Abstract

The effects of a two-dimensional, cylindrically symmetric periodic structure on the radiation from a source located inside the structure are considered. The coupling of the source radiation to the cylindrical cavity is analyzed classically by consideration of the interaction of a current line source with its own radiated field supported by the cavity. The analysis predicts variations in the radiative damping rate (inverse lifetime) and associated shifts in the oscillation frequency of the source. It is found that significant enhancement and inhibition of the radiation are possible even when the source is not at the center of the structure. Frequency shifts induced by the cavity are found to be negligible relative to the size of the band gap of the Bragg structure. The class of periodic structures analyzed has potential application to planar waveguide devices, such as concentric-circle gratings, surface-emitting lasers, and cylindrically symmetric lasers and amplifiers.

© 1993 Optical Society of America

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  1. E. M. Purcell, Phys. Rev. 69, 681 (1946).
    [CrossRef]
  2. D. Kleppner, Phys. Rev. Lett. 47, 233 (1981).
    [CrossRef]
  3. R. G. Hulet, E. S. Hilfer, and D. Kleppner, Phys. Rev. Lett. 55, 2137 (1985).
    [CrossRef] [PubMed]
  4. K. H. Drexhage, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1974), Vol. 12, Chap. 4.
    [CrossRef]
  5. D. G. Deppe and C. Lei, J. Appl. Phys. 70, 3443 (1991).
    [CrossRef]
  6. Y. Yamamoto, S. Machida, and G. Björk, Phys. Rev. A 44, 657 (1991), and G. Björk, S. Machida, Y. Yamamoto, and K. Igeta, Phys. Rev. A 44, 669 (1991).
    [CrossRef] [PubMed]
  7. H. Yokoyama, K. Nishi, T. Anan, H. Yamada, S. D. Brorson, and E. P. Ippen, Appl. Phys. Lett. 57, 2814 (1990).
    [CrossRef]
  8. E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
    [CrossRef] [PubMed]
  9. P. R. Villeneuve and M. Piché, J. Opt. Soc. Am. A 8, 1296 (1991).
    [CrossRef]
  10. M. Plihal and A. A. Maradudin, Phys. Rev. B 44, 8565 (1991).
    [CrossRef]
  11. T. Erdogan and D. G. Hall, IEEE J. Quantum Electron. 28, 612 (1992).
    [CrossRef]
  12. C. Wu, T. Makino, J. Glinski, R. Maciejko, and S. I. Najafi, IEEE J. Lightwave Technol. 9, 1264 (1991).
    [CrossRef]
  13. T. Erdogan, O. King, G. W. Wicks, D. G. Hall, C. L. Dennis, and M. J. Rooks, Appl. Phys. Lett. 60, 1773 (1992).
    [CrossRef]
  14. T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, Appl. Phys. Lett. 60, 1921 (1992).
    [CrossRef]
  15. G. W. Ford and W. H. Weber, Phys. Rep. 113, 195 (1984).
    [CrossRef]
  16. H. Kuhn, J. Chem. Phys. 53, 101 (1970).
    [CrossRef]
  17. R. R. Chance, A. Prock, and R. Silbey, in Advances in Chemical Physics, I. Prigogine and S. A. Rice, eds. (Wiley, New York, 1978), Vol. 37, pp. 1–65.
    [CrossRef]
  18. This statement is true only for a nonzero source position; for r0= 0, the source field is inherently singular at the center of the structure. However, this singularity does not affect the analysis, which is concerned only with the well-behaved reflected field at the source position.
  19. T. Erdogan and D. G. Hall, J. Appl. Phys. 68, 1435 (1990).
    [CrossRef]
  20. W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1956), Chap. 20 and Eq. (21-3).
  21. J. M. Wylie and J. E. Sipe, Phys. Rev. A 32, 2030 (1985).
    [CrossRef] [PubMed]

1992 (3)

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, C. L. Dennis, and M. J. Rooks, Appl. Phys. Lett. 60, 1773 (1992).
[CrossRef]

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, Appl. Phys. Lett. 60, 1921 (1992).
[CrossRef]

T. Erdogan and D. G. Hall, IEEE J. Quantum Electron. 28, 612 (1992).
[CrossRef]

1991 (5)

C. Wu, T. Makino, J. Glinski, R. Maciejko, and S. I. Najafi, IEEE J. Lightwave Technol. 9, 1264 (1991).
[CrossRef]

D. G. Deppe and C. Lei, J. Appl. Phys. 70, 3443 (1991).
[CrossRef]

Y. Yamamoto, S. Machida, and G. Björk, Phys. Rev. A 44, 657 (1991), and G. Björk, S. Machida, Y. Yamamoto, and K. Igeta, Phys. Rev. A 44, 669 (1991).
[CrossRef] [PubMed]

