Design of 100 / 300 GHz optical interleaver with IIR architectures

This paper presents an infinite impulse response (IIR) structure for designing a three-port optical interleaver. With the proposed IIR structure, the spectral responses of the interleaver can be designed in two steps to achieve simultaneously three channels of identical but 2π/3 shifted passband transmissions. Compared with a three-port finite impulse response (FIR) interleaver, the proposed IIR interleaver is superior in several aspects: it uses only two 3× 3 couplers, it provides a better form factor and has an improved side-lobe suppression. Simulation results are given to demonstrate the effectiveness of the proposed IIR interleaver design scheme. © 2005 Optical Society of America OCIS codes: (060.1810) Couplers, switches, and multiplexers; (060.2310) Fiber optics; (060.4230) Multiplexing; (230.1150) All-optical devices. References and links 1. C. K. Madsen and J. H. Zhao, Optical filter design and analysis: a signal processing approach (John Wiley & Sons, New York, 1999). 2. G. Keiser, Optical fibre communications (3rd Edition, McGraw-Hill, 2000). 3. A. V. Oppenheim and R. W. Schafter, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, 1975). 4. A. Jazairy, Wavelength-tunable fiber optic MEMs scanning Fabry-Perot interferometer (Mich. Ann Arbor, UMI, 2000). 5. C. K. Madsen, “Efficient architecture for exactly realizing optical filters with optium bandpass designs,” IEEE Photon. Technol. Lett. 10, 1136-1138 (1998). 6. B. B. Dingel and M. Izutsu, “Multifunction optical filter with a Michelson-Gires-Tournois interferometer for wavelength-division-multiplexed network system applications,” Opti. Lett. 23, 1099-1101 (1998). 7. S. Cao, C. Lin, C. Yang, E. Ning, J. Zhao, and G.Barbarossa, “ Birefringent Gires-Tournois interferometer (BGTI) for DWDM interleaving,” in OFC, (Anaheim, CA, 2002), pp. 395-396. 8. C. H. Hsieh, R. Wang, Z. 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, 242-244 (2003). 9. J. C. Chon, B. Jian, and J. R. Bautista, “High capacity and high speed DWDM and NWDM optical devices for telecom and datacom applications,” in SPIE Photonics West 2001, (San Jose CA, 2001), 4289-06. 10. K. Jinguji and M. Oguma, “Optical half-band filters,” IEEE J. Lightwave Technol. 18, 252-259 (2000). 11. Q. J. Wang, Y. Zhang and Y. C. Soh, “Efficient structure for optical interleavers using superimposed chirped fiber Bragg gratings,” IEEE Photon. Technol. Lett. 17, 387-389 (2005). 12. Q. J. Wang, Y. Zhang, and Y. C. Soh, “All-fiber 3× 3 interleaver design with flat-top passband,” IEEE Photon. Technol. Lett. 16, 168-170 (2004). 13. S. K. Sheem, “Optical fiber interferometers with [3×3] directional couplers: analysis,” J. Appl. Phys. 52, 38653872 (1981). 14. Y. H. Chew, T. T. Tjhung, and F. V. C. Mendis, “Performance of singleand double-ring resonators using 3× 3 optical fiber coupler,” IEEE J. Lightwave Technol. 11, 1998-2008 (1993). (C) 2005 OSA 4 April 2005 / Vol. 13, No. 7 / OPTICS EXPRESS 2643 #6540 $15.00 US Received 3 February 2005; revised 22 March 2005; accepted 23 March 2005 15. C. K. Madsen and G. Lenz, “Optical all-pass filters for phase response design with applications for dispersion compensation,” IEEE Photon. Technol. Lett. 10, 994-996 (1998). 16. S. K. Mitra and J. F. Kaiser, Handbook for Digital Signal Processing (Wiley, New York, 1993). 17. L. P. Ghislain, R. Sommer, R. J. Ryall, R. M. Fortenberry, D. Derickson, P. C. Egerton, M. R. Kozlowski, D. J. Poirier, S. DeMange, L. F. Stokes, and M. A. Scobey, “Miniature Solid Etalon Interleaver,” in NFOEC’2001, (Baltimore, MD, 2001), pp. 1397-1403. 18. Q. J. Wang, Y. Zhang and Y. C. Soh, “Design of spectrum equalization filter for SLED light source,” Opti. Commun. 229, 223-231 (2003).


