1. 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 single-input-single-output bandpass filters, an optical interleaver can offer several channels of phase-shifted 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.

2. 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 ΔL, 0, and -ΔL, respectively. The length difference ΔL 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 3ΔL 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.

 

Fig. 1. (a) Proposed configuration of IIR three-port interleaver; (b) Schematic configuration of the proposed interleaver.

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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 $E_1^I$ to the three output ports $E_1^o$, $E_2^o$, and $E_3^o$ 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 $E_m^o$ and $E_m^I$ 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^{j 2κ}+2e ^-jκ)/3 and δ=(e^{j 2κ}-e ^-jκ)/3, where j stands for √-1 and κ is the product of the coupling coefficient c and the length of 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.

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 ^-j2ΔLπn/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 $t_i=\sqrt{1-a_i}$ 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 $E_1^I$ to the three output ports $E_1^o$, $E_2^o$ and $E_3^o$, 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 |H_i |²=H_i$H_i^{*}$ (i=1,2,3) 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 S_i (z ³), we can solve (8) in an efficient way. It is shown in Appendix A that by selecting the design parameters of κ₁ and κ₂ as 2π/9 and 4π/9, respectively, we have

An immediate conclusion from this equation is that |H ₁|², |H ₂|² and |H ₃|² 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.

3. 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 two-step design scheme developed in the last section, the coupling ratios of the two 3×3 couplers are readily obtained as κ₁=2π/9 and κ₂=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-|$z_n^3$ |² and a phase of arg($z_n^3$ ). 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.

3.1. 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.

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

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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.

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3.2. 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 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.

 

Fig. 3. Comparison between one channel power spectrum of the proposed IIR three-port interleaver and that of the FIR interleaver proposed in [12].

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3.3. 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 substitutingηz ^-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.

4. 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.

 

Fig. 4. Interleaver responses with the design parameters bearing ±5% fabrication deviations.

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Appendix A: Derivation of equation

It is seen from Fig. 1(a) that the path length of all-pass ring resonators are chosen as three times that of the unit differential length in the proposed interleaver design structure. Such a structure implies that the frequency response of the ring resonators S_i (z ³) (i=1,2,3) will not change with a shift of 2π/3 or 4π/3 in frequency. This can be seen directly from

(A.1)$$S_i\left(z^3\right)=S_i\left({\left(z{e}^{-j\frac{2\pi }{3}}\right)}^3\right)=S_i\left({\left(z{e}^{-j\frac{4\pi }{3}}\right)}^3\right)\mathrm{for}i=1,2,3$$

Using this property, the first requirement of an interleaver can be easily fulfilled by choosing appropriate values for the design parameters κ_i (i=1, 2). To see this, we substitute Eqs. (5), (6) and (7) into Eq. (8). Then, the spectra in (8) can be expressed into the following forms:

(A.2)$${\mid H_1\left(z\right)\mid }^2=\sum _{i=1}^2\sum _{j=1}^2{a}_{\mathit{ij}}^{\left(1\right)}{z}^{2j-2i}{S}_i\left(z^3\right)S_j^{*}\left(z^3\right)+\sum _{j=1}^2{b}_j^{\left(1\right)}{z}^{3-2j}{S}_j\left(z^3\right)+\sum _{j=1}^2{c}_j^{\left(1\right)}{z}^{2j-3}{S}_j^{*}\left(z^3\right)+d^{\left(1\right)}$$

(A.3)$${\mid H_2\left(z{e}^{-j\frac{2\pi }{3}}\right)\mid }^2=\sum _{i=1}^2\sum _{j=1}^2{a}_{\mathit{ij}}^{\left(2\right)}{z}^{2j-2i}{S}_i\left(z^3\right)S_j^{*}\left(z^3\right)+\sum _{j=1}^2{b}_j^{\left(2\right)}{z}^{3-2j}{S}_j\left(z^3\right)+\sum _{j=1}^2{c}_j^{\left(2\right)}{z}^{2j-3}{S}_j^{*}\left(z^3\right)+d^{\left(2\right)}$$

(A.4)$${\mid H_3\left(z{e}^{-j\frac{4\pi }{3}}\right)\mid }^2=\sum _{i=1}^2\sum _{j=1}^2{a}_{\mathit{ij}}^{\left(3\right)}{z}^{2j-2i}{S}_i\left(z^3\right)S_j^{*}\left(z^3\right)+\sum _{j=1}^2{b}_j^{\left(3\right)}{z}^{3-2j}{S}_j\left(z^3\right)+\sum _{j=1}^2{c}_j^{\left(3\right)}{z}^{2j-3}{S}_j^{*}\left(z^3\right)+d^{\left(3\right)}$$

where the coefficients of $a_{\mathit{\text{ij}}}^{\left(1\right)}$ , $a_{\mathit{\text{ij}}}^{\left(2\right)}$ , $a_{\mathit{\text{ij}}}^{\left(3\right)}$ , $b_j^{\left(1\right)}$ , $b_j^{\left(2\right)}$ , $b_j^{\left(3\right)}$ , $c_j^{\left(1\right)}$ , $c_j^{\left(2\right)}$ , $c_j^{\left(3\right)}$ , d ⁽¹⁾, d ⁽²⁾, and d ⁽³⁾ are shown in Table 2.

