Finite size effects in surface emitting Terahertz quantum cascade lasers

Lukas Mahler; Alessandro Tredicucci; Fabio Beltram; Christoph Walther; Harvey E. Beere; David A. Ritchie

doi:10.1364/OE.17.006703

Finite size effects in surface emitting Terahertz quantum cascade lasers

Lukas Mahler, Alessandro Tredicucci, Fabio Beltram, Christoph Walther, Harvey E. Beere, and David A. Ritchie

Optics Express
Vol. 17,
Issue 8,
pp. 6703-6709
(2009)
•https://doi.org/10.1364/OE.17.006703

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Lukas Mahler, Alessandro Tredicucci, Fabio Beltram, Christoph Walther, Harvey E. Beere, and David A. Ritchie, "Finite size effects in surface emitting Terahertz quantum cascade lasers," Opt. Express 17, 6703-6709 (2009)

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Related Topics
Table of Contents Category
- Lasers and Laser Optics
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Infrared and far-infrared lasers (140.3070)
- Lasers, distributed-feedback (140.3490)
- Semiconductor lasers, quantum cascade (140.5965)
- Surface plasmons (240.6680)
- Vertical emitting lasers (250.7270)

About this Article
History
- Original Manuscript: January 23, 2009
- Revised Manuscript: March 26, 2009
- Manuscript Accepted: April 5, 2009
- Published: April 8, 2009

Accessible

Open Access

Abstract

We analyze surface-emitting distributed feedback resonators for Terahertz quantum cascade lasers fabricated from double-metal waveguides. We explain the influence on resonances and surface-emission properties of the finite length and width of the gratings in connection with absorbing boundary conditions, and show that, contrary to the infinite case, the modes on either side of the photonic band-gap have finite surface losses. The lateral design of the resonator is shown to be important to avoid transverse modes of higher order and anti-guiding effects. Experimental findings are indeed in excellent agreement with the simulations. Both modeling and fabrication can easily be applied to arbitrary gratings, of which we discuss here a first interesting example.

