Programmable wavelength filter with double ring loaded MZI

We propose a novel filter circuit that incorporates a double ring resonator with a balanced Mach–Zehnder interferometer (MZI). The circuit has a response equivalent to a conventional ring loaded MZI filter, but with added flexibility in terms of configurability. The second-order filter can also be cascaded to realize higher-order filters. The circuit incorporates a two-stage input and output coupler to further reduce the effect of dispersion. A combination of local and global optimization strategies to program the filter, using tailored objective functions, have been tested in simulation and experiments. To our best knowledge, this is the first time a global optimization strategy is directly used in ARMA filter synthesis and simulation without any additional requirement. We further extend the optimization strategy into experiments and demonstrated its use in practical case for programmable filter circuits.


FILTER SYNTHESIS OF AN MZI LOADED WITH SERIALLY COUPLED DOUBLE RING
The transmission of the filter from the input ports to the output ports can be calculated by the following transfer matrix multiplication in the z domain. Because we choose to use ring resonators with a fixed length L ring , we can apply the transfer matrix method (TMM) in the z domain, with .n e f f (λ).L ring /λ . The complex amplitude of the input and output electromagnetic waves in the z domain have the following relationship between each other: The coupling matrix, C i , is given by: where k i is the power cross-coupling coefficient, and 1 − k i is the straight-through coefficient for the directional coupler (assuming lossless operation). The propagation matrix P i of each phase delay section is given by: where we assume no loss, and equal length for waveguide propagation, so the common propagation terms can be ignored.
The propagation matrix D i of serially coupled double ring section is given by: where δ is written as: And X 1 and X 2 are written as: The final transfer function in the z notation can be written as the generic transfer function of a second-order filter: where a i and b i are a function of the coupling coefficients and phase shifts. The coefficients of the transfer function in Eq. S8 are given by:

