Abstract

We show explicitly how the commonly adopted prescription for calculating effective mode volumes is wrong and leads to uncontrolled errors. Instead, we introduce a generalized mode volume that can be easily evaluated based on the mode calculation methods typically applied in the literature, and which allows one to compute the Purcell effect and other interesting optical phenomena in a rigorous and unambiguous way.

© 2012 Optical Society of America

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References

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  1. K. J. Vahala, Nature 424, 839 (2003).
    [CrossRef]
  2. E. M. Purcell, Phys. Rev. 69, 674 (1946).
    [CrossRef]
  3. T. J. A. Kippenberg, “Nonlinear optics in ulta-high-Q whispering-gallery optical microcavities,” Ph.D. thesis, (Caltech, 2004).
  4. K. Srinivasan, P. E. Barclay, M. Borselli, and O. Painter, Phys. Rev. B 70, 081306R (2004).
    [CrossRef]
  5. A. F. Koenderink, Opt. Lett. 35, 4208 (2010).
    [CrossRef]
  6. K. M. Lee, P. T. Leung, and K. M. Pang, J. Opt. Soc. Am. B 16, 1409 (1999).
    [CrossRef]
  7. P. Martin, Multiple Scattering (Cambridge, 2006).
  8. P. Yao, V. S. C. Manga Rao, and S. Hughes, Laser Photon. Rev. 4, 499 (2010).
    [CrossRef]
  9. O. J. F. Martin and N. B. Piller, Phys. Rev. E 58, 3909 (1998).
    [CrossRef]
  10. P. T. Kristensen, P. Lodahl, and J. Mørk, J. Opt. Soc. Am. B 27, 228 (2010).
    [CrossRef]
  11. Lumerical FDTD Solutions: www.lumerical.com .

2010 (3)

2004 (1)

K. Srinivasan, P. E. Barclay, M. Borselli, and O. Painter, Phys. Rev. B 70, 081306R (2004).
[CrossRef]

2003 (1)

K. J. Vahala, Nature 424, 839 (2003).
[CrossRef]

1999 (1)

1998 (1)

O. J. F. Martin and N. B. Piller, Phys. Rev. E 58, 3909 (1998).
[CrossRef]

1946 (1)

E. M. Purcell, Phys. Rev. 69, 674 (1946).
[CrossRef]

Barclay, P. E.

K. Srinivasan, P. E. Barclay, M. Borselli, and O. Painter, Phys. Rev. B 70, 081306R (2004).
[CrossRef]

Borselli, M.

K. Srinivasan, P. E. Barclay, M. Borselli, and O. Painter, Phys. Rev. B 70, 081306R (2004).
[CrossRef]

Hughes, S.

P. Yao, V. S. C. Manga Rao, and S. Hughes, Laser Photon. Rev. 4, 499 (2010).
[CrossRef]

Kippenberg, T. J. A.

T. J. A. Kippenberg, “Nonlinear optics in ulta-high-Q whispering-gallery optical microcavities,” Ph.D. thesis, (Caltech, 2004).

Koenderink, A. F.

Kristensen, P. T.

Lee, K. M.

Leung, P. T.

Lodahl, P.

Manga Rao, V. S. C.

P. Yao, V. S. C. Manga Rao, and S. Hughes, Laser Photon. Rev. 4, 499 (2010).
[CrossRef]

Martin, O. J. F.

O. J. F. Martin and N. B. Piller, Phys. Rev. E 58, 3909 (1998).
[CrossRef]

Martin, P.

P. Martin, Multiple Scattering (Cambridge, 2006).

Mørk, J.

Painter, O.

K. Srinivasan, P. E. Barclay, M. Borselli, and O. Painter, Phys. Rev. B 70, 081306R (2004).
[CrossRef]

Pang, K. M.

Piller, N. B.

O. J. F. Martin and N. B. Piller, Phys. Rev. E 58, 3909 (1998).
[CrossRef]

Purcell, E. M.

E. M. Purcell, Phys. Rev. 69, 674 (1946).
[CrossRef]

Srinivasan, K.

K. Srinivasan, P. E. Barclay, M. Borselli, and O. Painter, Phys. Rev. B 70, 081306R (2004).
[CrossRef]

Vahala, K. J.

K. J. Vahala, Nature 424, 839 (2003).
[CrossRef]

Yao, P.

P. Yao, V. S. C. Manga Rao, and S. Hughes, Laser Photon. Rev. 4, 499 (2010).
[CrossRef]

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

Laser Photon. Rev. (1)

P. Yao, V. S. C. Manga Rao, and S. Hughes, Laser Photon. Rev. 4, 499 (2010).
[CrossRef]

Nature (1)

K. J. Vahala, Nature 424, 839 (2003).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. (1)

E. M. Purcell, Phys. Rev. 69, 674 (1946).
[CrossRef]

Phys. Rev. B (1)

K. Srinivasan, P. E. Barclay, M. Borselli, and O. Painter, Phys. Rev. B 70, 081306R (2004).
[CrossRef]

Phys. Rev. E (1)

O. J. F. Martin and N. B. Piller, Phys. Rev. E 58, 3909 (1998).
[CrossRef]

Other (3)

Lumerical FDTD Solutions: www.lumerical.com .

P. Martin, Multiple Scattering (Cambridge, 2006).

T. J. A. Kippenberg, “Nonlinear optics in ulta-high-Q whispering-gallery optical microcavities,” Ph.D. thesis, (Caltech, 2004).

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Figures (4)

Fig. 1.
Fig. 1.

Sketch of a photonic crystal (lattice constant a) in a membrane of high refractive index. A defect cavity is formed by the omission of a single hole. Right: absolute value of the cavity mode in the planes z=0 (top) and y=0 (bottom).

Fig. 2.
Fig. 2.

(a) Field along the x-axis of the cavity mode in the 2D crystallite for the case of N=1. Blue solid line shows the Fredholm type solution, and black circles show the calculation using FDTD. Inset shows long distance behavior on a logarithmic scale. (b) Field along the x-axis of the cavity mode for the case of N=2. Inset shows the field distribution in the xy-plane.

Fig. 3.
Fig. 3.

Effective mode volumes VeffN (thick lines) and VeffQ (thin lines) for N=1 (red dash-dotted), N=2 (green dashed), and N=3 (blue solid) as a function of radius R of the calculation domain. Circles indicate reference mode volumes Vefftot from independent Green’s tensor calculations [10].

Fig. 4.
Fig. 4.

Effective mode volume VeffN (red dashed) and VeffQ (blue solid) for the cavity in Fig. 1 as a function of height of the calculation domain Lz. Circles indicate reference mode volumes Vefftot derived from independent Green’s tensor calculations [8].

Equations (6)

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FP=34π2(λcnc)3(QVeff),
VeffN=Vϵr(r)|f~c(r)|2ϵr(rc)|f~c(rc)|2dr
××E(r,ω)k02ϵr(r)E(r,ω)=0,
f~μ|f~λ=limVVϵr(r)f~μ(r)·f~λ(r)dr+iϵrcω~μ+ω~λVf~μ(r)·f~λ(r)dr=δμ,λ,
1VeffQ=Re{1vQ},vQ=f~c|f~cϵr(rc)f~c2(rc),
E(r,ω)=(ωc)2VGB(r,r,ω)Δϵ(r)E(r,ω)dr,

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