P. R. Villeneuve and M. Piché, J. Opt. Soc. Am. A 8, 1296 (1991).
[CrossRef]

M. Plihal and A. A. Maradudin, Phys. Rev. B 44, 8565 (1991).
[CrossRef]

1990 (2)

H. Yokoyama, K. Nishi, T. Anan, H. Yamada, S. D. Brorson, and E. P. Ippen, Appl. Phys. Lett. 57, 2814 (1990).
[CrossRef]

T. Erdogan and D. G. Hall, J. Appl. Phys. 68, 1435 (1990).
[CrossRef]

1987 (1)

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef] [PubMed]

1985 (2)

J. M. Wylie and J. E. Sipe, Phys. Rev. A 32, 2030 (1985).
[CrossRef] [PubMed]

R. G. Hulet, E. S. Hilfer, and D. Kleppner, Phys. Rev. Lett. 55, 2137 (1985).
[CrossRef] [PubMed]

1984 (1)

G. W. Ford and W. H. Weber, Phys. Rep. 113, 195 (1984).
[CrossRef]

1981 (1)

D. Kleppner, Phys. Rev. Lett. 47, 233 (1981).
[CrossRef]

1970 (1)

H. Kuhn, J. Chem. Phys. 53, 101 (1970).
[CrossRef]

1946 (1)

E. M. Purcell, Phys. Rev. 69, 681 (1946).
[CrossRef]

Anan, T.

H. Yokoyama, K. Nishi, T. Anan, H. Yamada, S. D. Brorson, and E. P. Ippen, Appl. Phys. Lett. 57, 2814 (1990).
[CrossRef]

Anderson, E. H.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, Appl. Phys. Lett. 60, 1921 (1992).
[CrossRef]

Björk, G.

Y. Yamamoto, S. Machida, and G. Björk, Phys. Rev. A 44, 657 (1991), and G. Björk, S. Machida, Y. Yamamoto, and K. Igeta, Phys. Rev. A 44, 669 (1991).
[CrossRef] [PubMed]

Brorson, S. D.

H. Yokoyama, K. Nishi, T. Anan, H. Yamada, S. D. Brorson, and E. P. Ippen, Appl. Phys. Lett. 57, 2814 (1990).
[CrossRef]

Chance, R. R.

R. R. Chance, A. Prock, and R. Silbey, in Advances in Chemical Physics, I. Prigogine and S. A. Rice, eds. (Wiley, New York, 1978), Vol. 37, pp. 1–65.
[CrossRef]

Dennis, C. L.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, C. L. Dennis, and M. J. Rooks, Appl. Phys. Lett. 60, 1773 (1992).
[CrossRef]

Deppe, D. G.

D. G. Deppe and C. Lei, J. Appl. Phys. 70, 3443 (1991).
[CrossRef]

Drexhage, K. H.

K. H. Drexhage, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1974), Vol. 12, Chap. 4.
[CrossRef]

Erdogan, T.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, Appl. Phys. Lett. 60, 1921 (1992).
[CrossRef]

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, C. L. Dennis, and M. J. Rooks, Appl. Phys. Lett. 60, 1773 (1992).
[CrossRef]

T. Erdogan and D. G. Hall, IEEE J. Quantum Electron. 28, 612 (1992).
[CrossRef]

T. Erdogan and D. G. Hall, J. Appl. Phys. 68, 1435 (1990).
[CrossRef]

Ford, G. W.

G. W. Ford and W. H. Weber, Phys. Rep. 113, 195 (1984).
[CrossRef]

Glinski, J.

C. Wu, T. Makino, J. Glinski, R. Maciejko, and S. I. Najafi, IEEE J. Lightwave Technol. 9, 1264 (1991).
[CrossRef]

Hall, D. G.

T. Erdogan and D. G. Hall, IEEE J. Quantum Electron. 28, 612 (1992).
[CrossRef]

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, C. L. Dennis, and M. J. Rooks, Appl. Phys. Lett. 60, 1773 (1992).
[CrossRef]

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, Appl. Phys. Lett. 60, 1921 (1992).
[CrossRef]

T. Erdogan and D. G. Hall, J. Appl. Phys. 68, 1435 (1990).
[CrossRef]

Hilfer, E. S.

