## Abstract

Optical interleavers based on Michelson Gires-Tournois interferometer (MGTI) with arbitrary cascaded reflectors for symmetrical or asymmetrical periodic frequency response with arbitrary duty cycles are defined as universal MGTI optical interleaver (UMGTIOI). It can significantly enhance flexibility and applicability of optical networks. A novel and simple method based on digital signal processing is proposed for the design of UMGTIOI. Different kinds of design examples are given to confirm effectiveness of the method.

© 2010 OSA

## 1. Introduction

Optical interleavers are important building blocks of optical communication systems [1–13]. They are periodic optical filters capable of combining or separating a comb of dense wavelength-division multiplexed (DWDM) signals. The original design of interleavers separates (or combines) even channels from odd channels across a DWDM comb with two symmetrical output spectra.

In recent years, as optical networks become more complicated, various new demands for signal interleaving have emerged [14–19]. For example, there have been increasing demands for optical devices which add/drop only a few channels among multiples [20]. It is difficult to use symmetrical interleavers in such applications. Interleavers with asymmetrical output spectra should be a perfect solution in this case, in which limitations due to the symmetrical output no longer exist. In fact, a symmetrical interleaver can be considered as a special case of asymmetrical interleavers. Asymmetrical interleavers with arbitrary duty cycles can significantly enhance flexibility and applicability of optical networks.

Several techniques have been proposed to implement asymmetrical interleavers, for example, those based on Michelson Gires-Tournois interferometer (MGTI) [20], birefringent Michelson Gires-Tournois interferometer (BMGTI) [21], double stage symmetrical interleaver with wavelength shifting capability [22], etc. The MGTI structure is characterized by compact size, low cost, simple parameters and high performance, therefore often used in the design of symmetrical interleavers [9,23]. It is possible to implement a universal optical interleaver based on MGTI that can be programmed to produce symmetrical or asymmetrical output periodic spectra with arbitrary duty cycles.

MGTI optical interleavers can be designed using digital signal processing or optical interferometry analysis. As we known, to achieve a high performance response function with a flat top and steep roll-off, high-order finite impulse response (FIR) filters are usually necessary. However, the requirement of too many cascade stages is a serious draw back in practical applications. Compared to FIR filters, Infinite impulse response (IIR) filter can be made compact and requires significantly fewer stages. Spectral transmittance with high rectangle degree is obtained for MGTI interleaver with two Gires-Tournois etalons (GTE) composed of two cascaded reflectors [24]. Expressions of two output spectra are worked out based on an IIR filter model with requirements of passband and stopband parameters, making the phase change due to tiny variation of mirror reflectivity and cavity length of GTE as the circulation parameters. All the design parameters can be obtained with programmed hunting. Wei et al. investigated the MGTI interleaver by two-beam interferometry analysis [25]. The position and number of wave peak and trough were derived. Under the flatness conditions, all the design parameters were achieved. It should be noted that, when the structure of interleaver is complex with many design parameters and large numbers of peaks or troughs, the two methods become very complex and the computation complexity is very high.

In this paper, we propose a novel and simple design method for the universal MGTI optical interleaver (UMGTIOI) with arbitrary cascaded reflectors to achieve symmetrical or asymmetrical periodic spectra response with arbitrary duty cycles. The transfer functions of the two output ports are simplified using an IIR model. With an elliptical filter and the pole value, all design parameters can be obtained directly. The proposed method is simpler than the existing methods, especially for interleavers with complex structure. Optimized structure parameters for high performance can be obtained simultaneously.

The paper is organized as follows. Section 2 establishes a digital filter model for an MGTI interleaver with arbitrary cascaded reflectors. Section 3 presents the principle and method of MGTI interleaver design. Section 4 gives two specific examples of the method, a symmetrical interleaver and an asymmetrical interleaver. In Section 5, the design principle of interleavers with arbitrary duty cycles is proposed, and examples are given. In Section 6, steepness and bandwidth of the output spectrum is investigated for an interleaver with different duty cycles. In Section 7, the influence of design parameters on the output intensity spectra is discussed. Conclusions are drawn in Section 8.

