A Novel Working Scheme for Robust Optical Coupler Unit for Programable Unitary Matrices

Programmable optical unitary matrices (POUMs) in photonic integrated circuits (PICs) technology play a critical role in modern information technology. With the expansion of their scale and integration density, POUMs have their performance limited by fabrication errors and tuning phase errors of their building blocks, i.e., tunable 2 × 2 optical couplers. Here, we provide an in-depth theoretical analysis of a prior robust equivalent 2 × 2 optical coupler and demonstrate a general requirement of its passive beam splitters' split ratios, under which an arbitrary tunable transmission ratio can always be achieved. More importantly, we analyze the impact of the phase shifters' tuning phase errors which is always caused by the limited voltage/current resolution of the electronic driving interface or thermal fluctuation. By the analyzed result, we propose a novel working scheme for the first time by discovering an unconventional configuration with large beam split ratio errors, under which robust POUMs against fabrication and tuning phase errors can be achieved.

Unfortunately, it is almost impossible to keep all the optical beam splitters perfect during manufacturing process. Besides, optical tuning phase errors due to phase shifters will accumulate and significantly reduce the performance of the POUMs for many applications [8]. A common practice for reducing the impact of those errors in as-fabricated POUMs is via algorithm driven closed loop optimization such as nonlinear optimization [9], gradient descent [10], and self-configuration method [11], [12]. Recently, adding an additional beam splitter on the original MZI structure to obtain certain robustness of POUMs against fabrication errors has attracted the interest of researchers [13]. However, the effect of the tuning phase error from the phase modulators on the POUMs has yet to be studied. Understanding it helps discover new schemes where the performance degradation attributed to the phase error can be avoided or alleviated. Here, based on the device presented in [14], we discover its unconventional operation configuration which allows the POUMs have a more excellent fabrication and tuning phase error tolerance.
The original device proposed by Miller group has been demonstrated theoretically and experimentally [7], [15]. Almost at the same time, Suzuki group proposed a similar device but using only one VBS instead of two which has been experimentally shown to have an ultra-high extinction ratio [14]. Compare to the former structure by the Miller group, this structure has one less modulator. However, an in-depth theoretical analysis of the robustness of this structure has not been conducted so far. In this work, we analyze this particular device theoretically and demonstrate the requirement of the split ratio error of all passive beam splitters for achieving an arbitrary tunable transmission ratio. In addition, we analyze the impact of the tuning phase errors of the phase shifters on the stability of the power transmission ratio. Our result shows that larger split ratio error of beam splitters can allow for greater tuning phase error tolerances. This finding is beyond the conventional design methodology for MZIs. Based on our theoretical analysis, we propose a novel design configuration for designing POUMs with robust performance against fabrication and tuning phase errors.

