## Abstract

In this paper we investigate the variation of free spectral range (FSR) for the Fabry-Perot interferometer (FPI) consisting of mirrors with phase shift dispersion. The reflection phase shift on a mirror has been calculated employing the Transfer-Matrix Method and the values of FSR have been calculated under the condition of normal incidence of light beam. Fabry-Perot (FP) cavities have been fabricated employing bulk micromachining technology, and silicon wafers coated with multilayer dielectric films were used as mirrors. FSR of these FP cavities have been experimentally measured. The experimental data match the calculated results very well. The conclusion is that FSR shortening effect must be taken into account for the FPIs with a small plate gap, as the finesse and the tunable range of tunable FPI can be affected by the shortening effect greatly.

© 2003 Optical Society of America

## 1. Introduction

As an important multi-beam interference structure, FP cavity is widely used in applications such as spectral analysis, laser resonance cavities, and optical filters [1–4]. A FPI consists of two parallel partially reflective low-loss mirrors separated by a gap. The optical transmission spectrum of such a device shows a series of sharp peaks when the gap equals the multiples of a half wavelength of the incident light. These transmission peaks are caused by multiple reflections of the light beams between the two highly reflective mirrors.

In a conventional FPI the plate gap is large enough so that the phase shifts on the mirrors is negligible compared to the phase change between two successive transmitted light beams. However, the effect of the phase shift must be considered when accurate data of intensity, halfwidth, wavelength positioning and FSR of transmission peaks are expected for a FPI with a small plate separation. When a metal is used as reflective coating (with a finite conductance) the phase shift is small and almost invariant with wavelength. However, the phase shift is a function of wavelength when a dielectric multilayer film is used as reflective coating. So the dispersion of reflection phase shift has to be taken into account. Atherton reported that the halfwidth of the transmission peak is affected significantly by the dispersion of the reflection phase shift [5]. Troitski found that, for a two-mirror FPI, the dependence of transmission intensity on wavelength is very different from its dependence on plate separation. This is caused by the great reflection phase shift dispersion and reflection amplitude dispersion [6]. To reduce the interferometers dependence on the wavelength stability of the monochromatic light source, Troitski developed a new dispersion-free multi-beam interferometer with an unusual dielectric mutilayer design which can measure ultra small displacement (less than ~0.1λ) with a nonmonochromatic light source [7].

The FSR is the wavelength difference between two adjacent peaks of resonance, which play an important role in determining the finesse of FPI and the effective tunable range of tunable filters consisting of FPI. The FSR value with the phase shift and the dispersion of the phase shift is found to be different from the FSR value without reflection phase shift and the dispersion of phase shift. The reason is that the reflection phase and the dispersion of phase shift lengthen the return path of the beam, increasing the effective plate separation.

Analyzed in this paper is the shortening effect of FSR for FPI consisting of mirrors with reflection phase shift and the dispersion on the shift. The reflection phase shift on the dielectric mirror has been calculated employing the Transfer-Matrix Method [8]. The values of FSR have been calculated under the condition of normal incidence of the light beam. FP cavities have been fabricated employing bulk micromachining technology, and dielectric multilayer films are deposited on the silicon wafers used as mirrors [9]. Analysis shows that the reflection phase shift affects the value of FSR greatly if the order of interference is very small (not more than 10, i.e. the cavity length is not more than 5 times as large as incident wavelength). The difference caused by the reflection phase shift is a primary effect, and that caused by the dispersion of reflection phase shift is a secondary effect. However, if the order of interference is between 10 and 40 (as is considered in our experiment), the primary effect doesn’t play an important role any more. The difference of FSR is mainly caused by the dispersion of reflection phase shift. FSR of these FP cavities have been measured in our experiments and the results matched the theoretical results very well.