Introduction
Optical bandpass filters are important components in wavelength division multiplexing (WDM) systems because they provide various optical signal processing functions such as multiplexing/demultiplexing optical data, balancing the signal power, and adding/dropping channels [1] [2].The practical applications of optical bandpass filters require that the filters shall have flat passband transmission, wide bandwidth, and high channel isolation.From the signal processing perspective, the bandpass filters with infinite impulse response (IIR) can better achieve these requirements in an efficient way than the filters with a finite impulse response (FIR) [3].In the optical domain, however, the desired IIR transmission cannot be achieved by simply cascading several stages of IIR optical components such as the resonators and Fabry-Perot interferometers [4].The filter structure is an important consideration.Recently, this issue is addressed from the signal processing perspective in [5], where an efficient IIR structure has been proposed to realize Butterworth, Chebyshev, and elliptic bandpass filters.
As a class of important bandpass filters, optical interleavers which fundamentally function as optical filter banks have recently attracted significant amount of attention.Unlike the singleinput-single-output bandpass filters, an optical interleaver can offer several channels of phaseshifted bandpass transmissions simultaneously.Thus, in the design of an IIR filter to realize the interleaving transmissions, more constraints are imposed on the filter structure.So far, several approaches have been proposed to design IIR interleavers [6]- [11].However, all these results are obtained for two-port optical interleavers.In order to increase the channel counts with less stages and to achieve more flexible data multiplexing, it is of a great interest to explore an optical interleaver that is capable of providing more channels of passband transmissions.
More recently, a three-port interleaver using a lattice configuration of 3 × 3 directional couplers was reported in [12].The obtained interleaver is basically an FIR filter having only forward interference paths.In this paper, an efficient IIR architecture for the three-port interleaver is proposed to achieve an enhanced interleaver performance.The filter is formed by an 3 × 3 Mach-Zehnder interferometer (MZI) with a ring resonator on each arm.The length of each ring resonator in the differential delay line is chosen to be three times that of the differential delay of the 3 × 3 MZI.It is shown that by using the proposed structure, interleavers with desired specifications can be designed in two separate steps.In the first step, the coupling ratios of fiber couplers are designed to obtain three channels of transmissions which are identical but 2π/3 shifted in frequency.Then, in the second step, one channel of transmission spectrum is designed by choosing the parameters of ring resonators to satisfy the desired spectrum specifications on passband and isolation.
Compared with the FIR 100/300 GHz interleaver, the proposed IIR interleaver has increased the 0.5dB passband bandwidth from 40 GHz to 85 GHz and the channel isolation from 19 dB to 31 dB.It is also shown in the paper that although the major side-lobe in the transition bands cannot be totally eliminated, the area under the side-lobes can be reduced by proper selection of the design parameters such as the ring-to-path coupling ratios of the ring resonators.Thus, the crosstalk between channels is reduced when the side-lobes are narrowed down to a sufficiently small bandwidth as compared to that of the optical signals going through the interleaver.
The rest of the paper is organized into three sections.First, the proposed three-port IIR interleaver is illustrated in the next section.Then in Section 3, the performance of the proposed interleaver is studied via simulations.Finally, the paper is concluded in Section 4.

