Abstract

We show how an electromagnetic cavity with arbitrarily high Q and small (bounded) modal volume can be formed in two or three dimensions with a proper choice of dielectric constants. Unlike in previous work, neither a complete photonic bandgap nor a trade-off in mode localization for Q is required. Rather, scattering and radiation are prohibited by a perfect mode match of the TE-polarized modes in each subsection of a Bragg resonator. Q values in excess of 105 are demonstrated through finite-difference time-domain simulations of two- and three-dimensional structures with modal areas or volumes of the order of the wavelength squared or cubed.

© 2002 Optical Society of America

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References

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  1. E. M. Purcell, Phys. Rev. B 69, 681 (1946).
    [CrossRef]
  2. R. Coccioli, M. Boroditsky, K. W. Kim, Y. Rahmat-Samii, and E. Yablonovitch, IEE Proc. Optoelectron 145, 391 (1998).
    [CrossRef]
  3. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984), Chap. 7.
  4. P. R. Villeneuve, S. Fan, S. G. Johnson, and J. D. Joannopoulos, IEE Proc. Optoelectron. 145, 384 (1998).
    [CrossRef]
  5. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).
  6. P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, Phys. Rev. B 54, 7837 (1996).
    [CrossRef]
  7. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
    [CrossRef]
  8. H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Béraud, and C. Jouanin, Appl. Phys. Lett. 76, 532 (2000).
    [CrossRef]
  9. S. G. Johnson, S. Fan, A. Mekis, and J. D. Joannopoulos, Appl. Phys. Lett. 78, 3388 (2001).
    [CrossRef]
  10. S. G. Johnson, A. Mekis, S. Fan, and J. D. Joannopoulos, Computing Sci. Eng. 3, 38 (2001).
    [CrossRef]
  11. J. Vučković, M. Lončar, H. Mabuchi, and A. Scherer, Phys. Rev. E 65, 016608 (2002).
  12. J. Čtyroký, J. Opt. Soc. Am. A 18, 435 (2001).
    [CrossRef]
  13. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).
  14. K. S. Kunz and R. J. Luebbers, The Finite–Difference Time-Domain Method for Electromagnetics (CRC, Boca Raton, Fla., 1993)

2002 (1)

J. Vučković, M. Lončar, H. Mabuchi, and A. Scherer, Phys. Rev. E 65, 016608 (2002).

2001 (3)

J. Čtyroký, J. Opt. Soc. Am. A 18, 435 (2001).
[CrossRef]

S. G. Johnson, S. Fan, A. Mekis, and J. D. Joannopoulos, Appl. Phys. Lett. 78, 3388 (2001).
[CrossRef]

S. G. Johnson, A. Mekis, S. Fan, and J. D. Joannopoulos, Computing Sci. Eng. 3, 38 (2001).
[CrossRef]

2000 (1)

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Béraud, and C. Jouanin, Appl. Phys. Lett. 76, 532 (2000).
[CrossRef]

1998 (2)

R. Coccioli, M. Boroditsky, K. W. Kim, Y. Rahmat-Samii, and E. Yablonovitch, IEE Proc. Optoelectron 145, 391 (1998).
[CrossRef]

P. R. Villeneuve, S. Fan, S. G. Johnson, and J. D. Joannopoulos, IEE Proc. Optoelectron. 145, 384 (1998).
[CrossRef]

1996 (1)

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, Phys. Rev. B 54, 7837 (1996).
[CrossRef]

1946 (1)

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

Benisty, H.

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Béraud, and C. Jouanin, Appl. Phys. Lett. 76, 532 (2000).
[CrossRef]

Béraud, A.

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Béraud, and C. Jouanin, Appl. Phys. Lett. 76, 532 (2000).
[CrossRef]

Boroditsky, M.

R. Coccioli, M. Boroditsky, K. W. Kim, Y. Rahmat-Samii, and E. Yablonovitch, IEE Proc. Optoelectron 145, 391 (1998).
[CrossRef]

Cassagne, D.

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Béraud, and C. Jouanin, Appl. Phys. Lett. 76, 532 (2000).
[CrossRef]

Coccioli, R.

