Abstract

The lasing spectra and thresholds of a selectively pumped photonic molecule composed of two microdisks is investigated using effective index approximation and full-wave 2-D electromagnetic equations. The lasing eigenvalue problem formulation is used to find modal frequencies and threshold values of material gain. The influence of the optical coupling between active and passive microdisks on the lasing eigenvalues and directionalities of emission is studied. It is shown that for strong coupling the effect of making one of the resonators passive leads to the doubling of the threshold.

© 2013 Optical Society of America

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  1. P. W. Evans and N. Holonyak, Appl. Phys. Lett. 69, 2391 (1996).
    [CrossRef]
  2. S. C. Creagh and M. D. Finn, J. Phys. A 34, 3791 (2001).
    [CrossRef]
  3. S. Ishii, A. Nakagawa, and T. Baba, IEEE J. Sel. Top. Quantum Electron. 12, 71 (2006).
    [CrossRef]
  4. E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, IEEE J. Sel. Top. Quantum Electron. 12, 78 (2006).
    [CrossRef]
  5. E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, Opt. Lett. 31, 921 (2006).
    [CrossRef]
  6. J.-W. Ryu, S.-Y. Lee, C.-M. Kim, and Y.-J. Park, Phys. Rev. A 74, 013804 (2006).
    [CrossRef]
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    [CrossRef]
  8. S. Preu, H. G. L. Schwefel, S. Malzer, G. H. Dohler, L. J. Wang, M. Hanson, J. D. Zimmerman, and A. C. Gossard, Opt. Express 16, 7336 (2008).
    [CrossRef]
  9. X. Li, R. C. Myers, F. M. Mendoza, D. D. Awschalom, and N. Samarth, IEEE J. Quantum Electron. 45, 932 (2009).
    [CrossRef]
  10. C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, T. Pertsch, V. Shuvayev, and L. I. Deych, Phys. Rev. A 80, 043841 (2009).
    [CrossRef]
  11. M. Benyoucef, J.-B. Shim, J. Wiersig, and O. G. Schmidt, Opt. Lett. 36, 1317 (2011).
    [CrossRef]
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    [CrossRef]
  13. A. I. Nosich, E. I. Smotrova, S. V. Boriskina, T. M. Benson, and P. Sewell, Opt. Quantum Electron. 39, 1253 (2007).
    [CrossRef]
  14. E. I. Smotrova, V. O. Byelobrov, T. M. Benson, J. Ctyroky, R. Sauleau, and A. I. Nosich, IEEE J. Quantum Electron. 47, 20 (2011).
    [CrossRef]

2012

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Kasebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tunnermann, and T. Pertsch, Phys. Rev. A 85, 033827 (2012).
[CrossRef]

2011

E. I. Smotrova, V. O. Byelobrov, T. M. Benson, J. Ctyroky, R. Sauleau, and A. I. Nosich, IEEE J. Quantum Electron. 47, 20 (2011).
[CrossRef]

M. Benyoucef, J.-B. Shim, J. Wiersig, and O. G. Schmidt, Opt. Lett. 36, 1317 (2011).
[CrossRef]

2009

X. Li, R. C. Myers, F. M. Mendoza, D. D. Awschalom, and N. Samarth, IEEE J. Quantum Electron. 45, 932 (2009).
[CrossRef]

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, T. Pertsch, V. Shuvayev, and L. I. Deych, Phys. Rev. A 80, 043841 (2009).
[CrossRef]

2008

2007

S. V. Boriskina, Opt. Lett. 32, 1557 (2007).
[CrossRef]

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, T. M. Benson, and P. Sewell, Opt. Quantum Electron. 39, 1253 (2007).
[CrossRef]

2006

S. Ishii, A. Nakagawa, and T. Baba, IEEE J. Sel. Top. Quantum Electron. 12, 71 (2006).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, IEEE J. Sel. Top. Quantum Electron. 12, 78 (2006).
[CrossRef]

J.-W. Ryu, S.-Y. Lee, C.-M. Kim, and Y.-J. Park, Phys. Rev. A 74, 013804 (2006).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, Opt. Lett. 31, 921 (2006).
[CrossRef]

2001

S. C. Creagh and M. D. Finn, J. Phys. A 34, 3791 (2001).
[CrossRef]

1996

P. W. Evans and N. Holonyak, Appl. Phys. Lett. 69, 2391 (1996).
[CrossRef]

Awschalom, D. D.

