1. Introduction

Programmable photonic integrated circuits have grown rapidly into the established and promising technology platform [1]. The versatile functionality of the photonic integrated circuits (PICs) is enabled by programming the tunable elements such as phase shifters to perform an arbitrary linear optical transformation to a given set of input modes. To realize arbitrary linear operations with the PICs, it is essential to achieve lossless optical power-splitting and phase modulation. State-of-the-art linear optical circuits typically use MZI-based meshes with tunable phase shifters deployed at the interferometer arms. The first photonics-based architecture that enabled the implementation of an arbitrary N×N unitary matrix was the triangular photonic circuit arrangement proposed by Reck et al. [2]. Further improvement on the circuit fidelity of such an optical unitary matrix generator was accomplished by Clements et al., which features a more compact arrangement of the MZI-based building blocks [3]. Recently, self-configuring optics has been proposed with a straightforward algorithm employing universal linear optics, enabling the construction of non-unitary matrix [4].

MZI-based programmable PICs have mainly been utilized in three applications: optical neural network, mode-division (de)multiplexing, and linear optical quantum computation. Beginning with the first implementation of an optical neural network using an MZI network [5], efficient training algorithm for optical neural networks that involves adjoint variable method was proposed [6] to avoid the massive electronic computation-based training process. Recently, algorithms on initialization [7] and column-by-column programming [8] of the MZI-based architecture have also been developed, paving the path towards efficiently programmable and error-minimized photonic hardware for machine learning applications. Alongside the algorithmic development for linear optical matrix multiplication for neural networks, an optical component enabling arbitrary optical nonlinear activation functions was proposed theoretically [9] and was demonstrated based on silicon nitride waveguides [10]. Mode-division (de)multiplexing is another application that has gained interest, and 4×4 self-configuring universal linear optics was demonstrated [11,12] where all-optical mode reconstruction was shown for a specific operation wavelength [12]. Beyond the classical optics domain, the MZI-based architecture has also been applied to quantum photonic computation. For example, arbitrary linear operations of the high-dimensional quantum state (i.e., six photon processor) on the silica-on-silicon platform was successfully demonstrated [13], 16-dimensional entanglement generation and manipulation based on the path-encoded qubit was shown on the silicon-on-insulator (SOI) platform [14], and a two-qubit processor on SOI platform was also demonstrated [15], forecasting a possible probabilistic operation-based near-deterministic multi-qubit chip-scale quantum photonic processor. Recently, a programmable quantum photonic architecture involving waveguide-embedded quantum emitters that employ two-photon interactions was proposed, suggesting path-encoded qubit-based deterministic programmable quantum photonic computing [16]. Other interesting applications include quantum transport simulations [17] and recurrent Ising machines [18] that may shed light on solving NP-hard problems based on optics with supremacy over electronics-based hardware.

Such MZI array-based optical signal processors have mostly been based on FIR systems with symmetric MZI arm lengths for broadband operations. For channelized multiple wavelength operations, however, broadband universal linear optical systems based on symmetric MZIs might not be suitable for separate and independent manipulation of individual wavelength channels. To implement universal linear optics independently operating at M different input wavelengths using MZI architectures in PICs, M sets of N×N photonic circuits, as well as a wavelength demultiplexer, are necessary where some interconnect bottlenecks are expected. On the other hand, an infinite impulse response (IIR) optical signal processor based on cascaded resonators with different resonance wavelengths can modulate light only at a particular set of wavelengths (i.e., resonance wavelengths separated by the free spectral range (FSR)) while preserving the input light at all other non-resonant wavelengths [19]. Compared to the MZI-based FIR schemes, resonator-based IIR systems might enable independent and separate modulation of multiple wavelength channels within the FSR in a smaller footprint with lower energy consumption, where we may be able to consolidate multiple inputs to a wavelength-multiplexed waveguide and overcome some interconnect bottlenecks.

In this paper, we propose a resonator-based programmable universal linear optics circuit model consisting of cascaded resonators and access waveguides. Although most universal linear optics designs rely on MZI-based architectures, programmable beam splitting can also be achieved by employing the well-known add-drop configuration of an optical resonator with two access waveguides. By tuning the coupling coefficients between the resonator and the access waveguides, we show that it is possible to change the optical power split ratio in each resonator at resonance while off-resonance wavelength channels are not significantly influenced by the resonator’s add-drop configuration. Phase-shifting can also be implemented by taking advantages of the all-pass configuration of the resonator-waveguide system in the strongly coupled regime, exploiting the drastic phase shift near resonances while avoiding large-amplitude losses [20,21]. Starting from the theoretical modeling of a general optical resonator (e.g., photonic crystal cavity, micro-disk, and micro-ring) with two access waveguides, we present multi-wavelength versions of the achromatic universal linear optics construction algorithms. We adopt experimental values reported for micro-ring resonator to analyze the fidelities of different types of 2×2 unitary transformations and higher-dimensional unitary transformations was analyzed by employing multi-wavelength versions of the Reck and the Clements algorithm. We adopt experimental values reported for micro-ring resonator on the SOI platform [22] to examine the fidelities of different types of 2×2 unitary operators and higher-dimensional unitary transformations. The Reck and the Clements algorithm were investigated on our proposed multi-wavelength photonic integrated circuitry for the construction of higher-dimensional unitary transformations.