R. Coccioli, M. Boroditsky, K. W. Kim, Y. Rahmat-Samii, and E. Yablonovitch, IEE Proc. Optoelectron 145, 391 (1998).
[CrossRef]

Ctyroký, J.

Fan, S.

S. G. Johnson, S. Fan, A. Mekis, and J. D. Joannopoulos, Appl. Phys. Lett. 78, 3388 (2001).
[CrossRef]

S. G. Johnson, A. Mekis, S. Fan, and J. D. Joannopoulos, Computing Sci. Eng. 3, 38 (2001).
[CrossRef]

P. R. Villeneuve, S. Fan, S. G. Johnson, and J. D. Joannopoulos, IEE Proc. Optoelectron. 145, 384 (1998).
[CrossRef]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, Phys. Rev. B 54, 7837 (1996).
[CrossRef]

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984), Chap. 7.

Joannopoulos, J. D.

S. G. Johnson, A. Mekis, S. Fan, and J. D. Joannopoulos, Computing Sci. Eng. 3, 38 (2001).
[CrossRef]

S. G. Johnson, S. Fan, A. Mekis, and J. D. Joannopoulos, Appl. Phys. Lett. 78, 3388 (2001).
[CrossRef]

P. R. Villeneuve, S. Fan, S. G. Johnson, and J. D. Joannopoulos, IEE Proc. Optoelectron. 145, 384 (1998).
[CrossRef]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, Phys. Rev. B 54, 7837 (1996).
[CrossRef]

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).

Johnson, S. G.

S. G. Johnson, A. Mekis, S. Fan, and J. D. Joannopoulos, Computing Sci. Eng. 3, 38 (2001).
[CrossRef]

S. G. Johnson, S. Fan, A. Mekis, and J. D. Joannopoulos, Appl. Phys. Lett. 78, 3388 (2001).
[CrossRef]

P. R. Villeneuve, S. Fan, S. G. Johnson, and J. D. Joannopoulos, IEE Proc. Optoelectron. 145, 384 (1998).
[CrossRef]

Jouanin, C.

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Béraud, and C. Jouanin, Appl. Phys. Lett. 76, 532 (2000).
[CrossRef]

Kim, K. W.

R. Coccioli, M. Boroditsky, K. W. Kim, Y. Rahmat-Samii, and E. Yablonovitch, IEE Proc. Optoelectron 145, 391 (1998).
[CrossRef]

Krauss, T. F.

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Béraud, and C. Jouanin, Appl. Phys. Lett. 76, 532 (2000).
[CrossRef]

Kunz, K. S.

K. S. Kunz and R. J. Luebbers, The Finite–Difference Time-Domain Method for Electromagnetics (CRC, Boca Raton, Fla., 1993)

Labilloy, D.

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Béraud, and C. Jouanin, Appl. Phys. Lett. 76, 532 (2000).
[CrossRef]

Loncar, M.

J. Vučković, M. Lončar, H. Mabuchi, and A. Scherer, Phys. Rev. E 65, 016608 (2002).

Luebbers, R. J.

K. S. Kunz and R. J. Luebbers, The Finite–Difference Time-Domain Method for Electromagnetics (CRC, Boca Raton, Fla., 1993)

Mabuchi, H.

J. Vučković, M. Lončar, H. Mabuchi, and A. Scherer, Phys. Rev. E 65, 016608 (2002).

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).

Mekis, A.

S. G. Johnson, S. Fan, A. Mekis, and J. D. Joannopoulos, Appl. Phys. Lett. 78, 3388 (2001).
[CrossRef]

S. G. Johnson, A. Mekis, S. Fan, and J. D. Joannopoulos, Computing Sci. Eng. 3, 38 (2001).
[CrossRef]

Purcell, E. M.

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

Rahmat-Samii, Y.

R. Coccioli, M. Boroditsky, K. W. Kim, Y. Rahmat-Samii, and E. Yablonovitch, IEE Proc. Optoelectron 145, 391 (1998).
[CrossRef]

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[CrossRef]

Scherer, A.

