Nanophotonic computational design

Jesse Lu; Jelena Vučković

doi:10.1364/OE.21.013351

1. Introduction

Currently, almost all nanophotonic components are designed by hand-tuning a small number of parameters (e.g. waveguide widths and gaps, hole and ring sizes). However, the realization of increasingly complex, dense, and robust on-chip optical networks will require utilizing increasing numbers of parameters when designing nanophotonic components.

Opening the design space to include many more parameters allows for smaller footprint, higher performance devices by definition, since original designs are still included in this parameter space. Unfortunately, the lack of intuition for what such designs might look like and the inability to manually search such a large parameter space have greatly hindered the ability to achieve this.

To this end, increasingly sophisticated optimization strategies for specific nanophotonic devices have been developed which are able to search this parameter space. These range from the use of genetic algorithms[1], level-set methods[2], and the optimization of specific geometric parameters [3, 4]. Additionally, much success has been attained[5] through the use of topological design methods that have traditionally been applied in other fields [6, 7].

In this work, we have endeavored to extend the current art by developing and implementing a general computational method for nanophotonic design which extends across all linear nanophotonic devices. Specifically, our method uses the full parameter space to design linear nanophotonic components in three dimensions. Critically, our method requires no user intervention or manual tuning. Instead, a design-by-specification scheme is used to produce designs based solely on a user’s performance specification.

We show that our method can indeed produce designs which are extremely compact, and, at the same time, highly efficient. Furthermore, we demonstrate that devices with novel functionality are easily designed. We also show that our method can be used to produce designs with extreme robustness to wavelength and temperature shift, as well as fabrication error.

Lastly, all our results are produced by simply specifying the functionality and performance of the desired linear device in terms of mode conversion [9]. The variety of successful designs presented suggests that our method may indeed by able to design all linear nanophotonic devices.

2. Problem formulation

In order to produce designs which utilize the full parameter space, and are based solely on the user’s performance specification, we formulate the design problem in the following way:

minimize \sum_{i}^{M} {‖ A_{i} (z) x_{i} - b_{i} ‖}^{2}

subject to α_{i j} \leq | c_{i j}^{†} x_{i} | \leq β_{i j}, for i = 1, \dots, M and j = 1, \dots, N_{i}

z_{min} \leq z \leq z_{max}

The explanation for the various terms in Eq. (1) follows:

A_i(z)x_i − b_i is the physics residual for the ith mode. That is to say, A_i(z)x_i − b_i represents the underlying physics of the problem; namely, the electromagnetic wave equation $(\nabla \times μ_{0}^{- 1} \nabla \times - ω_{i}^{2} ε) E_{i} + i ω_{i} J_{i}$ .
The specific substitutions used in order to transform
$(\nabla \times μ_{0}^{- 1} \nabla \times - ω_{i}^{2} ε) E_{i} + i ω_{i} J_{i} \to A_{i} (z) x_{i} - b_{i}$
are
- E_i → x_i,
- ε → z,
- $\nabla \times μ_{0}^{- 1} \nabla \times - ω_{i}^{2} ε \to A_{i} (z)$ , and
- −iω_iJ_i → b_i.
In contrast to typical schemes for optimizing physical structures, our formulation actually allows for non-zero physics residuals; which can be deduced since A_i(z)x_i −b_i = 0 is not a hard constraint. Instead, this formulation is what we call an objective-first[8] formulation in that the design objective (explained below) is prioritized above satisfying physics.
The (field) design objective consist of the constraint $α_{i j} \leq | c_{i j}^{†} x_{i} | \leq β_{i j}$ . Physically, this constraint describes the performance specification of the device via a series of field overlap integrals at various output ports of the device. Specifically, the $c_{i j}^{†} x_{i}$ terms represents an overlap integral between the E-field of the ith mode (x_i) with an E-field of the user’s choice (c_ij), where the additional subscript j allows the user to include multiple such fields. The amplitude of the overlap integral is then forced to reside between α_ij and β_ij. This mechanism allows the user to express the desired performance of the device as a combination of field amplitudes in various output field patterns. These outputs would be in response to a predefined input excitation, which is determined by the current excitation b_i (−iω_iJ_i) in the physics residual of each mode.
As an example of a design objective for some mode 1 a user might choose to have the majority of the output power reside in some output pattern 1, while ensuring that only a small amount of power be transferred to some output pattern 2. In this case the user would use $0.9 \leq | c_{11}^{†} x_{1} | \leq 1.0$ for the former. and then $0.0 \leq | c_{12}^{†} x_{1} | \leq 0.01$ for the latter; where c₁₁ and c₁₂ are representative of output patterns 1 and 2 respectively.
We note that although the performance description of nearly all linear devices can be specified using such a constraint, the constraint itself is not linear (neither is it convex). Other performance objectives are indeed possible which are nonlinear and nonconvex (such as optimizing for energy density or Q-factor)—convex objectives would be straightforward to implement, while some non-convex objectives would require clever heurestics.
Finally, we note again that the design objective in our formulation is actually a hard constraint. This means that it is always satisfied, even to the extent of allowing for an unphysical field (since the physics residual will not be exactly 0). It is for this reason that we call such a formulation “objective-first”.
The final term in Eq. (1), z_min ≤ z ≤ z_max, is the structure design objective. It is used as a relaxation of the binary constraint, z ∈ {z_min, z_max}, which would ensure that the final design be composed of two discrete materials.

