1. Introduction

Rapid advances in photonic integrated circuits (PICs) have ushered in a new generation of devices capable of universal linear-optics transformations across a set of $N$ optical modes [1]. These ‘programmable photonic circuits’ are enabling new applications ranging from quantum information processing [2,3] to deep learning [4]. In leading implementations, the mode transformations are realized as an SU($N$) rotation, which can be factored into $N(N-1)/2$ SU(2) rotations [5]. Figure 1(a) illustrates the concept on input modes $A_1,\ldots, A_N$, where SU(2) rotations are programmed in Mach-Zehnder interferometers (MZIs), as shown in Fig. 1(b). A key advantage of this ‘MZI mesh’ (MZM) architecture is its compatibility with efficient programming algorithms [6,7]. But while tens of modes have been realized [1], the MZM architecture has inherent limitations – most importantly, the exacting fabrication requirements of 50:50 couplers, a large fraction of inter-waveguide ‘dead’ space, and architecture-constrained operating bandwidth and decomposition schemes [8,9].

 

Fig. 1. SU(2) mesh architectures (a) Schematics of an SU(N) transformation decomposed into a rectangular mesh of SU(2) transformations $T_{ij}$. (b) Mach-Zehnder interferometer (MZI) used in integrated photonic platforms for implementing individual $T_{ij}$s in the mesh. These building blocks consist of two phase shifters, where one of them is book-ended by directional couplers. (c) MMI approach to realizing SU(N) transformations. The latter are achieved by segmenting the waveguide into layers where programmed refractive index variations first alter the propagation constants $\beta _{m,n}$ of modes $m,n$ to $\tilde \beta ^{(l)}_{m,n}$ and subsequently couples them as parametrized by the coupling strength $\kappa ^{(l)}_{mn}$.

Download Full Size | PDF

Alternatively, could universal linear-optical transformations be deterministically programmed in a rectangular dielectric slab, a large ‘multimode interferometer’? We consider this question for the $N$ transverse spatial modes of a slab with an index contrast quantified by $\sqrt {\Delta \epsilon /\epsilon _0}$, as shown in Fig. 1(c). This slab has single-mode thickness $L_y= \lambda /2 \sqrt {\Delta \epsilon /\epsilon _0}$, length $L_z$, while the width $L_x$ accommodates $N\sim 2 L_x \sqrt {\Delta \epsilon /\epsilon _0}/\lambda$ transverse spatial modes. Recent experimental work did realize programmable $1\times 2$ switching in a 2-mode MMI fabricated in silicon [10–12], but programming the index perturbations relied on a randomized search algorithm that lacks the algorithmic determinism of the MZM approach [6,7,13]. On the other hand, programming SU($N$) transformations by cascading layers of phase masks and Fourier planes [14–17] requires a total length of $\mathcal {O}(N)$ elements [18], each of length $\mathcal {O}(N)$. This device’s length therefore scales as $L_z \propto N^{2}$, whereas the minimum necessary scaling is $L_z\propto N$ to accommodate the necessary $N(N-1)$ phase shifters.

Here, we show that the compactness of MMIs and the programmability and $\mathcal {O}(N)$ length scaling of MZMs are possible in one device: a programmable multimode interferometer (ProMMI) that relies on a mesh of $\mathcal {O}(N^{2})$ phase shifters distributed across a rectangular waveguide slab. Waveguide sidewall scattering loss is largely eliminated in the large cross-section waveguide; transverse dimensions are reduced by $\sim \! 1$ order of magnitude compared to MZMs; fabrication challenges of $2\times 2$ couplers are eliminated. The longitudinal dimension scales as $L_z=\textrm {max}(\alpha _1 N,\alpha _2 N^{2})$, where $\alpha _1,\alpha _2$ are coefficients that depend on the ProMMI material platform. We also translate recent hardware error correction methods from MZM [19] to ProMMI architectures, allowing substantial improvements in unitary transformation fidelity.

2. Results

Instead of operating on an array of meshed single mode waveguides, the ProMMI architecture implements SU(N) transformations on the eigenmodes of a slab. Here, the slab has a width $L_x=w$, allowing it to support $N$ modes. As in the MZM scheme [7], we cascade these transformations into $N$ layers performing SU(2) transformations on pairs of eigenmodes. But unlike the MZM, we apply these transformations via structured index perturbations that first modify a mode’s propagation constant from $\beta _m$ to $\tilde \beta _m$ and then couple modes $m$ and $n$ at a rate $\kappa _{mn}$. Each $(\tilde \beta _{m,n}, \kappa _{mn})$ has two DoFs as required to control, say, the independent variables of relative phase and amplitude. But, unlike MZMs, the ProMMI architecture overlays these perturbations. In this way, the ProMMI achieves the deterministic programmability and length scaling of the MZM, while collapsing its width to $w$, i.e., the minimum required for $N$ transverse modes.

