Abstract

The use of geometrical constraints exposes many new perspectives in photonics and in fundamental studies of nonlinear waves. By implementing surface structures in vertical cavity surface emitting lasers as manifolds for curved space, we experimentally study the impacts of geometrical constraints on nonlinear wave localization. We observe localized waves pinned to the maximal curvature in an elliptical-ring, and confirm the reduction in the localization length of waves by measuring near and far field patterns, as well as the corresponding energy-angle dispersion relation. Theoretically, analyses based on a dissipative model with a parabola curve give good agreement remarkably to experimental measurement on the reduction in the localization length. The introduction of curved geometry allows to control and design lasing modes in the nonlinear regime.

© 2017 Optical Society of America

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    [Crossref]

2016 (2)

A. Paredes and H. Michinel, “Interference of dark matter solitons and galactic offsets,” Phys. Dark. Univ. 12, 50 (2016).
[Crossref]

C. Conti, “Localization and shock waves in curved manifolds for the Gross-Pitaevskii equation,” Sci. Bull. 61, 570 (2016).
[Crossref]

2014 (2)

R. Bekenstein, J. Nemirovsky, I. Kaminer, and M. Segev, “Shape-preserving accelerating electromagnetic wave packets in curved space,” Phys. Rev. X 4, 011038 (2014)

C. Conti, “Linear and nonlinear Anderson localization in a curved potential,” Chin. Phys. Lett. 31, 30501 (2014).
[Crossref]

2013 (2)

2012 (1)

D. Faccio, T. Arane, M. Lamperti, and U. Leonhardt, “Optical black hole lasers,” Class. Quantum Grav. 29, 224009 (2012).
[Crossref]

2011 (1)

C.P. Jisha, Y.Y. Lin, T.-D. Lee, and R.-K. Lee, “Crescent waves in optical cavities,” Phys. Rev. Lett. 107, 183902 (2011).
[Crossref] [PubMed]

2010 (3)

W.-X. Yang, Y.Y. Lin, T.-D. Lee, R.-K. Lee, and Yu. S. Kivshar, “Nonlinear localized modes in bandgap microcavities,” Opt. Lett. 35, 3207 (2010).
[Crossref] [PubMed]

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tunnermann, S. Longhi, and U. Peschel, “Optics in curved space,” Phys. Rev. Lett. 105, 143901 (2010).
[Crossref]

S. Batz and U. Peschel, “Solitons in curved space of constant curvature,” Phys. Rev. A 81, 053806 (2010).
[Crossref]

2009 (2)

D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5, 687 (2009).
[Crossref]

A. Sacchetti, “Universal Critical Power for Nonlinear Schrödinger Equations with a Symmetric Double Well Potential,” Phys. Rev. Lett. 103, 194101 (2009).
[Crossref]

2008 (5)

Th. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. Konig, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science 319, 1367 (2008).
[Crossref] [PubMed]

Y. Tanguy, T. Ackemann, W. J. Firth, and R. Jager, “Realization of a semiconductor-based cavity soliton laser,” Phys. Rev. Lett. 100, 013907 (2008).
[Crossref] [PubMed]

T.-D. Lee, C.-Y. Chen, Y.Y. Lin, M.-C. Chou, T.-h. Wu, and R.-K. Lee, “Surface-structure-assisted chaotic mode lasing in vertical cavity surface emitting lasers,” Phys. Rev. Lett. 101, 084101 (2008).
[Crossref] [PubMed]

S. Batz and U. Peschel, “Linear and nonlinear optics in curved space,” Phys. Rev. A 78, 043821 (2008)
[Crossref]

Y. Y. Lin and R.-K. Lee, “Symmetry-breaking instabilities of generalized elliptical solitons,” Opt. Lett. 33, 1377 (2008).
[Crossref] [PubMed]

2007 (3)

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Rotating surface solitons,” Opt. Lett. 32, 2948 (2007).
[Crossref] [PubMed]

A. Kroner, F. Rinaldi, J. M. Ostermann, and R. Michalzik, “High-performance single fundamental mode AlGaAs VCSELs with mode-selective mirror reflectivities,” Opt. Commun. 270, 332 (2007).
[Crossref]

W. Wan, S. Jia, and J. W. Fleisher, “Dispersive superfluid-like shock waves in nonlinear optics,” Nat. Phys. 3, 46 (2007).
[Crossref]

2006 (2)

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777 (2006).
[Crossref] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780 (2006).
[Crossref] [PubMed]

2005 (1)

J.-W. Shi, C.-H. Jiang, K. M. Chen, J.-L. Yen, and Y.-J. Yang, “Single-mode vertical-cavity surface-emitting laser with ring-shaped light-emitting aperture,” Appl. Phys. Lett. 87, 031109 (2005).
[Crossref]

2004 (1)

2003 (1)

