Abstract

An exact and compact analytical formalism has been developed to calculate the Q factor for circular Bragg resonators. The electromagnetic fields, energy, and power flow have been expressed analytically relying on the transfer matrix coefficients. The Q factor has been derived for both TM and TE polarizations. The proposed formalism is then compared with two numerical methods.

© 2007 Optical Society of America

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  51. It is worth mentioning that finding a suitable physical measure to probe the spectral response of a leaky cavity accordingly is related to a more general issue than expected; in fact, the FWHM estimation of the cavity resonance essentially depends on the technical setup to access the corresponding field quantity and, hence, to the problem of proper normalization (or appropriate excitation) of the cavity mode. Thus the spectrally determined Q factors become context dependent. Given the normalization of the wave amplitudes in the cavity center, as proposed by condition , the resulting spectral response of the total field energy Wm(λ) yields a physically counterintuitive, but mathematically correct, drop at resonance.
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2006 (1)

2005 (5)

G. A. Turnbull, A. Carleton, G. F. Barlow, A. Tahraoui, T. F. Krauss, A. Shore, and I. D. W. Samuel, "Design and analysis of a low-threshold polymer circular-grating distributed- feedback laser," J. Appl. Phys. 98, 023105 (2005).
[CrossRef]

J. Scheuer, W. M. J. Green, G. A. DeRose, and A. Yariv, "InGaAsP annular Bragg lasers: theory, applications, and modal properties," IEEE J. Sel. Top. Quantum Electron. 11, 476-484(2005).
[CrossRef]

J. Scheuer, W. M. J. Green, G. A. DeRose, and A. Yariv, "Lasing from a circular Bragg nanocavity with an ultrasmall modal volume," Appl. Phys. Lett. 86, 251101 (2005).
[CrossRef]

D. Englund, I. Fushman, and J. Vuckovic, "General recipe for designing photonic crystal cavities," Opt. Express 13, 5961-5975 (2005).
[CrossRef] [PubMed]

S. Gulde, A. Jebali, and N. Moll, "Optimization of ultrafast all-optical resonator switching," Opt. Express 13, 9502-9515 (2005).
[CrossRef] [PubMed]

2004 (2)

A. Jebali, R. F. Mahrt, N. Moll, D. Erni, C. Bauer, E. B. Kley, G. L. Bona, and W. Bächtold, "Lasing in organic circular grating structures," J. Appl. Phys. 96, 3043-3049 (2004).
[CrossRef]

G. F. Barlow, A. Shore, G. A. Turnbu, and I. D. W. Samuel, "Design and analysis of a low-threshold polymer circular-grating distributed-feedback laser," J. Opt. Soc. Am. B 21, 2142-2150 (2004).
[CrossRef]

2003 (6)

A. M. Shams-Zadeh-Amiri, X. Li, and W.-P. Huang, "Hankel transform-domain analysis of scattered fields in multilayer planar waveguides and lasers with circular gratings," IEEE J. Quantum Electron. 39, 1086-1098 (2003).
[CrossRef]

J. Scheuer and A. Yariv, "Annular Bragg defect mode resonators," J. Opt. Soc. Am. B 20, 2285-2291 (2003).
[CrossRef]

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925-928 (2003).
[CrossRef] [PubMed]

J. Scheuer and A. Yariv, "Coupled-waves approach to the design and analysis of Bragg and photonic crystal annular resonators," IEEE J. Quantum Electron. 39, 1555-1561 (2003).
[CrossRef]

W. Bogaerts, P. Bienstmann, and R. Baets, "Scattering at sidewall roughness in photonic crystal slabs," Opt. Lett. 28, 689-691 (2003).
[CrossRef] [PubMed]

Y. Xu, W. Liang, A. Yariv, J. G. Fleming, and S.-Y. Lin, "High-quality-factor Bragg onion resonators with omnidirectional reflector cladding," Opt. Lett. 28, 2144-2146 (2003).
[CrossRef] [PubMed]

2002 (1)

N. Moll, C. Bauer, H. Giessen, B. Schnabel, E. B. Kley, U. Scherf, and R. F. Mahrt, "Evidence for bandedge lasing in a two-dimensional photonic bandgap polymer laser," Appl. Phys. Lett. 80, 734-736 (2002).
[CrossRef]

2001 (4)

P. L. Greene and D. G. Hall, "Effects on radiation on circular-grating DFB lasers--Part I: coupled-mode equations," IEEE J. Quantum Electron. 37, 353-364 (2001).
[CrossRef]

P. L. Greene and D. G. Hall, "Effects on radiation on circular-grating DFB lasers--Part II: device and pump-beam parameters," IEEE J. Quantum Electron. 37, 365-371 (2001).
[CrossRef]

C. Bauer, H. Giessen, B. Schnabel, E. B. Kley, C. Schmitt, U. Scherf, and R. F. Mahrt, "A surface-emitting circular grating polymer laser," Adv. Mater. (Weinheim, Ger.) 13, 1161-1164 (2001).
[CrossRef]

