Improved technique for measuring refractive index and thickness of a transparent plate

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Improved technique for measuring refractive index

and thickness of a transparent plate

Zhi-Cheng Jian, 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 10 April 2003; received in revised form 13 August 2003; accepted 3 September 2003

Abstract

The phase difference between s- and p-polarizations of a circularly polarized heterodyne light beam reflected from a transparent plate is measured. The measured data is substituted into the specially derived equations and the refractive index can be calculated. Next, the variations of phase difference between s- and p-polarizations due to the wavelength shift and the extraction of the plate in a modified Michelson interferometer are measured. Then, its thickness can be calculated based on the measured value of refractive index, the variations of phase difference, and the specified value of wavelength shift.

Keywords: Refractive index; Thickness; Phase measurement; Heterodyne interferometry

1. Introduction

A transparent plate such as glass plate is often used in optics and semiconductor industries. Its refractive index and its thickness are very impor-tant optical parameters, especially in the design and fabrication of optical components and opto-electronic devices. Its thickness is always extremely larger than the wavelength of the light beam. So it is diﬃcult to measure the thickness with conven-tional interferometric methods. Some optical methods, such as low-coherence interferometry

[1–3] and confocal microscopy [4,5], are proposed to measure the thickness of a transparent plate. However, they deliver not the geometry thickness, but the optical thickness, which is the product of the geometrical thickness and the refractive index. The associated optical thickness can be evaluated from the direct readouts of the division marks of the high-resolution stage. In addition, its numeri-cal aperture affects the measurement resolution of the confocal microscopy. On the other hand, some other methods [6–9] are reported for measuring the refractive index. Almost all of them are related to the measurement of light intensity variations. Consequently, the stability of the light source, the scattering light, the internal reflection, and other factors influence the accuracy of measurements and thus the resolution of the results is decreased. www.elsevier.com/locate/optcom

*

Corresponding author. Tel.: 573-1951; fax: +886-3-571-6631.

E-mail address:[email protected](D.-C. Su).

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Although several other papers [10–14] are pre-sented for the measurement of refractive index and the thickness of a transparent plate, they also need a high-resolution stage to displace the test plate, due to the introduction of the low-coherence interferometry or the confocal microscopy.

In this paper, an improved optical technique for measuring the refractive index and the thick-ness of a transparent plate is presented by using the heterodyne interferometry [15–17] and the two-wavelength interferometry [18]. Firstly, the phase difference between s- and p-polarizations of a circularly polarized heterodyne light beam re-flected from a transparent plate is measured. The measured data is substituted into specially de-rived equations and the refractive index can be calculated. Secondly, the variations of phase dif-ference between s- and p-polarizations due to the wavelength shift and the extraction of the plate in a modified Michelson interferometer [19] are

measured. Then, its thickness can be calculated based on the measured value of refractive index, the variation of phase diﬀerence, and the speciﬁed value of wavelength shift. Consequently, the measurements of the thickness and refractive in-dex can be operated precisely only with this single measurement system. This technique has merits such as a simple optical setup, easy operation, and rapid measurement. And the validity of this method is demonstrated.

2. Principle

The schematic diagram of this method is shown in Fig. 1. For convenience, theþz-axis is chosen to be along the light propagation direction and the x-axis is along the direction perpendicular to the paper plane. A light coming from a heterodyne light source [20] has an angular frequency diﬀerence x

Fig. 1. Schematic diagram for measuring the phase diﬀerences: EO, electro-optic modulator; BS, beam splitter; Q, quarter-wave plate; T, test plate; M, mirror; PBS, polarizing beam splitter; AN, analyzer; D, detector; FG, function generation; and PC, personal computer.

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between s- and p-polarizations, and its Jones vector [21] can be written as E¼ _EExexpðixt=2Þ yexpðixt=2Þ : ð1Þ

It is incident on a beam-splitter BS, and is divided into two parts: the transmitted light and the re-ﬂected light. The transmitted light passes through a quarter-wave plate Q. If the fast axis of Q is located at 45° relative to the x-axis, then the Jones vector of the light can be written as

Ei¼ Qð0°Þ E ¼ cosðxt 2Þ sinðxt 2Þ ¼1 2 1 i exp ixt 2 þ1 2 1 i exp ixt 2 : ð2Þ From Eq. (2), we can see that the right- and left-circular polarizations have frequency shifts x=2 and x=2, respectively. Thus there is an angular frequency diﬀerence x between them. The light, at the central wavelength k1 of the heterodyne light source, is incident at h on the test plate T with refractive index n1. The light reﬂected from this test plate passes through an analyzer AN1 and enters a photodetector D1. If the transmission axis of AN1is located at 45° with respect to the x-axis, then the Jones vector of the light arriving at D1 is E11¼ rpcos xt 2 h rssin xt 2 i 1=2 1=2 ; ð3Þ

where rp and rs are the reﬂection coeﬃcients for p- and s-polarizations, respectively. According to the FresnelÕs equations [22], we have

rp¼

cos½sin1ðsin h=n1Þ n1cos h cos½sin1ðsin h=n1Þ þ n1cos h

; ð4aÞ

and rs¼

cos h n1cos½sin1ðsin h=n1Þ cos hþ n1cos½sin1ðsin h=n1Þ

; ð4bÞ

respectively. Hence, the intensity measured by D1is I11¼ I0½1 þ c cosðxt þ /11Þ: ð5Þ