R. G. Hulet, E. S. Hilfer, and D. Kleppner, Phys. Rev. Lett. 55, 2137 (1985).
[CrossRef] [PubMed]

Hulet, R. G.

R. G. Hulet, E. S. Hilfer, and D. Kleppner, Phys. Rev. Lett. 55, 2137 (1985).
[CrossRef] [PubMed]

Ippen, E. P.

H. Yokoyama, K. Nishi, T. Anan, H. Yamada, S. D. Brorson, and E. P. Ippen, Appl. Phys. Lett. 57, 2814 (1990).
[CrossRef]

King, O.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, C. L. Dennis, and M. J. Rooks, Appl. Phys. Lett. 60, 1773 (1992).
[CrossRef]

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, Appl. Phys. Lett. 60, 1921 (1992).
[CrossRef]

Kleppner, D.

R. G. Hulet, E. S. Hilfer, and D. Kleppner, Phys. Rev. Lett. 55, 2137 (1985).
[CrossRef] [PubMed]

D. Kleppner, Phys. Rev. Lett. 47, 233 (1981).
[CrossRef]

Kuhn, H.

H. Kuhn, J. Chem. Phys. 53, 101 (1970).
[CrossRef]

Lei, C.

D. G. Deppe and C. Lei, J. Appl. Phys. 70, 3443 (1991).
[CrossRef]

Machida, S.

Y. Yamamoto, S. Machida, and G. Björk, Phys. Rev. A 44, 657 (1991), and G. Björk, S. Machida, Y. Yamamoto, and K. Igeta, Phys. Rev. A 44, 669 (1991).
[CrossRef] [PubMed]

Maciejko, R.

C. Wu, T. Makino, J. Glinski, R. Maciejko, and S. I. Najafi, IEEE J. Lightwave Technol. 9, 1264 (1991).
[CrossRef]

Makino, T.

C. Wu, T. Makino, J. Glinski, R. Maciejko, and S. I. Najafi, IEEE J. Lightwave Technol. 9, 1264 (1991).
[CrossRef]

Maradudin, A. A.

M. Plihal and A. A. Maradudin, Phys. Rev. B 44, 8565 (1991).
[CrossRef]

Najafi, S. I.

C. Wu, T. Makino, J. Glinski, R. Maciejko, and S. I. Najafi, IEEE J. Lightwave Technol. 9, 1264 (1991).
[CrossRef]

Nishi, K.

H. Yokoyama, K. Nishi, T. Anan, H. Yamada, S. D. Brorson, and E. P. Ippen, Appl. Phys. Lett. 57, 2814 (1990).
[CrossRef]

Panofsky, W. K. H.

W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1956), Chap. 20 and Eq. (21-3).

Phillips, M.

W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1956), Chap. 20 and Eq. (21-3).

Piché, M.

Plihal, M.

M. Plihal and A. A. Maradudin, Phys. Rev. B 44, 8565 (1991).
[CrossRef]

Prock, A.

R. R. Chance, A. Prock, and R. Silbey, in Advances in Chemical Physics, I. Prigogine and S. A. Rice, eds. (Wiley, New York, 1978), Vol. 37, pp. 1–65.
[CrossRef]

Purcell, E. M.

E. M. Purcell, Phys. Rev. 69, 681 (1946).
[CrossRef]

Rooks, M. J.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, C. L. Dennis, and M. J. Rooks, Appl. Phys. Lett. 60, 1773 (1992).
[CrossRef]

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, Appl. Phys. Lett. 60, 1921 (1992).
[CrossRef]

Silbey, R.

R. R. Chance, A. Prock, and R. Silbey, in Advances in Chemical Physics, I. Prigogine and S. A. Rice, eds. (Wiley, New York, 1978), Vol. 37, pp. 1–65.
[CrossRef]

Sipe, J. E.

J. M. Wylie and J. E. Sipe, Phys. Rev. A 32, 2030 (1985).
[CrossRef] [PubMed]

Villeneuve, P. R.

Weber, W. H.

G. W. Ford and W. H. Weber, Phys. Rep. 113, 195 (1984).
[CrossRef]

Wicks, G. W.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, Appl. Phys. Lett. 60, 1921 (1992).
[CrossRef]

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, C. L. Dennis, and M. J. Rooks, Appl. Phys. Lett. 60, 1773 (1992).
[CrossRef]

Wu, C.

C. Wu, T. Makino, J. Glinski, R. Maciejko, and S. I. Najafi, IEEE J. Lightwave Technol. 9, 1264 (1991).
[CrossRef]

Wylie, J. M.