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References

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R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature 417, 156–159 (2002).
[Crossref] [PubMed]
B. S. Williams, “Terahertz quantum-cascade lasers,” Nature Photonics 1, 517–525 (2007).
[Crossref]
H.-W. Hübers, S. G. Pavlov, H. Richter, A. D. Semenov, L. Mahler, A. Tredicucci, H. E. Beere, and D. A. Ritchie, “High-resolution gas phase spectroscopy with a distributed feedback terahertz quantum cascade laser,” Appl. Phys. Lett. 89, 061115 (2006).
[Crossref]
A. W. M. Lee, Q. Qin, S. Kumar, B. S. Williams, and Q. Hu “Real-time terahertz imaging over a standoff distance (>25 meters),” Appl. Phys. Lett. 89, 141125 (2006).
[Crossref]
S. Kumar and A. W. M. Lee “Resonant-phonon terahertz quantum-cascade lasers and video-rate terahertz imaging,” IEEE J. Sel. Top. Quantum Electron. 14, 333–344 (2008).
[Crossref]
H.-W. Hübers, S. Pavlov, A. Semenov, R. Köhler, L. Mahler, A. Tredicucci, H. Beere, D. Ritchie, and E. Linfield, “Terahertz quantum cascade laser as local oscillator in a heterodyne receiver,” Opt. Express 13, 5890–5896 (2005).
[Crossref] [PubMed]
B. S. Williams, S. Kumar, H. Callebaut, Q. Hu, and J. L. Reno “Terahertz quantum-cascade laser at λ ≈ 100 μm using metal waveguide for mode confinement,” Appl. Phys. Lett. 83, 2124–2126 (2003).
[Crossref]
B. S. Williams, S. Kumar, Q. Hu, and J. Reno, “Operation of terahertz quantum-cascade lasers at 164 K in pulsed mode and at 117 K in continuous-wave mode,” Opt. Express 13, 3331–3339 (2005).
[Crossref] [PubMed]
M. A. Belkin, J. A. Fan, S. Hormoz, F. Capasso, S. P. Khanna, M. Lachab, A. G. Davies, and E. H. Linfield, “Terahertz quantum cascade lasers with copper metal-metal waveguides operating up to 178 K,” Opt. Express 16, 3242–3248 (2008).
[Crossref] [PubMed]
M. I. Amanti, M. Fischer, C. Walther, G. Scalari, and J. Faist, “Horn antennas for terahertz quantum cascade lasers,” Electron. Lett. 43, 573–574, (2007).
[Crossref]
W. Maineult, P. Gellie, A. Andronico, P. Filloux, G. Leo, C. Sirtori, S. Barbieri, E. Peytavit, T. Akalin, J.-F. Lampin, H. E. Beere, and D. A. Ritchie, “Metal-metal terahertz quantum cascade laser with micro-transverse-electromagnetic-horn antenna,” Appl. Phys. Lett. 93183508 (2008).
[Crossref]
A. W. M. Lee, Q. Qin, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, “High-power and high-temperature THz quantum-cascade lasers based on lens-coupled metal-metal waveguides,” Opt. Lett. 32, 2840–2842, (2007).
[Crossref]
O. Demichel, L. Mahler, T. Losco, C. Mauro, R. Green, A. Tredicucci, J. Xu, F. Beltram, H. E. Beere, D. A. Ritchie, and V. Tamošinuas, “Surface plasmon photonic structures in terahertz quantum cascade lasers,” Opt. Express 14, 5335–5345 (2006).
[Crossref] [PubMed]
J. A. Fan, M. A. Belkin, F. Capasso, S. Khanna, M. Lachab, A. G. Davies, and E. H. Linfield, “Surface emitting terahertz quantum cascade laser with a double-metal waveguide,” Opt. Express 14, 11672–11680 (2006).
[Crossref] [PubMed]
S. Kumar, B. S. Williams, Q. Qin, A. W. Lee, Q. Hu, and J. L. Reno, “Surface-emitting distributed feedback terahertz quantum-cascade lasers in metal-metal waveguides,” Opt. Express 15, 113–128 (2007).
[Crossref] [PubMed]
C. Pflügl, M. Austerer, W. Schrenk, S. Golka, G. Strasser, R. P. Green, L. R. Wilson, J. W. Cockburn, A. B. Krysa, and J. S. Roberts, “Single-mode surface-emitting quantum-cascade lasers,” Appl. Phys. Lett. 86211102 (2005).
[Crossref]
M. Schubert and F. Rana, “Analysis of Terahertz Surface Emitting Quantum-Cascade Lasers,” IEEE J. Quantum Electron. 42, 257 (2006).
[Crossref]
M. A. Ordal, R. J. Bell, R. W. Alexander, L. L. Long, and M. R. Querry, “Optical properties of Au, Ni, and Pb at submillimeter wavelengths,” Appl. Opt. 26, 744 (1987).
[Crossref] [PubMed]
L. Mahler, A. Tredicucci, F. Beltram, C. Walther, J. Faist, B. Witzigmann, H. E. Beere, and D. A. Ritchie “Vertically emitting microdisk lasers,” Nat. Photonics 3, 46–49 (2009).
[Crossref]
L. Mahler, A. Tredicucci, R. Köhler, F. Beltram, H. E. Beere, E. H. Linfield, and D. A. Ritchie “High-Performance operation of single-mode terahertz quantum cascade lasers with metallic gratings,” Appl. Phys. Lett., 87, 181101 (2004).
[Crossref]
J. A. Fan, M. A. Belkin, F. Capasso, S. P. Khanna, M. Lachab, A. Giles Davies, and Edmund H. Linfield “Wide-ridge metal-metal terahertz quantum cascade lasers with high-order lateral mode suppression,” Appl. Phys. Lett. 92031106 (2008).
[Crossref]

2009 (1)

L. Mahler, A. Tredicucci, F. Beltram, C. Walther, J. Faist, B. Witzigmann, H. E. Beere, and D. A. Ritchie “Vertically emitting microdisk lasers,” Nat. Photonics 3, 46–49 (2009).
[Crossref]

2008 (4)

J. A. Fan, M. A. Belkin, F. Capasso, S. P. Khanna, M. Lachab, A. Giles Davies, and Edmund H. Linfield “Wide-ridge metal-metal terahertz quantum cascade lasers with high-order lateral mode suppression,” Appl. Phys. Lett. 92031106 (2008).
[Crossref]

S. Kumar and A. W. M. Lee “Resonant-phonon terahertz quantum-cascade lasers and video-rate terahertz imaging,” IEEE J. Sel. Top. Quantum Electron. 14, 333–344 (2008).
[Crossref]