Forcing $a_{\mathit{\text{ij}}}^{\left(1\right)}$ =$a_{\mathit{\text{ij}}}^{\left(2\right)}$ =$a_{\mathit{\text{ij}}}^{\left(3\right)}$ , $b_j^{\left(1\right)}$ =$b_j^{\left(2\right)}$ =$b_j^{\left(3\right)}$ , $c_j^{\left(1\right)}$ =$c_j^{\left(2\right)}$ =$c_j^{\left(3\right)}$ , and d ⁽¹⁾=d ⁽²⁾=d ⁽³⁾, it can be solved that when κ₁=2π/9 and κ₂=4π/9, equation (8) is satisfied regardless of S_i (z). With these solutions for κ₁ and κ₂, we have Eqn. (9), i.e.

(A.5)$$H_1{\left(z\right)\mid }_{{\kappa }_1=\frac{2\pi }{9},{\kappa }_2=\frac{4\pi }{9}}=e^{-j\frac{2\pi }{3}}{H}_2\left({\mathit{ze}}^{-j\frac{2\pi }{3}}\right){\mid }_{{\kappa }_1=\frac{2\pi }{9},{\kappa }_2=\frac{4\pi }{9}}=H_3\left({\mathit{ze}}^{-j\frac{4\pi }{3}}\right){\mid }_{{\kappa }_1=\frac{2\pi }{9},{\kappa }_2=\frac{4\pi }{9}}$$

Table 2. Expressions of the Coefficients of $a_{\mathit{\text{ij}}}^{\left(1\right)}$ , $a_{\mathit{\text{ij}}}^{\left(2\right)}$ , $a_{\mathit{\text{ij}}}^{\left(3\right)}$ , $b_j^{\left(1\right)}$ , $b_j^{\left(2\right)}$ , $b_j^{\left(3\right)}$ , $c_j^{\left(1\right)}$ , $c_j^{\left(2\right)}$ , $c_j^{\left(3\right)}$ , d⁽¹⁾, d⁽²⁾, and d⁽³⁾.

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Appendix B: Derivation of the maximum difference between the side-lobe peak and the transmission of the passband

Without loss of generality, we examine one of the transmissions, e.g. |H ₁|². To maximize the difference between the maximum transmission of the passband and the side-lobe peak in the transition band, we consider the following design target:

(B.1)$$max\left({\mid H_1\left(e^{\mathit{jw}}\right)\mid }_{w=\pi }^2-{\mid H_1\left(e^{\mathit{jw}}\right)\mid }_{w=\pi }^2\right)$$

We will first examine the stationary points of the formula |H ₁(e^jw )${|}_{w=\pi }^2$. Later study shows that the value of |H ₁(e^jw )${|}_{w=0}^2$ will reach its maximum simultaneously when |H ₁(e^jw )${|}_{w=\pi }^2$ achieves its minimal value by selecting optimal design parameters, such as ϕ_1,i and t_i (i=1,2, 3). Thus, the maximum value of Eqn.(B.1) can be obtained accordingly.

All the stationary points of |H ₁(e^jw )${|}_{w=\pi }^2$ with respect to the design parameters can be obtained by solving the following equations:

(B.2)$$\{\begin{array}{c}\frac{{\partial \mid H_1\left(e^{\mathit{jw}}\right)\mid }_{w=\pi }^2}{\partial {\varphi }_{1,1}}=\frac{{\partial \mid H_1\left(e^{\mathit{jw}}\right)\mid }_{w=\pi }^2}{\partial {\varphi }_{1,2}}=\frac{{\partial \mid H_1\left(e^{\mathit{jw}}\right)\mid }_{w=\pi }^2}{\partial {\varphi }_{1,3}}=0\\ \frac{{\partial \mid H_1\left(e^{\mathit{jw}}\right)\mid }_{w=\pi }^2}{\partial t_1}=\frac{{\partial \mid H_1\left(e^{\mathit{jw}}\right)\mid }_{w=\pi }^2}{\partial t_2}=\frac{{\partial \mid H_1\left(e^{\mathit{jw}}\right)\mid }_{w=\pi }^2}{\partial t_3}=0\end{array}$$

