## Abstract

Starting from the fully vectorial coupled local-mode theory, a general design rule for adiabatic waveguide design is derived. The design rule guarantees adiabaticity and puts an upper limit on the transmission loss of any guided wave transformation. The rule is applicable to any waveguide geometry, admits multi-dimensional optimization, and accounts for radiative loss to guided and radiation modes. Moreover, the design technique is enhanced with further optimization procedures that eliminate coupling between particular pairs of modes. The utility of this toolbox of design rules and methods is illustrated through the design of a 2×2 coupler and a bilayer waveguide transition. These design procedures can be used to optimize single-mode and multimode photonic devices by varying one or more waveguide parameters.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The emergence of fiber optics during the 1960s sparked investigation into guided wave optics [1]. The challenge of efficiently transmitting and manipulating optical modes along imperfect fibers led to rigorous analyses of modal propagation in perturbed dielectric waveguides and the further development of local mode theory [2,3].

In the early 1970s, Snyder showed that modal scattering is proportional to the rate at which a permittivity boundary is deformed [4,5]. Restating the problem of modal loss in terms of an engineerable quantity—refractive index profile—simplified device analysis and introduced new device engineering opportunities. Thereafter, adiabatic criteria for devices including fiber tapers, couplers, and splitters were established [6–10].

Advances in integrated optics over the past two decades have led to the miniaturization of photonic components and the further development of adiabatic devices [11–16]. Moreover, recent efforts in quantum technology have led photonics to borrow adiabatic concepts from quantum theory leading to adiabatic design protocols, including shortcuts to adiabaticity (STA), fast quasiadiabatic dynamics (FAQUAD), and coherent tunneling adiabatic passage (CTAP) [17–21]. Many of today’s adiabatic devices are optimized using a combination of algorithmic optimization, generally in a single dimension, and heuristic optimization. The literature shows a discrepancy in the choice of adiabatic design criteria: ray optics may be used to optimize low refractive index contrast devices, while the adiabatic criteria from quantum mechanics is mapped to photonics without rigor, and the phase relation between modes is often ignored with unclear repercussions.

In this work, we present the derivation and application of an adiabatic metric based on the well-established local-mode theory that is suitable for $m$-dimensional photonic device optimization. We use it to optimize devices in one and two dimensions to first order—neglecting phase interactions between modes. We then present a method for a further optimization whereby the phases of particular modes are aligned by minimally perturbing the pre-optimized design. The optimized designs are compared to alternate designs using eigenmode expansion (EME) and finite-difference time-domain (FDTD) modeling.

## 2. General adiabatic design method

#### 2.1 Coupled local-mode theory

In the coupled local-mode theory [6], the equation of modal amplitude propagation in a non-absorbing medium is

In the latter definition of $C_{mn}$, the evolution of the coupling coefficients is dictated by the changes in relative dielectric permittivity. In general, changes in the spatial distribution of the dielectric permittivity can take many forms (*e.g.*, adjustments to waveguide width, height, position, material, etc.). For this reason, it is convenient to parameterize such dependencies, in which case the partial derivative with respect to $z$ can be expanded by the chain rule

#### 2.2 Adiabatic solution and design rule

The preceding equations describing the transfer of power between different local modes are exact. To evaluate them in the adiabatic limit, it is convenient to make some simplifying assumptions. The first assumption is that it is reasonable to neglect power in all modes except a single forward-propagating mode of interest, labeled $\nu$. Under this assumption, the equations of evolution are simplified to

After integrating this equation and applying integration by parts once, the solution is

At this point, the adiabatic criterion is assumed to be valid, namely the $C_{m\nu } \left (z\right ) / \Delta \beta _{\nu m} \left (z\right )$ term is small and slowly varying or invariant, and the second term of Eq. (6) can be neglected. Then the amplitude in any particular mode is given by

Unlike previous analyses, we do not limit the analysis to coupling only between two modes, and instead we include coupling from the mode of interest to all other modes, guided modes and radiation modes alike. In this case, the total power lost to all modes is

