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Abstract
We consider waveguide lattices as the architecture to implement a wide range of multiport transformations. In this architecture, a particular transfer matrix is obtained by setting step-wise profiles of propagation constants experienced by a field evolving in a lattice. To investigate the capabilities of this architecture, we numerically study the implementation of random transfer matrices as well as several notable cases, such as the discrete Fourier transform, the Hadamard, and permutation matrices. We show that waveguide lattice schemes are more compact than their traditional lumped-parameter counterparts, thus the proposed architecture may be beneficial for photonic information processing systems of the future.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Linear transformations of multiple optical channels are often required for contemporary information processing systems. These include field mode unscramblers [1], matrix multiplication in photonic Ising machines [2] and neural networks [3,4], and cryptographic tasks [5]. Recently, a great deal of interest in linear multichannel photonics has been expressed by the quantum community, as it is a necessary tool for quantum information processing systems that are designed to outperform their classical counterparts in solving certain tasks. Among them are quantum simulators that sample discrete photons [6,7] or continuous variable states of light [8], and even more demanding quantum computers, wherein multichannel transformations are utilized to prepare resource quantum states [9,10] and to implement operations of quantum logic [11,12].
The established implementations of the programmable multichannel schemes exploit the lumped-parameter approach. An optical circuit is a mesh of building blocks, where each block performs a transformation over a subset of optical channels. These blocks are usually standard Mach-Zehnder interferometers (MZIs) equipped with two phase modulators which up to the global phase implement arbitrary two-mode linear transformations. Several mesh-based decompositions have been reported in the literature. The works [13,14] rely on the well-known mathematical representation of the $U(N)$ matrix as a product of effectively two-dimensional transformations. A simple programming algorithm and a straightforward implementation using well-known components made these two architectures the most widespread to date. However real-life implementation of the reconfigurable interferometers based on the architectures [13,14] is highly susceptible to fabrications defects [15–17]. The quest to relax stringent fabrication requirements and endow optical architectures with robustness to at least certain kinds of imperfections led to the development of the multiport [18] and beamsplitter (BS) based error-tolerant [19] circuit designs. All discussed architectures share the same feature - mode-mixing and phase-shifting elements are independent of each other. All elements are connected by optical waveguides according to the architecture of choice. Thus the layout of the interferometer circuit on a chip is sparse. The circuit effectively covers a large area even though each functional element is small. Growing interest in reconfigurable photonics from the industry [20] and the machine learning community [4] imply that insights increasing the device miniaturization capabilities on the architectural level may boost the development of practical devices.
In recent years, evanescently coupled waveguide lattices have become an object of active research, as they were adopted to the implementation of complex transformations, such as angular momentum rotation [21], discrete fractional Fourier transform [22], generation of W-states [23], and photonic quantum gates [24]. At the same time, it is the platform for large scale optical simulation experiments that cannot be performed with conventional photonic circuits [25,26]. The waveguide lattices are more compact and have lower losses, than lumped-parameter circuits since they lack bent segments of single-mode waveguides. Recent experiments [27–29] have demonstrated the potential of the waveguide lattices to serve as the basis of reconfigurable photonic devices.
In this work, we study the programmable waveguide lattice architecture. Reconfigurability is enabled by variable profiles of propagation constants experienced by optical fields during their evolution inside lattice channels. We study transformation capabilities of the waveguide lattice architecture by analyzing fidelity of the transformation for a set of random transfer matrices as well as some particular cases: the discrete Fourier transform, the Hadamard and permutation matrices.