X. Li, R. C. Myers, F. M. Mendoza, D. D. Awschalom, and N. Samarth, IEEE J. Quantum Electron. 45, 932 (2009).
[CrossRef]

Baba, T.

S. Ishii, A. Nakagawa, and T. Baba, IEEE J. Sel. Top. Quantum Electron. 12, 71 (2006).
[CrossRef]

Benson, T. M.

E. I. Smotrova, V. O. Byelobrov, T. M. Benson, J. Ctyroky, R. Sauleau, and A. I. Nosich, IEEE J. Quantum Electron. 47, 20 (2011).
[CrossRef]

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, T. M. Benson, and P. Sewell, Opt. Quantum Electron. 39, 1253 (2007).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, IEEE J. Sel. Top. Quantum Electron. 12, 78 (2006).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, Opt. Lett. 31, 921 (2006).
[CrossRef]

Benyoucef, M.

Boriskina, S. V.

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, T. M. Benson, and P. Sewell, Opt. Quantum Electron. 39, 1253 (2007).
[CrossRef]

S. V. Boriskina, Opt. Lett. 32, 1557 (2007).
[CrossRef]

Byelobrov, V. O.

E. I. Smotrova, V. O. Byelobrov, T. M. Benson, J. Ctyroky, R. Sauleau, and A. I. Nosich, IEEE J. Quantum Electron. 47, 20 (2011).
[CrossRef]

Chipouline, A.

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Kasebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tunnermann, and T. Pertsch, Phys. Rev. A 85, 033827 (2012).
[CrossRef]

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, T. Pertsch, V. Shuvayev, and L. I. Deych, Phys. Rev. A 80, 043841 (2009).
[CrossRef]

Creagh, S. C.

S. C. Creagh and M. D. Finn, J. Phys. A 34, 3791 (2001).
[CrossRef]

Ctyroky, J.

E. I. Smotrova, V. O. Byelobrov, T. M. Benson, J. Ctyroky, R. Sauleau, and A. I. Nosich, IEEE J. Quantum Electron. 47, 20 (2011).
[CrossRef]

Deych, L. I.

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, T. Pertsch, V. Shuvayev, and L. I. Deych, Phys. Rev. A 80, 043841 (2009).
[CrossRef]

Dohler, G. H.

Etrich, C.

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Kasebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tunnermann, and T. Pertsch, Phys. Rev. A 85, 033827 (2012).
[CrossRef]

Evans, P. W.

P. W. Evans and N. Holonyak, Appl. Phys. Lett. 69, 2391 (1996).
[CrossRef]

Finn, M. D.

S. C. Creagh and M. D. Finn, J. Phys. A 34, 3791 (2001).
[CrossRef]

Gossard, A. C.

Hanson, M.

Holonyak, N.

P. W. Evans and N. Holonyak, Appl. Phys. Lett. 69, 2391 (1996).
[CrossRef]

Ishii, S.

S. Ishii, A. Nakagawa, and T. Baba, IEEE J. Sel. Top. Quantum Electron. 12, 71 (2006).
[CrossRef]

Janunts, N.

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Kasebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tunnermann, and T. Pertsch, Phys. Rev. A 85, 033827 (2012).
[CrossRef]

Kasebier, T.

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Kasebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tunnermann, and T. Pertsch, Phys. Rev. A 85, 033827 (2012).
[CrossRef]

Käsebier, T.

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, T. Pertsch, V. Shuvayev, and L. I. Deych, Phys. Rev. A 80, 043841 (2009).
[CrossRef]

Kim, C.-M.

J.-W. Ryu, S.-Y. Lee, C.-M. Kim, and Y.-J. Park, Phys. Rev. A 74, 013804 (2006).
[CrossRef]

Klein, A.

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Kasebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tunnermann, and T. Pertsch, Phys. Rev. A 85, 033827 (2012).
[CrossRef]

Kley, E.-B.

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Kasebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tunnermann, and T. Pertsch, Phys. Rev. A 85, 033827 (2012).
[CrossRef]

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, T. Pertsch, V. Shuvayev, and L. I. Deych, Phys. Rev. A 80, 043841 (2009).
[CrossRef]

Lederer, F.

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Kasebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tunnermann, and T. Pertsch, Phys. Rev. A 85, 033827 (2012).
[CrossRef]

Lee, S.-Y.