M. A. Kaliteevski, S. Brand, R. A. Abram, and V. V. Nikolaev, "Optical eigenmodes of a multilayered spherical microcavity," J. Mod. Opt. , 48, 1503-1516 (2001).
[CrossRef]

2000 (2)

D. Ochoa, R. Houdré, M. Ilegems, H. Benisty, T. F. Krauss, and C. J. M. Smith, "Diffraction of cylindrical Bragg reflectors surrounding an in-plane semiconductor microcavity," Phys. Rev. B 61, 4806-4812 (2000).
[CrossRef]

T. Kawanishi and M. Izutsu, "Coaxial periodic optical waveguide," Opt. Express 7, 10-22 (2000).
[CrossRef] [PubMed]

1999 (3)

M. A. Kaliteevski, R. A. Abram, V. V. Nikolaev, and G. S. Sokolovski, "Bragg reflectors for cylindrical waves," J. Mod. Opt. 46, 875-890 (1999).
[CrossRef]

C. C. Wang and Z. Ye, "Spontaneous emission in cylindrical periodically-layered structures," Phys. Status Solidi A 174, 527-540 (1999).
[CrossRef]

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanley, R. Houdre, and U. Oesterle, "Lasing properties of disk microcavity based on a circular Bragg reflector," Appl. Phys. Lett. 75, 3051-3053 (1999).
[CrossRef]

1998 (2)

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Houdré, and U. Oesterle, "High-finesse disk microcavity based on a circular Bragg reflector," Appl. Phys. Lett. 73, 1314-1316 (1998).
[CrossRef]

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, and S. Rishton, "High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers," Appl. Phys. Lett. 72, 1284-1286 (1998).
[CrossRef]

1997 (1)

1995 (1)

M. Fallahi, F. Chatenoud, M. Dion, I. Templeton, R. Barber, and J. Thompson, "Circular-grating surface-emitting distributed Bragg reflector lasers on an InGaAs-GaAs structure for 0.98-μm applications," IEEE J. Sel. Top. Quantum Electron. 1, 382-386 (1995).
[CrossRef]

1994 (2)

E. X. Ping, "Transmission of electromagnetic waves in planar, cylindrical, and spherical dielectric layer systems and their applications," J. Appl. Phys. 76, 7188-7194 (1994).
[CrossRef]

X. M. Gong, A. K. Chan, and H. F. Taylor, "Lateral mode discrimination in surface emitting DBR lasers with cylindrical symmetry," IEEE J. Quantum Electron. 30, 1212-1218 (1994).
[CrossRef]

1993 (1)

1992 (3)

C. Wu, T. Makino, R. Maciejko, S. I. Najafi, and M. Svilans, "Simplified coupled-wave equations for cylindrical waves in circular grating planar waveguides," J. Lightwave Technol. 10, 1575-1589 (1992).
[CrossRef]

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, "Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser," Appl. Phys. Lett. 60, 1921-1923 (1992).
[CrossRef]

T. Erdogan and D. G. Hall, "Circularly symmetric distributed feedback laser: coupled mode treatment of TE vector fields," IEEE J. Quantum Electron. 28, 612-623 (1992).
[CrossRef]

1991 (2)

C. Wu, T. Makino, J. Glinski, R. Maciejko, and S. I. Najafi, "Self-consistent coupled-wave theory for circular gratings on planar dielectric waveguides," J. Lightwave Technol. 9, 1264-1277 (1991).
[CrossRef]

C. Wu, M. Svilans, M. Fallahi, T. Makino, J. Glinski, C. Maritan, and C. Blaauw, "Optically pumped surface-emitting DFB GalnAsP/InP lasers with circular grating," Electron. Lett. 27, 1819-1821 (1991).
[CrossRef]

1990 (3)

T. Erdogan and D. G. Hall, "Circularly symmetric distributed feedback semiconductor laser: An analysis," J. Appl. Phys. 68, 1435-1444 (1990).
[CrossRef]

M. Toda, "Single-mode behavior of a circular grating for potential disk-shaped DFB lasers," IEEE J. Quantum Electron. 26, 473-481 (1990).
[CrossRef]

X. H. Zheng and S. Lacroix, "Mode coupling in circular-cylindrical system and its application to fingerprint resonators," J. Lightwave Technol. 8, 1509-1516 (1990).
[CrossRef]

1989 (1)

X. H. Zheng, "Theory of two-dimensional 'fingerprint' resonators," Electron. Lett. 25, 1311-1312 (1989).
[CrossRef]

1978 (1)

Abram, R. A.