C. Barcelo, S. Liberati, and M. Visser, “Probing semiclassical analog gravity in Bose-Einstein condensates with widely tunable interactions,” Phys. Rev. A 68, 053613 (2003).
[Crossref]

2002 (2)

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jager, “Cavity solitons as pixels in semiconductor microcavities,” Nature 419, 699 (2002).
[Crossref] [PubMed]

J. Scheuer, M. Orenstein, and D. Arbel, “Nonlinear switching and modulational instability of wave patterns in ring-shaped vertical-cavity surface-emitting lasers,” J. Opt. Soc. Am. B 19, 2384 (2002).
[Crossref]

1997 (1)

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, “Spatial soliton pixels in semiconductor devices,” Phys. Rev. Lett. 79, 2042 (1997).
[Crossref]

1996 (1)

D. L. Huffaker, H. Deng, Q. Deng, and D. G. Deppe, “Ring and stripe oxide-confined vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 69, 3477 (1996).
[Crossref]

1995 (1)

D. V. Kuksenkov, H. Temkin, and S. Swirhun, “Frequency modulation characteristics of gain-guided AlGaAs/GaAs vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 66, 3239 (1995).
[Crossref]

1991 (1)

G. R. Olbright, R. P. Bryan, W. S. Fu, R. Apte, D. M. Bloom, and Y. H. Lee, “Linewidth, tunability, and VHF-millimeter wave frequency synthesis of vertical-cavity GaAs quantum-well surface-emitting laser diode arrays,” IEEE Photonics Technol. Lett. 3, 779 (1991).
[Crossref]

1981 (1)

C. H. Henry, R. A. Logan, and K. A. Bertness, “Spectral dependence of the change in refractive index due to carrier injection in GaAs lasers,” J. Appl. Phys. 52, 4457 (1981).
[Crossref]

Ackemann, T.

Y. Tanguy, T. Ackemann, W. J. Firth, and R. Jager, “Realization of a semiconductor-based cavity soliton laser,” Phys. Rev. Lett. 100, 013907 (2008).
[Crossref] [PubMed]

Agrawal, G. P.

Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals, (Academic, 2003).

Apte, R.

G. R. Olbright, R. P. Bryan, W. S. Fu, R. Apte, D. M. Bloom, and Y. H. Lee, “Linewidth, tunability, and VHF-millimeter wave frequency synthesis of vertical-cavity GaAs quantum-well surface-emitting laser diode arrays,” IEEE Photonics Technol. Lett. 3, 779 (1991).
[Crossref]

Arane, T.

D. Faccio, T. Arane, M. Lamperti, and U. Leonhardt, “Optical black hole lasers,” Class. Quantum Grav. 29, 224009 (2012).
[Crossref]

Arbel, D.

Arima, V.

Balle, S.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jager, “Cavity solitons as pixels in semiconductor microcavities,” Nature 419, 699 (2002).
[Crossref] [PubMed]

Bandres, M. A.

Barcelo, C.

C. Barcelo, S. Liberati, and M. Visser, “Probing semiclassical analog gravity in Bose-Einstein condensates with widely tunable interactions,” Phys. Rev. A 68, 053613 (2003).
[Crossref]

Barland, S.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jager, “Cavity solitons as pixels in semiconductor microcavities,” Nature 419, 699 (2002).
[Crossref] [PubMed]

Batz, S.

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tunnermann, S. Longhi, and U. Peschel, “Optics in curved space,” Phys. Rev. Lett. 105, 143901 (2010).
[Crossref]

S. Batz and U. Peschel, “Solitons in curved space of constant curvature,” Phys. Rev. A 81, 053806 (2010).
[Crossref]

S. Batz and U. Peschel, “Linear and nonlinear optics in curved space,” Phys. Rev. A 78, 043821 (2008)
[Crossref]

Bekenstein, R.

R. Bekenstein, J. Nemirovsky, I. Kaminer, and M. Segev, “Shape-preserving accelerating electromagnetic wave packets in curved space,” Phys. Rev. X 4, 011038 (2014)

Bertness, K. A.

C. H. Henry, R. A. Logan, and K. A. Bertness, “Spectral dependence of the change in refractive index due to carrier injection in GaAs lasers,” J. Appl. Phys. 52, 4457 (1981).
[Crossref]

Bloom, D. M.

G. R. Olbright, R. P. Bryan, W. S. Fu, R. Apte, D. M. Bloom, and Y. H. Lee, “Linewidth, tunability, and VHF-millimeter wave frequency synthesis of vertical-cavity GaAs quantum-well surface-emitting laser diode arrays,” IEEE Photonics Technol. Lett. 3, 779 (1991).
[Crossref]

Brambilla, M.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jager, “Cavity solitons as pixels in semiconductor microcavities,” Nature 419, 699 (2002).
[Crossref] [PubMed]

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, “Spatial soliton pixels in semiconductor devices,” Phys. Rev. Lett. 79, 2042 (1997).
[Crossref]

Bryan, R. P.