We now impose our adiabatic design rule. The constraint is to demand that the sum of all modes’ adiabatic criterion squared is constant. We assign this constant the symbol $\epsilon ^2$, and it is explicitly written as

By the Cauchy-Schwarz inequality

Using the expressions for the coupling coefficient in Eqs. (2) and (3) and the chain rule expansion in Eq. (4), we can write the more explicit expression in terms of parameters that describe the waveguide

In the case of only a single parameter, the equation is rearranged to produce the differential equation

Thus, we are left with an ordinary differential equation that can be integrated to solve for the optimal $p\left (z\right )$. Integrating over the entire device length gives the theoretical lower bound for transmitted power of the target mode, which is

where $L_0$ is a characteristic length, defined asTherefore, the maximum loss of an optimized transition is proportional to the inverse square of the transition length, which has been found previously [11,13]. Moreover, the characteristic length, $L_0$, indicates the length of a transition for a particular loss, *e.g.*, a transition guaranteed to have ${\leq }$1% loss must be a length ${\geq }10L_0$.

#### 2.3 Application to adiabatic 2$\times$2 coupler

To illustrate the method, consider an adiabatic 2$\times$2 coupler consisting of two silicon nitride (Si$_3$N$_4$) rib waveguides buried in silicon dioxide (SiO$_2$). The two Si$_3$N$_4$ waveguides are 200 nm thick and have a constant edge-to-edge separation of 250 nm. The remaining parameters that can be varied along the device length are the waveguide widths and the structure center (see Fig. 1). To reduce this to a single parameter, the center is assumed to be constant and the width changes are complementary, *i.e.*, an increase in one waveguide width is accompanied by a commensurate reduction in the other waveguide width. The waveguide widths are varied from 1000 nm at the symmetric output port to $\pm 300$ nm wider/narrower at the asymmetric input port.

The mode fields and propagation constants at 1550 nm wavelength are calculated numerically using the Lumerical [23] MODE Finite Difference Eigenmode (FDE) solver. The “box” modes that are confined by the non-absorbing simulation boundaries are used to approximate the radiation modes, which is found to be a good approximation if the boundaries are far enough from the structure edges and enough modes are included in the calculations. Since the structure only supports two guided TE modes, the first-order mode is used for the optimization (*i.e.*, $\nu = 2$ where mode indexing begins at 1). Using the first-order mode ensures that the effects of scattering to radiation are taken into account, since the first-order mode has strong coupling to both the fundamental mode and the radiation modes. The solved modes are used to calculate the coupling coefficients for the first-order mode using Eq. (16), and the optimal width variation as a function of length is calculated by numerically integrating Eq. (17). The resulting waveguide profile is shown in Fig. 2, and the calculated $L_0 = 3.77$ μm.

The performance of the calculated optimal waveguide profile is investigated using the Lumerical MODE EigenMode Expansion (EME) numerical propagation solver. The propagation problem is solved for a wavelength of 1550 nm, and the length is swept from ${<}L_0$ to ${>}20 L_0$. Both Mode 1 (wider mode at input, symmetric mode at output) and Mode 2 (narrower mode at input, anti-symmetric mode at output) are separately launched into the input port and measured at the output. A measure of the adiabaticity of a structure is the fractional transmission of the launched mode through the output. Figure 3 is a plot of the transmission of the two modes as a function of the structure length. At a length of around 35 μm, the transition is nearly lossless for both modes. In addition, as indicated by the dotted theoretical limit line, the loss of both modes remains below the $L_0^2 / L^2$ threshold.

