Michelson interferometer based interleaver design using classic IIR filter decomposition

Chi-Hao Cheng; Shasha Tang

doi:10.1364/OE.21.031330

1. Introduction

The wavelength interleaver has become a standard optical communications component [1–4] and the Michelson interferometer based interleaver is a popular implementation approach among optical engineers due to its compact size [3,4]. People have long recognized that the Michelson interferometer based interleaver can be modeled as a digital infinite impulse response (IIR) filter [5,6]. Some work proposes to design a Michelson interferometer based interleaver using elliptic filter; however, no general form is provided [7].

In this paper, we demonstrate how to convert a classic IIR filter design including Butterworth, Chebyshev, and elliptic filter to a Michelson interferometer based interleaver. The proposed design method allows engineers to generate an IIR filter design based on given specifications and convert the resulting IIR filter design to physical parameters of the Michelson interferometer based interleaver. The proposed design algorithm can use a Butterworth, Chebyshev, or elliptic filter with odd-number filter order as a starting point and it allows optical engineers to apply mature digital filter design techniques to interleaver design seamlessly thus improving productivity.

The rest of this article is organized as followings. The second section describes the decomposition of a classic IIR filter into two parallel all-pass filters (APF) that can be implemented with multi-mirror Fabry-Perot interferometers. The third section describes how to convert an APF design to a multi-mirror Fabry-Perot interferometer used to implement the Michelson interferometer based interleaver. Simulation results are presented in the fourth section to demonstrate the validity of proposed algorithm and the fifth section concludes this paper.

2. Decomposition of an IIR filter into two all-pass filters

The Michelson interferometer based interleaver is illustrated in Fig. 1(a). A 50:50 coupler splits the incoming light into two ways, which are totally reflected back by two Fabry-Perot interferometers whose last mirror’s reflectivity is 100%. The two reflected signals interfere with each other at the coupler and generate interleaving output spectra. Since there is no power loss in an ideal interleaver, its two input-output functions can be considered as doubly complementary transfer functions in theory [8]. Because a Fabry-Perot interferometer whose last mirror’s reflectivity is 100% is an allpass filter (APF), the interleaver structure shown in Fig. 1(a) can be represented as the doubly-complementary transfer function pair consisting of parallel APFs (A₀(z) and A₁(z)) as illustrated in in Fig. 1(b). It is well known that highpass-lowpass doubly-complementary transfer functions of classic Butterworth, Chebyshev, and elliptic filter can be implemented with the structure shown in Fig. 1(b) provided that the filter-order is an odd-number [8]. Therefore, for a given interleaver specification, we can use popular software such as Matlab^® to generate a classic IIR filter design to satisfy the specification and decompose the resulting IIR filter into two parallel APFs as shown in Fig. 1(b). The Matlab^® function used to decompose the classic IIR filter into two APF functions is ‘tr2ca’ [9]. The resulting APFs can then be implemented with Fabry-Perot interferometers and this conversion will be described in the next section.

Fig. 1 (a) The Michelson interferometer based interleaver (L is the distance between the coupler and Fabry-Perot interferometers) (b) doubly-complementary transfer function pair consists of parallel APFs.

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3. Converting an allpass filter to a multi-mirror Fabry-Perot interferometer

A (N + 1)-mirror Fabry-Perot interferometer is illustrated in Fig. 2. In our design, every cavity length, l_i, is assumed to be the same and determined by the interleaver’s channel spacing. The input-output relation of a specific mirror i in z-domain is given below [10].

[\begin{matrix} E_{i}^{+} \\ E_{i}^{-} \end{matrix}] = \frac{z^{0.5}}{t_{i}} [\begin{matrix} z^{- 1} & - r_{i} \\ - r_{i} \cdot z^{- 1} & 1 \end{matrix}] [\begin{matrix} E_{i + 1}^{+} \\ E_{i + 1}^{-} \end{matrix}]

Where z⁻¹ is the unit delay representing phase shift, e^−j4πli/λ (λ: signal wavelength), signal experiences when propagating over a distance of 2l_i,

E_{i}^{+}

is the input signal at mirror i,

E_{i}^{-}

is the reflected signal at mirror i,

E_{i + 1}^{+}

is the input signal at mirror i + 1,

E_{i + 1}^{-}

is the reflected signal at mirror i + 1, r_i is the reflectivity of mirror i, and t_i can be determined as:

Fig. 2 A (N + 1)-mirror Fabry-Perot interferometer.

