Abstract

We present a method for directly analyzing photonic nano-devices and apply it to photonic crystal cavities. Two-dimensional photonic crystals are scanned and reproduced in computer memory for Finite Difference Time Domain simuations. The results closely match experimental observations, with a fidelity far beyond that for idealized structures. This analysis allows close examination of error mechanisms and analytical error models.

© 2006 Optical Society of America

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References

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  1. S. G. Johnson, M. I. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B: Lasers and Optics 81, 283 – 293 (2005).
    [Crossref]
  2. S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic Crystal waveguides: role of fabrication disorder and photon Group velocity.” Phys. Rev. Lett. 94, 033,903 – 4 (2005).
    [Crossref]
  3. A. F. Koenderink, A. Lagendijk, and W. L. Vos, “Optical extinction due to intrinsic structural variations of photonic crystals,” Phys. Rev. B. 72, 153,102 – (2005).
    [Crossref]
  4. P. Vukusic and J. R. Sambles, “Photonic structures in biology,” Nature 424, 852–55 (2003).
    [Crossref] [PubMed]
  5. J. Vuckovic and Y. Yamamoto, “Photonic crystal microcavities for cavity quantum electrodynamics with a single quantum dot.” Appl. Phys. Lett. 82, 2374 – 6 (2003).
    [Crossref]
  6. D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. J. VučkoviĆ “Controlling the Spontaneous Emission Rate of Single Quantum Dots in a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett. 95(013904) (2005).
    [Crossref] [PubMed]
  7. K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis: Solving Maxwells Equation and the Schrödinger Equation (Wiley-Interscience Publications, New York, 2001).
    [PubMed]
  8. S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwells equations with shifting material boundaries,” Physical Review E 65(066611) (2002).
    [Crossref]
  9. D. Englund, I. Fushman, and J. VučkoviĆ, “General Recipe for Designing Photonic Crystal Cavities,” Opt. Express 12, 5961–75 (2005).
    [Crossref]
  10. J. H.C. Casey, D. D. Sell, and K. W. Wecht, “,” Journal of Applied Physics 46, 250 (1975).
    [Crossref]
  11. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley Interscience, 2003).
  12. G. C. D.et al., “Wet Chemical Digital Etching of GaAs at Room Temperature,” J. Electrochemical Soc. 143, 3652–56 (1996).
    [Crossref]
  13. J. Schilling, F. Muller, S. Matthias, R. B. Wehrspohn, U. Gosele, and K. Busch, “Three-dimensional photonic crystals based on macroporous silicon with modulated pore diameter,” Appl. Phys. Lett. 78, 1180–2 (2001).
    [Crossref]

2005 (5)

S. G. Johnson, M. I. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B: Lasers and Optics 81, 283 – 293 (2005).
[Crossref]

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic Crystal waveguides: role of fabrication disorder and photon Group velocity.” Phys. Rev. Lett. 94, 033,903 – 4 (2005).
[Crossref]

A. F. Koenderink, A. Lagendijk, and W. L. Vos, “Optical extinction due to intrinsic structural variations of photonic crystals,” Phys. Rev. B. 72, 153,102 – (2005).
[Crossref]

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. J. VučkoviĆ “Controlling the Spontaneous Emission Rate of Single Quantum Dots in a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett. 95(013904) (2005).
[Crossref] [PubMed]

D. Englund, I. Fushman, and J. VučkoviĆ, “General Recipe for Designing Photonic Crystal Cavities,” Opt. Express 12, 5961–75 (2005).
[Crossref]

2003 (2)

P. Vukusic and J. R. Sambles, “Photonic structures in biology,” Nature 424, 852–55 (2003).
[Crossref] [PubMed]

J. Vuckovic and Y. Yamamoto, “Photonic crystal microcavities for cavity quantum electrodynamics with a single quantum dot.” Appl. Phys. Lett. 82, 2374 – 6 (2003).
[Crossref]

2002 (1)

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwells equations with shifting material boundaries,” Physical Review E 65(066611) (2002).
[Crossref]

2001 (1)

J. Schilling, F. Muller, S. Matthias, R. B. Wehrspohn, U. Gosele, and K. Busch, “Three-dimensional photonic crystals based on macroporous silicon with modulated pore diameter,” Appl. Phys. Lett. 78, 1180–2 (2001).
[Crossref]

1996 (1)

G. C. D.et al., “Wet Chemical Digital Etching of GaAs at Room Temperature,” J. Electrochemical Soc. 143, 3652–56 (1996).
[Crossref]

1975 (1)

J. H.C. Casey, D. D. Sell, and K. W. Wecht, “,” Journal of Applied Physics 46, 250 (1975).
[Crossref]

Arakawa, Y.

