Abstract

It is known that the spectra of optical modes supported by microresonators are shape and size dependent. We have introduced an additional parameter, i.e., spatially varying refractive index, to tailor the spectra of optical modes in a microdisk resonator while keeping size and shape intact. A new class of whispering gallery mode microresonators, referred to as graded index (GRIN) microresonators, is proposed. The GRIN profile belongs to a special inhomogeneous medium known as Maxwell’s fish eye. The modal analysis of the structure is investigated both analytically and numerically. Solution of the Helmholtz equation is attempted under the spatially varying refractive index scenario and numerical verification of the analytical findings is tested by the finite-difference time-domain method. It is found that the two approaches support each other. It is expected that the findings of the GRIN microresonator may open up new research and device opportunities in photonics.

© 2014 Optical Society of America

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  4. V. Sandoghdar, F. Treussart, J. Hare, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, “Very low threshold whispering-gallery-mode microsphere laser,” Phys. Rev. A 54, R1777–R1780 (1996).
    [CrossRef]
  5. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317, 783–787 (2007).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  14. H. Rosu and M. Reyes, “Electromagnetic modes of Maxwell fisheye lens,” Nuovo Cimento D 16, 517–522 (1994).
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  15. S. Liu, L. Li, Z. Lin, H. Y. Chen, J. Zi, and C. T. Chan, “Graded index photonic hole: analytical and rigorous full wave solution,” Phys. Rev. B 82, 054204 (2010).
    [CrossRef]
  16. E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional light absorber,” Appl. Phys. Lett. 95, 041106 (2009).
    [CrossRef]
  17. C. Sheng, H. Liu, Y. Wang, S. N. Zhu, and D. A. Genov, “Trapping light by mimicking gravitational lensing,” Nat. Photonics 7, 902–906 (2013).
    [CrossRef]
  18. Y. Wang, C. Sheng, H. Liu, Y. J. Zheng, C. Zhu, S. M. Wang, and S. N. Zhu, “Transformation bending device emulated by graded-index waveguide,” Opt. Express 20, 13006–13013 (2012).
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    [CrossRef]
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    [CrossRef]
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  24. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
    [CrossRef]
  25. L. Prkna, J. Ctyroky, and M. Hubalek, “Ring microresonator as a photonic structure with complex eigenfrequency,” Opt. Quantum Electron. 36, 259–269 (2004).
    [CrossRef]
  26. A. D. Falco, S. C. Kehr, and U. Leonhardt, “Luneburg lens in silicon photonics,” Opt. Express 19, 5156–5162 (2011).
    [CrossRef]
  27. L. H. Gabrielli and M. Lipson, “Integrated Luneburg lens via ultra-strong index gradient on silicon,” Opt. Express 19, 20122–20127 (2011).
    [CrossRef]

2013 (3)

J. Liu, R. Mendis, and D. M. Mittleman, “A Maxwell’s fish eye lens for the terahertz region,” Appl. Phys. Lett. 103, 031104 (2013).
[CrossRef]

C. Sheng, H. Liu, Y. Wang, S. N. Zhu, and D. A. Genov, “Trapping light by mimicking gravitational lensing,” Nat. Photonics 7, 902–906 (2013).
[CrossRef]

E. F. Franchimon, K. R. Hiremath, R. Stoffer, and M. Hammer, “Interaction of whispering gallery modes in integrated optical microring or microdisk circuits: hybrid coupled mode theory model,” J. Opt. Soc. Am. B 30, 1048–1057 (2013).
[CrossRef]

2012 (2)

2011 (3)

2010 (3)

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

S. Liu, L. Li, Z. Lin, H. Y. Chen, J. Zi, and C. T. Chan, “Graded index photonic hole: analytical and rigorous full wave solution,” Phys. Rev. B 82, 054204 (2010).
[CrossRef]

U. Leonhardt and T. G. Philbin, “Perfect imaging with positive refraction in three dimensions,” Phys. Rev. A 81, 011804 (2010).
[CrossRef]

2009 (3)

U. Leonhardt, “Perfect imaging without negative refraction,” New J. Phys. 11, 093040 (2009).
[CrossRef]

E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional light absorber,” Appl. Phys. Lett. 95, 041106 (2009).
[CrossRef]

