Abstract

The availability of microlasers with highly directional far-field characteristics is crucial for future applications. To this end we study the far-field emission of active microcavities with a spiral shape using the Schrödinger–Bloch model. We find that they can provide directional emission under the conditions of (i) pumping along the resonator boundary and (ii) for specific resonator geometries. We systematically study the far-field characteristics under variation of the pumped area and the cavity geometry and identify a directionality optimized regime. Our results consistently explain previously obtained experimental results.

© 2009 Optical Society of America

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  1. S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, Appl. Phys. Lett. 60, 289 (1992).
    [CrossRef]
  2. R.K.Chang and A.J.Campillo, eds., Optical Processes in Microcavities (World Scientific, 1996).
    [CrossRef]
  3. K.J.Vahala, ed., Optical Microcavities (World Scientific, 2004).
    [CrossRef]
  4. G. D. Chern, H. E. Tureci, A. Douglas Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, Appl. Phys. Lett. 83, 1710 (2003).
    [CrossRef]
  5. M. Kneissl, M. Teepe, N. Miyashita, N. M. Johnson, G. D. Chern, and R. K. Chang, Appl. Phys. Lett. 84, 2485 (2004).
    [CrossRef]
  6. T. Ben-Messaoud and J. Zyss, Appl. Phys. Lett. 86, 241110 (2005).
    [CrossRef]
  7. A. Fujii, T. Nishimura, Y. Yoshida, K. Yoshino, and M. Ozaki, Jpn. J. Appl. Phys. Part 2 44, L1091 (2005).
    [CrossRef]
  8. Ch.-M. Kim, J. Cho, J. Lee, S. Rim, S. H. Lee, K. R. Oh, and J. H. Kim, Appl. Phys. Lett. 92, 131110 (2008).
    [CrossRef]
  9. R. Audet, M. A. Belkin, J. A. Fan, B. G. Lee, K. Lin, F. Capasso, E. E. Narimanov, D. Bour, S. Corzine, J. Zhu, and G. Höfler, Appl. Phys. Lett. 91, 131106 (2007).
    [CrossRef]
  10. M. Hentschel, T.-Y. Kwon, M. A. Belkin, R. Audet, and F. Capasso, “Characteristics of quantum cascade spiral microlasers: a consistent picture in terms of rays, waves, and experiments,” (submitted to Opt. Express).
  11. C. Gmachl, E. E. Narimanov, F. Capasso, J. N. Baillargeon, and A. Y. Cho, Opt. Lett. 27, 824 (2002).
    [CrossRef]
  12. J. Wiersig and M. Hentschel, Phys. Rev. Lett. 100, 033901 (2008).
    [CrossRef] [PubMed]
  13. T. Harayama, S. Sunada, and K. Ikeda, Phys. Rev. A 72, 013803 (2005).
    [CrossRef]
  14. S. Sunada, T. Harayama, and K. Ikeda, Phys. Rev. E 71, 046209 (2005).
    [CrossRef]
  15. S.-Y. Lee, S. Rim, J.-W. Ryu, T.-Y. Kwon, M. Choi, and Ch.-M. Kim, Phys. Rev. Lett. 93, 164102 (2004).
    [CrossRef] [PubMed]

2008 (2)

J. Wiersig and M. Hentschel, Phys. Rev. Lett. 100, 033901 (2008).
[CrossRef] [PubMed]

Ch.-M. Kim, J. Cho, J. Lee, S. Rim, S. H. Lee, K. R. Oh, and J. H. Kim, Appl. Phys. Lett. 92, 131110 (2008).
[CrossRef]

2007 (1)

R. Audet, M. A. Belkin, J. A. Fan, B. G. Lee, K. Lin, F. Capasso, E. E. Narimanov, D. Bour, S. Corzine, J. Zhu, and G. Höfler, Appl. Phys. Lett. 91, 131106 (2007).
[CrossRef]

2005 (4)

T. Harayama, S. Sunada, and K. Ikeda, Phys. Rev. A 72, 013803 (2005).
[CrossRef]

S. Sunada, T. Harayama, and K. Ikeda, Phys. Rev. E 71, 046209 (2005).
[CrossRef]

T. Ben-Messaoud and J. Zyss, Appl. Phys. Lett. 86, 241110 (2005).
[CrossRef]

A. Fujii, T. Nishimura, Y. Yoshida, K. Yoshino, and M. Ozaki, Jpn. J. Appl. Phys. Part 2 44, L1091 (2005).
[CrossRef]

2004 (2)

S.-Y. Lee, S. Rim, J.-W. Ryu, T.-Y. Kwon, M. Choi, and Ch.-M. Kim, Phys. Rev. Lett. 93, 164102 (2004).
[CrossRef] [PubMed]

M. Kneissl, M. Teepe, N. Miyashita, N. M. Johnson, G. D. Chern, and R. K. Chang, Appl. Phys. Lett. 84, 2485 (2004).
[CrossRef]

2003 (1)

G. D. Chern, H. E. Tureci, A. Douglas Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, Appl. Phys. Lett. 83, 1710 (2003).
[CrossRef]

2002 (1)

1992 (1)

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, Appl. Phys. Lett. 60, 289 (1992).
[CrossRef]

Audet, R.

