# 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

In the following we describe our main results and publications. The results are structured so that initially, we discuss those close to the project's core, i.e., the study of photonic wirebonds (PWB) losses. Afterward, we discuss different numerical methods developed to predict the losses in PWBs. Finally, we discuss the results of research done to enhance the functionality of PWBs.

#### Shape dependent optical losses of photonic wirebonds

In a publication by M. Blaicher et al. entitled Hybrid multi-chip assembly of optical communication engines by in situ 3d nano-lithography, three-dimensional nano-printing of freeform optical waveguides, also referred to as PWBs, allows for efficient coupling between different photonic chips and can significantly simplify optical system assembly. As a key advantage, the shape and the trajectory of PWBs can be adapted to the mode-field profiles and the positions of the chips. With dedicated test chips, the automated mass production of PWBs with insertion losses of (0.7 $\pm$ 0.15) dB and their resilience in environmental-stability tests and at high optical power was demonstrated [Bea20]. That was an important finding for our project. Furthermore, it clearly shows the reliability and reproducibility of PWBs, which motivates their further optimization.

#### A parallel solver for the waveguide in 3D

To better predict the losses of PWBs, efficient numerical tools are needed. The main results of this part of our work are documented in the doctoral thesis by Pascal Kraft entitled A Hierarchical Solver for Time-Harmonic Maxwell's Equations. This work aimed to solve Maxwell's equations in 3D for a long waveguide trajectory (length/width ratio up to 100), connecting two ports as a building block, and developing an optimization algorithm to minimize the loss for PWBs. Finite Nedéléc's elements were used, and the resulting linear system of equations was solved by the GMRES method. A hierarchical sweeping preconditioner was introduced to cope with the slowdown of the convergence of the method that comes with increasing length. An efficient parallel solver for the waveguide in 3D was implemented that allows to simulate significantly longer waveguides than before [Kra22]. This work also made use of transformation optics explained next.

#### Transformation optics approach

In a further part of our research, we exploited the concept of transformation optics to treat the prediction of losses in PWBs better. The results were published in a manuscript by Nesic et al. entitled Transformation-optics modeling of 3D-printed freeform waveguides [Nea22]. The basic idea of the transformation-optics approach is to transform curved waveguides in the original space into straight waveguides in a virtual space with a spatially inhomogeneous permittivity and permeability profile. Maxwell solvers can efficiently simulate these straight waveguides in a significantly reduced computational volume. Computational complexity was compared to the reference simulations using a commercially available Maxwell solver (CST Studio Suite) on a series of freeform waveguides with planar trajectories. As a result, the transformation optics approach is 3-6 times more efficient than the commercially available Maxwell solver, see Figure 1 [Nea22].

#### Fundamental mode approximation

In that part of our work, we pursue to optimize waveguide trajectories with a reduced model that does not permanently require a computationally expensive full wave solution to Maxwell's equations. For this purpose, it was assumed that waveguides support only a fundamental mode, i.e., that's why it is called the fundamental mode approximation. Under this assumption, the transmission loss of a given waveguide trajectory is derived from bending losses, i.e., the imaginary part of the propagation constant of a mode in a bent waveguide, and from transition losses, i.e., reflection losses that occur at the intersection between waveguides with different curvatures (Figure 2). The complex shape of a PWB is split into small pieces, and we consider 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. It was demonstrated in [Nea18] that this fundamental mode approximation could be used to optimize waveguide trajectories efficiently.

#### Functionalities beyond the transport of light: 3D printed polarization beam splitter

We showed that the functionality of 3D-printed waveguides can be extended by integrating components to manipulate the optical polarization state of the guided modes. This is vital when integrating PWBs into complex optical systems, e.g., when interfacing fibers with degenerate polarization states to highly polarization-sensitive on-chip waveguides. Leveraging the unprecedented design freedom of 3D freeform waveguides, we demonstrated compact ultra-broadband waveguide-based polarization beam splitters (PBS) [Nea21].

We showed the application of 3D-printed PBS in a dual-polarization receiver for coherent communications (Figure 3). Signals with orthogonal polarization states from the single mode fiber (SMF) are first split by the PBS, subsequently rotated by the polarization rotators (PR), and finally coupled to highly polarization-sensitive waveguides on a photonic integrated circuit (PIC) by adiabatic mode-field adapters. The two data streams on the PIC are then independently detected using a pair of coherent optical receivers (Coh. Rx) fed by a joint local oscillator (LO).

The PBS, PR, and additional 3D freeform waveguide elements, e.g., mode-field adapters, form a monolithic structure that can be adapted to a wide range of optical device facets. The structures can be fabricated in situ by high-resolution 3D-laser lithography without high-precision alignment. For the 3D-printed PBS, polarization splitting is accomplished through adiabatic Y-branches of geometrically birefringent polymer waveguides with high-aspect-ratio cross sections and complemented by polarization rotation in waveguides twisted along the propagation direction. We simulated the PBS over a wavelength range between 1250 nm and 1650 nm, finding the transmission from the input of the PBS to the desired mode of the respective output to be better than -2.0 dB over the entire wavelength range, with a maximum of -1.6 dB near 1550 nm. The polarization extinction ratio (PER), i.e., the ratio of the maximum and the minimum power observed in both modes at an output port when varying the excitation at the input over all possible polarization states, reaches over 16 dB.