2. Constructing a 2×2 arbitrary optical unitary operator via cascaded resonators

2.1. Fundamental building blocks employing a general resonator with access waveguides

Optical resonators with access waveguides can be analyzed by various methods, such as spatial coupled-mode analysis [23] and lumped oscillator analysis [24], and we employ the temporal coupled-mode theory for our analysis.

Figure 1 illustrates the two fundamental building blocks to construct a 2×2 arbitrary optical unitary operator employed in this paper: the add-drop configuration with two access waveguides (i.e., 4 Port system) and the all-pass configuration with a single access waveguide (i.e., 2 Port system). The first block described in Fig. 1(a) functions as a 2×2 optical amplitude splitter at the resonance wavelength, and the second block shown in Fig. 1(b) operates as a phase shifter.

 

Fig. 1. Illustrations of (a) the 2×2 optical power-splitter ‘A’ and (b) the 2×2 optical phase-shifter ‘P’. The gray wires indicate optical waveguides and the purple blocks indicate general resonators. The matrix representations of both blocks at the operation wavelength are shown below the illustration. The phase introduced by the propagation in both blocks is assumed to be a multiple of 2π to solely consider the local operation of each block.

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A 2×2 optical power splitter can be implemented by an optical resonator at the add-drop configuration with two access waveguides. At resonance, the amplitude of every matrix components in the 2×2 optical power-splitting matrix becomes a function of the loaded quality factors and the intrinsic quality factor. If the intrinsic quality factor is about two orders larger than the loaded quality factors, nearly lossless optical power-splitting can be assumed. Under this condition, we can express the matrix components with variables p=(Q_eA−Q_eB)/(Q_eA+Q_eB) and q=−2(Q_eAQ_eB)^1/2/(Q_eA+Q_eB). Here, Q_o is the intrinsic quality factor of the resonator in Fig. 1(a), and Q_eA and Q_eB are the loaded quality factors introduced by the access waveguides transmitting the propagation mode A and mode B, respectively. Note that we operate under over-coupled regime where Q_o>>Q_eA, Q_eB is satisfied (see Supplement 1 Section A.1), and therefore ruled out the situation where the intrinsic and loaded Q factors are comparable. By cascading resonators of different resonance wavelengths, we can manipulate the optical power-splitting at different wavelengths independently, which could not be done by employing the MZI-based optical power splitters.

Optical phase shifter can be implemented by utilizing the all-pass configuration of an optical resonator that is strongly over-coupled to a single access waveguide [20,21]. By tuning the resonance wavelength by changing the refractive index of the resonator, we can manipulate the phase (i.e., θ in the matrix for the phase shifter in Fig. 1(b)) of the optical mode A at the particular wavelength near the resonance wavelength. Compared to the conventional optical phase-shifting mechanisms employed at the interferometric arms of the MZI (e.g., carrier depletion, carrier injection, thermo-optic effect) where they inevitably affect the optical phase regardless of the input wavelength, optical resonator-based phase-shifting allows operation only near the specific resonant wavelengths without affecting other wavelength channels. Non-resonant wavelengths would pass through the resonator-based phase shifters without additional phase delays. Therefore, independent phase shifting of multiple wavelength channels could be implemented simply by cascading resonators with different resonance wavelengths to the identical access waveguide. Resonators were generally not considered as a tool for acquiring a pure phase shift either due to the small phase change or the accompanied amplitude modulation. However, when the access waveguide is strongly coupled to the resonators, nearly 2π phase shift (i.e., θ varies from 0 to 2π) can be obtained with minimum power loss within a small spectral range [20,21]. The physical origin of this optical phenomenon can be explained by the Fano resonance, which can also be modeled by the temporal coupled-mode theory [25]. The experimental demonstration was recently performed with the silicon nitride optical micro-ring resonator, utilizing thermal heaters to tune the resonance wavelength of the optical micro-ring resonator.

2.2. 2×2 arbitrary optical unitary operator based on cascaded resonators

The starting point to implement universal linear optics is to build a 2×2 arbitrary optical unitary operator. In this section, we show that a 2×2 arbitrary optical unitary operator can be constructed with the resonator-based fundamental building blocks ‘A’ and ‘P’ that we introduced in the previous section (Fig. 1).

