Abstract
A major difficulty in applying computational design methods to nanophotonic devices is ensuring that the resulting designs are fabricable. Here, we describe a general inverse design algorithm for nanophotonic devices that directly incorporates fabrication constraints. To demonstrate the capabilities of our method, we designed a spatialmode demultiplexer, wavelength demultiplexer, and directional coupler. We also designed and experimentally demonstrated a compact, broadband 1 × 3 power splitter on a silicon photonics platform. The splitter has a footprint of only 3.8 × 2.5 μm, and is well within the design rules of a typical silicon photonics process, with a minimum radius of curvature of 100 nm. Averaged over the designed wavelength range of 1400–1700 nm, our splitter has a measured insertion loss of 0.642 ± 0.057 dB and power uniformity of 0.641 ± 0.054 dB.
Introduction
Nanophotonic devices are typically designed by starting with an analytically designed structure, and handtuning a few parameters^{1}. In recent years, it has become increasingly popular to automate this process with the use of powerful optimization algorithms. In particular, by searching the full space of possible structures, it is possible to design devices with higher performance and smaller footprints than traditional devices^{2,3,4,5,6,7,8}.
A major challenge when designing devices with arbitrary topologies is ensuring that the structures remain fabricable. Many of these computationally designed structures have excellent performance when fabricated using highresolution electronbeam lithography, but they have features which are difficult to resolve with industrystandard optical lithography^{3, 7, 8}.
Building on our previous work^{5, 7, 9}, we propose an inverse design method for nanophotonic devices that incorporates fabrication constraints. Our algorithm achieves an approximate minimum feature size by imposing curvature constraints on dielectric boundaries in the structure. We then demonstrate the capabilities of our method by designing a spatialmode demultiplexer, wavelength demultiplexer, and directional coupler, and experimentally demonstrating an ultrabroadband 1 × 3 power splitter. All of our designs are compact, have no small features, and should be resolvable using modern photolithography. Additionally, with the exception of the wavelength demultiplexer, all of our devices are well within the design rules of existing silicon photonics processes.
Design Method
Due to the complexity of accurately modelling lithography and etching processes, most attempts to incorporate fabrication constraints into computational nanophotonic design have focused on heuristic methods. One approach is to restrict the design to rectangular pixels which are larger than the mininum allowable feature size^{10}. The resulting Manhattan geometry, however, is restrictive and likely not optimal for optical devices. Another method involves applying a convolutional filter to the design followed by thresholding^{11,12,13}, which can introduce artifacts smaller than the desired feature size. The approach used in this work is to impose curvature constraints on the device boundaries, which avoids the aforementioned issues. Curvature limits have been successfully applied in earlier work^{4}, but were not described in detail nor validated with experimental demonstrations.
Level Set Formulation
We assume that our device is planar and consists of only two materials. We can represent our structure by constructing a continuous function \(\varphi (x,y):{{\mathbb{R}}}^{2}\to {\mathbb{R}}\) over our design region, and letting the boundaries between the materials lie on the level set ϕ = 0. The permittivity ε is then given by
The advantage of this implicit representation is that changes in topology, such as the merging and splitting of holes, are trivial to handle. We can also manipulate our structure by adding a time dependence, and evolving ϕ(x, y, t) as a function of time t with a variety of partial differential equations collectively known as level set methods^{14, 15}.
To design a device, we first choose some objective function f [ε] which describes how well the structure matches our electromagnetic performance constraints^{5, 7}. We then evolve our structure, represented by ϕ, in such a way that we minimize our objective f. We can achieve this by adapting gradient descent optimization to our level set representation. The level set equation for moving boundaries in the normal direction is
where \(\nabla \varphi ={\varphi }_{x}+{\varphi }_{y}\) is the spatial gradient of ϕ, and v(x, y) is the local velocity. To implement gradient descent, we choose the velocity field v(x, y) to correspond to the gradient of the objective function f [ε]^{15}. The gradient can be efficiently computed using adjoint sensitivity analysis^{3, 4, 6, 9}. As t → ∞, ϕ converges to a locally optimal structure.
