Abstract

In recent years, researchers have demonstrated negative refraction theoretically and experimentally by pumping optical power into photonic crystal (PhC) or waveguide structures. The concept of negative refraction can be used to create a perfect lens that focuses an object smaller than the wavelength. By inserting two-dimensional PhCs into the peripheral of a semiconductor light emitting structure, this study presents an electroluminescent device with negative refraction in the visible wavelength range. This approach produces polarization dependent collimation behavior in far-field radiation patterns. The modal dispersion of negative refraction results in strong group velocity modulation, and self-focusing and -defocusing behaviors are apparent from light extraction. This study further verifies experimental results by using theoretic calculations based on equifrequency contours.

© 2013 Optical Society of America

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2013

Y. T. Wang, Y. Chou, L. Y. Chen, Y. F. Yin, Y. C. Lin, and J. J. Huang, IEEE J Quantum Electron. 49, 11 (2013).
[CrossRef]

2011

S. C. Wang, Y. W. Cheng, Y. F. Yin, L. Y. Chen, L. Y. Su, Y. J. Hung, and J. J. Huang, J Lightwave Technol. 29, 3772 (2011).
[CrossRef]

X. H. Li, R. Song, Y. K. Ee, P. Kumnorkaew, J. F. Gilchrist, and N. Tansu, IEEE Photon. J 3, 489 (2011).
[CrossRef]

Y. W. Cheng, S. C. Wang, Y. F. Yin, L. Y. Su, and J. J. Huang, Opt. Lett. 36, 1611 (2011).
[CrossRef]

2010

M. Hofman, N. Fabre, X. Mélique, D. Lippens, and O. Vanbésien, Opt. Commun. 283, 1169 (2010).
[CrossRef]

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, Nature 466, 735 (2010).
[CrossRef]

S. P. Burgos, R. De Waele, A. Polman, and H. A. Atwater, Nat. Mater. 9, 407 (2010).
[CrossRef]

C. F. Lai, H. C. Kuo, P. Yu, T. C. Lu, C. H. Chao, H. H. Yen, and W. Y. Yeh, Appl. Phys. Lett. 97, 013108 (2010).
[CrossRef]

2009

J. J. Wierer, A. David, and M. M. Megens, Nat. Photonics 3, 163 (2009).
[CrossRef]

2008

K. McGroddy, A. David, E. Matioli, M. Iza, S. Nakamura, S. DenBaars, J. S. Speck, C. Weisbuch, and E. L. Hu, Appl. Phys. Lett. 93, 103502 (2008).
[CrossRef]

2007

K. Busch, G. Von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, Phys. Rep. 444, 101 (2007).
[CrossRef]

D. W. Prather, S. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. Chen, B. L. Miao, and R. Martin, J Phys. D 40, 2635 (2007).
[CrossRef]

2005

A. Martínez and J. Martí, Phys. Rev. B 71, 235115 (2005).
[CrossRef]

2002

C. Luo, S. G. Johnson, J. Joannopoulos, and J. Pendry, Phys. Rev. B 65, 201104 (2002).
[CrossRef]

2000

M. Notomi, Phys. Rev. B 62, 10696 (2000).
[CrossRef]

Atwater, H. A.

S. P. Burgos, R. De Waele, A. Polman, and H. A. Atwater, Nat. Mater. 9, 407 (2010).
[CrossRef]

Burgos, S. P.

S. P. Burgos, R. De Waele, A. Polman, and H. A. Atwater, Nat. Mater. 9, 407 (2010).
[CrossRef]

Busch, K.

K. Busch, G. Von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, Phys. Rep. 444, 101 (2007).
[CrossRef]

Chao, C. H.

C. F. Lai, H. C. Kuo, P. Yu, T. C. Lu, C. H. Chao, H. H. Yen, and W. Y. Yeh, Appl. Phys. Lett. 97, 013108 (2010).
[CrossRef]

Chen, C.

D. W. Prather, S. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. Chen, B. L. Miao, and R. Martin, J Phys. D 40, 2635 (2007).
[CrossRef]

Chen, L. Y.

Y. T. Wang, Y. Chou, L. Y. Chen, Y. F. Yin, Y. C. Lin, and J. J. Huang, IEEE J Quantum Electron. 49, 11 (2013).
[CrossRef]

S. C. Wang, Y. W. Cheng, Y. F. Yin, L. Y. Chen, L. Y. Su, Y. J. Hung, and J. J. Huang, J Lightwave Technol. 29, 3772 (2011).
[CrossRef]

Cheng, Y. W.

