Abstract

Radial Bragg distributed-feedback (DFB) lasers are designed and studied using the transfer matrix method, allowing an accurate analysis beyond the coupled-mode equations approach (small perturbations). Compared with conventional circular grating DFB lasers, incorporating periodic gratings, lower threshold levels, and enhanced mode discrimination are achieved by employing an optimal design strategy for the radial Bragg reflector.

© 2007 Optical Society of America

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References

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    [CrossRef]
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2005 (4)

H. Altug and J. Vuckovic, "Experimental demonstration of the slow group velocity of light in two-dimensional coupled photonic crystal microcavity arrays," Appl. Phys. Lett. 86, 111102 (2005).
[CrossRef]

K. Nozaki, T. Ide, J. Hashimoto, T. Baba, and W.-H. Zheng, "Photonic crystal point-shift nanolaser with ultimate small modal volume," Electron. Lett. 41, 843-845 (2005).
[CrossRef]

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

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

2004 (5)

2003 (4)

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, 1555-1562 (2003).
[CrossRef]

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

Y. Akahane, T. Asano, B. S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature (London) 425, 944-947 (2003).
[CrossRef]

C. Y. Chao and L. J. Guo, "Biochemical sensors based on polymer microrings with sharp asymmetrical resonance," Appl. Phys. Lett. 83, 1527-1529 (2003).
[CrossRef]

2001 (2)

1999 (1)

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, and I. Kim, "Two-dimensional photonic band-gap defect mode laser," Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

1998 (2)

Y. Fink, J. N. Winn, F. Shanhui, C. Chiping, J. Michel, J. D. Joannopoulos, and E. L. Thomas, "A dielectric omnidirectional reflector," Science 282, 1679-1682 (1998).
[CrossRef] [PubMed]

B. E. Little, "Second-order filtering and sensing with partially coupled traveling waves in a single resonator," Opt. Lett. 23, 1570-1572 (1998).
[CrossRef]

1997 (1)

1994 (1)

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)

C. Wu, T. Makino, S. I. Najafi, R. Maciejko, M. Svilans, J. Glinski, and M. Fallahi, "Threshold gain and threshold current analysis of circular grating DFB and DBR lasers," IEEE J. Quantum Electron. 29, 2596-2606 (1993).
[CrossRef]

1992 (3)

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]

T. Makino and C. Wu, "Circular grating DFB and DBR semiconductor lasers: threshold current analysis," Opt. Commun. 90, 297-300 (1992).
[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]

1991 (1)

C. Wu, M. Svilans, M. Fallahi, T. Makino, J. Glinski, C. Maritan and C. Blaauw, "Optically pumped surface-emitting DFB GaInAsP/InP lasers with circular grating," Electron. Lett. 27, 1819-1820 (1991).
[CrossRef]

1990 (1)

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

1978 (1)

Appl. Phys. Lett. (3)

C. Y. Chao and L. J. Guo, "Biochemical sensors based on polymer microrings with sharp asymmetrical resonance," Appl. Phys. Lett. 83, 1527-1529 (2003).
[CrossRef]

H. Altug and J. Vuckovic, "Experimental demonstration of the slow group velocity of light in two-dimensional coupled photonic crystal microcavity arrays," Appl. Phys. Lett. 86, 111102 (2005).
[CrossRef]

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

Electron. Lett. (2)

K. Nozaki, T. Ide, J. Hashimoto, T. Baba, and W.-H. Zheng, "Photonic crystal point-shift nanolaser with ultimate small modal volume," Electron. Lett. 41, 843-845 (2005).
[CrossRef]

C. Wu, M. Svilans, M. Fallahi, T. Makino, J. Glinski, C. Maritan and C. Blaauw, "Optically pumped surface-emitting DFB GaInAsP/InP lasers with circular grating," Electron. Lett. 27, 1819-1820 (1991).
[CrossRef]

IEEE J. Quantum Electron. (5)

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]

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]

C. Wu, T. Makino, S. I. Najafi, R. Maciejko, M. Svilans, J. Glinski, and M. Fallahi, "Threshold gain and threshold current analysis of circular grating DFB and DBR lasers," IEEE J. Quantum Electron. 29, 2596-2606 (1993).
[CrossRef]

P. L. Greene and D. G. Hall, "Effects of radiation on circular-grating DFB lasers. Part I. Coupled-mode equations," IEEE J. Quantum Electron. 37, 353-364 (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, 1555-1562 (2003).
[CrossRef]

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

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

IEICE Trans. Electron. (1)

T. Yoshie, O. B. Shchekin, H. Chen, D. G. Deppe, and A. Scherer, "Planar photonic crystal nanolasers (II): low-threshold quantum dot lasers," IEICE Trans. Electron. E87-C, 300-307 (2004).