Unfortunately, this approach tends to result in the formation of extremely small features. We can avoid this problem by periodically enforcing curvature constraints. The level set equation for smoothing out curved regions is
where the local curvature κ is given by
Although equation (3) removes highly curved regions more quickly^{14}, the boundaries are eventually reduced to a set of straight lines with zero curvature as t → ∞.
From a fabrication perspective, we only need to smooth regions which are above some maximum allowable curvature κ _{0}. We can do this by introducing a weighting function
and modifying equation (3) to be
If we evolve ϕ with equation (6) until it reaches steady state, the maximum curvature will be less than or equal to κ _{0}.
Although curvature limiting will eliminate the formation of most small features, it does not prevent the formation of narrow gaps or bridges. We detect these features by applying morphological dilation and erosion operations to the set ϕ > 0, and checking for changes in topology. Once detected, these narrow gaps and bridges can be eliminated by “cutting” them in half, and then applying curvature filtering to round out the sharp edges.
The final design algorithm is as follows:

1.
Initialize ϕ and δt.

2.
Repeat until δt < δt _{ min }.

(a)
Let ϕ′ ← ϕ.

(b)
Gradient descent: evolve ϕ′ with eqn. (2) for time δt.

(c)
Gap and bridge removal: detect any small gaps or bridges, and modify ϕ′ to remove them.

(d)
Curvature limit: evolve ϕ′ with eqn. (6) until convergence.

(e)
If f[ε[ϕ′]] < f[ε[ϕ]], then let ϕ ← ϕ′ and increase δt.
Otherwise, decrease δt.
A detailed description of the objective function f[ε] and implementation details can be found in the supplementary information.
Designed Devices
To demonstrate the capabilities of our design method, we designed a variety of threedimensional waveguidecoupled devices on a silicon photonics platform. All of the structures we show here consist of a single fullyetched 220 nm thick Si layer with SiO_{2} cladding. Refractive indices of n _{Si} = 3.48 and \({n}_{{{\rm{SiO}}}_{{\rm{2}}}}=1.44\) were used.
1 × 3 splitter
Our first device is a broadband 1 × 3 power splitter with 500 nm wide input and output waveguides. We constrained the minimum radius of curvature to be 100 nm, well within the typical design rules of a silicon photonics process, and enforced bilateral symmetry. To design the splitter, we specified that power in the fundamental traverseelectric (TE) mode of the input waveguide should be equally split into the fundamental TE mode of the three output waveguides, with at least 95% efficiency. Broadband performance was achieved by simultaneously optimizing at 6 equally spaced wavelengths from 1400–1700 nm.
The optimization process is illustrated in Fig. 1; the simulated fields and performance are presented later in this paper alongside experimental results. Starting with a starshaped geometry, the optimization process converged in 18 iterations. Each iteration required two electromagnetic simulations per design frequency (see supplementary information), resulting in a total of 216 simulations. The device was designed in approximately 2 hours on a single server with an Intel Core i75820 K processor, 64 GB of RAM, and three Nvidia Titan Z graphics cards. Since the computational cost of optimization is dominated by the electromagnetic simulations, we performed them using a graphical processing unit (GPU) accelerated implementation of the finitedifference frequencydomain (FDFD) method^{16, 17}, with a spatial step size of 40 nm. A single FDFD solve is considerably faster and less computationally expensive than a finitedifference timedomain (FDTD) simulation.
Interestingly, the splitter appears to be operate using the multimode interferometer (MMI) principle^{18}, with a geometry that resembles a boundaryoptimized MMI. This was without any human input or intervention throughout the design process, suggesting that MMIbased devices may be optimal for this particular application.
Spatial mode demultiplexer
Our second device is a spatialmode demultiplexer that takes the TE_{10} and TE_{20} modes of a 750 nm wide input waveguide, and routes them to the fundamental TE mode of two 400 nm wide output waveguides. To design this device, we specified that >90% of the input power should be transmitted to the desired output port, and <1% should be coupled into the other output. As with the 1 × 3 splitter, this device was designed to be broadband by optimizing at six evenly spaced wavelengths between 1400 nm and 1700 nm. To obtain an initial structure for the level set optimization, we started with a uniform permittivity in the design region, allowed the permittivity to vary continuously in the initial stage of optimization, and applied thresholding to obtain a binary structure^{5}. We used a minimum radius of curvature of 70 nm, and a minimum gap or bridge width of 90 nm.