## 2. Digital filter model

Figure 1 shows a schematic diagram of the MGTI interleaver. It is formed by two sets of multi-cavity GTEs (MC-GTEs), a 50:50 beam splitter to output even channels, and an optical circulator to output odd channels.

GTE1 and GTE2 consist of *m* + 1 and *n* + 1 reflectors respectively. Reflectance and reflection coefficients of the *m* + 1 reflectors are *R*
_{0}
^{1}, *R*
_{1}
^{1}, *R*
_{2}
^{1}, …, *R _{m}*

^{1}, and

*r*

_{0}

^{1},

*r*

_{1}

^{1},

*r*

_{2}

^{1}, …,

*r*

_{m}^{1}, respectively. Reflectance and reflection coefficients of the

*n*+ 1 reflectors are

*R*

_{0}

^{2},

*R*

_{1}

^{2},

*R*

_{2}

^{2}, …,

*R*

_{n}^{2}, and

*r*

_{0}

^{2},

*r*

_{1}

^{2},

*r*

_{2}

^{2}, …,

*r*

_{n}^{2}, respectively. Cavity length of GTE1 and GTE2 are

*d*

_{1}

^{1},

*d*

_{2}

^{1}, …,

*d*

_{m}^{1}and

*d*

_{1}

^{2},

*d*

_{2}

^{2}, …,

*d*

_{n}^{2}, respectively. The reflectance

*R*

_{0}

^{1}and

*R*

_{0}

^{2}of the rear-reflector are assumed to be unity so that MC-GTE is a lossless all-pass filter in theory.

*L*

_{1}and

*L*

_{2}are the arm length of the MI.

Figure 2
indicates amplitudes at reflectors *i* and *i–*1. *E _{i}^{+}* and

*E*

_{i-}_{1}

*are, respectively, amplitude of the electric-field vector on the left side of reflectors*

^{+}*i*and

*i*−1 for a wave front traveling to the right. E

_{i}^{−}and E

_{i}_{-1}

^{−}are, respectively, amplitude on the left side of reflectors

*i*and

*i*−1 for a wave front traveling to the left. The distance between reflectors

*i*to

*i*−1 is

*d*

_{i}.

Therefore

*ϕ*,

_{i}= 2πd_{i}/λ*λ*is optical wavelength, Θ phase shift of GTE, and

*r*

_{i}and

*t*

_{i}amplitude reflection and transmission coefficients for reflector

*i*. Phase shift of GTE with arbitrary reflectors can by obtained from Eq. (2).

Amplitude transmission and reflection coefficients of the two output ports of the interleaver are

_{1}(Z) and A

_{2}(Z) are the all-pass filter function corresponding to GTE1 and GTE2 respectively.

*d*can be defined as reference cavity length and can be determined by $d=C/(2\cdot \Delta f)$, where

*C*is the speed of light in vacuum and Δ

*f*is the period of optical spectrum.

Thus the transfer functions of the two output ports can be written as

## 3. Principle and method of MGTI interleaver design

Because *A*
_{1}(*z*) and *A*
_{2}(*z*) are functions of z with real coefficients, the coefficients of *H*
_{1}(*z*) and *H*
_{2}(*z*) must be real. From the view point of digital signal processing, imaginary poles/zeros of *H*
_{1}(*z*) and *H*
_{2}(*z*) are in conjugate pairs. Without loss of generality, a simplest case with one pair of conjugate imaginary pole/zero and one real pole/zero are analyzed. Assume that poles of *H*
_{1}(*z*) and *H*
_{2}(*z*) are all on the imaginary axis. Figure 4
shows a pole/zero plot. The zero *z*
_{0} is on the real axis, and zeros *z*
_{1} and *z*
_{1}* are a conjugate pair. Pole *p _{0}* is located in the base point, and poles