2. Methods
2.1 Field evolution in the waveguide lattices
The architecture studied in this work is shown in Fig. 1. The central part is the waveguide lattice providing $N$-channel interference required to realise a general $N \times N$ linear transformation. The lattice consists of layers of equal length $l$. Each layer differs by the propagation constants $\beta _m(z)$ experienced by optical fields, where $m$ denotes the waveguide index, and $z$ is the propagation coordinate. We describe the model in which the propagation constants are fixed within layers so that $\beta _m(z)$ is a step-wise function. The phase shifts, $\varphi ^{(in)}_m$ and $\varphi ^{(out)}_m$, are added at the input and output of the lattice. This work aims to study the architecture capabilities to implement various multichannel linear transformations by adjusting $\beta _m(z)$ sequences and $\varphi ^{(in)}_m$ and $\varphi ^{(out)}_m$ phases using the fixed value of layers length $l$ for the given $N$.
Fig. 1. The proposed architecture of the multichannel linear transformation. The $N$-channel scheme consists of a waveguide lattice and phase shifts placed at the input and the output stages. The lattice consists of evanescently coupled waveguides with static coupling constants $C_{m,m+1}$ and a variable set of propagation constants defined as stepwise functions along the z axis for each of the $N-1$ layers. Here, $\delta _{mj}$ is the propagation constant tuning in the waveguide $m$ and the layer $j$ relative to the last $N-th$ waveguide, where $\beta _N$ is assumed to be constant (see the main text for details). Parameters $\varphi ^{(in)}_m$ and $\varphi ^{(out)}_m$ are the variable phase shifts ($m=\overline {1,N-1};j=\overline {1,N-1}$).
Our analysis of the waveguide lattice architecture rests upon the following assumptions. First, we take into account only the nearest-neighbour coupling which is justified when the coupling strengths $C_{mn}$ between the waveguides with indices $m$ and $n$ fulfil the condition: $C_{m,m+1}\gg {}C_{m,m+q}$ for $q\ge 2$. Second, the characteristic variance of the refractive indices of the waveguide cores $\delta {}n_m(z)$, which induce $\beta _m(z)$ values. We assume that $\delta {}n(z)$ affects only the propagation constants $\beta _m(z)$, and the effect on the coupling strengths $C_{mn}$ is negligible. We set the coupling strengths $C_{mn}$ to be constant along the $z$ axis. In particular, this is justified in the case of weakly coupled waveguides, e.g., those fabricated with low refractive index contrast technologies, for example, femtosecond laser writing [30]. In addition, we neglect losses that inevitably occur due to scattering and material absorption and treat optical field evolution as unitary.
The coupled mode theory defines the system of equations governing the field amplitudes’ evolution in the waveguide lattice [31]:
where $\mathbf {A}(z)=(A_1(z)\ldots {}A_N(z))^T$ is the column of field amplitudes at the propagation distance $z$, $H(z)$ is the tridiagonal $N\times {}N$ matrix:The solution of (1) is expressed in the matrix form:
where $V(z)=\exp \left (-i\int _0^{z}H(z')dz'\right )$ is a transfer matrix of the lattice. As a global phase acquired by amplitudes $\mathbf {A}(z)$ is usually irrelevant for measurable quantities, the $N-1$ propagation constants $\beta _m(z)$ are enough to describe the phase shifts of $N$ field amplitudes in the layer. Therefore, without loss of generality, only $\beta$’s of the first $N-1$ waveguides are considered variable, while the last one is fixed through all layers: $\beta _N(z)=\beta _0$. In addition, it is convenient to eliminate $\beta _0$ from the equations by the substitution: $\mathbf {A}(z)\rightarrow {}\exp (-i\beta _0z)\mathbf {A}(z)$. In this framework, the parameters relevant for a field evolution are the propagating constant differences $\delta _m(z)=\beta _m(z)-\beta _0$ ($m=\overline {1,N-1}$). The same reasoning is applied to the phase shifts $\varphi ^{(in)}_m$ and $\varphi ^{(out)}_m$ ($m=\overline {1,N-1}$).The dependence $H(z)$ on the propagation coordinate $z$ is due to the piecewise-defined propagation constants. We use the notation in which $\delta _m(z)$ of the waveguide with index $m$ takes constant value $\delta _{mj}$ when $(j-1)l\le {}z<jl$ ($m=\overline {1,N-1}$, $j=\overline {1,N-1}$). The single system (1) gets split into $N-1$ systems, each one is described by the constant matrix $H_j$, and the transformation of the lattice is given by the product $V=V_{N-1}V_{N-2}\cdot \ldots \cdot {}V_{1}$, where