The final design and simulated performance are illustrated in Fig. 2. The spatial mode multiplexer has an average insertion loss of 0.826 dB, and a contrast better than 16 dB over the design bandwidth of 1400–1700 nm.
Wavelength demultiplexer
Our third device is a 3channel wavelength demultiplexer with a 40 nm channel spacing with 500 nm wide input and output waveguides. To design this device, we specified that >80% of the input power should be transmitted to the desired output port, and <1% should be coupled into the remaining outputs. The initial structure was a rectangular slab of silicon with a regular array of holes, which had a pitch of 400 nm and a diameter of 250 nm. We enforced a minimum radius of curvature of 40 nm, and a minimum gap or bridge width of 90 nm.
The final design and simulated performance are illustrated in Fig. 3. At the center of each channel, the insertion loss is approximately 1.5 dB, and the contrast is better than 16 dB. Each channel has a usable bandwidth >10 nm.
Directional coupler
Our final device is a relatively compact 5050 directional coupler, with 400 nm input and output waveguides. This device was designed by specifying that half the power in fundamental mode of the input waveguide should be coupled into each of the outputs, with >90% efficiency. As in the design of the spatial mode demultiplexer, we obtained an initial structure by starting with a uniform permittivity in the design region, allowing the permittivity to vary continuously in the initial stage of optimization, and applying thresholding. To achieve moderate broadband performance, the device was simultaneously optimized for 6 wavelengths between 1470–1630 nm. We enforced a minimum radius of curvature of 70 nm, and a minimum bridge width of 90 nm.
The final device and simulated performance are illustrated in Fig. 4. At the optimal operating point of 1520 nm, the device couples 90% of the input power into the desired output waveguides. The device structure appears to be a gratingassisted directional coupler.
Experimental Realization of 1 × 3 Splitter
Robust and efficient power splitters are essential building blocks for integrated photonics. A variety of 1 × 2 splitters with attractive performance have been demonstrated on the silicon photonics platform, ranging from conventional devices^{19, 20} to those designed using advanced optimization techniques^{4, 21, 22}. However, it is not possible to split power equally into an arbitrary number of waveguides by cascading 1 × 2 splitters, and efficient and compact devices that fill this gap are lacking in the literature. To help fill this gap, we fabricated and experimentally demonstrated the 1 × 3 splitter we presented in the previous section. Our 1 × 3 splitter is considerably smaller and more broadband than any existing device in the literature^{23, 24}.
Fabrication
The power splitters were fabricated on Unibond SmartCut silicononinsulator (SOI) wafers obtained from SOITEC, with a nominal 220 nm device layer, and 3.0 μm buried oxide layer. A JEOL JBX6300FS electronbeam lithography system was used to pattern a 330 nm thick layer of ZEP520A resist spun on the samples. A transformercoupled plasma etcher was used to transfer the pattern to the device layer, using a C_{2}F_{6} breakthrough step and BCl_{3}/Cl_{2}/O_{2} main etch. The mask was stripped by soaking in solvents, followed by a piranha (H_{2}SO_{4}/H_{2}O_{2}) clean. Finally, the devices were capped with 1.6 μm of LPCVD (low pressure chemical vapour deposition) oxide.
A multistep etchbased process was used to expose waveguide facets for edge coupling. First, a chrome mask was deposited using liftoff to protect the devices. Next, the oxide cladding, device layer, and buried oxide layer were etched in a inductivelycoupled plasma etcher using a C_{4}F_{8}/ArO_{2} chemistry. To provide mechanical clearance for the optical fibers, the silicon substrate was then etched to a depth of ~100 μm using the Bosch process in a deep reactiveion etcher (DRIE). Finally, the chrome mask was chemically stripped, and the samples were diced into convenientlysized pieces.
Characterization
The final splitter is illustrated in Fig. 5, showing both an scanningelectron micrograph (SEM) of the fabricated device, and simulated electromagnetic energy density at the center wavelength of 1550 nm. The simulated electric energy density \({U}_{E}=\frac{1}{2}\varepsilon {E}^{2}\) as a function of wavelength is shown in the supplementary video.
