Abstract

In contrast to designing nanophotonic devices by tuning a handful of device parameters, we have developed a computational method which utilizes the full parameter space to design linear nanophotonic devices. We show that our method may indeed be capable of designing any linear nanophotonic device by demonstrating designed structures which are fully three-dimensional and multi-modal, exhibit novel functionality, have very compact footprints, exhibit high efficiency, and are manufacturable. In addition, we also demonstrate the ability to produce structures which are strongly robust to wavelength and temperature shift, as well as fabrication error. Critically, we show that our method does not require the user to be a nanophotonic expert or to perform any manual tuning. Instead, we are able to design devices solely based on the user’s desired performance specification for the device.

© 2013 OSA

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References

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  1. A. Gondarenko and M. Lipson, “Low modal volume dipole-like dielectric slab resonator,” Opt. Express1617689–17694 (2008).
    [CrossRef] [PubMed]
  2. C.-Y. Kao, S. Osher, and E. Yablonovitch, “Maximizing band gaps in two-dimensional photonic crystals by using level set methods,” Appl. Phys. B81, 235–244 (2005).
    [CrossRef]
  3. P. Seliger, M. Mahvash, C. Wang, and A. Levi, “Optimization of aperiodic dielectric structures,” J. Appl. Phys.100, 34310–6 (2006).
    [CrossRef]
  4. A. Oskooi, A. Mutapcic, S. Noda, J. D. Joannopoulos, S. P. Boyd, and S. G. Johnson, “Robust optimization of adiabatic tapers for coupling to slow-light photonic-crystal waveguides,” Opt. Express20, 21558–21575 (2012).
    [CrossRef] [PubMed]
  5. Y. Elesin, B. S. Lazarov, J. S. Jensen, and O. Sigmund, “Design of robust and efficient photonic switches using topology optimization,” Photonics and Nanostructures - Fundamentals and Applications10, 153165 (2012).
    [CrossRef]
  6. L. Martinelli and A. Jameson, “Computational aerodynamics: solvers and shape optimization,” J. Heat Transfer135, 011002 (2013).
    [CrossRef]
  7. M. P. Bendsoe and O. Sigmund, “Material interpolation schemes in topology optimization,” Archive of Applied Mechanics69, 635–654 (1999).
    [CrossRef]
  8. J. Lu and J. Vuckovic, “Objective-first design of high-efficiency, small-footprint couplers between arbitrary nanophotonic waveguide modes,” Opt. Express20, 7221–7236 (2012)
    [CrossRef] [PubMed]
  9. D. A. B. Miller, “All linear optical devices are mode converters,” Opt. Express20, 23985–23993 (2012).
    [CrossRef] [PubMed]
  10. S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Foundations and Trends in Machine Learning3, 1–122 (2011).
    [CrossRef]
  11. W. Shin and S. Fan, “Choice of the perfectly matched layer boundary condition for frequency-domain Maxwells equations solvers,” J. of Comp. Phys231, 3406–3431 (2012).
    [CrossRef]
  12. S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces: 1st Edition (Springer, 2002).
  13. Y. Jiao, S. Fan, and D. A. B. Miller, “Demonstrations of systematic photonic crystal design and optimization by low rank adjustment: an extremely compact mode separator”, Opt. Lett.30, 140–142 (2005).
    [CrossRef]
  14. J. Castro, D. F. Geraghty, S. Honkanen, C. M. Greiner, D. Iazikov, and T. W. Mossberg, “Demonstration of mode conversion using anti-symmetric waveguide Bragg gratings,” Opt. Express13, 4180–4184 (2005).
    [CrossRef] [PubMed]
  15. E. Khoo, A. Liu, and J. Wu, “Nonuniform photonic crystal taper for high-efficiency mode coupling,” Opt. Express13, 7748–7759 (2005).
    [CrossRef] [PubMed]
  16. P. Sanchis, J. Marti, J. Blasco, A. Martinez, and A. Garcia, “Mode matching technique for highly efficient coupling between dielectric waveguides and planar photonic crystal circuits,” Opt. Express10, 1391–1397 (2002).
    [CrossRef] [PubMed]
  17. T. D. Happ, M. Kamp, and A. Forchel, “Photonic crystal tapers for ultracompact mode conversion,” Opt. Lett.26, 1102–1104 (2001)
    [CrossRef]
  18. F. Van Laere, G. Roelkens, M. Ayre, J. Schrauwen, D. Taillaert, D. Van Thourhout, T. F. Krauss, and R. Baets, “Compact and highly efficient grating couplers between optical fiber and nanophotonic waveguides,” J. of Light-wave Tech.25, 151–156 (2007).
    [CrossRef]
  19. F. Wang, J. S. Jensen, and O. Sigmund, “Robust topology optimization of photonic crystal waveguides with tailored dispersion properties,” J. Opt. Soc. Am. B28, 387–397 (2011).
    [CrossRef]

