1. Introduction

As the complexity of photonic circuits increases to meet the high data rates demanded by industry, the fabrication yield, insertion loss and footprint of photonic devices becomes crucial to the success of a photonic product [1,2]. Common photonic operations such as splitting, rotating and mode multiplexing light can be implemented via numerous techniques [3–6]. Adiabatic theory has the advantage of being inherently robust against typical fabrication variation as devices do not rely upon the relative phase difference between eigenmodes and they may be lengthened to increase robustness [7,8].

Adiabatic expressions for particular photonic devices have been derived in literature [9–13], though the complexity of devices that can be analyzed in this manner is limited. More generic adiabatic optimization methods have been developed including STA [14], FAQUAD [10], CTAP [15], SUSY [16] and RAC [17]. These methods are typically limited to one or two dimensions, one or two modes, do not optimize against radiation modes or are based on a heuristic adiabatic criterion. Recently, we presented multidimensional optimization of devices via adiabatic loss limiting (MODALL) [18–24], where we derived an adiabatic criterion and demonstrated adiabatic optimization in two arbitrary dimensions, verifying the results via simulation. We later presented experimental results for devices including an interlayer photonic transition and an adiabatic 2$\times$2.

In this work, we further develop MODALL and apply it to the design and mass fabrication of a TE$_0$/TE$_1$ adiabatic mode multiplexer. We derive MODALL expressions that enable multi-parameter device optimization for an arbitrary number of modes, optimizing against an extra degree of freedom that does not increase the mode computation space and engineering the crosstalk between two modes. The theory presented in this work is applicable to many photonic devices that support one or more modes including tapers, polarization splitter rotators, polarization splitters, adiabatic 3 dB splitters, adiabatic 2$\times$2s, mode multiplexers, waveguide crossings, edge couplers and interlayer transitions.

2. Adiabatic design method

The design methodology originates from the multiparameter adiabatic design expression, Eq. (29) in [18]:

2.1 Extension to optimize for multiple modes

In many cases, components require transmitting two modes; for example, a mode multiplexer or adiabatic 2$\times$2 might transmit TE$_0$ and TE$_1$, and a polarization rotator will typically transmit TE$_0$ and convert TM$_0$ to TE$_1$. To handle these multimode structures, we should extend the analysis to include an adiabatic limit for an arbitrary number of modes rather than just a single-mode. Using Eq. (13) from [18] and the analysis preceding it, we generate adiabatic loss constraints for a set of modes, $v_i$

2.2 Extension to center shifting

In our design equation, $\mathbf {p}_t$ may represent an arbitrary set of design parameters; however, a special case occurs for design parameters that result in an invariant cross-section. In integrated optics, one such parameter could be structure center, $C$. When $C$ changes in $z$, all waveguide boundaries are shifted perpendicular to the $z$-axis by the same amount and the permittivity cross-section does not change, requiring a single eigenmode computation to fully solve a cross-section for all $C$. We let $\mathbf {p}_t = (\mathbf {p}, C)^T$ to introduce the lateral center parameter to our design expression and differentiate it from parameters, $\mathbf {p}$, that do not have the permittivity cross-sectional invariance property. Eq. (1) becomes

In this section, we restrict ourselves to optimizing for two modes, $v_1$ and $v_2$, involving an arbitrary number of interacting modes, though it should be possible to extend the principles introduced in this section to optimize for more than two modes.

When $C$ is constant in $z$, the minimum $\epsilon ^2_v$ satisfying both target modes is $\epsilon _v^2 = \max _i \epsilon _{v_i}^2$ as shown in Section 2.1. For the general case when $\textrm{d} C / \textrm{d} z \neq 0$, satisfying the adiabatic design rule is more challenging as we need to find the $\textrm{d} C / \textrm{d} z = C'$ for each position along the device that satisfies $\epsilon _{v_1}^2$ and $\epsilon _{v_2}^2$, and minimizes $\epsilon _v^2$. To do so, we minimize $\epsilon _v^2$ in $\textrm{d} C / \textrm{d} z$ subject to the constraints $\epsilon _{v_1}^2 \leq \epsilon _v^2$ and $\epsilon _{v_2}^2 \leq \epsilon _v^2$ [25]. The Lagrange function for the minimization is

