Nanophotonics has emerged as a powerful tool for manipulating light on chips. Almost all of today's devices, however, have been designed using slow and ineffective brute-force search methods, leading in many cases to limited device performance. In this article, we provide a complete demonstration of our recently proposed inverse design technique, wherein the user specifies design constraints in the form of target fields rather than a dielectric constant profile and in particular we use this method to demonstrate a new demultiplexing grating. The novel grating, which has not been developed using conventional techniques, accepts a vertical-incident Gaussian beam from a free-space and separates O-band (1300 nm) and C-band (1550 nm) light into separate waveguides. This inverse design concept is simple and extendable to a broad class of highly compact devices including frequency filters, mode converters and spatial mode multiplexers.
Conventional integrated photonic devices1 include components such as waveguide directional couplers2, multimode interference couplers3, distributed Bragg reflectors4, micro-ring resonators5, adiabatic tapers6 and grating couplers7. In all of these cases, the design space spans a relatively small (~2–5) number of parameters such as structure widths, heights and periodicity that are tuned throughout the photonic device design stage. To design a device, the photonic engineer specifies a dielectric profile, computes the electromagnetic field response using Maxwells equations and compares the response to the device specifications. The process is then repeated, modifying the dielectric profile each iteration, until satisfactory performance is obtained. This brute force approach suffers from a long device design cycle and does not take full advantage of the available design space of fabricable devices.
To this end, a wide variety of increasingly sophisticated approaches have been developed to search through this parameter space and optimize specific nanophotonic structures. Several methods, namely genetic algorithms8,9,10 and particle swarm optimization11, ignore the underlying physics but have achieved considerable success in fine-tuning existing structures and designing photonic crystal devices with selectively removed holes and posts. These methods, however, are typically restricted to optimizing a relatively small number of geometric parameters and scale poorly with additional degrees of freedom. Other methods exploit the underlying physics to quickly converge on local optima, typically by computing the local gradient of a performance metric and using steepest-descent optimization12. Owing to their much faster convergence, they can be used to design more complex structures with arbitrary topologies13,14,15,16,17,18,19. These algorithms can search through the design space of complex, aperiodic structures beyond those we can come up with based on our intuition and experience. Such devices may be able to provide novel functionality, or higher performance and smaller footprints than traditional devices, due to the greatly expanded design space.
We have recently proposed an inverse design approach for linear optical components, where the user specified input is not the basic structure of the device, but rather a set of performance metrics19. These include the device area, the modes of input and output waveguides to use (e.g. TE, TM and mode order), extinction ratios (e.g. >20 dB) and insertion loss (e.g. <1 dB). More rigorously, the user only specifies the coupling efficiencies between a set of input and output modes at various frequencies. Any linear optical device can be specified in this fashion20, including mode converters, spatial mode multiplexers and wavelength demultiplexers.
Two of the most important functions in integrated photonics, for both chip-to-chip and intra-chip optical interconnects, are wavelength-division multiplexing (WDM) and vertical-incidence coupling21. A device combining these two functions in a single, compact layout could be particularly useful for coupling between silicon photonic layers in a stacked-die microprocessor, or for coupling on/off chip using optical fibers. Although a uniform grating coupler with a tilted incident beam will act as a wavelength-demultiplexing grating coupler for two wavelength bands22, the symmetry imposed by a vertically incident beam implies that a uniform grating coupler cannot split wavelengths. Indeed, an efficient vertically-incident wavelength-demultiplexing grating coupler cannot be designed using current analytic methods, or by tuning a small number of parameters by hand.
Here, we provide a full demonstration of the inverse design technique's power by experimentally demonstrating a vertical-incidence, wavelength-demultiplexing grating coupler fabricated in silicon-on-insulator (SOI). The grating accepts a vertically incident beam from free space or an optical fiber and splits O-band (~1300 nm) and C-band (~1550 nm) light into separate silicon photonic waveguides with high extinction ratios (>10 dB).