Three-port interleaver with IIR architectures
The proposed three-port IIR type interleaver is formed by embedding three ring resonators in a 3 × 3 MZI, as shown in Fig. 1(a).The MZI consists of two 3 × 3 directional couplers linked port-to-port by three differential delay paths.The three delays are chosen as ∆L, 0, and −∆L, respectively.The length difference ∆L determines the channel spacing of the interleaver, e.g., a length difference of 2 mm corresponds to a 0.8 nm channel spacing.In all differential arms of the MZI, all-pass ring resonators with a path length of 3∆L are embedded respectively.Each resonator couples into the differential delay path with a coupling ratio a i (i = 1, 2, 3).In each of the resonators, a phase shift φ 1,i (i = 1, 2, 3) is introduced to shape the spectral response.In this configuration, the three channels of transmissions can be obtained from one of the input ports to the three output ports.Without loss of generality, the transmissions from the input port E I 1 to the three output ports E o 1 , E o 2 , and E o 3 are considered.To obtain three channels of identical but 2π/3 spectral shifted transmissions satisfying desired specifications, the interleaver is designed by choosing appropriate parameters including the coupling ratios κ i (i = 1, 2) of three 3 × 3 fibre couplers, the phase shifts φ 1,i (i = 1, 2, 3) in ring resonators, and the ring-to-path coupling ratios a i (i = 1, 2, 3).To this end, the three channels of transmissions are expressed as functions of the design parameters.
It has been shown in [13][14] that for a 3 × 3 optical fiber coupler, the input optical fields and the output fields of a symmetric 3 × 3 coupler can be related by a 3 × 3 transfer matrix M(κ) as follows: where E o m and E I m represent respectively the output field and the input field at the m-th port for m = 1, 2, 3. |γ| 2 and |δ | 2 denote the through-port and the cross-port power coupling ratios of a 3 × 3 directional coupler.They are given by γ = (e j2κ + 2e − jκ )/3 and δ = (e j2κ − e − jκ )/3, where j stands for √ −1 and κ is the product of the coupling coefficient c and the length of Next, we derive the transfer function of the differential lines with ring resonators.Denote z −1 as a unit time delay given by z −1 = e − j2∆Lπn/λ , where λ is the wavelength of wave propagating through free space and n is the refractive index.For the convenience of derivation, we represent the transmissions of three ring resonators by S 1 (z), S 2 (z) and S 3 (z) respectively, as shown in Fig. 1(b).The lossless ring resonator with a single coupler and a phase shifter is an all-pass filter.Its transfer function can be written as [15]: where t i = √ 1 − a i and a i is the ring-to-path coupling ratio of the single coupler on the i-th (i = 1, 2, 3) ring resonator.
Considering the differential delay caused by delay lines themselves, the transmission of the differential delay lines with ring resonators is obtained by Therefore, the normalized electric field transfer functions from the three inputs to the three outputs can be expressed by the following transfer matrix: Denote H 1 (z), H 2 (z) and H 3 (z) as the transmissions from the input port E I 1 to the three output ports E o 1 , E o 2 and E o 3 , respectively.From Eq. ( 4), the transmissions H 1 (z), H 2 (z) and H 3 (z) can be expressed as the following equations: Then, the 3 × 3 optical interleaver design problem can be described as one of determining appropriate design parameters such that the power spectra of |H 1 | 2 , |H 2 | 2 and |H 3 | 2 have a shift of 2π/3 in frequency, but they have identical spectral shape, where ) is the power transfer function in the proposed configuration.In addition, it is highly desired that the passbands and stopbands of |H 1 | 2 , |H 2 | 2 and |H 3 | 2 are as well isolated as possible to achieve good channel isolation, and more importantly, they should be as flat as possible in order to reduce power variations resulting from channel wavelength shifts.
To meet the first requirement of identical passband/stopband spectral shape for all the three channels [12], the desired design parameters shall satisfy the following equations: Clearly, there is a great deal of difficulty in solving these equations due to the complicated expressions of S i (z 3 ) in ( 2) and H i (z) in ( 5)- (7).However, with the proposed structures of have identical shapes with only a shift of 2π/3 in frequency between any pair of them.Therefore, the problem of the three-port optical IIR interleaver design is converted to one of designing S i (z 3 ) such that one of the channel transfer functions, e.g., |H 1 | 2 , satisfies the desired spectrum response.The available filter design methods including those for the design of Butterworth, Chebyshev and Elliptic bandpass filters can then be employed to determine the remaining design parameters such that the interleaving transmissions have desired spectral responses such as high channel isolation, flat passband, and wide bandwidth.

Numerical performance of IIR three-port interleaver
As a design example, the proposed IIR structure of the optical interleaver is designed to deliver three channels of transmissions which are interleaved by 2π/3 with each other, and each passband bandwidth at -0.5 dB is chosen as 85% of the channel spacing.Following the twostep design scheme developed in the last section, the coupling ratios of the two 3 × 3 couplers are readily obtained as κ 1 = 2π/9 and κ 2 = 4π/9, respectively, so that the three channels of transmissions have identical spectral shapes but phase shifted by 2π/3.Then, one of the transmissions, H 1 (z), is designed to satisfy the passband requirement.It can be obtained from ( 5) that H 1 (z) is a 12th order IIR filter.The desired transfer function of H 1 (z) can be obtained by using the least-squares (LS) method [16] and the resultant numerator and denominator power expansion coefficients are listed in Table 1.After obtaining the power expansion coefficients of H 1 (z), we determine the parameters of the ring resonators.The coupling ratios and phases for each ring resonator are easily calculated from the all-pass functions.For example, to produce a pole at z n requires a coupling ratio of a n = 1 − |z 3  n | 2 and a phase of arg(z 3 n ).Thus, by using Eqs.( 2) and ( 5), the design parameters of t i and φ 1,i (i = 1, 2, 3) are obtained and shown in the last column in Table 1.

Bandwidth and band-ripple performance
With these design parameters, a three-port IIR interleaver is obtained with the specified performance.Figure 2(a) shows the interleaving transmission spectra of the designed three-port IIR interleaver.The group delay response in one typical transmission channel is also shown in Fig. 2(b) in which the top-portion of the passband is amplified.It is seen from Fig. 2(a) and Fig. 2(b) that for the proposed IIR interleaver, the passband bandwidth is about 85 GHz, the channel isolation is around 31 dB within the passband, and the group delay is about 60 ps in the passband.To compare the performance quantitatively with the FIR three-port interleaver proposed in [12], the figure of merit (FOM) of the transmission is calculated.FOM is a common index used to evaluate the performance of interleavers [17].It is defined as the ratio between the bandwidth at -25 dB and the bandwidth at -0.5 dB.A simple numerical calculation shows that the FOM of the IIR interleaver is 86.8%.It is increased by more than 50% when compared to the 56.2% FOM for the FIR interleaver.For a clearer comparison, Fig. 3 displays one channel of transmission of the proposed IIR three-port interleaver and the corresponding FIR interleaver.It can be clearly seen that the passband transmission bandwidth and the channel isolation of the IIR interleaver are improved greatly.Furthermore, only two 3 × 3 couplers are required in the IIR optical interleaver, while three are used in the FIR interleaver.Therefore, it can be concluded that the proposed three-port IIR interleaver can achieve superior performance when compared to the FIR three-port interleavers.

