A method for measuring two-dimensional refractive index distribution with the total internal reflection of p-polarized light and the phase-shifting interferometry

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A method for measuring two-dimensional refractive index

distribution with the total internal reﬂection of p-polarized light

and the phase-shifting interferometry

Zhi-Cheng Jian, Po-Jen Hsieh, Hung-Chih Hsieh, Huei-Wen Chen, Der-Chin Su

*

Department of Photonics and Institute of Electro-Optical Engineering, National Chiao Tung University, 1001 Ta-Hsueh Road, Hsin-Chu 300, Taiwan, ROC

Received 19 January 2006; received in revised form 20 June 2006; accepted 5 July 2006

Abstract

Based on the total internal reﬂection of p-polarized light and the phase-shifting interferometry, an alternative method for measuring the two-dimensional refractive index distribution of a material is presented. The p-polarized light is incident on the boundary between a right-angle prism and a tested material. When the total internal reﬂection occurs at the boundary, and the p-polarized light has a phase vari-ation. It depends on the refractive index of the tested material. Firstly, the two-dimensional phase variation distribution of the p-polarized light at the boundary is measured by the four-step phase shifting interferometric technique. Then, substituting the data into the special equations derived from Fresnel equations, the two-dimensional refractive index distribution of the tested material can be obtained. 2006 Elsevier B.V. All rights reserved.

Keywords: Total internal reﬂection; Phase-shifting interferometry; Fresnel equations; Two-dimensional refractive index distribution

1. Introduction

Refractive index is an important characteristic constant of an optical material, and it also determines the transpar-ency of the material. Some techniques such as the reﬂec-tance method [1], the critical angle method [2,3], the ellipsometry [4], the polarization analysis method [5]and the heterodyne interferometry[6]have been proposed for measuring the refractive index. Although they have good measurement results, they are often used to evaluate the refractive index at one point of a material [7,8]. To over-come this drawback, an alternative method for measuring the two-dimensional refractive index distribution of a tested material is presented in this paper, based on the total internal reﬂection and the phase-shifting interferometry

[9–12]. Firstly, special equations to estimate the phase var-iation of the p-polarized light under the total internal

reﬂection are derived based on Fresnel equations [13]. Next, the four-step interferometric technique [14] is used to measure the two-dimensional phase variation distribu-tion of the p-polarized light at the boundary between a right-angle prism and a tested material under the total internal reﬂection. Finally, the measured data are substi-tuted into special equations derived previously, and the two-dimensional refractive index distribution of a tested material can be estimated. To show the validity of this method, a mixed liquid of oils and water was tested. It has several merits such as easy operation, rapid measure-ment, and a simple optical setup, etc.,

2. Principle

2.1. Phase variation of p-polarized light under total internal reﬂection

A ray of p-polarized light in air is incident at hton the

one side surface of a right-angle prism with refractive index

*

Corresponding author. Tel.: +886 3 573 1951; fax: +886 3 571 6631. E-mail address:t7503@faculty.nctu.edu.tw(D.-C. Su).

www.elsevier.com/locate/optcom Optics Communications 268 (2006) 23–26

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n1, as shown in Fig. 1. The light ray is refracted into the

prism and it propagates toward the base surface of the prism. At the base surface of the prism, there is a boundary between the prism and the tested material TM of refractive index n2(x, y) where n1> n2. If hiis larger than the critical

angle, the light is totally reflected at the boundary. Accord-ing to Fresnel equations, the reflection coefficients of p-polarized light can be expresses as

rp¼ n2_{cos h} i i ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin2hi n2 p n2_{cos h} iþ i ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin2hi n2 p ¼ jrpj expði/pÞ; ð1aÞ where n¼ n2 n1

, /p(x, y) is the phase-variation and it is

gi-ven as /p¼ 2 tan1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin2hi n2 p n2_{cos h} i ! : ð1bÞ

In addition, Eq.(1b)can also be rewritten as

n¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ 4 tan/p 2 cos hi sin hi 2 r 2 tan/p 2 cos hi 2 v u u u u u t : ð2Þ

Hence, it is obvious from Eq.(2)that n2(x, y) can be

cal-culated with the measurement of /p(x, y) under the

exper-imental conditions in which hiand n1are speciﬁed.