R. Audet, M. A. Belkin, J. A. Fan, B. G. Lee, K. Lin, F. Capasso, E. E. Narimanov, D. Bour, S. Corzine, J. Zhu, and G. Höfler, Appl. Phys. Lett. 91, 131106 (2007).
[CrossRef]

M. Hentschel, T.-Y. Kwon, M. A. Belkin, R. Audet, and F. Capasso, “Characteristics of quantum cascade spiral microlasers: a consistent picture in terms of rays, waves, and experiments,” (submitted to Opt. Express).

Baillargeon, J. N.

Belkin, M. A.

R. Audet, M. A. Belkin, J. A. Fan, B. G. Lee, K. Lin, F. Capasso, E. E. Narimanov, D. Bour, S. Corzine, J. Zhu, and G. Höfler, Appl. Phys. Lett. 91, 131106 (2007).
[CrossRef]

M. Hentschel, T.-Y. Kwon, M. A. Belkin, R. Audet, and F. Capasso, “Characteristics of quantum cascade spiral microlasers: a consistent picture in terms of rays, waves, and experiments,” (submitted to Opt. Express).

Ben-Messaoud, T.

T. Ben-Messaoud and J. Zyss, Appl. Phys. Lett. 86, 241110 (2005).
[CrossRef]

Bour, D.

R. Audet, M. A. Belkin, J. A. Fan, B. G. Lee, K. Lin, F. Capasso, E. E. Narimanov, D. Bour, S. Corzine, J. Zhu, and G. Höfler, Appl. Phys. Lett. 91, 131106 (2007).
[CrossRef]

Capasso, F.

R. Audet, M. A. Belkin, J. A. Fan, B. G. Lee, K. Lin, F. Capasso, E. E. Narimanov, D. Bour, S. Corzine, J. Zhu, and G. Höfler, Appl. Phys. Lett. 91, 131106 (2007).
[CrossRef]

C. Gmachl, E. E. Narimanov, F. Capasso, J. N. Baillargeon, and A. Y. Cho, Opt. Lett. 27, 824 (2002).
[CrossRef]

M. Hentschel, T.-Y. Kwon, M. A. Belkin, R. Audet, and F. Capasso, “Characteristics of quantum cascade spiral microlasers: a consistent picture in terms of rays, waves, and experiments,” (submitted to Opt. Express).

Chang, R. K.

M. Kneissl, M. Teepe, N. Miyashita, N. M. Johnson, G. D. Chern, and R. K. Chang, Appl. Phys. Lett. 84, 2485 (2004).
[CrossRef]

G. D. Chern, H. E. Tureci, A. Douglas Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, Appl. Phys. Lett. 83, 1710 (2003).
[CrossRef]

Chern, G. D.

M. Kneissl, M. Teepe, N. Miyashita, N. M. Johnson, G. D. Chern, and R. K. Chang, Appl. Phys. Lett. 84, 2485 (2004).
[CrossRef]

G. D. Chern, H. E. Tureci, A. Douglas Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, Appl. Phys. Lett. 83, 1710 (2003).
[CrossRef]

Cho, A. Y.

Cho, J.

Ch.-M. Kim, J. Cho, J. Lee, S. Rim, S. H. Lee, K. R. Oh, and J. H. Kim, Appl. Phys. Lett. 92, 131110 (2008).
[CrossRef]

Choi, M.

S.-Y. Lee, S. Rim, J.-W. Ryu, T.-Y. Kwon, M. Choi, and Ch.-M. Kim, Phys. Rev. Lett. 93, 164102 (2004).
[CrossRef] [PubMed]

Corzine, S.

R. Audet, M. A. Belkin, J. A. Fan, B. G. Lee, K. Lin, F. Capasso, E. E. Narimanov, D. Bour, S. Corzine, J. Zhu, and G. Höfler, Appl. Phys. Lett. 91, 131106 (2007).
[CrossRef]

Douglas Stone, A.