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t_{i} = \sqrt{(1 - r_{i}^{2})} .

The input-output relation of a (N + 1)-mirror (e.g. N-cavity) Fabry-Perot interferometer can then be expressed as:

\begin{array}{l} [\begin{matrix} E_{1}^{+} \\ E_{1}^{-} \end{matrix}] = \frac{z^{0.5 N}}{t_{1} t_{2} \dots t_{N}} [\begin{matrix} z^{- 1} & - r_{1} \\ - r_{1} \cdot z^{- 1} & 1 \end{matrix}] \dots [\begin{matrix} z^{- 1} & - r_{N} \\ - r_{N} \cdot z^{- 1} & 1 \end{matrix}] [\begin{matrix} E_{N + 1}^{+} \\ E_{N + 1}^{-} \end{matrix}] \\ = \frac{z^{0.5 N}}{t_{1} t_{2} \dots t_{N}} [\begin{matrix} A_{N} & B_{N} \\ C_{N} & D_{N} \end{matrix}] [\begin{matrix} E_{N + 1}^{+} \\ E_{N + 1}^{-} \end{matrix}] \end{array}

The reflection function of a (N + 1)-mirror Fabry-Perot interferometer whose last mirror reflectivity is 100% (r_{N + 1} = 1) can be derived as

H_{N} (z) = \frac{E_{1}^{-}}{E_{1}^{+}} = \frac{{N u m}_{N} (z)}{{D e n}_{N} (z)} = \frac{C_{N} - D_{N} \cdot r_{N + 1}}{A_{N} - B_{N} \cdot r_{N + 1}} = \frac{C_{N} - D_{N}}{A_{N} - B_{N}} = - (\frac{r_{1 \cdot z^{- N} + \dots + 1}}{z^{- N} + \dots + r_{1}})

Notice that, the negative sign in Eq. (4) can be ignored when we determine mirror reflectivity since it only causes a 180-degree constant phase shift. As the last mirror’s reflectivity is 100%, no energy is lost. Equation (4) is an APF transfer function and Num_N(z) = Z^−N⋅(Den_N(z⁻¹)) [8]. The (N + 1)-mirror Fabry-Perot interferometer matrix in Eq. (3) can be decomposed as:

[\begin{matrix} A_{N} & B_{N} \\ C_{N} & D_{N} \end{matrix}] = [\begin{matrix} z^{- 1} & - r_{1} \\ - r_{1} {\cdot z}^{- 1} & 1 \end{matrix}] [\begin{matrix} A_{N - 1} & B_{N - 1} \\ C_{N - 1} & D_{N - 1} \end{matrix}] = [\begin{matrix} {z^{- 1} \cdot A}_{N - 1} - r_{1} \cdot C_{N - 1} & {z^{- 1} \cdot B}_{N - 1} - r_{1} {\cdot D}_{N - 1} \\ {- r_{1} \cdot z^{- 1} \cdot A_{N - 1} + C}_{N - 1} & {- r_{1} \cdot z^{- 1} \cdot B_{N - 1} + D}_{N - 1} \end{matrix}]

The H_N(z) can be re-written as

H_{N} (z) = \frac{{N u m}_{N} (z)}{{D e n}_{N} (z)} = \frac{C_{N} - D_{N}}{A_{N} - B_{N}} = \frac{- r_{1} {\cdot Z}^{- 1} (A_{N - 1} - B_{N - 1}) + (C_{N - 1} - D_{N - 1})}{Z^{- 1} (A_{N - 1} - B_{N - 1}) - r_{1} \cdot (C_{N - 1} - D_{N - 1})}

If we separate the first mirror from the (N + 1)-mirror Fabry-Perot interferometer under consideration, the remaining N-mirror Fabry-Perot interferometer is still an APF and its transfer function can be represented as:

H_{N - 1} (z) = \frac{{N u m}_{N - 1} (z)}{{D e n}_{N - 1} (z)} = \frac{C_{N - 1} - D_{N - 1}}{A_{N - 1} - B_{N - 1}}

From Eqs. (6) and (7), we can determine the Den_N-1(z) as

{D e n}_{N - 1} (z) = \frac{{N u m}_{N} (z) \cdot r_{1} + {D e n}_{N} (z)}{(- r_{1}^{2} + 1) z^{- 1}}

The numerator polynomial, Num_N−1(z), can be determined from Den_N−1(z) since H_N−1(z) is an APF transfer function.