The general expression for an arbitrary 2×2 unitary operator is given by [1]

We first replace the sine and cosine of the matrix in Eq. (1) with the p and q of our matrix expression for the building block ‘A’. Because p²+q²≈1 and the tunable range of p and q is from 0 to -1, we can set sin(α/2)=-p and cos(α/2)=-q. Then, we substitute the phase shifter matrix with two cascaded ‘P’ blocks. Although technical advancements might suggest a single resonator-based phase shifter that can cover the entire 2π phase shift, state-of-the-art experimental demonstration based on silicon nitride micro-ring resonator shows a 1.64π phase shift [20]. For this reason, we allocate two ‘P’ blocks for implementing the phase shifter matrix as shown in Fig. 2 to ensure that the entire 2π phase shift is covered. We define the total phase shift with a single parameter θ (i.e., θ=θ₁+θ₂), where we assume the tunable range of θ to be 0 to more than or equal to 2π. Now, the only difference between our cascaded resonator-based unitary matrix and the unitary matrix in Eq. (1) is the sign. This difference in the sign is nothing but an additional π phase shift to the global phase (i.e., coefficient e^jπ multiplied to the entire matrix), and thus we conclude that our cascaded-resonator-based expression represents an arbitrary 2×2 optical unitary operator without loss of generality.

 

Fig. 2. An arbitrary 2×2 optical unitary operator, T (i.e., T = PPA), constructed by cascading the fundamental building blocks, ‘P’ and ‘A’, employed in this paper. θ₁ and θ₂ indicate the phase shift of the first and second phase shifter block, respectively.

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For the following discussions in this paper, we will refer to this cascaded resonator-based unitary operator as block T. For K different operation wavelengths, we will use T(λ₁), T(λ₂), …, and T(λ_K) for the blocks functioning at K different resonance wavelengths. Note that the block T(λ_i) is independent of the block T(λ_j) where j can be any integer between 1 and K, satisfying j≠i. This condition can be met by assuming that λ_j falls into the FSR of the resonator with resonance wavelength of λ_i where the normalized transmission is purely one at λ_j when passing through the resonator with resonance wavelength λ_i. We will employ this scenario throughout this paper.

Maintenance of the coherence between multiple optical modes can be an issue in this type of cascaded resonator-based optical matrix multiplication scheme, due to the high intrinsic quality factor that is required to ensure lossless power-splitting condition. However, setting the values of the loaded quality factors to the range of several hundred to several thousand, the total quality factor falls into the range where linear operations can be implemented without deteriorating the coherence between the multiple optical modes.

By employing respective sets of T operating at different resonance wavelengths (e.g., T(λ₁), T(λ₂) for two different operation wavelengths λ₁ and λ₂) that are matched to the corresponding wavelengths of each photon in a multi-qubit system (e.g., λ₁ and λ₂ for the two-qubit system), independent universal linear optical operations can be implemented to the individual qubit. For example, in the two-qubit silicon quantum photonic processer developed recently [15], we can maintain the same functionality of the FIR-based architecture by simply cascading two sets of T to a single photonic circuit. This reduces the necessity of both the wavelength demultiplexer at the input to route the single photons of different wavelengths and the separate photonic circuit configuration allocated for the independent linear operation of each qubit.

2.3. Fidelity of the cascaded micro-ring resonator-based circuit for a 2×2 unitary matrix

We take experimental values of the optical micro-ring resonator in SOI platform reported in [22] into account to further analyze the optical circuit model we proposed in this paper. L in Eq. (7) of Supplement 1 Section A.2 is sampled from a non-negative Gaussian distribution with a mean (standard deviation) of 0.85 (0.1) cm^-1, and γ_t in Eq. (8) of Supplement 1 Section A.2 is set to have a non-negative Gaussian distribution with a mean (standard deviation) of 5.16% (2.84%) [26]. 2×2 unitary matrices at a single operation wavelength were first considered, and thus we set γ_c to be zero. Specific kinds of 2×2 unitary matrices (i.e., single qubit gates) were adopted for the study and average fidelity [3] over 100 iterations were calculated for each case as shown in Fig. 3.

 

Fig. 3. Simulation of constructing single qubit gates with our proposed model. The orange histogram indicates the simulation results when coupling coefficients κ_A and κ_B were set to be in the range [0.01 0.9], and the cobalt histogram represents the simulation results when κ_A and κ_B were set to be in the range [0.001 0.9]. (a) Hadamard (b) Pauli-X (c) Pauli-Y (d) Pauli-Z (e) Pauli-Z with κ_A, κ_B ∈ [0.001 0.9], (f) Phase, (g) Phase with κ_A, κ_B ∈ [0.001 0.9], (h) π/8 gate, and (i) π/8 gate with κ_A, κ_B ∈ [0.001 0.9].