2013 (1)

L. Martinelli and A. Jameson, “Computational aerodynamics: solvers and shape optimization,” J. Heat Transfer135, 011002 (2013).
[CrossRef]

2012 (5)

Y. Elesin, B. S. Lazarov, J. S. Jensen, and O. Sigmund, “Design of robust and efficient photonic switches using topology optimization,” Photonics and Nanostructures - Fundamentals and Applications10, 153165 (2012).
[CrossRef]

W. Shin and S. Fan, “Choice of the perfectly matched layer boundary condition for frequency-domain Maxwells equations solvers,” J. of Comp. Phys231, 3406–3431 (2012).
[CrossRef]

J. Lu and J. Vuckovic, “Objective-first design of high-efficiency, small-footprint couplers between arbitrary nanophotonic waveguide modes,” Opt. Express20, 7221–7236 (2012)
[CrossRef] [PubMed]

A. Oskooi, A. Mutapcic, S. Noda, J. D. Joannopoulos, S. P. Boyd, and S. G. Johnson, “Robust optimization of adiabatic tapers for coupling to slow-light photonic-crystal waveguides,” Opt. Express20, 21558–21575 (2012).
[CrossRef] [PubMed]

D. A. B. Miller, “All linear optical devices are mode converters,” Opt. Express20, 23985–23993 (2012).
[CrossRef] [PubMed]

2011 (2)

F. Wang, J. S. Jensen, and O. Sigmund, “Robust topology optimization of photonic crystal waveguides with tailored dispersion properties,” J. Opt. Soc. Am. B28, 387–397 (2011).
[CrossRef]

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Foundations and Trends in Machine Learning3, 1–122 (2011).
[CrossRef]

2008 (1)

2007 (1)

F. Van Laere, G. Roelkens, M. Ayre, J. Schrauwen, D. Taillaert, D. Van Thourhout, T. F. Krauss, and R. Baets, “Compact and highly efficient grating couplers between optical fiber and nanophotonic waveguides,” J. of Light-wave Tech.25, 151–156 (2007).
[CrossRef]

2006 (1)

P. Seliger, M. Mahvash, C. Wang, and A. Levi, “Optimization of aperiodic dielectric structures,” J. Appl. Phys.100, 34310–6 (2006).
[CrossRef]

2005 (4)

J. Castro, D. F. Geraghty, S. Honkanen, C. M. Greiner, D. Iazikov, and T. W. Mossberg, “Demonstration of mode conversion using anti-symmetric waveguide Bragg gratings,” Opt. Express13, 4180–4184 (2005).
[CrossRef] [PubMed]

E. Khoo, A. Liu, and J. Wu, “Nonuniform photonic crystal taper for high-efficiency mode coupling,” Opt. Express13, 7748–7759 (2005).
[CrossRef] [PubMed]

Y. Jiao, S. Fan, and D. A. B. Miller, “Demonstrations of systematic photonic crystal design and optimization by low rank adjustment: an extremely compact mode separator”, Opt. Lett.30, 140–142 (2005).
[CrossRef]

C.-Y. Kao, S. Osher, and E. Yablonovitch, “Maximizing band gaps in two-dimensional photonic crystals by using level set methods,” Appl. Phys. B81, 235–244 (2005).
[CrossRef]

2002 (1)

2001 (1)

1999 (1)

M. P. Bendsoe and O. Sigmund, “Material interpolation schemes in topology optimization,” Archive of Applied Mechanics69, 635–654 (1999).
[CrossRef]

Ayre, M.

