# Project C4 • Modeling, design and optimization of 3D waveguides

### Principal investigators

 Prof. Dr. Willy Dörfler (7/2015 - ) Prof. Dr. Christian Koos (7/2015 - ) Prof. Dr. Wolfgang Reichel (7/2015 - 6/2019) Prof. Dr. Carsten Rockstuhl (7/2015 - )

### Project summary

Our aim is to develop fast and easily applicable algorithms for the optimization of low loss 3D freeform waveguides. We are going to achieve this with complete and simplified models approaches.

## Introduction

How to connect two photonic components together such that one could transfer a signal carrying information from one subject to another? It turns out that if the oscillating field is bounded in two dimensions, it will be propagated in the third dimension in the form of waves. Thus, an electromagnetic wave can be transmitted with a small loss in a special structure called a waveguide. Integration of photonic components into opto-electronic semiconductor devices has made tremendous progress. The increase in on-chip integration density means efficient optical off-chip interconnects are becoming indispensable for further progress. Such off-chip interconnects can be realized as 3D freeform waveguides produced by direct-write two-photon lithography (Figure 1).

However, whereas techniques for modeling and designing planar lightwave circuits are well understood, it is still a great challenge to model, design, and optimize 3D freeform waveguide structures.
This is particularly important for so-called photonic wire bonds, which are used to connect integrated photonic circuits across chip boundaries. Figure 2 represents various possible shapes of photonic wire bonds which are connecting components together. Propagation losses of photonic wire bonds depend heavily on the shape of the structure and in particular on its curvature. Straight waveguides, a special case of photonic wire bonds, can be operated without losses.

While applications for optical waveguides range from long distances (such as the transatlantic cable system TAT-14) to very short connections (on the scales of several wavelengths of the transmitted signal) and we focus on short scales. Our aim is to develop fast and easily applicable algorithms for the optimization of low loss 3D freeform waveguides.

## Complete Model

In these technical applications it is necessary to compute the optimal for a freeform connection of two ports. While the long-term goal of this project is to come up with methods to find such shapes within few seconds, we first require a method to evaluate a set of shapes as close to the physical reality as possible.
This means to solve the complete 3D Maxwell system without making strong assumptions, such as a reduction of the space dimension or methods like the beam propagation method (BPM). We are therefore working on a Maxwell-solver based on the finite element method capable of simulating any relevant waveguide shape by computing the EM-field across as much as a hundred wavelengths.
Since such problems typically lead to large, indefinite and ill-conditioned matrices we seek to find parallelizable preconditioning schemes as to enable iterative solution by GMRES. Since this part of the project is supposed to serve as a means to examine the accuracy of other techniques, runtime is not a primary concern, however due to the computational effort of a full 3D Maxwell FEM solver, some effort has to be made to even make this feasible at all.
The goal of this project is not to simply compute the EM-field for a given shape, however, but to compute an optimal shape, which is to solve an optimization problem based on these simulations. A simplified model might compute the signal transmission correctly for any given shape and still converge to a suboptimal shape. So we also need a reference for the optimization part of any simplified model. To this goal we include an adjoint method in the aforementioned solver to facilitate gradient-based optimization.

## Simplified Models

Waveguides are also interesting from the perspective of mathematical modeling and optimization problems for wave transport. In this context it is the question to find the geometrical shape of a freeform waveguide to minimise reflections and losses. Besides the question to construct algorithms to solve such problems reliably, it is also a demand to find sufficient surrogate models that allow us to find practical solutions in a very short time, i.e., a time that allows to use the method in a production process (approximately 10 seconds).
For open and closed planar waveguides, J. Ott has in his master and doctoral thesis shown a way to formulate and solve such freeform problems in case of the Helmholtz equation. We would like to reconsider this case for a method that maps the waveguide into a rectangular structure following the arclength parameter. This is justified by results that show, that at least for closed waveguides, the solution is approximated well by a variable-separated product of simpler functions. In this form it would also support the project that deals with the full Maxwell equation in this way. Further issues are the treatment of the boundary conditions since one has to truncate the domain by some kind of transparent boundary condition to emulate the free infinite space. The most commonly used method is the perfectly matched layer (PML). A newer method is the Hardy-space infinite elements (HSIE) method which should be considered as a contrast.
In this setting we consider the question, whether one can reduce the problem to a Schrödinger-type equation along the arclength and whether this also would give valuable optimization results (in short time) when we optimize the potential. As mathematical results one would like to obtain a rigorous numerical analysis for the different tasks.

