resonance (SPR) sensor is a useful
analytical tool based on the detection of refractive index
changes due to speci®c adsorption or complexing of the guest
target analyte into the host ligand prepared on a metallic ®lm[1,2]. The most popular SPR sensor is commercially available
for detecting speci®c polymers such as proteins concerning
antibody±antigen interactions under label-free and real-time
conditions [3]. In order to improve the total performance of
the SPR sensor for the real-time measurement, reliable andhigh-speed data analysis is very important.
In typical SPR sensors, the SPR curves of re¯ectanceversus incident light angle are a fundamental concept foranalyzing signals caused by the refractive index changes.
Kretschmann [4] was the ®rst to provide a theoreticalequation of the SPR curve that contains a minimum re¯ectance whose angle is normally called the resonance angle(RA). According to Kretschmann's theory, SPR signals are
obtained by monitoring changes in the RA with some dataprocessing of the SPR curves. Unfortunately, Kretschmann's*
Corresponding author. Present address: Department of Applied
Chemistry, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 2238522, Japan. Fax: 81-45-564-5095.
E-mail address: suzuki@applc.keio.ac.jp (K. Suzuki).

theoretical equation is roughly approximated from the threelayer Fresnel equation and therefore is not suited for ®tting
to the SPR curves. The most precise curve-®tting procedureusing a multilayer Fresnel equation is acceptable in the static
measurement but is much too time-consuming in real-timemeasurement.
In actual data analysis with SPR curves in real-time measurement, simple methods for ®nding the RA in the SPR curve
are applied such as the quadratic ®t, the polynomial ®t, thecenter-of-mass method, the smoothed ®rst-derivative zerocrossing method, and so on. As improved methods, the locallyweighted parametric regression (LWPR) method [5,6] and the
optimal linear method [7] were proposed. Nevertheless, thesemethods are just within ``minima hunt'' techniques in the SPR
curve so that useful accurate data for SPR curves situated far
from the minimum point corresponding to the RA are not usedin these methods. An SPR curve-®tting equation applicable
over a wide range around the RA would confer a higher
sensitivity to the real-time SPR sensors.
We now propose a new SPR curve-®tting equation forreal-time measurement that ®ts much better to almost all the
SPR curves experimentally obtained. The new equation isapplicable to asymmetric SPR curves frequently obtainedwith a gold (Au)-based SPR sensor. Our asymmetric SPR
curve-®tting equation was derived by closer approximationof the three-layer Fresnel equation relating to p-polarization

0925-4005/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 5 - 4 0 0 5 ( 0 2 ) 0 0 1 4 6 - 6

50

K. Kurihara et al. / Sensors and Actuators B 86 (2002) 49±57

than Kretschmann's SPR curve equation. Interestingly, we
have theoretically found that the conventional RA that gives
the minimum peak in the SPR curves is not the mostappropriate variable to monitor linear changes in refractive
index but is only an approximate one. A modi®ed RAde®ned by parameter D in the asymmetric SPR curve-®tting
equation is the most suitable variable for the SPR analysis.
The effectiveness of the proposed curve-®tting equation was
estimated by numerical calculations using actual opticalparameters and was demonstrated by Au-based SPR sensor
experiments with sucrose solutions.
2. Theory
We assumed a simple SPR sensory system which consists ofthree layers of a prism±metal±sensing layer structure with a
Kretschmann attenuated total re¯ection (ATR) con®guration
as shown in Fig. 1. A glass prism, a metallic ®lm, and a sensing
layer are abbreviated to p, m and s, respectively. When the ppolarized light has an incident angle y in units of radians, there¯ectance R of the light is given by the three-layer Fresnel
equation relating to p-polarization as follows [8±10]:
R  jrpms j2

(1)

with
rpms 

rpm  rms exp 2ikmz d
1  rpm rms exp 2ikmz d

(4)


o2
kjz  ej 2
c
kx 

kx2

p oep sin y
c

2pc
o
l

1  rms rpm1 exp 2ikmz drpm
1  rms rpm exp 2ikmz d
f kx   g kx rpm1 exp 2ikmz d
f kx   g kx rpm exp 2ikmz d

for j  p; m; s

(5)
(6)
(7)

rpm

f kx   kmz es  ksz em

(11)

and
ksz em(12)

