Abstract

Efficient coupling between integrated optical waveguides and optical fibers is essential to the success of silicon photonics. While many solutions exist, perfectly vertical grating couplers that scatter light out of a waveguide in the direction normal to the waveguide’s top surface are an ideal candidate due to their potential to reduce packaging complexity. Designing such couplers with high efficiencies, however, has proven difficult. In this paper, we use inverse electromagnetic design techniques to optimize a high efficiency two-layer perfectly vertical silicon grating coupler. Our base design achieves a chip-to-fiber coupling efficiency of 99.2% (−0.035 dB) at 1550 nm. Using this base design as a starting point, we run subsequent constrained optimizations to realize vertical couplers with coupling efficiencies over 96% and back reflections of less than −40 dB which can be fabricated using 65 nm-resolution lithography. These results demonstrate a new path forward for designing fabrication-tolerant ultra high efficiency grating couplers.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2017 (2)

2016 (2)

2015 (3)

2013 (3)

2011 (2)

J. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev. 5(2), 308–321 (2011).
[Crossref]

A. Mekis, S. Gloeckner, G. Masini, A. Narasimha, T. Pinguet, S. Sahni, and P. D. Dobbelaere, “A grating-coupler-enabled CMOS photonics platform,” IEEE J. Sel. Top. Quantum Electron. 17(3), 597–608 (2011).
[Crossref]

2010 (1)

2008 (1)

X. Chen, C. Li, and H. K. Tsang, “Fabrication-tolerant waveguide chirped grating coupler for coupling to a perfectly vertical optical fiber,” IEEE Photon. Technol. Lett. 20(23), 1914–1916 (2008).
[Crossref]

2007 (1)

2006 (1)

C. Gunn, “CMOS photonics for high-speed interconnects,” IEEE Micro 26(2), 58–66 (2006).
[Crossref]

2005 (1)

B. Wang, J. Jiang, and G. P. Nordin, “Embedded slanted grating for vertical coupling between fibers and silicon-on-insulator planar waveguides,” IEEE Photon. Technol. Lett. 17(9), 1884–1886 (2005).
[Crossref]

2004 (1)

D. Taillaert, P. Bienstman, and R. Baets, “Compact efficient broadband grating coupler for silicon-on-insulator waveguides,” Opt. Letters 29(23), 2749–2751 (2004).
[Crossref]

1997 (1)

Absil, P.

Alloatti, L.

J. Notaros, F. Pavanello, M. T. Wade, C. Gentry, A. Atabaki, L. Alloatti, R. J. Ram, and M. Popović, “Ultra-efficient CMOS fiber-to-chip grating couplers,” in Optical Fiber Communication Conference (2016), (Optical Society of America, 2016), paper M2I.5.

Alonso-Ramos, C.

Andreani, L. C.

Atabaki, A.

J. Notaros, F. Pavanello, M. T. Wade, C. Gentry, A. Atabaki, L. Alloatti, R. J. Ram, and M. Popović, “Ultra-efficient CMOS fiber-to-chip grating couplers,” in Optical Fiber Communication Conference (2016), (Optical Society of America, 2016), paper M2I.5.

Ayazi, A.

L. B. Verslegers, A. Mekis, T. Pinguet, Y. Chi, G. Masini, P. Sun, A. Ayazi, K. Y. Hon, S. Sahni, S. Gloeckner, C. Baudot, F. Boeuf, and P. D. Dobbelaere, “Design of low-loss polarization splitting grating couplers,” in Advanced Photonics for Communications (2014) (Optical Society of America, 2014), paper JT4A.2.

Baehr-Jones, T.

L. He, Y. Liu, C. Galland, A. E. J. Lim, G. Q. Lo, T. Baehr-Jones, and M. Hochberg, “A high-efficiency nonuniform grating coupler realized with 248-nm optical lithography,” IEEE Photon. Technol. Lett. 25(14), 1358–1361 (2013).
[Crossref]

Baets, R.