Here, the average intensity I0, the visibility c, and the phase diﬀerence /11 are given as

I0¼ 1 4 rp 2 þ rj js 2 ; ð6aÞ c¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A2_{þ B}2 p I0 ; ð6bÞ and /11¼ tan1 B A ¼ tan1 rprs 1=2ð rp 2 rj js 2 Þ " # : ð6cÞ

Here I11is the ﬁrst test signal. Besides, the modu-lated electronic signal of the heterodyne light source is ﬁltered and becomes the reference signal. It has the form as

Ir¼ 1

2½1 þ cosðxtÞ: ð7Þ

Both of these two sinusoidal signals I11 and Irare sent to a lock-in amplifier, and then /11 can be obtained. Moreover, it is easily seen from Eqs. (4)– (7) that the refractive index n1 depends on /11. Consequently, n1 can be calculated with the mea-surement of the phase difference /11 under the condition in which h is specified.

On the other hand, the reflected light coming from BS is reflected again by a mirror M1and en-ters a modified Michelson interferometer [19]. It consists of a polarization beam-splitter PBS, two mirrors M2and M3, an analyzer AN2, and a pho-todetector D2. The test plate T is just located in one arm and the light beam passes through it perpen-dicularly. In the interferometer, PBS divides the light into two beams. The paths of these two beams are PBS! M2! PBS ! M1! BS ! AN2! D2 (for the reflected s-polarization light) and PBS! T! M3! T ! PBS ! M1! BS ! AN2 ! D2(for the transmitted p-polarization light). If the trans-mission axis of AN2 is 45° to the x-axis, then Jones vectors of p- and s-polarizations arriving at D2are Ep¼ 1 2pffiffiffi2 1 1 eiððxt=2Þþ/21Þ_; _ð8aÞ

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and Es¼ 1 2pffiffiffi2 1 1 eiðxt=2Þ; ð8bÞ

respectively. /21 is the phase diﬀerence due to the optical path diﬀerence and it can be expressed as /21¼

4pðn1dþ d0Þ k1

; ð9Þ

where d is the thickness of the glass plate, and d0is the length difference between the two arms except the thickness d in the interferometer. Hence the intensity measured by D2is I21¼ Ep þ Es 2 ¼1 2½ þ cosðxt þ /1 21Þ: ð10Þ Here I21 is the second test signal. As these two si-nusoidal signals I21and Irare sent to a lock-in am-plifier, and then /₂₁can be obtained as previously. If the central wavelength of the heterodyne light source is changed to k2, then the test signals I11and I21still have the forms of Eqs. (5) and (10) but with the phase differences /12and /22, respectively. The refractive index n2 of T at wavelength k2 can be obtained with the measurement of /12by using the same technique to estimate n1. The variation of phase difference of the second test signal due to the wavelength variation Dk, W1¼ /22 /21 ¼4pðn2dþ d0Þ k2 4pðn1dþ d0Þ k1 ¼4p½dðn2k1 n1k2Þ d0Dk k1k2 ; ð11Þ

can be obtained, where Dk equals k2 k1. Next the test plate T is removed from the interferometer.

The second test signal also still has the form of Eq. (10) but with another phase diﬀerence /20. Hence, the phase diﬀerence variation of the second test signal after the test plate has been removed can be obtained similarly, and it can be written as W2¼ /20 /21¼ 4pðd þ d0ÞDk k1k2 : ð12Þ Consequently, we have W¼ W2 W1¼ 4pdðn1k2 n2k1 DkÞ k1k2 : ð13Þ

It is obvious from Eq. (13) that the thickness d can be calculated with the measurement of W under the conditions in which n1, n2, k1, and k2are speciﬁed.

3. Experiments and results

In order to show the feasibility of this method, we measured the thickness and the refractive in-dices of one BK7 plate, one fused silica plate, and one BASF2 plate at 25 °C. The heterodyne light source consisting of a tunable diode laser (Model 6304, New Focus) and an electro-optic modulator EO driven by a function generator FG was used, as shown in Fig. 1. The frequency difference be-tween p- and s-polarizations was 1 kHz. A lock-in amplifier with resolution 0.01° (Model SR850, Stanford Research System) was used to measure the phase difference. And a personal computer was employed to record and analyze the data. The experimental conditions h¼ 45°, k1¼ 632:80 nm, k2¼ 632:81 nm, and Dk ¼ 0:01 nm were used, and the results are listed in Table 1. Because Dk is so small, the measured results of n1 and n2are equal

Table 1

Measured results and their associated reference data

BK7 Fused silica BASF2

n 1.5151 1.4498 1.6607 nref 1.51509a 1.45b 1.6606c d(mm) 1.3201 2.9299 5.0368 dref(mm) 1.31 2.93 5.03 jDnj 3.77 104 _3.16₁₀4 _5.27₁₀4 jDdj (lm) 0.66 0.68 0.62

Note 1. Superscripts a, b, and c represent the reference data coming from Coring Ltd., Scott Ltd., and Ohara Ltd., respectively. Note 2. The temperatures are ﬁxed at 25°C during the measurement procedure.