Here, the function f kx  is applied to a Taylor expansion at
kx  k0 while g kx , rpm and kmz are approximated as g k0 ,
rpm kx k0 and kmz kx k0 at kx  k0 , respectively, and thefollowing equations are then obtained:
f kx   f k0   f 0 k0  kx

k0  

e2m e2sp kx
em es

k0 
(13)

g kx   g k0  

o 2em esp
c em  esp
ep em  es  em es  eprpm  rpm kx k0  p
ep em  es  em es ep
Fig. 1. A Kretschmann configuration of a prism±metal±sensing layerinterface. The glass prism, the metallic film and the sensing layer arelabeled p, m and s, respectively. y is the incident angle while d is the
thickness of the metallic film.

(10)

with

g kx   kmz es

1=2

(9)

In the simplest approximation, the SPR phenomenonstrongly occurs when the variable parameter kx is equal to
k0. Because the value of rms becomes an anomaly whenkx  k0 , Eq. (2) can be transferred as follows:

kmz es ksz em
kmz es  ksz em(8)where k0 is then obtained byoem es 1=2
k0 c em  es

(2)
(3)

and

kmz es  ksz em  0rpms kx  

kpz em kmz eprpm 
kpz em  kmz ep
rms 

where rpm and rms are the amplitude re¯ectances given by
the Fresnel formula relating to p-polarization for prism±
metal and metal±sensing layer interfaces, respectively, ej andkjz are the dielectric constant and the wave-vector component perpendicular to the interface in medium j, respectively,
kx the component of the incident wave-vector parallel to theinterface, given as a real number, d the thickness of themetallic ®lm, o and l are the angular frequency and thewavelength of the incident light, respectively, and c thevelocity of light.
The complex wavenumber k0 of the surface plasmon wave
generated at the metal±sensing layer interface in the absenceof the prism is derived from the following equation [1] withthe variable parameter kx:

(14)

(15)

and
kmz  kmz kx k0 oemp
c em  es(16)

K. Kurihara et al. / Sensors and Actuators B 86 (2002) 49±57

These four equations, Eqs. (13)±(16), make Eq. (10) a
simple equation as follows:
rpms kx  withr

kx k0 rs 1
kx k0 s 2 kRsskx k0 rskx k0 kR

(17)

"
#



o
2
em es 3=2
4pd
em
exp i
c em esl em  es 1=2
em  ess  rpm kx k0
kR  rs

1 2
A  p
s(21)

Eq. (17) is a fundamental equation when the SPR curves are
discussed. In the approximation that leads to the symmetric
SPR curve equation given by Kretschmann, two conditions
are assumed as follows:s  expif

(22)

and
jr00 j ! jr0 j(23)

where f is a real number. Under these two conditions ofEqs. (22) and (23), the term rs 1 can be described asfollows:
rs 1  kR

(24)

where kR is the complex conjugate of kR. By substituting
Eqs. (20) and (24) into the second term in Eq. (17), we ®nallyobtain the symmetric SPR curve equation as follows:R  jrpms kx j2 
1

a
kx

kx
kx

k0
k0

2
kR
expifkR(25)b2 g2

witha  4k00 kR00

(26)

0b  k00  kR0  kSP

(27)

and
00
g  k000  kR00  kSP

(28)
0

00

A 1

(19)

where kR is the perturbation of k0 in the presence of theprism. Surface plasmon waves in the Kretschmann con®guration have the complex wavenumber kSP given by

where the mathematical descriptions of x and x mean thereal and imaginary parts of the complex number x, respectively. The SPR curves in Eq. (25) are symmetrical to the0axis of kx  kSP
because the equation contains only a0quadratic term of kx kSP.

kx k0 s 2 kRskx k0 kR!
B  C kx DR  jrpms kx j2 

with

(20)

kSP  k0  kR

In our case, no condition is imposed in Eq. (17), which canbe exactly transferred to the following equation with ®vereal parameters of A, B, C, D and E:

(18)

and

51

B  2s0
2 1
C  2f s0

kx

2

(29)

D2  E2(30)
1kR0 2  kR00 2
s0 k000 kR00
1kR0

2s00 kR0 kR00
2

jsj jkR j

s00 kR00 g

22s00 k000 kR0
(31)
(32)

