# At first, the light is incident on a right-angle side of the pris

At first, the light is incident on a sellekchem right-angle side of the prism that not metal coated. Next, the reflected light is incident on the other side that is coated with two metal layers. Finally, the light is detected by a linear photo-detector when it passes through the hypotenuse of the right-angle prism and an analyzer. As the mirror departs from the focal plane, the beam converges or diverges into the prism. The two marginal rays of the beam exiting the prism will induce different phase difference variations between the s- and p-polarizations. Some special equations are derived according to the optical configuration and Fresnel��s equations . By substituting the phase difference data into these equations, small displacements can be measured. The method has some practical advantages: e.g., a simple optical setup, high resolution, high sensitivity and rapid measurement. In addition its feasibility is demonstrated.2.?Principle2.1. The phase difference at the TIR effectA ray of light in air is incident with an angle �� on one side of a right-angle prism with refractive index n, as shown in Figure 1.Figure 1.A ray of light in air incident at �� on one side surface of a right-angle prism with refractive index n.The light ray is refracted into the prism and it propagates toward the hypotenuse surface of the prism. At that surface, there is a boundary between the prism and air. If the angle of incidence at the boundary is ��1, then we have:��1=45��+sin?1(sin ��n)(1)Here the signs of ��1 and �� are defined as positive if they are measured clockwise from a surface normal. If ��1 is larger than the critical angle ��C, the light is totally reflected at the boundary. According to Fresnel��s equations, the phase difference between s- and p-polarizations is given as:?1=2 tan?1sin2[45��+sin?1(sin ��n)]?1n2tan[45��+sin?1(sin ��n)] sin[45��+sin?1(sin ��n)].(2)From Equation (2), the variation ���� of the incident angle can be written as:����?(n2 tan2��1?1)(n2 sin2��1?1)122n sin ��1[2?(n2?1) tan2 ��1]��(n2?sin2 ��)12cos �Ȧ�?1=A(��)��?1(3)where ����1 is the phase difference variation and:A(��)=(n2 tan2 ��1?1)(n2 sin2 ��1?1)122n sin ��1[2?(n2?1)tan2 ��1]��(n2?sin2 ��)12cos ��..(4)2.2. The basic principle of SPR technologyIn this paper, a right-angle prism with a four-layer device [prism-titanium(Ti)-gold(Au)-air] in the Kretchmann��s configuration  is used. For the Kretchmann configuration of the four-layer system as shown in Figure 2, the surface plasmons are excited when �� equals to the resonant angle ��sp.Figure 2.Kretchmann��s configuration for the generation of SPR.From Maxwell��s equations, the reflection coefficients of p- and s-polarizations can be expressed as :r1234t=r12t+r234tei2kz2d21+r12tr234tei2kz2d2,(5)r234t=r23t+r34tei2kz3d31+r23tr34tei2kz3d3(6)where rijt=Eit?EjtEit+Ejt, d2 and d3 are the thicknesses of medium 2 and medium 3, respectively, and t = p, s,EIt={nI2/kzIt=pkzIt=s, I=i, j;i, j=1,2,3,4.

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