J.-W. Ryu, S.-Y. Lee, C.-M. Kim, and Y.-J. Park, Phys. Rev. A 74, 013804 (2006).
[CrossRef]

Li, X.

X. Li, R. C. Myers, F. M. Mendoza, D. D. Awschalom, and N. Samarth, IEEE J. Quantum Electron. 45, 932 (2009).
[CrossRef]

Liebsch, M.

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Kasebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tunnermann, and T. Pertsch, Phys. Rev. A 85, 033827 (2012).
[CrossRef]

Malzer, S.

Mendoza, F. M.

X. Li, R. C. Myers, F. M. Mendoza, D. D. Awschalom, and N. Samarth, IEEE J. Quantum Electron. 45, 932 (2009).
[CrossRef]

Myers, R. C.

X. Li, R. C. Myers, F. M. Mendoza, D. D. Awschalom, and N. Samarth, IEEE J. Quantum Electron. 45, 932 (2009).
[CrossRef]

Nakagawa, A.

S. Ishii, A. Nakagawa, and T. Baba, IEEE J. Sel. Top. Quantum Electron. 12, 71 (2006).
[CrossRef]

Nosich, A. I.

E. I. Smotrova, V. O. Byelobrov, T. M. Benson, J. Ctyroky, R. Sauleau, and A. I. Nosich, IEEE J. Quantum Electron. 47, 20 (2011).
[CrossRef]

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, T. M. Benson, and P. Sewell, Opt. Quantum Electron. 39, 1253 (2007).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, Opt. Lett. 31, 921 (2006).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, IEEE J. Sel. Top. Quantum Electron. 12, 78 (2006).
[CrossRef]

Park, Y.-J.

J.-W. Ryu, S.-Y. Lee, C.-M. Kim, and Y.-J. Park, Phys. Rev. A 74, 013804 (2006).
[CrossRef]

Pertsch, T.

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Kasebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tunnermann, and T. Pertsch, Phys. Rev. A 85, 033827 (2012).
[CrossRef]

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, T. Pertsch, V. Shuvayev, and L. I. Deych, Phys. Rev. A 80, 043841 (2009).
[CrossRef]

Preu, S.

Ryu, J.-W.

J.-W. Ryu, S.-Y. Lee, C.-M. Kim, and Y.-J. Park, Phys. Rev. A 74, 013804 (2006).
[CrossRef]

Samarth, N.

X. Li, R. C. Myers, F. M. Mendoza, D. D. Awschalom, and N. Samarth, IEEE J. Quantum Electron. 45, 932 (2009).
[CrossRef]

Sauleau, R.

E. I. Smotrova, V. O. Byelobrov, T. M. Benson, J. Ctyroky, R. Sauleau, and A. I. Nosich, IEEE J. Quantum Electron. 47, 20 (2011).
[CrossRef]

Schmidt, C.

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Kasebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tunnermann, and T. Pertsch, Phys. Rev. A 85, 033827 (2012).
[CrossRef]

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, T. Pertsch, V. Shuvayev, and L. I. Deych, Phys. Rev. A 80, 043841 (2009).
[CrossRef]

Schmidt, O. G.

Schwefel, H. G. L.

Sewell, P.

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, T. M. Benson, and P. Sewell, Opt. Quantum Electron. 39, 1253 (2007).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, IEEE J. Sel. Top. Quantum Electron. 12, 78 (2006).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, Opt. Lett. 31, 921 (2006).
[CrossRef]

Shim, J.-B.

Shuvayev, V.

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, T. Pertsch, V. Shuvayev, and L. I. Deych, Phys. Rev. A 80, 043841 (2009).
[CrossRef]

Smotrova, E. I.

E. I. Smotrova, V. O. Byelobrov, T. M. Benson, J. Ctyroky, R. Sauleau, and A. I. Nosich, IEEE J. Quantum Electron. 47, 20 (2011).
[CrossRef]

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, T. M. Benson, and P. Sewell, Opt. Quantum Electron. 39, 1253 (2007).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, Opt. Lett. 31, 921 (2006).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, IEEE J. Sel. Top. Quantum Electron. 12, 78 (2006).
[CrossRef]

Tunnermann, A.

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Kasebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tunnermann, and T. Pertsch, Phys. Rev. A 85, 033827 (2012).
[CrossRef]

Tünnermann, A.