0
D  kSP k00  kR0 

(33)

00
E  kSP k000  kR00 

(34)

and
ss 2

(35)

where we de®ne the value of s as the characteristic factor of
the asymmetric SPR curves. The meaning of the parametersin Eq. (29) for the SPR curves are: A gives the maximum
value of the SPR curves, B and C contribute to the symmetry
and asymmetry of the SPR curves, respectively, D is amodi®ed RA that is the most appropriate variable that monitors changes in the refractive index, and E is the imaginary
part of the complex wavenumber of surface plasmon waves in
the Kretschmann con®guration and is related to the dip-widthof SPR curves. Compared with Kretschmann's equation of
Eq. (25), our equation (Eq. (29)) is not symmetrical to the axis
0of kx  kSP
because our equation contains the linear term of0
kx kSP . Eq. (29) is more general and appropriate than
Kretschmann's equation because we impose no condition on
the fundamental equation (Eq. (17)); thus the equation is moresuitable for a ®tting function.
In Eq. (29), the re¯ectance has a dip at a conventional RA
ymin and a minimum re¯ectance Rmin given bypo
B  B2  C 2 E 2
D(36)
np sin ymin  D Ccp!
B  B2  C 2 E 2
BRmin  A 1

A
1
(37)
E2
2E2
with
np 

p
ep

(38)

where np is the refractive index of the prism; the approximations indicated by `' in Eqs. (36) and (37) are obtained
under the condition of B @ CE > 0.

52

K. Kurihara et al. / Sensors and Actuators B 86 (2002) 49±57

The principle of SPR analysis is based on the momentum
conservation in the real part between the evanescent wave
and the surface plasmon wave given by0kx  kSP(39)

where kx denotes the wavenumber of the evanescent wave
0
denotes the real part
generated by the incident light and kSPof the complex wavenumber of the surface plasmon wave.
Eq. (39) determines the speci®c angle ySPR given byonp sin ySPR  k00  kR0c(40)

Thus, one can measure changes in the refractive index of thesensing layer on the basis of the following useful formulathat is an approximation of Eq. (40):sin ySPR 

ns
np

(41)

with
ns 

p
es

(42)

where ns is the refractive index of the sensing layer. In
Kretschmann's approximate equation (Eq. (25)), the angle
ySPR is the conventional RA, because it makes the re¯ectance a minimum in Eq. (25). However, in our asymmetricequation (Eq. (29)), because the conventional RA presenting
the minimum re¯ectance, ymin, is given by Eq. (36), theangle ySPR is not the conventional RA, ymin. We call theangle ySPR a modi®ed RA in our asymmetric SPR equation
as a distinction from the conventional RA, ymin, that presents
the minimum re¯ectance. In a physical phenomenon, ySPR
and ymin are determined by the momentum conservation in
the real part and in the complex, respectively, between theevanescent wave and the surface plasmon wave. The modi®ed RA, ySPR, is more suitable than the conventional RA,
ymin, for measuring linear changes in the refractive index,
because the useful formula in Eq. (41) is a good approximation of ySPR rather than ymin. This is a new concept for the
SPR curve data analysis.
3. Numerical procedure
Numerical calculations were carried out using two software programs, Mathematica 4 (Wolfram Research) and
Igor Pro (Wave Metrics). The numerical data for theoretical
SPR curves obtained by the Mathematica calculations were®tted to our asymmetric SPR curve equation using a curve®tting routine of Igor Pro. The curve-®tting operation using
Igor Pro is designed for user-de®ned ®ts that allow one tode®ne the ®tting function, the weighting for each point andthe epsilon values to calculate the partial derivatives withrespect to each coef®cient. Igor Pro uses the Levenberg±
Marquardt algorithm [11] that is a standard resolution of the
problem to search for the coef®cient values in the nonlinear