2.1 Coupled mode theory

We approximate the step-index slab as infinitely thick such that its permittivity $\epsilon _s(\mathbf {r})=\epsilon _{\textrm {core}}$ for $|x|<w/2$ and $\epsilon _s(\mathbf {r})=\epsilon _{\textrm {clad}}$ elsewhere. The bounded monochromatic transverse electric (TE) field propagating along the waveguide in the $z$ direction thus takes the form

To optimize the selective coupling between modes $m$ and $n$, we therefore set $\epsilon _{L,(m,n)}^{(l)}(z)= \cos ((\beta _m-\beta _n)z)$ and use Eq. (5) to determine $\epsilon _{T,(m,n)}^{(l)}(x,y)$ that maximizes $\kappa ^{(l)}_{mn}$.

This $\epsilon ^{(l)}_{p,(m,n)}(\mathbf {r})$ in principle allows coupling between other pairs of modes $m',n'$ as indicated by the maximum transfer amplitude

We set the propagation constants $\tilde {\beta }_m^{(l)}$ in layer $l$ (Fig. 1(c)) with the transverse perturbations $\epsilon _{T,(m,n)}^{(l)}(x,y)$ while turning off the mode coupling term, i.e., $\epsilon _{L,(m,n)}^{(l)}(z)=1$. Across the full layer $l$ of length $L_{\textrm {layer}}$, the perturbation $\epsilon ^{(l)}_{p,(m,n)}(\mathbf {r})$ thus transforms modes $m$ and $n$ as

By this procedure, we determine all the necessary permittivity perturbations $\epsilon _{p,(m,n)}^{(l)}$ to construct the total perturbation $\epsilon _{p}^{(l)}(\mathbf {r}) = \sum _{(m,n)}\epsilon _{p,(m,n)}^{(l)}(\mathbf {r})$ in layer $l$. We repeat the process for all $l$ to complete the desired SU(N) on the slab’s eigenmodes.

2.2 4-mode transformation

Let’s consider an exemplary unitary transformation SU(4) on the TE modes of a $N=4$ slab waveguide, with matrix elements:

 

Fig. 2. ProMMI architecture of a 4-mode slab waveguide. (a) Structured refractive index perturbations used for arbitrary SU(4) transformations. The location of each perturbation matches that of the parameter it modifies according to Fig. 1(c). Inset: refractive index perturbation used to couple the waveguide’s TE2 mode to the TE3 mode, where $\Lambda _{23}=2\pi /(\beta _2-\beta _3)$. (b) Refractive index perturbations required to induce the SU(4) transformation provided in Eq. (9). (c) Intensity of the optical field upon propagating through the perturbed waveguide for an input mode defined as $(a_1,a_3,a_2,a_0)^{T}=(0,1,0,0)^{T}$, which in our architecture, consists of the waveguide’s TE3 mode. Refractive indices of $n_{\textrm {core}}=2.2886$ and $n_{\textrm {clad}}=1.44$ were assigned to the core and the cladding of the waveguide, respectively, and we considered an optical wavelength of 1.55 µm. (d) Weight of the field in (c) contained in the bounded modes of the device. (e) Absolute square and (f) phase values of the mode expansion coefficients, $a_{i,\textrm {N}}$, of the simulated field in (c) overlaid onto circular markers plotting the coefficients, $a_{i,M}$, expected from Eq. (8). (g) Optical transformations in each layer represented in $k$-space. The density plots show the absolute square of the field’s Fourier transform at the start of the phase shifting and coupling sections of each layer. Markers located at each mode’s central $\mathbf {k}$ value are colored based on their weight and the angular position of the dot on their outline corresponds to their phase. Vertical and horizontal arrows indicate the propagation constant shifts enabled by the $\epsilon _{L,(m,n)}^{(l)}$ and $\epsilon _{T,(m,n)}^{(l)}$ components of the permittivity perturbations $\epsilon _{p,(m,n)}^{(l)}$, respectively, and are colored based on their corresponding $\theta$ values.