W. Maineult, P. Gellie, A. Andronico, P. Filloux, G. Leo, C. Sirtori, S. Barbieri, E. Peytavit, T. Akalin, J.-F. Lampin, H. E. Beere, and D. A. Ritchie, “Metal-metal terahertz quantum cascade laser with micro-transverse-electromagnetic-horn antenna,” Appl. Phys. Lett. 93183508 (2008).
[Crossref]

M. A. Belkin, J. A. Fan, S. Hormoz, F. Capasso, S. P. Khanna, M. Lachab, A. G. Davies, and E. H. Linfield, “Terahertz quantum cascade lasers with copper metal-metal waveguides operating up to 178 K,” Opt. Express 16, 3242–3248 (2008).
[Crossref] [PubMed]

2007 (4)

S. Kumar, B. S. Williams, Q. Qin, A. W. Lee, Q. Hu, and J. L. Reno, “Surface-emitting distributed feedback terahertz quantum-cascade lasers in metal-metal waveguides,” Opt. Express 15, 113–128 (2007).
[Crossref] [PubMed]

A. W. M. Lee, Q. Qin, S. Kumar, B. S. Williams, Q. Hu, and J. L. Reno, “High-power and high-temperature THz quantum-cascade lasers based on lens-coupled metal-metal waveguides,” Opt. Lett. 32, 2840–2842, (2007).
[Crossref]

M. I. Amanti, M. Fischer, C. Walther, G. Scalari, and J. Faist, “Horn antennas for terahertz quantum cascade lasers,” Electron. Lett. 43, 573–574, (2007).
[Crossref]

B. S. Williams, “Terahertz quantum-cascade lasers,” Nature Photonics 1, 517–525 (2007).
[Crossref]

2006 (5)

H.-W. Hübers, S. G. Pavlov, H. Richter, A. D. Semenov, L. Mahler, A. Tredicucci, H. E. Beere, and D. A. Ritchie, “High-resolution gas phase spectroscopy with a distributed feedback terahertz quantum cascade laser,” Appl. Phys. Lett. 89, 061115 (2006).
[Crossref]

A. W. M. Lee, Q. Qin, S. Kumar, B. S. Williams, and Q. Hu “Real-time terahertz imaging over a standoff distance (>25 meters),” Appl. Phys. Lett. 89, 141125 (2006).
[Crossref]

M. Schubert and F. Rana, “Analysis of Terahertz Surface Emitting Quantum-Cascade Lasers,” IEEE J. Quantum Electron. 42, 257 (2006).
[Crossref]

O. Demichel, L. Mahler, T. Losco, C. Mauro, R. Green, A. Tredicucci, J. Xu, F. Beltram, H. E. Beere, D. A. Ritchie, and V. Tamošinuas, “Surface plasmon photonic structures in terahertz quantum cascade lasers,” Opt. Express 14, 5335–5345 (2006).
[Crossref] [PubMed]

J. A. Fan, M. A. Belkin, F. Capasso, S. Khanna, M. Lachab, A. G. Davies, and E. H. Linfield, “Surface emitting terahertz quantum cascade laser with a double-metal waveguide,” Opt. Express 14, 11672–11680 (2006).
[Crossref] [PubMed]

2005 (3)

B. S. Williams, S. Kumar, Q. Hu, and J. Reno, “Operation of terahertz quantum-cascade lasers at 164 K in pulsed mode and at 117 K in continuous-wave mode,” Opt. Express 13, 3331–3339 (2005).
[Crossref] [PubMed]

H.-W. Hübers, S. Pavlov, A. Semenov, R. Köhler, L. Mahler, A. Tredicucci, H. Beere, D. Ritchie, and E. Linfield, “Terahertz quantum cascade laser as local oscillator in a heterodyne receiver,” Opt. Express 13, 5890–5896 (2005).
[Crossref] [PubMed]

C. Pflügl, M. Austerer, W. Schrenk, S. Golka, G. Strasser, R. P. Green, L. R. Wilson, J. W. Cockburn, A. B. Krysa, and J. S. Roberts, “Single-mode surface-emitting quantum-cascade lasers,” Appl. Phys. Lett. 86211102 (2005).
[Crossref]

2004 (1)

L. Mahler, A. Tredicucci, R. Köhler, F. Beltram, H. E. Beere, E. H. Linfield, and D. A. Ritchie “High-Performance operation of single-mode terahertz quantum cascade lasers with metallic gratings,” Appl. Phys. Lett., 87, 181101 (2004).
[Crossref]

2003 (1)

B. S. Williams, S. Kumar, H. Callebaut, Q. Hu, and J. L. Reno “Terahertz quantum-cascade laser at λ ≈ 100 μm using metal waveguide for mode confinement,” Appl. Phys. Lett. 83, 2124–2126 (2003).
[Crossref]

2002 (1)

R. Köhler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature 417, 156–159 (2002).
[Crossref] [PubMed]

1987 (1)

M. A. Ordal, R. J. Bell, R. W. Alexander, L. L. Long, and M. R. Querry, “Optical properties of Au, Ni, and Pb at submillimeter wavelengths,” Appl. Opt. 26, 744 (1987).
[Crossref] [PubMed]

Akalin, T.