In these cases, $\textrm{d} C_{v_1}^\mathrm {(min)} / \textrm{d} z$ and $\textrm{d} C_{v_2}^\mathrm {(min)} / \textrm{d} z$ are the shifts of center that minimize $\epsilon ^2_{v_1}$ and $\epsilon ^2_{v_2}$ respectively, and $\left \lbrace \textrm{d} C_{q_1} / \textrm{d} z,\textrm{d} C_{q_2} / \textrm{d} z\right \rbrace$ are the pair of solutions resulting from the $\epsilon ^2_{v_1}=\epsilon ^2_{v_2}$ equality condition. The prevailing case at each point in $z$ will be the one with the smallest $\epsilon _v^2$ and where $\lambda _1, \lambda _2, s_1, s_2 \geq 0$.

To resolve cases C.1 and C.2 we must first find the $\textrm{d} C / \textrm{d} z$ that minimizes $\epsilon _{v_1}^2$ and $\epsilon _{v_2}^2$ by taking the derivative in $C'(z)$ of both sides and setting $\mathrm {d} \epsilon _{v_i}^2 / \textrm{d} C'(z) = 0$ to find the extrema. The differential equation describing $\textrm{d} C_{v_i}^\mathrm {(min)} / \textrm{d} z$, the shift in center that minimizes $\epsilon ^2_{v_i}$ is:

Case C.3 requires solving

2.3 Extension to crosstalk suppression

Our analysis so far has focused solely on minimizing the adiabatic loss of multiple target modes. When optimizing for two modes, it may be desirable for the scattering between the two modes of interest to be weighted more heavily than the scattering to other modes. For an adiabatic 2$\times$2, we would like to minimize scattering from the TE$_0$ and TE$_1$ modes to all modes; however, we may want to bias the calculation such that it favors minimizing scattering between the TE$_0$ and TE$_1$ modes themselves.

We introduce the crosstalk optimization parameters $F_\mathrm {XT}$ and $F_\mathrm {L}$ via the substitution

We shall now outline some details surrounding what can and cannot be achieved using $F_\mathrm {L}$ tuning. The crosstalk is bounded as follows for the three non-trivial $T_{v_1 v_2}(z)$ cases:

For some devices, including edge couplers and polarization splitter rotators, crosstalk between the two modes of interest may be of little importance. In this case, one should set $F_\mathrm {XT}=0$ and $F_\mathrm {L}=1$ to ensure minimum scattering to unwanted radiation modes, while ensuring the optimization is not restricted by minimizing scattering between the $v_1$ and $v_2$ modes. In cases where crosstalk is critical for most two-mode devices, one should choose $F_\mathrm {XT}=1$ as well as an adequate $F_\mathrm {L} \in [0,1]$.

2.4 Discretization

The equations presented cannot, in general, be resolved analytically, and most useful problems require the equations to be discretized to enable numerical computation. Discretization is implemented by changing the derivative operator to its discrete counterpart, $\textrm{d} \to \Delta$. As examples, Eq. (5) becomes

3. Application to a low crosstalk mode multiplexer

To illustrate the adiabatic optimization design process, we apply all three extensions introduced in Section 2 to a mode multiplexer ("modemux") that transmits TE$_1$ to TE$_0$ on waveguide one and transmits TE$_0$ to TE$_0$ on waveguide two. We require the modemux to have a low insertion loss for both modes and have low modal crosstalk across the O-band. Low crosstalk is critical to applications such a two-mode Bragg-based wavelength demultiplexer, where demultiplexed light passes through the modemux twice (once in the forward direction, and once in the reverse direction) [26].

3.1 Design and fabrication

To demonstrate the design strategy, we use a photonic platform comprising 200 nm fully etched SiN clad with SiO$_2$ and fabricated at a foundry on 300 mm Si substrates. The modemux has a 1.7 µm wide input multimode waveguide and a 100 nm tip 200 nm away that adiabatically transition to two single-mode waveguides 900 nm and 1 µm wide separated by a 2.5 µm gap—the slight difference in waveguide widths is needed to avoid degeneracy between the TE$_0$ and TE$_1$ modes. From the modemux cartoon above Fig. 3(b), when TE$_0$ is launched at $E$, ideally all the optical power should transmit to $G$ (no power should transmit to $F$), and when TE$_1$ is launched at $E$, all the optical power should transmit to $F$ (no power should transmit to $G$). At the output side of the modemux, the two single-mode waveguides have a large enough gap not to couple light between them and induce further crosstalk.