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. J. VučkoviĆ “Controlling the Spontaneous Emission Rate of Single Quantum Dots in a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett. 95(013904) (2005).
[Crossref] [PubMed]

Busch, K.

J. Schilling, F. Muller, S. Matthias, R. B. Wehrspohn, U. Gosele, and K. Busch, “Three-dimensional photonic crystals based on macroporous silicon with modulated pore diameter,” Appl. Phys. Lett. 78, 1180–2 (2001).
[Crossref]

Casey, J. H.C.

J. H.C. Casey, D. D. Sell, and K. W. Wecht, “,” Journal of Applied Physics 46, 250 (1975).
[Crossref]

D., G. C.

G. C. D.et al., “Wet Chemical Digital Etching of GaAs at Room Temperature,” J. Electrochemical Soc. 143, 3652–56 (1996).
[Crossref]

Englund, D.

D. Englund, I. Fushman, and J. VučkoviĆ, “General Recipe for Designing Photonic Crystal Cavities,” Opt. Express 12, 5961–75 (2005).
[Crossref]

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. J. VučkoviĆ “Controlling the Spontaneous Emission Rate of Single Quantum Dots in a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett. 95(013904) (2005).
[Crossref] [PubMed]

Fattal, D.

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. J. VučkoviĆ “Controlling the Spontaneous Emission Rate of Single Quantum Dots in a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett. 95(013904) (2005).
[Crossref] [PubMed]

Fink, Y.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwells equations with shifting material boundaries,” Physical Review E 65(066611) (2002).
[Crossref]

Fushman, I.

D. Englund, I. Fushman, and J. VučkoviĆ, “General Recipe for Designing Photonic Crystal Cavities,” Opt. Express 12, 5961–75 (2005).
[Crossref]

Gosele, U.

J. Schilling, F. Muller, S. Matthias, R. B. Wehrspohn, U. Gosele, and K. Busch, “Three-dimensional photonic crystals based on macroporous silicon with modulated pore diameter,” Appl. Phys. Lett. 78, 1180–2 (2001).
[Crossref]

Hughes, S.

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic Crystal waveguides: role of fabrication disorder and photon Group velocity.” Phys. Rev. Lett. 94, 033,903 – 4 (2005).
[Crossref]

Ibanescu, M.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwells equations with shifting material boundaries,” Physical Review E 65(066611) (2002).
[Crossref]

Jacobs, S.

S. G. Johnson, M. I. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B: Lasers and Optics 81, 283 – 293 (2005).
[Crossref]

Joannopoulos, J. D.

S. G. Johnson, M. I. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B: Lasers and Optics 81, 283 – 293 (2005).
[Crossref]

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwells equations with shifting material boundaries,” Physical Review E 65(066611) (2002).
[Crossref]

Johnson, S. G.

S. G. Johnson, M. I. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B: Lasers and Optics 81, 283 – 293 (2005).
[Crossref]

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwells equations with shifting material boundaries,” Physical Review E 65(066611) (2002).
[Crossref]

Karalis, A.

S. G. Johnson, M. I. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B: Lasers and Optics 81, 283 – 293 (2005).
[Crossref]

Kawano, K.

K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis: Solving Maxwells Equation and the Schrödinger Equation (Wiley-Interscience Publications, New York, 2001).
[PubMed]

Kitoh, T.

K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis: Solving Maxwells Equation and the Schrödinger Equation (Wiley-Interscience Publications, New York, 2001).
[PubMed]

Koenderink, A. F.