G. R. Olbright, R. P. Bryan, W. S. Fu, R. Apte, D. M. Bloom, and Y. H. Lee, “Linewidth, tunability, and VHF-millimeter wave frequency synthesis of vertical-cavity GaAs quantum-well surface-emitting laser diode arrays,” IEEE Photonics Technol. Lett. 3, 779 (1991).
[Crossref]

Chen, C.-Y.

T.-D. Lee, C.-Y. Chen, Y.Y. Lin, M.-C. Chou, T.-h. Wu, and R.-K. Lee, “Surface-structure-assisted chaotic mode lasing in vertical cavity surface emitting lasers,” Phys. Rev. Lett. 101, 084101 (2008).
[Crossref] [PubMed]

Chen, K. M.

J.-W. Shi, C.-H. Jiang, K. M. Chen, J.-L. Yen, and Y.-J. Yang, “Single-mode vertical-cavity surface-emitting laser with ring-shaped light-emitting aperture,” Appl. Phys. Lett. 87, 031109 (2005).
[Crossref]

Chou, M.-C.

T.-D. Lee, C.-Y. Chen, Y.Y. Lin, M.-C. Chou, T.-h. Wu, and R.-K. Lee, “Surface-structure-assisted chaotic mode lasing in vertical cavity surface emitting lasers,” Phys. Rev. Lett. 101, 084101 (2008).
[Crossref] [PubMed]

Conti, C.

C. Conti, “Localization and shock waves in curved manifolds for the Gross-Pitaevskii equation,” Sci. Bull. 61, 570 (2016).
[Crossref]

C. Conti, “Linear and nonlinear Anderson localization in a curved potential,” Chin. Phys. Lett. 31, 30501 (2014).
[Crossref]

N. Ghofraniha, I. Viola, A. Zacheo, V. Arima, G. Gigli, and C. Conti, “Transition from nonresonant to resonant random lasers by the geometrical confinement of disorder,” Opt. Lett. 38, 5043 (2013).
[Crossref] [PubMed]

Deng, H.

D. L. Huffaker, H. Deng, Q. Deng, and D. G. Deppe, “Ring and stripe oxide-confined vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 69, 3477 (1996).
[Crossref]

Deng, Q.

D. L. Huffaker, H. Deng, Q. Deng, and D. G. Deppe, “Ring and stripe oxide-confined vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 69, 3477 (1996).
[Crossref]

Deppe, D. G.

D. L. Huffaker, H. Deng, Q. Deng, and D. G. Deppe, “Ring and stripe oxide-confined vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 69, 3477 (1996).
[Crossref]

Dreisow, F.

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tunnermann, S. Longhi, and U. Peschel, “Optics in curved space,” Phys. Rev. Lett. 105, 143901 (2010).
[Crossref]

Faccio, D.

D. Faccio, T. Arane, M. Lamperti, and U. Leonhardt, “Optical black hole lasers,” Class. Quantum Grav. 29, 224009 (2012).
[Crossref]

Firth, W. J.

Y. Tanguy, T. Ackemann, W. J. Firth, and R. Jager, “Realization of a semiconductor-based cavity soliton laser,” Phys. Rev. Lett. 100, 013907 (2008).
[Crossref] [PubMed]

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, “Spatial soliton pixels in semiconductor devices,” Phys. Rev. Lett. 79, 2042 (1997).
[Crossref]

Fleisher, J. W.

W. Wan, S. Jia, and J. W. Fleisher, “Dispersive superfluid-like shock waves in nonlinear optics,” Nat. Phys. 3, 46 (2007).
[Crossref]

Fu, W. S.

G. R. Olbright, R. P. Bryan, W. S. Fu, R. Apte, D. M. Bloom, and Y. H. Lee, “Linewidth, tunability, and VHF-millimeter wave frequency synthesis of vertical-cavity GaAs quantum-well surface-emitting laser diode arrays,” IEEE Photonics Technol. Lett. 3, 779 (1991).
[Crossref]

Genov, D. A.

D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5, 687 (2009).
[Crossref]

Ghofraniha, N.

Gigli, G.

Giudici, M.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jager, “Cavity solitons as pixels in semiconductor microcavities,” Nature 419, 699 (2002).
[Crossref] [PubMed]

Gutierrez-Vega, J. C.

Henry, C. H.

C. H. Henry, R. A. Logan, and K. A. Bertness, “Spectral dependence of the change in refractive index due to carrier injection in GaAs lasers,” J. Appl. Phys. 52, 4457 (1981).
[Crossref]

Hill, S.

Th. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. Konig, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science 319, 1367 (2008).
[Crossref] [PubMed]

Huffaker, D. L.

D. L. Huffaker, H. Deng, Q. Deng, and D. G. Deppe, “Ring and stripe oxide-confined vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 69, 3477 (1996).
[Crossref]

Jager, R.