## Results

Below are the main results and publications related to the C4 project.

PhD thesis Julian Ott (April 2017) Continuation of the diploma thesis (2013) [Ott15], but now open waveguide in 2D. Definition of halfspace problems and definition of in/outgoing solutions in halfspaces. System of equations by coupling of halfspace problems. Allows treatment of 90-degree bend waveguides and junctions (restriction by a geometrical condition on the halfspaces). "Waveguides" are represented by their $\varepsilon$-value, minimisation with "material derivative", minimisers are "rough sets" [Ott17].

PhD thesis Fernando Negredo. Cheap waveguide models for wirebond connections. Splits the wirebond into small pieces and considers a local model for bending and transmission losses. Especially transmission losses between curves of constant curvature are computed in advance and inserted in a dynamic optimization scheme. First results are summarised in a paper [JOSAA18].

## Publications

1. and . Inverse design of nanophotonic devices with structural integrity. ACS Photonics, 7(8):2190–2196, July . [bibtex]

2. , , , , , , and . Superconducting-nanowire single-photon spectrometer exploiting cascaded photonic crystal cavities. Phys. Rev. Applied, 13(1):014061, January . [preprint] [bibtex]

3. , , , , , , , , , and . Photonic-integrated circuits with non-planar topologies realized by 3D-printed waveguide overpasses. Opt. Express, 27(12):17402–17425, June . [bibtex]

4. and . Discrete gradient flows for general curvature energies. SIAM J. Sci. Comput., 41(3):A2012–A2036, June . [preprint] [bibtex]

5. , , , , , , , , , , , , and . Hybrid integration of silicon photonics circuits and InP lasers by photonic wire bonding. Optica, 5(7):876–883, July . [bibtex]

6. , , , , , , , and . Fast and reliable method to estimate losses of single-mode waveguides with an arbitrary 2D trajectory. J. Opt. Soc. Am. A, 35(6):1063–1073, June . [bibtex]

7. , , , , , , and . Superconducting nanowire single-photon detector implemented in a 2D photonic crystal cavity. Optica, 5(5):658–665, May . [bibtex]

8. , , , , , , , , , , , , and . In situ 3D nanoprinting of free-form coupling elements for hybrid photonic integration. Nat. Photonics, 12(4):241–247, April . [bibtex]

9. , , , , , , , , , and . Subwavelength focusing of Bloch surface waves. ACS Photonics, 4(6):1477–1483, May . [bibtex]

10. , , , , , , and . Hybrid 2D/3D photonic integration for non-planar circuit topologies. In European Conference on Optical Communication (ECOC), pages 707–709, December . [bibtex]

11. , , , , , , , , , , and . Cavity-enhanced and ultrafast superconducting single-photon detectors. Nano Lett., 16(11):7085–7092, October . [bibtex]

12. , , , , , , , , , and . Image formation properties and inverse imaging problem in aperture based scanning near field optical microscopy. Opt. Express, 24(4):4128–4142, February . [preprint] [bibtex]

## Theses

1. . 3D freeform waveguides in integrated optics – concepts, modeling, and applications. PhD thesis, Karlsruhe Institute of Technology (KIT), March . [bibtex]

2. . Numerical methods for shape optimization of photonic nanostructures. PhD thesis, Karlsruhe Institute of Technology (KIT), November . [bibtex]

3. . Halfspace matching: a domain decomposition method for scattering by 2D open waveguides. PhD thesis, Karlsruhe Institute of Technology (KIT), May . [bibtex]

## Other references

1. . Domain optimization for an acoustic waveguide scattering problem. Appl. Math. Optim., 72(1):101–146, August . [bibtex]

2. , , , , , , , , and . Connecting silicon photonic circuits to multicore fibers by photonic wire bonding. J. Light. Technol., 33(4):755–760, February . [bibtex]

3. , , , , , and . Three-dimensional two-photon lithography: an enabling technology for photonic wire bonding and multi-chip integration. In H. Helvajian, A. Piqué, M. Wegener, and B. Gu, editors, SPIE Proceedings, volume 8970, page 897008, March . [bibtex]

4. , , , , , , , and . Photonic wire bonding: a novel concept for chip-scale interconnects. Opt. Express, 20(16):17667–17677, July . [bibtex]