J. Appl. Phys. (1)

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

J. Lightwave Technol. (1)

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]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (2)

Nature (London) (1)

Y. Akahane, T. Asano, B. S. Song, and S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature (London) 425, 944-947 (2003).
[CrossRef]

Opt. Commun. (1)

T. Makino and C. Wu, "Circular grating DFB and DBR semiconductor lasers: threshold current analysis," Opt. Commun. 90, 297-300 (1992).
[CrossRef]

Opt. Express (2)

Opt. Lett. (3)

Phys. Rev. E (1)

J. Scheuer and A. Yariv, "Circular photonic crystal resonators," Phys. Rev. E 70, 036603 (2004).
[CrossRef]

Science (2)

Y. Fink, J. N. Winn, F. Shanhui, C. Chiping, J. Michel, J. D. Joannopoulos, and E. L. Thomas, "A dielectric omnidirectional reflector," Science 282, 1679-1682 (1998).
[CrossRef] [PubMed]

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, and I. Kim, "Two-dimensional photonic band-gap defect mode laser," Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

Other (4)

C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (Wiley, 1999).

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

G. N. Watson, Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, 1952).

A. Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford Press, 1997).

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Figures (8)

Fig. 1
Fig. 1

SEM images of the circular Bragg nanocavities with (a) “ring” and (b) “disk” configurations and the corresponding modal field profiles for the (c) ring and for the (d) disk.

Fig. 2
Fig. 2

(a) Threshold resonance wavelengths and (b) gain levels of the modes of an CBL employing shallow gratings. The structure is designed to resonate in the zeroth-order angular mode ( m = 0 ) at λ = 1.55 μ m . The high and low-index layers have n = 3.51 and 3.49, respectively. The external Bragg reflector consists of 100 periods.

Fig. 3
Fig. 3

(a) Threshold resonance wavelengths and (b) gain levels of the modes of an CBL employing strong perturbations. The structure is designed to resonate in the zeroth-order angular mode ( m = 0 ) at λ = 1.55 μ m . The high and low-index layers have n = 2.8 and 1.5, respectively. The external Bragg reflector consists of seven periods.

Fig. 4
Fig. 4

Radial index profiles of the optimal (solid curve) curves and conventional (dashed curve) structures based on the high-index contrast design of Fig. 3.

Fig. 5
Fig. 5

Comparison of the lasing wavelengths and threshold levels and of conventional and optimal CBLs for the shallow grating case shown in Fig. 2.

Fig. 6
Fig. 6

Comparison of the lasing wavelengths and threshold levels and of conventional and optimal CBLs for the strong perturbations case shown in Fig. 3.

Fig. 7
Fig. 7

Radial field profiles of the m = 0 (solid curve), m = 1 (dashed curve) and m = 2 (dashed–dotted curve) modes of the optimal structure employing high-index contrast. The dark and light gray areas indicate, respectively, high and low refractive indices.

Fig. 8
Fig. 8

Threshold resonance wavelengths (triangles) and gain levels (squares) of the m = 0 mode of the CBL as a function of Fig. 3 as a function of fabrication errors.

Equations (7)

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

ρ 2 2 R H , E ρ 2 + ρ R H , E ρ + [ ( k 2 ( ρ ) β 2 ) ρ 2 m 2 ] R H , E = 0 ,
R E , H = A H m ( 1 ) ( k ρ ) + B H m ( 2 ) ( k ρ ) ,
( A j + 1 B j + 1 ) = M ̃ j + 1 1 ( ρ j + 1 ) M ̃ j ( ρ j ) ( A j B j ) ,
M ̃ j TM = ( H m ( 1 ) ( k j ρ ) H m ( 2 ) ( k j ρ ) n j 2 γ j [ H m ( 1 ) ( k j ρ ) ] n j 2 γ j [ H m ( 2 ) ( k j ρ ) ] ) ,
M ̃ j TE = ( H m ( 1 ) ( k j ρ ) H m ( 2 ) ( k j ρ ) 1 γ j [ H m ( 1 ) ( k j ρ ) ] 1 γ j [ H m ( 2 ) ( k j ρ ) ] ) ,
( A n + 1 0 ) = M ̂ ( λ , n i ) ( 1 1 ) ,
H m ( 1 , 2 ) ( x ) 2 π x exp [ ± i ( x 1 2 m π 1 4 π ) ] .

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