2.2. Phase variation measurements with phase-shifting interferometry

The schematic diagram of this method is shown in

Fig. 2. For convenience, the +z-axis is chosen to be along the light propagation direction and the y-axis is along the direction perpendicular to the paper plane. A light beam coming from a laser light source passes through a polarizer P and a beam-expander BE, and it becomes the collimating light. The collimating light enters a modified Twyman– Green interferometer. If the transmission axis of P is located at 0 with respect to the x-axis, then the light becomes the p-polarized light. In the interferometer, the collimating light is incident on a beam-splitter BS and divided into two parts: the transmitted light and the reflected light. The reflected light is normally reflected by a mirror M1driven by a piezo-transducer PZT and passes

through BS. Then it enters a CCD camera. Here, it acts as the reference light Er. On the other hand, the transmitted

light is reﬂected by the mirrors M2and M3, and enters a

right-angle prism. After it is totally reﬂected at the bound-ary between the prism and the tested material TM, it prop-agates out of the prism. Then, it is normally reﬂected by a mirror M4and comes back along the original path. It is

reﬂected by BS and also enters the CCD camera. Here, it acts as the test light Et. Therefore, the interference intensity

measured by the CCD can be written as Iðx; yÞ ¼ jEtþ Erj 2 ¼ at ei/t þ ar ei/r 2 ¼ a2 rþ a 2 t þ 2 ar at cosð/t /rÞ ¼ a2 rþ a 2 t þ 2 ar at cosð2 /pþ wÞ

¼ Aðx; yÞ þ Bðx; yÞ cosð/ðx; yÞÞ; ð3Þ

where aiand /i(i = t or r) represent the amplitude and the

phase of Ei, /p is the phase variation of the p-polarized

light under the total internal reflection in the prism, and wis the sum of the phase difference owing to the optical path difference between two interfering lights and extra 45° i θ n2(x,y) n1 t θ TM

Fig. 1. The total internal reﬂection at the boundary between a prism and a tested medium. CCD He-Ne Laser BS PC TM M1 M2 M3 BE M4 PZT P(0˚) Prism

Fig. 2. Schematic diagram for measuring the two-dimensional refractive index distribution of a material. P, polarizer; BE, beam expander; BS, beam splitter; M, mirror; TM, tested material; PC, personal computer.

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phase shifts produced by the reﬂections at BS and mirrors. Then, the phase-shifting interferometric technique is ap-plied to measure the two-dimensional phase distribution /(x, y). The CCD takes four interferograms as the PZT moves M1to change the phase of the reference light. An

ex-tra phasep

2is added between two successive interferograms.

So the intensities of these four interferograms can be writ-ten as

I1ðx; yÞ ¼ Aðx; yÞ þ Bðx; yÞ cosð/ðx; yÞÞ; ð4aÞ

I2ðx; yÞ ¼ Aðx; yÞ þ Bðx; yÞ cosð/ðx; yÞ þ p=2Þ; ð4bÞ

I3ðx; yÞ ¼ Aðx; yÞ þ Bðx; yÞ cosð/ðx; yÞ þ pÞ; ð4cÞ

and

I4ðx; yÞ ¼ Aðx; yÞ þ Bðx; yÞ cos /ðx; yÞ þ

3p 2

: ð4dÞ

By solving the simultaneous equations, we get /ðx; yÞ ¼ tan1 I4 I2

I1 I3

: ð5Þ

From Eq.(5) follows that Eq.(3), we have /pðx; yÞ ¼

1

2ð/ðx; yÞ wÞ: ð6Þ

In the second measurement let the base surface of the prism free without any tested material. We obtain

/0¼ 2 /aþ w; ð7Þ

where the phase variation /a can be calculated with the

refractive index n1 of the prism and n2= 1. Substituting

/a and /0 into Eq. (7), the data of w can be calculated.