Download Full Size | PDF

Figure 2(b) shows the index shift caused by the total perturbation $\sum _{(m,n),l}\epsilon ^{(l)}_{p,(m,n)}(\mathbf {r})$ implementing Eq. (9). Figure 2(c) plots the $|E_y(x,0,z)|^{2}$ of a trial input field $(a_1,a_3,a_2,a_0)^{T}=(0,1,0,0)^{T}$. Here, we used a split-step Fourier method [21] to numerically propagate the optical fields. As expected, $|E_y|^{2}$ is flat during the first half of layer 1 given that the only eigenmode propagating in the waveguide experiences a global phase shift. No coupling occurs in the second half because $\epsilon _{p,(3,n)}^{(1)}(\mathbf {r})=0$. In the second layer, the field remains flat in the first half for the same reasons as it did in layer 1, whereas it is entirely converted to $A_2$ in the second half since $\kappa ^{(2)}_{2,3}L_{\textrm {layer}}/2=\pi /2$. The first half of $l=3$ features a mostly flat field as in $l=1,2$ and partial conversion to $A_0$ as specified by $\kappa ^{(3)}_{0,2}L_{\textrm {layer}}/2=\pi /4$. The relative phase between $A_0$ and $A_2$ manifests in the beating pattern in the first half of $l=4$ followed by the full conversion of $A_2$ to $A_3$ according to $\kappa ^{(4)}_{2,3}L_{\textrm {layer}}/2=\pi /2$ in the second half. Fig. 2(d) plots the transmission of the ProMMI, as quantified by the absolute square of the bounded field, i.e. $T=\sum _{i=0}^{3}|a_i|^{2}$. The drops observed in the second halves of each layer indicate stronger scattering to the unbounded modes of the system, as expected from the more rapid spatial variations of permittivity in these regions. In Fig. 2(e), (f), we provide the expansion coefficients of the field and compare them to those obtained directly from Eq. (8).

For further insight, it helps to view the transformation problem in $k$-space. Figure 2(g) shows the Fourier transform of the field in Fig. 2(c) as it enters the phase-shifting and coupling sections of each layer. Markers located at the central $\mathbf {k}$ value of each eigenstate represent their relative weights. The perturbations $\epsilon _{p,(m,n)}^{(l)}$ couple modes $m,n$ at a rate $\kappa _{mn}^{(l)}$ — they are therefore the ‘$k$-space’ analog of the MZI SU(2) rotations. Thus, the ProMMI is the ‘$k$-space’ realization of an MZM programmable unitary that, however, is inherently more compact and does not need single-mode patterning.

2.3 Error correction

To further gauge the reliability of our architecture, we performed the mode propagation simulations shown in Fig. 2 for 50 Haar random SU(4) rotations, $U$. We extracted the corresponding transmission matrix of the ProMMI from these simulations, $U_{\textrm {exp}}$, and thereafter the fidelity of the transformation, $F(U_{\textrm {exp}},U) = |\textrm {Tr}(U^{\dagger } U_{\textrm {exp}})/(N \, \textrm {Tr}(U_{\textrm {exp}}^{\dagger } U_{\textrm {exp}}))^{1/2}|^{2}$ [7]. Limits in the fidelity of these transformations arise from perturbations $\epsilon _{p,(m,n)}^{(l)}(\mathbf {r})$ being partially phased-matched to couple modes $m'\neq m$ and $n' \neq n$. These conditions increase the maximum transfer amplitude $\eta _{m'n'}$ defined in Eq. (6), causing transformations $T_{mn}^{(l)}$ to deviate from the block-diagonal form in Eq. (8). For instance, $\eta _{23}\sim 0.005$ in the second half of layer 2 from Fig. 2, explaining the oscillation in $|a_1|^{2}$ (see Fig. 2(e)) with an amplitude $\delta _a$ of roughly 0.5% of $|a_2|^{2}$ in that region of the ProMMI. The cross-talk introduced by this process consequently weakens the validity of our proposed SU(N) decomposition. However, as shown in Fig. 3(a), we can use additional optimization methods to further increase the fidelity of these transformations by altering the relative strengths of $\epsilon _{p,(m,n)}^{(l)}$. The cross-talk can also be mitigated by globally decreasing the strengths of $\epsilon _{p,(m,n)}^{(l)}$, albeit at the expense of increasing the total length of the device.