F. Van Laere, G. Roelkens, M. Ayre, J. Schrauwen, D. Taillaert, D. Van Thourhout, T. F. Krauss, and R. Baets, “Compact and highly efficient grating couplers between optical fiber and nanophotonic waveguides,” J. of Light-wave Tech.25, 151–156 (2007).
[CrossRef]

Baets, R.

F. Van Laere, G. Roelkens, M. Ayre, J. Schrauwen, D. Taillaert, D. Van Thourhout, T. F. Krauss, and R. Baets, “Compact and highly efficient grating couplers between optical fiber and nanophotonic waveguides,” J. of Light-wave Tech.25, 151–156 (2007).
[CrossRef]

Bendsoe, M. P.

M. P. Bendsoe and O. Sigmund, “Material interpolation schemes in topology optimization,” Archive of Applied Mechanics69, 635–654 (1999).
[CrossRef]

Blasco, J.

Boyd, S.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Foundations and Trends in Machine Learning3, 1–122 (2011).
[CrossRef]

Boyd, S. P.

Castro, J.

Chu, E.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Foundations and Trends in Machine Learning3, 1–122 (2011).
[CrossRef]

Eckstein, J.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Foundations and Trends in Machine Learning3, 1–122 (2011).
[CrossRef]

Elesin, Y.

Y. Elesin, B. S. Lazarov, J. S. Jensen, and O. Sigmund, “Design of robust and efficient photonic switches using topology optimization,” Photonics and Nanostructures - Fundamentals and Applications10, 153165 (2012).
[CrossRef]

Fan, S.

W. Shin and S. Fan, “Choice of the perfectly matched layer boundary condition for frequency-domain Maxwells equations solvers,” J. of Comp. Phys231, 3406–3431 (2012).
[CrossRef]

Y. Jiao, S. Fan, and D. A. B. Miller, “Demonstrations of systematic photonic crystal design and optimization by low rank adjustment: an extremely compact mode separator”, Opt. Lett.30, 140–142 (2005).
[CrossRef]

Fedkiw, R.

S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces: 1st Edition (Springer, 2002).

Forchel, A.

Garcia, A.

Geraghty, D. F.

Gondarenko, A.

Greiner, C. M.

Happ, T. D.

Honkanen, S.

Iazikov, D.

Jameson, A.

L. Martinelli and A. Jameson, “Computational aerodynamics: solvers and shape optimization,” J. Heat Transfer135, 011002 (2013).
[CrossRef]

Jensen, J. S.

Y. Elesin, B. S. Lazarov, J. S. Jensen, and O. Sigmund, “Design of robust and efficient photonic switches using topology optimization,” Photonics and Nanostructures - Fundamentals and Applications10, 153165 (2012).
[CrossRef]

F. Wang, J. S. Jensen, and O. Sigmund, “Robust topology optimization of photonic crystal waveguides with tailored dispersion properties,” J. Opt. Soc. Am. B28, 387–397 (2011).
[CrossRef]

Jiao, Y.

Y. Jiao, S. Fan, and D. A. B. Miller, “Demonstrations of systematic photonic crystal design and optimization by low rank adjustment: an extremely compact mode separator”, Opt. Lett.30, 140–142 (2005).
[CrossRef]

Joannopoulos, J. D.

Johnson, S. G.

Kamp, M.

Kao, C.-Y.

C.-Y. Kao, S. Osher, and E. Yablonovitch, “Maximizing band gaps in two-dimensional photonic crystals by using level set methods,” Appl. Phys. B81, 235–244 (2005).
[CrossRef]

Khoo, E.

Krauss, T. F.

