## Abstract

We designed a universal linear circuit by using microring resonators instead of conventional Mach-Zehnder interferometers and phase shifters. We illustrated that the footprint of the universal linear circuit can be drastically reduced (∼ 1/10). In addition, power consumption can also be reduced by using the sensitivity of the phase change in the vicinity of the resonant peak. Furthermore, as an important example of the application for optical communication, MIMO compensation operation is numerically demonstrated by our proposed universal linear circuit. The proposed design can be adapted to other experimentally reported devices, which will accelerate the integration of the universal linear circuit.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Universal linear circuits are reconfigurable mode converters that have great potential for all-optical multi-input multi-output (MIMO) processing and quantum information processing [1–5]. Theoretically, any unitary transformation can be executed by combining *N*⋅(*N*−1)/2 Mach-Zehnder interferometers (MZIs) and *N*⋅(*N *+ 1)/2 phase shifters (PSs), where *N* is the order of input/output vectors; namely, the number of ports. Recently, various experimental demonstrations of universal linear circuits have been reported [6–8]. In these devices, it is necessary to control the phase change to a maximum of *π* and 2*π* in all MZIs and PSs, respectively, to ensure arbitrary unitary operation. Therefore, in these experimental demonstrations, a large refractive index change is required, which is induced by a thermo-optical (TO) effect with a wide modulation area and high power (i.e., a 100 µm TiN heater and consumption power of 70 mW in the Si platform [7]). Such a large device size limits the scalability of the universal linear circuit.

This study presents a miniaturized design of a universal linear circuit by using microring resonators, as displayed in Fig. 1. Instead of an MZI, we deploy an add-drop filter based on the four-port double ring resonator as the power divider. In addition, the PS is replaced by cascaded two-port single ring resonators. As illustrated in Fig. 1, the proposed universal linear circuit for a 3 × 3 unitary operation can be constructed with a footprint of ∼ 12*D* × 4*D* for the optical waveguide, where *D* is the diameter of the ring resonator. The typical diameter of a silicon microring resonator is approximately 10 µm; hence, the proposed design can drastically reduce the size of the device (approximately ∼ 1/10). To simplify the discussion, we take the first simple configuration by Reck *et al.* instead of the triangular scheme by Reck *et al.* [1] and the symmetry scheme by Clements *et al.* [4] . By applying these approaches, the footprint of the optical waveguide can be reduced to ∼ 9*D* × 4*D*, although the electrode footprint becomes dominant in the actual device. One may concern about the sensitivity of resonator-based operation instead of advantage of miniaturization. However, we suppose that such drawback can be compensated by electrically driven refractive index change.

This paper is organized as follows: Section 2 briefly explains the basic formalization of the unitary operation. Section 3 explains the design of the controllable PS and power divider based on the microring resonator, where we assume a silicon microring resonator in which the TiN heater is mounted to control the refractive index. Section 4 presents a numerical example of the MIMO compensation using the universal linear circuit, which demonstrates the validity of our configuration.

## 2. Basic formalization of unitary operation in a planar lightwave circuit

As stated in [1] in detail, any *N* × *N* unitary transfer matrix can be represented by a product of *N*⋅(*N*−1)/2 + 1 transfer matrices. A universal linear circuit can be constructed by a cascading 2 × 2 transfer system **v**_{out} = **Uv**_{in} where the unitary transfer matrix **U** is

**A**(

*φ*) and

**B**(

*θ*) correspond to the controllable power divider (typically, a MZI) and the PS for one optical path, respectively. Here, we define the

*N*×

*N*transfer matrix

**T**

_{ij}^{(N)}(

*φ*,

*θ*) that is the

*N*×

*N*identity matrix

**I**

^{(N)}, in which the

*ii*,

*ij*,

*ji*, and

*jj*components are replaced by 11, 12, 21, and 22 components of

**U**in Eq. (1). For example,

**T**

_{23}

^{(3)}is given by

**S**

_{ij}^{(N)}=

**T**

_{ij}^{(N)}(

*π*/2, 0). In the planar lightwave circuit, to make the interference between the

*i*th and

*j*th port (|

*i*−

*j*| > 1), the required switching components are shown by

**T**

_{13}in Fig. 1. For example, any 3 × 3 unitary transfer matrix

**T**

^{(3)}for a planar lightwave circuit can be generalized as

**D**

^{(N)}(

*θ*

_{1},

*…*,

*θ*) is a diagonal matrix for adjusting the phase. Any order universal linear circuit can be constructed in a similar way.