G. Roelkens, D. V. Thourhout, and R. Baets, “High efficiency grating coupler between silicon-on-insulator waveguides and perfectly vertical optical fibers,” Opt. Lett. 32(11), 1495–1497 (2007).
[Crossref] [PubMed]

D. Taillaert, P. Bienstman, and R. Baets, “Compact efficient broadband grating coupler for silicon-on-insulator waveguides,” Opt. Letters 29(23), 2749–2751 (2004).
[Crossref]

Baudot, C.

L. B. Verslegers, A. Mekis, T. Pinguet, Y. Chi, G. Masini, P. Sun, A. Ayazi, K. Y. Hon, S. Sahni, S. Gloeckner, C. Baudot, F. Boeuf, and P. D. Dobbelaere, “Design of low-loss polarization splitting grating couplers,” in Advanced Photonics for Communications (2014) (Optical Society of America, 2014), paper JT4A.2.

Benedikovic, D.

Bhargava, S.

Bienstman, P.

D. Taillaert, P. Bienstman, and R. Baets, “Compact efficient broadband grating coupler for silicon-on-insulator waveguides,” Opt. Letters 29(23), 2749–2751 (2004).
[Crossref]

Boeuf, F.

L. B. Verslegers, A. Mekis, T. Pinguet, Y. Chi, G. Masini, P. Sun, A. Ayazi, K. Y. Hon, S. Sahni, S. Gloeckner, C. Baudot, F. Boeuf, and P. D. Dobbelaere, “Design of low-loss polarization splitting grating couplers,” in Advanced Photonics for Communications (2014) (Optical Society of America, 2014), paper JT4A.2.

Bogaerts, W.

Bozzola, A.

Bräuer, A.

Carroll, L.

Chang-Hasnain, C.

Cheben, P.

Chen, E.

Chen, X.

X. Chen, C. Li, and H. K. Tsang, “Fabrication-tolerant waveguide chirped grating coupler for coupling to a perfectly vertical optical fiber,” IEEE Photon. Technol. Lett. 20(23), 1914–1916 (2008).
[Crossref]

Chen, Y.

Chi, Y.

L. B. Verslegers, A. Mekis, T. Pinguet, Y. Chi, G. Masini, P. Sun, A. Ayazi, K. Y. Hon, S. Sahni, S. Gloeckner, C. Baudot, F. Boeuf, and P. D. Dobbelaere, “Design of low-loss polarization splitting grating couplers,” in Advanced Photonics for Communications (2014) (Optical Society of America, 2014), paper JT4A.2.

Cristiani, I.

Dado, M.

Dai, M.

Dannberg, P.

Dobbelaere, P. D.

A. Mekis, S. Gloeckner, G. Masini, A. Narasimha, T. Pinguet, S. Sahni, and P. D. Dobbelaere, “A grating-coupler-enabled CMOS photonics platform,” IEEE J. Sel. Top. Quantum Electron. 17(3), 597–608 (2011).
[Crossref]

L. B. Verslegers, A. Mekis, T. Pinguet, Y. Chi, G. Masini, P. Sun, A. Ayazi, K. Y. Hon, S. Sahni, S. Gloeckner, C. Baudot, F. Boeuf, and P. D. Dobbelaere, “Design of low-loss polarization splitting grating couplers,” in Advanced Photonics for Communications (2014) (Optical Society of America, 2014), paper JT4A.2.

Galland, C.

L. He, Y. Liu, C. Galland, A. E. J. Lim, G. Q. Lo, T. Baehr-Jones, and M. Hochberg, “A high-efficiency nonuniform grating coupler realized with 248-nm optical lithography,” IEEE Photon. Technol. Lett. 25(14), 1358–1361 (2013).
[Crossref]

Gentry, C.

J. Notaros, F. Pavanello, M. T. Wade, C. Gentry, A. Atabaki, L. Alloatti, R. J. Ram, and M. Popović, “Ultra-efficient CMOS fiber-to-chip grating couplers,” in Optical Fiber Communication Conference (2016), (Optical Society of America, 2016), paper M2I.5.