F. Van Laere, G. Roelkens, M. Ayre, J. Schrauwen, D. Taillaert, D. Van Thourhout, T. F. Krauss, and R. Baets, “Compact and highly efficient grating couplers between optical fiber and nanophotonic waveguides,” J. of Light-wave Tech.25, 151–156 (2007).
[CrossRef]

Lazarov, B. S.

Y. Elesin, B. S. Lazarov, J. S. Jensen, and O. Sigmund, “Design of robust and efficient photonic switches using topology optimization,” Photonics and Nanostructures - Fundamentals and Applications10, 153165 (2012).
[CrossRef]

Levi, A.

P. Seliger, M. Mahvash, C. Wang, and A. Levi, “Optimization of aperiodic dielectric structures,” J. Appl. Phys.100, 34310–6 (2006).
[CrossRef]

Lipson, M.

Liu, A.

Lu, J.

Mahvash, M.

P. Seliger, M. Mahvash, C. Wang, and A. Levi, “Optimization of aperiodic dielectric structures,” J. Appl. Phys.100, 34310–6 (2006).
[CrossRef]

Marti, J.

Martinelli, L.

L. Martinelli and A. Jameson, “Computational aerodynamics: solvers and shape optimization,” J. Heat Transfer135, 011002 (2013).
[CrossRef]

Martinez, A.

Miller, D. A. B.

D. A. B. Miller, “All linear optical devices are mode converters,” Opt. Express20, 23985–23993 (2012).
[CrossRef] [PubMed]

Y. Jiao, S. Fan, and D. A. B. Miller, “Demonstrations of systematic photonic crystal design and optimization by low rank adjustment: an extremely compact mode separator”, Opt. Lett.30, 140–142 (2005).
[CrossRef]

Mossberg, T. W.

Mutapcic, A.

Noda, S.

Osher, S.

C.-Y. Kao, S. Osher, and E. Yablonovitch, “Maximizing band gaps in two-dimensional photonic crystals by using level set methods,” Appl. Phys. B81, 235–244 (2005).
[CrossRef]

S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces: 1st Edition (Springer, 2002).

Oskooi, A.

Parikh, N.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Foundations and Trends in Machine Learning3, 1–122 (2011).
[CrossRef]

Peleato, B.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Foundations and Trends in Machine Learning3, 1–122 (2011).
[CrossRef]

Roelkens, G.

F. Van Laere, G. Roelkens, M. Ayre, J. Schrauwen, D. Taillaert, D. Van Thourhout, T. F. Krauss, and R. Baets, “Compact and highly efficient grating couplers between optical fiber and nanophotonic waveguides,” J. of Light-wave Tech.25, 151–156 (2007).
[CrossRef]

Sanchis, P.

Schrauwen, J.

F. Van Laere, G. Roelkens, M. Ayre, J. Schrauwen, D. Taillaert, D. Van Thourhout, T. F. Krauss, and R. Baets, “Compact and highly efficient grating couplers between optical fiber and nanophotonic waveguides,” J. of Light-wave Tech.25, 151–156 (2007).
[CrossRef]

Seliger, P.

P. Seliger, M. Mahvash, C. Wang, and A. Levi, “Optimization of aperiodic dielectric structures,” J. Appl. Phys.100, 34310–6 (2006).
[CrossRef]

Shin, W.

W. Shin and S. Fan, “Choice of the perfectly matched layer boundary condition for frequency-domain Maxwells equations solvers,” J. of Comp. Phys231, 3406–3431 (2012).
[CrossRef]

Sigmund, O.

Y. Elesin, B. S. Lazarov, J. S. Jensen, and O. Sigmund, “Design of robust and efficient photonic switches using topology optimization,” Photonics and Nanostructures - Fundamentals and Applications10, 153165 (2012).
[CrossRef]

F. Wang, J. S. Jensen, and O. Sigmund, “Robust topology optimization of photonic crystal waveguides with tailored dispersion properties,” J. Opt. Soc. Am. B28, 387–397 (2011).
[CrossRef]

M. P. Bendsoe and O. Sigmund, “Material interpolation schemes in topology optimization,” Archive of Applied Mechanics69, 635–654 (1999).
[CrossRef]

Taillaert, D.