Y. Tanguy, T. Ackemann, W. J. Firth, and R. Jager, “Realization of a semiconductor-based cavity soliton laser,” Phys. Rev. Lett. 100, 013907 (2008).
[Crossref] [PubMed]

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jager, “Cavity solitons as pixels in semiconductor microcavities,” Nature 419, 699 (2002).
[Crossref] [PubMed]

Jia, S.

W. Wan, S. Jia, and J. W. Fleisher, “Dispersive superfluid-like shock waves in nonlinear optics,” Nat. Phys. 3, 46 (2007).
[Crossref]

Jiang, C.-H.

J.-W. Shi, C.-H. Jiang, K. M. Chen, J.-L. Yen, and Y.-J. Yang, “Single-mode vertical-cavity surface-emitting laser with ring-shaped light-emitting aperture,” Appl. Phys. Lett. 87, 031109 (2005).
[Crossref]

Jisha, C.P.

C.P. Jisha, Y.Y. Lin, T.-D. Lee, and R.-K. Lee, “Crescent waves in optical cavities,” Phys. Rev. Lett. 107, 183902 (2011).
[Crossref] [PubMed]

Kaminer, I.

R. Bekenstein, J. Nemirovsky, I. Kaminer, and M. Segev, “Shape-preserving accelerating electromagnetic wave packets in curved space,” Phys. Rev. X 4, 011038 (2014)

Kartashov, Y. V.

Kivshar, Yu. S.

Knodl, T.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jager, “Cavity solitons as pixels in semiconductor microcavities,” Nature 419, 699 (2002).
[Crossref] [PubMed]

Konig, F.

Th. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. Konig, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science 319, 1367 (2008).
[Crossref] [PubMed]

Kroner, A.

A. Kroner, F. Rinaldi, J. M. Ostermann, and R. Michalzik, “High-performance single fundamental mode AlGaAs VCSELs with mode-selective mirror reflectivities,” Opt. Commun. 270, 332 (2007).
[Crossref]

Kuklewicz, C.

Th. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. Konig, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science 319, 1367 (2008).
[Crossref] [PubMed]

Kuksenkov, D. V.

D. V. Kuksenkov, H. Temkin, and S. Swirhun, “Frequency modulation characteristics of gain-guided AlGaAs/GaAs vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 66, 3239 (1995).
[Crossref]

Kuo, K.-H.

Lamperti, M.

D. Faccio, T. Arane, M. Lamperti, and U. Leonhardt, “Optical black hole lasers,” Class. Quantum Grav. 29, 224009 (2012).
[Crossref]

Lee, R.-K.

K.-H. Kuo, Y.Y. Lin, and R.-K. Lee, “Thresholdless crescent waves in an elliptical ring,” Opt. Lett. 38, 1077 (2013).
[Crossref] [PubMed]

C.P. Jisha, Y.Y. Lin, T.-D. Lee, and R.-K. Lee, “Crescent waves in optical cavities,” Phys. Rev. Lett. 107, 183902 (2011).
[Crossref] [PubMed]

W.-X. Yang, Y.Y. Lin, T.-D. Lee, R.-K. Lee, and Yu. S. Kivshar, “Nonlinear localized modes in bandgap microcavities,” Opt. Lett. 35, 3207 (2010).
[Crossref] [PubMed]

Y. Y. Lin and R.-K. Lee, “Symmetry-breaking instabilities of generalized elliptical solitons,” Opt. Lett. 33, 1377 (2008).
[Crossref] [PubMed]

T.-D. Lee, C.-Y. Chen, Y.Y. Lin, M.-C. Chou, T.-h. Wu, and R.-K. Lee, “Surface-structure-assisted chaotic mode lasing in vertical cavity surface emitting lasers,” Phys. Rev. Lett. 101, 084101 (2008).
[Crossref] [PubMed]

Lee, T.-D.

C.P. Jisha, Y.Y. Lin, T.-D. Lee, and R.-K. Lee, “Crescent waves in optical cavities,” Phys. Rev. Lett. 107, 183902 (2011).
[Crossref] [PubMed]

W.-X. Yang, Y.Y. Lin, T.-D. Lee, R.-K. Lee, and Yu. S. Kivshar, “Nonlinear localized modes in bandgap microcavities,” Opt. Lett. 35, 3207 (2010).
[Crossref] [PubMed]

T.-D. Lee, C.-Y. Chen, Y.Y. Lin, M.-C. Chou, T.-h. Wu, and R.-K. Lee, “Surface-structure-assisted chaotic mode lasing in vertical cavity surface emitting lasers,” Phys. Rev. Lett. 101, 084101 (2008).
[Crossref] [PubMed]

Lee, Y. H.

G. R. Olbright, R. P. Bryan, W. S. Fu, R. Apte, D. M. Bloom, and Y. H. Lee, “Linewidth, tunability, and VHF-millimeter wave frequency synthesis of vertical-cavity GaAs quantum-well surface-emitting laser diode arrays,” IEEE Photonics Technol. Lett. 3, 779 (1991).
[Crossref]

Leonhardt, U.

