Abstract

We derive the quasi-orthogonality condition of optical modes in a dispersive and inhomogeneous medium. This condition leads to the proper normalization rule of the optical field in a general linear medium with inhomogeneity. The derived rule also provides a physically meaningful method in estimating the magnitude of the field-enhanced spontaneous and stimulated emission rates, for example, in a microcavity or nanocavity.

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References

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  1. J. M. Gerard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, Phys. Rev. Lett. 81, 1110 (1998).
    [CrossRef]
  2. G. S. Solomon, M. Pelton, and Y. Yamamoto, Phys. Rev. Lett. 86, 3903 (2001).
    [CrossRef] [PubMed]
  3. Y. Xu, R. K. Lee, and A. Yariv, Phys. Rev. A 61, 033808 (2000).
    [CrossRef]
  4. R. J. Glauber and M. Lewenstein, Phys. Rev. A 43, 467 (1991).
    [CrossRef] [PubMed]
  5. L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).
  6. E. M. Lifshitz, L. D. Landau, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Elsevier Butterworth-Heinemann, 1984).
  7. P. D. Drummond, Phys. Rev. A 42, 6845 (1990).
    [CrossRef] [PubMed]
  8. B. Huttner and S. M. Barnett, Phys. Rev. A 46, 4306 (1992).
    [CrossRef] [PubMed]
  9. R. Matloob, R. Loudon, S. M. Barnett, and J. Jeffers, Phys. Rev. A 52, 4823 (1995).
    [CrossRef] [PubMed]
  10. R. Matloob and R. Loudon, Phys. Rev. A 53, 4567 (1996).
    [CrossRef] [PubMed]

2001 (1)

G. S. Solomon, M. Pelton, and Y. Yamamoto, Phys. Rev. Lett. 86, 3903 (2001).
[CrossRef] [PubMed]

2000 (1)

Y. Xu, R. K. Lee, and A. Yariv, Phys. Rev. A 61, 033808 (2000).
[CrossRef]

1998 (1)

J. M. Gerard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, Phys. Rev. Lett. 81, 1110 (1998).
[CrossRef]

1996 (1)

R. Matloob and R. Loudon, Phys. Rev. A 53, 4567 (1996).
[CrossRef] [PubMed]

1995 (1)

R. Matloob, R. Loudon, S. M. Barnett, and J. Jeffers, Phys. Rev. A 52, 4823 (1995).
[CrossRef] [PubMed]

1992 (1)

B. Huttner and S. M. Barnett, Phys. Rev. A 46, 4306 (1992).
[CrossRef] [PubMed]

1991 (1)

R. J. Glauber and M. Lewenstein, Phys. Rev. A 43, 467 (1991).
[CrossRef] [PubMed]

1990 (1)

P. D. Drummond, Phys. Rev. A 42, 6845 (1990).
[CrossRef] [PubMed]

Barnett, S. M.

R. Matloob, R. Loudon, S. M. Barnett, and J. Jeffers, Phys. Rev. A 52, 4823 (1995).
[CrossRef] [PubMed]

B. Huttner and S. M. Barnett, Phys. Rev. A 46, 4306 (1992).
[CrossRef] [PubMed]

Brillouin, L.

L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).

Costard, E.

J. M. Gerard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, Phys. Rev. Lett. 81, 1110 (1998).
[CrossRef]

Drummond, P. D.

P. D. Drummond, Phys. Rev. A 42, 6845 (1990).
[CrossRef] [PubMed]

Gayral, B.

J. M. Gerard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, Phys. Rev. Lett. 81, 1110 (1998).
[CrossRef]

Gerard, J. M.

J. M. Gerard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, Phys. Rev. Lett. 81, 1110 (1998).
[CrossRef]

Glauber, R. J.

R. J. Glauber and M. Lewenstein, Phys. Rev. A 43, 467 (1991).
[CrossRef] [PubMed]

Huttner, B.