Gerace, D.

Gloeckner, S.

A. Mekis, S. Gloeckner, G. Masini, A. Narasimha, T. Pinguet, S. Sahni, and P. D. Dobbelaere, “A grating-coupler-enabled CMOS photonics platform,” IEEE J. Sel. Top. Quantum Electron. 17(3), 597–608 (2011).
[Crossref]

L. B. Verslegers, A. Mekis, T. Pinguet, Y. Chi, G. Masini, P. Sun, A. Ayazi, K. Y. Hon, S. Sahni, S. Gloeckner, C. Baudot, F. Boeuf, and P. D. Dobbelaere, “Design of low-loss polarization splitting grating couplers,” in Advanced Photonics for Communications (2014) (Optical Society of America, 2014), paper JT4A.2.

Gunn, C.

C. Gunn, “CMOS photonics for high-speed interconnects,” IEEE Micro 26(2), 58–66 (2006).
[Crossref]

Halir, R.

He, L.

L. He, Y. Liu, C. Galland, A. E. J. Lim, G. Q. Lo, T. Baehr-Jones, and M. Hochberg, “A high-efficiency nonuniform grating coupler realized with 248-nm optical lithography,” IEEE Photon. Technol. Lett. 25(14), 1358–1361 (2013).
[Crossref]

Hochberg, M.

L. He, Y. Liu, C. Galland, A. E. J. Lim, G. Q. Lo, T. Baehr-Jones, and M. Hochberg, “A high-efficiency nonuniform grating coupler realized with 248-nm optical lithography,” IEEE Photon. Technol. Lett. 25(14), 1358–1361 (2013).
[Crossref]

Hon, K. Y.

L. B. Verslegers, A. Mekis, T. Pinguet, Y. Chi, G. Masini, P. Sun, A. Ayazi, K. Y. Hon, S. Sahni, S. Gloeckner, C. Baudot, F. Boeuf, and P. D. Dobbelaere, “Design of low-loss polarization splitting grating couplers,” in Advanced Photonics for Communications (2014) (Optical Society of America, 2014), paper JT4A.2.

Janz, S.

Jensen, J.

J. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev. 5(2), 308–321 (2011).
[Crossref]

Jiang, J.

B. Wang, J. Jiang, and G. P. Nordin, “Embedded slanted grating for vertical coupling between fibers and silicon-on-insulator planar waveguides,” IEEE Photon. Technol. Lett. 17(9), 1884–1886 (2005).
[Crossref]

Karthe, W.

Kley, E.-B.

Lalau-Keraly, C. M.

Lapointe, J.

Lepage, G.

Li, C.

C. Li, H. Zhang, M. Yu, and G. Q. Lo, “CMOS-compatible high efficiency double-etched apodized waveguide grating coupler,” Opt. Express 21(7), 7868–7874 (2013).
[Crossref] [PubMed]

X. Chen, C. Li, and H. K. Tsang, “Fabrication-tolerant waveguide chirped grating coupler for coupling to a perfectly vertical optical fiber,” IEEE Photon. Technol. Lett. 20(23), 1914–1916 (2008).
[Crossref]

Lim, A. E. J.

L. He, Y. Liu, C. Galland, A. E. J. Lim, G. Q. Lo, T. Baehr-Jones, and M. Hochberg, “A high-efficiency nonuniform grating coupler realized with 248-nm optical lithography,” IEEE Photon. Technol. Lett. 25(14), 1358–1361 (2013).
[Crossref]

Liu, X.

Liu, Y.

L. He, Y. Liu, C. Galland, A. E. J. Lim, G. Q. Lo, T. Baehr-Jones, and M. Hochberg, “A high-efficiency nonuniform grating coupler realized with 248-nm optical lithography,” IEEE Photon. Technol. Lett. 25(14), 1358–1361 (2013).
[Crossref]

Lo, G. Q.

