1. Introduction

Optical unitary converters (OUCs) can convert mutually orthogonal input optical modes into arbitrary output modes. OUCs are attracting increasing interest in various application areas, including optical communication [1–4], neural networks [5–8], and quantum computing [9–11]. To this end, various types of integrated OUCs have been demonstrated on the Si [2–5,10–13], silica [9,14], and InP [15–17] photonic integration platforms. The conventional architecture is based on the meshed Mach-Zehnder interferometers (MZIs) [18,19], where arbitrary transformation is achieved by configuring the phase shifters. These MZI-based OUCs, however, face an issue that each MZI needs to operate with precise 50:50 splitters [6,20]. This severe fabrication tolerance may limit the scaling of these OUCs, especially on the InP platforms, where integration of compact and accurate MZIs are challenging. On the other hand, the monolithic InP platform provides a unique advantage of integrating various active and nonlinear optical components, such as lasers, semiconductor optical amplifiers (SOAs), and high-speed photodetectors (PDs), with minimal insertion losses [21–23].

To solve this problem, we have recently demonstrated an integrated monolithic InP OUC with cascaded N×N mode-mixing layers and phase shifter arrays [17]. The basic mechanism relies on the concept of multi-plane light conversion (MPLC) [24], which has originally been demonstrated using free-space optics [25,26]. In our photonic integrated implementation, the device consists of N-port multimode interferometers (MMIs) and phase shifter arrays arranged in N stages, which could be fabricated with a significantly less complexity even on InP, compared with the MZI-based scheme, requiring dense integration of N(N-1)/2 MZIs.

Here, we propose and experimentally demonstrate a novel type of OUC that can further reduce the footprint and complexity of the MPLC-based InP OUC. We utilize the unique redundancy of the MPLC method that the mode-mixing layer may be implemented by other types of linear device, provided that its operation is described by a sufficiently dense unitary matrix [12,24–26]. Therefore, instead of using a uniform MMI as in the previous work [4,17,27], we propose to employ a so-called half-integer MMI or an over-lapping-image MMI [28,29], which provides non-uniform splitting among the ports, but still is a unitary device. Since this half-integer MMI has half the length of the uniform MMI, the footprint of the mixing layer can be reduced by a factor of two without sacrificing the OUC performance. For proof of concept, we fabricate a 4×4 OUC with the half-integer MMIs on InP and experimentally demonstrate reconfigurable unitary conversion.

2. Principle of the MPLC-based OUC using MMIs

The MPLC-based OUC using MMIs is illustrated schematically in Fig. 1(a). It consists of cascaded phase shifter arrays and N-port MMIs. The matrix U describing the transformation of the complex amplitudes from N input ports to N output ports of the entire OUC chip is expressed as

It is important to note that the unitary matrix M, responsible for scrambling the complex amplitudes, does not necessarily have to be a specific transformation such as Fourier transform or uniform N×N splitter as demonstrated in the previous work [4,17,27]. Instead, it is only required to be a sufficiently dense unitary transformation [12,24–26]. Therefore, we expect that the uniform N-port MMI used in the previous work may be replaced with a much shorter MMI, as long as it provides sufficient coupling among all N ports.

Figure 1(b) describes a schematic of a general N × N MMI, where we define x and z coordinates as shown in the figure and W is the effective MMI width. For simplicity of explanation, we assume N to be an even number, but a similar result can be derived for the odd case as well [28,29]. The position of the input and output ports are expressed as ${x_m} \equiv W({2m - 1} )/2N$ ($m = 1,\; 2,\; \ldots ,\; N$). We define ${L_0} \equiv 4{n_{eff}}{W^2}/\lambda $ to represent the self-imaging length of the MMI, where ${n_{eff}}$ and $\lambda $ are the effective refractive index and the wavelength, respectively. At $z = {L_0}$, input optical field $E({x,0} )$ is reproduced within the range of $0 \le x \le W$, corresponding to the real MMI region [29]. More generally, at $z = {L_0}/q$, where q is a positive integer, the field is represented as [29]

 

Fig. 1. (a) Schematic of the MPLC-based OUC using cascaded MMIs. (b) Schematic of general N×N MMI.

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In the conventional N × N MMIs, where the input light from all N ports is split equally among the N output ports, the MMI length is set to L₀/N, so that q = N. In this case, two terms inside the summation of Eq. (2) do not overlap and constitutes N equally spaced copies at the output ports. On the other hand, if we do not have to split the power uniformly, the MMI length can be reduced to half by setting q = 2N. In this case, two terms in Eq. (2) overlap at the output ports and interfere to generate non-uniform power distributions [28,29]. Nevertheless, the total power from all N outputs is constant so that we can achieve unitary coupling, which is exactly what we need in this work. We can therefore expect that the MMI length can be reduced to half without causing essential penalty to the OUC performance.

3. Numerical simulation

To confirm these characteristics, first, we numerically simulate wave propagation inside an MMI by the eigenmode expansion (EME) method [30]. Figures 2(a) and 2(b) show the top and cross-sectional designs of the 4×4 MMI on InP, considered in this work. Figures 2(c) and 2(d) show the transmittance to all output ports for various MMI length L, simulated at 1550-nm wavelength. We can see that when L = 252 µm and 504 µm, the light is split without excess loss, corresponding respectively to the non-uniform and uniform unitary splitters. Figures 2(e)-(h) show the field distribution inside the MMI for L = 504 µm [Fig. 2(e) and (f)] and 252 µm [Fig. 2(g) and (h)]. We can confirm that when L = 252 µm [Fig. 2(g) and (h)], although the input power is split non-uniformly, all power is coupled to the four output ports without loss. In this case, the MMI is a half-integer MMI and its footprint can be reduced to half compared with the uniform MMI.