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Brute-force search algorithm was chosen for our paper when constructing the unitary operator where we set the coupling coefficients (i.e., κ_A and κ_B in Eq. (7)) to have 90 different values with equal intervals in the range [0.01 0.9] and the amount of phase shift (i.e., θ in Eq. (8)) to have 200 different values with equal intervals in the range [0.01π 2π], respectively. The results showed average fidelity of over 99% for Hadamard, Pauli-X, and Pauli-Y, and over 95% for Pauli-Z, Phase, and π/8 gate. Average fidelity was increased to over 99% for Pauli-Z, Phase, and π/8 gate when we set the coupling coefficients κ_A and κ_B to be in the range [0.001 0.9]. We also note that there exists plenty of room in achieving better fidelity when appropriate optimization strategies are adopted.

In [27], we reported a compact 2×2 optical circuit design based on cascaded micro-ring resonators. The footprint of the circuit model is at least comparable to [5,12], or an order of magnitude smaller than the unit cell of MZI-based model [15]. Further optimization of the design can be achieved by utilizing other types of optical resonators and devising a more compact layout of the components. This may enable large scale optical matrix multiplication in a smaller footprint compared to MZI-based architectures.

3. Multiple wavelength universal linear optical circuits

Generally, the mathematical relation between the N input modes and the M output modes of any linear optical device can be represented as E_out=RE_in, where E_out is the M×1 matrix of the output modes, R is the M×N matrix for the linear optical operation of the device, and E_in is the N×1 matrix of the input modes.

In linear algebra, any M×N matrix R can be decomposed into the multiplication of an M×M unitary matrix U, an M×N diagonal matrix D, and the conjugate transpose of an N×N unitary matrix V by the singular value decomposition (SVD) (i.e., R = UDV^†). Arbitrary linear functions can be implemented in optics provided that we can construct the arbitrary M×M (or N×N) unitary matrix and the M×N diagonal matrix. For the construction of an arbitrary M×M unitary matrix, the “Reck” encoding scheme was first proposed [2], and the “Clements” encoding scheme was recently developed for achieving better fidelity under lossy devices [3]. The architecture for the implementation of the operator R by the sequential adaptation of the phase shifters and the beamsplitters was also introduced recently with MZI-based building blocks [4]. Until now, these algorithms for the universal linear optics were developed for achromatic purposes.

In this section, we develop the multi-wavelength versions of the up-to-date algorithms for universal linear optics and propose the corresponding photonic circuit configurations. We employ our cascaded resonator-based building blocks to the algebraic derivations previously developed in the relevant work for each scheme and propose the multiple wavelength versions of the respective universal linear optics encoding algorithms.

3.1. Multi-wavelength versions of the “Reck” algorithm and the “Clements” algorithm

Both the “Reck” encoding algorithm and the “Clements” encoding algorithm are based on the decomposition of the M×M unitary matrix into the product of the T_m,n matrices [2,3]. The T_m,n matrix is an M-dimensional identity matrix with the matrix elements, I_mm, I_mn, I_nm, and I_nn substituted by the corresponding matrix elements of block T in Fig. 2. Thus, the T_m,n matrix consists of M-2 ones at the diagonal, M²-M-2 zeros, and the four matrix elements of T. T_m,n performs a unitary operation to the two-dimensional subspace of the M dimensional Hilbert space (i.e., the transformation between channels m and n), leaving the M-2 dimensional Hilbert space intact [2]. We use the same notation T_m,n used in [2,3] for our cascaded resonator-based building block, as shown in Eq. (2). Note that the Eq. (2) represents a particular case where m = n-1, and in this case I_mm, I_mn, I_nm, and I_nn are replaced by e^jθp, e^jθq, q, and –p, respectively.

To implement unitary operators for K different wavelengths, we employ T_m,n(λ₁), T_m,n(λ₂), …, and T_m,n(λ_K) that are independent to each other as explained in the previous section. The general expression for both the “Reck” encoding and “Clements” encoding developed previously involves the multiplication of T_m,n blocks at a single wavelength (e.g., T_m,n(λ₁)). Adding an additional degree of freedom (i.e., wavelength) to the achromatic expression is mathematically identical to introducing a new orthonormal basis $|{\Psi _i}$ (i=1, 2, …, K) to each M×M unitary operator at λ_i. The bra-ket notation, $|{\Psi _i}$ and $ {{\Psi _i}} |$ are the K×1 and 1×K vector respectively, where all matrix elements are zero except for the i^th matrix element that has the value of one. Thus, the general expression for M×M unitary matrices for K different wavelength can be written with a tensor product,

We now move onto the multiple wavelength versions of the “Reck” encoding algorithm and the “Clements” encoding algorithm to construct a multiple wavelength M×M unitary operator.

