## 1.

## Introduction

One of the key components in DWDM systems is the multi-channel periodic optical filter. In order to avoid loss penalties, a filter with a flattop spectrum is highly desirable. Devices based on waveguides^{1} and optical fibers, including fiber delay lines^{2} and fiber Bragg gratings^{3} have been proposed to create flattop periodic filters. Recently, Dingle et al.;^{4}
^{5} proposed a robust approach by using a two-mirror Gire-Tournois resonator (2MGTR) in one of the interferometer arms to achieve flattop spectral response, while Hsieh et al.;^{6} used two 2MGTRs, one in each interferometer arm. In this paper, we present a periodic flattop filter, based on a multi-cavity, three-mirror Gire-Tournois resonator (3MGTR), in a Michelson interferometer (MI). We present the analytical expression for the optimum conditions for the occurrence of flattop spectral response. By using a multi-cavity etalon, the filter has many sets of optimum values, greatly relaxing the design conditions needed to achieve superior device performance in terms of flattop bandwidth and steepness. We note that while optical filters may be analyzed by using a Z-transform in a signal processing approach,^{7} our approach is based on two-beam interferometry analysis with physical considerations to obtain analytical expressions for the optimum design.

## 2.

## Filter Architecture and Optimal Design

The periodic optical filter under study is shown in Fig. 1. It is a modified MI, consisting of one 50:50 beam splitter, one 100 reflective mirror (M
_{3}), and a 3MGTR. The lengths of the arms of the interferometer are L_{1}
and L_{2}. The 3MGTR is a special two-cavity Fabry-Perot etalon with two partially reflective mirrors (M
_{1}
and
M
_{2}) and one 100 reflective mirror (M
_{0}) at rear. The cavity length of the etalon is *d*, and the lengths of the two cavities are identical. The reflective coefficients of
M
_{1}
and
M
_{2}
are r_{1}
and r_{2}, respectively. To find the complex reflection coefficient of a 3MGTR, the 3MGTR can be considered as a 2MGTR constructed with a partially reflecting mirror
M
_{3}
and an equivalent reflecting mirror 2MGTR with a relative phase shift of θ_{2M}, given in Refs. 8 and 4. Thus, the reflection coefficient r_{3M}
and the relative phase shift θ_{3M}
of a 3MGTR can be obtained by using the results of the 2MGTR:

## (1)

$${r}_{3M}=\frac{-{r}_{2}+{e}^{-i(2\delta +{\theta}_{2M})}}{1-{r}_{2}{e}^{-i(2\delta +{\theta}_{2M})}}={e}^{i{\theta}_{3M}},$$_{1}and a

_{2}are defined as reflective factors;

*n*is the refractive index of the material of the cavity; β=2π/λ is the propagation constant in free space with λ the wavelength of light in vacuum. The 3MGTR is an all-pass filter and the resultant amplitude of the electrical field is unity while the resultant phase is determined by the reflective coefficients of the mirrors. This is the property that gives the filter its specific characteristics.

In general, to ensure that the spectrum of the transmitted output will have a flattop response, the one-way phase factor δ has to be equal to twice the phase difference between the two interferometer arms, i.e., δ=2βΔL,
^{4}
^{5}
^{6} here ΔL_{1}−L_{2}
is the difference of the lengths from the beam splitter to the two arms. Then the normalized output intensity can be written as

## (6)

$$-2\text{}{\mathrm{tan}}^{-1}\left\{{a}_{2}\mathrm{tan}[\delta -{\mathrm{tan}}^{-1}({a}_{1}\mathrm{tan}\delta )]\right\}+\delta =2m\text{}\pi .$$_{2}as a parameter and r

_{1}fixed at −0.68. The optimum value of r

_{2}is −0.24 if we choose r

_{1}to be −0.68 according to Eq. (8). Figure 2(a) clearly shows that with the optimal design, the transmission curve has a flattop spectrum shown as a solid line. When the reflectivity of M

_{2}is lower or higher than the optimum value, the spectrum has triple peaks shown as a dotted line, or a single peak with a rounded top shown as a dashed line.

Equation (8) tells us that there are numerous pairs of r_{1}
and r_{2}, which would give the flattop spectral responses for 3MGTR based filters. Of specific interest are the following two special cases when the 3MGTR becomes a 2MGTR. The first situation is when r_{1}=−1, i.e., a_{1}=0. The 3MGTR becomes a 2MGTR. By using Eq. (8), we can find that the general optimum condition for a 2MGTR-based periodic filter is a_{2}=1/2, which gives rise to r_{2}
=−1/3. Another extreme situation is when r_{1}=−1/3, i.e., a_{1}=1/2, which results in r_{2}=0. This again gives a 2MGTR-based periodic filter. Both results are consistent with the one shown in Ref. 5. The 3MGTR allows much more suitable choices for optimal designs compared with only one optimal value for the 2MGTR. More importantly, the 3MGTR-based design may enhance the optical performance of the filters in terms of the bandwidth of the flattop region and steepness, as can be seen in Fig. 2(b). Figure 2(b) gives the transmission spectra for three different configurations: standard MI, 2MGTR-based MI, and 3MGTR based MI. In Fig. 2(b), the reflective coefficients of the mirrors are chosen to be 1 for the standard MI, −1/3 for the partially reflective mirror of 2MGTR, and −0.63 and −0.214 for the two partially reflective mirrors of 3MGTR. Figure 2(b) clearly displays that the 3MGTR-based MI has an improved performance compared with the 2MGTR-based MI or the standard MI in terms of bandwidth and slope.

## 3.

## Conclusions

To summarize, we have theoretically studied periodic optical filters based on a multi-cavity GTR-based MI. The general condition for the occurrence of flattop transmission passband for a 3MGTR-based periodic filter is derived. The result shows that a 3MGTR-based filter is superior to a 2MGTR since it could provide a much wider bandwidth and very steep slope when reflective coefficients are suitably chosen.