1. Introduction

In recent years, there has been a growing interest in the application of optical delay-line circuits in optical signal processing. These circuits are composed of directional couplers, phase shifters and delay-lines [1–3]. Optical filters are a good test vehicle for photonic technology since filtering is a fundamental aspect of any form of photonic signal processing. Narrow, box-like, bandpass optical filters are of particular importance for channelizing in RF photonic applications as well as for monitoring optical signals in dense wavelength-division-multiplexed (DWDM) systems [4]. Unlike digital filters, optical filters are roughly classified as FIR and IIR type and they demonstrate similar filter characteristics like digital filters [5, 6].

There exist a number of methods for lattice-form IIR digital filters. Synthesis algorithms were proposed for 1×M IIR digital filters [7]. These methods are based on division of total transfer matrix into unit blocks. It is well known that, optical systems and digital systems are differentiated on two major points. Optical paths necessarily cause phase change whereas the signal paths in digital systems can connect two points without phase change. Directional couplers used in optical circuits are expressed by complex transfer matrices whereas the Givens rotation used in digital systems are expressed by real transfer matrices [8]. Due to these differences, the circuit configurations and synthesis theories developed for digital filters are not applicable to optical filters.

A general design algorithm is presented to implement an IIR optical filter using all-pass ring resonators in a Mach-Zehnder configuration [9]. However, this circuit is a single layer two-port presentation and unable to offer 1×M IIR type optical filter. Recently, 1×3 IIR type optical delay-line circuit that can offer three-port arbitrary IIR filter characteristics is proposed [10]. Synthesis algorithm is also confirmed by design example. Circuit configuration and synthesis mechanism of a 1×3 IIR type optical filter is a foundation for M – channel IIR optical filter. In recent times, optical interleave filters attracted significant amount of attention in optical communication [11, 12]. Researchers in this field already reported the circuit structure and synthesis algorithm for two-port and three-port optical interleave filters with IIR architecture [13, 14]. But, these methods are dedicated to interleave filter characteristics and unable to arbitrary filter characteristics.

In this correspondence, a novel circuit configuration for 1×M optical delay-line circuit with ring waveguides is presented in this paper that can offer multi-port arbitrary filter characteristics. Each unit element is composed of one symmetric Mach-Zehnder interferometer and one ring waveguide. The symmetric Mach-Zehnder interferometer includes (M - 1) directional couplers and (M - 1) phase shifters. The lossless ring resonator with a single coupler and a phase shifter is an all-pass filter. Synthesis algorithm is based on the repeated size-reduction. A set of recursion equations are derived to obtain all unknown circuit parameters. It is assumed in this paper that, the synthesis algorithm for M — channel IIR filter is lossless, which implies that the sum of output powers is unity.

In this paper following notations are used. Boldfaced characters are used to denote vectors and matrices, A*, A ^T and A ^† denote the conjugate, transpose and transpose conjugate of A respectively. The notation Ã(z) represents the para-conjugate of polynomial A(z) which is defined as $\tilde{A}\left(z\right)=A^{*}\left(\frac{1}{z^{*}}\right)$. Ã(z) denotes the para-conjugate of polynomial matrix A(z) and is defined by $\tilde{\mathbf{A}}\left(z\right)=\sum _{k=1}^N{\mathbf{a}}_k^{†}{z}^k$ when $\mathbf{A}\left(z\right)=\sum _{k=1}^N{\mathbf{a}}_k{z}^{-k}$, where a _k is a coefficient matrix.

An outline of this paper is as follows. Section 2 describes the circuit configuration with transfer function. Section 3 demonstrates synthesis algorithm. Design examples are presented in Section 4. Concluding remarks are written in Section 5.

2. Circuit formulation

The circuit configuration of a 1×M (M ≥ 2) lattice-form optical delay-line circuit with ring waveguides is presented in this section. This circuit includes (M + 1) optical waveguides, (MN + M - 1) directional couplers, (MN + M - 1) phase shifters and an external phase shifterφ_ex. Delay-line with delay time difference Δτ is maintained by the ring waveguides. All optical waveguides are considered lossless with negligible bending loss in this paper. A novel 1×M optical delay-line circuit with ring waveguides that can realize power-complementary multi-port outputs is shown in Fig. 1. This proposed circuit can offer 100% power transmittance.