A. F. Koenderink, A. Lagendijk, and W. L. Vos, “Optical extinction due to intrinsic structural variations of photonic crystals,” Phys. Rev. B. 72, 153,102 – (2005).
[Crossref]

Lagendijk, A.

A. F. Koenderink, A. Lagendijk, and W. L. Vos, “Optical extinction due to intrinsic structural variations of photonic crystals,” Phys. Rev. B. 72, 153,102 – (2005).
[Crossref]

Matthias, S.

J. Schilling, F. Muller, S. Matthias, R. B. Wehrspohn, U. Gosele, and K. Busch, “Three-dimensional photonic crystals based on macroporous silicon with modulated pore diameter,” Appl. Phys. Lett. 78, 1180–2 (2001).
[Crossref]

Muller, F.

J. Schilling, F. Muller, S. Matthias, R. B. Wehrspohn, U. Gosele, and K. Busch, “Three-dimensional photonic crystals based on macroporous silicon with modulated pore diameter,” Appl. Phys. Lett. 78, 1180–2 (2001).
[Crossref]

Nakaoka, T.

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. J. VučkoviĆ “Controlling the Spontaneous Emission Rate of Single Quantum Dots in a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett. 95(013904) (2005).
[Crossref] [PubMed]

Povinelli, M. I.

S. G. Johnson, M. I. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B: Lasers and Optics 81, 283 – 293 (2005).
[Crossref]

Ramunno, L.

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic Crystal waveguides: role of fabrication disorder and photon Group velocity.” Phys. Rev. Lett. 94, 033,903 – 4 (2005).
[Crossref]

Sambles, J. R.

P. Vukusic and J. R. Sambles, “Photonic structures in biology,” Nature 424, 852–55 (2003).
[Crossref] [PubMed]

Schilling, J.

J. Schilling, F. Muller, S. Matthias, R. B. Wehrspohn, U. Gosele, and K. Busch, “Three-dimensional photonic crystals based on macroporous silicon with modulated pore diameter,” Appl. Phys. Lett. 78, 1180–2 (2001).
[Crossref]

Sell, D. D.

J. H.C. Casey, D. D. Sell, and K. W. Wecht, “,” Journal of Applied Physics 46, 250 (1975).
[Crossref]

Sipe, J. E.

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic Crystal waveguides: role of fabrication disorder and photon Group velocity.” Phys. Rev. Lett. 94, 033,903 – 4 (2005).
[Crossref]

Skorobogatiy, M. A.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwells equations with shifting material boundaries,” Physical Review E 65(066611) (2002).
[Crossref]

Soljacic, M.

S. G. Johnson, M. I. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B: Lasers and Optics 81, 283 – 293 (2005).
[Crossref]

Solomon, G.

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. J. VučkoviĆ “Controlling the Spontaneous Emission Rate of Single Quantum Dots in a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett. 95(013904) (2005).
[Crossref] [PubMed]

Vos, W. L.

A. F. Koenderink, A. Lagendijk, and W. L. Vos, “Optical extinction due to intrinsic structural variations of photonic crystals,” Phys. Rev. B. 72, 153,102 – (2005).
[Crossref]

VuckoviC, J.

D. Englund, I. Fushman, and J. VučkoviĆ, “General Recipe for Designing Photonic Crystal Cavities,” Opt. Express 12, 5961–75 (2005).
[Crossref]

J. Vuckovic and Y. Yamamoto, “Photonic crystal microcavities for cavity quantum electrodynamics with a single quantum dot.” Appl. Phys. Lett. 82, 2374 – 6 (2003).
[Crossref]

VuckoviC, J. J.

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. J. VučkoviĆ “Controlling the Spontaneous Emission Rate of Single Quantum Dots in a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett. 95(013904) (2005).
[Crossref] [PubMed]

Vukusic, P.

P. Vukusic and J. R. Sambles, “Photonic structures in biology,” Nature 424, 852–55 (2003).
[Crossref] [PubMed]

Waks, E.

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. J. VučkoviĆ “Controlling the Spontaneous Emission Rate of Single Quantum Dots in a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett. 95(013904) (2005).
[Crossref] [PubMed]

Wecht, K. W.