least-squares method. In all the ®ts, the weighting and theepsilon values were automatically selected according to theuser's manual for Igor Pro. The weighting is unity for all theraw data while the epsilon values for each coef®cient, K, are
10 10  K.
4. Experimental
In the SPR sensor response experiments, the refractive
indices of the sucrose solutions with ®ve concentrations of 0,1, 4, 10 and 20 wt.% were basically measured with acommercial SPR instrument (DKK, Japan), which is ¯exible
in the optical elements such as a prism and a light source.
The optical system was the typical Kretschmann ATR
con®guration. The prism was a hemicylindrical prism madefrom SFL6 glass (OHARA, Tokyo, Japan) with a highrefractive index of 1.80 at 587 nm. The light source was
two light-emitting diodes (LEDs) that had central wavelengths of 630 and 880 nm with half widths of 20 and40 nm, respectively. The light was focused into a converging beam through a cylindrical lens so that the sensor
surface was simultaneously observed over a range of incident angles. A charge-coupled device (CCD; SONY XC-77,
Tokyo, Japan) was used to measure the re¯ected lightintensity as a function of the incident angle. A linear
polarizer was placed between the prism and the CCD toselectively observe the re¯ected light of the p- or s-polarizations. The SPR curve was obtained by dividing there¯ected light intensity of the p-polarization by that of
the s-polarization because the non-homogeneous distribution of the light intensity was removed.
The gold substrates for the SPR experiments had a
45 nm thick gold ®lm with an area of 20  20 mm deposited onto 51 mm  20 mm  1 mm thick SFL6 glass plates.
A 5 nm thick layer of chromium was placed between
the gold ®lm and the glass plate to improve the adhesionof the gold to the glass. A minimum amount of matching¯uid was used to reduce the re¯ection at the glass surfaces
when the gold substrate was ®xed to the hemicylindricalprism.
5. Results and discussion
The asymmetric SPR curve equation given by Eq. (29)was examined for its effectiveness as an approximate equation using numerical calculations and for its usefulness as a®tting function using actual SPR experiments. In the SPR
numerical simulations, the refractive index of a water solution was measured by an Au-based SPR sensor with a prism
having a high refractive index, np  1:80. The typicalrefractive index of water, nw, is given by nw  1:3330 at
a wavelength of 589.3 nm and a temperature of 20 8C. The
dispersion effect is included in the gold metal, but not in
the prism and the water solutions. As dispersive optical

K. Kurihara et al. / Sensors and Actuators B 86 (2002) 49±57

parameters of Au, the Drude model was assumed for the
dielectric constant of the metal layer, em, as
em  e1mo2po o  iot (43)where op and ot are the plasma frequency and the damping
frequency, respectively, e1
m is the background frequency ofthe metal at a frequency of in®nity. According to the curve®tting of reliable data in the literature [12], the following
16
values can be used: e1
m  9:75, op  1:36  10 rad=s and14
ot  1:45  10 rad=s in the range 600±900 nm.
In Figs. 2 and 3, the asymmetric SPR curve equation from
Eq. (29) was examined at wavelengths of 600 and 800 nm,
respectively. In part (a) of Figs. 2 and 3, the solid lines
represent the original SPR curves obtained by the three-layer
Fresnel equation of p-polarization while the broken lines are
the SPR curves obtained by the asymmetric SPR curve
equation of approximation using Eqs. (29)±(35). The SPR

53

curves were measured with ®ve different metallic thicknesses of 30, 40, 50, 60 and 70 nm. The approximate SPR
curves denoted by the broken lines correspond well with theoriginal SPR curves indicated by the solid line when themetallic thickness is 50, 60 and 70 nm in Fig. 2 and 40, 50,
60 and 70 nm in Fig. 3. The approximate SPR curves agree
closely but not exactly with the original SPR curves. In part(b) of Figs. 2 and 3, the original SPR curves denoted by thesolid lines were ®tted to the asymmetric SPR curve equation
with ®ve parameters A, B, C, D and E in Eq. (29). The ®ts
were done in the range 108 from 528 to 628 in Fig. 2b andfrom 488 to 588 in Fig. 3b. The ®tting curves denoted by thebroken lines cannot be seen in these ®gures because of thevery good agreement between the ®tting curves and theoriginal SPR curves. This indicates that the asymmetric SPR
curve equation is very useful as a ®tting function of the SPR
curve. Part (c) of Figs. 2 and 3 shows the ratios of the ®veparameters A±E obtained by the ®tting procedures to thosegiven by Eqs. (30)±(34) in the asymmetric SPR curve