In general, we can specify the performance of a linear optical device by defining the mode conversion efficiency between a set of input modes and output modes20. These modes are specified by the user and kept fixed during the optimization process. The input modes i = 1 … M are at frequencies ωi and can be represented by equivalent current density distributions Ji. The generated electric fields Ei should satisfy Maxwell's equations in the frequency domain,
where is the electric permittivity and μ0 is the magnetic permeability of free space.
We can then specify Ni output modes of interest for each input mode i. The output mode electric fields are given over output surfaces Sij, where j = 1 … Ni. The amplitude of each output mode should be bounded between αij and βij, which can be expressed as
for i = 1 … M and j = 1 … Ni. We are thus interested in finding and Ei which simultaneously satisfy (1) and (2).
The WDM grating was designed by specifying the input mode to be a 4.4 μm diameter vertically-incident Gaussian beam and the output modes to be the fundamental TE mode of the output silicon slab waveguides, as shown in figure 1a. The algorithm was directed to maximize power into the left waveguide and minimize power into the right waveguide at 1300 nm and the converse at 1550 nm.
Given the design specifications, our algorithm iteratively optimizes the structure using a close analogue of steepest-descent optimization to meet the constraints given in equation (2), as detailed in the Supplementary Information. We use finite-difference frequency domain (FDFD) simulations to calculate the local gradient during each step23,24. As illustrated in figure 1b, the design procedure consists of two stages. In the first stage, the permittivity is allowed to smoothly vary within the design region. In the second stage, we convert the structure to a level-set representation25 and fine-tune the final structure. We did not apply any design rules such as a minimum feature size, although such constraints could be incorporated into the design process19. The entire inverse design process for this grating took only ~15 minutes on a single Intel Core i7 processor.
The final device was fabricated in SOI with a 220 nm thick Si device layer and a 3 μm BOX (buried oxide) layer by fully etching the Si device layer, as described in figure 2a. The dimensions and locations of the grating trenches are detailed in Supplementary Table 1. The broadband performance of the device was verified by using 2D finite-difference time domain (FDTD) simulations. The phasor fields at 1293 nm and 1540 nm for a 4.4 um diameter Gaussian beam, obtained using 2D-FDTD simulations, are plotted in figure 2b. The fabricated device is presented in figure 2c.
The final fabricated structures also incorporate a waveguide and curved broadband output grating on either side of the wavelength-demultiplexing grating, as shown in figure 3a. Scanning electron microscopy (SEM) images of the fabricated structures are shown in figure 3b. The output gratings were strongly chirped to provide broadband performance. To minimize Fabry-Perot resonances due to back-reflections, the output gratings were slightly curved and placed far from the ends of the waveguides (inset). In future, the output waveguides could be edge-coupled to optical fibers to obtain a well-characterized out-coupling efficiency26.
The WDM grating was excited in the vertical direction by a focused Gaussian beam with a diameter of ~4 μm and a fraction of the coupled light was out-coupled by the output gratings. To ensure a clean Gaussian input beam, the source was passed through a length of single-mode optical fiber. Spectroscopic data was measured using a broadband LED source and narrowband images were taken using tunable lasers. The structure was both excited and measured through a single plan-apochromat microscope objective integrated into a custom confocal microscopy setup. The collected light was either directly imaged using an InGaAs near-infrared camera, or spatially filtered by a pinhole at a focal plane and analyzed using a grating spectrometer with an InGaAs detector.
Images of the device broadly illuminated with white light and excited by a focused tunable laser are presented in figure 3c–e. At 1320 nm, light is only coupled to the left output grating and at 1540 nm, light is only coupled to the right output grating, clearly demonstrating the basic functionality of the device.
