This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

A small-displacement sensor based on total-internal reflection theory and surface plasmon resonance technology is proposed for use in heterodyne interferometry. A small displacement can be obtained simply by measuring the variation in phase difference between s- and p-polarization states with the small-displacement sensor. The theoretical displacement resolution of the small-displacement sensor can reach 0.45 nm. The sensor has some additional advantages, e.g., a simple optical setup, high resolution, high sensitivity and rapid measurement. Its feasibility is also demonstrated.

As is well known, small displacement measurement plays an important role in high technology industries, especially in the liquid crystal display (LCD) manufacturing process and semiconductor production. Over the past few decades several articles have proposed ways of increasing the resolution of small displacement measurements [

The methods for measuring small displacement of the research papers [

As a heterodyne light source focuses on a mirror driven by a piezoelectric transducer (PZT), the reflected light passes through an objective lens and is incident on a beam-splitter. Afterwards its reflected light from the beam-splitter is refracted into the hypotenuse of the right-angle prism. At first, the light is incident on a 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 [

A ray of light in air is incident with an angle

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:

Here the signs of _{1} and _{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:

From _{1} is the phase difference variation and:

In this paper, a right-angle prism with a four-layer device [prism-titanium(Ti)-gold(Au)-air] in the Kretchmann’s configuration [_{sp}

From Maxwell’s equations, the reflection coefficients of p- and s-polarizations can be expressed as [_{2} and _{3} are the thicknesses of medium 2 and medium 3, respectively, and t = p, s,

In _{zi}_{(}_{j}_{)} is the component of the wave vector in a medium _{1} is the refractive index of the prism, _{2} is the refractive index of Ti metal, _{3} is the refractive index of Au metal, _{4} is the refractive index of air and _{0} is the wave vector in vacuum. If the amplitude reflection coefficients

Besides, the reflectivities of p- and s- polarization components are

Because the phase difference variation _{2} due to the SPR effect is a function of the rotation angle or deviation angle

From

Suppose that the defocusing amount _{1} = _{0} + _{2} = _{0} − _{0} is the initial incident angle as _{0} is chosen when the incident angle _{sp}_{0} = sin^{−1}[_{sp}

We can then obtain the incident angles of the two marginal rays that are incident at one side of the right-angle prism (the surface of the side is uncoated metal) are
_{1} for one marginal ray indent on the side of the right-angle prism at the incident angle _{1}, then we can obtain the phase difference variation _{1}^{′} = −_{1} for the other marginal ray at the incident angle _{2}. Thus we can achieve the phase difference variation on account of the TIR effect is:

Combining

Similarly, the incidence angles of the two marginal rays that are incident at the other side (coated with two metal layers) are

If the phase difference variation is _{2} for one marginal ray incident on the side of the right-angle prism at the incidence angle _{1}, then we can obtain the phase difference variation _{2}^{′} = −_{2} for the other marginal ray at the incidence angle _{2}. Thus we can achieve the phase difference variation on account of the SPR phenomenon as:

From _{t}_{2} can be expressed by:

Therefore, the total phase difference variation _{t}_{2} due to the displacement

_{2} of 2 nm, the Au film thickness _{3} of 45.5 nm, and the wavelength λ of 632.8 nm. The permittivities of Bk-7 glass prism (_{1} = _{1}^{2}), Ti film (_{2} = _{2}^{2}), Au metal (_{2} = _{3}^{2}), and air (_{4} = _{4}^{2}) are _{1} = (1.51509)^{2}, _{2} = −3.84+12.5_{3} = −12+1.26_{4} = (1.0003)^{2}, respectively.

As shown in

In addition, I must design a band-pass filter in order to filter out the high and low frequency noises. Owing to the use of the beat frequency of 2-KHz in the experimental setup, this band-pass filter can be easily designed. For the sake of convenience, the band-pass filter was designed using a resister tunable filter (model: RT-3BP1/2, manufactured by NF Corporation) and some electronic components.

The experimental and theoretical curves of the total phase difference variation _{t}

At this moment, let me discuss the sensitivity of the small-displacement sensor of small displacement measurement by use of multiple internal reflections in heterodyne interferometry. The sensitivity

In this article, a heterodyne interferometer is used to measure a small displacement. Owing to its common-path configuration, it has the advantages of high resolution, high stability and real-time measurement. It also has some inherent nonlinear errors due to imperfect optical components, misalignment, or operating defects that will reduce the system performance. Generally speaking, the inherent nonlinear errors accompanying the phase differences during the phase range of 0 ∼ 2

The first order error accompanying the phase range 0 ∼ 2

The light source is not complete linear polarization [

Missing alignment of the optical setup [^{,};

Using imperfect optical components, such as, the poor extinction ratio of a polarizer or polarization beam-splitter [

Suppose that the polarizer or polarization beam-splitter is imperfect, it can be found that a small amount of the TE state exists in the TM arm and vice versa. Thus, the electric fields of the TM and TE arms are given by, respectively [_{0} are the angular frequencies of heterodyne light and optical light source, respectively; ^{iϕA} and ^{iϕB} are the amplitudes of p- and s-polarizations, respectively; and ^{iϕα} and ^{iββ} are the polarization mixing amplitudes of s- and p-polarizations, respectively.

If the transmission axis of analyzer with respect to x-axis is equal to 45° and all phase are the same, such as _{α} = ϕ_{A}_{β}_{B}_{A}_{B}_{r}_{t}

Thus, the first order error due to polarization mixing is Δ_{m}^{−5}, i.e.

Second harmonic error is due to the polarization rotation between s- and p-polarizations. And the second harmonic error Δ_{r}_{r}_{r}_{r}_{r}

Besides, if the phase difference error only results from the resolution of the lock-in amplifier, the phase difference error Δ

Because the resolution Δ

As a matter of fact, the

In the paper, a small displacement can be measured by simply measuring the variation in phase difference between s- and p-polarization states with the small-displacement sensor. Besides, the optical structure is designed as a common-path structure and the principle is based on the TIR theory and SPR technology in heterodyne interferomery. Thus, it is stable against the turbulences of the environment such as air turbulences or mechanical vibrations. The new instrument has some distinct advantages: e.g., high resolution, high sensitivity, rapid measurement.

This research was partially supported by the National Science Council in Taiwan through Grant NSC 95-2221-E-231-027 & NSC 97-2221-E-231 -001 -MY2.

A ray of light in air incident at

Kretchmann’s configuration for the generation of SPR.

Rays back from the DP (_{0}

The incident angles of the two marginal rays that lie on the plane perpendicular to the hypotenuse of the right-angle prism are _{1} = _{0} + _{2} = _{0} −

The experimental configuration (AN: analyzer; PZT: piezoelectric transducer, LIA: lock-in amplifier; BS: beam splitters, PC: personal computer).

Two marginal rays passing through an analyzer detected by a linear photo-detector.

The experimental and theoretical curves of the total phase difference variation versus the displacement

The sensitivity

The first order error as a function of the phase difference

The second harmonic error Δ_{r}

The theoretical displacement resolution of the small-displacement sensor.