M. A. Kaliteevski, S. Brand, R. A. Abram, and V. V. Nikolaev, "Optical eigenmodes of a multilayered spherical microcavity," J. Mod. Opt. , 48, 1503-1516 (2001).
[CrossRef]

M. A. Kaliteevski, R. A. Abram, V. V. Nikolaev, and G. S. Sokolovski, "Bragg reflectors for cylindrical waves," J. Mod. Opt. 46, 875-890 (1999).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1972).

Anderson, E. H.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, "Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser," Appl. Phys. Lett. 60, 1921-1923 (1992).
[CrossRef]

Armani, D. K.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925-928 (2003).
[CrossRef] [PubMed]

Asano, T.

Bächtold, W.

A. Jebali, R. F. Mahrt, N. Moll, D. Erni, C. Bauer, E. B. Kley, G. L. Bona, and W. Bächtold, "Lasing in organic circular grating structures," J. Appl. Phys. 96, 3043-3049 (2004).
[CrossRef]

Baets, R.

Barber, R.

M. Fallahi, F. Chatenoud, M. Dion, I. Templeton, R. Barber, and J. Thompson, "Circular-grating surface-emitting distributed Bragg reflector lasers on an InGaAs-GaAs structure for 0.98-μm applications," IEEE J. Sel. Top. Quantum Electron. 1, 382-386 (1995).
[CrossRef]

Barlow, G. F.

G. A. Turnbull, A. Carleton, G. F. Barlow, A. Tahraoui, T. F. Krauss, A. Shore, and I. D. W. Samuel, "Design and analysis of a low-threshold polymer circular-grating distributed- feedback laser," J. Appl. Phys. 98, 023105 (2005).
[CrossRef]

G. F. Barlow, A. Shore, G. A. Turnbu, and I. D. W. Samuel, "Design and analysis of a low-threshold polymer circular-grating distributed-feedback laser," J. Opt. Soc. Am. B 21, 2142-2150 (2004).
[CrossRef]

Bauer, C.

A. Jebali, R. F. Mahrt, N. Moll, D. Erni, C. Bauer, E. B. Kley, G. L. Bona, and W. Bächtold, "Lasing in organic circular grating structures," J. Appl. Phys. 96, 3043-3049 (2004).
[CrossRef]

N. Moll, C. Bauer, H. Giessen, B. Schnabel, E. B. Kley, U. Scherf, and R. F. Mahrt, "Evidence for bandedge lasing in a two-dimensional photonic bandgap polymer laser," Appl. Phys. Lett. 80, 734-736 (2002).
[CrossRef]

C. Bauer, H. Giessen, B. Schnabel, E. B. Kley, C. Schmitt, U. Scherf, and R. F. Mahrt, "A surface-emitting circular grating polymer laser," Adv. Mater. (Weinheim, Ger.) 13, 1161-1164 (2001).
[CrossRef]

Benisty, H.

D. Ochoa, R. Houdré, M. Ilegems, H. Benisty, T. F. Krauss, and C. J. M. Smith, "Diffraction of cylindrical Bragg reflectors surrounding an in-plane semiconductor microcavity," Phys. Rev. B 61, 4806-4812 (2000).
[CrossRef]

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanley, R. Houdre, and U. Oesterle, "Lasing properties of disk microcavity based on a circular Bragg reflector," Appl. Phys. Lett. 75, 3051-3053 (1999).
[CrossRef]

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Houdré, and U. Oesterle, "High-finesse disk microcavity based on a circular Bragg reflector," Appl. Phys. Lett. 73, 1314-1316 (1998).
[CrossRef]

Bienstmann, P.

Blaauw, C.

C. Wu, M. Svilans, M. Fallahi, T. Makino, J. Glinski, C. Maritan, and C. Blaauw, "Optically pumped surface-emitting DFB GalnAsP/InP lasers with circular grating," Electron. Lett. 27, 1819-1821 (1991).
[CrossRef]

Bogaerts, W.

Bona, G. L.

A. Jebali, R. F. Mahrt, N. Moll, D. Erni, C. Bauer, E. B. Kley, G. L. Bona, and W. Bächtold, "Lasing in organic circular grating structures," J. Appl. Phys. 96, 3043-3049 (2004).
[CrossRef]

Brady, D. J.

Brand, S.

M. A. Kaliteevski, S. Brand, R. A. Abram, and V. V. Nikolaev, "Optical eigenmodes of a multilayered spherical microcavity," J. Mod. Opt. , 48, 1503-1516 (2001).
[CrossRef]

Campillo, A. J.

R. K. Chang and A. J. Campillo, Optical Processes in Microcavities (World Scientific, 1996), pp. 27-34.

Carleton, A.

G. A. Turnbull, A. Carleton, G. F. Barlow, A. Tahraoui, T. F. Krauss, A. Shore, and I. D. W. Samuel, "Design and analysis of a low-threshold polymer circular-grating distributed- feedback laser," J. Appl. Phys. 98, 023105 (2005).
[CrossRef]

Chan, A. K.