3. Method of solution

We employed the alternating directions method of multipliers (ADMM) algorithm [10] in order to solve Eq. (1). The ADMM algorithm solves Eq. (1) by iteratively solving for x_i, z, and a dual variable u_i.

Since we are working in three dimensions, solving Eq. (1) for x_i is non-trivial in that it involves millions of variables and requires solving for the ill-conditioned A_i(z) matrix. For this reason, we use a home-built finite-difference frequency-domain (FDFD) solver which implements a hardware-accelerated iterative solver[11] on Amazon’s Elastic Compute Cloud. Critically, our cloud-based solver allows us to scale to solve problems with arbitrarily-large number of modes, with no significant penalty in runtime.

In contrast to solving for x_i, solving for z is much simpler since we only consider planar structures; thereby limiting z to have only thousands of variables.

Lastly, in order to arrive at fully discrete, manufacturable structures, we convert z to a boundary parameterization[12] and tune our structure using a steepest-descent method.

4. Results

We demonstrate the effectiveness of our design method by producing designs for a variety of nanophotonic devices.

All of our results are in three dimensions and are planar structures, consisting of a 250 nm etched silicon slab completely surrounded by silica. The permittivity values of silicon and silica used were ε_Si = 12.25 and ε_SiO₂ = 2.25 respectively.

Many, if not all, of the produced designs exhibit novel functionality, high efficiency, and very compact footprints of only a few square vacuum wavelengths, while still remaining manufacturable. We also show that many devices can be designed to exhibit different functionality for different input excitations. Additionally, we show that devices can be designed with large tolerances for errors in wavelength, temperature and fabrication.

4.1. Mode converters

Our first devices consisted of waveguide mode converters. Such devices are simple in that they are single-input, single-output devices. At the same time, such a device is significant because it demonstrates the feasibility of multi-mode on-chip optical networks by showing that high-efficiency mode conversion can readily be achieved in planar on-chip nanophotonic structures.

We show, through the design of mode converters for both the TE- and TM-polarized waveguide modes, that our method is indeed fully three-dimensional. Additionally, the devices also demonstrate very small device footprints (1.6 × 2.4 microns for this device in particular operating at 1550 nm wavelength). Lastly, our devices convert between modes of opposing symmetry.

This is in contrast to prior art[14, 15, 16, 17], which is largely limited to two-dimensional designs, where mode-coupling is between same-symmetry modes, and devices are often larger in the transverse direction because of the need for photonic crystal mirrors.

4.1.1. TE mode converter

Our first result is a mode conversion device operating in the TE polarization, where the primary E-field component of the waveguide mode is polarized in the plane of the structure.

Our performance specification (Fig. 1) for the device was for ≥ 90% of the input power to be transferred from the fundamental waveguide mode, to the second-order waveguide mode. At the same time, we specified that no more than 1% of the input power was to remain in the transmitted fundamental mode.

Fig. 1 Perfomance specification of the TE mode converter. Input mode is the fundamental TE-polarized mode on the left. Primary output mode is the second-order mode on the right. Output power in the transmitted fundamental mode should be no more than 1%. The structure shown is the final three-dimensional design (the same holds for all the following figures in the article).

Abstract

1. Introduction

2. Problem formulation

3. Method of solution

4. Results

4.1. Mode converters

4.1.1. TE mode converter

4.1.2. TM mode converter

4.2. Mode splitters

4.2.1. Spatial mode splitter

4.2.2. TE/TM splitter

4.2.3. Wavelength splitter

4.3. Hubs

4.3.1. 3×3 hub

4.3.2. 4×4 hub

4.3.3. 2×2×2 hub

4.4. Fiber couplers

4.4.1. Compact fiber coupler

4.4.2. Mode-splitting fiber coupler

4.4.3. Wavelength-splitting fiber coupler

4.5. Broadband wavelength splitter

4.5.1. Temperature-robustness of broadband wavelength splitter

4.5.2. Fabrication-robustness of broadband wavelength splitter

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (28)

Equations (4)

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