The design begins by choosing the parameters to define and optimize the adiabatic structure. The parameters must enable a structure with the aforementioned waveguide width and gap endpoints. Figure 1(a) shows the relevant widths, gap and center used in the design: $p_{w_1}$ is the width of the narrower waveguide, $p_{w_2}$ is the width of the wider waveguide, $p_g$ is the gap between the waveguides and $C$ is the center point of the structure. We let $\mathbf {p}=[p_w,p_g]^T$ where $p_{w_1}=p_w$ and $p_{w_2}=-0.875 p_w + 1.7875$. These parameters let us solve a $\dim {\mathbf {p}} = 2$ ($\dim {\mathbf {p}_t} = 3$) problem that will satisfy our design endpoints.

 

Fig. 1. (a) Cartoon of the design structure with the critical parameters labeled. (b) The shortest path through the optimization space calculated using a discrete shortest path algorithm where the steps between nodes are minimized in $\mathrm {d}C/\mathrm {d}z$. (c) The parameters the shortest path in (b) correspond to along the length of the device. (d) The case that was used to minimize the path in $\mathrm {d}C/\mathrm {d}z$ at each node along the shortest path as defined in Section 2.2.

Download Full Size | PDF

Using Section 2.1, we set up adiabatic loss conditions for both modes of interest: $v_1=\mathrm {TE}_0$ and $v_2=\mathrm {TE}_1$. Section 2.2 allows us to add an extra dimension, $C$, to our design space to obtain a more optimized structure without performing additional eigenmode computations. Finally, Section 2.3 lets us introduce the parameter $F_\mathrm {L}$ that will allow us to further reduce the crosstalk at the cost of increased insertion loss for a given device length. We discretize Eqs. (5)–(10) and use cases C.1, C.2 and C.3 to determine $\mathbf {p}(z)$ and $C(z)$ via a shortest path algorithm similar to that described in Section 4.1 of [18]. We conservatively chose $F_\mathrm {L}=0.2$ for our design as it appeared to slightly improve the simulated crosstalk while negligibly impacting the simulated transmission.

The $p_w$–$p_g$ space (where each node is minimized in $\textrm{d} C / \textrm{d} z$) is shown in Fig. 1(b), with the shortest path to traverse our two target endpoints shown in green. We note that to adhere to the foundry’s manufacturing requirements, the domain of the $p_w$–$p_g$ space is restricted to $p_w \geq {100 \; \textrm{nm}}$ and $p_g \geq {150\; \textrm{nm}}$. The optimized parameters $p_w(z/L)$, $p_g(z/L)$ and $C(z/L)$ where $L$ is the length of the modemux are shown in Fig. 1(c) along with a spline fit represented by a lighter colored line. Figure 1(c) shows the $\textrm{d} C / \textrm{d} z$ case that was used at each position along the design: towards the start of the device, the TE$_1$ loss limit governs $\textrm{d} C / \textrm{d} z$ and towards the end of the device, equality between the TE$_0$ and TE$_1$ loss limits governs $\textrm{d} C / \textrm{d} z$. This is expected as TE$_1$ is more sensitive to bending than TE$_0$ in a multimode waveguide. Towards the end of the device, TE$_0$ and TE$_1$ become increasingly similar and have similar bending properties.

The results shown in Fig. 2 show the simulated TE$_1$ transmission and the simulated crosstalk of various modemux designs for various $F_\mathrm {L}$, as well as the case without center shifting. Figure 2(a) shows how $L_0$—a length that characterizes the adiabatic loss limit [18]—varies in $F_\mathrm {L}$, reveals decreasing $F_\mathrm {L}$ results in a longer design as expected. Figures 2(b) and 2(c) confirm that a smaller $F_\mathrm {L}$ has improved crosstalk but requires a longer device to maintain a high TE$_1$ transmission. For this particular design, the optimization in $\textrm{d} C / \textrm{d} z$ was critical to achieving a low crosstalk.