L. He, Y. Liu, C. Galland, A. E. J. Lim, G. Q. Lo, T. Baehr-Jones, and M. Hochberg, “A high-efficiency nonuniform grating coupler realized with 248-nm optical lithography,” IEEE Photon. Technol. Lett. 25(14), 1358–1361 (2013).
[Crossref]

C. Li, H. Zhang, M. Yu, and G. Q. Lo, “CMOS-compatible high efficiency double-etched apodized waveguide grating coupler,” Opt. Express 21(7), 7868–7874 (2013).
[Crossref] [PubMed]

Lu, M.

Ma, L.

Masini, G.

A. Mekis, S. Gloeckner, G. Masini, A. Narasimha, T. Pinguet, S. Sahni, and P. D. Dobbelaere, “A grating-coupler-enabled CMOS photonics platform,” IEEE J. Sel. Top. Quantum Electron. 17(3), 597–608 (2011).
[Crossref]

L. B. Verslegers, A. Mekis, T. Pinguet, Y. Chi, G. Masini, P. Sun, A. Ayazi, K. Y. Hon, S. Sahni, S. Gloeckner, C. Baudot, F. Boeuf, and P. D. Dobbelaere, “Design of low-loss polarization splitting grating couplers,” in Advanced Photonics for Communications (2014) (Optical Society of America, 2014), paper JT4A.2.

Mekis, A.

A. Mekis, S. Gloeckner, G. Masini, A. Narasimha, T. Pinguet, S. Sahni, and P. D. Dobbelaere, “A grating-coupler-enabled CMOS photonics platform,” IEEE J. Sel. Top. Quantum Electron. 17(3), 597–608 (2011).
[Crossref]

L. B. Verslegers, A. Mekis, T. Pinguet, Y. Chi, G. Masini, P. Sun, A. Ayazi, K. Y. Hon, S. Sahni, S. Gloeckner, C. Baudot, F. Boeuf, and P. D. Dobbelaere, “Design of low-loss polarization splitting grating couplers,” in Advanced Photonics for Communications (2014) (Optical Society of America, 2014), paper JT4A.2.

Michaels, A.

A. Michaels and E. Yablonovitch, “Gradient-based inverse electromagnetic design using continuously-smoothed boundaries,” Arxiv (2017).

Miller, O. D.

Na, N.

Narasimha, A.

A. Mekis, S. Gloeckner, G. Masini, A. Narasimha, T. Pinguet, S. Sahni, and P. D. Dobbelaere, “A grating-coupler-enabled CMOS photonics platform,” IEEE J. Sel. Top. Quantum Electron. 17(3), 597–608 (2011).
[Crossref]

Nordin, G. P.

B. Wang, J. Jiang, and G. P. Nordin, “Embedded slanted grating for vertical coupling between fibers and silicon-on-insulator planar waveguides,” IEEE Photon. Technol. Lett. 17(9), 1884–1886 (2005).
[Crossref]

Notaros, J.

J. Notaros and M. Popović, “Band-structure approach to synthesis of grating couplers with ultra-high coupling efficiency and directivity,” in Optical Fiber Communication Conference (2015), (Optical Society of America, 2015), p. Th3F.2.

J. Notaros, F. Pavanello, M. T. Wade, C. Gentry, A. Atabaki, L. Alloatti, R. J. Ram, and M. Popović, “Ultra-efficient CMOS fiber-to-chip grating couplers,” in Optical Fiber Communication Conference (2016), (Optical Society of America, 2016), paper M2I.5.

Ortega-Moñux, A.

Pavanello, F.

J. Notaros, F. Pavanello, M. T. Wade, C. Gentry, A. Atabaki, L. Alloatti, R. J. Ram, and M. Popović, “Ultra-efficient CMOS fiber-to-chip grating couplers,” in Optical Fiber Communication Conference (2016), (Optical Society of America, 2016), paper M2I.5.

Petykiewicz, J.

A. Y. Piggott, J. Petykiewicz, L. Su, and J. Vučkovié, “Fabrication-constrained nanophotonic inverse design,” Sci. Rep. 7(1), 1786 (2017).
[Crossref] [PubMed]

Piggott, A. Y.