X. M. Gong, A. K. Chan, and H. F. Taylor, "Lateral mode discrimination in surface emitting DBR lasers with cylindrical symmetry," IEEE J. Quantum Electron. 30, 1212-1218 (1994).
[CrossRef]

Chang, R. K.

R. K. Chang and A. J. Campillo, Optical Processes in Microcavities (World Scientific, 1996), pp. 27-34.

Chatenoud, F.

M. Fallahi, F. Chatenoud, M. Dion, I. Templeton, R. Barber, and J. Thompson, "Circular-grating surface-emitting distributed Bragg reflector lasers on an InGaAs-GaAs structure for 0.98-μm applications," IEEE J. Sel. Top. Quantum Electron. 1, 382-386 (1995).
[CrossRef]

DeRose, G. A.

J. Scheuer, W. M. J. Green, G. A. DeRose, and A. Yariv, "InGaAsP annular Bragg lasers: theory, applications, and modal properties," IEEE J. Sel. Top. Quantum Electron. 11, 476-484(2005).
[CrossRef]

J. Scheuer, W. M. J. Green, G. A. DeRose, and A. Yariv, "Lasing from a circular Bragg nanocavity with an ultrasmall modal volume," Appl. Phys. Lett. 86, 251101 (2005).
[CrossRef]

Dion, M.

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C. Wu, M. Svilans, M. Fallahi, T. Makino, J. Glinski, C. Maritan, and C. Blaauw, "Optically pumped surface-emitting DFB GalnAsP/InP lasers with circular grating," Electron. Lett. 27, 1819-1821 (1991).
[CrossRef]

Tahraoui, A.

G. A. Turnbull, A. Carleton, G. F. Barlow, A. Tahraoui, T. F. Krauss, A. Shore, and I. D. W. Samuel, "Design and analysis of a low-threshold polymer circular-grating distributed- feedback laser," J. Appl. Phys. 98, 023105 (2005).
[CrossRef]

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X. M. Gong, A. K. Chan, and H. F. Taylor, "Lateral mode discrimination in surface emitting DBR lasers with cylindrical symmetry," IEEE J. Quantum Electron. 30, 1212-1218 (1994).
[CrossRef]

Templeton, I.

M. Fallahi, F. Chatenoud, M. Dion, I. Templeton, R. Barber, and J. Thompson, "Circular-grating surface-emitting distributed Bragg reflector lasers on an InGaAs-GaAs structure for 0.98-μm applications," IEEE J. Sel. Top. Quantum Electron. 1, 382-386 (1995).
[CrossRef]

Thompson, J.

M. Fallahi, F. Chatenoud, M. Dion, I. Templeton, R. Barber, and J. Thompson, "Circular-grating surface-emitting distributed Bragg reflector lasers on an InGaAs-GaAs structure for 0.98-μm applications," IEEE J. Sel. Top. Quantum Electron. 1, 382-386 (1995).
[CrossRef]

Toda, M.

M. Toda, "Single-mode behavior of a circular grating for potential disk-shaped DFB lasers," IEEE J. Quantum Electron. 26, 473-481 (1990).
[CrossRef]

Tricomi, F. G.

A. Erdélyi, M. F. Oberhettinger, and F. G. Tricomi, Tables of Integral Transforms. Based, in Part, on Notes Left by Harry Bateman and Compiled by the Staff of the Bateman Manuscript Project (McGraw-Hill, 1954).
[PubMed]

Turnbu, G. A.

Turnbull, G. A.

G. A. Turnbull, A. Carleton, G. F. Barlow, A. Tahraoui, T. F. Krauss, A. Shore, and I. D. W. Samuel, "Design and analysis of a low-threshold polymer circular-grating distributed- feedback laser," J. Appl. Phys. 98, 023105 (2005).
[CrossRef]

Vahala, K. J.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925-928 (2003).
[CrossRef] [PubMed]

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[CrossRef]

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Houdré, and U. Oesterle, "High-finesse disk microcavity based on a circular Bragg reflector," Appl. Phys. Lett. 73, 1314-1316 (1998).
[CrossRef]

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C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, and S. Rishton, "High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers," Appl. Phys. Lett. 72, 1284-1286 (1998).
[CrossRef]

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[CrossRef]

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C. Wu, T. Makino, R. Maciejko, S. I. Najafi, and M. Svilans, "Simplified coupled-wave equations for cylindrical waves in circular grating planar waveguides," J. Lightwave Technol. 10, 1575-1589 (1992).
[CrossRef]

C. Wu, T. Makino, J. Glinski, R. Maciejko, and S. I. Najafi, "Self-consistent coupled-wave theory for circular gratings on planar dielectric waveguides," J. Lightwave Technol. 9, 1264-1277 (1991).
[CrossRef]

C. Wu, M. Svilans, M. Fallahi, T. Makino, J. Glinski, C. Maritan, and C. Blaauw, "Optically pumped surface-emitting DFB GalnAsP/InP lasers with circular grating," Electron. Lett. 27, 1819-1821 (1991).
[CrossRef]

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Yariv, A.