The “Reck” encoding algorithm is based on the idea of nullifying the off-diagonal components by multiplying T_m,n matrices. We take an M×M unitary operator and multiply the T_m,n matrices at the right to make all off-diagonal elements of the M^th column, and the M^th row zero. We repeat the process for the (M-1)^th column and row to the 1^st column and row to make the matrix a diagonal matrix, and finally, by multiplying the diagonal matrix D, we can obtain an M-dimensional identity matrix (i.e., UT_M,M-1T_M,M-2⋅⋅⋅T_2,1D = I). Multiplying the inverse matrices of T and D, we arrive at the unitary matrix U. Thus, the final expression for the “Reck” encoding theorem can be written as U=(T_M,M-1T_M,M-2⋅⋅⋅T_2,1D)^-1, where the inverse operation is simply equivalent to performing the experiment in reverse (i.e., taking the output ports as inputs and performing the same procedure). The resulting photonic circuit features the triangular configuration. We can add an additional degree of freedom, wavelength, to the “Reck” algorithm simply by replacing the single block T_m,n with the cascaded blocks of T_m,n(λ₁), T_m,n (λ₂), …, and T_m,n(λ_K). These blocks are independent of each other, and so we can employ the “Reck” encoding separately and independently to each channel with different operation wavelengths. The multiple wavelength version of the “Reck” algorithm and the corresponding photonic circuit configuration is shown in Figs. 4(a) and (b), respectively.

 

Fig. 4. The multiple wavelength version of the “Reck” encoding algorithm. H(λ) in the algorithm table is introduced to represent the summation in Eq. (3). (b) illustrates the photonic circuit configuration that employs the algorithm in (a), respectively, to implement a 5×5 (M=5) unitary matrix for K different wavelength channels. Each pink block represents T_m,n at each operation wavelength, i.e., T_m,n(λ₁), T_m,n(λ₂), …, and T_m,n(λ_K). I₁, I₂, …, I₅ indicate the input modes and O₁, O₂, …, O₅ indicate the output modes. Each input and output consists of K different values for K different wavelengths.

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The “Clements” encoding algorithm takes the same fundamental idea as the “Reck” encoding algorithm to nullify the off-diagonal matrix elements of the M×M unitary matrix, except this approach exploits both T_m,n blocks and T^-1_m,n blocks. The “Clements” encoding algorithm nullifies the successive diagonals of the M×M unitary matrix U from the bottom left (i.e., the matrix element U_M1) to the diagonal right before the diagonal of the M×M matrix by alternating the T_m,n blocks and the T^-1_m,n blocks to make an M-dimensional diagonal matrix (e.g., for a 5×5 unitary matrix, T_4,5T_3,4T_2,3T_1,2T_4,5T_3,4UT^-1_1,2T^-1_3,4T^-1_2,3T^-1_1,2=D). Thus, the unitary matrix can be obtained by U = T^-1_3,4T^-1_4,5T^-1_1,2T^-1_2,3T^-1_3,4T^-1_4,5DT_1,2T_2,3T_3,4T_1,2. If D consists of single-mode phase-shifts, we can always find D’ that satisfies T^-1_m,nD = D'T_m,n, and therefore, the expression for the unitary matrix U becomes U = D'T_3,4T_4,5T_1,2T_2,3T_3,4T_4,5T_1,2T_2,3T_3,4T_1,2. The corresponding photonic circuit configuration of the “Clements” encoding scheme exhibits a symmetric design, and such photonic circuit arrangement allows a better tolerance to the unbalanced loss between the T blocks [3]. Multiple wavelength version of the “Clements” encoding algorithm can be implemented by taking the same approach we took at the “Reck” encoding, and the algorithm and the corresponding photonic circuit configuration is introduced in Fig. 5(a) and (b).

 

Fig. 5. The multiple wavelength version of the “Clements” encoding algorithm. (b) illustrates the photonic circuit configuration that employs the algorithm in (a), respectively, to implement a 5×5 (M=5) unitary matrix for K different wavelength channels.

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3.2. Fidelities of the “Reck” algorithm and the “Clements” algorithm and issues for large scale optical matrix multiplications

Programmable photonic integrated circuits are often utilized for applications that require large scale optical matrix multiplication. Optical interchannel crosstalk and cascading losses are the two fundamental constraints for scaling up our system [28]. Although nonlinear crosstalk is expected in the conventional usage of optical resonators pursuing high Q factors, we set our ring resonators to operate in the relatively low loaded Q regime (∼1000), and therefore assume that the nonlinear effects are negligible in this paper. To further investigate the tolerance of our system to these two factors in large scale optical signal processing, we study the two canonical reconstruction algorithms (i.e., Reck and Clements) with the incorporation of the effect of optical interchannel crosstalk between different wavelength channels and cascading losses to large scale systems.