*p*

_{1}and

*p*

_{1}* are a conjugate pair on the imaginary axis. Suppose $z={e}^{j\omega}$, so

From Eq. (9), it is clear that $\left|{H}_{1}({e}^{-j\omega})\right|=\left|{H}_{1}({e}^{j\omega})\right|$, meaning that $\left|{H}_{1}({e}^{j\omega})\right|$ is symmetrical about ω = 0 and ω = π. Assume *d*
_{1}
^{1} = *p*
_{11}⋅*d*, *d*
_{2}
^{1} = *p*
_{12}⋅*d*, …, *d _{m}*

^{1}=

*p*

_{1}

*⋅*

_{m}*d*,

*d*

_{1}

^{2}=

*p*

_{21}⋅

*d*,

*d*

_{2}

^{2}=

*p*

_{22}⋅

*d*, …,

*d*

_{n}^{2}=

*p*

_{2}

*⋅*

_{n}*d*, and

*ΔL*=

*q*⋅

*d*, where

*p*

_{11},

*p*

_{12}, …,

*p*

_{1}

*,*

_{m}*p*

_{21},

*p*

_{22}, …,

*p*

_{2}

*and*

_{n}*q*are integers. The parameters

*p*

_{11},

*p*

_{12}, …,

*p*

_{1}

*,*

_{m}*p*

_{21},

*p*

_{22}, …,

*p*

_{2}

*and*

_{n}*q*are the ratio of the cavity length of the GTE1 (

*d*

_{1}

^{1},

*d*

_{2}

^{1}, …,

*d*

_{m}^{1}), GTE2 (

*d*

_{1}

^{2},

*d*

_{2}

^{2}, …,

*d*

_{n}^{2}) and the length difference of the two arms (

*ΔL*) to the reference cavity length (

*d*), respectively. Because

*A*

_{1}(

*z*) and

*A*

_{2}(

*z*) are all-pass filter functions, polynomials in the numerator and the denominator of

*A*

_{1}(

*z*) and

*A*

_{2}(

*z*) have the same coefficients, with the sequence in a reversed order. When

*p*

_{11},

*p*

_{12}, …,

*p*

_{1}

*,*

_{m}*p*

_{21},

*p*

_{22}, …,

*p*

_{2}

*are even and*

_{n}*q*is odd, the following relation is obtained:

It can be seen from Eq. (10) that ω = π/2 and ω = -π/2 are 3dB frequencies. Therefore the 3dB bandwidth is π that just equals half of the period. From the above discussion, it can be concluded that poles of *H*
_{1}(*z*) (or *H*
_{2}(*z*)) must be conjugate pairs on the imaginary axis to form a symmetric interleaver, otherwise an asymmetrical spectrum is obtained.

As design of analog filter has been fully developed, one can design a digital filter by first designing an analog filter and then converting to a digital filter. Elliptical filters have equal ripple behavior in both the passband and stopband, and no other filter of an equal order can have a faster transition in gain between the passband and stopband [27]. Therefore the design method of elliptical filters is adopted here for the design of interleavers.

Taking into account the above facts, we propose the following method to design interleavers with different bandwidth requirements. First, calculate *H*
_{1}(*z*) and *H*
_{2}(*z*) using Eq. (7) and (8). Second, choose values of *p*
_{11}, *p*
_{12}, …, *p*
_{1}
* _{m}*,

*p*

_{21},

*p*

_{22}, …,

*p*

_{2}

*according to the requirements that the poles of*

_{n}*H*

_{1}(

*z*) (or

*H*

_{2}(

*z*)) are conjugate pairs on the imaginary axis for a symmetrical interleaver, and otherwise for an asymmetrical interleaver. Choose values of

*q*to make

*H*

_{1}(

*z*) and

*H*

_{2}(

*z*) satisfy the format of the elliptic filter’s transfer function. Thus the order of the interleaver,

*N*, is determined. Third, design an

*N-*th order elliptic filter which satisfies the design index of the interleaver. Finally, reflectance of all reflectors can directly be obtained by making the poles of