S. C. Wang, Y. W. Cheng, Y. F. Yin, L. Y. Chen, L. Y. Su, Y. J. Hung, and J. J. Huang, J Lightwave Technol. 29, 3772 (2011).
[CrossRef]

Y. W. Cheng, S. C. Wang, Y. F. Yin, L. Y. Su, and J. J. Huang, Opt. Lett. 36, 1611 (2011).
[CrossRef]

Chettiar, U. K.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, Nature 466, 735 (2010).
[CrossRef]

Chou, Y.

Y. T. Wang, Y. Chou, L. Y. Chen, Y. F. Yin, Y. C. Lin, and J. J. Huang, IEEE J Quantum Electron. 49, 11 (2013).
[CrossRef]

David, A.

J. J. Wierer, A. David, and M. M. Megens, Nat. Photonics 3, 163 (2009).
[CrossRef]

K. McGroddy, A. David, E. Matioli, M. Iza, S. Nakamura, S. DenBaars, J. S. Speck, C. Weisbuch, and E. L. Hu, Appl. Phys. Lett. 93, 103502 (2008).
[CrossRef]

De Waele, R.

S. P. Burgos, R. De Waele, A. Polman, and H. A. Atwater, Nat. Mater. 9, 407 (2010).
[CrossRef]

DenBaars, S.

K. McGroddy, A. David, E. Matioli, M. Iza, S. Nakamura, S. DenBaars, J. S. Speck, C. Weisbuch, and E. L. Hu, Appl. Phys. Lett. 93, 103502 (2008).
[CrossRef]

Drachev, V. P.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, Nature 466, 735 (2010).
[CrossRef]

Ee, Y. K.

X. H. Li, R. Song, Y. K. Ee, P. Kumnorkaew, J. F. Gilchrist, and N. Tansu, IEEE Photon. J 3, 489 (2011).
[CrossRef]

Fabre, N.

M. Hofman, N. Fabre, X. Mélique, D. Lippens, and O. Vanbésien, Opt. Commun. 283, 1169 (2010).
[CrossRef]

Gilchrist, J. F.

X. H. Li, R. Song, Y. K. Ee, P. Kumnorkaew, J. F. Gilchrist, and N. Tansu, IEEE Photon. J 3, 489 (2011).
[CrossRef]

Hofman, M.

M. Hofman, N. Fabre, X. Mélique, D. Lippens, and O. Vanbésien, Opt. Commun. 283, 1169 (2010).
[CrossRef]

Hu, E. L.

K. McGroddy, A. David, E. Matioli, M. Iza, S. Nakamura, S. DenBaars, J. S. Speck, C. Weisbuch, and E. L. Hu, Appl. Phys. Lett. 93, 103502 (2008).
[CrossRef]

Huang, J. J.

Y. T. Wang, Y. Chou, L. Y. Chen, Y. F. Yin, Y. C. Lin, and J. J. Huang, IEEE J Quantum Electron. 49, 11 (2013).
[CrossRef]

Y. W. Cheng, S. C. Wang, Y. F. Yin, L. Y. Su, and J. J. Huang, Opt. Lett. 36, 1611 (2011).
[CrossRef]

S. C. Wang, Y. W. Cheng, Y. F. Yin, L. Y. Chen, L. Y. Su, Y. J. Hung, and J. J. Huang, J Lightwave Technol. 29, 3772 (2011).
[CrossRef]

Hung, Y. J.

S. C. Wang, Y. W. Cheng, Y. F. Yin, L. Y. Chen, L. Y. Su, Y. J. Hung, and J. J. Huang, J Lightwave Technol. 29, 3772 (2011).
[CrossRef]

Iza, M.

K. McGroddy, A. David, E. Matioli, M. Iza, S. Nakamura, S. DenBaars, J. S. Speck, C. Weisbuch, and E. L. Hu, Appl. Phys. Lett. 93, 103502 (2008).
[CrossRef]

Joannopoulos, J.

C. Luo, S. G. Johnson, J. Joannopoulos, and J. Pendry, Phys. Rev. B 65, 201104 (2002).
[CrossRef]

Joannopoulos, J. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton, 2008).

Johnson, S. G.

C. Luo, S. G. Johnson, J. Joannopoulos, and J. Pendry, Phys. Rev. B 65, 201104 (2002).
[CrossRef]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton, 2008).

