## Abstract

The analytic spectral transmittance of lattice-form and birefringent interleaver filters are revealed to be equivalent mathematically. The corresponding relationship between structural parameters of the two kinds of interleaver filters is also presented. With this mathematical equivalence relationship, we can easily obtain the optimum circuit parameters for designing a lattice-form interleaver filter by using the structural parameters of birefringent interleaver filter obtained by a simple numerical method developed by us recently instead of the complex algorithm based on scattering matrix factorization. More choice of structural parameters with any required spectral transmittance (channel spacing, flatness, isolation and ripple) can be obtained using this method when compared with results presented in references.

©2003 Optical Society of America

## 1. Introduction

Interleaver filters, which can divide dense WDM signals into two groups of even and odd channels, is very useful because it can relax the specification requirements for demultiplexers in WDM systems [1]. For practical applications, one of the most important features this filter must be possessed of is wide flat passband (stopband) and high isolation [2]. Otherwise, the combined effects of laser spectral width, drift and mode partition would cause great degradation on system performance.

There are many kinds of interleaver filter schemes based on different principles [3–6]. Of these, lattice-form interleaver filter composed of cascaded Mach-Zehnder interferometers (MZI) is considered to be promising with the development of planar lightwave circuit (PLC) technology [6–8]. It can be mass-produced cost-effectively and is both highly reliable and capable of high density integration. Through controlling the coupling ratio of directional couplers and optical path difference of delay lines, the odd and even channel signals can be divided into two groups of signals. Oguma *et al.* [6] fabricated an interleaver filter with 200GHz channel spacing based on a two-stage lattice structure. The coupling ratio of the couplers and the optical pass differences of the delay lines were derived by using a filter synthesis algorithm based on scattering matrix factorization [9]. But this algorithm is not only complex but also only one group of circuit parameter with a certain desired transfer function can be derived.

Birefringent interleaver is another kind of promising schemes. It is based on birefringence of crystal and interference of polarized light [10]. Carlsen *et al.* [5] obtained a flattening spectral transmittance by utilizing three cascaded birefringent plates. The azimuth angles of the plates were derived by using the Ammann’s method [11]. This method is also very complex and only one group of the azimuth angle of the plates was obtained when the ripples was limited to a uniform 1% in both the passband and the stopband. When the ripples were limited to a uniform 0.1%, the ratio of the passband and the stopband width to period is not greater than 3% with this method.

In our recent work, a new method, which simplifies the calculation of the azimuth angle of the plates and greatly improved the spectral transmittance of birefringent interleaver filter, has been proposed [12]. By using this method we can obtain all the numerical solutions of the azimuth angles of the plates when ripples are relatively small in both the relatively wide passband and stopband. Moreover, it is very simple.

In this paper, the analytic spectral transmittance of lattice-form and birefringent interleaver filters are compared and analyzed. The mathematical equivalence relationship between structural parameters of the two kinds of interleaver filters is also presented. With this equivalence relationship, we can easily obtain the optimum circuit parameters for designing a lattice-form interleaver filter by using the structural parameters of birefringent interleaver filter obtained by a simple numerical method developed by us recently instead of the complex algorithm based on scattering matrix factorization. Several design examples are presented and analyzed. More choice of structural parameters with any required spectral transmittance (channel spacing, flatness, isolation and ripple) can be obtained using this method when compared with results presented in references.

## 2. Theoretical analysis

The schematic configuration of a lattice-form interleaver filter is shown in Fig. 1(a). It consists of *i*+*1* directional couplers and *i* delay lines (phase shifters is deposited on the delay lines to correct any optical-path-difference errors). The working principle of the lattice-form interleaver filter is based on interference effect of multiple beams, just as that of cascaded MZI [13–15].

It is assumed that all device losses such as waveguide propagation loss and bending loss are negligible and frequency dependence in the coupling coefficients of the directional couplers is not considered here.