J. M. Wylie and J. E. Sipe, Phys. Rev. A 32, 2030 (1985).
[CrossRef] [PubMed]

Yablonovitch, E.

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef] [PubMed]

Yamada, H.

H. Yokoyama, K. Nishi, T. Anan, H. Yamada, S. D. Brorson, and E. P. Ippen, Appl. Phys. Lett. 57, 2814 (1990).
[CrossRef]

Yamamoto, Y.

Y. Yamamoto, S. Machida, and G. Björk, Phys. Rev. A 44, 657 (1991), and G. Björk, S. Machida, Y. Yamamoto, and K. Igeta, Phys. Rev. A 44, 669 (1991).
[CrossRef] [PubMed]

Yokoyama, H.

H. Yokoyama, K. Nishi, T. Anan, H. Yamada, S. D. Brorson, and E. P. Ippen, Appl. Phys. Lett. 57, 2814 (1990).
[CrossRef]

Appl. Phys. Lett. (3)

H. Yokoyama, K. Nishi, T. Anan, H. Yamada, S. D. Brorson, and E. P. Ippen, Appl. Phys. Lett. 57, 2814 (1990).
[CrossRef]

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, C. L. Dennis, and M. J. Rooks, Appl. Phys. Lett. 60, 1773 (1992).
[CrossRef]

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, and M. J. Rooks, Appl. Phys. Lett. 60, 1921 (1992).
[CrossRef]

IEEE J. Lightwave Technol. (1)

C. Wu, T. Makino, J. Glinski, R. Maciejko, and S. I. Najafi, IEEE J. Lightwave Technol. 9, 1264 (1991).
[CrossRef]

IEEE J. Quantum Electron. (1)

T. Erdogan and D. G. Hall, IEEE J. Quantum Electron. 28, 612 (1992).
[CrossRef]

J. Appl. Phys. (2)

D. G. Deppe and C. Lei, J. Appl. Phys. 70, 3443 (1991).
[CrossRef]

T. Erdogan and D. G. Hall, J. Appl. Phys. 68, 1435 (1990).
[CrossRef]

J. Chem. Phys. (1)

H. Kuhn, J. Chem. Phys. 53, 101 (1970).
[CrossRef]

J. Opt. Soc. Am. A (1)

Phys. Rep. (1)

G. W. Ford and W. H. Weber, Phys. Rep. 113, 195 (1984).
[CrossRef]

Phys. Rev. (1)

E. M. Purcell, Phys. Rev. 69, 681 (1946).
[CrossRef]

Phys. Rev. A (2)

Y. Yamamoto, S. Machida, and G. Björk, Phys. Rev. A 44, 657 (1991), and G. Björk, S. Machida, Y. Yamamoto, and K. Igeta, Phys. Rev. A 44, 669 (1991).
[CrossRef] [PubMed]

J. M. Wylie and J. E. Sipe, Phys. Rev. A 32, 2030 (1985).
[CrossRef] [PubMed]

Phys. Rev. B (1)

M. Plihal and A. A. Maradudin, Phys. Rev. B 44, 8565 (1991).
[CrossRef]

Phys. Rev. Lett. (3)

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef] [PubMed]

D. Kleppner, Phys. Rev. Lett. 47, 233 (1981).
[CrossRef]

R. G. Hulet, E. S. Hilfer, and D. Kleppner, Phys. Rev. Lett. 55, 2137 (1985).
[CrossRef] [PubMed]

Other (4)

K. H. Drexhage, in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1974), Vol. 12, Chap. 4.
[CrossRef]

R. R. Chance, A. Prock, and R. Silbey, in Advances in Chemical Physics, I. Prigogine and S. A. Rice, eds. (Wiley, New York, 1978), Vol. 37, pp. 1–65.
[CrossRef]

This statement is true only for a nonzero source position; for r0= 0, the source field is inherently singular at the center of the structure. However, this singularity does not affect the analysis, which is concerned only with the well-behaved reflected field at the source position.

W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1956), Chap. 20 and Eq. (21-3).

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Figures (7)

Fig. 1
Fig. 1

Scanning electron micrographs of a typical grating resonator for a concentric-circle grating, surface-emitting semiconductor laser.

Fig. 2
Fig. 2

Illustration of the geometry of the current line source and the cylindrical periodic structure. Both source and structure extend uniformly to z = ±∞.

Fig. 3
Fig. 3

Normalized radiative damping rate versus grating phase for a full range of phases (0 → 2π) and source placement at the center.