Kildishev, A. V.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, Nature 466, 735 (2010).
[CrossRef]

Kumnorkaew, P.

X. H. Li, R. Song, Y. K. Ee, P. Kumnorkaew, J. F. Gilchrist, and N. Tansu, IEEE Photon. J 3, 489 (2011).
[CrossRef]

Kuo, H. C.

C. F. Lai, H. C. Kuo, P. Yu, T. C. Lu, C. H. Chao, H. H. Yen, and W. Y. Yeh, Appl. Phys. Lett. 97, 013108 (2010).
[CrossRef]

Lai, C. F.

C. F. Lai, H. C. Kuo, P. Yu, T. C. Lu, C. H. Chao, H. H. Yen, and W. Y. Yeh, Appl. Phys. Lett. 97, 013108 (2010).
[CrossRef]

Li, X. H.

X. H. Li, R. Song, Y. K. Ee, P. Kumnorkaew, J. F. Gilchrist, and N. Tansu, IEEE Photon. J 3, 489 (2011).
[CrossRef]

Lin, Y. C.

Y. T. Wang, Y. Chou, L. Y. Chen, Y. F. Yin, Y. C. Lin, and J. J. Huang, IEEE J Quantum Electron. 49, 11 (2013).
[CrossRef]

Linden, S.

K. Busch, G. Von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, Phys. Rep. 444, 101 (2007).
[CrossRef]

Lippens, D.

M. Hofman, N. Fabre, X. Mélique, D. Lippens, and O. Vanbésien, Opt. Commun. 283, 1169 (2010).
[CrossRef]

Lu, T. C.

C. F. Lai, H. C. Kuo, P. Yu, T. C. Lu, C. H. Chao, H. H. Yen, and W. Y. Yeh, Appl. Phys. Lett. 97, 013108 (2010).
[CrossRef]

Luo, C.

C. Luo, S. G. Johnson, J. Joannopoulos, and J. Pendry, Phys. Rev. B 65, 201104 (2002).
[CrossRef]

Martí, J.

A. Martínez and J. Martí, Phys. Rev. B 71, 235115 (2005).
[CrossRef]

Martin, R.

D. W. Prather, S. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. Chen, B. L. Miao, and R. Martin, J Phys. D 40, 2635 (2007).
[CrossRef]

Martínez, A.

A. Martínez and J. Martí, Phys. Rev. B 71, 235115 (2005).
[CrossRef]

Matioli, E.

K. McGroddy, A. David, E. Matioli, M. Iza, S. Nakamura, S. DenBaars, J. S. Speck, C. Weisbuch, and E. L. Hu, Appl. Phys. Lett. 93, 103502 (2008).
[CrossRef]

McGroddy, K.

K. McGroddy, A. David, E. Matioli, M. Iza, S. Nakamura, S. DenBaars, J. S. Speck, C. Weisbuch, and E. L. Hu, Appl. Phys. Lett. 93, 103502 (2008).
[CrossRef]

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton, 2008).

Megens, M. M.

J. J. Wierer, A. David, and M. M. Megens, Nat. Photonics 3, 163 (2009).
[CrossRef]

Mélique, X.

M. Hofman, N. Fabre, X. Mélique, D. Lippens, and O. Vanbésien, Opt. Commun. 283, 1169 (2010).
[CrossRef]

Miao, B. L.

D. W. Prather, S. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. Chen, B. L. Miao, and R. Martin, J Phys. D 40, 2635 (2007).
[CrossRef]

Mingaleev, S.

K. Busch, G. Von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, Phys. Rep. 444, 101 (2007).
[CrossRef]

Murakowski, J.

D. W. Prather, S. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. Chen, B. L. Miao, and R. Martin, J Phys. D 40, 2635 (2007).
[CrossRef]

Nakamura, S.

K. McGroddy, A. David, E. Matioli, M. Iza, S. Nakamura, S. DenBaars, J. S. Speck, C. Weisbuch, and E. L. Hu, Appl. Phys. Lett. 93, 103502 (2008).
[CrossRef]

Ni, X.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, Nature 466, 735 (2010).
[CrossRef]

Notomi, M.

M. Notomi, Phys. Rev. B 62, 10696 (2000).
[CrossRef]

Pendry, J.

C. Luo, S. G. Johnson, J. Joannopoulos, and J. Pendry, Phys. Rev. B 65, 201104 (2002).
[CrossRef]

Polman, A.