D. Faccio, T. Arane, M. Lamperti, and U. Leonhardt, “Optical black hole lasers,” Class. Quantum Grav. 29, 224009 (2012).
[Crossref]

Th. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. Konig, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science 319, 1367 (2008).
[Crossref] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777 (2006).
[Crossref] [PubMed]

Liberati, S.

C. Barcelo, S. Liberati, and M. Visser, “Probing semiclassical analog gravity in Bose-Einstein condensates with widely tunable interactions,” Phys. Rev. A 68, 053613 (2003).
[Crossref]

Lin, Y. Y.

Lin, Y.Y.

K.-H. Kuo, Y.Y. Lin, and R.-K. Lee, “Thresholdless crescent waves in an elliptical ring,” Opt. Lett. 38, 1077 (2013).
[Crossref] [PubMed]

C.P. Jisha, Y.Y. Lin, T.-D. Lee, and R.-K. Lee, “Crescent waves in optical cavities,” Phys. Rev. Lett. 107, 183902 (2011).
[Crossref] [PubMed]

W.-X. Yang, Y.Y. Lin, T.-D. Lee, R.-K. Lee, and Yu. S. Kivshar, “Nonlinear localized modes in bandgap microcavities,” Opt. Lett. 35, 3207 (2010).
[Crossref] [PubMed]

T.-D. Lee, C.-Y. Chen, Y.Y. Lin, M.-C. Chou, T.-h. Wu, and R.-K. Lee, “Surface-structure-assisted chaotic mode lasing in vertical cavity surface emitting lasers,” Phys. Rev. Lett. 101, 084101 (2008).
[Crossref] [PubMed]

Logan, R. A.

C. H. Henry, R. A. Logan, and K. A. Bertness, “Spectral dependence of the change in refractive index due to carrier injection in GaAs lasers,” J. Appl. Phys. 52, 4457 (1981).
[Crossref]

Longhi, S.

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tunnermann, S. Longhi, and U. Peschel, “Optics in curved space,” Phys. Rev. Lett. 105, 143901 (2010).
[Crossref]

Lugiato, L. A.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jager, “Cavity solitons as pixels in semiconductor microcavities,” Nature 419, 699 (2002).
[Crossref] [PubMed]

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, “Spatial soliton pixels in semiconductor devices,” Phys. Rev. Lett. 79, 2042 (1997).
[Crossref]

Maggipinto, T.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jager, “Cavity solitons as pixels in semiconductor microcavities,” Nature 419, 699 (2002).
[Crossref] [PubMed]

Michalzik, R.

A. Kroner, F. Rinaldi, J. M. Ostermann, and R. Michalzik, “High-performance single fundamental mode AlGaAs VCSELs with mode-selective mirror reflectivities,” Opt. Commun. 270, 332 (2007).
[Crossref]

Michinel, H.

A. Paredes and H. Michinel, “Interference of dark matter solitons and galactic offsets,” Phys. Dark. Univ. 12, 50 (2016).
[Crossref]

Miller, M.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jager, “Cavity solitons as pixels in semiconductor microcavities,” Nature 419, 699 (2002).
[Crossref] [PubMed]

Nemirovsky, J.

R. Bekenstein, J. Nemirovsky, I. Kaminer, and M. Segev, “Shape-preserving accelerating electromagnetic wave packets in curved space,” Phys. Rev. X 4, 011038 (2014)

Nolte, S.

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tunnermann, S. Longhi, and U. Peschel, “Optics in curved space,” Phys. Rev. Lett. 105, 143901 (2010).
[Crossref]

Olbright, G. R.

G. R. Olbright, R. P. Bryan, W. S. Fu, R. Apte, D. M. Bloom, and Y. H. Lee, “Linewidth, tunability, and VHF-millimeter wave frequency synthesis of vertical-cavity GaAs quantum-well surface-emitting laser diode arrays,” IEEE Photonics Technol. Lett. 3, 779 (1991).
[Crossref]

Orenstein, M.

Ostermann, J. M.

A. Kroner, F. Rinaldi, J. M. Ostermann, and R. Michalzik, “High-performance single fundamental mode AlGaAs VCSELs with mode-selective mirror reflectivities,” Opt. Commun. 270, 332 (2007).
[Crossref]

Paredes, A.

A. Paredes and H. Michinel, “Interference of dark matter solitons and galactic offsets,” Phys. Dark. Univ. 12, 50 (2016).
[Crossref]

Pendry, J. B.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780 (2006).
[Crossref] [PubMed]

Peschel, U.

S. Batz and U. Peschel, “Solitons in curved space of constant curvature,” Phys. Rev. A 81, 053806 (2010).
[Crossref]

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tunnermann, S. Longhi, and U. Peschel, “Optics in curved space,” Phys. Rev. Lett. 105, 143901 (2010).
[Crossref]

S. Batz and U. Peschel, “Linear and nonlinear optics in curved space,” Phys. Rev. A 78, 043821 (2008)
[Crossref]

Philbin, Th. G.