We generate 100 random unitary matrices [29], and calculate the average fidelity for the two different reconstruction algorithms. We adopted brute-force search algorithm for setting each tunable components (coupling coefficients and the amount of phase shifts) where the conditions for the value of coupling coefficients, the amount of phase shifts, the distribution of the loss terms L and γ_t were set identically to Section 2.3 (i.e., κ_A, κ_B ϵ [0.01 0.9]). We assume that γ_c in Eq. (7) and Eq. (8) of Supplement 1 Section A.2 is identical for every component in this paper. Figure 6 indicates that the Clements algorithm shows more robustness to scaling up the size of the optical circuit because of the balanced loss assured from the symmetric circuit configuration [3]. γ_c was assigned to have three different values (i.e., γ_c=0, 0.1, 0.2). As we increase the amount of loss induced by the optical interchannel crosstalk, drastic deterioration of fidelity occurred for both Reck and Clements algorithm. Furthermore, in the presence of optical interchannel crosstalk, fidelity gets worsened as we scale up the size of the optical circuit. Thus, optical interchannel crosstalk between different wavelength channels is a crucial factor to consider for high fidelity multi-wavelength operation based on our proposal. In order to implement simultaneous and independent multiple wavelength operations successfully, we must avoid the interplay between different wavelength channels by carefully locating the resonance regime of each set of optical resonators (fixed to a particular operation wavelength, e.g., λ_i) at the free spectral regime of another set of optical resonators (fixed to a different operation wavelength, e.g., λ_j≠i). For optical micro-ring resonators with free spectral range (FSR) of 18 nm and the total quality factor of 20000, negligible optical interchannel crosstalk was reported for the spacing of 14 channels, where an increase up to 50 channels was expected to be possible with micro-rings of 3 μm radius [19].

 

Fig. 6. Average fidelity of cascaded micro-ring-based universal optical circuit implementing (a) the Clements algorithm and (b) the Reck algorithm to construct random unitary matrices. Optical circuit size were varied from n=3 to n=10. Optical interchannel crosstalk was assigned to have three different values: γ_c=0 (red histogram), γ_c=1 (green histogram), and γ_c=2 (blue histogram).

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3.3. Multi-wavelength self-configuring universal linear optics

Another interesting and useful configuration scheme developed for the universal linear optics is the self-configuring optics, where the phase shifters and the optical power splitters are sequentially adapted to implement a general operator R for arbitrary linear optic devices [4]. The main advantage of this approach is that the implementation of arbitrary optical linear functions with high circuit fidelity for a large number of channels can be done without the detailed knowledge of the global phase (i.e., knowledge of the relative phase between the MZI blocks). Therefore, the coherence of the photonic circuit can be well-maintained even under propagation length mismatches between the blocks. With our cascaded resonator-based building block T, we show that the multiple wavelength version of the self-configuring optics can be implemented.

Each T(λ), operating at the wavelength λ, can be decomposed into two components, P(λ) and A(λ). P(λ) represents the two phase-shifting blocks at the wavelength λ, and A(λ) represents the optical power-splitting block in Fig. 2 at the wavelength λ, respectively. The phase θ (i.e., θ=θ₁+θ₂) of the block P(λ) will determine the relative phase between the two output channels and the amplitude coefficient p (note that p²+q²≈1, thus the optical power split ratio can be determined by the single parameter p) in A(λ) will determine the relative amplitude between the two output channels. Photodetector D can also be defined for each wavelength, and we will denote the photodetector at λ as D(λ).

We first show that the cascaded resonator model is capable of implementing the self-configuring universal beam coupling [30] for multiple wavelength channels. The phase shifters and the optical resonator-waveguide couplings implemented for different operation wavelengths are independently and sequentially adapted to form a single-mode beam for each wavelength without fundamental losses. For simplicity, we take an example for the case of three input modes (M=3) at two different wavelengths (K=2) (i.e., a total of 6 input channels), although the extension for M input modes is straightforward.

To couple the arbitrary input vector with three input modes operating at two different wavelength λ₁ and λ₂, we sequentially tune the parameters of the block T from the bottom left (i.e., T_1,1) to the top right (i.e., T_1,3) to make the optical powers detected at the photodetectors minimum (see Fig. 7 for the overall circuit diagram). Let us focus on the lossless beam combining of the input mode vector at λ₁. At the first stage, θ of P_1,1(λ₁) and p of A_1,1(λ₁) are sequentially tuned, to minimize the power detected at the photodetector D_1,1(λ₁) (this stage is optional because it is just equivalent to making the reflectivity to 1, i.e., simply pass the input I₁ to T_1,2). Then, the phase θ of P_1,2(λ₁) is adjusted to make the power at D_1,2(λ₁) to the global minimum, and next, we tune the amplitude splitting coefficient p of A_1,2(λ₁) to make the power detected at D_1,2(λ₁) zero. We repeat the same process for P_1,3(λ₁) and A_1,3(λ₁) sequentially to make the power at D_1,3(λ₁) zero, and consequently, we form a collected single-mode beam (i.e., O) from arbitrary two-wavelength input beams without generating loss at λ₁. The same process can be done simultaneously and independently for λ₂ because the two input mode vector at different wavelengths are orthogonal to each other. The mathematical proof for the lossless combining of a monochromatic input mode vector can be shown by the identical process introduced in [30], and thus we do not discuss the relevant content in further detail in this paper.