J. H.C. Casey, D. D. Sell, and K. W. Wecht, “,” Journal of Applied Physics 46, 250 (1975).
[Crossref]

Wehrspohn, R. B.

J. Schilling, F. Muller, S. Matthias, R. B. Wehrspohn, U. Gosele, and K. Busch, “Three-dimensional photonic crystals based on macroporous silicon with modulated pore diameter,” Appl. Phys. Lett. 78, 1180–2 (2001).
[Crossref]

Weisberg, O.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwells equations with shifting material boundaries,” Physical Review E 65(066611) (2002).
[Crossref]

Yamamoto, Y.

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. J. VučkoviĆ “Controlling the Spontaneous Emission Rate of Single Quantum Dots in a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett. 95(013904) (2005).
[Crossref] [PubMed]

J. Vuckovic and Y. Yamamoto, “Photonic crystal microcavities for cavity quantum electrodynamics with a single quantum dot.” Appl. Phys. Lett. 82, 2374 – 6 (2003).
[Crossref]

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley Interscience, 2003).

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley Interscience, 2003).

Young, J. F.

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic Crystal waveguides: role of fabrication disorder and photon Group velocity.” Phys. Rev. Lett. 94, 033,903 – 4 (2005).
[Crossref]

Zhang, B.

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. J. VučkoviĆ “Controlling the Spontaneous Emission Rate of Single Quantum Dots in a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett. 95(013904) (2005).
[Crossref] [PubMed]

Appl. Phys. B: Lasers and Optics (1)

S. G. Johnson, M. I. Povinelli, M. Soljacic, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B: Lasers and Optics 81, 283 – 293 (2005).
[Crossref]

Appl. Phys. Lett. (2)

J. Vuckovic and Y. Yamamoto, “Photonic crystal microcavities for cavity quantum electrodynamics with a single quantum dot.” Appl. Phys. Lett. 82, 2374 – 6 (2003).
[Crossref]

J. Schilling, F. Muller, S. Matthias, R. B. Wehrspohn, U. Gosele, and K. Busch, “Three-dimensional photonic crystals based on macroporous silicon with modulated pore diameter,” Appl. Phys. Lett. 78, 1180–2 (2001).
[Crossref]

J. Electrochemical Soc. (1)

G. C. D.et al., “Wet Chemical Digital Etching of GaAs at Room Temperature,” J. Electrochemical Soc. 143, 3652–56 (1996).
[Crossref]

Journal of Applied Physics (1)

J. H.C. Casey, D. D. Sell, and K. W. Wecht, “,” Journal of Applied Physics 46, 250 (1975).
[Crossref]

Nature (1)

P. Vukusic and J. R. Sambles, “Photonic structures in biology,” Nature 424, 852–55 (2003).
[Crossref] [PubMed]

Opt. Express (1)

D. Englund, I. Fushman, and J. VučkoviĆ, “General Recipe for Designing Photonic Crystal Cavities,” Opt. Express 12, 5961–75 (2005).
[Crossref]

Phys. Rev. B. (1)

A. F. Koenderink, A. Lagendijk, and W. L. Vos, “Optical extinction due to intrinsic structural variations of photonic crystals,” Phys. Rev. B. 72, 153,102 – (2005).
[Crossref]

Phys. Rev. Lett. (2)

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, “Extrinsic optical scattering loss in photonic Crystal waveguides: role of fabrication disorder and photon Group velocity.” Phys. Rev. Lett. 94, 033,903 – 4 (2005).
[Crossref]

D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. J. VučkoviĆ “Controlling the Spontaneous Emission Rate of Single Quantum Dots in a Two-Dimensional Photonic Crystal,” Phys. Rev. Lett. 95(013904) (2005).
[Crossref] [PubMed]

Physical Review E (1)

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwells equations with shifting material boundaries,” Physical Review E 65(066611) (2002).
[Crossref]

Other (2)

K. Kawano and T. Kitoh, Introduction to Optical Waveguide Analysis: Solving Maxwells Equation and the Schrödinger Equation (Wiley-Interscience Publications, New York, 2001).
[PubMed]

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley Interscience, 2003).

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Figures (7)

Fig. 1.
Fig. 1.

