Measurement of wavelength shift by using surface
plasmon resonance heterodyne interferometry
Kun-Huang Chen, Cheng-Chih Hsu, Der-Chin Su
*Institute of Electro-Optical Engineering, National Chiao Tung University, 1001 Ta-Hsueh Road, Hsin-Chu 300, Taiwan, ROC Received 8March 2002; received in revised form 13 May 2002; accepted 30 May 2002
Abstract
A linearly polarized light is incident on a surface plasmon resonance (SPR) apparatus at the resonant angle, the surface plasmons are excited. Small wavelength shifts will introduce phase difference variations between s- and p-polarizations of the reflected light. These phase difference variations can be measured accurately by using heterodyne interferometry. Based on these facts, a novel method for measuring small wavelength shifts is proposed. It has the advantages of both common-path interferometry and heterodyne interferometry. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction
A monochromatic interferometer is used to differentiate the test wave with a reference wave in the unit of the light wavelength. It is necessary to measure the wavelength variations of a light source to insure the measurement resolution. In addition, an optical interferometric sensor tech-nique for measuring small wavelength shift is be-coming important [1–4]. A common spectrometer and an unequal-path interferometer [4] are always used to measure the wavelength shifts. Although the latter has a better resolution than the former, it needs an additional stabilized light source and a
feedback control system. So it becomes more complicated and is difficult to operate.
In this paper, a novel method for measuring small wavelength shifts is presented. A linearly polarized light enters a surface plasmon resonance (SPR) [5,6] apparatus. If the incident angle is just equivalent with the resonant angle, the surface plasmons are excited. At this time, phase differ-ences between s- and p-polarizations of the re-flected light is changed with the variation of the refractive index of the thin metal film of SPR ap-paratus. The refractive index is related to the wavelength. And the phase difference variations can be accurately measured by using heterodyne interferometry. Based on these effects, a novel method for measuring small wavelength shifts is presented by using the specified dispersive equa-tion of the thin metal film. It has the advantages of both common-path interferometry and heterodyne interferometry.
Optics Communications 209 (2002) 167–172
www.elsevier.com/locate/optcom
*
Corresponding author. Tel.: 573-1951; fax: +886-3-571-6631.
E-mail address:[email protected](D.-C. Su).
0030-4018/02/$ - see front matterÓ 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8( 0 2 ) 0 1 6 4 1 - 3
2. Principle
2.1. Phase difference resulting from reflection of SPR apparatus
A ray of light in the air is incident at h on one side surface of the SPR [5,6] apparatus of a Krestschmann configuration [7] as shown in Fig. 1. This apparatus is an isosceles right-prism with a thin metal film of thickness d2 deposited on the
hypotenuse surface. The refractive indices of the prism and the thin metal film are n1 and n2,
re-spectively. As h equals the resonant angle hsp,
surface plasmons are excited. Then the reflection coefficients of p- and s-polarization components can be expressed as [5] rq¼ rq12þ rq 23e i2kz2d2 1þ rt 12rt23ei2kz2d2 q¼ p; s; ð1Þ where rijq is the Fresnel reflection coefficient
be-tween the ith and jth media and is given as rqij¼X q i X q j Xiqþ Xjq ; ð2aÞ and Xiq¼ n 2 i=kzi q¼ p; kzi q¼ s; ð2bÞ where kzi is the component of the wave vector in
medium i in the z direction and is given as kzi¼ k0ðn2i n
2 1sin
2
hÞ1=2; ð2cÞ and k0is the free-space wave vector. The amplitude
reflection coefficients rp and rscan be written as
rp¼ rp
ei/p; r
s¼ rj jes i/s; ð3Þ
then the phase difference variations / between p and s polarization components is
/¼ /p /s: ð4Þ
It is obvious from Eqs. (1)–(4) that the phase dif-ference is strongly dependent on n1 and n2. In
general the dispersion equations of an absorption material are given as [8]
nðkÞ ¼ a0þ a1kþ a2k2þ a3k3þ a4k4þ ; ð5aÞ
and
kðkÞ ¼ b0þ b1kþ b2k2þ b3k3þ b4k4þ ; ð5bÞ
where n and k are the real and imaginary indices; a0; a1; a2; a3; . . ., and b0; b1; b2; b3; . . .are the
coeffi-cients and k is the wavelength. If the wavelength has small variation Dk, then the variation in phase difference is
D/¼ o/ðnðkÞ; kðkÞÞ ok
Dk: ð6Þ
Eq. (6) can be rewritten as Dk¼ ok
o/ðnðkÞ; kðkÞÞ
D/: ð7Þ
If the thin film with specified dispersion equation is used, then it is seen that from Eqs. (1), (4), (5a), (5b) and (7) that the small wavelength variation Dk can be calculated with the measurement of the phase difference variation D/.