F. Van Laere, G. Roelkens, M. Ayre, J. Schrauwen, D. Taillaert, D. Van Thourhout, T. F. Krauss, and R. Baets, “Compact and highly efficient grating couplers between optical fiber and nanophotonic waveguides,” J. of Light-wave Tech.25, 151–156 (2007).
[CrossRef]

Van Laere, F.

F. Van Laere, G. Roelkens, M. Ayre, J. Schrauwen, D. Taillaert, D. Van Thourhout, T. F. Krauss, and R. Baets, “Compact and highly efficient grating couplers between optical fiber and nanophotonic waveguides,” J. of Light-wave Tech.25, 151–156 (2007).
[CrossRef]

Van Thourhout, D.

F. Van Laere, G. Roelkens, M. Ayre, J. Schrauwen, D. Taillaert, D. Van Thourhout, T. F. Krauss, and R. Baets, “Compact and highly efficient grating couplers between optical fiber and nanophotonic waveguides,” J. of Light-wave Tech.25, 151–156 (2007).
[CrossRef]

Vuckovic, J.

Wang, C.

P. Seliger, M. Mahvash, C. Wang, and A. Levi, “Optimization of aperiodic dielectric structures,” J. Appl. Phys.100, 34310–6 (2006).
[CrossRef]

Wang, F.

Wu, J.

Yablonovitch, E.

C.-Y. Kao, S. Osher, and E. Yablonovitch, “Maximizing band gaps in two-dimensional photonic crystals by using level set methods,” Appl. Phys. B81, 235–244 (2005).
[CrossRef]

Appl. Phys. B (1)

C.-Y. Kao, S. Osher, and E. Yablonovitch, “Maximizing band gaps in two-dimensional photonic crystals by using level set methods,” Appl. Phys. B81, 235–244 (2005).
[CrossRef]

Archive of Applied Mechanics (1)

M. P. Bendsoe and O. Sigmund, “Material interpolation schemes in topology optimization,” Archive of Applied Mechanics69, 635–654 (1999).
[CrossRef]

Foundations and Trends in Machine Learning (1)

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Foundations and Trends in Machine Learning3, 1–122 (2011).
[CrossRef]

J. Appl. Phys. (1)

P. Seliger, M. Mahvash, C. Wang, and A. Levi, “Optimization of aperiodic dielectric structures,” J. Appl. Phys.100, 34310–6 (2006).
[CrossRef]

J. Heat Transfer (1)

L. Martinelli and A. Jameson, “Computational aerodynamics: solvers and shape optimization,” J. Heat Transfer135, 011002 (2013).
[CrossRef]

J. of Comp. Phys (1)

W. Shin and S. Fan, “Choice of the perfectly matched layer boundary condition for frequency-domain Maxwells equations solvers,” J. of Comp. Phys231, 3406–3431 (2012).
[CrossRef]

J. of Light-wave Tech. (1)

F. Van Laere, G. Roelkens, M. Ayre, J. Schrauwen, D. Taillaert, D. Van Thourhout, T. F. Krauss, and R. Baets, “Compact and highly efficient grating couplers between optical fiber and nanophotonic waveguides,” J. of Light-wave Tech.25, 151–156 (2007).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Express (7)

Opt. Lett. (2)

T. D. Happ, M. Kamp, and A. Forchel, “Photonic crystal tapers for ultracompact mode conversion,” Opt. Lett.26, 1102–1104 (2001)
[CrossRef]

Y. Jiao, S. Fan, and D. A. B. Miller, “Demonstrations of systematic photonic crystal design and optimization by low rank adjustment: an extremely compact mode separator”, Opt. Lett.30, 140–142 (2005).
[CrossRef]

Photonics and Nanostructures - Fundamentals and Applications (1)

Y. Elesin, B. S. Lazarov, J. S. Jensen, and O. Sigmund, “Design of robust and efficient photonic switches using topology optimization,” Photonics and Nanostructures - Fundamentals and Applications10, 153165 (2012).
[CrossRef]

Other (1)

S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces: 1st Edition (Springer, 2002).