COMPARISON WITH DOUBLE RING LOADED MZI WITHOUT COUPLING BETWEEN THE RINGS
Fig. S1. Schematic drawing of the classical ring loaded MZI is shown in (a); the schematic drawing of coupled ring loaded MZI is shown in (b). The main difference between our design given in (b) and the classical double ring loaded MZI given in (a) is that the two rings are connected with a tunable coupler. It is obvious that the design in (b) could be configured to an MZI with delay length of twice the ring circumference, which is not possible with the design in (a). The phase shifters are shown as a pink box, and the tunable coupler is shown as a normal DC with a pink dot in the middle, the single add drop ring is constructed by connecting the input and output port of the tunable coupler on the same side. The input and output tunable coupler of the design in (b) is realized by a broadband balanced MZI [1] on the actual chip.
In this section, we will compare the pole-zero diagram of the classical ring loaded MZI and of our design. We demonstrate that with the same loss characteristics of the couplers, both design could cover the same area of the unit circle in the z-plane, and thus both have the potential to exactly realize second-order bandpass filter.
The ring loaded MZI in Fig. S1(a) is a complex filter where its a i and b i coefficients are complex numbers, however the coupled ring loaded MZI in Fig. S1(b) can be configured both as a real filter and as a complex filter.
In the following, we would configure our tunable filter design into the double ring loaded MZI as shown in Fig. S1(b) and ring loaded MZI as shown in Fig. S1(a) separately.
The coupled ring loaded MZI in Fig. S1(b) has 6 free parameters, which are k 0 , k 1 , k 2 , k 3 , k 4 and φ 1 . In simulation we directly modify k i , but in the experiment the coupling factors k i are realized using a balanced MZI with a phase shifter in each arm, and thus the tuning parameter becomes another phase shift. The coupled ring loaded MZI in Fig. S1(b) can be configured to be a real filter where its coefficients are real numbers. In order to force the coefficients to be real, the phase shifters inside the rings need to be set to either zero (configuration I) or π (configuration II). These two situations are equivalent with each other in the spectrum response, we will discuss these two situations in Fig. S2.
The ring loaded MZI in Fig. S1(a) can also be a real filter, but then its configurability is limited. To implement a generic second-order bandpass filter, it needs to be configured as a complex filter where its a i and b i coefficients are complex numbers. The classical ring loaded MZI has 5 free parameters in simulation, which are k 1 , k 2 , φ 1 , φ 2 , φ 3 , where the input and output couplers are set to k 0 = k 3 = 0.5.This architecture is proposed for realizing a bandpass filter with two all-pass filters in [2]. For most cases, a symmetric spectral response is desired, which requires k 1 = k 2 and The filter synthesis equations have been accordingly rewritten for the ring loaded MZI in Fig. S1(a), the equations Eq. (S4) -Eq. (S14) for the ring loaded MZI could be rewritten by setting the coupling k 2 to 0, thus C 2 = 0 and S 2 = 1. Then we compare the two types of filter in simulation. We will illustrate the programmability space of the two filter types using the Latin Hypercube sampling (LHS) method. LHS is a statistical method for generating a near-random sample of parameter values from a multidimensional distribution. In our following analysis, we use this method to explore the relationship between the parameter space and the distribution of the filter's poles and zeros in the z-plane.
For both cases, 500 sampling points are used. We plot the pole-zero map of both structures in Fig. S2. From Fig. S2, we can see that the poles of Fig. S1(a) are homogeneously distributed in the z-plane. The poles within the same pair are not necessarily symmetrically positioned with respect to the Real (x) axis. The zeros are mostly located in the unit circle. The poles for configuration I (φ 1 = 0) of Fig. S1(b) are homogeneously distributed in the right half-plane of the unit circle, while the zeros are scattered homogeneously, also in the right half of the z-plane. The poles of configuration II (φ 1 = π) of Fig. S1(b) are homogeneously distributed in the left half-plane of the unit circle, the zeros are scattered homogeneously also in the left-half of the z plane.
These two designs (coupled ring-loaded MZI and ring-loaded MZI ) show full configurability. In the following, we would show how to get the desired coupling values for a certain filter with both complex and real filter coefficients. We will use three examples to show that these two designs are equivalent with each other, however one may perform better than the other in certain cases.
The following target function is used for optimizing a complex filter: Likewise, the target function for real filter is: Let's build a first example filter with a second-order elliptical low-pass filter with 1 dB in-band ripple, 20 dB extinction ratio and the normalized edge frequencies of 0.1 π rad/sample. The a i and b i coefficients for such filter can be calculated with Matlab: We will now compare the filter responses of configuration I of Fig. S1(b) and complex filter in Fig. S1(a). After we apply the synthesis algorithm as described before, the pole-zero plot and     Fig. S3(b) achieve. The corresponding parameters for Fig. S3 are given in Table S1.According to our experiments, we found out that for normal filters such as elliptical filter, the structure is actually symmetrical, thus, certain constrains could be applied for the two designs when we try to solve for the couplings and phases. For our design shown in Fig. S1(a), the constrains are k 0 = k 4 , k 1 = k 3 and phi 1 = 0. For the ring loaded MZI, the constrains are k 0 = k 3 = 0.5, k 1 = k 2 and phi 2 = −phi 3 . There are more solutions for our design when a given elliptical filter is synthesized, four solutions for the same elliptical filter in example one is shown in Table S2.As we can see, solution 1 is the most symmetrical result, and solution 2 will simplify our design since no output coupler is needed. Filter designer might select the most stable design or the one with less wavelength dispersion.
The pole-zero plot and corresponding transfer function for the coupled ring-loaded MZI are shown in Fig. S3(a) and Fig. S3(b). In Fig. S3(a), the target pole and zero positions overlap with the fitted ones, and we see a perfect correspondence for their spectrum response. The pole-zero plot and corresponding transfer function for the conventional ring loaded MZI are shown in Fig. S3(c-d). In Fig. S3(c), the target poles and zeros are in the right half of the z plane, while the fitted one is in the left half plane. This corresponds to a shift of the spectral response with half a free spectral range (FSR), or a mirroring on the Nyquist frequency axis. Since the spectrum response is repeated with the FSR, so the final spectrum response of the targeted result and the fitted result show no difference with each other. Now we consider a second filter example, let's construct a Chebyshev type II filter cheby2(2 ,20, 0.9, 'low'). The a i and b i coefficients are as following: The pole-zero diagram and the corresponding spectrum response of the coupled ring filter in configuration II (φ 1 = π) are shown in Fig. S4(a-b).The pole-zero diagram and the corresponding spectrum response of the conventional ring loaded MZI is shown in Fig. S4(c-d). The corresponding parameters for Fig. S4 is given in Table S3.
The two circuits perform equivalently. The spectrum responses are the same if we would like to systhesize a given bandpass filter, and the only difference would lie in the configuration of the filter and its corresponding a i and b i coefficients.
For the third example, the a i and b i coefficients are not calculated from any bandpass filters. We try to set the following values to our design, k 0 = 0.6, k 1 = 0.9, k 2 = 0.5, k 3 = 0.1, k 4 = 0.4 cheby2(2,20,0.9,'low'),   The pole-zero diagram and the corresponding spectrum response of the coupled ring-loaded MZI in configuration II are shown in Fig. S5(a-b).The pole-zero diagram and the corresponding spectrum response of ring loaded MZI is shown in Fig. S5(c-d). The parameters for Fig. S5 are given in Table S4. In this case, the fitted curve and the targeted curve in Fig. S5(d) did not overlap with each other very well, which indicates that the ring loaded MZI could not be configured to match the double ring loaded MZI. The double ring loaded MZI thus is a better candidate for a fully programmable filter. This is important when considering uses other than bandpass filters, such as arbitrary waveform generation.   Fig. S1(b), while (c) and (d) correspond to the classical ring-loaded MZI of Fig. S1(a).
As we can see from the three examples, the a i and b i coefficients in our design are always real numbers. Actually if the φ 1 is set not be 0 or π, the a i and b i coefficients would also be complex number for our design. If we try to fit such complex a i and b i by the given ring loaded MZI, we can not get a good fit, either. Such situation is very similar to the third example.