Fig. 2. Numerical calculations of asymmetric SPR curve equation at an excitation wavelength of 600 nm. It is assumed that water with a refractive index of
1.3330 is measured by an Au-based SPR with the prism having a high refractive index of 1.80: (a) SPR curve comparison of the asymmetric SPR curve
equation of Eq. (29) (broken line) with the complete equation given by the three-layer Fresnel equation of p-polarization (solid line). Five SPR curves with
Au-film thicknesses of 30, 40, 50, 60 and 70 nm are compared; (b) SPR curves (solid line) obtained by the complete equation are fitted to asymmetric SPR
curve equation of Eq. (29) with unknown parameters A, B, C, D and E; (c) ratios of five parameters A, B, C, D and E obtained by the fitting procedure to thosegiven by Eqs. (30)±(35) in the asymmetric SPR curve equation (parameter D is unity with an error of 0.2%).

Fig. 3. Numerical calculations of asymmetric SPR curve equation at an excitation wavelength of 800 nm. Numerical values for calculations are the same as
those used in Fig. 2 except for the excitation wavelength: (a) SPR curve comparison of the asymmetric SPR curve equation of Eq. (29) (broken line) with thecomplete equation given by three-layer Fresnel equation of p-polarization (solid line); (b) SPR curves (solid line) obtained by the complete equation are fitted
to the asymmetric SPR curve equation of Eq. (29); (c) ratios of five parameters A, B, C, D and E obtained by the fitting procedure to those given by Eqs. (30)±(35) in the asymmetric SPR curve equation (parameter D is unity with an error of 0.2%).

54

K. Kurihara et al. / Sensors and Actuators B 86 (2002) 49±57

equation. Of ®ve parameters, only parameter D is unity
within an error of about 0.2%. Parameter D is especially
important in the SPR experiments because it is the modi®ed
RA that provides correct information on linear changes in
the refractive index of the sensing layer. The exact agreement in parameter D between the theory and the ®ttingmeans that the approximation is highly effective and also
shows that the asymmetric SPR curve equation of Eq. (29) isuseful as a ®tting function.
In the actual SPR experiments, the real-time change in there¯ectance dip in the SPR curves is monitored in which therelative but not the absolute measurements of the angle arenormally carried out. In this case, the asymmetric SPR curve
equation given by Eq. (29) is unsuitable for numericalcalculations in data processing because some informationon the absolute value of the incident angle y is necessary.
This problem can be resolved by applying the function kx yto the ®rst-order expansion of a Taylor series at y  ySPR as
follows:
kx y p oep fsin ySPR  ycySPR cos ySPR g(44)

Using this equation, the asymmetric SPR curve equation of
Eq. (29) is further transferred to the following equation:
R X  A 1

BC X
X

D

D2  E2!
(45)

where X is the incident angle; the replacements are given by
A ) A, L2 B ) B, LC ) C, ySPR sin ySPR  LD ) D,pLE ) E and y ) X with L  c=o eP . When the unit of
X is degrees, the replacements are given by A ) A,
L2 B ) B, LC ) C, 180=p ySPR sin p=180ySPR  LD ) D, LE ) E, 180=py ) X with L  180=pc=po eP . As referred to in Eqs. (36) and (37), the R X of
Eq. (45) has a minimum, Rmin, at X  Xmin given byp!
B  B2  C 2 E 2Rmin Xmin   A 1
(46)
2E2
and
Xmin  D 

Bp
B2  C 2 E 2C(47)

When these procedures are applied to Eq. (25), Eq. (48) canbe obtained:a
RK X  1
(48)
X b2 g2
where X is the incident angle; descriptions of the replacements are omitted here. This is a ®tting function of Kretschmann's SPR equation for the actual experiments.
In Figs. 4 and 5, the usefulness of the asymmetric SPR
curve equation of Eq. (45) as a ®tting function for SPR
curves is assessed. As the original SPR curves were ®tted tothe asymmetric SPR curve equation, we used SPR curves
calculated by the three-layer Fresnel equation relating to thep-polarization at the excitation wavelengths of 600 and
800 nm in Figs. 4 and 5, respectively. In the calculations,water solutions with refractive indexes of 1.333±1.368 by
0.005 increments were used that were obtained with the Aubased SPR sensor that had an Au-®lm thickness of 50 nm
and a prism with a refractive index of 1.80. In part (a) ofFigs. 4 and 5, the original SPR curves are indicated by thesolid lines while the ®tting SPR curves are shown by thebroken lines. The ®ts were also done in a range of 108 from
528 to 628 in Fig. 4a and from 488 to 588 in Fig. 5a. The®tting curves are hardly seen in these ®gures because theoriginal curves and the ®tting curves almost overlapped. Part