S. P. Burgos, R. De Waele, A. Polman, and H. A. Atwater, Nat. Mater. 9, 407 (2010).
[CrossRef]

Prather, D. W.

D. W. Prather, S. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. Chen, B. L. Miao, and R. Martin, J Phys. D 40, 2635 (2007).
[CrossRef]

Schneider, G. J.

D. W. Prather, S. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. Chen, B. L. Miao, and R. Martin, J Phys. D 40, 2635 (2007).
[CrossRef]

Shalaev, V. M.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, Nature 466, 735 (2010).
[CrossRef]

Sharkawy, A.

D. W. Prather, S. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. Chen, B. L. Miao, and R. Martin, J Phys. D 40, 2635 (2007).
[CrossRef]

Shi, S.

D. W. Prather, S. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. Chen, B. L. Miao, and R. Martin, J Phys. D 40, 2635 (2007).
[CrossRef]

Song, R.

X. H. Li, R. Song, Y. K. Ee, P. Kumnorkaew, J. F. Gilchrist, and N. Tansu, IEEE Photon. J 3, 489 (2011).
[CrossRef]

Speck, J. S.

K. McGroddy, A. David, E. Matioli, M. Iza, S. Nakamura, S. DenBaars, J. S. Speck, C. Weisbuch, and E. L. Hu, Appl. Phys. Lett. 93, 103502 (2008).
[CrossRef]

Su, L. Y.

S. C. Wang, Y. W. Cheng, Y. F. Yin, L. Y. Chen, L. Y. Su, Y. J. Hung, and J. J. Huang, J Lightwave Technol. 29, 3772 (2011).
[CrossRef]

Y. W. Cheng, S. C. Wang, Y. F. Yin, L. Y. Su, and J. J. Huang, Opt. Lett. 36, 1611 (2011).
[CrossRef]

Tansu, N.

X. H. Li, R. Song, Y. K. Ee, P. Kumnorkaew, J. F. Gilchrist, and N. Tansu, IEEE Photon. J 3, 489 (2011).
[CrossRef]

Tkeshelashvili, L.

K. Busch, G. Von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, Phys. Rep. 444, 101 (2007).
[CrossRef]

Vanbésien, O.

M. Hofman, N. Fabre, X. Mélique, D. Lippens, and O. Vanbésien, Opt. Commun. 283, 1169 (2010).
[CrossRef]

Von Freymann, G.

K. Busch, G. Von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, Phys. Rep. 444, 101 (2007).
[CrossRef]

Wang, S. C.

S. C. Wang, Y. W. Cheng, Y. F. Yin, L. Y. Chen, L. Y. Su, Y. J. Hung, and J. J. Huang, J Lightwave Technol. 29, 3772 (2011).
[CrossRef]

Y. W. Cheng, S. C. Wang, Y. F. Yin, L. Y. Su, and J. J. Huang, Opt. Lett. 36, 1611 (2011).
[CrossRef]

Wang, Y. T.

Y. T. Wang, Y. Chou, L. Y. Chen, Y. F. Yin, Y. C. Lin, and J. J. Huang, IEEE J Quantum Electron. 49, 11 (2013).
[CrossRef]

Wegener, M.

K. Busch, G. Von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, Phys. Rep. 444, 101 (2007).
[CrossRef]

Weisbuch, C.

K. McGroddy, A. David, E. Matioli, M. Iza, S. Nakamura, S. DenBaars, J. S. Speck, C. Weisbuch, and E. L. Hu, Appl. Phys. Lett. 93, 103502 (2008).
[CrossRef]

Wierer, J. J.

J. J. Wierer, A. David, and M. M. Megens, Nat. Photonics 3, 163 (2009).
[CrossRef]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton, 2008).

Xiao, S.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, Nature 466, 735 (2010).
[CrossRef]

Yeh, W. Y.

C. F. Lai, H. C. Kuo, P. Yu, T. C. Lu, C. H. Chao, H. H. Yen, and W. Y. Yeh, Appl. Phys. Lett. 97, 013108 (2010).
[CrossRef]

Yen, H. H.

C. F. Lai, H. C. Kuo, P. Yu, T. C. Lu, C. H. Chao, H. H. Yen, and W. Y. Yeh, Appl. Phys. Lett. 97, 013108 (2010).
[CrossRef]

Yin, Y. F.

