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. Ó 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 h t on the one side surface of a right-angle prism with refractive index 0030-4018/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.07.009 * 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