In figure 4, we present both the simulated and experimentally measured coupling efficiency spectra of the WDM grating. The coupling efficiencies of the WDM grating computed using 2D FDTD are plotted in figure 4a. At 1293 nm, the simulated coupling efficiencies into the left and right waveguides are 0.2937 and 0.0008 respectively, whereas at 1540 nm, the coupling efficiencies are 0.0027 and 0.4544. The measured signal intensity, normalized to source brightness, from the two output gratings is plotted in figure 4b. The output grating coupling efficiencies were not measured, but due to the symmetry of the experimental setup, measurements of the two output ports are directly comparable. The experimental data from 1350 nm–1450 nm is shaded to due to the presence of strong atmospheric water absorption lines27. Fabry-perot fringing, arising from reflections between the WDM grating and output gratings, is visible in the measured spectra, with fringe spacing corresponding to the spacing between the gratings. The measured signals broadly match the simulated coupling efficiencies of the WDM grating.
The splitting ratio, defined as the ratio of power emitted from the two output ports, is plotted in figure 4c. The fabricated device has a measured splitting ratio of 17 ± 2 dB at 1310 nm and 12 ± 2 dB at 1540 nm, whereas the designed values at these wavelengths were 19.6 dB and 22.2 dB and respectively. The discrepancy between the simulated and measured splitting ratio is likely due to fabrication imperfections.
In conclusion, we have provided the first experimental demonstration of a nanophotonic device designed using our inverse design algorithm. In particular, we have implemented an efficient vertical-incidence wavelength-demultiplexing grating coupler, which cannot be designed by hand or by using parameter sweeps. By vastly opening up the parameter space for nanophotonic devices, inverse-design has broad implications for the future design of novel and compact nanophotonic components with full three-dimensional freedom.
Optimization algorithm and electromagnetic simulations
Our inverse design algorithm uses a close analogue of steepest-descent optimization and incorporates the MaxwellFDFD finite-difference frequency-domain solver to calculate local gradients23,24, as detailed in the Supplementary Information.
After completing the design process, the broadband performance of the device was calculated using 2D finite-difference time-domain (FDTD) simulations.
The detailed grating dimensions are listed in Supplementary Table 1. The devices were fabricated on Unibond™ SmartCut™ silicon-on-insulator (SOI) wafers obtained from SOITEC, with a nominal 220 nm device layer and 3.0 μm BOX layer. A JEOL JBX-6300FS electron beam lithography system was used to pattern 330 nm of ZEP-52A electron beam resist spun on the samples. The pattern was then transferred to the Si device layer with a magnetically-enhanced reactive-ion etcher using a HBr/Cl2 chemistry. Finally, the mask was stripped by sonicating in Microposit remover 1165.
Sun, J., Timurdogan, E., Yaacobi, A., Hosseini, E. S. & R. Watts, M. Large-scale nanophotonic phased array. Nature 493, 195–199 (2013).
Trinh, P., Yegnanarayanan, S. & Jalali, B. Integrated optical directional couplers in silicon-on-insulator. Elect. Lett. 31, 2097–2098 (1995).
Soldano, L. B. & Pennings, E. C. M. Optical multi-mode interference devices based on self-imaging: principles and applications. J. Lightw. Technol. 13, 615–627 (1995).
Murphy, T. E., Hastings, J. T. & Smith, H. I. Fabrication and characterization of narrow-band bragg-reflection filters in silicon-on-insulator ridge waveguides. J. Lightw. Technol. 19, 1938–1942 (2001).
Dumon, P. et al. Low-loss SOI photonic wires and ring resonators fabricated with deep UV lithography. IEEE Photon. Technol. Lett. 16, 1328–1330 (2004).
Shani, Y., Henry, C. H., Kistler, R. C., Kazarinov, R. F. & Orlowsky, K. J. Integrated optic adiabatic devices on silicon. IEEE J. Quantum Electron. 27, 556–566 (1991).
Taillaert, D. et al. Grating couplers for coupling between optical fibers and nanophotonic waveguides. Jpn. J. Appl. Phys. 45, 6071–6077 (2006).
Gondarenko, A. & Lipson, M. Low modal volume dipole-like dielectric slab resonator. Opt. Express 16, 17689–17694 (2008).