Structure S1. (a) SEM image. Inset: Ideal structure, simulated (periodicity a = 256nm, target hole radius r = 0.3a, slab thickness d = 0.65a.) (b) Difference between fabricated and target structures. (c) Measured spectrum shows dipole mode. (d) The simulated (by 2D-FDTD) cavity spectrum of the fabricated structure closely matches the center-frequencies of observed resonances. The digitized structure supports the confined dipole-mode, shown here as the Bz field component at the slab center and Q ~ 2861 (both calculated with 3D-FDTD).

Fig. 2.
Fig. 2.

(a) Simulated field Bz at center of optimal single-defect cavity. For this simulation, each unit cell contains a = 20 program units. Standard hole radius r = 0.3a, slab thickness d = 0.65a; defect hole parameters for holes H1 r 1 = 0.25ax 1 = ±0.05ay 1 = ±0.1a; holes H2 have r 2 = 0.3ax 2 = ±0.2ay 1 = 0. The resonance occurs at λ 0 = 0.259a. (b) Parametric search optimization of single-defect cavity. Two 2D slices are shown: on top, r 1/a = 0.3,r 2/a = 0.3,Δx 2 = 0.2ay 2 = 0, and x 1,y 1 are varied; on the bottom, r 1 = 0.3a,r 2 = 0.25ax 2 = 0.2ay 2 = 0, and x 1,y 1 varied. All calculations done by 3D-FDTD.

Fig. 3.
Fig. 3.

Structure S2. (a) PL spectrum shows resonance of Q=1850 at 924.5nm. (b) The SEM image is converted into the dielectric structure. (c) The simulated (by 2D-FDTD) structure shows the dipole resonance at 946nm (error 2% compared to actual resonance) with Qsim = 1837 (by 3D-FDTD). Inset: The wide spectrum shows the bandgap extending from 800nm to 1050 nm. (d) Simulated dipole mode inside fabricated structure, given as Bz (r⃗) just above the slab.

Fig. 4.
Fig. 4.

(a) Calibration curve for SEM-conversion: the threshold value for differentiating between GaAs and air is varied to match observed and simulated resonances. Simulations done by 2-D FDTD. (b) Image intensity histogram for thresholding. (c) The measured dipole mode resonances of samples 1-7 agree far better with simulations of the fabricated than the idealized structures. (d) The Q values also show good agreement between measured and simulated values.

Fig. 5.
Fig. 5.

Ideal cavity Q vs. cavity size σx . (a) Only intrinsic losses present. (b) Material loss γM ~ 10 cm-1 included.

Fig. 6.
Fig. 6.

Extrinsic scattering loss in structure S3. Columns from left to right correspond to the real structure, ‘polished’ structure, and the difference. Top row: The dipole mode (Bz just above the membrane surface) for the original (left) and ‘corrected’ (center) structures. The difference in the fields show the effect of the defect (right). Center Row: 2D Fourier-transforms, shown as log plots, of the fields show that the defect scatters into a wider range of angle than the cavities. Bottom Row: Poynting vector component Sz gives the scattering contribution due to the defect (right).

Fig. 7.
Fig. 7.

(a) Design with binary refractive index: ε(r⃗),FT 2(ε), and F.T. of resonant dipole mode. (b) The design with smoothed refractive index (left) shows decreased scattering of high k- components in the F.T. of the structure (center) and resonant dipole mode (right).

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

γ tot = γ + γ + γ S + γ M
Q tot 1 = Q 1 + Q 1 + Q S 1 + Q M 1
Δ ω ω 2 Δ ε E 2 ε E 2
K k x k y = η k z 2 2 λ k 2 [ 1 η 2 F T 2 ( E z ) 2 + F T 2 ( H z ) 2 ] ,
k z = ( 2 π λ ) 2 k x 2 k y 2 ,
k × [ k × A ( k ) ] + ω 2 μ G A ( k G ) ε G = 0
ε ( r ) = G ε G exp ( i G · r )
F cav λ r μ = F cav 0 ( E ( r A ) · μ E max μ ) 2 1 1 + 4 Q 2 ( λ λ cav 1 ) 2

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