*H*

_{1}(

*z*) or

*H*

_{2}(

*z*) equal that of the

*N*-th order elliptic filter. It should be pointed out that in order to make

*r*

_{0}

^{1},

*r*

_{1}

^{1},

*r*

_{2}

^{1}, …,

*r*

_{m}^{1}and

*r*

_{0}

^{2},

*r*

_{1}

^{2},

*r*

_{2}

^{2}, …,

*r*

_{n}^{2}to be real values, the

*p*

_{11},

*p*

_{12}, …,

*p*

_{1}

*,*

_{m}*p*

_{21},

*p*

_{22}, …,

*p*

_{2}

*and*

_{n}*q*should be integer. If

*p*

_{11},

*p*

_{12}, …,

*p*

_{1}

*,*

_{m}*p*

_{21},

*p*

_{22}, …,

*p*

_{2}

*and*

_{n}*q*are not integer, the above mentioned conditions cannot be satisfied simultaneously, and the optimum design of the interleaver cannot be accomplished.

## 4. Design examples

#### 4.1. Symmetrical interleaver

As an example, GTE1 and GTE2 consisting of two reflectors are used to obtain a 50 GHz symmetrical interleaver with isolation ≤ −30dB. Assume that ${e}^{-j\raisebox{1ex}{$4\pi \cdot d$}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.}={e}^{-j\omega}={z}^{-1}$. *R*
_{0}
^{1} and *R*
_{0}
^{2} equal 1. The transfer functions of output ports can be obtained from Eqs. (2), (7) and (8) as follows:

To assure that the poles of Eqs. (11) (and (12)) are conjugate on the imaginary axis, *p*
_{11} and *p*
_{21} can equal to 2.When *q* = 1, the transfer function of the output port satisfies the format of half-band elliptical filter. Thus the order of the interleaver is five. A fifth-order elliptic filter with isolation of −30dB is designed, with a pole/zero plot shown in Fig. 5
. Reflectance *R*
_{1}
^{1} and *R*
_{1}
^{2} is then obtained by making the poles of *H*
_{1}(*z*) or *H*
_{2}(*z*) equal that of the fifth-order elliptic filter. Its output spectrum is shown as the dot line in Fig. 6
. The solid and dash lines are output spectra obtained using the same method with isolation of −38 dB and −47 dB respectively. Design parameters corresponding to the three different output spectra are shown in Table 1
. Figure 7
is an enlarged drawing in the passband of Fig. 6. From Figs. 6 and 7, it can be seen that isolation and passband width at 1 dB increase and decrease with the decrease of *R*
_{1}
^{1} and *R*
_{1}
^{2} respectively. Furthermore, a change on *R*
_{1}
^{1} and *R*
_{1}
^{2} has greater influence on isolation than on passband width. In practical applications, reflectance *R*
_{1}
^{1} and *R*
_{1}
^{2} should be determined according to practical demand.

Interleavers with other GTE structures can also be designed using the method in a similar manner as shown in Fig. 8
. Structures of the schemes in Fig. 8 are (A) holophote GTE1 and GTE2, (B) two reflectors GTE1 and holophote GTE2, (C) two reflectors GTE1 and GTE2, (D) three reflectors GTE1 and two reflectors GTE2, and (E) three reflectors GTE1 and GTE2. Figure 8 (a) shows the output spectral transmittance of each scheme. Figures 8(b) and 8(c) are enlarged drawing in the passband and stopband respectively in one period. Table 2
gives values of *q*, the elliptical filter order, and the peak number in one period of schemes A, B, C, D and E.

From Table 2, it can be seen that *q* changes alternately between 1 and −1 to satisfy the format of elliptical filter function. The passband width increases with the increase of the elliptical filter order, as shown in Figs. 8(b) and 8(c). We can see from Fig. 8(c) and Table 2 that the peak number in one period increases with the increase of the filter order. This agrees with the results presented in [17]. It can also be observed from Fig. 8 that contradiction of isolation (ripple) and passband width can be solved by increase of the reflector number in GTE. Interleavers with higher order can also be conveniently designed using this method.