Then substituting the data of w into Eq.(6), /p(x, y) can

be estimated. Finally, the two-dimensional refractive index distribution n2(x, y) of a tested material can be calculated

by using Eq.(1b).

3. Experiments and results

In order to show the feasibility of this method, we tested a mixed liquid of ricinus oil, olive oil, baby oil, and water. Their refractive indices are 1.513, 1.474, 1.463, and 1.33, respectively. A He–Ne laser with a 632.8 nm wavelength, a right-angle prism made of SF8 glass with refractive index n1= 1.68894 [15], the PZT (PZ-91, Burleigh Instruments,

Inc.) with motion sensitivity 0.002 lm/V, and the CCD camera (TM-545, PULNiX Inc.) with 510· 492 pixels and 8-bit gray levels were used in this test. It is necessary to choose a suitable incident angle that is larger than the critical angles of all tested materials. In our experiment, the maximum refractive index of the tested material is

1.513, and its critical angle is 63.615. For our convenience, we choose the condition hi= 63.62. Besides, the incident

light intensity and the dynamic range of the CCD camera are so adjusted that they can match with each other. Con-sequently, the output of the CCD camera is linearly pro-portional to the light intensity.

The interferograms were sent to a personal computer PC and they were analyzed with the software IntelliWaveTM

(Engineering Synthesis Design Inc.). The results were depicted as shown in Figs. 3 and 4 by using the software Matlab (MathWorks Inc.). They are the two-dimensional phase variation distribution /p(x, y) and the associated

two-dimensional refractive index distribution n2(x, y) of

the tested material, respectively. 4. Discussion

From Eq. (2)we get

where C¼ cos2 hi sin 2 hi tan /p 2 ; ð8bÞ

Dn2and D/pare the errors in n2and /p, respectively. The

error D/p may be inﬂuenced by the phase-resolution of a

phase-shifting interferometry and the polarization-mixing error[16–18]. The gray levels of the minima and the max-ima of the interferograms are 0 and 255, respectively, as the phase-shifting interferometry is fully utilized. The tested light is totally reflected twice at the boundary between the prism and the tested material, the theoretical resolution of this method is about 180/(256) (ffi0.703). In our exper-iments, the extinction ratio of the polarizer (Newport Inc.) Dn2¼ 1 n1 on2 o/_p D/p¼ 1 n1 csc /p 2 2 sec /p 2 sin2 hiþ cot /p 2 csc /p 2 sec2_h i 1 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ 4C p h i 2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2þ 8C ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cot2 /p 2 sec2_h i 1 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ 4C p r D/p; ð8aÞ

Fig. 3. The two-dimensional phase variation distribution /p(x, y) of the

tested material.

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is 1· 103. So the polarization-mixing error is about 0.028. Hence, the total error of D/pis 0.74. Substituting

the experimental conditions hi= 63.62, n1= 1.68894, and

D/p= 0.74 into Eq.(8)we can obtain the relation curves

of Dn2 versus n2 as shown in Fig. 5. It is obvious from

Fig. 5 that Dn2 is related to n2. The measurement error

Dn2can be decreased to 3.6· 105when n2is near 1.513.

Although the s-polarized light has also a phase varia-tion, it is smaller than that of the p-polarized light. Hence, we choose to measure the phase variation of the p-polar-ized light in our method to enhance the measurement reso-lution. The measurable range of this method is limited by the refractive index of the prism. To expand the measurable

range, it is better to use a prism with a high refractive index.

5. Conclusion

An alternative method for measuring the two-dimen-sional refractive index distribution of a material is proposed. First, the phase variation distribution of the p-polarized of the reﬂected light under the total internal reﬂection in a prism, whose base is contacted with the tested material, can be measured accurately with the phase-shifting interferometry. Then it is substituted into special equations derived from Fresnel equations, and the two dimensional refractive index distribution of the tested material can be estimated. Its validity has been demon-strated. It has some merits such as simple optical setup, easy operation and rapid measurement.

Acknowledgement

This study was supported in part by the National Science Council, Taiwan, ROC, under Contract NSC 94-2215-E-009-002.

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