 

Fig. 2. Eigenmode expansion simulations of various modemuxes tuned using $F_\mathrm {XT}$. (a) Sweep of $F_\mathrm {L}$ showing how $L_0$ increases as $F_\mathrm {L} \to 0$. (b) Crosstalk between the TE$_0$ and TE$_1$ modes through the modemuxes. (c) Transmission of the TE$_1$ mode through the modemuxes.

Download Full Size | PDF

3.2 Measurements

Measurements were conducted across four 300 mm wafers each containing 64 reticles with conservatively sized modemuxes of lengths 190 µm, 280 µm and 400 µm. The 190 µm and 280 µm lengths approximately correspond to nulls in the length–crosstalk simulations shown in Fig. 2(b), and the 400 µm length was chosen as an extreme case against which the shorter length devices could be benchmarked. A fiber array angled at 8° was used to couple light into and out of each reticle via grating couplers spaced 127 µm. The grating couplers exhibited minimum reticle-to-reticle variation at 1310 nm, with variation increasing significantly closer to the edges of the O-band. Figure 3 presents transmission and crosstalk measurements for the 280 µm long modemux, though no significant performance differences were seen between the various modemux lengths as expected from simulation. A face-to-face modemux configuration was used to measure the transmission of a single modemux by calculating $(C+D)/2$ and the results are shown in Fig. 3(a). Figure 3(b) shows the crosstalk measurements made using just a single modemux calculated via $F/(F+G)$. The solid lines show the median crosstalk across all reticles in a particular wafer at each measured wavelength, while the boxplots show the spread in transmission for the center wavelengths of the four CWDM bands.

 

Fig. 3. Measurement macro description and measurement results for transmission and crosstalk across four wafers each containing 64 reticles. (a) Boxplots for the transmission for the TE$_0$ and TE$_1$ mode at 1310 nm. (b) Crosstalk measurements showing the TE$_0$ to TE$_1$ crosstalk with boxplots showing the the variation across the four CWDM bands. The solid lines show the median crosstalk in wavelength across the 64 reticles. The cartoons above (a) and (b) show the insertion loss and crosstalk macros respectively with the arrows to the left left indicating optical inputs (from a swept laser) and the arrows to the right indicating optical outputs (to off-chip photodetectors).

Download Full Size | PDF

Nearly every reticle measured had a crosstalk $<\! {-40}\; \textrm{db}$ across the O-band with the mean crosstalk being approximately −45 dB. Accurate measurement of modemux transmission was not a priority, but was measured at 1310 nm (to minimize reticle-to-reticle variation) to be $< {-0.1}\; \textrm{db}$ with the limitations being the test macro and measurement fiber alignment accuracy. The characterization results show that the modemux operates well with extremely low crosstalk across the whole O-band and does not have significant insertion loss for either the TE$_0$ or the TE$_1$ modes, verifying the theory outlined in Section 2. There is good agreement between the simulated crosstalk at 1310 nm shown in Fig. 2(b) (−46 dB at $L= {280}\; \mathrm{\mu m}$) and the median crosstalk measurements at 1310 nm that lie between −45 dB and −50 dB.

4. Conclusion

We extend MODALL theory to enable adiabatic device optimization of many modes, optimization against a particular dimension that does not require extra eigenmode computation and optimization of crosstalk in two-mode devices. The theory was applied to a TE$_0$/TE$_1$ adiabatic mode multiplexer designed using all the theoretical extensions introduced, mass produced at a photonic foundry and characterized across multiple 300 mm wafers. The adiabatic mode multiplexer demonstrated robust performance with 100% of reticles having $< {-38}\; \textrm{db}$ crosstalk at 1310 nm and median crosstalk at 1310 nm matching well with the simulation performed at 1310 nm. The proposed MODALL extensions pave the way towards highly optimized adiabatic photonic devices including power splitters, mode multiplexers and polarization rotators. Future work will focus on applying the theory presented in this work to other adiabatic devices critical to photonic transceivers.