H. Quan and Z. Guo, “Analyses of whispering-gallery modes in small resonators,” J. Micro/Nanolith. MEMS MOEMS 8, 033060 (2009).
[CrossRef]

2007 (2)

T. Wilk, S. C. Webster, A. Kuhn, and G. Rempe, “Single-atom single-photon quantum interface,” Science 317, 488–490 (2007).
[CrossRef]

M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317, 783–787 (2007).
[CrossRef]

2004 (1)

L. Prkna, J. Ctyroky, and M. Hubalek, “Ring microresonator as a photonic structure with complex eigenfrequency,” Opt. Quantum Electron. 36, 259–269 (2004).
[CrossRef]

2003 (1)

K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
[CrossRef]

2002 (1)

K. Djordjev, S. J. Choi, and P. D. Dapkus, “Microdisk tunable resonant filters and switches,” IEEE Photon. Technol. Lett. 14, 828–830 (2002).
[CrossRef]

2000 (1)

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamogùlu, “A quantum dot single-photon turnstile device,” Science 290, 2282–2285 (2000).
[CrossRef]

1996 (1)

V. Sandoghdar, F. Treussart, J. Hare, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, “Very low threshold whispering-gallery-mode microsphere laser,” Phys. Rev. A 54, R1777–R1780 (1996).
[CrossRef]

1994 (1)

H. Rosu and M. Reyes, “Electromagnetic modes of Maxwell fisheye lens,” Nuovo Cimento D 16, 517–522 (1994).
[CrossRef]

1958 (1)

C. T. Tai, “Maxwell fish-eye treated by Maxwell equations,” Nature 182, 1600–1601 (1958).
[CrossRef]

Arfken, G. B.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Elsevier, 2005).

Armani, M.

M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317, 783–787 (2007).
[CrossRef]

Bao, C.

C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer, 2002).

Becher, C.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamogùlu, “A quantum dot single-photon turnstile device,” Science 290, 2282–2285 (2000).
[CrossRef]

Bermel, P.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

Chan, C. T.

S. Liu, L. Li, Z. Lin, H. Y. Chen, J. Zi, and C. T. Chan, “Graded index photonic hole: analytical and rigorous full wave solution,” Phys. Rev. B 82, 054204 (2010).
[CrossRef]

Chen, H. Y.

S. Liu, L. Li, Z. Lin, H. Y. Chen, J. Zi, and C. T. Chan, “Graded index photonic hole: analytical and rigorous full wave solution,” Phys. Rev. B 82, 054204 (2010).
[CrossRef]

Choi, S. J.

K. Djordjev, S. J. Choi, and P. D. Dapkus, “Microdisk tunable resonant filters and switches,” IEEE Photon. Technol. Lett. 14, 828–830 (2002).
[CrossRef]

Ctyroky, J.

L. Prkna, J. Ctyroky, and M. Hubalek, “Ring microresonator as a photonic structure with complex eigenfrequency,” Opt. Quantum Electron. 36, 259–269 (2004).
[CrossRef]

Dapkus, P. D.

K. Djordjev, S. J. Choi, and P. D. Dapkus, “Microdisk tunable resonant filters and switches,” IEEE Photon. Technol. Lett. 14, 828–830 (2002).
[CrossRef]

Djordjev, K.

K. Djordjev, S. J. Choi, and P. D. Dapkus, “Microdisk tunable resonant filters and switches,” IEEE Photon. Technol. Lett. 14, 828–830 (2002).
[CrossRef]

Falco, A. D.

Flagan, R. C.

M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317, 783–787 (2007).
[CrossRef]

Franchimon, E. F.

Fraser, S. E.

M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317, 783–787 (2007).
[CrossRef]

Gabrielli, L. H.

Genov, D. A.

C. Sheng, H. Liu, Y. Wang, S. N. Zhu, and D. A. Genov, “Trapping light by mimicking gravitational lensing,” Nat. Photonics 7, 902–906 (2013).
[CrossRef]

Gomez-Reino, C.

C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer, 2002).

Guo, Z.

H. Quan and Z. Guo, “Analyses of whispering-gallery modes in small resonators,” J. Micro/Nanolith. MEMS MOEMS 8, 033060 (2009).
[CrossRef]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

Hammer, M.