CASCADE FOR HIGHER ORDER FILTER
The roll-off ( measured in dB/decade ) reduces by 20 dB/decade per filter order increment. Thus, if a second-order filter function has a roll-off of 20 dB/decade, the fourth-order filter would have a roll-off of 60 dB/decade. This results in a more ideal amplitude frequency response. On the other hand, higher-order filters are complex to configure and design, and they are used only when a sharp transition between the passband and stopband is required.
We take a fourth-order Chebyshev type II filter with 30 dB stop band and 0.5π rad/dB as an example. The power response of the Chebyshev filter will have a 60 dB extinction ratio. The  Fig. S6. Schematic of a two stage filter and the corresponding coupling values and phase response for a fourth-order Chebyshev type II filter with 30 dB stop band and 0.5π rad/dB. transfer function of such a filter is given by: Eq. S20 can be decomposed to two second-order filters, H 1 (z) and H 2 (z). The corresponding second-order transfer functions are: Then we use least-square fitting to find the corresponding coupling value for the each subcircuit. Fig. S6 is the schematic drawing of a two-stage filter and the corresponding coupling values and phase response from the least square fitting. We then build the Caphe circuit model for the circuit in Fig. S6 and we can see the corresponding spectrum response in Fig. S7. This shows that the designed filter can reach an extinction ratio of around 60 dB, which meets our requirement.