J. Scheuer, W. M. J. Green, G. A. DeRose, and A. Yariv, "Lasing from a circular Bragg nanocavity with an ultrasmall modal volume," Appl. Phys. Lett. 86, 251101 (2005).
[CrossRef]

J. Scheuer, W. M. J. Green, G. A. DeRose, and A. Yariv, "InGaAsP annular Bragg lasers: theory, applications, and modal properties," IEEE J. Sel. Top. Quantum Electron. 11, 476-484(2005).
[CrossRef]

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C. C. Wang and Z. Ye, "Spontaneous emission in cylindrical periodically-layered structures," Phys. Status Solidi A 174, 527-540 (1999).
[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, and S. Rishton, "High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers," Appl. Phys. Lett. 72, 1284-1286 (1998).
[CrossRef]

A. Shaw, B. Roycroft, J. Hegarty, D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Stanley, R. Houdre, and U. Oesterle, "Lasing properties of disk microcavity based on a circular Bragg reflector," Appl. Phys. Lett. 75, 3051-3053 (1999).
[CrossRef]

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, C. J. M. Smith, R. Houdré, and U. Oesterle, "High-finesse disk microcavity based on a circular Bragg reflector," Appl. Phys. Lett. 73, 1314-1316 (1998).
[CrossRef]

J. Scheuer, W. M. J. Green, G. A. DeRose, and A. Yariv, "Lasing from a circular Bragg nanocavity with an ultrasmall modal volume," Appl. Phys. Lett. 86, 251101 (2005).
[CrossRef]

Electron. Lett. (2)

C. Wu, M. Svilans, M. Fallahi, T. Makino, J. Glinski, C. Maritan, and C. Blaauw, "Optically pumped surface-emitting DFB GalnAsP/InP lasers with circular grating," Electron. Lett. 27, 1819-1821 (1991).
[CrossRef]

X. H. Zheng, "Theory of two-dimensional 'fingerprint' resonators," Electron. Lett. 25, 1311-1312 (1989).
[CrossRef]

IEEE J. Quantum Electron. (7)

A. M. Shams-Zadeh-Amiri, X. Li, and W.-P. Huang, "Hankel transform-domain analysis of scattered fields in multilayer planar waveguides and lasers with circular gratings," IEEE J. Quantum Electron. 39, 1086-1098 (2003).
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[CrossRef]

P. L. Greene and D. G. Hall, "Effects on radiation on circular-grating DFB lasers--Part I: coupled-mode equations," IEEE J. Quantum Electron. 37, 353-364 (2001).
[CrossRef]

P. L. Greene and D. G. Hall, "Effects on radiation on circular-grating DFB lasers--Part II: device and pump-beam parameters," IEEE J. Quantum Electron. 37, 365-371 (2001).
[CrossRef]

J. Scheuer and A. Yariv, "Coupled-waves approach to the design and analysis of Bragg and photonic crystal annular resonators," IEEE J. Quantum Electron. 39, 1555-1561 (2003).
[CrossRef]

X. M. Gong, A. K. Chan, and H. F. Taylor, "Lateral mode discrimination in surface emitting DBR lasers with cylindrical symmetry," IEEE J. Quantum Electron. 30, 1212-1218 (1994).
[CrossRef]

M. Toda, "Single-mode behavior of a circular grating for potential disk-shaped DFB lasers," IEEE J. Quantum Electron. 26, 473-481 (1990).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

J. Scheuer, W. M. J. Green, G. A. DeRose, and A. Yariv, "InGaAsP annular Bragg lasers: theory, applications, and modal properties," IEEE J. Sel. Top. Quantum Electron. 11, 476-484(2005).
[CrossRef]

M. Fallahi, F. Chatenoud, M. Dion, I. Templeton, R. Barber, and J. Thompson, "Circular-grating surface-emitting distributed Bragg reflector lasers on an InGaAs-GaAs structure for 0.98-μm applications," IEEE J. Sel. Top. Quantum Electron. 1, 382-386 (1995).
[CrossRef]

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A. Jebali, R. F. Mahrt, N. Moll, D. Erni, C. Bauer, E. B. Kley, G. L. Bona, and W. Bächtold, "Lasing in organic circular grating structures," J. Appl. Phys. 96, 3043-3049 (2004).
[CrossRef]

T. Erdogan and D. G. Hall, "Circularly symmetric distributed feedback semiconductor laser: An analysis," J. Appl. Phys. 68, 1435-1444 (1990).
[CrossRef]

G. A. Turnbull, A. Carleton, G. F. Barlow, A. Tahraoui, T. F. Krauss, A. Shore, and I. D. W. Samuel, "Design and analysis of a low-threshold polymer circular-grating distributed- feedback laser," J. Appl. Phys. 98, 023105 (2005).
[CrossRef]