Crosstalk performance
It can be seen from Fig. 2(a) that the channel isolation of the three-port interleaver is more than 30 dB within the adjacent passband.However, in the transition band of one channel of passband transmission, a high side-lobe is observed.The area included by this side-lobe defines the leakage power of a signal passing through this channel of passband.The leakage power would result in the crosstalk among channels.As shown in Appendix B, the isolation from the Fig. 3. Comparison between one channel power spectrum of the proposed IIR three-port interleaver and that of the FIR interleaver proposed in [12].
side-lobe peak to the maximum transmission of the passband is at most 8/9.This result shows that no matter how the design parameters are selected, the side-lobe peak in the transition band cannot be totally eliminated.This limitation results from the structure of the interleaver.However, only the power of the crosstalk is of concern in practice, hence the suppression of the area under the side-lobe equivalently reduces the crosstalk.It is shown that a thinner side-lobe in the transition band can be achieved by choosing appropriate design parameters t i (i = 1, 2, 3), as shown in Fig. 2(a).In addition, by adding more ring resonators on the differential arms of the MZI, the area of the side-lobe of the interleaver in the transition band can be further minimized.

Performance in the presence of parameter variations
In practice, the design parameters are subject to deviations during fabrication and the fiber coupling may suffer from certain loss [18].It is therefore necessary to conduct simulation studies to investigate the sensitivity of the interleaver performance with respect to the design parameters and the insertion loss of the couplers.In the simulation, 5% tolerant deviations in the design parameters κ i (i = 1, 2), t i and φ 1,i (i = 1, 2, 3) are considered, respectively.The coupling loss is considered by substituting ηz −3 for z −3 , where loss in decibels for one feedback path is −20log10(η).Figure 4 shows the spectral responses of the interleaver with the design parameters and the coupling loss varying within the tolerant range.The results clearly show that the performance of the proposed IIR interleaver is tolerant to deviations in both the design parameters and the coupling loss.

Conclusion
This paper studied the design of IIR optical filters to deliver three channels of interleaving passband transmissions while satisfying desired specifications.An efficient structure using all-pass ring resonators in an 3 × 3 MZI interferometer is proposed.In the proposed structure, the path length of the ring resonator is chosen to be three times that of the unit time-delay line linking the two 3 × 3 couplers.With the proposed structure, the design of the desired IIR interleaver can be simplified into two steps such that the phase-shifts and the spectral responses are designed separately.It is shown that if the coupling ratios of the 3 × 3 couplers of the proposed interleaver structure are selected appropriately, the property of 2π/3 shift in frequency among the three output channels can be satisfied automatically.Then, the desired interleaving transmissions can be achieved by designing just one of the channel of transmissions to satisfy the desired specifications.It is also shown in the paper that the main side-lobe in the transition band can be efficiently squeezed by proper selection of the design parameters.Simulation results have been conducted to verify the analysis.

(C) 2005 OSA 4
April 2005 / Vol. 13, No. 7 / OPTICS EXPRESS 2645 the coupling region L, i.e., κ = c × L. As the three optical fibers are laid out symmetrically, the coupling coefficients between any pair of optical fibers are equal.

Fig. 2 .
Fig. 2. (a) Power spectra of three outputs of the designed three-port IIR interleaver; (b) Power spectrum and group delay of the proposed IIR interleaver in one typical channel.

2 (B. 6 )
t i sin φ 1,i 1 + t i sin φ 1,i = −β + 2kπ + πFrom Eqns.(B.5) and (B.6), we can obtain that φ 1,i shall take the values of φ 1,i = pπ (i = 1, 2, 3 and p is an integer).That means no matter what value t i (i = 1, 2, 3) takes, as long as φ 1,i = pπ, the corresponding point is always the stationary point with respect to the design parameter.It can be further examined by calculating the associated Hessian matrix that the stationary point of |H 1 (e jw )| 2 w=π obtained by choosing φ 1,i = pπ is the minimal value in the transition band.

Table 1 .
Order of z −k , Numerator Amplitude Expansion Coefficients, Denominator Amplitude Expansion Coefficients, and Circuit Parameters for 12th Order IIR Three-Port Interleaver.