1 Introduction

Multifunction optical filters (Dingel and Izutsu 1998; Dingel 2014) are key components in the next-generation intelligent optical networks, which provide configurable and programmable features (Modarrerssi and Mohan 2000). Optical filters with simple configurations are necessary for a compact, versatile and economical dense wavelength division multiplexed (DWDM) system. Optical ring resonator is the most widely used basic block in various application systems, such as optical interconnect (Xia et al. 2007), optical communication (Little et al. 2004), optical sensing (Sun and Fan 2011), optical signal processing and so on. Due to their ability to provide wavelength selectivity, and easily cascadable, the optical ring resonators have great potential for allowing the realization of various filter responses, which is the basis of multifunctional filtering.

The synthesis and analysis of ring-resonator type multifunction filters is still a rather challenging task. Numerous phases, delays, and coupling parameters need to be controlled. In recent years, researchers in optics have paid increasing attentions on the digital signal processing (DSP) technology because it brings about new solutions to some important optical signal processing issues (Cheng 2006; Zhang 2016). Some optical filters can be designed based on DSP methods due to the similar feedback characteristics of optical and digital filters (Madsen and Zhao 1999; Yang and Zhang 2008; Cheng and Tang 2013; Madsen 2000; Zhang and Yang 2010; Zhang et al. 2012, 2014; Pinzon et al. 2013). Compared with the complex optical interferometric analysis, DSP-based design methods are more simple and effective because many mature computer-based DSP algorithms and technologies can be directly used.

In this paper, we demonstrate the use of a DSP method to the synthesis of a ring-resonator based multifunctional optical filter. The proposed structure is an unbalanced Mach–Zehnder interferometer (MZI) with coupled ring resonators. The multifunction optical filter can function as symmetrical interleaver, asymmetrical interleaver with arbitrary duty cycles, notch filter with a single notch point, and notch filter with two notch points, depending on the parameters of the structure.

2 Optical structure and digital signal model

Figure 1 shows the schematic structure of the multifunction filter. It is formed by a Mach–Zehnder interferometer (MZI) with two cascaded serially coupled double-ring resonators and a single-ring resonator coupled on the arm 1 and arm 2 respectively. On arm 1, the coupling coefficients of the first double-ring resonators are K11, K12 and the corresponding ring lengths are l 11, l 12, respectively. The coupling coefficients of the second double-ring resonators are K21, K22 and the corresponding ring lengths are l 21, l 22, respectively. On arm 2, the coupling coefficient and ring length of the single-ring resonator are KL and L L respectively. ΔL (= L1 − L2) is the length difference of the two arms of the MZI. K a and K b (K a  = K b  = 0.5) are the coupling coefficients of the 3 dB couplers in the input and output ports of the MZI respectively.

Fig. 1
figure 1

Schematic of the proposed multifunction filter

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The optical structure shown in Fig. 1 can be transformed into a digital signal model by the Z-transform method, as shown in Fig. 2. The equivalence of the optical and signal parameters can be shown as c a  = (1 − K a )1/2, s a  = (K a )1/2, c b  = (1 − K b )1/2 and s b  = (K b )1/2. e −jφ = e −jβΔL is the phase shift produced by the MZI, where β = 2πn e /λ, n e is the effective refractive index of medium, and λ is the wavelength in vacuum. H 1 dou (z) and H 2 dou (z) are the transfer functions of the first and second double-ring resonator in the upper arm respectively. H sin (z) is the transfer functions of the single-ring resonator in the lower arm.

Fig. 2
figure 2

Digital signal model of the proposed multifunction filter

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The proposed model of the multifunction filter is different from the available published models of interleavers or notch filters in the following two aspects. First, it uses two serially coupled ring resonators, rather than commonly-used (cascaded) single rings in one arm of the MZI. Second, no phase shifter is needed on each ring. Multifunction filter can be realized only by choosing appropriate values of the path length differences of the MZI, ring length of the resonators, and the coupling coefficients of the couplers. Figure 3 shows the Z-transform schematics of the single-ring and serially coupled double-ring resonator respectively.

Fig. 3
figure 3

Z-transform schematics of a single-ring resonator, b serially coupled double-ring resonators

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The transfer function of the single-ring resonator can be obtained from Fig. 3a as

$$H_{d} (z) = H_{sin} (z) = \frac{{z^{ - 1} - c_{L} }}{{c_{L} z^{ - 1} - 1}}.$$
(1)

here c L  = (1 − KL)1/2. From Fig. 3b, we can get the transfer function of the double-ring resonator by recursion as (Zhang 2016)

$$H_{dou}^{i} (z) = \frac{{z^{ - 2} - c_{i1} \left( {1 + c_{i2} } \right)z^{ - 1} + c_{i2} }}{{c_{i2} z^{ - 2} - c_{i1} \left( {1 + c_{i2} } \right)z^{ - 1} + 1}},\;\;\left( {i = 1, \, 2} \right)$$
(2)

here c i1 = (1 − K i1)1/2 and c i2 = (1 − K i2)1/2, thus the transfer function of two cascaded double-ring resonators can be denoted as

$$H_{u} (z) = H_{dou}^{1} (z) \times H_{dou}^{2} (z),$$
(3)

where z −1 = e −jω = e −j(2πneL)/λ, L = C/(Δf · n e ), L is the unit ring length, Δf is the spectral period and C is the speed of light in vacuum.