The transfer matrix of directional coupler with an amplitude coupling coefficient of sinφ (coupling ratio *C*=*sin ^{2}φ*) can be expressed as [14]

The transfer matrix of delay line composed of a pair of waveguides with optical path length difference of Δ*l* can be expressed as

where *β*(=*k _{0}n*) is the propagation constant of the waveguide. The time delay

*t*or frequency spacing Δ

_{i}*f*of the delay lines can be expressed by

_{i}where *c* is speed of light. Multiplying the transfer matrix of all the basic components, we can obtain the output spectral transmittance as

where *a _{0}, a_{1}, a_{2}…a_{n}* are the coefficients relative to the phase factor of the coupling ratio

*φ*. And

_{1}, φ_{2}…φ_{i}, φ_{i+1}*t′*are the individual term or combined sum, difference and sum-and-difference term of

_{1}…t′_{n}*t*. For a two-stage lattice-form interleaver filter,

_{1}, t_{2}…t_{i}*t′*(

_{n}*n*=1,2,3,4) is, respectively, expressed as

while the corresponding coefficient *a _{n}*(

*n*=0,1,…4) can be expressed as

Figure 1(b) is the schematic configuration of a birefringent interleaver filter. It is composed of several cascaded birefringent plates located between a polarizar and an analyzer. The azimuth angles of the plates are defined as the angle between the slow-axis of plates and the polarization direction of the polarizer. The working principle of it is based on interference effect of polarized light.

According to Jones-matrix method, we assume that there is no reflection of light from either surface of the birefringent plate and the light is totally transmitted through the plate surfaces, and we can write the Jones-matrix of each optical component and then obtain the spectral transmittance of the system [12]

where *T _{0}, T_{1}, T_{2}…T_{n}* are the coefficients relative to the crystals’ azimuth angles

*θ*and the analyzer’s azimuth angle

_{1}, θ_{2}, θ_{3}, …θ_{i}*θ*(the angle between the polarization direction of analyzer and polarizer).

_{P}*γ′*are the individual term or combined sum, difference and sum-and-difference term of

_{1}…γ′_{n}*γ*is time delay of the

_{1}, γ_{2}…γ_{i}. γ_{i}*i*th birefringent plate and can be expressed as

where *Δn _{i}*(=

*n*-

_{i,o}*n*) are the difference of ordinary and extraordinary refractive indices of the

_{i,e}*i*th plate.

*L*is the length of the ith plate. For a birefringent interleaver filter composed of two cascaded birefringent plates,

_{i}*γ′*(

_{n}*n*=1,2,3,4) is, respectively, expressed as

while the corresponding coefficient *T _{n}* (

*n*=0,1,…4) can be expressed as

From the analysis above, it can be understood that the lattice-form interleaver filter and the birefringent interleaver filter are almost equivalent mathematically. The equivalence relationship of the structural parameters of the two kinds of interleaver filters can be expressed as

From the equations above, we can see that the circuit parameters used for designing a lattice-form interleaver filter with a desired transfer function can be easily obtained from the available structural parameters of a birefringent interleaver filter.

However, the equivalence is only mathematical and not full from the physics point of view. It is important to mention also the difference between the two types of interleavers. The splitting of light and production of phase delay in the lattice form interleaver are accomplished by directional coupler and delay line respectively, while for the birefringent interleaver, both of the two functions are accomplished by birefringent plates. Birefringent interleaver has relatively higher insertion loss, bigger size and larger polarization dependence when compared with lattice form interleaver. The polarization dependence of birefringent interleaver can be greatly reduced by replacing the polarizers with polarization beam splitters or polarization beam displacers. We have investigated the influence of the change of the azimuth angles and thickness of the crystals on the output waveform in detail [12]. Both the fabrication tolerance dependency of the lattice form interleaver on the inaccuracy of coupling ratios of directional couplers and that of birefringent interleaver on the inaccuracy in angular alignment are very strong. With the development of semiconductor technology, the manufacturing procedure of lattice form interleaver based on PLC can be accurately controlled.

## 3. Design examples

The structural parameters of birefringent interleaver filter can be obtained by a simple numerical method developed by us recently [12]. The main idea of this method is briefly described here. In order to obtain the rectangular spectral transmittance with a certain opening ratio (1/2 for general interleaver), the azimuth angles and thickness of the crystals must satisfy certain conditions. We obtain it by comparing the expression of the spectral transmittance of the birefringent crystal chain system calculated from Jones theory, as shown in Eq.(7), with the Fourier series representation of periodical rectangular function [16], and making them close to each other through optimizing the azimuth angles and thickness of the crystals. Then the spectral transmittance becomes closer to the square response.

The thickness of crystals can be determined during comparison. Our numerical searching method is to calculate the spectral transmittance at regular angle intervals in the whole range of *θ _{i}* and

*θ*, and at the same time estimate whether the spectral transmittance satisfy the desired flattening requirement or not. Thus the desired angles of

_{P}*θ*and

_{i}*θ*are all searched out finally. Although we can’t obtain ideal periodic rectangular function for the limited crystal number in practice, a flattening passband and stopband can be obtained.