J. Vučković, M. Lončar, H. Mabuchi, and A. Scherer, Phys. Rev. E 65, 016608 (2002).

Smith, C. J. M.

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Béraud, and C. Jouanin, Appl. Phys. Lett. 76, 532 (2000).
[CrossRef]

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[CrossRef]

Villeneuve, P. R.

P. R. Villeneuve, S. Fan, S. G. Johnson, and J. D. Joannopoulos, IEE Proc. Optoelectron. 145, 384 (1998).
[CrossRef]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, Phys. Rev. B 54, 7837 (1996).
[CrossRef]

Vuckovic, J.

J. Vučković, M. Lončar, H. Mabuchi, and A. Scherer, Phys. Rev. E 65, 016608 (2002).

Weisbuch, C.

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Béraud, and C. Jouanin, Appl. Phys. Lett. 76, 532 (2000).
[CrossRef]

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).

Yablonovitch, E.

R. Coccioli, M. Boroditsky, K. W. Kim, Y. Rahmat-Samii, and E. Yablonovitch, IEE Proc. Optoelectron 145, 391 (1998).
[CrossRef]

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

Appl. Phys. Lett. (2)

H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Béraud, and C. Jouanin, Appl. Phys. Lett. 76, 532 (2000).
[CrossRef]

S. G. Johnson, S. Fan, A. Mekis, and J. D. Joannopoulos, Appl. Phys. Lett. 78, 3388 (2001).
[CrossRef]

Computing Sci. Eng. (1)

S. G. Johnson, A. Mekis, S. Fan, and J. D. Joannopoulos, Computing Sci. Eng. 3, 38 (2001).
[CrossRef]

IEE Proc. Optoelectron (1)

R. Coccioli, M. Boroditsky, K. W. Kim, Y. Rahmat-Samii, and E. Yablonovitch, IEE Proc. Optoelectron 145, 391 (1998).
[CrossRef]

IEE Proc. Optoelectron. (1)

P. R. Villeneuve, S. Fan, S. G. Johnson, and J. D. Joannopoulos, IEE Proc. Optoelectron. 145, 384 (1998).
[CrossRef]

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

Phys. Rev. B (2)

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, Phys. Rev. B 54, 7837 (1996).
[CrossRef]

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

Phys. Rev. E (1)

J. Vučković, M. Lončar, H. Mabuchi, and A. Scherer, Phys. Rev. E 65, 016608 (2002).

Other (5)

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984), Chap. 7.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[CrossRef]

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

K. S. Kunz and R. J. Luebbers, The Finite–Difference Time-Domain Method for Electromagnetics (CRC, Boca Raton, Fla., 1993)

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Figures (4)

Fig. 1
Fig. 1

Schematic junction of two step-index waveguides.

Fig. 2
Fig. 2

Two dimensional x-z Fabry–Perot cavity, with in gray scale, confining a TE mode whose field Ey is shown as blue (red) for negative (positive). The cavity consists of alternating index-guided waveguides with core/cladding indices n1/n2 and n˜1/n˜2, where the core has width a and the n2 regions have finite width T. The indices satisfy Eq. (2), which ensures zero radiation losses at the waveguide interfaces.

Fig. 3
Fig. 3

Q as a function of cladding thickness T for the two-dimensional cavity of Fig. 2 and the three-dimensional cavity for Fig. 4, below, for different numbers N of Bragg periods on either side of the cavity. Q increases exponentially with T or N, depending on which one is limiting the Q.

Fig. 4
Fig. 4

Schematic of a three-dimensional Fabry–Perot cavity consisting of alternating index-guide cylindrical waveguides stacked in the z direction with core/cladding indices n1/n2 and n˜1/n˜2, where the core (seen in cutaway at the bottom) has diameter a and the cladding has diameter T. When the indices satisfy Eq. (2), the TE01 mode does not radiate at the waveguide interfaces.

Equations (6)

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

T2+μ0iω2-β2E=0,
1-2=˜1-˜2.
zˆz×ET=-jωμ0HT=-jβzˆ×ET.
ET1+r=t·E˜T,
HT1-r=t·H˜T,
r=neff-n˜effneff+n˜eff,

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