According to Fig. 2, we can obtain the transfer function of the whole structure as follows:

$$H_{1} (z) = \frac{{Y_{1} }}{{X_{1} }} = \frac{1}{2}\left[ {H_{u} (z) \cdot z^{{ - \frac{\Delta L}{L}}} - H_{d} (z)} \right],$$
(4)
$$H_{2} (z) = \frac{{Y_{2} }}{{X_{1} }} = - j\frac{1}{2}\left[ {H_{u} (z) \cdot z^{{ - \frac{\Delta L}{L}}} + H_{d} (z)} \right].$$
(5)

Different filter responses to achieve multi-functions can be obtained by changing the transfer functions. Then the corresponding structural parameters can be deduced. The schematic process for synthesizing the multifunction filters is shown in Fig. 4.

Fig. 4
figure 4

Schematic process of synthesizing the multifunction filter based on a DSP method

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3 Multifunction filtering characteristics

3.1 Optical interleavers

The poles of H 1(z) (or H 2(z)) must be conjugate pairs on the imaginary axis to form a symmetrical interleaver, otherwise asymmetrical spectra can be obtained (Zhang and Yang 2010). According to this design theory, symmetrical interleaver and asymmetrical interleaver with arbitrary duty cycles can be synthesized.

Firstly, the structure can function as a symmetrical interleaver when K21 = K22 = 0, ΔL = L and l 11 = l 12 = L L = 2ΔL. Choosing appropriate coupling coefficients of the first double-ring resonator and the single-ring resonator, a flat-top spectral response can be obtained. Figure 5a shows the transmission spectrum as the function of frequency with K 11 = 0.08358, K 12 = 0.93041 and K L = 0.51442. The detailed passband spectrum is shown in Fig. 5b. The ripple of the flat-top equals to 0.00432 dB, i.e. the isolation equals to 30.04 dB. The rectangle degree, defined as the ratio of the channel width at 3 dB to 20 dB, is calculated to be 0.970. As we know, the larger the rectangle degree, the better the filtering performance of the interleaver.

Fig. 5
figure 5

Transmission spectra of a symmetrical interleaver. a transmission spectrum as the function of frequency. b zoomed passband

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Secondly, the structure can also function as an asymmetrical interleaver with arbitrary duty cycles. It is known that an optical interleaver can be viewed as a lowpass-highpass doubly-complementary filter pair. In DSP theory, an odd order lowpass-highpass doubly-complementary filter pair can always be decomposed into sum or difference of two all-pass filters with adjacent orders (Mitra 2006). Therefore, from Eqs. (4) and (5), we can see that interleavers with arbitrary duty cycles can be synthesized by decomposing transfer functions into sum or difference of the all-pass filters with adjacent orders when ΔL equals to zero. According to this design theory, the structure can function as asymmetrical interleaver with arbitrary duty cycle when K21 = K22 = 0, ΔL = 0 and l 11 = l 12 = l 21 = l 22 = L L = L.

Figure 6 shows the output spectra of the asymmetrical interleavers as the function of frequency with four different duty cycles. The corresponding structural parameters are shown in Table 1 in detail. Table 2 shows the 3 dB, 20 dB bandwidth and the rectangle degree of the asymmetrical interleavers for the two output ports. It can be seen that flat-top profile and high isolation can be obtained for the two output ports at the same time. Moreover, the rectangle degree for the wide output port is always larger than that for the narrow output port with the same duty cycle. The rectangle degree of the symmetrical interleaver is higher than that of the asymmetrical interleaver due to the higher filter order of the former. The symmetrical interleaver is a seven-order filter, whereas the asymmetrical interleaver is a three-order filter. Asymmetrical interleavers with higher rectangle degrees can be obtained by increasing the number of coupled optical rings.

Fig. 6
figure 6

Output spectra of the two output ports with the duty cycle of a 1:3 b 1:4 c 1:5 d 1:6

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Table 1 Coupling coefficients of the resonators for the asymmetrical interleavers with different duty cycles
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Table 2 Bandwidths and rectangle degrees of the asymmetrical interleavers with different duty cycles for the narrow and wide output ports
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3.2 Optical notch filters

The optical notch filter can be transformed into an allpass filter as H(z) = 1/2(1 + A(z)), where A(z) is an allpass filter (Pei and Tseng 1997). Therefore, the structure can function as a notch filter when KL = 0, and ΔL = 0.