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[CrossRef]

J. Lightwave Technol. (3)

C. Wu, T. Makino, J. Glinski, R. Maciejko, and S. I. Najafi, "Self-consistent coupled-wave theory for circular gratings on planar dielectric waveguides," J. Lightwave Technol. 9, 1264-1277 (1991).
[CrossRef]

C. Wu, T. Makino, R. Maciejko, S. I. Najafi, and M. Svilans, "Simplified coupled-wave equations for cylindrical waves in circular grating planar waveguides," J. Lightwave Technol. 10, 1575-1589 (1992).
[CrossRef]

X. H. Zheng and S. Lacroix, "Mode coupling in circular-cylindrical system and its application to fingerprint resonators," J. Lightwave Technol. 8, 1509-1516 (1990).
[CrossRef]

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M. A. Kaliteevski, R. A. Abram, V. V. Nikolaev, and G. S. Sokolovski, "Bragg reflectors for cylindrical waves," J. Mod. Opt. 46, 875-890 (1999).
[CrossRef]

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[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (4)

Nature (1)

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Ultra-high-Q toroid microcavity on a chip," Nature 421, 925-928 (2003).
[CrossRef] [PubMed]

Opt. Express (4)

Opt. Lett. (2)

Phys. Rev. B (1)

D. Ochoa, R. Houdré, M. Ilegems, H. Benisty, T. F. Krauss, and C. J. M. Smith, "Diffraction of cylindrical Bragg reflectors surrounding an in-plane semiconductor microcavity," Phys. Rev. B 61, 4806-4812 (2000).
[CrossRef]

Phys. Status Solidi A (1)

C. C. Wang and Z. Ye, "Spontaneous emission in cylindrical periodically-layered structures," Phys. Status Solidi A 174, 527-540 (1999).
[CrossRef]

Other (12)

A. Yariv, ed. Optical Electronics in Modern Communications (Oxford U. Press, 1997).

R. K. Chang and A. J. Campillo, Optical Processes in Microcavities (World Scientific, 1996), pp. 27-34.

Note that, in reality, the coefficients ξmj and ηmj have units of Vm−1.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1972).

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, 1987).

K. Zhang and D. Li, Electromagnetic Theory for Microwaves and Optoelectronics (Springer, 1998).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 1980).

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, 1962).

A. Erdélyi, M. F. Oberhettinger, and F. G. Tricomi, Tables of Integral Transforms. Based, in Part, on Notes Left by Harry Bateman and Compiled by the Staff of the Bateman Manuscript Project (McGraw-Hill, 1954).
[PubMed]

It is worth mentioning that finding a suitable physical measure to probe the spectral response of a leaky cavity accordingly is related to a more general issue than expected; in fact, the FWHM estimation of the cavity resonance essentially depends on the technical setup to access the corresponding field quantity and, hence, to the problem of proper normalization (or appropriate excitation) of the cavity mode. Thus the spectrally determined Q factors become context dependent. Given the normalization of the wave amplitudes in the cavity center, as proposed by condition , the resulting spectral response of the total field energy Wm(λ) yields a physically counterintuitive, but mathematically correct, drop at resonance.

COMSOL Multiphysics, http://www.comsol.com

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

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Figures (7)

Fig. 1
Fig. 1

Representation of the structure geometry, with period a, duty cycle d c defined as p a , refractive indices n I and n II , inner-cavity radius r 1 and outer radii r 2 , , r N 1 . L represents the contour of the outermost ring.

Fig. 2
Fig. 2

(a) Example of the spectral function R m for the two first azimuthal orders m = 0 and 1, inner-cavity radius r 1 = 0.83 a , indices n I = 1.6 and n II = 1.96 , number of layers N = 32 , and duty cycle d c = 0.45 . (b) Zoom of the resonance peak and the FWHM of the linewidth δ ν at the resonance frequency ν Res .

Fig. 3
Fig. 3

(a) Two-dimensional and (b) 3D graphs of the E z j , 0 field in the case of ( n I , n II ) = ( 1.6 , 1.96 ) , N = 32 layers, cavity radius of r 1 = 0.83 a , with a being the lattice constant and d c = 0.45 , for the monopole mode ( m = 0 ) at resonance.

Fig. 4
Fig. 4

Cavity design: (a) typical power-ratio plot for a dipole mode m = 1 at three different inner-cavity radii, i.e., r 1 = 1.0 a , 1.35 a , and 1.8 a . Resonances Res. 1, Res. 2, and Res. 3 are located at the lower band edge, at the middle of the PBG, and at the higher band edge, respectively. (b) Scan over the cavity radii of the resonances for the azimuthal orders m = 0 (monopole) and m = 1 (dipole). Maximum Q is expected at radii where the resonance is in the middle of the PBG, which, in our case, are R 0 = 0.83 a and R 1 = 1.35 a . For clarity, Res. 1, Res. 2, and Res. 3 of (a) are also indicated here.