G. D. Chern, H. E. Tureci, A. Douglas Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, Appl. Phys. Lett. 83, 1710 (2003).
[CrossRef]

Fan, J. A.

R. Audet, M. A. Belkin, J. A. Fan, B. G. Lee, K. Lin, F. Capasso, E. E. Narimanov, D. Bour, S. Corzine, J. Zhu, and G. Höfler, Appl. Phys. Lett. 91, 131106 (2007).
[CrossRef]

Fujii, A.

A. Fujii, T. Nishimura, Y. Yoshida, K. Yoshino, and M. Ozaki, Jpn. J. Appl. Phys. Part 2 44, L1091 (2005).
[CrossRef]

Gmachl, C.

Harayama, T.

T. Harayama, S. Sunada, and K. Ikeda, Phys. Rev. A 72, 013803 (2005).
[CrossRef]

S. Sunada, T. Harayama, and K. Ikeda, Phys. Rev. E 71, 046209 (2005).
[CrossRef]

Hentschel, M.

J. Wiersig and M. Hentschel, Phys. Rev. Lett. 100, 033901 (2008).
[CrossRef] [PubMed]

M. Hentschel, T.-Y. Kwon, M. A. Belkin, R. Audet, and F. Capasso, “Characteristics of quantum cascade spiral microlasers: a consistent picture in terms of rays, waves, and experiments,” (submitted to Opt. Express).

Höfler, G.

R. Audet, M. A. Belkin, J. A. Fan, B. G. Lee, K. Lin, F. Capasso, E. E. Narimanov, D. Bour, S. Corzine, J. Zhu, and G. Höfler, Appl. Phys. Lett. 91, 131106 (2007).
[CrossRef]

Ikeda, K.

S. Sunada, T. Harayama, and K. Ikeda, Phys. Rev. E 71, 046209 (2005).
[CrossRef]

T. Harayama, S. Sunada, and K. Ikeda, Phys. Rev. A 72, 013803 (2005).
[CrossRef]

Johnson, N. M.

M. Kneissl, M. Teepe, N. Miyashita, N. M. Johnson, G. D. Chern, and R. K. Chang, Appl. Phys. Lett. 84, 2485 (2004).
[CrossRef]

G. D. Chern, H. E. Tureci, A. Douglas Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, Appl. Phys. Lett. 83, 1710 (2003).
[CrossRef]

Kim, Ch.-M.

Ch.-M. Kim, J. Cho, J. Lee, S. Rim, S. H. Lee, K. R. Oh, and J. H. Kim, Appl. Phys. Lett. 92, 131110 (2008).
[CrossRef]

S.-Y. Lee, S. Rim, J.-W. Ryu, T.-Y. Kwon, M. Choi, and Ch.-M. Kim, Phys. Rev. Lett. 93, 164102 (2004).
[CrossRef] [PubMed]

Kim, J. H.

Ch.-M. Kim, J. Cho, J. Lee, S. Rim, S. H. Lee, K. R. Oh, and J. H. Kim, Appl. Phys. Lett. 92, 131110 (2008).
[CrossRef]

Kneissl, M.

M. Kneissl, M. Teepe, N. Miyashita, N. M. Johnson, G. D. Chern, and R. K. Chang, Appl. Phys. Lett. 84, 2485 (2004).
[CrossRef]

G. D. Chern, H. E. Tureci, A. Douglas Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, Appl. Phys. Lett. 83, 1710 (2003).
[CrossRef]

Kwon, T.-Y.

S.-Y. Lee, S. Rim, J.-W. Ryu, T.-Y. Kwon, M. Choi, and Ch.-M. Kim, Phys. Rev. Lett. 93, 164102 (2004).
[CrossRef] [PubMed]

M. Hentschel, T.-Y. Kwon, M. A. Belkin, R. Audet, and F. Capasso, “Characteristics of quantum cascade spiral microlasers: a consistent picture in terms of rays, waves, and experiments,” (submitted to Opt. Express).

Lee, B. G.

R. Audet, M. A. Belkin, J. A. Fan, B. G. Lee, K. Lin, F. Capasso, E. E. Narimanov, D. Bour, S. Corzine, J. Zhu, and G. Höfler, Appl. Phys. Lett. 91, 131106 (2007).
[CrossRef]

Lee, J.

Ch.-M. Kim, J. Cho, J. Lee, S. Rim, S. H. Lee, K. R. Oh, and J. H. Kim, Appl. Phys. Lett. 92, 131110 (2008).
[CrossRef]

Lee, S. H.

Ch.-M. Kim, J. Cho, J. Lee, S. Rim, S. H. Lee, K. R. Oh, and J. H. Kim, Appl. Phys. Lett. 92, 131110 (2008).
[CrossRef]

Lee, S.-Y.