is the transformation of each layer $j$ ($j=\overline {1,N-1}$).The phase shifts added to the input and output of the scheme are described by diagonal matrices:
2.2 Relevant parameters of the architecture
Using (4) and (5), the transfer matrix of the $N$-channel scheme is given by a product of the transfer matrices, describing the layers of the lattice and the input and output phase shifts:
We consider the lattice with equal coupling coefficients ($C_{m,m+1}=C$), which is attained in practice by fabricating a lattice with equally spaced waveguides. As a result, $C$ can be extracted from the matrices $H_j$. We then replace the layer length with the dimensionless value $\tilde {l}=Cl$. Following that, the $H_j$ matrix gets converted to
2.3 Optimization procedure
The following fidelity measure quantifies the performance of the multichannel interferometer
The function $F$ compares a target transfer matrix $U_0$ and an actual transfer matrix $U$ realized by the lattice, where $N$ is the size of the lattice. When the matrices $U$ and $U_0$ are equal up to a complex multiplier, the fidelity (8) gets its maximum value of $F=1$.
The analysis aims to check whether the proposed lattice-based interferometer accurately reproduces different target transfer matrices. No analytical solution to this problem is available, thus, we relied on a numerical optimization procedure that searched for a global maximum of $F$ over the space of $N^2-1$ parameters $\left \{\delta _{mj}\right \}$, $\left \{\varphi ^{(in)}_m\right \}$, $\left \{\varphi ^{(out)}_m\right \}$ ($m=\overline {1,N-1};j=\overline {1,N-1}$). To do this, we used the basin-hopping global optimization algorithm implemented with the BFGS local optimizer provided by the SciPy library in Python. It should be noted that the arbitrary length $\tilde {l}$ does not guarantee the optimal performance of the interferometer, hence we included the layer length in the parameter space of the optimization as well.
3. Results
The target unitary matrix set $U_0$ includes the Haar-random matrices [32] and three specific types of matrices often applied to information processing: discrete Fourier transform (DFT) matrices [33], Hadamard matrices [34] and permutation matrices.
Figure 2 summarizes the optimization results. The histograms of the lowest achieved infidelities $1-F$ for matrices with a different size $N$ are shown in the left column. In practice, $\delta _{mj}$ cannot take arbitrary large values due to adopted assumptions and, more importantly, because they are limited by the strength of an effect exploited to tune the propagation constants, e.g. the thermo- or electro-optical effects (see Sec. 4 for estimates). Therefore, we monitor the values of $\delta _{mj}$ that implement target matrices. The $\tilde {\delta }_{mj}$ distributions are displayed in the middle column of Fig. 2. Also, corresponding distributions of optimal layer lengths $\tilde {l}$ are shown in Fig. 2 on the right.
Fig. 2. The histograms of the infidelity values $1-F$ (left), dimensionless quantities $\tilde {\delta }_{mj}$ (middle) and layer length $\tilde {l}$ (right) in cases of different target matrices: a) a set of $100$ Haar-random unitary matrices for each $N$, b) DFT matrices at multiple runs of the optimization algorithm, c) Hadamard matrices at multiple runs of the optimization algorithm, d) a set of $100$ random permutation matrices.
The case of the random matrices is shown in Fig. 2(a). A set of $100$ Haar random target matrices with the size ranging from $N=2$ to $N=12$ were generated. For larger $N$ values the approximation precision decreases, which may be caused by increasing complexity of the optimized function landscape. Nevertheless, the resulting configurations approximate sampled unitaries within the range of experimentally reachable precision.