A. Y. Piggott, J. Petykiewicz, L. Su, and J. Vučkovié, “Fabrication-constrained nanophotonic inverse design,” Sci. Rep. 7(1), 1786 (2017).
[Crossref] [PubMed]

Pinguet, T.

A. Mekis, S. Gloeckner, G. Masini, A. Narasimha, T. Pinguet, S. Sahni, and P. D. Dobbelaere, “A grating-coupler-enabled CMOS photonics platform,” IEEE J. Sel. Top. Quantum Electron. 17(3), 597–608 (2011).
[Crossref]

L. B. Verslegers, A. Mekis, T. Pinguet, Y. Chi, G. Masini, P. Sun, A. Ayazi, K. Y. Hon, S. Sahni, S. Gloeckner, C. Baudot, F. Boeuf, and P. D. Dobbelaere, “Design of low-loss polarization splitting grating couplers,” in Advanced Photonics for Communications (2014) (Optical Society of America, 2014), paper JT4A.2.

Popovic, M.

J. Notaros, F. Pavanello, M. T. Wade, C. Gentry, A. Atabaki, L. Alloatti, R. J. Ram, and M. Popović, “Ultra-efficient CMOS fiber-to-chip grating couplers,” in Optical Fiber Communication Conference (2016), (Optical Society of America, 2016), paper M2I.5.

J. Notaros and M. Popović, “Band-structure approach to synthesis of grating couplers with ultra-high coupling efficiency and directivity,” in Optical Fiber Communication Conference (2015), (Optical Society of America, 2015), p. Th3F.2.

Ram, R. J.

J. Notaros, F. Pavanello, M. T. Wade, C. Gentry, A. Atabaki, L. Alloatti, R. J. Ram, and M. Popović, “Ultra-efficient CMOS fiber-to-chip grating couplers,” in Optical Fiber Communication Conference (2016), (Optical Society of America, 2016), paper M2I.5.

Roelkens, G.

Rong, H.

Sahni, S.

A. Mekis, S. Gloeckner, G. Masini, A. Narasimha, T. Pinguet, S. Sahni, and P. D. Dobbelaere, “A grating-coupler-enabled CMOS photonics platform,” IEEE J. Sel. Top. Quantum Electron. 17(3), 597–608 (2011).
[Crossref]

L. B. Verslegers, A. Mekis, T. Pinguet, Y. Chi, G. Masini, P. Sun, A. Ayazi, K. Y. Hon, S. Sahni, S. Gloeckner, C. Baudot, F. Boeuf, and P. D. Dobbelaere, “Design of low-loss polarization splitting grating couplers,” in Advanced Photonics for Communications (2014) (Optical Society of America, 2014), paper JT4A.2.

Schmid, J. H.

Schnabel, B.

Selvaraja, S.

Sigmund, O.

J. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev. 5(2), 308–321 (2011).
[Crossref]

Song, Q.

Su, L.

A. Y. Piggott, J. Petykiewicz, L. Su, and J. Vučkovié, “Fabrication-constrained nanophotonic inverse design,” Sci. Rep. 7(1), 1786 (2017).
[Crossref] [PubMed]

Sun, P.

L. B. Verslegers, A. Mekis, T. Pinguet, Y. Chi, G. Masini, P. Sun, A. Ayazi, K. Y. Hon, S. Sahni, S. Gloeckner, C. Baudot, F. Boeuf, and P. D. Dobbelaere, “Design of low-loss polarization splitting grating couplers,” in Advanced Photonics for Communications (2014) (Optical Society of America, 2014), paper JT4A.2.

Taillaert, D.

D. Taillaert, P. Bienstman, and R. Baets, “Compact efficient broadband grating coupler for silicon-on-insulator waveguides,” Opt. Letters 29(23), 2749–2751 (2004).
[Crossref]

Thourhout, D. V.

Tsang, H. K.