Transmission through the device was measured by edgecoupling to the input and output waveguides using lensed fibers. A polarizationmaintaining fiber was used on the input side to ensure that only the TE mode of the waveguide was excited. To obtain consistent coupling regardless of the transmission spectra of the devices, the fibers were aligned by optimizing the transmitted power of a 1570 nm laser. The transmission spectrum was then measured by using a supercontinuum source and a spectrum analyzer. The device characteristics were obtained by normalizing the transmission with respect to a waveguide running parallel to the device.
The simulated and measured transmission spectra of the device are plotted in Fig. 6. The simulations and measurements match reasonably well, although the measured devices have slightly higher losses and exhibit a spectral shift with respect to simulations. The device performance is highly consistent across all 4 measured devices, indicating that they are robust to fabrication error. The spectral shifts are likely due to slight overetching or underetching errors, as indicated by simulations we present in the supplementary information.
The two key criteria for a power splitter are low insertion loss, and excellent power uniformity. The power uniformity is defined as the ratio between the maximum and minimum output powers. Averaged over the designed wavelength range of 1400–1700 nm, our 1 × 3 splitter has a measured insertion loss of 0.642 ± 0.057 dB, and a power uniformity of 0.641 ± 0.054 dB. Here, the uncertainty refers to the variability between different measured devices.
Conclusion
In summary, we have incorporated fabrication constraints into an inverse design algorithm for nanophotonic devices. Using this method, we designed a spatial mode demultiplexer, a 3channel wavelength demultiplexer, and 50–50 directional coupler. We also designed and experimentally demonstrated a broadband 1 × 3 splitter. Critically, our devices have no small features which would be difficult to resolve with photolithography, paving the way for inverse designed structures to become practical components of integrated photonics systems.
Additional information
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References
 1.
Reed, G. T. Silicon Photonics: The State of the Art (John Wiley & Sons, Chichester, West Sussex, U.K. 2008).
 2.
Mutapcica, A., Boyd, S., Farjadpour, A., Johnson, S. G. & Avnielb, Y. Robust design of slowlight tapers in periodic waveguides. Eng. Optimiz. 41, 365–384, doi:10.1080/03052150802576797 (2009).
 3.
Jensen, J. S. & Sigmund, O. Topology optimization for nanophotonics. Laser Photonics Rev 5, 308–321, doi:10.1002/lpor.201000014 (2011).
 4.
LalauKeraly, C. M., Bhargava, S., Miller, O. D. & Yablonovitch, E. Adjoint shape optimization applied to electromagnetic design. Opt. Express 21, 21693–21701, doi:10.1364/OE.21.021693 (2013).
 5.
Lu, J. & Vučković, J. Nanophotonic computational design. Opt. Express 21, 13351–13367, doi:10.1364/OE.21.013351 (2013).
 6.
Niederberger, A. C. R., Fattal, D. A., Gauger, N. R., Fan, S. & Beausoleil, R. G. Sensitivity analysis and optimization of subwavelength optical gratings using adjoints. Opt. Express 22, 12971–12981, doi:10.1364/OE.22.012971 (2014).
 7.
Piggott, A. Y. et al. Inverse design and demonstration of a compact and broadband onchip wavelength demultiplexer. Nature Photonics 9, 374–377, doi:10.1038/nphoton.2015.69 (2015).
 8.
Frellsen, L. F., Ding, Y., Sigmund, O. & Frandsen, L. H. Topology optimized mode multiplexing in silicononinsulator photonic wire waveguides. Opt. Express 24, 16866–16873, doi:10.1364/OE.24.016866 (2016).
 9.
Piggott, A. Y. et al. Inverse design and implementation of a wavelength demultiplexing grating coupler. Sci. Rep. 4, 7210, doi:10.1038/srep07210 (2014).
 10.
Shen, B., Wang, P., Polson, R. & Menon, R. An integratednanophotonics polarization beamsplitter with 2.4 × 2.4 μm footprint. Nature Photonics 9, 378–382, doi:10.1038/nphoton.2015.80 (2015).