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Figures (28)

Fig. 1
Fig. 1

Perfomance specification of the TE mode converter. Input mode is the fundamental TE-polarized mode on the left. Primary output mode is the second-order mode on the right. Output power in the transmitted fundamental mode should be no more than 1%. The structure shown is the final three-dimensional design (the same holds for all the following figures in the article).

Fig. 2
Fig. 2

Structure and E-field at the central plane of the TE mode converter. The conversion efficiency into the second-order mode is 86.4%, while the power into the rejection mode (fundamental) is 0.7%. Device footprint is 1.6×2.4 microns. Operating wavelength is 1550 nm.

Fig. 3
Fig. 3

Perfomance specification of the TM mode converter. Input mode is the fundamental TM-polarized mode on the left. Primary output mode is the second-order mode on the right. Output power in the transmitted fundamental mode on the right above 1% is to be rejected. The structure shown is the final three-dimensional design.

Fig. 4
Fig. 4

Structure and E-field at the central plane of the TM mode converter. The conversion efficiency into the second-order mode is 76.9%, while the power into the rejection mode (fundamental) is 1.0%. Device footprint is 1.6 × 2.4 microns.

Fig. 5
Fig. 5

Spatial mode splitter performance specification. Input mode is either the fundamental or second-order TE-polarized mode on the left. Output modes are the fundamental waveguide modes of either output waveguide on the right. Output power into the desired output arm is specified to be greater than 90%, while power into the opposing arm is set to below 1%.

Fig. 6
Fig. 6

Spatial mode splitter final result. The conversion efficiencies into the upper and lower output arms are 88.7% and 77.4% respectively, while the rejection powers for the same modes are 0.27% and 0.20%. Device footprint is 2.8 × 2.8 microns.

Fig. 7
Fig. 7

TE/TM splitter performance specification. Input mode is either the fundamental TE-polarized (Ey dominant, top left) and TM-polarized (Ez dominant, top right). Output power into the desired output arm is specified to be greater than 90%, while power into the opposing arm is set to below 1%.

Fig. 8
Fig. 8

TE/TM splitter final result. The conversion efficiencies into the upper and lower output arms are 87.6% and 88.8% respectively, while the rejection powers for the same modes are 1.06% and 0.58%. Device footprint is 2.8 × 2.8 microns.

Fig. 9
Fig. 9

Wavelength splitter performance specification. Input mode is the fundamental TE-polarized mode on the left at a wavelength of either 1550 nm or 1330 nm. Output modes are the fundamental waveguide modes of either output waveguide on the right; however, the 1550 nm wavelength is directed into the top output, while the 1310 nm wavelength is directed into the bottom output. Output power into the desired output arm is specified to be greater than 90%, while power into the opposing arm is set to below 1%.

Fig. 10
Fig. 10

Wavelength splitter final result. The conversion efficiencies into the upper and lower output arms are 83.2% and 78.7% respectively, while the rejection powers for the same modes are 0.49% and 1.66%. Device footprint is 2.8 × 2.8 microns.

Fig. 11
Fig. 11

3×3 hub performance specification. Input and output modes all consist of the fundamental TE-polarized mode. Output power into the desired output arm is specified to be greater than 90%, no rejection modes are used for computational efficiency. This hub directs input power from input ports 1, 2, and 3 (from top to bottom) into output ports 2, 3, and 1 respectively.

Fig. 12
Fig. 12

3×3 hub final result. The conversion efficiencies into the selected output arms are 88.6%, 90.6%, and 87.3% for input arms 1, 2, and 3 respectively (top to bottom).

Fig. 13
Fig. 13

4×4 hub performance specification. Input and output modes all consist of the fundamental TE-polarized mode. Output power into the desired output arm is specified to be greater than 90%, no rejection modes are used for computational efficiency. This hub directs input power from input ports 1, 2, 3, and 4 into output ports 3, 2, 4, and 1 respectively.