X. Chen, C. Li, and H. K. Tsang, “Fabrication-tolerant waveguide chirped grating coupler for coupling to a perfectly vertical optical fiber,” IEEE Photon. Technol. Lett. 20(23), 1914–1916 (2008).
[Crossref]

Tseng, H.-L.

Verheyen, P.

Vermeulen, D.

Verslegers, L. B.

L. B. Verslegers, A. Mekis, T. Pinguet, Y. Chi, G. Masini, P. Sun, A. Ayazi, K. Y. Hon, S. Sahni, S. Gloeckner, C. Baudot, F. Boeuf, and P. D. Dobbelaere, “Design of low-loss polarization splitting grating couplers,” in Advanced Photonics for Communications (2014) (Optical Society of America, 2014), paper JT4A.2.

Vivien, L.

Vuckovié, J.

A. Y. Piggott, J. Petykiewicz, L. Su, and J. Vučkovié, “Fabrication-constrained nanophotonic inverse design,” Sci. Rep. 7(1), 1786 (2017).
[Crossref] [PubMed]

Wade, M. T.

J. Notaros, F. Pavanello, M. T. Wade, C. Gentry, A. Atabaki, L. Alloatti, R. J. Ram, and M. Popović, “Ultra-efficient CMOS fiber-to-chip grating couplers,” in Optical Fiber Communication Conference (2016), (Optical Society of America, 2016), paper M2I.5.

Waldhäusl, R.

Wang, B.

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[Crossref]

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D. Vermeulen, S. Selvaraja, P. Verheyen, G. Lepage, W. Bogaerts, P. Absil, D. V. Thourhout, and G. Roelkens, “High-efficiency fiber-to-chip grating couplers realized using an advanced CMOS-compatible Silicon-On-Insulator platform,” Opt. Express 18(17), 18278–18283 (2010).
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[Crossref] [PubMed]

M. Dai, L. Ma, Y. Xu, M. Lu, X. Liu, and Y. Chen, “Highly efficient and perfectly vertical chip-to-fiber dual-layer grating coupler,” Opt. Express 23(2), 1691–1698 (2015).
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L. B. Verslegers, A. Mekis, T. Pinguet, Y. Chi, G. Masini, P. Sun, A. Ayazi, K. Y. Hon, S. Sahni, S. Gloeckner, C. Baudot, F. Boeuf, and P. D. Dobbelaere, “Design of low-loss polarization splitting grating couplers,” in Advanced Photonics for Communications (2014) (Optical Society of America, 2014), paper JT4A.2.

J. Notaros and M. Popović, “Band-structure approach to synthesis of grating couplers with ultra-high coupling efficiency and directivity,” in Optical Fiber Communication Conference (2015), (Optical Society of America, 2015), p. Th3F.2.

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Figures (10)