Alexander, R. W.

M. A. Ordal, R. J. Bell, R. W. Alexander, L. L. Long, and M. R. Querry, “Optical properties of Au, Ni, and Pb at submillimeter wavelengths,” Appl. Opt. 26, 744 (1987).
[Crossref] [PubMed]

Amanti, M. I.

M. I. Amanti, M. Fischer, C. Walther, G. Scalari, and J. Faist, “Horn antennas for terahertz quantum cascade lasers,” Electron. Lett. 43, 573–574, (2007).
[Crossref]

Andronico, A.

Austerer, M.

Barbieri, S.

Beere, H.

Beere, H. E.

L. Mahler, A. Tredicucci, F. Beltram, C. Walther, J. Faist, B. Witzigmann, H. E. Beere, and D. A. Ritchie “Vertically emitting microdisk lasers,” Nat. Photonics 3, 46–49 (2009).
[Crossref]

Belkin, M. A.

Bell, R. J.

M. A. Ordal, R. J. Bell, R. W. Alexander, L. L. Long, and M. R. Querry, “Optical properties of Au, Ni, and Pb at submillimeter wavelengths,” Appl. Opt. 26, 744 (1987).
[Crossref] [PubMed]

Beltram, F.

L. Mahler, A. Tredicucci, F. Beltram, C. Walther, J. Faist, B. Witzigmann, H. E. Beere, and D. A. Ritchie “Vertically emitting microdisk lasers,” Nat. Photonics 3, 46–49 (2009).
[Crossref]

Callebaut, H.

Capasso, F.

Cockburn, J. W.

Davies, A. G.

Demichel, O.

Faist, J.

L. Mahler, A. Tredicucci, F. Beltram, C. Walther, J. Faist, B. Witzigmann, H. E. Beere, and D. A. Ritchie “Vertically emitting microdisk lasers,” Nat. Photonics 3, 46–49 (2009).
[Crossref]

M. I. Amanti, M. Fischer, C. Walther, G. Scalari, and J. Faist, “Horn antennas for terahertz quantum cascade lasers,” Electron. Lett. 43, 573–574, (2007).
[Crossref]

Fan, J. A.

Filloux, P.

Fischer, M.

M. I. Amanti, M. Fischer, C. Walther, G. Scalari, and J. Faist, “Horn antennas for terahertz quantum cascade lasers,” Electron. Lett. 43, 573–574, (2007).
[Crossref]

Gellie, P.

Giles Davies, A.

Golka, S.

Green, R.

Green, R. P.

Hormoz, S.

Hu, Q.

A. W. M. Lee, Q. Qin, S. Kumar, B. S. Williams, and Q. Hu “Real-time terahertz imaging over a standoff distance (>25 meters),” Appl. Phys. Lett. 89, 141125 (2006).
[Crossref]

Hübers, H.-W.

Iotti, R. C.

Khanna, S.

Khanna, S. P.

Köhler, R.

Krysa, A. B.

Kumar, S.

S. Kumar and A. W. M. Lee “Resonant-phonon terahertz quantum-cascade lasers and video-rate terahertz imaging,” IEEE J. Sel. Top. Quantum Electron. 14, 333–344 (2008).
[Crossref]

A. W. M. Lee, Q. Qin, S. Kumar, B. S. Williams, and Q. Hu “Real-time terahertz imaging over a standoff distance (>25 meters),” Appl. Phys. Lett. 89, 141125 (2006).
[Crossref]

Lachab, M.

Lampin, J.-F.

Lee, A. W.

Lee, A. W. M.

S. Kumar and A. W. M. Lee “Resonant-phonon terahertz quantum-cascade lasers and video-rate terahertz imaging,” IEEE J. Sel. Top. Quantum Electron. 14, 333–344 (2008).
[Crossref]

A. W. M. Lee, Q. Qin, S. Kumar, B. S. Williams, and Q. Hu “Real-time terahertz imaging over a standoff distance (>25 meters),” Appl. Phys. Lett. 89, 141125 (2006).
[Crossref]

Leo, G.