Hare, J.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, “Very low threshold whispering-gallery-mode microsphere laser,” Phys. Rev. A 54, R1777–R1780 (1996).
[CrossRef]

Haroche, S.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, “Very low threshold whispering-gallery-mode microsphere laser,” Phys. Rev. A 54, R1777–R1780 (1996).
[CrossRef]

Hiremath, K. R.

Hu, E.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamogùlu, “A quantum dot single-photon turnstile device,” Science 290, 2282–2285 (2000).
[CrossRef]

Hubalek, M.

L. Prkna, J. Ctyroky, and M. Hubalek, “Ring microresonator as a photonic structure with complex eigenfrequency,” Opt. Quantum Electron. 36, 259–269 (2004).
[CrossRef]

Ibanescu, M.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

Imamogùlu, A.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamogùlu, “A quantum dot single-photon turnstile device,” Science 290, 2282–2285 (2000).
[CrossRef]

Joannopoulos, J. D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

Johnson, S. G.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

Kehr, S. C.

Kildishev, A. V.

E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional light absorber,” Appl. Phys. Lett. 95, 041106 (2009).
[CrossRef]

Kiraz, A.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamogùlu, “A quantum dot single-photon turnstile device,” Science 290, 2282–2285 (2000).
[CrossRef]

Kuhn, A.

T. Wilk, S. C. Webster, A. Kuhn, and G. Rempe, “Single-atom single-photon quantum interface,” Science 317, 488–490 (2007).
[CrossRef]

Kulkarni, R. P.

M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317, 783–787 (2007).
[CrossRef]

Lefèvre-Seguin, V.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, “Very low threshold whispering-gallery-mode microsphere laser,” Phys. Rev. A 54, R1777–R1780 (1996).
[CrossRef]

Leonhardt, U.

A. D. Falco, S. C. Kehr, and U. Leonhardt, “Luneburg lens in silicon photonics,” Opt. Express 19, 5156–5162 (2011).
[CrossRef]

U. Leonhardt and T. G. Philbin, “Perfect imaging with positive refraction in three dimensions,” Phys. Rev. A 81, 011804 (2010).
[CrossRef]

U. Leonhardt, “Perfect imaging without negative refraction,” New J. Phys. 11, 093040 (2009).
[CrossRef]

Li, L.

S. Liu, L. Li, Z. Lin, H. Y. Chen, J. Zi, and C. T. Chan, “Graded index photonic hole: analytical and rigorous full wave solution,” Phys. Rev. B 82, 054204 (2010).
[CrossRef]

Lin, Z.

S. Liu, L. Li, Z. Lin, H. Y. Chen, J. Zi, and C. T. Chan, “Graded index photonic hole: analytical and rigorous full wave solution,” Phys. Rev. B 82, 054204 (2010).
[CrossRef]

Lipson, M.

Liu, H.

C. Sheng, H. Liu, Y. Wang, S. N. Zhu, and D. A. Genov, “Trapping light by mimicking gravitational lensing,” Nat. Photonics 7, 902–906 (2013).
[CrossRef]

Y. Wang, C. Sheng, H. Liu, Y. J. Zheng, C. Zhu, S. M. Wang, and S. N. Zhu, “Transformation bending device emulated by graded-index waveguide,” Opt. Express 20, 13006–13013 (2012).
[CrossRef]

Liu, J.

J. Liu, R. Mendis, and D. M. Mittleman, “A Maxwell’s fish eye lens for the terahertz region,” Appl. Phys. Lett. 103, 031104 (2013).
[CrossRef]

Liu, S.

S. Liu, L. Li, Z. Lin, H. Y. Chen, J. Zi, and C. T. Chan, “Graded index photonic hole: analytical and rigorous full wave solution,” Phys. Rev. B 82, 054204 (2010).
[CrossRef]

Liu, Y.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. 6, 151–155 (2011).
[CrossRef]

Luan, F.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California, 1964).

Mendis, R.

J. Liu, R. Mendis, and D. M. Mittleman, “A Maxwell’s fish eye lens for the terahertz region,” Appl. Phys. Lett. 103, 031104 (2013).
[CrossRef]

Michler, P.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamogùlu, “A quantum dot single-photon turnstile device,” Science 290, 2282–2285 (2000).
[CrossRef]

Mikkelsen, M. H.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. 6, 151–155 (2011).
[CrossRef]

Mittleman, D. M.