II. ARCHITECTURE ROBUSTNESS
A conventional tunable 2 × 2 optical coupler usually consists of two beam splitters (LBS and RBS, respectively) and a phase shifter (PS), as shown in Fig. 1(a). Following the method in [7], we theoretically analyze this device under a nonideal circumstance: both beam splitters have transmission loss and split ratio error due to fabrication errors. We further define a variable ρ L , This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ representing the normalized total power transmittance of the LBS, which satisfies where t L and k L represent the optical field transmission and cross-coupling coefficient of the LBS, respectively. ρ R of the RBS has the a similar definition as ρ L , i.e., ρ R = |t R | 2 + |k R | 2 ≤ 1.
Using transfer matrix method, when light is coupled into the top left port of LBS, the output power from the top right port of RBS, P T , is (2) where θ is the phase difference between the two optical paths of the optical couplers, including an intrinsic part determined in fabrication and a tunable part controlled after fabrication by the PS. φ is the constant phase of t L t R k L k R .
We further define Δ L = 1/2 − |t L | 2 /ρ L and Δ R = 1/2 − |t R | 2 /ρ R for LBS and RBS, which represent the split ratio error, i.e., the difference between 1/2 and the normalized power transmission ratio, e.g., |t L | 2 /ρ L , respectively. Based on (1), the range of Δ L and Δ R ∈ [−0.5, 0.5]. Using this definition, the normalized power transmission ratio of the optical coupler, T , i.e., T = P T /ρ R ρ L , can be expressed as where Notice that T is maximized only when Θ = π. We here first assume that there are some physical methods to adjust the split ratio of LBS and RBS which we will discuss later. To achieve a normalized-one power transmission ratio, i.e., T = 1, (3) needs to satisfy max{T Θ=π } = 1. Based on the above definition of Δ L and Δ R , we have Hence, max{T Θ=π } = 1 happens when and only when (4) takes the equal sign, i.e., Δ L = Δ R . Similarly, T is minimized only when Θ = 0. To achieve a normalized-zero power transmission ratio, i.e., T = 0, (3) needs to satisfy min{T Θ=0 } = 0. Since min{T Θ=0 } = 0 happens when and only when Δ L = −Δ R . From the above analysis, achieving an arbitrary normalized power transmission ratio from 0 to 1 requires Δ L = Δ R = 0, meaning both LBS and RBS must have a zero split ratio error or, equivalently speaking, a perfect 50:50 split ratio which is the requirement of a conventional tunable coupler. In practice, the inevitable fabrication error will discourage this strict condition. Therefore, such a configuration is difficult to implement reliably in reality A robust architecture for arbitrary tunable optical couplers is to replace one of the two non-adjustable beam splitters with a variable beam splitter (VBS) [14]. Here, we use one VBS to replace the LBS as shown in Fig. 1(b), and notice that the above analysis still hold by replacing Δ L with Δ V , which is the effective variable split ratio error of VBS. Since, the VBS is equivalent to the conventional 2 × 2 optical coupler analyzed above, Δ V is derived from (3) as follows, where the definition of Δ V,L and Δ V,R is similar to Δ L and Δ R , and Θ V is similar to Θ above. The operating principle of this device is to adjust the split ratio of the VBS and make it equal to Δ R for the normalized-one transmission ratio and −Δ R for the normalized-zero transmission ratio. The device architecture is robust under a configuration where the following two conditions are satisfied: (6) into the two expressions of Δ V = ±Δ R , shifting the terms and squaring both sides yield the following expression, (7) takes positive sign for Δ V = Δ R and negative sign for Δ V = −Δ R , which can be rearranged as Condition (ii), from (3), for an arbitrary Θ, To be able to obtain all normalized power transmission ratio values from 0 to 1, the two range of T above must overlap, meaning i.e.,|Δ R | ≤ 1/8. In summary, if the split ratio errors of all passive beam splitters in this robust architecture satisfy (8) and |Δ R | ≤ 1/8, an arbitrary normalized transmission ratio from 0 to 1 can always be achieved.

III. IMPACT OF TUNING PHASE ERRORS
The stability of the transmission ratio is a key parameter affecting the performance of POUMs composed of arbitrary tunable optical couplers. The significant factor affecting it is the tuning phase errors of the PSs, which could be caused, for instance, by the limited voltage/current resolution of the electronic driving interface or thermal fluctuation. We next analyze the impact of the tuning phase errors of the two PSs in Fig. 1(b) on the stability of the power transmission ratio, respectively.
First, the impact of the tuning phase error of the PS1, δ Θ V , can be expressed as here S and P represent the value of Θ V and Θ with which the power transmission ratio is T 0 , i.e., T 0 = T (Θ V = S, Θ = P ).
From (3), (6), considering the two conditions, Δ V = ±Δ R , we can get here To minimize the impact of the tuning phase error of PS1, the right side of (10) needs to be equal to zero. This condition is satisfied when either Δ R = 0 or Next, the impact of the tuning phase errors of the PS2, δ Θ , can be described similarly as here S, P , T 0 are the similar definition as above. From (3), (6), we can find here M has the same definition as above. Eq (13) indicates that the stability of the transmission ratio of the tunable optical coupler is less affected by the tuning phase error of PS2 for Δ R with a big value. When taking into account the impact of the tuning phase errors of both PS1 and PS2 simultaneously, Δ R should be maximized while (11) is satisfied. Notice that |Δ R | ≤ 1/8 (derived at the end of above Section). Substitute |Δ R | = 1/8 into (11), and its solutions are Δ V,R = 0, Δ V,L = Δ R = 1/8 and Δ V,L = 0, Δ V,R = Δ R = 1/8.
The two solutions, corresponding to two device configurations, are equivalent, so we use the first configuration as an illustration for the following analysis. The robustness of the device under this configuration is analyzed by studying the deviation of the transmission ratio from a nominal value under various phase errors. As a comparison, a conventional design with Δ V,R = Δ V,L = Δ R = 0 (corresponding to zero split ratio error or three perfect 50:50 power splitters) is also studied. The corresponding numerical simulation result, for a targeted normalized power transmission ratio of 0.3 as an illustration, are shown in Fig. 2(a) and (b). To compare the phase robustness of these two configurations, the area enclosed by boundary curves for the same transmission ratio error is calculated. It is found that the area for our configuration is always larger than that of configuration one, meaning a large phase-error space is allowed. This observation agrees well with our theoretical analysis above, indicating our new design is much more tolerant against tuning phase errors. If, for example, a transmission ratio error of 0.01 can be tolerated, our design would be 80% more tolerant than the conventional design, as indicated by the area enclosed by the two yellow lines in Fig. 2(a) and (b). It is also worth noting that the transmission ratio error is less insensitive to the tuning phase error of PS1, Θ V , for both designs since they both make the value of (10) equal to zero. However, our design is much more tolerant against the tuning phase error of PS2 because the value of (13) is much smaller than that of the conventional design configuration. According to the above analysis, the best theoretical design is for split ratio errors of PSs derived from the boundary values of the solution of (8). However, considering practical fabrication errors, their designed value should be placed slightly inside the boundary to ensure that the actual values of the as-fabricated devices are close to the boundary but do not fall out of the boundary. For instance, a possible set of designed split ratio errors could be Δ V,R = 0, Δ V,L = Δ R = 1/8 − 0.05. In this case, Eq.(8) will still hold if the deviation errors of the split ratio errors of the as-fabricated beam splitters lie in the range of ±0.04, which is an exaggerated value compared with the current state-of-the-art fabrication error for beam splitters [16], i.e., ±0.025. Within this range, the worst possible case for our configuration is when Δ V,R = 0.04, Δ V,L = Δ R = 1/8 − 0.09, under which the (10) and (13) is maximized, and the simulation result is shown in Fig. 2(c). The phase error space, enclosed by the yellow lines in Fig. 2(c), is still 33.2% larger than that in Fig. 2(b) for a transmission ratio error of 0.01, implying that the worst case of our novel design configuration is still much more robust than the conventional design configuration.