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[23]. We used the symbol n to represent them in the Table. In addition, the thickness of these test plates was measured with a conventional micrometer. Their associated results are also ad-ded to Table 1 for comparison. It is clear that they show good agreement.

4. Discussion

Let Dni, D/1i, Dd, and DW be the errors in ni, /1i, d, W, respectively, where i is either 1 or 2. From Eqs. (3)–(6) we get

Dni j j ¼ðB þ CÞ D/j 1ij A ; ði ¼ 1 or 2Þ; ð14aÞ where A¼ 8ð1 þ n2 iÞ cos h cos½sin 1_{ðsin h=n} iÞ

fcos 2h cos½2 sin1ðsin h=niÞg; ð14bÞ B¼ 2ð1 þ n2

iÞ 2

cos 2hþ n2 icos 4h þ cos 2½h sin1ðsin h=niÞ þ 2ð1 n2 i þ n 4 i þ cos½2 sin 1_{ðsin h=n} iÞÞ þ cos 2½h þ sin1ðsin h=niÞ; ð14cÞ and C¼ n2 if2½2 þ n 2 i þ ð4 þ n 2 iÞ cos 2h cos½2 sin1ðsin h=niÞ

þ cos½4 sin1ðsin h=niÞg: ð14dÞ In our experiments, Eq. (13) can be rewritten as W¼4pdðn 1ÞDk

k1k2

¼4pdðn 1Þ keq

; ð15Þ

where keq is the equivalent wavelength. In addi-tion, from Eq. (15) we also get

Dd

j j ¼ keq

4pðn 1ÞjDWj: ð16Þ

Angular resolution of the lock-in amplifier, second harmonic error, and polarization-mixing errors are the factors that may influence the accuracy in the phase difference errors in this method. So the total phase difference errors ofjD/11j and jDWj can be decreased to 0.03° [24] in our experiments.

Substituting the conditions jD/1ij ¼ jDWj ¼ 0:03° and Dk¼ 0:01 nm into Eqs. (14) and (16), the measurement errors of each plate are calculated and listed in Table 1. In this table, jDnj is used to represent jDn1j and jDn2j And we also obtain keqﬃ 40 mm.

To avoid phase wrapping [25], it is necessary to let the data of W be smaller than p. So our ex-perimental conditions are suitable for the trans-parent plate with the thickness smaller than keq/ 4(n 1), which is about 20 mm. From Eqs. (15) and (16), it can be seen that as keq increases, the measurable range of the thickness becomes wider and the associated measurement resolution decreases.

The phase difference between s- and p-polar-izations of the light reflected from the surface of the test plate is measured to evaluate the re-fractive index and the thickness in this method. So, surface contamination will distort the phase difference, the associated errors will introduce to the measured results. In industry, a transparent plate must be cleaned and its homogeneity tested with a conventional Twyman–Green interfero-meter or Mach–Zehnder interferointerfero-meter before it is fabricated. Hence, the test plate should be done with the same procedures as in industrial fabrication before it is tested with this method. If not, we would obtain the refractive index of the surface material on the test plate and its nominal thickness.

Moreover, both the refractive index and the thickness of the test plate depend on the ambient temperature. Here, BK7 is taken as an example. Its temperature coeﬃcient of refractive index [26] is about 2.89 106_{/°C in the range 20–40 °C,} and its coeﬃcient of linear expansion [26] is 7.1 106_{/°C in the range 30–70 °C. If the shift} in the ambient temperature is 10 °C in our ex-periments, then the induced errors in the mea-surements of refractive index and thickness are 2.89 105 _{and 0.094 lm, respectively. Both of} them are so small compared with the associated measurement errors listed in Table 1 that they may be neglected under our experimental con-ditions. Other test plates give similar results. In our experiments, the ambient temperature was kept at 25 °C.

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5. Conclusion

Based on the heterodyne interferometry and the two-wavelength interferometry, an improved technique for measuring the refractive index and the thickness of a transparent plate has been pre-sented. These two optical parameters can be esti-mated in just one optical conﬁguration. This technique has merits such as a simple optical set-up, easy operation, and rapid measurement. Its validity has been demonstrated. It is suitable for a transparent plate with the thickness smaller than 20 mm in our experiments.

Acknowledgements

This study was supported in part by the National Science Council, Taiwan, ROC, under contract NSC 91-2215-E-009-020.

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