J. Liu, R. Mendis, and D. M. Mittleman, “A Maxwell’s fish eye lens for the terahertz region,” Appl. Phys. Lett. 103, 031104 (2013).
[CrossRef]

Narimanov, E. E.

E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional light absorber,” Appl. Phys. Lett. 95, 041106 (2009).
[CrossRef]

Oskooi, A. F.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

Perez, M. V.

C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer, 2002).

Petroff, P. M.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamogùlu, “A quantum dot single-photon turnstile device,” Science 290, 2282–2285 (2000).
[CrossRef]

Philbin, T. G.

U. Leonhardt and T. G. Philbin, “Perfect imaging with positive refraction in three dimensions,” Phys. Rev. A 81, 011804 (2010).
[CrossRef]

Prkna, L.

L. Prkna, J. Ctyroky, and M. Hubalek, “Ring microresonator as a photonic structure with complex eigenfrequency,” Opt. Quantum Electron. 36, 259–269 (2004).
[CrossRef]

Quan, H.

H. Quan and Z. Guo, “Analyses of whispering-gallery modes in small resonators,” J. Micro/Nanolith. MEMS MOEMS 8, 033060 (2009).
[CrossRef]

Raimond, J.-M.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, “Very low threshold whispering-gallery-mode microsphere laser,” Phys. Rev. A 54, R1777–R1780 (1996).
[CrossRef]

Rempe, G.

T. Wilk, S. C. Webster, A. Kuhn, and G. Rempe, “Single-atom single-photon quantum interface,” Science 317, 488–490 (2007).
[CrossRef]

Reyes, M.

H. Rosu and M. Reyes, “Electromagnetic modes of Maxwell fisheye lens,” Nuovo Cimento D 16, 517–522 (1994).
[CrossRef]

Rosu, H.

H. Rosu and M. Reyes, “Electromagnetic modes of Maxwell fisheye lens,” Nuovo Cimento D 16, 517–522 (1994).
[CrossRef]

Roundy, D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

Sandoghdar, V.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, “Very low threshold whispering-gallery-mode microsphere laser,” Phys. Rev. A 54, R1777–R1780 (1996).
[CrossRef]

Schoenfeld, W. V.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamogùlu, “A quantum dot single-photon turnstile device,” Science 290, 2282–2285 (2000).
[CrossRef]

Sheng, C.

C. Sheng, H. Liu, Y. Wang, S. N. Zhu, and D. A. Genov, “Trapping light by mimicking gravitational lensing,” Nat. Photonics 7, 902–906 (2013).
[CrossRef]

Y. Wang, C. Sheng, H. Liu, Y. J. Zheng, C. Zhu, S. M. Wang, and S. N. Zhu, “Transformation bending device emulated by graded-index waveguide,” Opt. Express 20, 13006–13013 (2012).
[CrossRef]

Shum, P.

Stoffer, R.

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

Tai, C. T.

C. T. Tai, “Maxwell fish-eye treated by Maxwell equations,” Nature 182, 1600–1601 (1958).
[CrossRef]

Treussart, F.

V. Sandoghdar, F. Treussart, J. Hare, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, “Very low threshold whispering-gallery-mode microsphere laser,” Phys. Rev. A 54, R1777–R1780 (1996).
[CrossRef]

Vahala, K. J.

M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317, 783–787 (2007).
[CrossRef]

K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
[CrossRef]

Valentine, J.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. 6, 151–155 (2011).
[CrossRef]

Wang, S. M.

Wang, Y.

C. Sheng, H. Liu, Y. Wang, S. N. Zhu, and D. A. Genov, “Trapping light by mimicking gravitational lensing,” Nat. Photonics 7, 902–906 (2013).
[CrossRef]

Y. Wang, C. Sheng, H. Liu, Y. J. Zheng, C. Zhu, S. M. Wang, and S. N. Zhu, “Transformation bending device emulated by graded-index waveguide,” Opt. Express 20, 13006–13013 (2012).
[CrossRef]

Weber, H. J.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Elsevier, 2005).

Webster, S. C.

T. Wilk, S. C. Webster, A. Kuhn, and G. Rempe, “Single-atom single-photon quantum interface,” Science 317, 488–490 (2007).
[CrossRef]

Wilk, T.

T. Wilk, S. C. Webster, A. Kuhn, and G. Rempe, “Single-atom single-photon quantum interface,” Science 317, 488–490 (2007).
[CrossRef]

Yu, X.