Th. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. Konig, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science 319, 1367 (2008).
[Crossref] [PubMed]

Prati, F.

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, “Spatial soliton pixels in semiconductor devices,” Phys. Rev. Lett. 79, 2042 (1997).
[Crossref]

Rayleigh, J. W. S.

J. W. S. Rayleigh, The Theory of Sound, (Macmillan, 1894).

Rinaldi, F.

A. Kroner, F. Rinaldi, J. M. Ostermann, and R. Michalzik, “High-performance single fundamental mode AlGaAs VCSELs with mode-selective mirror reflectivities,” Opt. Commun. 270, 332 (2007).
[Crossref]

Robertson, S.

Th. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. Konig, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science 319, 1367 (2008).
[Crossref] [PubMed]

Sacchetti, A.

A. Sacchetti, “Universal Critical Power for Nonlinear Schrödinger Equations with a Symmetric Double Well Potential,” Phys. Rev. Lett. 103, 194101 (2009).
[Crossref]

Scheuer, J.

Schultheiss, V. H.

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tunnermann, S. Longhi, and U. Peschel, “Optics in curved space,” Phys. Rev. Lett. 105, 143901 (2010).
[Crossref]

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780 (2006).
[Crossref] [PubMed]

Segev, M.

R. Bekenstein, J. Nemirovsky, I. Kaminer, and M. Segev, “Shape-preserving accelerating electromagnetic wave packets in curved space,” Phys. Rev. X 4, 011038 (2014)

Shi, J.-W.

J.-W. Shi, C.-H. Jiang, K. M. Chen, J.-L. Yen, and Y.-J. Yang, “Single-mode vertical-cavity surface-emitting laser with ring-shaped light-emitting aperture,” Appl. Phys. Lett. 87, 031109 (2005).
[Crossref]

Smith, D. R.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780 (2006).
[Crossref] [PubMed]

Spinelli, L.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jager, “Cavity solitons as pixels in semiconductor microcavities,” Nature 419, 699 (2002).
[Crossref] [PubMed]

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, “Spatial soliton pixels in semiconductor devices,” Phys. Rev. Lett. 79, 2042 (1997).
[Crossref]

Swirhun, S.

D. V. Kuksenkov, H. Temkin, and S. Swirhun, “Frequency modulation characteristics of gain-guided AlGaAs/GaAs vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 66, 3239 (1995).
[Crossref]

Szameit, A.

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tunnermann, S. Longhi, and U. Peschel, “Optics in curved space,” Phys. Rev. Lett. 105, 143901 (2010).
[Crossref]

Tanguy, Y.

Y. Tanguy, T. Ackemann, W. J. Firth, and R. Jager, “Realization of a semiconductor-based cavity soliton laser,” Phys. Rev. Lett. 100, 013907 (2008).
[Crossref] [PubMed]

Temkin, H.

D. V. Kuksenkov, H. Temkin, and S. Swirhun, “Frequency modulation characteristics of gain-guided AlGaAs/GaAs vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 66, 3239 (1995).
[Crossref]

Tissoni, G.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jager, “Cavity solitons as pixels in semiconductor microcavities,” Nature 419, 699 (2002).
[Crossref] [PubMed]

Torner, L.

Tredicce, J. R.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jager, “Cavity solitons as pixels in semiconductor microcavities,” Nature 419, 699 (2002).
[Crossref] [PubMed]

Tunnermann, A.

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tunnermann, S. Longhi, and U. Peschel, “Optics in curved space,” Phys. Rev. Lett. 105, 143901 (2010).
[Crossref]

Vahala, K. J.

K. J. Vahala, Optical Microcavities, (World Scientific, 2004).
[Crossref]

Viola, I.

Visser, M.

C. Barcelo, S. Liberati, and M. Visser, “Probing semiclassical analog gravity in Bose-Einstein condensates with widely tunable interactions,” Phys. Rev. A 68, 053613 (2003).
[Crossref]

Vysloukh, V. A.

Wan, W.

W. Wan, S. Jia, and J. W. Fleisher, “Dispersive superfluid-like shock waves in nonlinear optics,” Nat. Phys. 3, 46 (2007).
[Crossref]

Wu, T.-h.

T.-D. Lee, C.-Y. Chen, Y.Y. Lin, M.-C. Chou, T.-h. Wu, and R.-K. Lee, “Surface-structure-assisted chaotic mode lasing in vertical cavity surface emitting lasers,” Phys. Rev. Lett. 101, 084101 (2008).
[Crossref] [PubMed]

Yang, W.-X.

Yang, Y.-J.