 

Fig. 3. ProMMI error correction. (a) Device infidelity $1-F(U_{\textrm {exp}}, U)$ before and after correcting for cross-talk due to partial phase matching with gradient descent optimization. (b) Infidelity in the unitaries constructed from phase mismatched versions of Eq. (8) before and after correcting for imperfect phase matching. To emulate the effect of variations in the waveguide width from one layer to another, each SU(2) considered in the unitary’s reconstruction was affected by a mismatch $\delta$ drawn from a normal distribution with a standard deviation of $\delta = \alpha \beta _p$. For the waveguide parameters considered in Fig. 2, $\alpha$ values of 0.1% and 0.2% correspond to width discrepancies of 1.5 nm and 3 nm, respectively.

Download Full Size | PDF

As in the case of MZMs, hardware errors can further reduce fidelity. For MZMs, the dominant source of errors is faulty couplers that do not have a 50:50 splitting ratio. For the ProMMI, such errors occur when a perturbation $\epsilon _{p,(m,n)}^{(l)}(\mathbf {r})$ has a period $\Lambda _p \neq 2\pi /|\beta _m - \beta _n|$ due to fabrication errors in the MMI waveguide that shift $\beta _{m,n}$ from their target values. In this event, the coupling between modes $m$ and $n$ is not perfectly phase-matched, which increases the coupling rate between these modes and constrains the off-diagonal terms of $T_{mn}^{(l)}$ to absolute values below unity. For instance, under these mismatched conditions, the bottom left element of $T_{mn}^{(l)}$ becomes

2.4 Scaling

The ProMMI dimensions scale favorably compared to the MZM. First, in both approaches, the device width scales linearly with $N$, $w=\alpha _{\textrm {MZM,ProMMI}} N$, as seen in Fig. 4(a). However, $\alpha _{\textrm {MZM}}\gg \alpha _{\textrm {ProMMI}}$ to allow for spacing between the MZM waveguides. Second, as shown in Fig. 4(b), both approaches have a length $L = \alpha _1 N$, where $\alpha _1=2 L_\pi$ for the ProMMI and $L_\pi$ is the distance for one phase-degree of freedom. $L/N$ is higher for the MZM as this architecture must also include 50:50 couplers. However, to keep the cross-talk specified in Eq. (6) below a maximum value, $\eta _{m',n'}^{\textrm {max}}$, the device length $L_z$ must be lower-bounded. With increasing $N$, the propagation constants $\beta _{m}$ become more and more closely spaced, reducing the denominator in Eq. (6). To counter this increase, $\kappa _{m',n'}^{(l)}$ must be commensurately reduced, thereby increasing the device length. As detailed in Supplement 1, by solving for $\kappa _{m',n'}^{(l)}$, the maximum cross-talk constraint $\eta _{m',n'}^{\textrm {max}}$ implies a device length $L_z \geq \alpha _2 N^{2}$, where

 

Fig. 4. Scaling properties of ProMMI vs MZM devices. (a) Device width in units of $w_0 = \lambda /\pi \sqrt {(n_{\textrm {core}}^{2}-n_{\textrm {clad}}^{2})}$. (b) Device length for both implementations in units of $L_\pi$. (c) Optical transmission expressed in units of $\Gamma _1 L_0$, where $\Gamma _1$ is the loss rate attributed to a single mode waveguide and $L_0$ is the layer length of the mode transformer. State-of-the art (SoA) values are attributed to SOI platforms equipped with thermo-optic phase shifters. The minimum width scaling in (a) is attributed to the waveguide spacing that leads to less than 1% cross-coupling between the waveguides of an MZM over its entire length.

Download Full Size | PDF

The total device length $L_z$ is the greater of the $\mathcal {O}(N)$ and $\mathcal {O}(N^{2})$ length requirements, yielding $L_z=\textrm {max}(\alpha _1 N, \alpha _2 N^{2})$. Figure 4(b) shows this linear to quadratic cross-over for various values of $\eta _{m',n'}^{\textrm {max}}$ and a nominal $L_\pi$ value used in state-of-the-art integrated lithium niobate modulators [23]. The cross-over happens when $\alpha _1 N= \alpha _2 N^{2}$, i.e., at $N=\alpha _1/\alpha _2$.

Some individual layers may need to be extended slightly to keep the total dielectric perturbation in layer $l$, $\sum _{(m,n)}\epsilon _{p,(m,n)}^{(l)}(\mathbf {r})$, below a practical maximum — for example, the power supply voltage. In such instances, we can lower all perturbation amplitudes by the same factor $x_l$ and extend that layer’s length by $1/x_l$.