Fig. 1
Fig. 1 Diagrams of the possible modes that an input (blue dashed arrow) can couple to in a grating coupler. The left diagram depicts the chip-to-fiber coupling case while the right depicts the fiber-to-chip coupling case. In both situations, the input mode can couple to four modes of the grating. By controlling the amplitude and phase of three of these modes (represented by red arrows), the input is automatically coupled to the output mode (blue solid arrow) in the desired way. This tells us that at least six degrees of freedom are required to independently control all of the variables (amplitudes and phases) of the grating coupler.
Fig. 2
Fig. 2 Description of the operation of two-layer grating couplers. The two-layer grating resembles two fully-etched grating couplers which have been stacked one on top of the other with the top layer shifted forward by a small amount relative to the bottom layer. When each grating layer is one quarter of a wavelength thick and the spacing between grooves in each layer is a quarter wavelength, then constructive interference in the upwards direction and destructive interference in the downwards direction can be achieved. Furthermore, under these conditions, waves reflected back into the input of the grating destructively interfere leading to an inherently low back reflection.
Fig. 3
Fig. 3 The grating coupler is parameterized in terms of local “periods” and duty factors which evolve along the length of the grating. The distance between the onset of one gap and the next gap of the grating is the period Λ while the duty factor describes the fraction of this period that is unetched. Each period of the grating is assigned an index n and the way in which the period and duty factor evolve along the grating is defined using a smooth chirp function of this index n.
Fig. 4
Fig. 4 Plot of the initial starting geometry used in the optimization of a two layer SiO2-clad silicon grating coupler. A uniform grating with a duty factor of 80% and a period of 586 nm is chosen for both the top and bottom layers. This choice of period and duty factor results in a nearly vertical beam with a high directionality.
Fig. 5
Fig. 5 Optimization results for our perfectly vertical two layer grating coupler. The real part of Ez, which has been overlayed with an outline of the optimized refractive index, shows perfectly vertical emission with an extremely high directionality and flat wavefronts. This optimized structure is very well mode-matched to the mode of a 10 μm mode field diameter single mode fiber located 2 μm above the grating surface which is reflected by a chip-to-fiber efficiency of 99.2% (−0.035 dB).
Fig. 6
Fig. 6 Plots comparing the simulated electric field amplitude (top) and phase (bottom) of the optimal grating coupler design to the desired amplitude and phase. The visible deviation in the simulated phase is inconsequential as it occurs only when the field amplitude is very small.
Fig. 7
Fig. 7 Plot of the chirp functions for the optimized two-layer grating. The blue curves show the the period as a function of the position (index) along the grating. The red curves, meanwhile, show the duty factor along the grating. In both sets of curves, the solid trace corresponds to the top layer while the dotted curve corresponds to the bottom layer of the grating.
Fig. 8
Fig. 8 Plot of the coupling efficiency for perfectly vertical grating couplers optimized with a minimum feature size constraint. Using the “ideal” optimized result as a starting point, additional constrained optimizations are performed in order to design gratings that can be fabricated with lithography that has a limited resolution.
Fig. 9
Fig. 9 Plots of the coupling efficiency (top) and back reflection (bottom) as a function of wavelength for the ideal optimized result (blue dashed line) and the 65 nm constrained minimum feature size result (red solid line). Both cases achieve a high peak coupling efficiency at 1550 nm as well as a modest 1 dB bandwidth of 24 nm. The reflection in both cases is exceptionally low, reaching below −40 dB at 1550 nm.
Fig. 10
Fig. 10 Diagrams of a two-layer grating with large duty factor (left) and smaller duty factor (right). In both diagrams, d denotes layer offset and Λ denotes grating period (which for the sake of simplicity are assumed to be equal in both layers). The lines marked ne represent the effective index in each section of the grating. Although depicted here as being approximately equal, the thicknesses of the two layers need not be equal, and thus ne2 and ne3 will not necessarily be the same.

Equations (12)

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η = 1 4 P m P src | A d A E × H m * | 2
Λ ( n ) = a 0 + m = 1 M a m sin ( π 2 m N n ) + m = 1 M b m cos ( π 2 m N n )
Λ = m λ ( 1 D ) ( n e 2 + n e 3 ) + ( 2 D 1 ) n e 1 n 0 sin θ .
F ( E , H , p ) = η ( E , H ) f penalty ( p )
k g 2 π m Λ = k 0 sin θ .
φ g = 2 π m + Λ k 0 sin θ
φ g = Λ [ ( 1 D ) ( k 2 + k 3 ) + ( 2 D 1 ) k 1 ]
Λ = m λ ( 1 D ) ( n e 2 + n e 3 ) + ( 2 D 1 ) n e 1 n 0 sin θ
d = 1 n e 1 [ λ 4 + Λ ( 1 D ) ( n e 1 n e 2 ) ]
D > ( 4 m 1 ) n e 2 n e 3 + n 0 sin θ + n e 1 2 n e 1 + ( 4 m 1 ) n e 2 n e 3 .
Λ = m λ λ 4 n e 2 ( n e 2 + n e 3 n e 1 n 0 ) D n e 1 + n 0 ( 1 D ) n 0 sin θ
d = λ 4 n e 2 .

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