_{N}## 3. Controllable phase shifters and power dividers based on ring resonators

To date, the MZI is often used to realize the transfer matrix given by Eq. (2). In this study, we use the ring resonator as the power divider and a PS for miniaturizing the device. Using a resonator as a PS has already been reported [9,10]. Figures 2(a) and (b) present a comparison of the phase shift and transmission that occur by changing the refractive index in three waveguides: a wire waveguide, a single two-port ring resonator, and cascaded two-port ring resonators. Here, we perform an electromagnetic analysis based on the 2D finite element method [11], using the effective index approximation, and the TM mode is solved (corresponding to the quasi-TE mode for a 3D wire waveguide). The effective refractive indices of the core and cladding for 1550 nm are *n*_{Si-2D} = 2.76 − *j*1.174 × 10^{−5} and *n*_{SiO2} = 1.44, respectively. Here, the imaginary value is determined so that the propagation loss becomes 4 dB/cm in wire waveguide. The change in refractive index is applied only to the core; namely, *n*_{Si-2D} (Δ*n*) = 2.76 + Δ*n* − *j*1.174 × 10^{−5}. Other structural parameter details are provided in the legend of Fig. 2. The input wavelength is fixed to 1549.46 nm that is the resonant wavelength of the ring resonator for Δ*n* = 0. As depicted in Fig. 2(a), refractive index changes (Δ*n*) of 0.016 and 0.012 are necessary to obtain a 2*π* phase shift in the wire waveguide and the single two-port ring resonator, respectively. It is slightly advantageous to use a ring resonator instead of a simple wire waveguide. In the resonant condition, the phase of transmittance is sensitive to the refractive index. However, to obtain a phase shift of 2*π* in the single two-port ring resonator, the round-trip phase shift *β*⋅(2*πR*) must be increased to 2*π* by changing the refractive index, where *β* and *R* are the propagation constant in the ring waveguide and the radius of the ring, respectively. By cascading the two-port ring resonators, an extremely large phase shift can be obtained by a small change in the refractive index because of the sensitivity of the phase, which appears in the resonant condition. In the cascaded resonators, a 2*π* phase shift can be obtained with very small Δ*n* of 0.002. This means that the required power drastically reduces (∼1/10) by using cascaded resonators instead of a simple wire waveguide. We also should take care about the propagation loss in the wire and ring resonator. As depicted in Fig. 2(b), although the loss in the ring resonator increases to 0.5 dB (that is roughly 10 times larger than 100 µm wire) at maximum, where the loaded *Q*-factor is ∼1700. Such loss can be reduced by lowering *Q*-factor.

Furthermore, the power divider is also constructed by the ring resonator. It is known that the four-port ring resonator can be used as an add-drop filter [12], and the spectrum becomes a box-like shape by the coupling rings [13]. A change in the refractive index changes the resonant wavelength. Such behavior indicates that an arbitrary power ratio can be realized by controlling the refractive index changes Δ*n*. If we attempt to use the four-port single ring resonator as the MZI, the input and add ports are separated, and an additional waveguide crossing is required. Therefore, the four-port double ring resonator is used in the proposed device, as shown in Fig. 1. Figure 3(a) shows the transmission spectra for the through ports (blue lines) and drop ports (red lines) in the four-port double ring resonator. The solid lines correspond to the original refractive index, whereas the dashed lines depict the spectra when the refractive index is changed with a Δ*n* of 0.0055, indicating the change induced by the TO effect in silicon with + 30 K [14] (noting that, the assumed index change by TO effect is actual value, but the effective refractive index is used in the simulation). To form the box-like spectrum, we chose structural parameters of *g*_{1 }= 160 nm and *g*_{2} = 360 nm; the other parameters are identical to the single two-port ring resonator in Fig. 2. We observed a sufficient resonant peak shift with a change in the practical refractive index. Figure 3(b) displays the transmission for a wavelength of 1.5496 µm as a function of the refractive index change Δ*n*. For Δ*n* = 0, the input lightwave completely transmits to the drop port, which means that the four-port double ring resonator can function as a switching component. Whereas, for Δ*n* > 0.005, the input lightwave transmits to the thorough port with an extinction ratio greater than 20 dB. Noting that, if the fabrication error gives the mismatch between the resonant wavelengths of two resonators, it may be required to heat two rings individually or replace the four-port double resonator with the four-port single resonator and crossing waveguide.