S.-Y. Lee, S. Rim, J.-W. Ryu, T.-Y. Kwon, M. Choi, and Ch.-M. Kim, Phys. Rev. Lett. 93, 164102 (2004).
[CrossRef] [PubMed]

Levi, A. F. J.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, Appl. Phys. Lett. 60, 289 (1992).
[CrossRef]

Lin, K.

R. Audet, M. A. Belkin, J. A. Fan, B. G. Lee, K. Lin, F. Capasso, E. E. Narimanov, D. Bour, S. Corzine, J. Zhu, and G. Höfler, Appl. Phys. Lett. 91, 131106 (2007).
[CrossRef]

Logan, R. A.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, Appl. Phys. Lett. 60, 289 (1992).
[CrossRef]

McCall, S. L.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, Appl. Phys. Lett. 60, 289 (1992).
[CrossRef]

Miyashita, N.

M. Kneissl, M. Teepe, N. Miyashita, N. M. Johnson, G. D. Chern, and R. K. Chang, Appl. Phys. Lett. 84, 2485 (2004).
[CrossRef]

Narimanov, E. E.

R. Audet, M. A. Belkin, J. A. Fan, B. G. Lee, K. Lin, F. Capasso, E. E. Narimanov, D. Bour, S. Corzine, J. Zhu, and G. Höfler, Appl. Phys. Lett. 91, 131106 (2007).
[CrossRef]

C. Gmachl, E. E. Narimanov, F. Capasso, J. N. Baillargeon, and A. Y. Cho, Opt. Lett. 27, 824 (2002).
[CrossRef]

Nishimura, T.

A. Fujii, T. Nishimura, Y. Yoshida, K. Yoshino, and M. Ozaki, Jpn. J. Appl. Phys. Part 2 44, L1091 (2005).
[CrossRef]

Oh, K. R.

Ch.-M. Kim, J. Cho, J. Lee, S. Rim, S. H. Lee, K. R. Oh, and J. H. Kim, Appl. Phys. Lett. 92, 131110 (2008).
[CrossRef]

Ozaki, M.

A. Fujii, T. Nishimura, Y. Yoshida, K. Yoshino, and M. Ozaki, Jpn. J. Appl. Phys. Part 2 44, L1091 (2005).
[CrossRef]

Pearton, S. J.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, Appl. Phys. Lett. 60, 289 (1992).
[CrossRef]

Rim, S.

Ch.-M. Kim, J. Cho, J. Lee, S. Rim, S. H. Lee, K. R. Oh, and J. H. Kim, Appl. Phys. Lett. 92, 131110 (2008).
[CrossRef]

S.-Y. Lee, S. Rim, J.-W. Ryu, T.-Y. Kwon, M. Choi, and Ch.-M. Kim, Phys. Rev. Lett. 93, 164102 (2004).
[CrossRef] [PubMed]

Ryu, J.-W.

S.-Y. Lee, S. Rim, J.-W. Ryu, T.-Y. Kwon, M. Choi, and Ch.-M. Kim, Phys. Rev. Lett. 93, 164102 (2004).
[CrossRef] [PubMed]

Slusher, R. E.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, Appl. Phys. Lett. 60, 289 (1992).
[CrossRef]

Sunada, S.

S. Sunada, T. Harayama, and K. Ikeda, Phys. Rev. E 71, 046209 (2005).
[CrossRef]

T. Harayama, S. Sunada, and K. Ikeda, Phys. Rev. A 72, 013803 (2005).
[CrossRef]

Teepe, M.

M. Kneissl, M. Teepe, N. Miyashita, N. M. Johnson, G. D. Chern, and R. K. Chang, Appl. Phys. Lett. 84, 2485 (2004).
[CrossRef]

Tureci, H. E.

G. D. Chern, H. E. Tureci, A. Douglas Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, Appl. Phys. Lett. 83, 1710 (2003).
[CrossRef]

Wiersig, J.

J. Wiersig and M. Hentschel, Phys. Rev. Lett. 100, 033901 (2008).
[CrossRef] [PubMed]

Yoshida, Y.

A. Fujii, T. Nishimura, Y. Yoshida, K. Yoshino, and M. Ozaki, Jpn. J. Appl. Phys. Part 2 44, L1091 (2005).
[CrossRef]

Yoshino, K.

A. Fujii, T. Nishimura, Y. Yoshida, K. Yoshino, and M. Ozaki, Jpn. J. Appl. Phys. Part 2 44, L1091 (2005).
[CrossRef]

Zhu, J.