 

Fig. 7. Self-configuring universal beam coupler for three input modes (M=3) at two operation wavelengths (K=2). I₁, I₂, and I₃ are the input modes each consisting of two values for λ₁ and λ₂. O is the collected single-mode beam consisting of two values for λ₁ and λ₂. The blocks, P(λ), A(λ), and D(λ) represent the phase shifter, the optical power splitter, and the photodetector at the wavelength λ.

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The process can also be extended to more than one orthogonal (e.g., orthogonal polarization) mode sets [4]. The discussion for the self-configuring of multiple orthogonal beams at multiple wavelengths will be provided in the subsequent paragraphs. Thus, we can take arbitrary input mode vectors of multiple orthogonalities at multiple wavelengths and combine vector at each wavelength into a single-mode beam without any information of the inputs, but by just following simple serial processes.

We now examine how the proposed multiple wavelength N×N self-configuring universal linear optical components operate. To implement the multiple wavelength output coupler of the self-configuring optical circuits, we consider the reverse process (i.e., inserting monochromatic single-mode beam from the opposite direction) to show that the cascaded model in Fig. 7 is capable of taking a single-mode input beam and generating the specific output mode vector that we seek for. This is accomplished by first shining a phase conjugated version of the desired output mode vector to the input mode vector (I₁, I₂, and, I₃). Performing self-configuring beam coupling to the input mode vector of the phase conjugated version and then shining a single-mode beam from the opposite direction will result in the desired output mode vector [4]. This process can be extended to the multiple wavelength channels with our cascaded resonator-based model. For example, for the case of two operation wavelengths (K=2), we can shine the phase conjugated version of the desired output mode vectors for each wavelength and proceed with the self-configuring process for the output coupler simultaneously and independently.

We can cascade more sets of identical T blocks of up to K different operational wavelengths. For example, T(λ₃), T(λ₄), …, T(λ_K) can be added next to the T(λ₁) and T(λ₂) blocks, and we can apply the same mechanism to each set independently. This process can be done simultaneously because operations at one wavelength (λ_i) do not affect the operation at the other (λ_j≠i), and the component set for each wavelength can operate independently to the allocated wavelength. As a result, universal linear optical operation of multiple channels with K different wavelengths can be done at the same time by sequentially matching the parameters. The MZI-based architectures would require a wavelength demultiplexer at the input, and K different photonic circuits are necessary for K different wavelengths. On the other hand, our approach allows us to have another degree of freedom (i.e., wavelength) by simply cascading optical resonators with different resonance wavelengths to a single photonic circuit.

Self-configuring optics for multiple orthogonal beams at K different wavelength channels applies for the architecture in Fig. 8(b), which is the cascaded optical resonator version of the MZI-based model of 3×3 unitary operator in [4] for K different wavelength channels. Beams that are orthogonal to the beam coupled into the single beam at the first row (i.e., row with components T_1,1, T_1,2, and T_1,3) will pass through the first row and arrive at the second row (i.e., row with components T_2,1, and T_2,2). Thus, we can repeat the self-configuring beam coupling at the second row for a beam that passes through the first row. For the output coupler, as discussed, we can shine the phase-conjugated version of the desired output mode vector with multiple orthogonal beams at K different wavelength channels and perform the self-configuring optics. Therefore, we can conclude that the photonic circuit configuration in Fig. 8(b) enables the implementation of an arbitrary 3×3 linear optical operation for K different wavelength channels, where the extension to the arbitrary M×M linear optical operation for K different wavelengths is straightforward. Transparent photodetector used for the MZI-based universal linear optical component [31] can also be used in the proposed architecture. We allocate individual photodetector for each wavelength, and monitor the power detected at the corresponding wavelength independently to carry out the self-configuring process simultaneously.

 

Fig. 8. The multiple wavelength version of the self-configuring optics. (a) Algorithm of self-configuring optics to implement arbitrary optical linear functions of M input modes and M output modes for K different wavelength channels. (b) The photonic circuit configuration for implementing arbitrary 3×3 linear optical operations for K different wavelength channels. P(λ) (P^o(λ)), A(λ) (A^o(λ)), and D(λ) (D^o(λ)) represent the phase shifter, the optical power splitter, and the photodetector at the wavelength λ for the input (output) coupler. The modulation part is to implement the overall amplitude and phase of the output modes.