Fig. 4. Curve-fitting to the asymmetric SPR curve equation when the original SPR curves are calculated at an excitation wavelength of 600 nm by the threelayer Fresnel equation relating to p-polarization. It is assumed that water solutions with refractive indices of 1.333±1.368 by 0.005 increment are measured byan Au-based SPR that has an Au-film thickness of 50 nm and a prism refractive index of 1.80. (a) Calculated SPR curves indicated by solid lines are fitted tothe asymmetric SPR curve equation whose curves are shown by broken lines but are hardly seen because the original curves cover the fitting curves. (b)Dependence of the fitting parameters A, B, C, D and E on the refractive index of the sensing layer. The fitting parameters are, respectively, normalized bythose obtained when the refractive index in the sensing layer is 1.333 as shown in Table 1. (c) Responses to refractive index changes in the modified
resonance angle (mod.RA) defined as parameter
D and the conventional resonance angle (con.RA) that precisely gives the minimum point in the reflectance.
p
The con.RA is described as D  B  B2  C2 E2 =C in Eq. (47).

K. Kurihara et al. / Sensors and Actuators B 86 (2002) 49±57

55

Fig. 5. Curve-fitting to the asymmetric SPR curve equation when the original SPR curves are calculated at an excitation wavelength of 800 nm. Numericalcalculations are the same as those used in Fig. 4 except for the excitation wavelength: (a) Calculated SPR curves indicated by solid lines are fitted to theasymmetric SPR curve equation whose curves are shown by a broken line; (b) Dependence of the fitting parameters A, B, C, D and E on the refractive index
of the sensing layer. The fitting parameters are, respectively, normalized by those obtained when the refractive index in the sensing layer is 1.333 as shown in
Table 1. (c) Responses to refractive index changes in the mod.RA and con.RA.

(b) of Figs. 4 and 5 shows the dependence of the ®tting
parameters A, B, C, D and E on the refractive index of thesensing layer. These ®ve ®tting parameters were, respectively, normalized by those obtained when the refractive
index in the sensing layer is 1.333 as shown in Table 1. The
parameters with a linear correlation coef®cient, r, of
1 jrj < 10 3 are A, D and E in Fig. 4b, and D in
Fig. 5b. The linear correlation coef®cients of A, B, C, D
and E in Figs. 4b and 5b are summarized in Table 2. Onlyparameter D satis®es the condition of 1 jrj < 10 4 in thetwo cases. This means that parameter D has a more linear
relation with the refractive index than the other parameters.
In part (c) of Figs. 4 and 5, the modi®ed RA, exactly equal toparameters D, is compared with the conventional RA whosemathematical formula is given by Eq. (47). Lines of theconventional and modi®ed RAs have a parallel relation witheach other. All four lines in part (c) of Figs. 4 and 5 havelinear correlation coef®cients, r, of 1 jrj < 10 4 . The
results of Figs. 4 and 5 indicate that the asymmetric SPR
curve formula of Eq. (45) is very useful as a ®tting function
Table 1
Actual coefficients of A, B, C, D and E used for the normalization in
Figs. 4b and 5b (the coefficients are obtained when the refractive index of
the sensing layer is 1.333)

Fig. 4bFig. 5b

A

B

C

D

E

0.87980.89393.066
0.16910.88000.2015

55.26
50.39

1.808
0.4311

Table 2
Linear correlation coefficients of A, B, C, D and E with respect to therefractive index in Figs. 4b and 5b
A
Fig. 4bFig. 5b0.99827
0.99036