_{P}By using the method mentioned above, we obtain the azimuth angles of a birefringent interleaver filter composed of two-cascaded birefringent plates as shown in table 1. These parameters are obtained when the length of the second plate is two times that of the first one and the ripple is not greater than 0.3% (isolation <-25 dB) in both the passband and stopband with band width of greater than 2/13 period. Table 2 shows the azimuth angles of a birefringent interleaver filter composed of three cascaded birefringent plates. The ratio of the length of the three plates equals to 1:2:4 and the ripple is not greater than 0.1% (isolation <-30 dB) in both the passband and stopband with band width of greater than 1/5 period. The coupling ratios of the directional couplers corresponding to the azimuth angles obtained by the use of Eq. (11) are also shown in the two tables.

Figures 2 and 3 show the transmitted spectra of lattice-form interleaver filters with different circuit parameters. The black, red and green curves (curve A, B and C) are corresponding to two-stage interleaver filters while the blue and magenta curves (curve D and E) three-stage ones. The green and blue curves (curve C and D) show the spectral transmittance when the coupling ratios are *C _{1}*=50%,

*C*=72%,

_{2}*C*=92% with

_{3}*Δl*=1/2 (listed in table 1) and

_{1}/Δl_{2}*C*=50%,

_{1}*C*=50%,

_{2}*C*=98%,

_{3}*C*=2% with

_{4}*Δl*=1/2/4 (listed in table 2) respectively. The black curve (curve A) is obtained by using the parameters of

_{1}/Δl_{2}/Δl_{3}*C*=50%,

_{1}*C*=68%,

_{2}*C*=12% and

_{3}*Δl*=1/2 (not listed in tables). The red curve (curve B) is obtained by using the parameters given by M. Oguma

_{1}/Δl_{2}*et al*[6], i.e.

*C*=50%,

_{1}*C*=70%,

_{2}*C*=10% and

_{3}*Δl*=

_{1}/Δl_{2}*Δl*(2

_{1}/*Δl*+

_{1}*λ*/2). The magenta curve (curve E) is obtained by using the parameters given by T. Chiba

*et al*[8], i.e.

*C*=50%,

_{1}*C*=50%,

_{2}*C*=2%,

_{3}*C*=2% and

_{4}*Δl*=-1/2,

_{1}/Δl_{2}*Δl*=

_{1}/Δl_{3}*Δl*(4

_{1}/*Δl*-

_{1}*λ*/2). The above two curves are drawn in the same Figures for comparison.

When comparing the red curve with the green and black curves, we can clearly see that the green curve has higher isolation and narrower 1dB passband width, whereas the black curve has wider 1 dB passband width and lower isolation. Among the five curves, the blue curve has the widest 1 dB passband width and nearly highest isolation (flatness) simultaneously. So by using the structure parameters of birefringent interleaver filter and Eq. (11), we can easily design the lattice-form interleaver filter with any required spectral transmittance. Moreover, more choice of structural parameters can be obtained. Table 3 shows four groups of structure parameters obtained by this method satisfying the same requirement (ripple≤1.5% and flat band width greater than 9/40 period). The parameters given by M. Oguma *et al.* [6] are included in it.

Because the structural parameters of birefringent interleaver filters with any channel spacing (200GHz, 100GHz, 50GHz or smaller) and flattening requirements can be obtained by this numerical method [12], so the circuit parameters of lattice-form interleaver filters with the same desired spectral characteristics can also be obtained easily.

## 4. Conclusion

In this paper, the analytic spectral transmittance of lattice-form and birefringent interleaver filters are compared and analyzed. The mathematical equivalence relationship between structural parameters of the two kinds of interleaver filters is also presented. With this equivalent relationship, we can easily obtain the optimum circuit parameters for designing a lattice-form interleaver filter by using the structural parameters of birefringent interleaver filter obtained by a simple method we proposed recently instead of the complex algorithm based on scattering matrix factorization. Design examples are presented and analyzed. More choice of structural parameters with any required spectral transmittance (channel spacing, flat band width and ripple or isolation) can be obtained using this method when compared with results presented in references.

## Acknowledgments

The authors gratefully acknowledge the support of the Ministry of Science and Technology of China and the National Natural Science Foundation of China.

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