A critical and unique feature of this optimization approach is that it takes into account coupling to radiation modes. This is essential for the design of adiabatic transitions in single-mode waveguides (as will be explored in an upcoming section), but it is also important for optimization of multimode guiding structures such as the 2$\times$2 coupler. In the 2$\times$2 coupler case, if only intermodal coupling between the fundamental and first-order modes is considered, then the optimized transition can still experience significant radiative loss. For example, the optimization equation, Eq. (17), can be used while only considering Modes 1 and 2 and thereby eliminating the summation. This produces a different waveguide profile that is less optimal than the one that includes all modes. Consequently, as shown in Fig. 3, there is lower transmission of the two modes due to the higher radiative scattering. Thus, by incorporating all modes into the optimization procedure, it is possible to design an adiabatic waveguide transition that avoids radiative scattering and consequently suffers lower loss at a shorter length.

## 3. Perturbative optimization for a particular transition length

#### 3.1 Derivation of the optimization approach

As shown in the preceding sections, the derived adiabatic design rule guarantees that the loss at any point along the transition is less than a theoretical $L_0^2 / L^2$ limit, where $L_0$ is a characteristic length of the unoptimized structure. However, the design rule does not guarantee that the transmission at the output is maximized for a chosen length. To achieve this, the oscillatory power transfer between local modes must be taken into account. Adjusting the power transfer oscillations between local modes appropriately is a complicated problem that does not guarantee adiabaticity, *i.e.*, the transition can still be sensitive to small variations in wavelength, waveguide dimensions, etc. Still, if one uses the design rule above as a starting point and only varies one parameter, it is straightforward to make perturbations to the design to achieve the desired local-mode power transfer that greatly increases the output transmission.

To perform this perturbative optimization, it is useful to reformulate the solution for the amplitude lost to an unwanted mode in terms of the parameter being varied:

*e.g.*, the first-order mode of the 2$\times$2 coupler) and the mode that most strongly interacts with it (

*e.g.*, the fundamental mode).

The optimization approach proceeds as follows. The $\left (\textrm {d}p/\textrm {d}z\right )^{-1}$ found using Eq. (17) is perturbed by some unknown function $\delta \left (p\right )$, and the first constraint of the optimization problem is to seek a $\delta \left (p\right )$ that satisfies

In addition, to preserve the chosen length of the adiabatic transition, another constraint is given by

There are a multitude of solutions that satisfy these conditions, so it is useful to cast this as an optimization problem. Since we are seeking to make small changes to the design that maintain adiabaticity, it is natural to choose to minimize the $L^2$-norm of the perturbations to $\left (\textrm {d}p/\textrm {d}z\right )^{-1}$:

The result is a problem of finding the extremal of the above functional subject to three equality constraints. From this, the Euler-Lagrange equation with Lagrange multipliers can be derived as

By solving this equation and the constraint equations, one finds $\phi \left (p\right )$, $\lambda _1$, $\lambda _2$, and $\lambda _3$ that produce an adiabatic transition that eliminates power in an unwanted mode while preserving the length and minimizing the changes from the initial adiabatic design.

Note that this is one of many options for canceling power to an unwanted mode, as other metrics can be minimized besides the one chosen in Eq. (25). However, the choice presented here has given the best results to date when compared with other alternatives. Moreover, the optimization need not be performed by solving the constrained Euler-Lagrange equations. There are a multitude of numerical nonlinear optimization algorithms for constrained problems that can be used. Thus, this outlines just one approach for further optimizing an adiabatic design, albeit in a manner that appears to be generally compatible with any waveguide geometry supporting two guided modes.

#### 3.2 Perturbative optimization of the 2$\times$2 coupler

The perturbative optimization approach outlined above can be applied to the 2$\times$2 coupler design explored previously. As discussed in the previous section, the output transmission of an unwanted mode can be canceled for a particular transition length through small changes to the adiabatic design. The lengths 35 μm (near the first high transmission point in Fig. 3) and 50 μm (near the first transmission dip in Fig. 3) are chosen for further optimization for the 2$\times$2 coupler. Solving the optimization problem of the preceding section results in the new taper shapes illustrated in Fig. 2.