Zentgraf, T.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnol. 6, 151–155 (2011).
[CrossRef]

Zhang, L.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamogùlu, “A quantum dot single-photon turnstile device,” Science 290, 2282–2285 (2000).
[CrossRef]

Zhang, X.

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Supplementary Material (1)

» Media 1: AVI (1277 KB)     

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Figures (5)

Fig. 1.
Fig. 1.

(a) Schematic view of a 2D Maxwell’s fish eye MFE coupled to an optical waveguide. (b) 3D representation of refractive index distribution of the complete configuration, which includes a bus waveguide and an MFE resonator.

Fig. 2.
Fig. 2.

Normalized transmission spectum of the waveguide in port P2 showing resonance modes of the MFE.

Fig. 3.
Fig. 3.

E-field intensity profiles for (a) n=1, m=29; (b) n=2, m=27; (c) n=4, m=27; and (d) n=5, m=22, and E-field distribution of an MFE coupled optical waveguide for (e) n=4, m=23 (FDTD simulations can be seen in Media 1).

Fig. 4.
Fig. 4.

(a) Graphical solution of Eq. (17). (b) Radial component of the electric field inside of the MFE.

Fig. 5.
Fig. 5.

Electric field distribution inside the 2D MFE for n=4 and m=23.

Equations (27)

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n(r)=n0/(1+(r/R)2),
2E+k2ε(r)E=0,
2Ezr2+1rEzr+1r22Ezφ2+k2ε(r)Ez=0.
Ez(r,φ)=Ψ(r)Φ(φ)
d2Φ(φ)dφ2+m2Φ(φ)=0,
d2Ψ(r)dr2+1rdΨ(r)dr+[k2n02[1+(r/R)2]2m2r2]Ψ(r)=0rR,
d2Ψ(r)dr2+1rdΨ(r)dr+[k2m2r2]Ψ(r)=0rR.
Φm(φ)=m(Amcosmφ+Bmsinmφ)m=1,2,3,
Φm(φ)=mAmcosmφm=1,2,3.
Ψmin(ξ)=m{Cm+Rm+2ηξm(1+ξ2)ηF([η,η+m],[m+1],ξ2)+CmRm+2ηξm(1+ξ2)ηF([η,ηm],[m+1],ξ2)},
Ψmout(r)=mreal{Dm(Jm(kr)±jYm(kr))}.
Cm+Rm+2η2ηF([η,η+m],[m+1],1)=Dm(Jm(kR)±jYm(kR)),
Dm=Cm+Rm+2η2ηF([η,η+m],[m+1],1)(Jm(kR)±jYm(kR)).
Ezin(r,φ)Ezin(ξ,φ)=m{AmCm+Rm+2ηξm(1+ξ2)ηF([η,η+m],[m+1],ξ2)cos(mφ)},rR,
Ezout(r,φ)=m{AmCm+Rm+2η2ηF([η,η+m],[m+1],1)×(Jm(kR)Jm(kr)+Ym(kR)Ym(kr))(Jm2(kR)+Ym2(kR))cos(mφ)}rR.
Cm+Rm+2η·2η·1R{mF([η,η+m],[m+1],1)+ηF([η,η+m],[m+1],1)2η(m+η)m+1F([η+1,η+m+1],[m+2],1)}=Cm+Rm+2η2ηF([η,η+m],[m+1],1)×Jm(kR)Jm(kR)+Ym(kR)Ym(kR)(Jm2(kR)+Ym2(kR)).
mξm1(1+ξ2)ηF([η,η+m],[m+1],ξ2)+2ξm+1(1+ξ2)η1F([η,η+m],[m+1],ξ2)=2η(η+m)m+1ξm+1F([η+1,η+m+1],[m+2],ξ2).
Ψmin(ξ)=m{ξm(1+ξ2)ηF([η,η+m],[m+1],ξ2)}.
×E=iωμ0H,
×H=iωε0ε(r)E,
·[ε(r)E]=0,
·H=0.
××Ek2ε(r)E=0,
(·E)2Ek2ε(r)E=0.
[ε(r)]·E+ε(r)·E=0.
[ε(r)]·E=ε(r)rEr+ε(r)rθEθ+ε(r)rsinθϕEϕ.
2E+k2ε(r)E=0.

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