Linfield, E.

Linfield, E. H.

Linfield, Edmund H.

Long, L. L.

M. A. Ordal, R. J. Bell, R. W. Alexander, L. L. Long, and M. R. Querry, “Optical properties of Au, Ni, and Pb at submillimeter wavelengths,” Appl. Opt. 26, 744 (1987).
[Crossref] [PubMed]

Losco, T.

Mahler, L.

L. Mahler, A. Tredicucci, F. Beltram, C. Walther, J. Faist, B. Witzigmann, H. E. Beere, and D. A. Ritchie “Vertically emitting microdisk lasers,” Nat. Photonics 3, 46–49 (2009).
[Crossref]

Maineult, W.

Mauro, C.

Ordal, M. A.

M. A. Ordal, R. J. Bell, R. W. Alexander, L. L. Long, and M. R. Querry, “Optical properties of Au, Ni, and Pb at submillimeter wavelengths,” Appl. Opt. 26, 744 (1987).
[Crossref] [PubMed]

Pavlov, S.

Pavlov, S. G.

Peytavit, E.

Pflügl, C.

Qin, Q.

A. W. M. Lee, Q. Qin, S. Kumar, B. S. Williams, and Q. Hu “Real-time terahertz imaging over a standoff distance (>25 meters),” Appl. Phys. Lett. 89, 141125 (2006).
[Crossref]

Querry, M. R.

M. A. Ordal, R. J. Bell, R. W. Alexander, L. L. Long, and M. R. Querry, “Optical properties of Au, Ni, and Pb at submillimeter wavelengths,” Appl. Opt. 26, 744 (1987).
[Crossref] [PubMed]

Rana, F.

M. Schubert and F. Rana, “Analysis of Terahertz Surface Emitting Quantum-Cascade Lasers,” IEEE J. Quantum Electron. 42, 257 (2006).
[Crossref]

Reno, J.

Reno, J. L.

Richter, H.

Ritchie, D.

Ritchie, D. A.

L. Mahler, A. Tredicucci, F. Beltram, C. Walther, J. Faist, B. Witzigmann, H. E. Beere, and D. A. Ritchie “Vertically emitting microdisk lasers,” Nat. Photonics 3, 46–49 (2009).
[Crossref]

Roberts, J. S.

Rossi, F.

Scalari, G.

M. I. Amanti, M. Fischer, C. Walther, G. Scalari, and J. Faist, “Horn antennas for terahertz quantum cascade lasers,” Electron. Lett. 43, 573–574, (2007).
[Crossref]

Schrenk, W.

Schubert, M.

M. Schubert and F. Rana, “Analysis of Terahertz Surface Emitting Quantum-Cascade Lasers,” IEEE J. Quantum Electron. 42, 257 (2006).
[Crossref]

Semenov, A.

Semenov, A. D.

Sirtori, C.

Strasser, G.

Tamošinuas, V.

Tredicucci, A.

L. Mahler, A. Tredicucci, F. Beltram, C. Walther, J. Faist, B. Witzigmann, H. E. Beere, and D. A. Ritchie “Vertically emitting microdisk lasers,” Nat. Photonics 3, 46–49 (2009).
[Crossref]

Walther, C.

L. Mahler, A. Tredicucci, F. Beltram, C. Walther, J. Faist, B. Witzigmann, H. E. Beere, and D. A. Ritchie “Vertically emitting microdisk lasers,” Nat. Photonics 3, 46–49 (2009).
[Crossref]

M. I. Amanti, M. Fischer, C. Walther, G. Scalari, and J. Faist, “Horn antennas for terahertz quantum cascade lasers,” Electron. Lett. 43, 573–574, (2007).
[Crossref]

Williams, B. S.

B. S. Williams, “Terahertz quantum-cascade lasers,” Nature Photonics 1, 517–525 (2007).
[Crossref]

A. W. M. Lee, Q. Qin, S. Kumar, B. S. Williams, and Q. Hu “Real-time terahertz imaging over a standoff distance (>25 meters),” Appl. Phys. Lett. 89, 141125 (2006).
[Crossref]

Wilson, L. R.

Witzigmann, B.