By applying Eqns.(2) and (5) into Eqn.(B.2), we have

(B.3)$$A_1=A_1^{*}{A}_2=A_2^{*}{A}_3=A_3^{*}$$

(B.4)$$B_1={-B}_1^{*}{B}_2={-B}_2^{*}{B}_3={-B}_3^{*}$$

where $A_i=\frac{e^{j{\varphi }_{1,i}}}{{\left(1+t_i{e}^{j{\varphi }_{1,i}}\right)}^2}\left(\frac{t_1+e^{-j{\varphi }_{\mathrm{1,1}}}}{1+t_1{e}^{-j{\varphi }_{\mathrm{1,1}}}}-\frac{t_2+e^{-j{\varphi }_{\mathrm{1,2}}}}{1+t_2{e}^{-j{\varphi }_{\mathrm{1,2}}}}+\frac{t_3+e^{-j{\varphi }_{\mathrm{1,3}}}}{1+t_3{e}^{-j{\varphi }_{\mathrm{1,3}}}}\right)$, $B_i=\frac{{1-e}^{2j{\varphi }_{1,i}}}{{\left(1+t_i{e}^{j{\varphi }_{1,i}}\right)}^2}\left(\frac{t_1+e^{-j{\varphi }_{\mathrm{1,1}}}}{1+t_1{e}^{-j{\varphi }_{\mathrm{1,1}}}}-\frac{t_2+e^{-j{\varphi }_{\mathrm{1,2}}}}{1+t_2{e}^{-j{\varphi }_{\mathrm{1,2}}}}+\frac{t_3+e^{-j{\varphi }_{\mathrm{1,3}}}}{1+t_3{e}^{-j{\varphi }_{\mathrm{1,3}}}}\right)$, $A_i^{*}$ and $B_i^{*}$ are the conjugates of A _i and B_i (i=1,2, 3), respectively. From (B.3), it is clear that A_i (i=1,2,3) are real numbers, while from (B.4), B_i (i=1,2,3) are imaginary numbers.

We denote $\left(\frac{t_1+e^{-j{\varphi }_{\mathrm{1,1}}}}{1+t_1{e}^{-j{\varphi }_{\mathrm{1,1}}}}-\frac{t_2+e^{-j{\varphi }_{\mathrm{1,2}}}}{1+t_2{e}^{-j{\varphi }_{\mathrm{1,2}}}}+\frac{t_3+e^{-j{\varphi }_{\mathrm{1,3}}}}{1+t_3{e}^{-j{\varphi }_{\mathrm{1,3}}}}\right)=\alpha e^{j\beta }$, then the angle of $\left(\frac{e^{j{\varphi }_{1,i}}}{{\left(1+t_i{e}^{j{\varphi }_{1,i}}\right)}^2}\right)(i=1,2,3)$ should be equal to -β+2kπ to meet the requirement that A_i (i=1,2,3) are real numbers, where k is an integer. Similarly, the angle of $\left(\frac{{1-e}^{2j{\varphi }_{1,i}}}{{\left(1+t_i{e}^{j{\varphi }_{1,i}}\right)}^2}\right)(i=1,2,3)$ should be equal to -β+2kπ+π/2 to meet the requirement that B_i (i=1,2,3) are imaginary numbers. Thus, from Eqns.(B.3) and (B.4), we have

(B.5)$$\arg \left(\frac{e^{j{\varphi }_{1,i}}}{{\left(1+t_i{e}^{j{\varphi }_{1,i}}\right)}^2}\right)={\varphi }_{1,i}-2\arctan \frac{t_i\sin {\varphi }_{1,i}}{1+t_i\sin {\varphi }_{1,i}}=-\beta +2k\pi$$

(B.6)$$\arg \left(\frac{1-e^{2j{\varphi }_{1,i}}}{{\left(1+t_i{e}^{j{\varphi }_{1,i}}\right)}^2}\right)=-{\varphi }_{1,i}+\frac{\pi }{2}-2\arctan \frac{t_i\sin {\varphi }_{1,i}}{1+t_i\sin {\varphi }_{1,i}}=-\beta +2k\pi +\frac{\pi }{2}$$

From 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 )${|}_{w=\pi }^2$ obtained by choosing ϕ_1,i=pπ is the minimal value in the transition band.

By applying the optimal design parameters ϕ_1,i=pπ and Eqn.(2) into Eqn.(9), it is verified that |H ₁|², at w=0 and ϕ_1,i=pπ, takes the value of 1, which is the maximal transmission that the passive interleaver can achieve. Therefore, the design target (B.1) is achieved at ϕ_1,i=pπ where the maximal difference between the transmission in the passband and the side-lobe peak in the transition band is 8/9.

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