#### 4.2. Asymmetrical interleaver

Take the structure of three reflectors GTE1 and two reflectors GTE2 as an example to design an interleaver with duty cycle 1:3. *R*
_{0}
^{1} and *R*
_{0}
^{2} equal 1. The transfer function of output ports can be obtained from Eqs. (2), (7) and (8) as follows:

From the design principle, *p*
_{11}, *p*
_{21} and *p*
_{12} are equal to 1 and *q* is zero. Therefore, a third order elliptical filter should be designed. Such a filter that satisfies the requirement of spectral characteristics is designed, and its pole/zero plot shown in Fig. 9
. Making the poles of *H*
_{1}(*z*) or *H*
_{2}(*z*) equal to that of the third-order elliptic filter, the reflectance of reflectors can be directly obtained. Figure 10
shows intensity of the two output ports. The parameters are *R*
_{1}
^{1} = 0.2037, *R*
_{2}
^{1} = 0.2468, and *R*
_{1}
^{2} = 0.0709.

From Fig. 10, the two output spectra have isolation more than 35 dB. The passband width of wide and narrow port at 1dB is greater than 31GHz and 14GHZ respectively. An asymmetrical interleaver with other GTE structure can also be designed with the method. Figure 11 is the two output spectra of scheme A' (three reflectors GTE1 and holophote GTE2), C' (three reflectors GTE1 and two reflectors GTE2), and D' (three reflectors GTE1 and three reflectors GTE2). The output spectrum of [12] with two reflectors GTE1 and GTE2 structure (labeled B') is also drawn in Fig. 11. In order to evaluate the spectral transmittance, passband bandwidth ratio is defined in [17]. It is the ratio of passband width at certain isolation to that at −3dB. According to the definition, passband bandwidth ratio for the four different schemes is shown in Table 3 .

From Table 3 and Fig. 11, passband width ratio increases with the increase of the reflector number. Scheme A' has the lowest passband band width ratio, whereas scheme D' has the highest. The passband bandwidth ratio of schemes B' and C' for wide port are similar, whereas that of the narrow port for schemes B' and C' is greatly different. That means the passband bandwidth ratio of narrow port can be greatly improved by replacing the two reflectors GTE with three reflectors GTE. For both wide port and narrow port, the change of passband bandwidth ratio at three different isolations is the largest for scheme A' and the smallest for scheme D'. Also, the rectangle degree of scheme A' is the lowest, and that of scheme D' is the highest.

## 5. Interleavers with arbitrary duty cycles

An optical interleaver can be viewed as a lowpass-highpass doubly-complementary filter pair. In digital signal processing, an odd order lowpass-highpass doubly-complementary filter pair can always be decomposed into sum or difference of two all-pass filter with adjacent orders [28]. For example, a first-order filter can be decomposed into sum or difference of all-pass filter with order of 1 and 0, and a third-order filter can be decomposed into sum or difference of all-pass filter with order of 2 and 1. Therefore, from Eqs. (7) and (8), interleavers with arbitrary duty cycle can be accomplished by decomposing transfer functions into sum or difference of all-pass filter with adjacent orders when *q* equals zero. Parameters *m* and *n* related to the number of reflectors in GTE1 and GTE2 are determined by the orders of the two decomposed all-pass filter. Finally, reflectance of reflectors in the two GTEs can be obtained using the method described in Section 3.

In the previous subsection, a design example of third-order interleaver with duty cycle of 1:3 has been given with a structure of three reflectors GTE1and two reflectors GTE2. From the principle, it can be designed to realize an interleaver with arbitrary duty cycles. Figure 12 shows the output spectrum of the interleaver with duty cycles 1:4, 1:5 and 1:6. The result of duty cycle 1:3 is also given for comparison. The design parameters are shown in Table 4 .