PROGRAMMABLE FILTER DESIGN
Programmable photonic circuits have been well studied in recent years [3], and there have been several demonstrations to achieve a programmable optical filter [4]. The reconfigurability is desired in many applications such as optical interleavers, spectral slicers, or filtering and equalization circuits for microwave photonics [5]. The ability to change its response according to the requirements would on one hand lower the price of the communication channel, on the other hand, it would allow the software to correct or adjust itself with feedback , which is beneficial to the signal performance.
For a programmable filter, we need to be able to adjust the different phases and coupling ratios in our filter circuit. Instead of passive directional couplers, we can use a tunable coupler, as shown in Fig. S8(a). In this work, the tunable coupler is composed of a balanced Mach-Zehnder interferometer, but other types of tunable coupler designs have also been proposed to compensate for fabrication variation or dispersion [1].  The phases φ 1 or φ 2 are used to adjust the power coupling. This also introduces a phase shift (φ 1 + φ 2 )/2 at the outputs. So in order to maintain the same phase response as a static directional coupler, we need to use φ 3 and φ 4 to compensate this additional phase change induced by φ 1 and/or φ 2 . This need for compensation complicates the control of the entire circuit somewhat. If we set the phase shifter of the upper arm of the MZI to be φ 1 = x, and lower arm of the phase shifter to be zero, in order to induce no phase change compared to a passive coupler, then, φ 3 and φ 4 should have the following value, φ 3 = φ 4 = 2mπ − x/2, with m an integer number. For phase shifters that can induce both a positive and negative phase shift, m is preferably zero, but for phase shifters that can only operate in a single direction, such as heater-based phase shifters, m = 1.
We replaced all the couplers in Fig. S6 with the tunable coupler in Fig. S8(a), and built a circuit model for the two-stage filter in Fig. S9(b). We then translated the desired coupling in Fig. S6(b) to the corresponding phase values in Fig. S9(a), including the compensation phases. These parameters of phases are shown in Table S5. The spectrum response simulated with the Caphe model of Fig. S9(a) is then shown in Fig. S10.

ESTIMATION OF Q FACTOR
In this section, we will have an estimation of the Q factor [6]. We assume the loss for the tunable coupler is 0.12 dB, n g is 4.3, the loss for the phase shifter is around 0.02 dB. The resonance dc:phase dc6: φ 6 φ 6 = 0.26π wg6:φ 6 -(φ 6 + φ 7 )/2 Arm2u2:φ 6 -φ 5 /2 dc7: φ 7 φ 7 = 0.279π wg7:φ 7 0 Arm2d1:φ 7 -φ 6 /2 + π dc8: φ 8 φ 8 = 0.2755π wg8:φ 8 -(φ 8 + φ 7 )/2 Arm2d2:φ 8 -φ 8 /2 dc9: φ 9 φ 9 = φ 5 Table S5. Parameters For Fig. S9   Fig. S10. Simulation result for the Butterworth filter using the two-stage design with the corresponding caphe model wavelength is 1.55 um, and the round trip length L of the ring is calculated according to the FSR measured and it is 712 um, the length of the tunable coupler is 173.4 um and the length of the heater is 50 um. The waveguide loss is estimated to be 1.3 dB/cm. r is the critical coupling and g is the amplitude attenuation coefficient. The loss of power of the single ring is composed of waveguide loss, loss for two tunable couplers and the loss of the phase shifter, the roundtrip loss of the single ring is 0.300976 dB.
Loss dB = −10 log 10 (P out /P in ) = −20 log 10 (e −gL ) = 20gL · log 10 e (S22) The Q factor of the single ring is calculated by the following equation: The Q at the critical coupling (i.e. r = g) is estimated to be 18184.

TOLERANCE ANALYSIS
In experiments, the dispersion of the building blocks, as well as the fabrication variation and thermal cross-talk need to be considered, as they move the filter away form its nominal operating point and complicate the configuration and control. Fig. S11 is a Monte-Carlo simulation performed for two filter designs in Fig. S6 for a given filter response, and it is hard to tell which filter is better in terms of fabrication variation. It is certain, however, that the phase shifters inside the ring contribute the most significantly to the deformation of the spectrum response.