B

C

D

E

0.999190.99897

0.993070.982820.99995
0.99998

0.999540.99731

for the SPR curves. On the other hand, our theory shows thatthe modi®ed RA is better in linearity than the conventional
RA. In fact, the modi®ed RA is better in linearity than theconventional RA by 1:9  10 5 for the linear correlationcoef®cient, while the modi®ed and conventional RAs appear
almost equal in linearity in Fig. 5c. This is an example that
supports our theory.
Finally, the asymmetric SPR curve equation was examined by actual SPR experiments using sucrose solutions withthe ®ve concentrations of 0, 1, 4, 10 and 20 wt.%. The solidlines of part (a) in Figs. 6 and 7 were obtained by the SPR
experiments with excitation wavelengths of 630 and
880 nm, respectively, while the broken lines are the ®tting
curves due to the asymmetric SPR curve equation. Table 3
shows two sets of ®ve parameters obtained by ®tting the SPR
curves denoted by 0 wt.% in part (a) of Figs. 6 and 7. The
experimental SPR curves largely differed from the theoretical SPR curves because of the optical distortion due to thehemicylindrical prism and cylindrical lens, spectral line
broadening of the light source, the adhesive thin metal layer,and so on. In spite of that, the ®tting curves closely agreed
with the experimental SPR curves in Figs. 6a and 7a. This,
therefore, suggests that close ®ts to the experimental SPR
curves are impossible with less than ®ve parameters while®ts with more than ®ve parameters give meaningless information on the SPR curves. Part (b) of Figs. 6 and 7 shows
plots of the modi®ed RA equal to parameter D and theconventional RA as a function of refractive indices ofthe sucrose solutions that are values at a temperature of
20 8C and at a wavelength of 589 nm [13]. Although theTable 3
Coefficients of A, B, C, D and E used for fitting SPR curves denoted by
0 wt.% in Figs. 6a and 7aFig. 6a (0 wt.%)
Fig. 7a (0 wt.%)

A

B

C

D

E

0.72070.85271.382
0.10061.078
0.2054

53.70
49.24

1.612
0.6428

56

K. Kurihara et al. / Sensors and Actuators B 86 (2002) 49±57

Fig. 6. Experimental examination of the asymmetric SPR curve equation by fitting of the SPR curves obtained at an excitation wavelength of 630 nm.
Sucrose solutions with concentrations of 0, 1, 4, 10 and 20 wt.% are measured: (a) The solid lines are the experimental SPR curves while the broken lines are
fitted to the asymmetric SPR curve equation given by Eq. (45). Parameters used for fitting SPR curves denoted by 0 wt.% are shown in Table 3. (b) Responses
to refractive index
changes of the mod.RA and con.RA. The mod.RA is given as parameter D by fitting to the asymmetric SPR equation. The con.RA is givenp
as D  B  B2  C 2 E2 =C in Eq. (47) by the fitting.

Fig. 7. Experimental examination of the asymmetric SPR curve equation by fitting of the SPR curves obtained at an excitation wavelength of 880 nm.
Sucrose solutions with concentrations of 0, 1, 4, 10 and 20 wt.% are measured: (a) The solid lines are the experimental SPR curves while the broken lines are
fitted to the asymmetric SPR curve equation given by Eq. (45). Parameters used for fitting SPR curves denoted by 0 wt.% are shown in Table 3. (b) Responses
to refractive index changes of the mod.RA and con.RA determined by fitting to the asymmetric SPR equation.

measurement conditions are different in wavelength fromthe referred data [13], the relative changes in the refractive
index of the sucrose solutions relative to that of water arealmost the same irrespective of the wavelength. In Figs. 6band 7b, the modi®ed RA is better in linearity than theconventional RA by 2:0  10 4 and 1:4  10 4 for the
linear correlation coef®cient, respectively. In our case, there
is a minor difference between the modi®ed and conventionalRAs, but in the high-performance SPR sensors, the modi®ed
RA would be more useful than the conventional RA.
Further experimental results and discussion will be
required for proving the practical usefulness of the asymmetric SPR curve-®tting equation, though the numericalsimulations in Figs. 2±5 and the experimental analyses in
Figs. 6 and 7 are demonstrated. An important issue fordiscussing the practical usefulness of the asymmetric SPR
curve-®tting equation is the real-time monitoring, which isplanned for experiments and will be described elsewhere.