## 4. Operation of the universal linear circuit

In this section, we numerically demonstrate the operation of the universal linear circuit shown in Fig. 1. The MIMO compensation that divides the mixed signals is an important application for universal linear circuit. Figure 4 presents a schematic of the MIMO compensation using the proposed universal linear circuit, where we assume that the signals are linearly mixed by the unitary transfer matrix **T**. Hence, the mixed signals can be successfully divided by the unitary operation of **T**^{†}. Figure 5 shows the field distribution of the universal linear circuit for MIMO compensation by a 3 × 3 unitary operation. The structural parameters are identical to the previous section and the operation wavelength is 1549.46 nm that is resonant wavelength for Δ*n* = 0. The three different lightwaves, mixed by the unitary transformation of **T**, are input from the left side, and they are successfully divided among the three right ports. The minimum transmission is −1.2 dB, and the maximum crosstalk is −29 dB. In this example, we set the unitary transfer matrix as **T **= (**v**_{1}, **v**_{2}, **v**_{3}), where **v**_{1} = (−0.019, +0.719, −0.695)* ^{T}*,

**v**

_{2}= (−0.509, −0.605, −0.612)

*, and*

^{T}**v**

_{3}= (−0.860, +0.342, +0.378)

*. According to the Gauss elimination method, we can obtain the transfer matrix of*

^{T}**T**

^{†}that has nine variables in Eq. (5) as follows:

*φ*

_{23}= 0.823,

*θ*

_{23}= 1.571,

*φ*

_{13}= 0.598,

*θ*

_{13}= 0.000,

*φ*

_{12}= 1.064,

*θ*

_{12}= 1.571,

*θ*

_{1}= 0,

*θ*

_{2}= 1.571, and

*θ*

_{3}= 1.571. These values correspond to refractive index changes of Δ

*n*

_{1}= 2.94 × 10

^{−4}, Δ

*n*

_{2}= 1.37 × 10

^{−3}, Δ

*n*

_{3}= 7.48 × 10

^{−4}, Δ

*n*

_{4}= 9.94 × 10

^{−4}, Δ

*n*

_{5}= 2.20 × 10

^{−4}, and Δ

*n*

_{6}= 2.77 × 10

^{−4}, where Δ

*n*corresponds to the label in the ring in Fig. 5 (here,

_{i}*θ*

_{1}∼

*θ*

_{3}are omitted). This numerical result, obtained by 2D finite element analysis, demonstrates the validity of our configuration. For a 3 × 3 unitary operation, the footprint for the optical waveguide is only ∼ 0.1 mm × ∼ 0.05 mm. Although the actual device size becomes larger by mounting the electrode and TiN heater that changes the refractive index, it is indicated that a drastic miniaturization can be realized by using microring resonators as the controllable power dividers and PSs. We would like to emphasize that, even if the lightwaves mixed by the other unitary transfer matrix are input, the MIMO compensation can be realized by appropriately adjusting the refractive index change.

## 5. Conclusion

An ultrasmall design for a universal linear circuit is realized by using microring resonators, and the validity of this design is confirmed. In the proposed design, we construct the power divider and phase shifter by using a four-port double ring resonator and cascaded two-port single ring resonators, respectively. They can work with the refractive index change of ∼ 0.005 and in the footprint of ∼100 µm × ∼40 µm, resulting in scale reduction ∼1/10. Such a miniaturized design helps to improve the scalability of integrated universal linear circuits, and it becomes an important step forward to realistic application for optical communication and computing.

## Funding

Japan Society for the Promotion of Science (19K15037).

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