Based on a given specification, a classic IIR filter can be designed and decomposed into parallel APFs. Assume the transfer function of one resulting real-valued Nth-order APF is represented as following

H' (z) = \frac{{N u m}_{N}^{'} (z)}{{D e n}_{N}^{'} (z)} = \frac{1 + d_{1} z^{- 1} + \dots + d_{N - 1} z^{- (N - 1)} {+ d}_{N} Z^{- N}}{d_{N} + d_{N - 1} z^{- 1} + \dots + d_{1} z^{- (N - 1)} + Z^{- N}}

We can then determine the corresponding Fabry-Perot interferometer implementation’s mirror reflectivity iteratively by following procedures below.

1. Comparing Eqs. (4) and (9), we set r₁ = d_N.
2. Reduce the order of the APF by one using Eq. (8)
3. Repeat Steps 1 and 2, until every mirror’s reflectivity is determined.

It is worthy of notice that, when a classic IIR filter is decomposed into an APF pair, the resulting APF transfer functions might generate negative reflectivity when we determine its corresponding multi-mirror Fabry-Perot interferometer. This issue can be solved by changing the APF’s polynomial coefficients to their absolute values before conversion. Such a change places zeros and poles of APFs in the left half of Z-plane and does not change resulting IIR filter magnitude spectrum.

4. Design examples and simulation results

To demonstrate effectiveness of the proposed interleaver algorithm, we applied it to generate two interleaver designs: symmetrical and asymmetrical interleavers. The channel spacing of interleaver is 50GHz.

4.1 Symmetrical interleaver

Four parameters are necessary to design a classic digital IIR filter: (1) passband edge (range: 0~π) (2) stopband edge (range: 0~π) (3) maximum passband attenuation, and (4) minimum stopband attenuation. Software such as Matlab^® can generate classic IIR filter designs from these parameters. Butterworth filter has no ripple in either passband or stopband, Chebyshev type-1 filter has ripple in passband but not in stopband, Chebyshev type-2 filter has ripple in stopband but not in passband, and elliptic filter has ripple in both stopband and passband. We use Butterworth, Chebyshev type-1, and elliptic filters to design interleavers. For an interleaver one of whose output ports is a Chebyshev type-1 filter, its other output port will be a Chebyshev type-2 filter because two output ports of interleaver are power complimentary.

The two output ports of a symmetrical interleaver have the same passband bandwidth. To use the digital IIR filter to design a symmetrical interleaver, two conditions need to be satisfied (i) (passband edge + stopband edge) = π and (ii) (maximum passband attenuation + minimum stopband attenuation) = 1 (in linear scale). We would like to design a 50-GHz symmetrical interleaver with 40GHz passband bandwidth and 30dB stopband attenuation. These requirements correspond to (1) passband edge: 0.4π (2) stopband edge: 0.6π (3) maximum passband attenuation: 0.0043dB and (4) minimum stopband attenuation: 30dB. We applied the proposed design method to design the symmetrical interleaver and the resulting interleaver spectra based on different classic IIR filters are illustrated in Fig. 3. As shown in Fig. 3, the elliptic filter has ripples in both passband and stopband, the Chebyshev filter has stopband ripple in one output port, and the Butterworth filter has no ripple in either passband or stopband. These spectra are consistent with filter characteristics. The filter order of Butterworth, Chebyshev, and elliptic filter are 11 (interferometer cavity: 6 and 5), 7 (interferometer cavity: 4 and 3), and 5 (interferometer cavity: 3 and 2) respectively. Each design’s Fabry-Perot interferometer reflectivity is listed in Table 1.

Fig. 3 Output spectra of symmetric interleaver based on (a) elliptic filter (b) Chebyshev filter (c) Butterworth filter.