In our proof-of-concept experiments, we use 3D-printed PBS/PR as part of a receiver in a dual-polarization data-transmission experiment. In this experiment, we launched a dual-polarization 16-state quadrature amplitude modulation (16QAM) signal at a symbol rate of 80 GBd into the input fiber, and we fed the two output signals to a pair of coherent receivers. We swept the optical signal-to-noise ratio (OSNR) at the input of the PBS and recorded the constellation diagrams along with corresponding bit error ratios (BER). For comparison, we replaced the PBS/PR assembly with a commercially available fiber-coupled PBS with a PER over 30 dB and repeated the experiment. The results suggest that the quality of the received signals obtained by the 3D-printed PBS/PR assembly is on par with that obtained by a conventional high-performance fiber-coupled PBS.

#### Inversely designed couplers between chips and photonic wirebonds

We also developed suitable computational tools for the inverse design of structures that can be integrated into PWBs to enhance their functionality. These results were published in the manuscript by X. García-Santiago et al. entitled Bayesian optimization with improved scalability and derivative information for efficient design of nanophotonic structures [GSea21]. The purpose of that initial manuscript was to optimize the structure where light is coupled from the photonic chips into the PWBs. Traditionally, this is done by some adiabatic couplers, but they tend to occupy an extended spatial domain, and shorter couplers are much more desirable. For the optimal design, one needs to find the combination of materials and geometry that results in a device with the best performance. An optimized final structure was found with a Bayesian approach to the shape optimization of the PWB in the coupling region. Optimized couplers showed a similar performance as those of adiabatic couplers, but the necessary length was roughly a factor of four smaller. Transmission could be significantly improved with the optimized devices for the same device length [GSea21].

#### Optimization of an open waveguide junction

The main results are published in the Ph.D. thesis of Julian Ott, [Ott17]. These results are an extension of the results from his diploma work [Ott17] to the case of open waveguides in 2D. The main ingredients of the method are the definition of halfspace problems and the definition of in-/out-going solutions in halfspaces. In the halfspace method we seek the freefrom waveguide in a compact domain that is the intersection of halfspaces. This inner solution is extended beyond each individual halfspace by an explicit solution formula when in- and outlets are perpendicular to the plane or the exterior halfspace is material free. Since the halfspaces define a decomposition of the full plane into overlapping sub-domains, one has to formulate compatibility equations on all the pairwise intersections of halfspaces. It is possible to write this as a set of integral equations and it was shown, and numerically approved, that this problem can be solved with a fixed point iteration. In this setting, "waveguides" are defined by their absorption parameter distribution and not by a prescribed class of geometrical forms. Minimization is thus done using the "material derivative", i.e., the derivative of the cost functional with respect to the material parameter in the underlying differential equation [Ott17]. As a consequence of that, and minimizers are "rough sets", even not simply connected. Examples were the 90-degree bend waveguide and Y-junctions.

#### Advanced functional elements contained in photonic wirebonds

Finally, in a manuscript by Y. Augenstein and C. Rockstuhl entitled Inverse design of nanophotonic devices with structural integrity, we optimized multiple components that can be integrated into PWB to enhance their functionality [AR20]. Unique to our approach was the additional consideration of the mechanical stability of the optimized devices. Therefore, not just the optical function was a part of the objective function but also the stiffness of the devices. That is necessary to reach feasible devices that can be fabricated with state-of-the-art nanofabrication technology (Figure 5).

## Publications

1. , , , , , , , , , , , , and . Transformation-optics modeling of 3D-printed freeform waveguides. Opt. Express, 30(21):38856–38879, October . [bibtex]

2. , , , and . Bayesian optimization with improved scalability and derivative information for efficient design of nanophotonic structures. J. Light. Technol., 39(1):167–177, January . [bibtex]

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

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

5. , , , , , , , , , , , , , , , , , , and . Hybrid multi-chip assembly of optical communication engines by in situ 3D nano-lithography. Light Sci. Appl., 9:71, . [bibtex]

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

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

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

9. , , , , , , , 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]

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

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

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

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

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

15. , , , , , , , , , 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]

## Preprints

1. , , , , , , and . Ultra-broadband polarization beam splitter and rotator based on 3D-printed waveguides. Preprint, Karlsruhe Institute of Technology (KIT), July . [bibtex]

## Theses

1. . A hierarchical solver for time-harmonic Maxwell's equations. PhD thesis, Karlsruhe Institute of Technology (KIT), April . [bibtex]

2. . Photonic packaging enabled by three-dimensional micro-printing. PhD thesis, Karlsruhe Institute of Technology (KIT), July . [bibtex]

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

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

5. . Coherent terabit/s communications using chip-scale optical frequency comb sources. PhD thesis, Karlsruhe Institute of Technology (KIT), July . [bibtex]

6. . Silicon-organic hybrid electro-optic modulators for high-speed communication systems. PhD thesis, Karlsruhe Institute of Technology (KIT), March . [bibtex]

7. . 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]