Fig. 5
Fig. 5

Q-factor calculation with both methods: analytically and with FWHM (power ratio) for N = 32 layers, m = 0 (circles), and m = 1 (triangles). (a) Scan over the cavity radii: maximum Q value is the one located in the center of the PBG as illustrated in Fig. 4, i.e., R 0 = 0.83 a for m = 0 and R 1 = 1.35 a for m = 1 . (b) Difference percentage Δ = ( Q FWHM Q Analytic ) Q Analytic is plotted for both m orders.

Fig. 6
Fig. 6

Comparison of the Q values obtained with the different methods for N = 32 , 40, 48, and 64. (a) Monopole ( m = 0 ) : example of the central-PBG cavity radius, R 0 = 0.83 a , and an edge-PBG one, R 0 = 1.2 a . Ratio between the FWHM values and the analytical ones is 20 % for R 0 = 0.83 a and 8 % (mean value) for R 0 = 1.2 a . (b) Dipole ( m = 1 ) : same as (a) but for two cavity radii R 1 = 1.35 a and R 1 = 1.8 a . Ratio between the FWHM values and the analytical ones is 57 % for R 1 = 1.35 a and 13 % (mean value) for R 1 = 1.8 a . Here, we also added some FEMLAB calculations of the Q factor (black diamonds and error bars), only for the R 1 = 1.35 a case.

Fig. 7
Fig. 7

E z 2 field calculated with FEMLAB for the dipole mode ( R 1 = 1.35 a , N = 32 layers, λ = 850 nm , a = 240 nm ). Distance between the two excitations, δ = 320 nm , is illustrated in the close-up of the cavity (right). The two circle excitations have a radius of 10 nm .

Tables (1)

Tables Icon

Table 1 Recapitulation of Energy, Power, and Q Factor a

Equations (51)