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4. Discussion

Universal linear optics has applications in large-scale optical matrix multiplication, and therefore it is important to match the resonance wavelengths of multiple resonators to the wavelength of the input channel. In practice, however, fabrication errors would cause unpredictable shifts in resonance wavelengths and therefore make this difficult without post-tuning procedures. One of the most popular optical resonators that are utilized in photonic integrated circuits is the micro-ring resonator, and several post-tuning mechanisms to lock the resonance wavelengths of multiple resonators have been proposed in SOI platform [28]. Two particular approaches have experimentally proved the feasibility of locking multiple micro-ring resonances in large-scale PICs. The first approach is to leverage the DC component of the signal from the monitoring photodetector (MPD) located at the combined through-port to provide a closed-loop feedback control over multiple micro-ring resonators [32]. Multi-wavelength resonance locking can be implemented with this approach by minimizing the power detected at the MPD for different operation wavelengths. The other approach is to utilize the photoconductive heater for resonance alignment of multiple micro-rings in large-scale systems [33]. Multiple wavelength locking could be implemented by repeating this approach for every operation wavelength, where in each step we inject a specific operation wavelength and each set of rings allocated for the specific operation wavelength is locked to the input.

When we take the micro-ring resonator as a building block, adjusting the power split ratio (i.e., coupling coefficient) can be achieved by either varying the gap using MEMS structure such as comb actuators [34] and vertical adiabatic couplers [35] or tuning the refractive index of the arms of interferometer [36,37]. The maximum value of the coupling coefficient can be somewhat limited in the last case [36,37], but this would not still violate the condition required for our model (i.e., Q_e<<Q_o). MEMS-based actuation within the silicon photonic platform could also be an excellent choice for implementing tunable couplers. Employing the MEMS approach for cascaded resonator-based universal linear optics allows dynamic tuning of the coupling coefficients by more than two orders of magnitude [35], but might require fine synchronization of the resonance wavelengths between identical resonators, and the effect of coupling-induced resonance frequency shifts (CIFS) must be taken into consideration [38]. If we want to avoid the resonance shift while keeping the possible power-splitting range wide, it would be reasonable to exploit the interferometric coupling between the micro-ring resonator and the access waveguide [36,37]. With the approach, we can fix the gap between the micro-ring resonator and increase the coupling coefficient to the over-coupling level while avoiding the CIFS. Recently, a hybrid photonic-plasmonic structure for the optical switching based on whispering gallery mode micro-resonator was demonstrated. This approach enables rapid switching (tens of nanoseconds) with low optical loss (0.1 dB) and low operation voltage (CMOS-level of ∼1.4 V) in a compact footprint (10 μm²), which features many promising aspects for the implementation of our proposed approach [39].

For the input that contains modes of multiple wavelengths, the independent phase shifting to the individual wavelength is required. Phase shifter taking the all-pass configuration of the optical resonator features this characteristic. This type of optical phase shifter requires three attributes: strong coupling of the access waveguide and the optical resonator, sufficiently low loss cavity, and resonance frequency tuning device. Strong coupling by shortening the gap between the optical resonator and access waveguide involves the tradeoff between the mode conversion loss and the magnitude of the coupling coefficient [40]. Interferometric coupling between the access waveguide and the optical resonator can minimize this effect [36,37].

Leveraging the linear characteristics of the resonators to implement a particular quantum gate (i.e., CNOT gate) has been introduced recently [41]. The results suggest that the approach can allow a more robust quantum gate operation against external noise. Applying our approach to the quantum realm, we can cover arbitrary unitary operations of multiple wavelengths in a single photonic circuit, opening up a new perspective for quantum photonic computing. For the KLM protocol that turns the probabilistic operation into near a deterministic regime at the price of additional resources (e.g., waveguides, beamsplitters, etc.) [14,15], our scheme may allow chances for operations of multitudinous qubits in a relatively small PIC footprint. Compared to the MZI-based approach where one needs separate circuits to perform single qubit operations to the signal and idler photons generated at different wavelengths, our approach allows one to utilize only a single circuit, possibly leading us to the chip-scale 100 logical qubit regime.

5. Conclusion

In this work, we have shown that multi-wavelength universal linear optics can be implemented with cascaded resonator-based photonic integrated circuits. This scheme provides an additional degree of freedom (i.e., wavelength) to the MZI-based achromatic version, broadening the functionality of universal linear optics circuits to the multiple wavelength channels. We believe that the proposed model can open a new perspective for the universal linear optics and quantum photonic computing.