Fig. 14
Fig. 14

4×4 hub final result. The conversion efficiencies into the selected output arms are 85.9%, 88.1%, 85.4%, and 84.3% for input arms 1, 2, 3, and 4 respectively (top to bottom).

Fig. 15
Fig. 15

2×2×2 hub performance specification. Input and output modes all consist of the fundamental TE-polarized mode at either 1550 nm or 1310 nm wavelengths. Output power into the desired output arm is specified to be greater than 80%, no rejection modes are used for computational efficiency. This hub directs input arms 1 and 2 into output arms 1 and 2 at 1550 nm, but swaps them at 1310 nm.

Fig. 16
Fig. 16

2×2×2 hub final result. The conversion efficiencies at the 1550 nm wavelength are 77.6% and 73.7% respectively for the top and bottom inputs. At 1310 nm, the respective efficiencies are 75.7% and 75.2%.

Fig. 17
Fig. 17

Compact fiber coupler performance specification. Input mode consists of the Ey-polarized fundamental fiber mode. Output mode is the fundamental TE-polarized mode of the in-plane waveguide. Output power into the desired output arm is specified to be greater than 90%.

Fig. 18
Fig. 18

Compact fiber coupler final result. The conversion efficiency into the in-plane waveguide mode is 51.5%.

Fig. 19
Fig. 19

Mode-splitting fiber coupler performance specification. Input mode consists of the fundamental fiber mode or the third-order, circularly polarized fiber mode. Output mode is the fundamental TE-polarized mode of either in-plane waveguide. The device is designed to couple the fundamental fiber mode into the upper output arm, while the third-order fiber mode is coupled into the lower output arm. Output power into the desired output arm is specified to be greater than 90%.

Fig. 20
Fig. 20

Mode-splitting fiber coupler final result. The conversion efficiency for the fundamental fiber mode input is 32.6% (top plot). The conversion efficiency for the third-order fiber mode input is 22.7% (bottom plot).

Fig. 21
Fig. 21

Wavelength-splitting fiber coupler performance specification. Input mode consists of the Ey-polarized fundamental fiber mode at either the 1310 nm or 1550 nm wavelengths. Output mode is the fundamental TE-polarized mode of either in-plane waveguide. The structure is designed to guide light at the 1550 nm wavelength into the upper output arm, while 1310 nm light is guided into the lower output arm. Output power into the desired output arm is specified to be greater than 90%.

Fig. 22
Fig. 22

Wavelength-splitting fiber coupler final result. The conversion efficiency into the in-plane waveguide mode at 1550 nm is 31.6%. The conversion efficiency into the in-plane waveguide mode at 1310 nm is 28.6%.

Fig. 23
Fig. 23

Broadband analysis of previously design wavelength splitter (Fig. 10). Although high-efficiency operation is achieved, the performance quickly drops off away from the target wavelengths (denoted by arrows).

Fig. 24
Fig. 24

Broadband analysis of broadband wavelength splitter (final design shown in Fig. 25. The addition of multiple target wavelengths (vertical arrows) allows for high-efficiency operation is achieved across a wide bandwidth.

Fig. 25
Fig. 25

Broadband wavelength splitter final result. The efficiencies at the central target wavelengths of 1550 nm and 1310 nm exceed those of its narrowband counterpart (Fig. 10).

Fig. 26
Fig. 26

Temperature analysis of the broadband wavelength splitter. Stable operating points (defined as efficiency ≥ 80%) exist over a temperature shift of 905K.

Fig. 27
Fig. 27

Analysis of fabrication-error on the performance of the broadband wavelength splitter. Original central wavelengths are shown to hold greater than 70% efficiency, in spite of up to 8 nm of over- or under-etch error.

Fig. 28
Fig. 28

Comparison of under-etched, as-designed, and over-etched structures. Differences are subtle since the pixel size is 40 nm and the fabrication error is 8 nm.

Equations (4)

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minimize i M A i ( z ) x i b i 2
subject to α i j | c i j x i | β i j , for i = 1 , , M and j = 1 , , N i
z min z z max
( × μ 0 1 × ω i 2 ε ) E i + i ω i J i A i ( z ) x i b i

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