For the DFT and the Hadamard transformations for each particular matrix size, we have run $100$ optimization procedures, which enabled us to explore the quality of the optimization. The results are shown in Fig. 2(b) and Fig. 2(c), respectively. It should be noted that the optimizer finds different solutions with almost the same precision but with different lattice characteristics. The histograms in Fig. 2(b) and Fig. 2(c) show that the Fourier transformation accepts a much wider variety of equivalent solutions than the Hadamard transformation.
Lastly, we performed an optimization of the lattice structure to fit the permutation matrices (Fig. 2(d)). For this, a set of 100 permutation matrices was generated prior to the optimization run. The lattice configuration was then tuned to reproduce the permutation matrices from this set. This class of transformations imposes the most strict requirements on the architecture parameters since the perfect result corresponds to the totally constructive interference at the given output channels.
Figure 3(a) illustrates the field pattern that propagates in the lattice implementing DFT of $N=8$ channels with optimized infidelity $1-F \sim 10^{-11}$ when a single input channel is exited. The corresponding field pattern for the approximated permutation matrix with infidelity $1-F \sim 10^{-3}$ is displayed in Fig. 3(b). Obvious solutions correspond to much simpler light distributions for these kinds of transformations. The optimizer finds a representative of a set of solutions that might correspond to a less trivial light propagation pattern.
Fig. 3. The simulated pattern of light propagating in two lattices optimized for: a) the DFT matrix ($N = 8$, $1-F \sim 10^{-11}$) and b) for the permutation matrix ($N = 8$, $1-F \sim 10^{-3}$). In both cases, a single input channel is excited; $\xi =Cz$ is the normalized propagation coordinate. c) The illustration of the effect of inaccuracies in $\delta _{mj}$ and $l$ on these two transformations quantified by the width of the error distributions $\Delta =\Delta _{\tilde {\delta }}=\Delta _{\tilde {l}}$, where $\Delta _{\tilde {\delta }}$ and $\Delta _{\tilde {l}}$ are the distribution widths of $\tilde {\delta }$ and $\tilde {l}$ inaccuracy, respectively.
To investigate the tolerance of the transformation quality against inaccuracy in $\delta _{mj}$ and $l$, we added normally distributed random errors to the optimal values, characterized by distribution widths $\Delta _{\tilde {\delta }}$ and $\Delta _{\tilde {l}}$. For simplicity, the widths were set equal: $\Delta _{\tilde {\delta }}=\Delta _{\tilde {l}}=\Delta$. The corresponding fidelities are then calculated. Figure 3(c) compares the effect of the errors in $\delta _{mj}$ and $l$ for the DFT and the permutation matrices of $N=8$. Here, the corresponding optimal infidelities in case of no errors are $\sim 10^{-11}$ and $\sim 10^{-3}$, respectively. Expectedly, elevating the degree of inaccuracy corrupts both transformations. However, while the error-free fidelity for the DFT matrix is much higher, it catches up with the fidelity for the permutation matrix at $\Delta \sim 10^{-1}$.