Three-dimensional calculations performed with the Lumerical Finite-Difference Time-Domain (FDTD) solver are used to evaluate the performance of the new adiabatic 2$\times$2 designs. The wavelength range explored is centered at 1550 nm with an optical bandwidth of 100 nm. Perfectly matched layer (PML) boundary conditions are used to accurately treat radiative loss. To consider the worst case, the first-order mode is launched at the input port and the coupling to the two output modes is calculated. Using the power in the symmetric and anti-symmetric output modes, the worst-case split ratio is calculated by assuming the relative phase between the two modes is matched. Any other phase condition resulting from a mismatch at the output or further propagation in a coupled state after the output port will result in a split ratio no worse than this value, and the ideal 50%/50% split case will only be guaranteed if either Mode 1 or Mode 2 has no power. The excess loss is also calculated as consisting of any power not contained in the first two modes, *i.e.*, power in unguided radiation modes.

Figure 4 shows plots of the worst-case split ratio and excess loss for the nominal 2$\times$2 design and the designs optimized for 35 μm and 50 μm transition lengths. The plots show the calculated values at the 1550 nm central wavelength (solid lines) as well as the extrema over the full 100 nm bandwidth (shaded area). In all cases, the chosen taper shapes have excess loss less than approximately 0.02 dB for tapers 35 μm long or longer. The primary influence of the perturbative optimization for particular lengths is to improve the worst-case split ratio, since canceling of power in one of the first two modes results in a more ideal split. In particular, at the lengths of optimization (35 μm and 50 μm), the worst-case split ratio is approximately 50% $\pm$ 1% over the entire 100 nm bandwidth. At lengths other than the optimized ones, the worst-case split ratio degrades slightly, but there is still a general trend of improved split ratio and excess loss as length is increased, as is to be expected for adiabatic designs. Thus, this perturbative optimization approach tends to maintain adiabatic behavior while allowing for better performance targeted at a particular shorter length.

## 4. Multi-dimensional design

#### 4.1 Extension of design rule to multiple parameters

Up to this point, variation of only a single structure parameter has been considered for optimization of an adiabatic transition. However, the optimization procedure is generalizable to an arbitrary number of parameters. For multiple parameters, the optimization approach is similar to that already outlined in Section 2.2, except now the mode evolution can take many different paths through an $N$-dimensional space (for $N$ parameters) where an optimal path can be found.

The general design rule for an adiabatic transition is given by Eqs. (14)–(16). Unlike earlier, the problem is no longer reduced to a single parameter. In vector notation, the design expression is

There are multiple options for finding the shortest path that satisfies Eq. (29). One is to parameterize the variables $\mathbf {p}$ by a variable $t$, such that the equation becomes

The problem can be solved by using the calculus of variations to minimize the functional in Eq. (31) with respect to $\mathbf {p}$ subject to Eq. (30). This results in a complicated system of nonlinear differential equations. Although these equations can be solved using numerical nonlinear solvers, the procedure can be quite cumbersome and inefficient.

An alternative approach, and the one used here, is to discretize Eq. (29) and use it to create a weighted connected graph. The resulting discrete equation is

*Dijkstra’s algorithm*, to $S$, the shortest path, $\vec {\mathbf {p}}_{\textrm {des}}$, from any initial $\mathbf {p}_{\textrm {i}}$ to any final $\mathbf {p}_{\textrm {f}}$ may be calculated.

#### 4.2 Canceling of coupling coefficient

As is evident from inspecting Eq. (32), it may be possible to choose a path through the multidimensional space such that the coupling between two modes partially or even entirely cancels over the whole transition length. Recently, it has been proposed and demonstrated that such a canceling can be achieved by appropriately bending a 2$\times$2 coupler [24]. This design, coined a “Rapid Adiabatic Coupler,” is a special case of traversing a multidimensional space to cancel the largest coupling coefficients (in the case of [24], the two parameters are waveguide width and structure center). Clearly this can be extended to many other parameters that describe a waveguiding structure, and for the case of two parameters the only strict requirement is to satisfy the equation

When this canceling rule is applied for two parameters, the problem is immediately reduced again to a single parameter problem since $p_j$ will depend explicitly on $p_i$. In this case, the optimization procedure introduced first in this paper can be applied (Eq. (17)). Moreover, for a two-mode structure, coupling to radiation modes becomes the dominant source of loss since the coupling between the two guided modes is negligibly small. Thus, even for a two-mode structure, incorporation of many more modes (and the radiation modes in particular) is critically important to determine the optimal adiabatic design.