The number of free parameters of an optical delay line circuit is already reported [6] [10]. In this paper the complex expansion coefficients are C_1,k ~ C_M,k (k = 0 ~ N) and D_k (k = 1 ~ N). Therefore, the degree of freedom for N stages is 2{M(N + 1)+N}. However, this circuit includes (MN + M - 1) directional couplers and (MN + M - 1) phase shifters. In addition, restriction conditions take away (2N + 1) degree of freedom. Hence, Eq. (1) confirms that an external phase shifter φ_ex is requisite in this circuit configuration as below:

The presented multi-port optical delay-line circuit has a number of cascaded unit elements. Each unit element is composed of one symmetric Mach-Zehnder interferometer and one ring waveguide. Transfer function of the rth directional coupler and phase shifter of nth stage MZI can be written as [16],

Where K_r,n, K̂_r,n and P_r,n represents cos θ_r,n, -j sin θ_r,n and $e^{-j{\phi }_{r,n}}$ respectively. The lossless ring waveguide with a single coupler and a phase shifter is an all-pass filter. Its nth stage transfer function can be written by the following rational function [10]:

where α_n indicates the nth pole of the transfer function with α_n = K_a,n P_a,n. θ_a,n is the coupling angle of the directional coupler connecting the nth stage ring waveguide and the symmetric MZI and φ_a,n indicates the phase shifter value on the nth ring waveguide. The transfer function of an optical delay-line with a delay time difference Δτ is written as e^-jωΔτ. In terms of Z transform e^-jωΔτ can be replaced by the term z ⁻¹. The transfer function of a nth unit element that includes one MZI and one ring waveguide can be written as follows [10, 15]:

Where, U_a,n and W_a,n represent (1-αn z^-1) and $\left(\cos {\theta }_{a,n}-e^{-j{\phi }_{a,n}}{z}^{-1}\right)$ respectively.

 

Fig. 1. Circuit configuration of a 1×M(M=5) lattice-form optical delay-line circuit with Ring waveguides proposed in this paper.

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The input of this proposed delay-line circuit is considered as[1,0, ……0,0]^T. Therefore, the output can be expressed as follows,

The vector elements R₁(z) ~ R_M(z) and Q(z) of a multi-port optical delay-line circuit with ring waveguides can be expressed by complex expansion coefficients C_1,k ~ C_M,k (k = 0 ~ N) and D_k(k = 1 ~ N) as follows [10, 15]:

3. Synthesis Algorithm

This section presents a novel synthesis approach to design an 1×M optical delay-line circuit with ring waveguides. The aim of this algorithm is to calculate the unknown expansion coefficients C_1,k ~ C_M,k (k = 0 ~ N) and D_k (k = 1 ~ N), coupling coefficient angles θ_1,k ~ θ_M-1,k (k = 0 ~ N) and θ_a,k (k = 1 ~ N) of (MN + M - 1) directional couplers, phase shift values φ_1,k ~ φ_M-1,k (k = 0 ~ N) and φ_a,k (k = 1 ~ N) of (MN + M - 1) phase shifters. The whole synthesis algorithm is composed of three stages as illustrated in Fig. 2.

Step 1: The purpose of this step is to find the unit delay time difference (Δτ) from a desired frequency period (f_p). It is calculated by,

Step 2: In step 2 optimum expansion coefficients of R₁(z) ~ R_M (z) and Q(z) are determined. Several methods for designing IIR digital filters can be considered in this regard. For example, Eigenfilter design method, least square approximation method and bilinear transformation methods are mostly used. Note that, all poles of the rational function needs to be inside the unit circle (|z|=1) in order to be filter stable. Since, the optical delay-line circuits are passive filters, maximum transmission cannot exceed 100%.

Restriction condition mentioned in Eq. (8) expressed in details by the complex expansion coefficients [5, 10].

Step 3: In this step a set of recursion equations are derived to obtain all unknown circuit parameters. The entire transfer matrix S(z) is decomposed into N + 1 unit blocks and finally all coupling angles θ_1,k ~ θ_M-1,k (k = 0 ~ N) and θ_a,k (k = 1 ~ N) of (MN + M - 1) directional couplers, phase values φ_1,k ~ φM_-1,k (k = 0 ~ N) and φ_a,k (k = 1 ~ N) of (MN + M - 1) phase shifters are obtained. Applying factorization, total vector T(z) can be decomposed into the following form,

Here it is assumed that the decomposition processes have proceeded successfully until the n + 1 stages as follows. Therefore Eq. (9) can be written as,

where T[ⁿ] (z) indicates the remaining part after the (n + 1)th stage decomposition and the output at the nth stage is defined as,

Q ^[n] (z) of Eq. (11) can be expressed as below,

From the definition of the nth zero α_n of Q ^[n] (z), θ_a,n and φ_a,n of the nth block can be obtained as below,

The unknown circuit parameters θ_1,n ~ θ_M-1,n and φ_1,n ~ φ_M-1,n of the nth block can be acquired by separating S_n(z) (transfer matrix of nth block) fromT ^[n] (z). Property of paraunitary ensures that S̃_n(z)S_n(z)=I _n where I _n is a (n×n) unit matrix. Applying the paraunitary of S_n(z), T ^[n-1] (z) can be written as,

With this decomposition, on, T ^[n-1] (z) can be expressed as follows,

Since $R_M^{[n-1]}(z)~R_1^{[n-1]}(z)$ must be (n - 1) th order polynomials, it is required that the numerator of each function is capable of division by the denominator. This required condition concerning the polynomials of $R_M^{[n-1]}(z)~R_1^{[n-1]}(z)$ can be expressed in Eq. (17).