2.2. Phase-difference measurements with heterodyne interferometry
Chiu et al. [9] proposed a method for measuring the refractive index of a transparent material by using total-internal-reflection heterodyne interfer-ometry. A schematic diagram of the optical ar-rangement of our method, which is based on similar considerations, was designed and is shown in Fig. 2. A linearly polarized light passes through a half-wave plate H and its polarization plane is at a with respective to the horizontal axis. Then it passes through an electro-optic modulator (EO), and is incident at hsp upon a surface plasma
reso-nance apparatus. The reflected light passes an
analyzer AN with the transmission axis at 45° to the horizontal axis and is detected by a photode-tector D. If the fast axis of EO under an applied electric field is in the horizontal direction and a sawtooth signal of angular frequency x and am-plitude Vk=2, the half-wave voltage of the EO, is
applied to the EO, the intensity measured by D can be derived as [9] It1¼ Ej jt 2 ¼1 4 r 2 pcos 2 a h þ r2 ssin 2 a
þ 2rprscos asin a cos xtð þ /1Þ
i : ð8Þ Here It1 is the test signal. On the other hand, the
electrical signal generated by the function genera-tor FG is filtered and becomes the reference signal. It has the form as
Ir¼
1
2½1 þ cosðxtÞ : ð9Þ Both of these two sinusoidal signals are sent to the phase meter PM, then /1 can be obtained. In
the second measurement let wavelength be changed to kþ Dk, then the test signal has the form It2¼ Ej jt 2 ¼1 4 r 2 pcos 2a h þ r2 ssin 2 a
þ 2rprscos a sin a cos xtð þ /2Þ
i : ð10Þ
/2 can be obtained in a similar way. Finally, by
substituting the value of D/¼ /2 /1 into Eq.
(7), we can evaluate the small wavelength shift Dk.
3. Experiments and results
In order to show the feasibility of this method, we used an SPR apparatus with thin gold film of thickness 35 nm to measure the wavelength vari-ation in the range 632.6 and 633.9 nm, with 633.3 nm as the initial wavelength. This apparatus con-sists of a BK7 glass prism and the thin gold film deposited by the commercial sputtering system (Model BA510, Balzers) with the1 nm thickness accuracy. To insure its quality, an ellipsometer (Model eta, Steag) was used to measure the re-fractive indices and the thickness of the thin gold film and the refractive index of the prism in situ. The refractive index of prism is 1.5151 and it does not change in this wavelength range. And the measured results of the thin gold film are shown in Fig. 3. If these data are substituted into Eqs. (5a) and (5b), we can obtain the coefficients of the dispersion equations by using polynomial fitting technique that is calculated by the software ‘‘OR I G I N’’. They are a0¼ 65:00949, a1¼
0:36354, a2¼ 7:70989 104, a3¼ 7:3235 107, a 4¼ 2:6242 1010, b0¼ 14:32305, b1¼ 0:05643, b2¼ 6:13036 105, b3¼ 2:65412 108, respectively. Then h sp¼ 43:9° can be
calcu-lated [10]. It could be achieved by using a high-resolution rotation stage with an angular resolution 0.005° (Model PS-h-90, Japan Chuo Precision Industrial Company), and was made sure by measuring the critical minimum reflectance [5]. A phase meter with an angular resolution of 0.01° was used to measure the phase difference. A velocity laser (Model 6304, New Focus) and an EO (Mode 4002, New Focus) with a half-voltage of 125 V were used in this test. The frequency of the sawtooth signal that was applied to the EO was 1 kHz. In addition, we used a personal computer to record and analyze the data. The experimental results of the phase difference variation D/ versus the wavelength shift Dk calculated with Eqs. (1), (4) and (7), are shown in Fig. 4. For comparison
Fig. 2. Schematic diagram of measurement of the phase dif-ference of the reflected light: H, half-wave plate; EO, electro-optic modulator; FG, function generator; PM, phase meter; AN, analyzer; D, detector; PC: personal computer.