 11.
Elesin, Y., Lazarov, B., Jensen, J. & Sigmund, O. Design of robust and efficient photonic switches using topology optimization. Phot. Nano. Fund. Appl. 10, 153–165, doi:10.1016/j.photonics.2011.10.003 (2012).
 12.
Deng, Y. & Korvink, J. G. Topology optimization for threedimensional electromagnetic waves using an edge elementbased finiteelement method. Proc. R. Soc. A 472, 20150835, doi:10.1098/rspa.2015.0835 (2016).
 13.
Frandsen, L. H. & Sigmund, O. Inverse design engineering of allsilicon polarization beam splitters. Proc. SPIE 9756, 97560Y–1–97560Y–6, doi:10.1117/12.2210848 (2016).
 14.
Osher, S. & Fedkiw, R. Level Set Methods and Dynamic Implicit Surfaces (Springer, New York, USA 2003).
 15.
Burger, M. & Osher, S. J. A survey on level set methods for inverse problems and optimal design. Eur. J. Appl. Math. 16, 263–301, doi:10.1017/S0956792505006182 (2005).
 16.
Shin, W. & Fan, S. Choice of the perfectly matched layer boundary condition for frequencydomain Maxwell’s equations solvers. J. Comput. Phys. 231, 3406–3431, doi:10.1016/j.jcp.2012.01.013 (2012).
 17.
Shin, W. & Fan, S. Accelerated solution of the frequencydomain Maxwell’s equations by engineering the eigenvalue distribution. Opt. Express 21, 22578–22595, doi:10.1364/OE.21.022578 (2013).
 18.
Besse, P. A., Bachmann, M., Melchior, H., Soldano, L. B. & Smit, M. K. Optical bandwidth and fabrication tolerances of multimode interference couplers. J. Lightwave Technol. 12, 1004–1009, doi:10.1109/50.296191 (1994).
 19.
Sakai, A., Fukuzawa, T. & Baba, T. Low loss ultrasmall branches in a silicon photonic wire waveguide. IEICE Trans. Electron. E85C, 1033–1038 (2002).
 20.
Tao, S. H. et al. Cascade wideangle yjunction 1 × 16 optical power splitter based on silicon wire waveguides on silicononinsulator. Opt. Express 16, 21456–21461, doi:10.1364/OE.16.021456 (2008).
 21.
Borel, P. I. et al. Topology optimised broadband photonic crystal ysplitter. Electron. Lett. 41, 69–71, doi:10.1049/el:20057717 (2005).
 22.
Zhang, Y. et al. A compact and low loss yjunction for submicron silicon waveguide. Opt. Express 21, 1310–1316, doi:10.1364/OE.21.001310 (2013).
 23.
Besse, P. A., Gini, E., Bachmann, M. & Melchior, H. New 2 × 2 and 1 × 3 multimode interference couplers with free selection of power splitting ratios. J. Lightwave Technol. 14, 2286–2293, doi:10.1109/50.541220 (1996).
 24.
Zhang, M., Malureanu, R., Krüger, A. C. & Kristensen, M. 1 × 3 beam splitter for te polarization based on selfimaging phenomena in photonic crystal waveguides. Opt. Express 18, 14944–14949, doi:10.1364/OE.18.014944 (2010).
Acknowledgements
This work was funded by the AFOSR MURI for Aperiodic Silicon Photonics, grant number FA95501510335, the Gordon and Betty Moore Foundation, and GlobalFoundries Inc. All devices were fabricated at the Stanford Nanofabrication Facility (SNF) and Stanford Nano Shared Facilities (SNSF).
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Ginzton Laboratory, Stanford University, Stanford, California, 94305, USA
 Alexander Y. Piggott
 , Jan Petykiewicz
 , Logan Su
 & Jelena Vučković
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Contributions
A.Y.P. designed, simulated, fabricated and measured the devices. A.Y.P., J.P., and L.S. developed the design and simulation software. J.V. supervised the project. All members contributed to the discussion and analysis of the results.
Competing Interests
The authors declare that they have no competing interests.
Corresponding author
Correspondence to Alexander Y. Piggott.
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