Håkansson, A. & Sánchez-Dehesa, J. Inverse designed photonic crystal demultiplex waveguide coupler. Opt. Express 13, 5440–5449 (2005).
Minkov, M. & Savona, V. Automated optimization of photonic crystal slab cavities. Sci. Rep. 4, 5124 (2014).
Ma, Y. et al. Ultralow loss single layer submicron silicon waveguide crossing for SOI optical interconnect. Opt. Express 21, 29374–29382 (2013).
Boyd, S. & Vandenberghe, L. Convex Optimization (Cambridge University Press, Cambridge, U.K., 2004).
Jensen, J. S. & Sigmund, O. Systematic design of photonic crystal structures using topology optimization: Lowloss waveguide bends. Appl. Phys. Lett. 84, 2022 (2004).
Borel, P. I. et al. Topology optimization and fabrication of photonic crystal structures. Opt. Express 12, 1996–2001 (2004).
Mutapcica, A., Boyd, S., Farjadpour, A., Johnson, S. G. & Avnielb, Y. Robust design of slow-light tapers in periodic waveguides. Eng. Optimiz. 41, 365384 (2009).
Jensen, J. S. & Sigmund, O. Topology optimization for nano-photonics. Laser Photonics Rev. 5, 308–321 (2011).
Lalau-Keraly, C. M., Bhargava, S., Miller, O. D. & Yablonovitch, E. Adjoint shape optimization applied to electromagnetic design. Opt. Express 21, 21693–21701 (2013).
Niederberger, A. C. R., Fattal, D. A., Gauger, N. R., Fan, S. & Beausoleil, R. G. Sensitivity analysis and optimization of sub-wavelength optical gratings using adjoints. Opt. Express 22, 12971–12981 (2014).
Lu, J. & Vučković, J. Nanophotonic computational design. Opt. Express 21, 13351–13367 (2013).
Miller, D. A. B. All linear optical devices are mode converters. Opt. Express 20, 23985–23993 (2012).
Reed, G. T. Silicon Photonics: The State of the Art (John Wiley & Sons, Chichester, West Sussex, U.K., 2008).
Roelkens, G., Thourhout, D. V. & Baets, R. Silicon-on-insulator ultra-compact duplexer based on a diffractive grating structure. Opt. Express 15, 10091–10096 (2007).
Shin, W. & Fan, S. Choice of the perfectly matched layer boundary condition for frequency-domain Maxwell's equations solvers. J. Comput. Phys. 231, 34063431 (2012).
Shin, W. MaxwellFDFD webpage. (2014). URL web.stanford.edu/~wsshin/maxwellfdfd. Date of access: 2014-09-11.
Osher, S. & Fedkiw, R. Level Set Methods and Dynamic Implicit Surfaces (Springer, New York, U.S.A., 2003).
Kopp, C. et al. Silicon photonic circuits: On-CMOS integration, fiber optical coupling and packaging. IEEE J. Sel. Topics Quantum Electron. 17, 498–509 (2011).
McClatchey, R. A., Fenn, R. W., Selby, J. E. A., Volz, F. E. & Garing, J. S. Optical Properties of the Atmosphere (third edition). AFCRL-72-0497 (Air Force Cambridge Research Laboratories, L. G. Hanscom Field, Bedford, Massachusetts, U.S.A., 1972).
This work has been supported by the AFOSR MURI for Complex and Robust On-chip Nanophotonics (Dr. Gernot Pomrenke), grant number FA9550-09-1-0704. A.Y.P. also acknowledges support from the Stanford Graduate Fellowship. KGL acknowledges support from the Swiss National Science Foundation.
The authors declare no competing financial interests.
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Piggott, A., Lu, J., Babinec, T. et al. Inverse design and implementation of a wavelength demultiplexing grating coupler. Sci Rep 4, 7210 (2014). https://doi.org/10.1038/srep07210
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