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, T. Pertsch, V. Shuvayev, and L. I. Deych, Phys. Rev. A 80, 043841 (2009).
[CrossRef]

Wang, L. J.

Wiersig, J.

Zimmerman, J. D.

Appl. Phys. Lett.

P. W. Evans and N. Holonyak, Appl. Phys. Lett. 69, 2391 (1996).
[CrossRef]

IEEE J. Quantum Electron.

X. Li, R. C. Myers, F. M. Mendoza, D. D. Awschalom, and N. Samarth, IEEE J. Quantum Electron. 45, 932 (2009).
[CrossRef]

E. I. Smotrova, V. O. Byelobrov, T. M. Benson, J. Ctyroky, R. Sauleau, and A. I. Nosich, IEEE J. Quantum Electron. 47, 20 (2011).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

S. Ishii, A. Nakagawa, and T. Baba, IEEE J. Sel. Top. Quantum Electron. 12, 71 (2006).
[CrossRef]

E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, IEEE J. Sel. Top. Quantum Electron. 12, 78 (2006).
[CrossRef]

J. Phys. A

S. C. Creagh and M. D. Finn, J. Phys. A 34, 3791 (2001).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Quantum Electron.

A. I. Nosich, E. I. Smotrova, S. V. Boriskina, T. M. Benson, and P. Sewell, Opt. Quantum Electron. 39, 1253 (2007).
[CrossRef]

Phys. Rev. A

C. Schmidt, A. Chipouline, T. Käsebier, E.-B. Kley, A. Tünnermann, T. Pertsch, V. Shuvayev, and L. I. Deych, Phys. Rev. A 80, 043841 (2009).
[CrossRef]

C. Schmidt, M. Liebsch, A. Klein, N. Janunts, A. Chipouline, T. Kasebier, C. Etrich, F. Lederer, E.-B. Kley, A. Tunnermann, and T. Pertsch, Phys. Rev. A 85, 033827 (2012).
[CrossRef]

J.-W. Ryu, S.-Y. Lee, C.-M. Kim, and Y.-J. Park, Phys. Rev. A 74, 013804 (2006).
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

In-plane geometry of two optically coupled circular disk microresonators, active (right) and passive (left).

Fig. 2.
Fig. 2.

Threshold values of material gain and the normalized lasing frequencies (inset) for the Hz-polarized supermodes of the family (Hz)7,1 versus the relative distance, w/a, between microdisks; α=2.63 and N=70.

Fig. 3.
Fig. 3.

Fading factors F for the Hz-polarized supermodes built on the WG modes (Hz)7,1 versus the relative distance, w/a, between microdisks; α=2.63 and N=70.

Fig. 4.
Fig. 4.

Near- and far-field patterns |Hz| for the Hz-polarized supermodes of the OqE (a) and EqO (b) types, built on the WG modes (Hz)7,1 for widely spaced microdisks, w/a=1.75.

Fig. 5.
Fig. 5.

Directivities for the Hz-polarized supermodes built on the WG modes (Hz)7,1 versus the relative distance, w/a, between microdisks; α=2.63 and N=70.

Fig. 6.
Fig. 6.

Near- and far-field patterns |Hz| for the Hz-polarized supermodes of the OqE (a) EqO (b) types, built on the WG modes (Hz)7,1 for closely spaced microdisks, w/a=0.57.

Equations (7)

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

U1,2(ρ,φ)=p=(0)1Ap1,2Jp(kν1,2ρ1,2)Sp(ϕ1,2),ρ1,2,<a,
U(ρ,φ)=p=0(1)Bp1Hp(1)(kρ1)Sp(φ1)+p=0(1)Bp2Hp(1)(kρ2)Sp(φ2),
Fm(κ,νj)=Jm(κνj)Hm(1)(κ)νjβE,HJm(κνj)Hm(1)(κ)
xm±+p=0(1)μpyp±Kmp(κ,ν2)[Hmp(1)(κl)±(1)pHm+p(1)(κl)]=0,
ym±+p=0(1)μpxp±Kmp(κ,ν1)[Hmp(1)(κl)±(1)pHm+p(1)(κl)]=0,
Kmp(κ,νj)=Jm(κ)Vp(κ,νj)[Fp(κ,νj)Jp(κ)]1,
Vm(κ,νj)=Jm(κνj)Jm(κ)νjβjE,HJm(κνj)Jm(κ),j=1,2,

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