with the readout data of the wavelength shift on the controller of the velocity laser, the relation curve between the readout data Dkrand the
mea-sured data Dk was depicted and shown in Fig. 5. In this figure, the symbols and represent the readout data and the measured data, respectively. Here the quantity of the error bar is 0.04215 nm which is the standard deviation of the difference between the measured data and readout data. Because the relation curve is nearly a straight line, it can be seen that the measured data show good correspondence with the readout data. Hence it can be realized that if the SPR apparatus consist-ing of the thin metal film with known dispersion equations is used, as described in this method, then Dk can be estimated from the measured D/.
4. Discussion
The thin gold film was deposited on BK7 prism by sputtering processes. Owing to the associated adhesion, its dispersion equation will change slightly. Although the dispersion equation of gold material could be obtained from reference book [8], it is necessary to measure its dispersion equa-tion in situ to enhance its accuracy.
As h equals the resonance angle hsp, the
reflec-tion coefficient rp is very small. To enhance the
contrast of the test signal, the component of p-polarization of the incident linearly polarized light should be increased. So a half-wave plate is located before the EO to rotate the azimuth angle of the polarization plane of the incident light. In our
(a) (b)
Fig. 3. Measurement results for (a) n and (b) k of gold film in the wavelength range 600–700 nm.
Fig. 4. Experimental results of D/ versus Dk.
experiments, the angle between its fast axis and the horizontal axis is set to 10°, the contrast of the test signal is about 0.88.
Angular resolution of the phase meter, second harmonic error, and polarization-mixing errors are factors that may influence the accuracy in the phase difference in this method. So the total phase difference errors D/err can be decrease to 0.03° in
our experiments [11]. Assuming the measured wavelength shift range Dk is 0.1 nm, then the phase difference variation is about 0.37° according to Eqs. (1)–(5b) and (7). Substituting these data into the following equation
Dkerr¼ Dk D/err D/ ; ð11Þ
a resolution Dkerr of 0.00853 nm can be obtained.
If the thickness d changes and the complex re-fractive index of the thin gold film remains un-changed, then the relation curve of the Dkerrversus
d can be obtained as shown in Fig. 6. Although theoretically the resolution gets better as d in-creases, the reflection coefficient rp will be too
small to detect. Compromising these conditions, d¼ 35 nm is chosen in our experiments.
If there are damages, such as, aging, contami-nation scratching, and dirt, on the thin metal film, hsp will be changed. So the condition h¼ hspis no
more valid, and the measurement resolution de-creases obviously. To avoid these damages, the SPR apparatus is located into a protection box as shown with dotted lines in Fig. 1. The condition h¼ hsp can be obtained by measuring the critical
minimum reflectance, and every time it should be operated in advance. In general the cross-section of the laser beam is about 4 mm2, and the area on
the thin metal film illuminated by the laser beam is nearly 7 mm2. Within this small area, the thin
metal film has a good uniformity.
5. Conclusions
In this paper, a novel method for measuring a small wavelength shift is presented. A linearly polarized light enters a surface plasmon resonance apparatus of Krestschmann configuration. If the incident angle is exactly equivalent with the reso-nant angle, the surface plasmons are excited. At this time, small wavelength shifts will introduce phase difference variations between s- and p-po-larizations of the reflected light. They can be measured accurately by using heterodyne inter-ferometry. A surface plasmon resonance appara-tus with thin gold thin film of thickness 35 nm was used to measure the wavelength shift in the range 632.6 and 633.9 nm at hsp¼ 43:9°. The measured
data show good correspondence with the readout data on the controller of the light source, its res-olution is 0.00853 nm. It has the advantages of both common-path interferometry and heterodyne interferometry. And it may have some applications such as the determination of wavelength variations in a wavelength division multiplexing system or the measurement of the wavelength change in-duced in optical sensors used for monitoring gas exhausts, temperature fluctuations, and mechani-cal vibration [2,4].
Acknowledgement
This study was support in part by National Science Council, Taiwan, under contract NSC 89-2112-M-009-022.
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