Y. T. Wang, Y. Chou, L. Y. Chen, Y. F. Yin, Y. C. Lin, and J. J. Huang, IEEE J Quantum Electron. 49, 11 (2013).
[CrossRef]

Y. W. Cheng, S. C. Wang, Y. F. Yin, L. Y. Su, and J. J. Huang, Opt. Lett. 36, 1611 (2011).
[CrossRef]

S. C. Wang, Y. W. Cheng, Y. F. Yin, L. Y. Chen, L. Y. Su, Y. J. Hung, and J. J. Huang, J Lightwave Technol. 29, 3772 (2011).
[CrossRef]

Yu, P.

C. F. Lai, H. C. Kuo, P. Yu, T. C. Lu, C. H. Chao, H. H. Yen, and W. Y. Yeh, Appl. Phys. Lett. 97, 013108 (2010).
[CrossRef]

Yuan, H. K.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, Nature 466, 735 (2010).
[CrossRef]

Appl. Phys. Lett.

K. McGroddy, A. David, E. Matioli, M. Iza, S. Nakamura, S. DenBaars, J. S. Speck, C. Weisbuch, and E. L. Hu, Appl. Phys. Lett. 93, 103502 (2008).
[CrossRef]

C. F. Lai, H. C. Kuo, P. Yu, T. C. Lu, C. H. Chao, H. H. Yen, and W. Y. Yeh, Appl. Phys. Lett. 97, 013108 (2010).
[CrossRef]

IEEE J Quantum Electron.

Y. T. Wang, Y. Chou, L. Y. Chen, Y. F. Yin, Y. C. Lin, and J. J. Huang, IEEE J Quantum Electron. 49, 11 (2013).
[CrossRef]

IEEE Photon. J

X. H. Li, R. Song, Y. K. Ee, P. Kumnorkaew, J. F. Gilchrist, and N. Tansu, IEEE Photon. J 3, 489 (2011).
[CrossRef]

J Lightwave Technol.

S. C. Wang, Y. W. Cheng, Y. F. Yin, L. Y. Chen, L. Y. Su, Y. J. Hung, and J. J. Huang, J Lightwave Technol. 29, 3772 (2011).
[CrossRef]

J Phys. D

D. W. Prather, S. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. Chen, B. L. Miao, and R. Martin, J Phys. D 40, 2635 (2007).
[CrossRef]

Nat. Mater.

S. P. Burgos, R. De Waele, A. Polman, and H. A. Atwater, Nat. Mater. 9, 407 (2010).
[CrossRef]

Nat. Photonics

J. J. Wierer, A. David, and M. M. Megens, Nat. Photonics 3, 163 (2009).
[CrossRef]

Nature

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, Nature 466, 735 (2010).
[CrossRef]

Opt. Commun.

M. Hofman, N. Fabre, X. Mélique, D. Lippens, and O. Vanbésien, Opt. Commun. 283, 1169 (2010).
[CrossRef]

Opt. Lett.

Phys. Rep.

K. Busch, G. Von Freymann, S. Linden, S. Mingaleev, L. Tkeshelashvili, and M. Wegener, Phys. Rep. 444, 101 (2007).
[CrossRef]

Phys. Rev. B

M. Notomi, Phys. Rev. B 62, 10696 (2000).
[CrossRef]

A. Martínez and J. Martí, Phys. Rev. B 71, 235115 (2005).
[CrossRef]

C. Luo, S. G. Johnson, J. Joannopoulos, and J. Pendry, Phys. Rev. B 65, 201104 (2002).
[CrossRef]

Other

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton, 2008).

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Figures (4)

Fig. 1.
Fig. 1.

Device structure of a PhC LED with surrounding nanohole arrays. The right sides of the figure shows a scanning electron microscopic image of the PhC region with corresponding nanohole geometry labeled.

Fig. 2.
Fig. 2.

(a) Electroluminescent radiation profiles under an injection current of 20 mA. Both the intensities of TE and TM are plotted in their absolute values, without any normalization. (b) Measured TE photonic band. (c) Measured TM. (d) Principle of the negative refraction and light collimation of PhC nanoholes under TE light.

Fig. 3.
Fig. 3.

Far-field radiation of the PhCLED measured (a) for TE and (b) for TM at a wavelength of 454 nm in a hemisphere with a radius 10 cm away from the device under an injection current of 20 mA.

Fig. 4.
Fig. 4.

Simulated EFCs. Visualization of (a) TE and (b) TM polarization band surfaces within the first Brillouin zone. (c) and (d) Corresponding group velocity distributions (in speed of light in vacuum, c0). The white and black arrows, respectively represent positive and negative group velocities.

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