L. Mahler, A. Tredicucci, F. Beltram, C. Walther, J. Faist, B. Witzigmann, H. E. Beere, and D. A. Ritchie “Vertically emitting microdisk lasers,” Nat. Photonics 3, 46–49 (2009).
[Crossref]

Xu, J.

Appl. Opt (1)

M. A. Ordal, R. J. Bell, R. W. Alexander, L. L. Long, and M. R. Querry, “Optical properties of Au, Ni, and Pb at submillimeter wavelengths,” Appl. Opt. 26, 744 (1987).
[Crossref] [PubMed]

Appl. Phys. Lett (7)

A. W. M. Lee, Q. Qin, S. Kumar, B. S. Williams, and Q. Hu “Real-time terahertz imaging over a standoff distance (>25 meters),” Appl. Phys. Lett. 89, 141125 (2006).
[Crossref]

Electron. Lett (1)

M. I. Amanti, M. Fischer, C. Walther, G. Scalari, and J. Faist, “Horn antennas for terahertz quantum cascade lasers,” Electron. Lett. 43, 573–574, (2007).
[Crossref]

IEEE J. Quantum Electron (1)

M. Schubert and F. Rana, “Analysis of Terahertz Surface Emitting Quantum-Cascade Lasers,” IEEE J. Quantum Electron. 42, 257 (2006).
[Crossref]

IEEE J. Sel. Top. Quantum Electron (1)

S. Kumar and A. W. M. Lee “Resonant-phonon terahertz quantum-cascade lasers and video-rate terahertz imaging,” IEEE J. Sel. Top. Quantum Electron. 14, 333–344 (2008).
[Crossref]

Nat. Photonics (1)

L. Mahler, A. Tredicucci, F. Beltram, C. Walther, J. Faist, B. Witzigmann, H. E. Beere, and D. A. Ritchie “Vertically emitting microdisk lasers,” Nat. Photonics 3, 46–49 (2009).
[Crossref]

Nature (1)

Nature Photonics (1)

B. S. Williams, “Terahertz quantum-cascade lasers,” Nature Photonics 1, 517–525 (2007).
[Crossref]

Opt. Express (6)

Opt. Lett (1)

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Fig. 1. Three dimensional modeling of the laser ridge. Only one unit cell of the grating is considered, and a top view of the electric field distribution of the eigenmodes is plotted. The propagation direction is indicated by the arrow. a), b) Lowest loss modes of a design taken from [14], c),d) lowest loss modes of the design in this work.

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Fig. 2. (a). Top view of a fabricated device, the dark grey at the end of the grating metallization are the Cr pads that act as end absorbers. (b). Enlarged view of the grating, the slits are kept narrow, in order to maintain uniform pumping, and extend over a large part of the waveguide to avoid anti-guiding effects. One period is 27.3 μm.

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Fig. 3. (a). Computed (columns) and measured (red line) spectra of the device with a period of 27.3 μm and a grating duty-cycle of 90 % , depicted in Fig. 2. The computed spectrum is obtained by a two dimensional eigenfrequency analysis. The height of the green bars indicates the radiative efficiency and that of the blue ones the value of the quality factor. The right panel shows the magnetic field of the computed eigenmode corresponding to the lasing resonance in panel a), at the center of the resonator (b), and at the end of the resonator (c). The small shift in the mode with respect to the grating at the end leads to the surface emission.

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Fig. 4. (a). Vertically emitted peak power vs. current of a device operated with 800 ns pulses at 50 kHz repetition rate. No collection optics is employed, and the values are not corrected for the cryostat window. For the current density, the device area was assumed to be 200×1800 μm. (extension of the metallization). The inset shows the measured spectrum on a logarithmic scale. A side-mode suppression ratio of more than 20 dB is observed. (b). Far-field pattern of the device. The blue and red line show computed and measured far-field, respectively. The angle is measured with respect to the surface normal of the device, in the plane formed by surface normal and waveguide direction.

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Fig. 5. (a). Comparison between a standard and a modified grating, each with a period of 30 μm and a grating duty-cycle of 80 %. Left and right panel show a standard and a modified grating respectively. By introducing two small phase-shifts, the surface losses can be increased by a factor of 5, while maintaining a favorable spectrum. Resonances above the photonic band-gap are not plotted, as they have low quality factors. (b). Computed far-field of the highest quality factor mode of the modified grating.

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Equations (1)

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\nabla \cdot (\frac{1}{n^{2} (x, y)} \nabla H_{z}) = \frac{ω^{2}}{c^{2}} H_{z}