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Table 1. Fabry-Perot Interferometer Reflectivity of 3 Symmetric Interleaver Designs

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4.2 Asymmetrical interleaver

The asymmetrical interleaver whose two output ports have different passband bandwidths attracts significant attentions in recent years because it is more suitable for a hybrid optical system with different transmission rates among channels [11–14]. We would like to design a 50GHz channel spacing asymmetrical interleaver with 20GHz passband in one output port, 70GHz passband in another output port, and 30dB stopband attenuation. These requirements correspond to (1) passband edge: 0.2π (2) stopband edge: 0.3π (3) maximum passband attenuation: 0.0043dB and (4) minimum stopband attenuation: 30dB.

With the same design method for symmetrical interleaver we generated asymmetrical interleaver designs and the resulting interleaver spectra based on different classic IIR filters are illustrated in Fig. 4. The filter orders of Butterworth, Chebyshev, and elliptic filters are 17 (interferometer cavity: 8 and 9), 9 (interferometer cavity: 4 and 5), and 5 (interferometer cavity: 3 and 2) respectively. Each design’s Fabry-Perot interferometer reflectivity is listed in Table 2.

Fig. 4 Output spectra of asymmetric interleaver based on (a) elliptic filter (b) Chebyshev filter (c) Butterworth filter.

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Table 2. Fabry-Perot Interferometer Reflectivity of 3 Asymmetric Interleaver Designs

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As shown by two examples presented in this section, a Michelson interferometer based interleaver design can be achieved by (1) generating a classic IIR filter design from a given specification, (2) decomposing the resulting IIR filter into parallel APFs, and (3) converting APFs to multi-mirror Fabry-Perot interferometers. The first two tasks can be accomplished by commercial software tools such as Matlab^® and the last task can be accomplished by the procedure detailed in Section III. The proposed method can greatly expedite design of a Michelson interferometer based interleaver. Moreover, the proposed method also provides some design alternatives for engineers to choose from based on other requirements such as chromatic dispersion and spectrum ripples.

5. Conclusion

In this paper, we present an interleaver design method based on classic IIR filters. An IIR filter design can be generated from desired specifications with commercial software such as Matlab^®. The resulting IIR filter can be decomposed into two APFs, which can be implemented with multi-mirror Fabry-Perot interferometers with which an interleaver can be built. The proposed method allows engineers to use mature digital signal processing techniques to design Michelson interferometer based interleaver seamlessly. As shown by our simulation results, the proposed method can be applied to design symmetric and asymmetrical interleavers.

References and links

1. S. Cao, J. Chen, J. N. Damask, C. R. Doerr, L. Guiziou, G. Harvey, Y. Hibino, H. Li, S. Suzuki, K.-Y. Wu, and P. Xie, “Interleaver Technology: Comparisons and Applications Requirement,” J. Lightwave Technol. 22(1), 281–289 (2004). [CrossRef]

2. K.-Y. Wu and J.-Y. Liu, “Switchable Wavelength Router” US Patent No. 5,694,233.

3. B. B. Dingel and T. Aruga, “Properties of a Novel Noncascaded Type, Easy-to-Design, Ripple-Free Optical Bandpass Filter,” J. Lightwave Technol. 17(8), 1461–1469 (1999). [CrossRef]

4. C.-H. Hsieh, R. Wang, Z. J. Wen, I. McMichael, P. Yeh, C.-W. Lee, and W.-H. Cheng, “Flat-Top Interleavers Using Two Gires-Tournois Etalons as Phase-Dispersive Mirrors in a Michelson Interferometer,” IEEE Photon. Technol. Lett. 15(2), 242–244 (2003). [CrossRef]

5. C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: a Signal Processing Approach (John Wiley & Sons, 1999.)

6. C. K. Madsen, “General IIR Optical Filter Design for WDM Applications Using All-Pass Filters,” J. Lightwave Technol. 18(6), 860–868 (2000). [CrossRef]

7. J. Zhang and X. Yang, “Universal Michelson Gires-Tournois Interferometer Optical Interleaver Based On Digital Signal Processing,” Opt. Express 18(5), 5075–5088 (2010). [CrossRef] [PubMed]