From Eq. (17) φ_M-1,n and θ_M-1,n can be derived as follows,

Substitute the value of P_M-1,n ensure the term $\left\{\frac{j{R}_M^{\left[n\right]}\left({\alpha }_n\right)P_{M-1,n}}{R_{M-1}^{\left[n\right]}\left({\alpha }_n\right)}\right\}$ is real and so θ_M-1,n is real as well. Note that, circuit parameters obtained from Eq. (17c) and Eq. (17d) are equal [10] [15]. Consequently, all the circuit parameters φ_M-1,n ~ φ_1,n and θ_M-1,n ~ θ_1,n can be found by successively in the order n = (N ~ 0) as shown in Appendix A. The external phase shifter value can be obtained as follows

Where $C_{1,0}^0~C_{M,0}^0$ are the 0th stage expansion coefficients.

Thus, all the circuit parameters θ_M-1,n ~ θ_1,n, φ_M-1,n ~ φ_1,n (n=N ~ 0) and θ_a,n, φ_a,n(n=N ~ 1) can be obtained by performing the third synthesis steps described above.

4. Synthesis Example

In this section, two examples of IIR optical frequency filters are demonstrated which are synthesized by design data of IIR optical filters.

A. Optical Elliptic Filter

A fifth-order optical elliptic filter was synthesized using the design data obtained in [15]. Table 1(a) shows the normalized expansion coefficients. Circuit parameters calculated by the present synthesis algorithm are presented in Table 1(b). Fig. 3 shows the cross and through port power transmittance. The transmittance at the stop band was less than -13 dB. It can be confirmed that a maximum transmittance of 100% is realized as expected.

B. Optical IIR Interleave Filter

A three-port optical IIR interleave filter is demonstrated by referring the design data [14]. In this example, the number of expansion coefficients is N = 12. Table 2(a) shows the

Table 1. (a). Normalized expansion coefficients of a two-port fifth-order optical elliptic filter

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Table 1. (b). Calculated circuit parameters of a two-port fifth-order optical elliptic filter.

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middle band expansion coefficients. Calculated circuit parameters values are listed in Table (2b). It is seen that the last two stages of this design example approaches FIR type that is synthesized by the synthesis algorithm [5] [6]. However, the rest part is synthesized by the synthesis algorithm presented in this paper. Fig. 4 shows the synthesized power frequency response of a three-port IIR optical interleave filter. The 1 dB-down bandwidth of the passband is approximately 0.31, the transmittance at the stopband is less than -30 dB. Presented design example also satisfies the law of energy conservation.

 

Fig. 2. Flowchart diagram of the present synthesis algorithm.

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Fig. 3. Power frequency response of a two-port fifth order optical elliptic filter.

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Table 2. (a). Middle band numerator and denominator amplitude expansion coefficients of a three-port optical IIR interleave filter.

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Table 2. (b). Calculated circuit parameters of a three-port optical IIR interleave filter.

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External phase shifter value (φ_ex): 7.8121 e - 001×π

 

Fig. 4. Power frequency response of a three-port optical IIR interleave filter.

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In practically fabricated optical filters optical waveguides inevitably exhibit propagation loss which is less than 1.0 dB/cm [9,10]. Therefore, the impact of waveguide loss is examined for the proposed optical IIR interleave filter as shown in Fig. 5. The passband transmission decreases proportional to the feedback path loss, but the stopband response is maintained and the filter spectra are in fairly good agreement with the ideal one. With current PLC technology, the minimum allowable diameter for ring waveguides is about 3 mm. It can be estimated that the realizable maximum frequency period is about 20 GHz. The proposed IIR optical filter is capable of higher-functional optical processing in a small number of stages because of the feedback effect.

 

Fig. 5. Impact of waveguide loss on the attenuation spectrum of a three-port optical IIR interleave filter.

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5. Conclusion

This paper presented a one-input M-output (1×M) circuit configuration for realizing optical IIR lattice circuits with M-output channels (M ≥ 2). The circuit configuration has a multilayer structure composed of MZIs with delay time difference Δτ for designing output responses. Furthermore, synthesis algorithm is also briefly discussed. It is expected that they will be employed as various optical filters and optical adaptive filters in FDM and WDM optical communication.