Figure 7a shows the output spectra of a notch filter with an equal 3 dB rejection bandwidth when K12 = 0.2460, K11 = 0.4769 and K22 = 0.2528, K21 = 0.9483. The two notch points are located at λ 1 = 1550.2 nm, and λ 2 = 1550.3 nm. The 3 dB rejection bandwidth of the two notch peaks are Δλ1 = Δλ2 = 0.008 nm.

Fig. 7
figure 7

The output spectra of a notch filter with a equal, and b unequal 3 dB rejection bandwidth

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Figure 8 shows the output spectra of the optical notch filter when K12, K11, K22 and K21 shift by a certain value independently. It can be seen that the parameters K12 and K11 determine the characteristics of the first notch point (1550.2 nm), while the parameters K22 and K21 determine the characteristics of the second notch point (1550.3 nm). The 3 dB bandwidth of the first and the second notch point can be tuned by the parameters K12 and K22 respectively. When K12 and K22 was decreased, the 3 dB bandwidth of the notch filter becomes narrower. The location of the first and the second notch point can be controlled by the parameters K11 and K21 respectively. When K11 and K21 was decreased, the location of the notch points shifts to the long wavelength direction. The independent tunabilities of the bandwidth and notch locations make the structure more flexible in applications. Specific filtering properties can be obtained through adjusting corresponding parameters. For example, Figs. 7b and 8c shows the spectral response of a notch filter with unequal 3 dB rejection bandwidth (Δλ1 = 0.008 nm, Δλ2 = 0.016 nm) by adjusting K22.

Fig. 8
figure 8

The spectra response of a notch filter when parameters a K12; b K11; c K22; d K21 shift independently

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If only one double-ring resonator, such as the first double-ring resonator, is used in the structure (i.e. K21 = K22 = 0), a single notch point located at 1550.2 nm can be obtained, as shown in Fig. 9a. Similarly, when only the second double-ring resonator is used in the structure (i.e. K11 = K12 = 0), a single notch point located at 1550.3 nm can be obtained, as shown in Fig. 9b. More notch points can be obtained by coupling more double-ring resonators in the horizontal direction on the upper arm.

Fig. 9
figure 9

The spectra response of the notch filters with notch points located at a 1550.2 nm; b 1550.3 nm

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3.3 Poles/zeros figures for different filter responses

As we know, the poles of the transfer function amplify the frequency response while the zeros attenuate it. We can place poles and zeros on the unit circle and quickly evaluate the filter’s frequency response. The poles/zeros plots of the multifunction filters are shown in Fig. 10. The corresponding frequency (spectral) response was also shown.

Fig. 10
figure 10

The poles/zeros and frequency (spectral) responses of the multifunction filters. a symmetrical interleaver, b asymmetrical interleaver, c notch filter with a single notch point, d notch filter with two notch points

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4 Fabrication analysis

The proposed multifunction filter can be fabricated based on planar light circuit (PLC) technology or micro-fiber technique. PLC technology is the most often used method for making micro-ring resonator. It is particularly suited for monolithic integration with other components. The detailed description of fabrication technique can be easily found in published literatures (such as Kohtoku et al. 2000).

The path length differences of the MZI, ring length of the resonators, and the coupling coefficients of the couplers are the key structural parameters of the proposed multifunction filter. Different parameters have different influences on the filtering characteristics. For example, there are five coupling coefficients, i.e. K a , K b , K11, K12, KL, two length parameters, i.e. the length difference of the two arms of the MZI ΔL and the ring length of ring resonator l, in a symmetrical interleaver structure. The deviation of K11, K12 and KL from the ideal value will have more influence on the output spectra than that of K a , K b . Moreover, among K11, K12 and KL, the influence of K12 is the largest, KL is less, and K11 is the least. For the length parameters, the condition l/ΔL = 2:1 should be met in fabrication. Otherwise, the passband and stopband will not be flat any more. If ΔL and l changed synchronously keeping l/ΔL = 2:1, the center frequency will shift although the output passband and stopband are still flat. The change of l is more sensitive than that of ΔL.

5 Conclusion

In this paper, a DSP method was used to synthesis a ring-resonator based multifunctional optical filter. The transfer function of the unbalanced Mach–Zehnder interferometric structure was deduced from its digital signal model obtained by the Z-transform method. This structure can function as a symmetrical interleaver, an asymmetrical interleaver with arbitrary duty cycles, a notch filter with a single notch point and a notch filter with two notch points, by simply tuning the structural parameters, such as the path length differences of the MZI, ring length of the resonators, and the coupling coefficients of the couplers. Computer-based DSP algorithms and technologies exhibit much advantage in the design of complex interferometric structures, such as multifunction filters used in the future intelligent DWDM networks.