8. S. K. Mitra, Digital Signal Processing: a Computer-Based Approach, 4th ed. (McGraw-Hill, 2011.)

9. MathWorks Document Center, “tf2ca,” http://www.mathworks.com/help/dsp/ref/tf2ca.html

10. H. van de Stadt and J. M. Muller, “Multimirror Fabry-Perot Interferometers,” J. Opt. Soc. Am. A 2(8), 1363–1370 (1985). [CrossRef]

11. C.-H. Cheng, “Asymmetrical Interleaver Structure Based on the Modified Michelson Interferometer,” Opt. Eng. 44(11), 115003 (2005). [CrossRef]

12. H.-W. Lu, K.-J. Wu, Y. Wei, B.-G. Zhang, and G.-W. Luo, “Study of all-fiber asymmetric interleaver based on two-stage cascaded Mach–Zehnder Interferometer,” Opt. Commun. 285(6), 1118–1122 (2012). [CrossRef]

13. J. X. Li and K. X. Chen, “An Interleaver with Arbitrary Passband Width Ratio Based on Hybrid Structure of Microring and Mach-Zehnder Interferometer,” J. Lightwave Technol. 31(10), 1538–1543 (2013). [CrossRef]

14. P. J. Pinzon, C. Vazquez, I. Perez, and J. M. S. Pena, “Synthesis of Asymmetric Flat-Top Birefringent Interleaver Based on Digital Filter Design and Genetic Algorithm,” IEEE Photon. J. 5(3), 7100113 (2013).

	Reflectivity of 1st Interferometer	Reflectivity of 2nd Interferometer
Butterworth	0.0032, 0.0010, 0.1733, 0.0083, 0.8349, 0.0073, 1.	0.0001, 0.0356, 0.0039, 0.4821, 0.0098, 1.
Chebyshev Type 1	0.1797, 0.4241, 0.8001, 0.2993, 1.	0.1712, 0.4994, 0.4904, 1.
elliptic	0.2985, 0.1141, 1	0.0443, 0.7756, 0.1269, 1.

	Reflectivity of 1st Interferometer	Reflectivity of 2nd Interferometer
Butterworth	0.0126, 0.1067, 0.3585, 0.6399, 0.7970, 0.8642, 0.9019, 0.8822, 1.	0.0126, 0.1067, 0.3585, 0.6399, 0.7969, 0.8656, 0.8915, 0.9420, 0.7999, 1.
Chebyshev Type 1	0.4228, 0.8246, 0.9166, 0.9248, 1.	0.4228, 0.8246, 0.9155, 0.9422, 0.8620, 1.
elliptic	0.4554, 0.7256, 1.	0.3601, 0.8916, 0.7624, 1.

	Reflectivity of 1st Interferometer	Reflectivity of 2nd Interferometer
Butterworth	0.0032, 0.0010, 0.1733, 0.0083, 0.8349, 0.0073, 1.	0.0001, 0.0356, 0.0039, 0.4821, 0.0098, 1.
Chebyshev Type 1	0.1797, 0.4241, 0.8001, 0.2993, 1.	0.1712, 0.4994, 0.4904, 1.
elliptic	0.2985, 0.1141, 1	0.0443, 0.7756, 0.1269, 1.

	Reflectivity of 1st Interferometer	Reflectivity of 2nd Interferometer
Butterworth	0.0126, 0.1067, 0.3585, 0.6399, 0.7970, 0.8642, 0.9019, 0.8822, 1.	0.0126, 0.1067, 0.3585, 0.6399, 0.7969, 0.8656, 0.8915, 0.9420, 0.7999, 1.
Chebyshev Type 1	0.4228, 0.8246, 0.9166, 0.9248, 1.	0.4228, 0.8246, 0.9155, 0.9422, 0.8620, 1.
elliptic	0.4554, 0.7256, 1.	0.3601, 0.8916, 0.7624, 1.

Michelson interferometer based interleaver design using classic IIR filter decomposition

Abstract

1. Introduction

2. Decomposition of an IIR filter into two all-pass filters

3. Converting an allpass filter to a multi-mirror Fabry-Perot interferometer

4. Design examples and simulation results

4.1 Symmetrical interleaver

4.2 Asymmetrical interleaver

5. Conclusion

References and links

Cited By

Figures (4)

Tables (2)

Equations (9)

Optics Express