#### 4.3 Two-parameter optimization of the 2$\times$2 coupler

The multi-parameter formulation of the adiabatic design method allows for optimization of the 2$\times$2 coupler over a larger parameter space. As suggested in the previous section and by preceding literature, one such added parameter is the lateral position of the structure center. In the following, the structure center is allowed to vary in addition to the waveguide width considered earlier.

Applying the design rule summarized by Eq. (32), a two-dimensional connected graph is constructed over the width-center space. Each node (point in width-center space) is assigned a weight for each path through it, *i.e.*, a particular path length between it and its neighboring nodes in multiple directions. Figure 5 illustrates the shortest step direction through each node with a collection of line segments. The orientation of the line segment indicates the direction of lowest weight, and thus the representation is essentially a vector field plot of the shortest paths at any given point.

Since the waveguide center can be freely chosen, this leaves the endpoint along that axis a free variable. Consequently, it is possible to choose to optimize with different targets in mind. In particular, it is possible to optimize for the path that minimizes crosstalk between the two guided modes, which produces the best split ratio, or for the shortest total path, which produces lower excess loss for a given transition length. Both cases are considered here, and the vector field plot of Fig. 5 also shows the paths for shortest total path and for coefficient canceling for the two guided modes. In particular, the shortest path algorithm is used to solve for the former, while the condition in Eq. (33) is solved for the latter. The bend and taper profiles are constructed from the path and corresponding $\Delta z$ weights, and the results are illustrated in Fig. 6.

FDTD calculations verify the performance of these 2$\times$2 coupler designs. The simulation conditions are the same as described previously, and the results are plotted in Fig. 7. The 2$\times$2 coupler designed to cancel the coupling coefficient between the two guided modes displays the desired characteristic of nearly ideal split ratio over a wide range of transition lengths. The split ratio is better than approximately 50% $\pm$ 1% over the 100 nm bandwidth for all lengths greater than 50 μm, and almost all lengths show a 50% $\pm$ 5% split ratio. Moreover, since the two-parameter problem is reduced to a single parameter by the coefficient canceling condition, this design can be further improved using the perturbative optimization technique described earlier. In contrast, the second 2$\times$2 design does not show as good of a split ratio, although lengths greater than 30 μm exhibit worst-case splits better than 50% $\pm$ 5%. However, as is the target of this design, the excess loss is reduced as compared to the coefficient-canceling design. Although this is less critical for a device such as a 2$\times$2 coupler, it is critically important for many other designs, particularly single-mode designs, as will be elucidated next.

#### 4.4 Two-parameter optimization of single-mode waveguide transitions

The previous examples have focused on devices that predominantly radiate power to one other mode. However, the adiabatic design method can, in theory, minimize the loss to any arbitrary number of guided or radiation modes. In this section, the multi-parameter adiabatic design method is employed to design a bilayer transition consisting two SiO$_2$ clad waveguides, namely a Si$_3$N$_4$ layer 300 nm above a lower Si layer. The transition should adiabatically transfer TE-polarized optical power at a wavelength of 1310 nm from a single mode waveguide in the lower Si layer (nominal width 450 nm) to a single mode waveguide in the upper Si$_3$N$_4$ layer (nominal width ${1}\;\mathrm{\mu} \textrm {m}$).

Given the significant asymmetries between the two waveguide layers including refractive index, thickness, and nominal single mode waveguide width, the upper- and lower-layer waveguides are not expected to follow the same width profiles. Naturally, this is a 2D problem with $\mathbf {p} = [w_{\textrm {up}}, w_{\textrm {low}}]$ where $w_{\textrm {up}}$ and $w_{\textrm {low}}$ are the variable widths of the waveguides in the upper and lower layers respectively.