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Q = ω Res W P ,
× E = μ 0 i ω H ,
× H = ε 0 n j 2 i ω E ,
E = H = 0 ,
E z j ( r , φ ) = m = 0 [ ξ m j H m ( n j k 0 r ) + η m j H m * ( n j k 0 r ) ] e ± i m φ .
E z j , m ( r ) = ξ m j H m ( n j k 0 r ) + η m j H m * ( n j k 0 r ) .
H r j ( r , φ ) = 1 μ 0 ω m = 0 [ ξ m j H m ( n j k 0 r ) + η m j H m * ( n j k 0 r ) ] m e i m φ ,
= m = 0 H r j , m ( r ) e i m φ ,
H φ j ( r , φ ) = n j μ 0 i c m = 0 [ ξ m j H m ( n j k 0 r ) + η m j H m * ( n j k 0 r ) ] e i m φ ,
= m = 0 H φ j , m ( r ) e i m φ ,
( ξ m N η m N ) = [ A m N B m N B m N * A m N * ] ( ξ m 1 η m 1 ) ,
R m = ξ m 1 2 + η m 1 2 ξ m N 2 + η m N 2 = 1 A m N + B m N 2
P m N = L R ( S r N , m ) d L .
ξ m 1 = 1 , η m N = 0 ,
( ξ m N 0 ) = [ A m N B m N B m N * A m N * ] ( 1 η m 1 ) ,
{ ξ m N = ( A m N 2 B m N 2 ) A m N * η m 1 = B m N * A m N * .
S r N , m = 1 2 E z N , m ( r ) e i m φ H φ N , m * ( r ) e i m φ = n N 2 μ 0 i c [ ξ m N H m ( n N k 0 r ) + η m N H m * ( n N k 0 r ) ] [ ξ m N * H m * ( n N k 0 r ) + η m N * H m ( n N k 0 r ) ]
= η m N = 0 i n N 2 μ 0 c ξ m N 2 H m ( n N k 0 r ) H m * ( n N k 0 r ) ,
R ( S r N , m ) = n N 2 μ 0 c ξ m N 2 R ( i H m H m * ) = n N 2 μ 0 c ξ m N 2 I ( H m H m * ) = 1 π μ 0 ω r ξ m N 2 ,
R ( i H m H m * ) = J m ( n N k 0 r ) N m ( n N k 0 r ) J m ( n N k 0 r ) N m ( n N k 0 r ) = W { J m ( n N k 0 r ) , N m ( n N k 0 r ) } = 2 π n N k 0 r ,
P m N = 0 2 π 1 π μ 0 ω r ξ m N 2 r d φ = 2 μ 0 ω ξ m N 2 = 2 μ 0 ω ( A m N 2 B m N 2 ) 2 A m N 2 .
w j , m = 1 2 ε j E z j , m ( r ) 2 .
ξ m 1 = η m 1 = 1 .
( ξ m j η m j ) = [ A m j B m j B m j * A m j * ] ( 1 1 ) ,
{ ξ m j = A m j + B m j η m j = B m j * + A m j * = ξ m j * .
E z j , m ( r ) 2 = [ E z j , m ( r ) ] 2 = 4 [ R 2 ( A m j + B m j ) J m 2 ( n j k 0 r ) + I 2 ( A m j + B m j ) N m 2 ( n j k 0 r ) 2 R ( A m j + B m j ) I ( A m j + B m j ) J m ( n j k 0 r ) N m ( n j k 0 r ) ] .
W j , m = A j w j , m d A j = 1 2 φ r ε j E z j , m ( r ) 2 r d r d φ = π ε 0 n j 2 r j 1 r j E z j , m ( r ) 2 r d r .
W m = 4 π ε 0 j = 1 N n j 2 W j , m ,
W j , m = γ J W J m + γ N W N m + γ J N W J N m .
Q TM m = ω Res W m P m N = 2 π k Res 2 A m N 2 ( A m N 2 B m N 2 ) 2 j = 1 N n j 2 W j , m ,
Q = ν Res δ ν ,
E z j ( r j , φ ) = E z j + 1 ( r j , φ ) ,
H φ j ( r j , φ ) = H φ j + 1 ( r j , φ ) ,
( ξ m j + 1 η m j + 1 ) = [ a m j b m j b m j * a m j * ] ( ξ m j η m j ) ,
a m j = 1 D m j [ H m ( n j r j k 0 ) H m * ( n j + 1 r j k 0 ) n j + 1 n j H m * ( n j + 1 r j k 0 ) H m ( n j r j k 0 ) ] ,
b m j = 1 D m j [ H m * ( n j r j k 0 ) H m * ( n j + 1 r j k 0 ) n j + 1 n j H m * ( n j + 1 r j k 0 ) H m * ( n j r j k 0 ) ] ,
D m j = H m ( n j + 1 r j k 0 ) H m * ( n j + 1 r j k 0 ) H m * ( n j + 1 r j k 0 ) H m ( n j + 1 r j k 0 ) .
( ξ m N η m N ) = [ A m N B m N B m N * A m N * ] ( ξ m 1 η m 1 ) ,
[ A m N B m N B m N * A m N * ] = [ a m N 1 b m N 1 b m N 1 * a m N 1 * ] [ a m 1 b m 1 b m 1 * a m 1 * ] .
W J m = 1 2 r 2 [ J m ( n j k 0 r ) 2 J m 1 ( n j k 0 r ) J m + 1 ( n j k 0 r ) ] r j 1 r j ,
W N m = 1 2 r 2 [ N m ( n j k 0 r ) 2 N m 1 ( n j k 0 r ) N m + 1 ( n j k 0 r ) ] r j 1 r j ,
W J N m = 1 4 r 2 [ J m ( n j k 0 r ) N m ( n j k 0 r ) 2 J m + 1 ( n j k 0 r ) N m 1 ( n j k 0 r ) 2 J m 1 ( n j k 0 r ) N m + 1 ( n j k 0 r ) ] r j 1 r j .
W 1 , m = 2 2 2 m π r 1 m + 1 ( n 1 k 0 ) 4 m Γ ( m + 3 2 ) Γ ( m + 2 ) Γ ( m + 1 ) S m ( 2 n 1 k 0 r 1 ) ,
H z j ( r , φ ) = m = 0 [ ξ m j H m ( n j k 0 r ) + η m j H m * ( n j k 0 r ) ] e i m φ = m = 0 H z j , m ( r ) e i m φ ,
E r j ( r , φ ) = 1 ε 0 n j 2 ω m = 0 [ ξ m j H m ( n j k 0 r ) + η m j H m * ( n j k 0 r ) ] m e i m φ = m = 0 E r j , m ( r ) e i m φ ,
E r j ( r , φ ) = 1 ε 0 n j c m = 0 [ ξ m j H m ( n j k 0 r ) + η m j H m * ( n j k 0 r ) ] e i m φ = m = 0 E φ j , m ( r ) e i m φ .
S r N , m = 1 2 H z N , m * ( n N k 0 r ) E φ N , m ( n N k 0 r ) = i 2 ε 0 n N c ξ m N 2 H m * H m ,
R ( S τ N , m ) = 1 2 ε 0 n N c ξ m N 2 I ( H m H m * ) = 1 π ε 0 n N 2 ω r ξ m N 2 ,
P m N = 2 ε 0 n N 2 ω ξ m N 2 = 2 ε 0 n N 2 ω ( A m N 2 B m N 2 ) 2 A m N 2 .
W m = 1 2 A μ 0 H z j , m ( r ) 2 d A = π μ 0 r H z j , m ( r ) 2 r d r = 4 π μ 0 j = 1 N W j , m ,
Q TE m = 2 π n N 2 k Res 2 A m N 2 ( A m N 2 B m N 2 ) 2 j = 1 N W j , m .

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