Fig. 4
Fig. 4

Normalized radiative damping rate versus source position for various index modulation strengths: (a) grating phase Ω chosen to give maximum damping rate (maximized enhancement) in Fig. 3, (b) grating phase Ω chosen to give minimum damping rate (maximized inhibition) in Fig. 3, (c) grating phase Ω = 3π/2.

Fig. 5
Fig. 5

Normalized radiative damping rate versus source position for various inner radii of a 10-period Bragg region.

Fig. 6
Fig. 6

Normalized radiative damping rate versus detuning of the source frequency from the Bragg frequency defined by the Bragg structure. The line source is placed at the center, and the grating phase is chosen to give inhibition at the Bragg frequency.

Fig. 7
Fig. 7

Cavity-induced frequency shift of the line source versus grating phase for a full range of phases (0 → 2π) and source placement at the center.

Equations (30)

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p ¨ + ω 0 2 p = ( q 2 / m ) E R - b 0 p ˙ ,
p = p 0 exp [ - ( i ω + b / 2 ) t ] ,
E R = E 0 exp [ - ( i ω + b / 2 ) t ] ,
b b 0 = 1 + q 2 m ω p 0 b 0 Im ( E 0 ) ,
ω 2 - ω 0 2 = b 2 4 - b b 0 2 - q 2 m p 0 Re ( E 0 ) .
Δ ω b 2 8 ω 0 - b b 0 4 ω 0 - q 2 2 m ω 0 p 0 Re ( E 0 ) .
E 0 = E S Γ .
J ( r , ϕ ) = - i ω 0 p 0 δ ( r - r 0 ) z ^ = - i ω 0 p 0 ( 1 / r ) δ ( r - r 0 ) δ ( ϕ - ϕ 0 ) z ^ ,
× × E ( r , ϕ ) - μ 0 ω 0 2 E ( r , ϕ ) = i μ 0 ω 0 J ( r , ϕ ) ,
( 2 + k 0 2 ) E z ( r , ϕ ) = - μ 0 ω 0 2 p 0 δ ( r - r 0 ) ,
E z ( r , ϕ ) = E S H 0 ( 1 ) ( k 0 r - r 0 ) ,
E S i p 0 ω 0 2 4 0 c 2
E z S ( r , ϕ ) = E S m = - exp [ i m ( ϕ - ϕ 0 ) ] × { J m ( k 0 r ) H m ( 1 ) ( k 0 r 0 ) r < r 0 J m ( k 0 r 0 ) H m ( 1 ) ( k 0 r ) r > r 0 ,
E z S ( r , ϕ ) = E S m = - C m H m ( 1 ) ( k 0 r ) exp ( i m ϕ )             r > r 0 ,
C m exp ( - i m ϕ 0 ) J m ( k 0 r 0 ) .
E z T = E z S + E z R ,
E z R = m = - [ a m H m ( 1 ) ( k 0 r ) + b m H m ( 2 ) ( k 0 r ) ] exp ( i m ϕ ) ,
a m = b m .
b m H m ( 2 ) ( k 0 r 1 ) = ρ m [ a m H m ( 1 ) ( k 0 r 1 ) + E S C m H m ( 1 ) ( k 0 r 1 ) ] .
d A m d r = i π k 0 r 2 K ( r ) H m ( 2 ) ( k 0 r ) × [ A m ( r ) H m ( 1 ) ( k 0 r ) + B m ( r ) H m ( 2 ) ( k 0 r ) ] ,
d B m d r = - i π k 0 r 2 K ( r ) H m ( 1 ) ( k 0 r ) × [ A m ( r ) H m ( 1 ) ( k 0 r ) + B m ( r ) H m ( 2 ) ( k 0 r ) ] .
K ( r ) = 2 κ cos ( 2 π Λ r - Ω ) ,
κ = k 0 4 n 2 Δ ( n 2 ) .
ρ m = B m ( r 1 ) A m ( r 1 ) .
Γ = m = - 2 ρ m H m ( 1 ) ( k 0 r 1 ) H m ( 2 ) ( k 0 r 1 ) - ρ m H m ( 1 ) ( k 0 r 1 ) [ J m ( k 0 r 0 ) ] 2
P = 2 μ 0 ω 0 E S 2 .
b 0 = q 2 ω 0 4 0 m c 2 .
b / b 0 = 1 + Re ( Γ ) ,
Δ ω b 0 2 8 ω 0 { [ Re ( Γ ) ] 2 - 1 } + b 0 2 Im ( Γ ) .
Δ ω ( b 0 / 2 ) Im ( Γ ) .

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