Interleavers of other orders with arbitrary duty cycle can also be designed in the same way. Figure 13 shows spectral transmittance of first-order and fifth-order interleavers with duty cycle 1:4. The design parameters are given in Table 5 . The result of a third-order interleaver is also shown for comparison.

From Fig. 13, the rectangle degree of output spectrum can be improved by increasing the filter order. Output spectrum with higher order and better rectangle degree can be designed using the method. In the mean time, however, the fabrication complexity will increase. Therefore an appropriate structure should be chosen in practical applications.

## 6. Steepness and passband width

Steepness is introduced to describe the characteristic of transition region. It is defined as the ratio between the difference of isolation and its corresponding frequency, or wavelength. The greater the steepness, the more obvious the change of transition region is. Table 6
gives parameters of output spectrum of a third-order interleaver with five different duty cycles. Let duty cycle be 1:*N*. Figures 14(a)
and 14(b) show the passband width at 1 dB and steepness as a function of *N*, respectively.

From Table 6 and Fig. 14 (a), the passband width at 1 dB for narrow port decreased with the increase of *N*, whereas the passband width at 1 dB for wide port increased with the increase of *N*. The sum of passband width at 1 dB for narrow and wide port increases with the increase of *N*, i.e., the bandwidth usage becomes more efficient. It can be seen from Fig. 14(b) that steepness of output spectrum for wide and narrow port increases with the increase of *N*. Moreover, the changed value for a wide port is greater than that of a narrow port. Interleavers of other orders can be analyzed in the same way, with results similar to the third-order interleaver.

## 7. Design robustness

As we known, the MGTI system is very sensitive to any deviations of design parameters [29]. It is a critical issue to discuss the influence of design parameters on the output intensity spectra, which will be helpful to understand the design robustness of this type of interleaver. The dependences of output intensity spectra on the design parameters for scheme a in Tab.1 are investigated as examples, as shown in Fig. 15 .

Some results can be summarized as follows:

- (1) The change of output intensity spectra mainly shown in the change of passband width and isolation. The change of isolation is much larger than that of passband width. The influences of reflectance of reflectors (
*R*_{1}^{1}and*R*_{1}^{2}) on output intensity spectra are much smaller than that of the length of GTE1 (*d*_{1}^{1}), GTE2 (*d*_{1}^{2}) and the length difference between the two arms (*ΔL*) on output intensity spectra. - (2) The passband width is kept almost unchanged and the isolation is changed only by about 2 dB when
*R*_{1}^{1}and*R*_{1}^{2}deviated by 1% from the ideal values. While the output intensity spectra will be seriously deteriorated when*d*_{1}^{1},*d*_{1}^{2}or*ΔL*deviated by the same percentage from the ideal values (not shown in figures). - (3) To obtain an almost unchanged bandwidth and a small change of isolation (in several dB scale), the diviations from the ideal values for
*d*_{1}^{1},*d*_{1}^{2}and*ΔL*should be in 10^{−8}m scale (~0.001%). Positioners with nanometer resolution should be needed for practical applications.

## 8. Conclusions

A novel method for designing UMGTIOI is proposed based on digital signal processing. Symmetrical and asymmetrical periodic spectra response with arbitrary duty cycles can be obtained for MGTI optical interleavers having an arbitrary number of cascaded reflectors. The output spectra have high isolation and wide flat passband and stopband. Compared to other methods, the proposed approach is simpler, especially for interleavers with a complicated interference structure. Design examples of symmetrical and asymmetrical interleavers with arbitrary duty cycles are given. The results are analyzed and compared. Although the interleaver studied in this paper is based on the MGTI structure, the proposed method can be extended to interference filters consisting of other all-pass filter structures.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China under the grant No. 10804070, Innovation Program of Shanghai Municipal Education Commission under the grant No. 09YZ06 and Shanghai Leading Academic Discipline Project and STCSM under the grant No. S30108 and 08DZ2231100. The authors are grateful to Professor Shuozhong Wang for his assistance in improving the language usage.

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