R. Audet, M. A. Belkin, J. A. Fan, B. G. Lee, K. Lin, F. Capasso, E. E. Narimanov, D. Bour, S. Corzine, J. Zhu, and G. Höfler, Appl. Phys. Lett. 91, 131106 (2007).
[CrossRef]

Zyss, J.

T. Ben-Messaoud and J. Zyss, Appl. Phys. Lett. 86, 241110 (2005).
[CrossRef]

Appl. Phys. Lett. (6)

G. D. Chern, H. E. Tureci, A. Douglas Stone, R. K. Chang, M. Kneissl, and N. M. Johnson, Appl. Phys. Lett. 83, 1710 (2003).
[CrossRef]

M. Kneissl, M. Teepe, N. Miyashita, N. M. Johnson, G. D. Chern, and R. K. Chang, Appl. Phys. Lett. 84, 2485 (2004).
[CrossRef]

T. Ben-Messaoud and J. Zyss, Appl. Phys. Lett. 86, 241110 (2005).
[CrossRef]

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, Appl. Phys. Lett. 60, 289 (1992).
[CrossRef]

Ch.-M. Kim, J. Cho, J. Lee, S. Rim, S. H. Lee, K. R. Oh, and J. H. Kim, Appl. Phys. Lett. 92, 131110 (2008).
[CrossRef]

R. Audet, M. A. Belkin, J. A. Fan, B. G. Lee, K. Lin, F. Capasso, E. E. Narimanov, D. Bour, S. Corzine, J. Zhu, and G. Höfler, Appl. Phys. Lett. 91, 131106 (2007).
[CrossRef]

Jpn. J. Appl. Phys. Part 2 (1)

A. Fujii, T. Nishimura, Y. Yoshida, K. Yoshino, and M. Ozaki, Jpn. J. Appl. Phys. Part 2 44, L1091 (2005).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

T. Harayama, S. Sunada, and K. Ikeda, Phys. Rev. A 72, 013803 (2005).
[CrossRef]

Phys. Rev. E (1)

S. Sunada, T. Harayama, and K. Ikeda, Phys. Rev. E 71, 046209 (2005).
[CrossRef]

Phys. Rev. Lett. (2)

S.-Y. Lee, S. Rim, J.-W. Ryu, T.-Y. Kwon, M. Choi, and Ch.-M. Kim, Phys. Rev. Lett. 93, 164102 (2004).
[CrossRef] [PubMed]

J. Wiersig and M. Hentschel, Phys. Rev. Lett. 100, 033901 (2008).
[CrossRef] [PubMed]

Other (3)

M. Hentschel, T.-Y. Kwon, M. A. Belkin, R. Audet, and F. Capasso, “Characteristics of quantum cascade spiral microlasers: a consistent picture in terms of rays, waves, and experiments,” (submitted to Opt. Express).

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

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

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Figures (3)

Fig. 1
Fig. 1

Far-field radiation characteristics for spiral cavities of three different geometries ( ϵ = 0.3 , 0.2 , and 0.1 from top to bottom) and (a)–(c) three different boundary pumping schemes, as well as for (d) uniform pumping. The pumped areas are shown in red (gray) in the insets; the parameters are R R 0 = 0.99 , 0.9, 0.8, and 0 from (a) to (d). The intensity is obtained as the time average after a transient regime when a stable lasing operation is established. It is normalized to one in each panel. The size parameter (gain center) is n k R 0 62.2 ; similar values yield comparable results.

Fig. 2
Fig. 2

Far-field characteristics for boundary-pumped spiral microcavities ( R R 0 = 0.9 ) as in Fig. 1b, but now for geometry parameters ϵ around 0.2. The chaotic dynamics in the spiral billiards results in a strong sensitivity against variations of the geometry, causing shifts in the relative weight of the emission peaks. The directed emission from the notch under an angle of ϕ 45 ° persists for most ϵ near 0.2.

Fig. 3
Fig. 3

(a) Total internal light intensity in the lasing regime as a function of time for near-threshold external pumping strengths of W = 0.0002 (solid curve) and W = 0.0004 (dashed curve; this value is used in all other figures) for R R 0 = 0.9 and ϵ = 0.2 . The oscillations resemble those known from two-mode beating. Oscillations take place between a high-Q cavity state, see snapshot (b), and a low-Q cavity state, see snapshot (c). The directed output from the notch originates from cavity states similar to snapshot (c). The output is pulsed and highest at the circles marked in (a). In snapshots (b) and (c) the intensity is normalized to 1, values larger than 0.3 are shown in the same color to highlight the cavity’s emission properties.

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