J.-W. Shi, C.-H. Jiang, K. M. Chen, J.-L. Yen, and Y.-J. Yang, “Single-mode vertical-cavity surface-emitting laser with ring-shaped light-emitting aperture,” Appl. Phys. Lett. 87, 031109 (2005).
[Crossref]

Yen, J.-L.

J.-W. Shi, C.-H. Jiang, K. M. Chen, J.-L. Yen, and Y.-J. Yang, “Single-mode vertical-cavity surface-emitting laser with ring-shaped light-emitting aperture,” Appl. Phys. Lett. 87, 031109 (2005).
[Crossref]

Zacheo, A.

Zhang, S.

D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5, 687 (2009).
[Crossref]

Zhang, X.

D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5, 687 (2009).
[Crossref]

Appl. Phys. Lett. (3)

D. L. Huffaker, H. Deng, Q. Deng, and D. G. Deppe, “Ring and stripe oxide-confined vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 69, 3477 (1996).
[Crossref]

J.-W. Shi, C.-H. Jiang, K. M. Chen, J.-L. Yen, and Y.-J. Yang, “Single-mode vertical-cavity surface-emitting laser with ring-shaped light-emitting aperture,” Appl. Phys. Lett. 87, 031109 (2005).
[Crossref]

D. V. Kuksenkov, H. Temkin, and S. Swirhun, “Frequency modulation characteristics of gain-guided AlGaAs/GaAs vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 66, 3239 (1995).
[Crossref]

Chin. Phys. Lett. (1)

C. Conti, “Linear and nonlinear Anderson localization in a curved potential,” Chin. Phys. Lett. 31, 30501 (2014).
[Crossref]

Class. Quantum Grav. (1)

D. Faccio, T. Arane, M. Lamperti, and U. Leonhardt, “Optical black hole lasers,” Class. Quantum Grav. 29, 224009 (2012).
[Crossref]

IEEE Photonics Technol. Lett. (1)

G. R. Olbright, R. P. Bryan, W. S. Fu, R. Apte, D. M. Bloom, and Y. H. Lee, “Linewidth, tunability, and VHF-millimeter wave frequency synthesis of vertical-cavity GaAs quantum-well surface-emitting laser diode arrays,” IEEE Photonics Technol. Lett. 3, 779 (1991).
[Crossref]

J. Appl. Phys. (1)

C. H. Henry, R. A. Logan, and K. A. Bertness, “Spectral dependence of the change in refractive index due to carrier injection in GaAs lasers,” J. Appl. Phys. 52, 4457 (1981).
[Crossref]

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

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

Nat. Phys. (2)

D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5, 687 (2009).
[Crossref]

W. Wan, S. Jia, and J. W. Fleisher, “Dispersive superfluid-like shock waves in nonlinear optics,” Nat. Phys. 3, 46 (2007).
[Crossref]

Nature (1)

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jager, “Cavity solitons as pixels in semiconductor microcavities,” Nature 419, 699 (2002).
[Crossref] [PubMed]

Opt. Commun. (1)

A. Kroner, F. Rinaldi, J. M. Ostermann, and R. Michalzik, “High-performance single fundamental mode AlGaAs VCSELs with mode-selective mirror reflectivities,” Opt. Commun. 270, 332 (2007).
[Crossref]

Opt. Lett. (5)

Phys. Dark. Univ. (1)

A. Paredes and H. Michinel, “Interference of dark matter solitons and galactic offsets,” Phys. Dark. Univ. 12, 50 (2016).
[Crossref]

Phys. Rev. A (3)

S. Batz and U. Peschel, “Linear and nonlinear optics in curved space,” Phys. Rev. A 78, 043821 (2008)
[Crossref]

S. Batz and U. Peschel, “Solitons in curved space of constant curvature,” Phys. Rev. A 81, 053806 (2010).
[Crossref]

C. Barcelo, S. Liberati, and M. Visser, “Probing semiclassical analog gravity in Bose-Einstein condensates with widely tunable interactions,” Phys. Rev. A 68, 053613 (2003).
[Crossref]

Phys. Rev. Lett. (6)

Y. Tanguy, T. Ackemann, W. J. Firth, and R. Jager, “Realization of a semiconductor-based cavity soliton laser,” Phys. Rev. Lett. 100, 013907 (2008).
[Crossref] [PubMed]

T.-D. Lee, C.-Y. Chen, Y.Y. Lin, M.-C. Chou, T.-h. Wu, and R.-K. Lee, “Surface-structure-assisted chaotic mode lasing in vertical cavity surface emitting lasers,” Phys. Rev. Lett. 101, 084101 (2008).
[Crossref] [PubMed]

V. H. Schultheiss, S. Batz, A. Szameit, F. Dreisow, S. Nolte, A. Tunnermann, S. Longhi, and U. Peschel, “Optics in curved space,” Phys. Rev. Lett. 105, 143901 (2010).
[Crossref]

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, “Spatial soliton pixels in semiconductor devices,” Phys. Rev. Lett. 79, 2042 (1997).
[Crossref]