IV. PROGRAMABLE UNITARY MATRICES
The immediate application of our robust tunable 2 × 2 optical coupler is an optical network for analog optical computing. As an illustrative example, consider the matrix decomposition proposed in [17], which is widely used today to construct physical POUMs. The corresponding schematic layout is illustrated in Fig. 3(a) for N × N (N = 8 in the figure) POUMs. Its main building block, the blue part in Fig. 3(a), usually consists of a conventional tunable optical coupler and a phase shifter, shown in Fig. 3(b). The robust scheme for POUMs takes advantage of our robust tunable optical coupler, as shown in Fig. 3(c), instead of the conventional tunable optical coupler.
The performance of the POUMs is quantified by a fidelity function that evaluates the fidelity between the target unitary matrix U 0 and the constructed unitary matrix U with the optical network. It is defined as [17] where   To study the robustness of the POUMs against the deviation of split ratio errors, a numerical optimization method named basin-hopping in the SciPy python library is used mainly because it is particularly useful when the function has many minima/maximum separated by large barriers. It searches for a global maximum of the fidelity function over the space of the phase vectors under the different maximum allowed deviation from the nominal split ratio errors. The corresponding simulation result is shown in Fig. 4(a). It indicates that the fidelity of the two structures with robust building blocks (RBBC and RBBN) is very close to 1 and thus hardly compromised by the potential deviation of split ratio errors caused by fabrication errors. In addition, to study the impact of tuning phase errors of the POUMs, the deviation of split ratio errors of the beam splitters is first assumed to be as large as 0.04, corresponding to the worst case in Fig. 4(a). Fig. 4(b) shows the variation of the fidelity function as a function of tuning phase errors. It is evident that the fidelity of the structure with RBBN is greater than that of RBBC under the same tuning phase error. The above analysis suggests that our proposed novel design configuration is robust for both fabrication and tuning phase errors in implementing the POUMs. Additional studies are conducted to examine the possibility of scaling up POUMs to higher dimensions using our proposed design configuration. The corresponding results are summarized in Table I. The F SRE represents the average fidelity under a maximum split ratio error of 0.04, and the F P E+SRE represents the average fidelity under a maximum tuning phase error of 0.04π and split ratio error of 0.04. The data in Table I demonstrate that our novel design configuration continues to be more robust against tuning phase errors, even for very large-scale POUMs, while remaining robust to fabrication errors.

V. CONCLUSION
An in-depth theoretical analysis of a tunable 2 × 2 couplers is given in this study. From the obtained results, a novel design configuration with large split ratio errors of passive beam splitters or imperfect beam splitters is proposed, under which great tuning-phase-error tolerance for the tunable coupler is demonstrated. Based on this novel configuration, a robust programable optical unitary matrice is presented, which inherits the advantage of its building blocks with robustness against tuning phase errors while maintaining excellent fabrication tolerance.