4. Discussion
Let us summarize the simulation results and figure out the guidelines for a possible experimental implementation using currently available integrated photonic technologies. We discuss optical circuits formed by weakly-guiding waveguide lattices, such as those fabricated by femtosecond direct laser writing in glasses and proton-exchanged waveguides in a lithium niobate crystal. We take the value of the coupling strength $C = 0.1$ mm$^{-1}$, which is typical for lattices fabricated using these technologies [28,35] and small enough to ensure the nearest-neighbour coupling. The results discussed above yield the ranges $-2.5\le \tilde {\delta }_{mj}\le 2.5$ and $1.0\le \tilde {l}\le 2.0$ (with an average layer length $\tilde {l} \approx 1.6$), hence the layer length $l$ for these lattices can be estimated to fall into the $10\le l \le 20$ mm window. The following are the estimates of the propagation constant tuning related to the implementation of the programmable waveguide lattices by thermo-optic and electro-optic effects:
- 1. Thermo-optic tuning. The thermo-optic phenomenon is widely exploited in reconfigurable photonic chips created by femtosecond direct laser writing in glasses [36–38]. In the case of fused silica, typically chosen to be the material of the chip, the refractive index contrast between the inscribed core ($n_{co}$) and the cladding ($n_{cl}$) amounts to $\sim 10^{-3}$. To achieve the coupling strength $C=0.1$ mm$^{-1}$ with this index contrast, an interwaveguide distance $d_{wg} \approx 20$ $\mu m$ is required [35]. The reprogramming by thermo-optics happens due to the material’s local heating, which induces positive change of the refractive index. Thus, we assume that $\delta$ can cover a positive range from $0$ to $5C$ corresponding to values $0-0.5$ mm$^{-1}$ to meet the requirements mentioned above. This corresponds to the induced change of the refractive index and the temperature rise, taking the values $\Delta n \approx 6.4 \times 10^{-5}$ and $\Delta T \approx 6.4$ $K$, respectively. The coupling constant modulation is of the order of $\Delta C/C$ = $\Delta n / (n_{co} - n_{cl})$ = $\lambda \Delta \beta / 2 \pi (n_{co} - n_{cl}) = 1.3\%$, which is small enough to be neglected. However, the femtosecond laser written thermo-optical tuners suffer from a large amount of dissipated heat associated with the refractive index tuning and from intrinsically significant cross-talks. Both of these issues can be diminished by fabricating isolating trenches at sides and the bottom of the waveguides as well as operating the chip in a vacuum [39].
- 2. Electro-optic tuning. The modulation of propagation constants using an electro-optic effect is utilized in proton-exchanged waveguides typically created in lithium niobate crystals. For example, recent work by Youssry et al. [28] explores the evanescently coupled waveguide lattice reconfigured by applying an electric field to the individual waveguides of the lattice. Utilizing electro-optics rather than thermo-optics comes with the advantage of much lower cross-talks between tuners and ultra-fast reconfiguration of the scheme. The coupling constant $C$ = $0.1$ mm$^{-1}$ is achieved at a distance of $d_{wg}=10$ $\mu m$ between adjacent waveguides [28]. Notice that the electro-optical effect allows both positive and negative changes of the refractive index controlled by voltage polarity. The tuning range $-2.5C\le \delta \le {}2.5C$ requires minimum and maximum values of $\delta =-0.25$ mm$^{-1}$ and $\delta =+0.25$ mm$^{-1}$. The magnitudes of the corresponding refractive index and voltage modulations are $\Delta n$ = 3.2 $\approx$ 10$^{-5}$ and $\Delta {}V \approx 6.4$ $V$, respectively. The calculated variations of the coupling constants induced by the maximal $\delta$ are $\Delta {}C/C=0.5\%$, which is smaller than for thermo-optically reconfigurable lattices.
Let us demonstrate the miniaturization advantage of the waveguide lattice over known lumped-parameter architectures with universal capabilities. Reasonable architectures to compare with are the universal MZI-based architecture by Clements et al. [14] and the BS-based architecture by Fldzhyan et al. [19] enabling the implementation of the range of arbitrary transfer matrices. Table 1 shows the result of the comparative analysis for $N=10$. Estimates are given for femtosecond direct laser written circuits in fused silica [35,38,40], where the average propagation loss is about 0.5 dB/cm.