## 2. Theory

The transmission peak of FPI is generally determined by Airy functions [10]:

and

where *T* and *R* are the intensity transmissivity and reflectivity of each FPI mirror, respectively. And in Eq. (2), *µ* is the refractive index and *h* the gap distance between two plates (i.e. the length of the FP cavity), and *θ* the incident angle of the light beam on the mirror. The first term in Eq. (2) is caused by the gap distance, and the second term *ε(λ)* is caused by the reflection on one plate (Let us consider symmetric mirrors). Generally, the gap of a FPI is large enough (1mm~1m) so that the first term is much larger than the second term. Under the conditions of *Φ(λ _{n})*=

*nπ*and

*Φ(λ*

_{n+1}

*)*=

*(n*+

*1)π*, there are the

*n*th order and

*(n*+

*1)*th order transmission peaks. The wavelength difference,

*λ*-

_{n}*λ*

_{n+1}(i.e. the

*n*th order FSR without considering phase shift and dispersion on the mirrors), is:

With the development of MEMS technology, FPIs with the cavity length of only a few microns or tens of microns have been achieved [11–13]. In this case, the second term *ε(λ)* is comparable with the first term in Eq. (2), and the FSR of the *n*th order is:

Equation (4) is an accurate expression for FSR. However, it is not convenient to calculate FSR from it because the wavelength positions of transmission peaks, *λ _{n}* and

*λ*

_{n+1}, have to be found by self-consistent calculation. Now let us consider the normal incident condition, which is important for practical applications. As the incident angle

*θ*is zero, the phase shift caused by the reflective plate coated with multilayer dielectric films can be approximated as a linear function of the wavelength. With the approximation, we have:

If *k* is not equal to zero, Eq. (5) is a function of linear dispersion. By substituting the second term of Eq. (2) for Eq. (5), for *Φ(λ _{n})*=

*nπ*and

*Φ(λ*

_{n+1}

*)*=

*(n*+

*1)π*, we find the

*n*th order FSR considering linear dispersion on the mirrors:

## 3. Experiment

A new method has been used in this work to fabricate FP cavities with bulk micromachining technology. FP cavities have been achieved with a cavity length from several microns to tens of microns [9]. By employing wet etching process of silicon, spacers with the same height are formed on a double-polished (100) silicon wafer (bottom wafer) as shown in Fig. 1(a). An oxidation polishing process follows to improve the smoothness of the etched surface [14]. High-reflective (HR) film is deposited on the polished surface and another polished (100) silicon wafer (top wafer) and used as the mirrors of the FP cavity. In order to decrease the reflectivity, anti-reflective (AR) film is deposited on the back sides of the top and bottom wafers. The FP cavity is formed by bonding the two wafers together with adhesive, so that the two HR mirrors are face to face. The cavity length is determined by the step height of the spacers on the bottom wafer. The parallelism of the two mirrors is controlled by the uniformity of the spacers. The device formed is schematically shown in Fig. 1(b).

Spacers with different heights from 8µm to 20µm have been made on several wafers. The uniformity of heights on each wafer as well as the root-mean-square (RMS) deviation of the surface roughness is measured before and after the polish process. The standard deviations (SD) of the spacers’ heights on each wafer are less than 0.05µm. Since the distance between two adjacent spacers is larger than 10mm, the included angle of the two mirrors comprising the FP cavity is less than 10^{-5}rad so that the parallelism of the mirrors is good enough.

The RMS of roughness of the original silicon wafer is measured to be about 2nm. The RMS of roughness of the etched surface increases rapidly with the etch depth. However, it can be decreased by a half if an oxidation polish process is applied. By using the polish process, several wafers with different heights spacers are processed simultaneously with a RMS of roughness from 4nm to 11nm.

FP cavities with the finesse of about 50 have been achieved. It is worth to point out that the performances of the fabricated FP cavities are mainly determined by the quality of etched surfaces. The effort to further improve the quality of etched surface is ongoing.

Figure 2 is the sketch of the optical test system. Light source with the wavelength tunable range from 1520nm to 1620nm is used. Two collimators with a light beam diameter of 350µm are employed to couple the light beams. The light beam transmitted from the FP cavity is analyzed by an optical spectral analyzer (OSA). The FSRs of FP cavity samples are listed in Table 1.

## 4. Approach and discussion

For AR film, zirconium oxide is chosen as the coating material. There is only one layer of zirconium oxide with a thickness of 200nm in our experiment.