Also, the asymmetric SPR curve-®tting equation is usefulfor SPR curves of absorption-based SPR sensors [14±16]that monitor changes in a complex refractive index caused
by absorption changes. The asymmetric SPR curve-®tting
equation is open for every real-time SPR sensory system
personally developed or commercially available.
6. Conclusions
The proposed asymmetric SPR curve equation offers apowerful ®tting function for real-time SPR sensors in termsof close ®ts to almost all SPR curves experimentallyobtained. The new curve-®tting equation has only ®veparameters of the ®ts to be suitable for a fast procedurein computer-assisted real-time monitoring. The new equation has a theoretical background of a closer approximationof the three-layer Fresnel equation regarding p-polarization

K. Kurihara et al. / Sensors and Actuators B 86 (2002) 49±57

than Kretschmann's SPR curve equation. It is theoreticallyfound that the modi®ed RA de®ned as parameter D in the
proposed equation is a better parameter in monitoringrefractive index changes than the conventional RA presenting the re¯ectance minimum in the SPR curve. The usefulness of the new equation in the ®ts to SPR curves wasnumerically and experimentally demonstrated.
Acknowledgements
We thank Dr. O. Niwa of the NTT Lifestyle and Environmental Technology Laboratories, Mr. T. Tobita of the NTT
Advanced Technology Corporation and Mr. H. Hashimotoof the Airport Environment Improvement Foundation for
their cooperation in the SPR experiments. Special thanks
also go to Dr. Y. Iwasaki of the NTT Lifestyle and Environmental Technology Laboratories for fruitful discussions
about the asymmetric SPR curve-®tting equation.
References[1] J.R. Sambles, G.W. Bradbery, F. Yang, Contemp. Phys. 32 (1991)
173±183.
[2] J. Homora, S.S. Yee, G. Gauglitz, Sens. Actuators B 54 (1999) 3±15.
[3] http://www.biacore.com/.
[4] E. Kretschmann, Z. Physik 241 (1971) 313±324.
[5] K.S. Johnston, S.S. Yee, K.S. Booksh, Anal. Chem. 69 (1997) 1851±
1884.
[6] K.S. Johnston, K.S. Booksh, T.M. Chinowsky, S.S. Yee, Sens.
Actuators B 54 (1999) 80±88.

57[7] T.M. Chinowsky, L.S. Jung, S.S. Yee, Sens. Actuators B 54 (1999)
89±97.
[8] I. Pockrand, Surf. Sci. 72 (1978) 577±588.
[9] W.P. Chen, J.M. Chen, Surf. Sci. 91 (1980) 601±617.
[10] W.P. Chen, J.M. Chen, J. Opt. Soc. Am. 71 (1981) 189±191.
[11] W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling (Eds.),
Numerical Recipes in C, Cambridge University Press, Cambridge, 1988.
[12] R.A. Innes, J.R. Sambles, J. Phys. F 17 (1987) 277±287.
[13] D.R. Lide (Ed.), Handbook of Chemistry and Physics, CRC Press,
Boca Raton, FL, 1996.
[14] K. Kurihara, E. Hirayama, K. Nakamura, S. Sasaki, D. Citterio, K.
Suziki, Proc. Pittcon 2000 (2000) 1323.
[15] K. Kurihara, K. Nakamura, E. Fujii, H. Hashimoto, D. Citterio, K.
Suzuki, Proc. Pacifichem. 2000 (2000) 139.
[16] K. Kurihara, K. Suzuki, Anal. Chem. 74 (2002) 696±701.

Biographies
Kazuyoshi Kurihara received his BS and DS degrees in physics from the
University of Tokyo in 1989 and 1994, respectively. He joined the photon
control project of Kanagawa Academy of Science and Technology (KAST)
from 1994 to 1998. He has worked for Collaboration of Regional Entities
for the Advancement of Technological Excellence (CREATE) in KAST
since 1999. His current work involves the development of new SPR
instruments for chemical and biochemical sensing.
Kaori Nakamura received her BE degree in 2000 from Keio University.
She is now a master course student in the same university. Her current
interest is the development of absorption-based SPR sensors for chemicaland biochemical sensing.
Koji Suzuki received his BE and DE degrees from Keio University in 1977
and 1982, respectively. He is Professor in Keio University as well as groupleader of CREATE in KAST. His major is analytical chemistry withchemical and biochemical sensing/sensors.