- 1. MZI-, BS-based. In both architectures, the $N$-channel schemes require $2N$ layers of static BSs with balanced and more transmissive characteristics, respectively. The length of the single BS $l_{bs}$ depends on the separation distance $d_{wg}$ between the pairs of waveguides of the BS and a bend radius $R$. Standard v-groove fiber optic assemblies are available with the pitch $250$ $\mu m$ or $127$ $\mu m$ between fibers so that we can use these values as $d_{wg}$. A typical bend radius is 80 mm with extra bending losses of 0.1 dB/cm. In addition, one can consider another compromise bend radius of 40 mm with extra bending losses of 0.5 dB/cm. Also, the lengths of the waveguide segments designed for active phase-shifts $l_{ps}$ should be added to obtain the total layer length $l=l_{bs} + l_{ps}$. Taking $l_{ps}=2.5$ mm [38], the value $l$ lies between $8.7-15$ mm. Therefore, the total length of the $10$-channel lumped-parameter interferometer is $L_{scheme} = 2Nl = 17.4-30$ cm. The length is too large for the interferometer to be practical, not to mention large total losses $13-20.1$ dB that come with it.
- 2. Lattice-based. Here, the $N$-channel schemes require $N-1$ lattice layers and input and output phase-shifters. In general, there is only a condition for the product of the propagation constant tuning and the layer length $\delta l \approx 8$ (for the phase shifter in MZIs this value must reach 2$\pi$), so $l \approx 1.3 l_{ps}$ and the total length of the lattice-based interferometer can be $2N(l_{bs} + l_{ps})/((N-1)l + 2l_{ps}) \approx 5 - 9$ times shorter than MZI-, BS-based. However, the cross-talks prevents the waveguides from being placed closer, and one has to work at a lower coupling constant, which, of course, increases the layer length. The layer length depends on the interwaveguide distance, and for the mentioned above $d_{wg} = 20$ $\mu m$ it equals $l=15.7$ mm. Though this value is comparable to the MZI-, BS-based case, the total length of the $10$-channel lattice-based is $L_{scheme}=(N-1)l + 2l_{ps} = 19.1$ cm with a total loss estimate 9.6 dB. If we consider closer waveguide spacing $d_{wg} = 10$ and 15 $\mu m$, and the respective values of the coupling strength $C = 0.4$ and 0.2 mm$^{-1}$, the total length and loss substantially decrease. On the other hand, the fan-out section at the input and the output of the lattice will still occupy a large area if the standard v-groove assemblies with 127 $\mu m$ pitch are employed. The non-standard v-groove with pitch reducing solutions exist on the market [41] with much narrower fiber spacing of $30-40$ $\mu m$ or even down to $12$ $\mu m$, therefore, solving the issue.
Table 1. Comparative analysis of the MZI-, BS-based and lattice-based architectures
We emphasize that only rough estimates, and improvements can be made by devising the circuit layout judiciously. For example, the lengths of the lumped-parameter schemes can be shortened by using bent waveguide sections for phase shifting. However, even in this case, our architecture retains its advantage.
It should be noted that although we analyzed the lattice architecture using the model of coupled-mode theory with nearest-neighbour interactions and considered only weakly-guiding waveguides, we expect that our results can be generalized to more miniature integrated photonics platforms, such as silicon photonics within the framework of other more complex models [42].
5. Conclusion
In this work, we have considered the programmable architecture based on evanescently coupled waveguide lattices in order to investigate its capability to implement multiport transformations. Using numerical simulation results, we infer that this lattice architecture is well suited for the efficient implementation of unitary transformations for up to $\sim {}10$ channels. The lattice architecture might substantially decrease the size and the insertion loss of reconfigurable integrated photonic circuits. Our work provides the pathway to further miniaturization of reconfigurable linear photonic elements. The results may be of particular interest for applications with stringent requirements to optical loss and without the necessity of universal reconfiguration. For instance, the state-of-the-art linear optical quantum computer architecture implies concatenation of numerous identical multichannel elements and most of those elements are static and do not even require reconfiguration. The low-loss implementation of these elements will highly benefit overall optical processor performance. We believe that our results may serve as the basis for developing of ultra low-loss reconfigurable multiport integrated photonic devices.
Funding
Foundation for the Advancement of Theoretical Physics and Mathematics (20-1-3-31-1); Russian Foundation for Basic Research (19-52-80034).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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