 

Fig. 2. Numerical analyses of 4×4 MMI on InP. (a) Top and (b) cross-sectional structures. (c, d) The transmittance to all output ports (Out 1, 2, 3, 4) and their sum (Total) when the light is input from port 1 (c) and port 2 (d), calculated for various MMI length L. At L = 252 µm and 504 µm, the light is split without excess loss, corresponding respectively to the non-uniform and uniform splitters. (e-h) The wave propagation inside the MMI for L = 504 µm (e, f) and 252 µm (g, h) from the input port 1 (e, g) and port 2 (f, h).

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We then simulate the performance of the entire 4×4 OUC as shown in Fig. 1(a) with the uniform and half-integer MMIs, designed in Fig. 2. We set a random 4×4 complex unitary matrix as the target matrix ${\mathbf U^{\prime}}$ that we want to realize. Then, all phase shifters in the OUC are optimized through the simulated annealing algorithm, so that ${\mathbf U}$ in Eq. (1) approaches ${\mathbf U^{\prime}}$. More specifically, we minimize the mean-squared error (MSE), defined as

 

Fig. 3. Simulated performance of 4 × 4 OUC with uniform (blue) and half-integer (red) MMIs. Average mean-square error (MSE) of the obtained unitary operation after optimizing the OUCs with different number of stages K. The bars represent the distributed results for 100 random unitary matrixes, tested as the target unitary operations.

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4. Experimental results

To investigate the validity of the proposed concept experimentally, we fabricated a proof-of-concept 4 × 4 OUC on InP and tested its performance in several mode-sorting scenarios as the example cases of unitary operation.

4.1 Device fabrication

A 4×4 OUC with K = 4 and the half-integer MMI designed in Fig. 2 was fabricated on InP for experimental demonstration. We employed 800-µm-long p-i-n InP/InGaAsP phase shifters, operated under carrier injection to induce efficient phase shift through the band-filling and free-carrier plasma effects in InGaAsP. The designed OUC was fabricated by electron-beam lithography and inductively-coupled-plasma reactive-ion etching followed by SiO₂ deposition. The electrodes of the phase shifters were patterned by evaporating Ti/Au and lift-off process. Figure 4 shows the micrograph and scanning electron microscope (SEM) images of the fabricated OUC. The entire circuit (excluding the electrode pads) fits inside a 2×13 mm² footprint.

 

Fig. 4. (a) Fabricated OUC using half-integer MMI on InP platform. (b-d) The SEM image of (b) cross-sectional image at passive waveguide, (c) top-view of the 4×4 MMI, and (d) cross-sectional image of the phase shifter.

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4.2 Measurement

The experimental setup and procedures were similar to those reported in [4]. In addition, we employed silica-based pitch converters to reduce the array pitch from 127 μm at the fiber array to 30 μm at the InP chip for efficient coupling [32]. A microcontroller was employed to sequentially select one of the four input ports and to collect the signals from four photodetectors to measure the transmittance to all output ports. After convergence, all phase shifters were set to the optimized condition. A continuous-wave light at 1550 nm wavelength and transverse-electric (TE) polarization mode was employed. The total fiber-to-fiber loss was measured to be around 35 dB, which consists of the coupling loss of 9.5 dB/facet, the propagation loss of 5.8 dB/cm, excess loss at the MMI of 1.5 dB/interferometer, and additional loss of around 2 dB including the phase shifter insertion loss.

Figure 5(a) shows the measured transmittance from all input ports to all output ports before tuning the phase shifters. Due to the strong coupling inside the chip, severe crosstalk is observed at the output ports. In contrast, Fig. 5(b) shows the result when the phase shifters are set to switch {In1, In2, In3, In4} to {Out 1, Out 2, Out3, Out4}. Similarly, Fig. 5(c)–5(f) show the results under different configurations. The crosstalk and MSE are suppressed below -5.9 dB and -14.6 dB in all cases, respectively. The discrepancy from the numerical simulation is attributed to the large insertion loss and the insufficient resolution of the driver circuit, which can be improved in future. We should note that the obtained performance in this work is comparable to the previous work using the uniform MMIs [17].

 

Fig. 5. Measured transmittance at the wavelength 1550 nm from each input and output port (a) before optimization, after optimization from {In1, In2, In3, In4} to (b) {Out1, Out2, Out3, Out4}, (c) {Out1, Out3, Out2, Out4}, (d) {Out1, Out4, Out2, Out3}, (e) {Out2, Out1, Out3, Out4}, and (f) {Ou41, Out3, Out2, Out1}. The crosstalk and MSE after optimization in the respective cases are (b) (-6.1 dB, -15.8 dB), (c) (-6.1 dB, -14.9 dB), (d) (-6.1 dB, -14.8 dB), (e) (-6.2 dB, -14.7 dB), and (f) (-5.9 dB, -14.6 dB).

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5. Conclusion

In conclusion, we have numerically and experimentally demonstrated a novel reconfigurable OUC using half-integer MMIs and phase shifter arrays on InP. By optimizing the phase shifters, we experimentally realized 4-mode sorting as well as mode switching. Since the MPLC method only requires a dense unitary matrix, the OUC with half-integer MMIs allows us to shrink the MMI size by half without inducing a fundamental penalty to the OUC performance.