Finally, the ProMMI architecture sharply curtails edge roughness losses due to reduced mode overlap. Since edge roughness commonly dominates the overall propagation loss rate $\Gamma$, we consider this advantage in detail. From coupled mode theory, the loss rate of the $m^{\textrm {th}}$ eigenmode due to coupling to radiative modes is given by

3. Discussion

Material platforms for the ProMMI should be selected based on function. Applications demanding high-speed programmability would favor electro-optic (EO) waveguide materials, such as thin-film lithium niobate [23] or barium titanate [24]. These materials have exceptionally low material losses [26], which the ProMMI architecture can reach by suppressing otherwise dominant edge roughness losses. Moreover, residual $\propto 1/N$ edge roughness losses are actually further suppressed in otherwise problematic trapezoidal waveguide profiles [23,25]. Applications with relaxed modulation speed requirements could combine passive waveguide layers with existing large-scale phase modulators. For example, the modes in a slab of silicon nitride (SiN) covered in liquid crystals (LCs) could reach $\epsilon _{p,(m,n)}^{(l)}(\mathbf {r})/\epsilon _0 \sim 0.3$, combining excellent modulation contrast with the CMOS scalability of LC-on-silicon (LCOS) displays with millions of pixels.

The ProMMI operating spectrum is determined by the phase-matching requirement as seen from Eq. (6). Though a lower index modulation $\Delta n$ reduces bandwidth, it reduces cross-talk due to partial phase-matching. This limitation is analogous to the bandwidth limitation due to modal dispersion in the couplers of MZMs. But whereas the MZM couplers are fixed, the ProMMI can be reprogrammed for different operating wavelengths – e.g., by using a regular electrode array of $>N^{2}$ pixels. A CMOS driver backplane is well suited to the task as commercial LCOS systems have millions of electrodes across an area of $\sim \!1$ cm$^{2}$. Alternatively, holographic techniques could efficiently program the $N(N-1)$ coupling perturbations into photorefractive crystals (such as lithium niobate) with $N(N-1)$ projected laser fields [27,28].

The ProMMI architecture is remarkable in another aspect: it saturates the compactness bound of programmable optical transformations in packing $N^{2}$ degrees of freedom as tightly as the material allows. Each ProMMI modal degree of freedom requires an area of $A_{\textrm {mode}}= N L_\pi \lambda / \sqrt {n_{\textrm {core}}^{2}-n_{\textrm {clad}}^{2}}$. Thus, assuming a lithium niobate ProMMI with an applied voltage of 0-1.5 V, a $1~\textrm {cm}^{2}$ chip area would enable full programmability of $N\sim 90$ modes. The universality of the ProMMI programmability naturally extends to other waveguide geometries — for example, a MM waveguide bend, described by an $N$-dimensional unitary evolution $U_{\textrm {bend}}$, can be incorporated into the ProMMI transformation as $U = \prod _{(m,n), l=1}^{N} T_{m,n}^{(l)} \rightarrow (\prod _{(m,n), l=n+1}^{N} T_{m,n}^{(l)})U_{\textrm {bend}}(\prod _{(m,n), l=1}^{n} T_{m,n}^{(l)})$.

4. Conclusion

In conclusion, we introduced an architecture for programmable mode transformations that combines the best attributes of MZM and MMI constructions: (i) a constructive programming algorithm requiring the minimum $N^{2}$ control degrees of freedom; (ii) length scaling $\propto N$ for $N<\alpha _1/\alpha _2$ and otherwise $\propto N^{2}$; (iii) width scaling $\propto N$ but without the "dead-space" between MZM waveguides; (iv) edge roughness loss rates reduced by $\mathcal {O}(1/N)$; (v) far easier waveguide fabrication as 50:50 couplers and bends are eliminated. Due to its large packing density, CMOS electronics are well suited to control the coupling perturbations. Holographic programming enables $N(N-1)$ control fields to control the $N(N-1)$ coupling perturbations. When combined with optical nonlinearities in the waveguide, the ProMMI architecture should be of great use in applications spanning deep learning [4,29] to physics-based simulators [30–32], to general-purpose quantum computing in photonic random walks [33]. Moreover, the ProMMI saturates ‘control density’: i.e., it is as compact as possible to fit $\mathcal {O}(N(N-1))$ control degrees of freedom. This high control density, together with the potential for ultra-low optical loss and constructive programmability, will benefit applications ranging from LIDAR to virtual- and augmented- reality displays, to high-density telecom optical switches, machine learning accelerators, and optical quantum computing devices. We anticipate moreover that our approach extends beyond electromagnetic to other propagating bosonic fields, such as programmable multimode acoustic modes in a slab or magnons in 2D spin ensembles.