A. Sacchetti, “Universal Critical Power for Nonlinear Schrödinger Equations with a Symmetric Double Well Potential,” Phys. Rev. Lett. 103, 194101 (2009).
[Crossref]

C.P. Jisha, Y.Y. Lin, T.-D. Lee, and R.-K. Lee, “Crescent waves in optical cavities,” Phys. Rev. Lett. 107, 183902 (2011).
[Crossref] [PubMed]

Phys. Rev. X (1)

R. Bekenstein, J. Nemirovsky, I. Kaminer, and M. Segev, “Shape-preserving accelerating electromagnetic wave packets in curved space,” Phys. Rev. X 4, 011038 (2014)

Sci. Bull. (1)

C. Conti, “Localization and shock waves in curved manifolds for the Gross-Pitaevskii equation,” Sci. Bull. 61, 570 (2016).
[Crossref]

Science (3)

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777 (2006).
[Crossref] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780 (2006).
[Crossref] [PubMed]

Th. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. Konig, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science 319, 1367 (2008).
[Crossref] [PubMed]

Other (4)

J. W. S. Rayleigh, The Theory of Sound, (Macmillan, 1894).

Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals, (Academic, 2003).

R. K. Chang and A. J. Campillo, eds., Optical Processes in Microcavities, (World Scientific, 1996).
[Crossref]

K. J. Vahala, Optical Microcavities, (World Scientific, 2004).
[Crossref]

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Figures (6)

Fig. 1
Fig. 1

(a) Schematic diagram of our electrically driven GaAs-based vertical cavity surface emitting laser (VCSEL) with a curved potential on the emission aperture defined by p-Pad metal. The emission aperture is protected by a SiNx layer. The current aperture and built-in optical confinement are defined by an AlOx layer right above the active layers, which is composited by multiple quantum wells (MQWs). Top view of scanning electron microscope (SEM) images for (b) a circular-ring; (c) an elliptical-ring, and (d) a cold cavity used as the reference sample. Experimental setups, both for near-/far-field and AREL measurements, are illustrated in (e) and (f), respectively.

Fig. 2
Fig. 2

Near field images of our electrically driven GaAs-based VCSELs operated below (the first row) and above (the second row) for a circular-ring (a, d), an elliptical-ring (b, e), and the reference cold cavity (c, f), respectively. Here, the white dashed-curves represent the edge profiles of the corresponding emission aperture defined by p-Pad metal. Panels (g–i) show the L-I curves, output intensity versus current, for the cavities in (d–f), with lasing spectra in the insets.

Fig. 3
Fig. 3

Comparison of near field patterns for (a) circular-ring and (b) elliptical-ring potentials below and above the threshold currents. In (c), we show the localization factor Lm defined in Eq. (1) as a function of the injection current I, normalized to the threshold current Ith. One clearly sees that for a circular-ring potential, the localization factor is nearly independent of I; on the contrary, for the elliptical-ring, the localization length changes abruptly when the injection current is larger than the threshold Ith. Here, the abbreviation (a.u.) stands for arbitrary unit.

Fig. 4
Fig. 4

Far field patterns (the first row) and angle resolved electroluminescence (AREL) spectra measuring the energy at different angles (the second row) for (a) circular-ring, (b) elliptical-ring, and (c) cold cavities, operated above the threshold currents, corresponding to the near field patterns shown in Figs. 2(d)2(f), respectively. Insets in the second row show the corresponding AREL spectra operated below the threshold currents Here, all the dotted-curves are fitting curves as parabola.

Fig. 5
Fig. 5

(a) Profile of solutions for Eq. (4) in the linear case, and for two nonlinear cases (γ = −0.1 and −0.2). The inset shows the shape of an elliptical-ring (green-line) and its parabolic approximation (dashed-line) with the adopted curvilinear coordinate q. (b) Localization length (solid-line) and peak intensity (dashed-line) for the localized wave for different curvature radii versus injection current I (C = 1).

Fig. 6
Fig. 6

Dependence of the localization factor Lm on the inverse line-width enhancement factor 1/α. The top panel gives the relation for a circular-ring, the ratio = 1.0; while lower panel shows the curves for elliptical-rings, but with a different ellipticity (ratio). The corresponding mode profiles shown in the inset are marked with a given line-width enhancement factor. The shadowed region corresponding to the reported value of line-width enhancement factor in the literature [35–37].

Equations (4)

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

L m 1 L C L C U ( Ω ) U max d Ω ,
α t Ψ + δ Ψ ( α + i ) [ ( 1 + η ) + 2 C ( I 1 ) 1 + | Ψ | 2 ] Ψ + ( α i d ) 2 Ψ = 0 .
t Ψ = 2 Ψ + 2 C ( I 1 ) 1 + | Ψ | 2 Ψ .
d 2 Ψ d η 2 + V G ( η ) Ψ = γ Ψ 1 + | Ψ | 2 ,