In practice, multilayer dielectric films are widely used as the high reflective mirrors. Let us now consider a dielectric multilayer in the form of air-(LH)^{m}-substrate, where H and L represent quarter wavelength layers of high and low refractive index materials, respectively. In the HR film coating process, silicon and silicon dioxide are chosen as high and low refractive index materials, respectively. Five layers of silicon dioxide film and silicon film are deposited on the silicon substrate alternately to form the dielectric multilayer HR film (with *m*=5). The optical thickness of one layer of film (silicon dioxide or silicon) should be a quarter of working wavelength. In our experiment, the physical thickness for silicon dioxide and silicon are 270nm and 130nm, respectively.

Using the Transfer-Matrix Method [8], we have calculated the reflective phase shift of the normal incident light on the HR film. The dependence of phase shift (*ε(λ)*) on wavelength for each HR film is shown in Fig. 3.

From the linear Eq. of phase shift, i.e., Eq. (5), we have:

Therefore, we obtain:

and

By substituting the values of *k* and *d* into Eq. (6), we can calculate the FSR for different cavity lengths. The results are shown in Fig. 4. The FSR calculated from Eq. (3) is also given in Fig. 4. The FSR are calculated for the cavity length from 6*µ*m to 30*µ*m. Since the cavity lengths used in the experiments are very small, the orders of interference are very small (between 10 and 40). The order of the FSR is shown by the scale top in Fig. 4. The difference between two curves represents the shortening effect of FSR. As is shown in the figure, shortening effect of FSR decreases with the interference order.

When the cavity length is small, both of the curves in Fig. 4 decline rapidly with the increase of the cavity length. The smaller the cavity length, the larger the slopes of the curves are. When compared with the FSR calculated by Eq. (3), the FSR calculated by Eq. (6) is much smaller. And the smaller the cavity length, the more significantly the shortening effect of FSR. When the cavity length larger than 30µm, the difference of FSR from Eq. (3) and that from Eq. (6) is diminishing.

The FSRs measured from the samples shown in Table. 1 are also shown in Fig. 4 by dot marks. The data match the FSR curve calculated from Eq. (6) very well.

In order to clarify the main factor that causes the change in FSR, we analyze the condition of the dispersionless phase shift firstly. By assuming that *ε(λ)* is dispersionless (i.e. *ε(λ _{n})*=

*ε(λ*

_{n+1}

*)*=

*ψ*), Eq. (4) becomes:

For *n* small (in some degree, less than 10), the term *ψ/π* is comparable with the order of interference, and the value of FSR is influenced significantly by the phase shift term *ψ/π*. For *n* larger than 10, the value of FSR calculated by Eq. (7) is only a little smaller than that calculated by Eq. (2), because the interference order *n* is much larger than the term *ψ/π*. As the order of interference in our experiment is between 10 and 40, as shown in Fig. 4, the FSR is shortened significantly. So we conclude that the shortening effect is caused mainly by the dispersion of reflection phase shift.

It may be mentioned here that in our experiment, the phase shift dispersion is positive, (i.e. the phase shift increases with the increase of wavelength,) therefore, a shortening effect is caused. On the contrary, it is a possibility that the value of the FSR can be lengthened if the dispersion is negative. We will discuss this issue in another paper.

## 5. Conclusion

We have investigated the shortening effect of FSR of FPI consisting of mirrors with phase shift dispersion. The values of FSR are calculated in this condition employing the Transfer-Matrix Method. FP cavities are fabricated employing bulk micromachining technology, and multilayer dielectric films are deposited on silicon wafers to form mirrors with phase shift dispersion. The FSRs of these FP cavities are measured experimentally. The results match the theoretical results very well. As, in our experiment, the order of interference is between 10 and 40, and the shortening effect of FSR is mainly caused by the positive dispersion of reflection phase shift.

## Acknowledgment

This work is supported by the Major State Basic Research Development Program of China (G1999033104), the National High Technology Research and Development Program of China (2002AA312070), and the Natural Science Foundation of China (60377030).

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