Because both waveguides are single mode, it is anticipated that the accurate computation of Eq. (32) will depend on the inclusion of a sufficiently large number of box modes. Significant underestimation in the solution to Eq. (32) may occur should the simulation window be too small, or an insufficient number of modes included in the calculation—this type of computational inaccuracy is analogous to that encountered when verifying a traditional EME propagation simulation. For this example, the first 100 modes of the waveguide system are solved and used to compute Eq. (32), and a visualization analogous to Fig. 5 is shown in Fig. 8—again, the slope of each node indicates the most efficient direction to traverse that node, and the redness indicates a longer step size. The shortest path to adiabatically transition between a Si waveguide of nominal width in the lower layer and a Si$_3$N$_4$ waveguide of nominal width in the upper layer is overlayed in violet.

The design path naturally avoids the deep red region in Fig. 8 where both waveguide widths are relatively narrow. In the red region, the fundamental mode is more closely phase velocity matched to the radiation modes. The vertical node lines, which are present throughout most of the design space, indicate that changes in the Si waveguide width are more costly than Si$_3$N$_4$ waveguide width changes. Thus, a design that more slowly varies the lower waveguide width than the upper is expected.

Figure 9, which shows the number of modes required to achieve ${\geq } 0.9 \Delta z(\vec {\mathbf {p}}_{\textrm {des}})$, provides insight into the locations along the design that are more or less radiative and helps to confirm that the number of modes used to compute Eq. (32) is sufficient. Towards the beginning of the transition, between nodes #0 and #7 (Si waveguide width reduces from $450$ to ${\sim }{400}\;\textrm {nm}$), ${\geq } 24$ modes significantly contribute to the step size, while between nodes #7 and #73 (tapering the Si$_3$N$_4$ waveguide), far fewer modes contribute significantly. Therefore, any radiation from the Si waveguide towards the beginning of the device is expected to have a larger radiation angle than that of the Si$_3$N$_4$ waveguide.

The transition design is generated by extracting the waveguide width profiles from the design path, as shown in Fig. 10. A differentiable function is fit to the width profiles to ensure the derivative of Eq. (3) remains well behaved along the length of the design. To verify the design, the transition is loaded into Lumerical MODE EME, and the transmission is simulated over a range of device lengths. Note that the designed transition slightly increases the width of the Si$_3$N$_4$ waveguide between the middle and end of the device, resulting in a substantially different width profile than that of a *conventional* (linear taper/inverse taper width profile) design. Figure 10 also shows the transmission results of the designed transition, along with the transmission results of a *conventional* transition. The designed transition achieves ${>}99\%$ transmission at a device length of ${19.5}\;\mathrm{\mu} \textrm {m}$ ($L_0={3.26}\;\mathrm{\mu} \textrm {m}$), outperforming the conventional transition, which requires a device length of ${25.5}\;\mathrm{\mu} \textrm {m}$ to achieve the same transmission.

## 5. Conclusion

Starting from well-established coupled local-mode theory, we have derived an adiabatic criterion suitable for single and multi-dimensional optimization. The criterion determines the expected worst case radiation between a target mode and an arbitrary number of unwanted modes for a particular variation in waveguide geometry—it is limited neither to optimizations in a single parameter nor to optimizations against a single mode. A 1D design algorithm based on the established criterion is applied to a 2$\times$2 adiabatic coupler, which is then further optimized using proposed second round perturbative optimization. Multi-parameter designs are investigated for both a 2$\times$2 adiabatic coupler and a bilayer single mode transition, highlighting the broad applicability of the design process.

## Acknowledgments

The authors wish to acknowledge the support and feedback provided by Prakash Gothoskar and Mark Webster of Cisco Systems, Inc.

## Disclosures

The authors declare no conflicts of interest.

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