B. Huttner and S. M. Barnett, Phys. Rev. A 46, 4306 (1992).
[CrossRef] [PubMed]

Jeffers, J.

R. Matloob, R. Loudon, S. M. Barnett, and J. Jeffers, Phys. Rev. A 52, 4823 (1995).
[CrossRef] [PubMed]

Landau, L. D.

E. M. Lifshitz, L. D. Landau, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Elsevier Butterworth-Heinemann, 1984).

Lee, R. K.

Y. Xu, R. K. Lee, and A. Yariv, Phys. Rev. A 61, 033808 (2000).
[CrossRef]

Legrand, B.

J. M. Gerard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, Phys. Rev. Lett. 81, 1110 (1998).
[CrossRef]

Lewenstein, M.

R. J. Glauber and M. Lewenstein, Phys. Rev. A 43, 467 (1991).
[CrossRef] [PubMed]

Lifshitz, E. M.

E. M. Lifshitz, L. D. Landau, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Elsevier Butterworth-Heinemann, 1984).

Loudon, R.

R. Matloob and R. Loudon, Phys. Rev. A 53, 4567 (1996).
[CrossRef] [PubMed]

R. Matloob, R. Loudon, S. M. Barnett, and J. Jeffers, Phys. Rev. A 52, 4823 (1995).
[CrossRef] [PubMed]

Matloob, R.

R. Matloob and R. Loudon, Phys. Rev. A 53, 4567 (1996).
[CrossRef] [PubMed]

R. Matloob, R. Loudon, S. M. Barnett, and J. Jeffers, Phys. Rev. A 52, 4823 (1995).
[CrossRef] [PubMed]

Pelton, M.

G. S. Solomon, M. Pelton, and Y. Yamamoto, Phys. Rev. Lett. 86, 3903 (2001).
[CrossRef] [PubMed]

Pitaevskii, L. P.

E. M. Lifshitz, L. D. Landau, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Elsevier Butterworth-Heinemann, 1984).

Sermage, B.

J. M. Gerard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, Phys. Rev. Lett. 81, 1110 (1998).
[CrossRef]

Solomon, G. S.

G. S. Solomon, M. Pelton, and Y. Yamamoto, Phys. Rev. Lett. 86, 3903 (2001).
[CrossRef] [PubMed]

Thierry-Mieg, V.

J. M. Gerard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, Phys. Rev. Lett. 81, 1110 (1998).
[CrossRef]

Xu, Y.

Y. Xu, R. K. Lee, and A. Yariv, Phys. Rev. A 61, 033808 (2000).
[CrossRef]

Yamamoto, Y.

G. S. Solomon, M. Pelton, and Y. Yamamoto, Phys. Rev. Lett. 86, 3903 (2001).
[CrossRef] [PubMed]

Yariv, A.

Y. Xu, R. K. Lee, and A. Yariv, Phys. Rev. A 61, 033808 (2000).
[CrossRef]

Phys. Rev. A (6)

Y. Xu, R. K. Lee, and A. Yariv, Phys. Rev. A 61, 033808 (2000).
[CrossRef]

R. J. Glauber and M. Lewenstein, Phys. Rev. A 43, 467 (1991).
[CrossRef] [PubMed]

P. D. Drummond, Phys. Rev. A 42, 6845 (1990).
[CrossRef] [PubMed]

B. Huttner and S. M. Barnett, Phys. Rev. A 46, 4306 (1992).
[CrossRef] [PubMed]

R. Matloob, R. Loudon, S. M. Barnett, and J. Jeffers, Phys. Rev. A 52, 4823 (1995).
[CrossRef] [PubMed]

R. Matloob and R. Loudon, Phys. Rev. A 53, 4567 (1996).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

J. M. Gerard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, Phys. Rev. Lett. 81, 1110 (1998).
[CrossRef]

G. S. Solomon, M. Pelton, and Y. Yamamoto, Phys. Rev. Lett. 86, 3903 (2001).
[CrossRef] [PubMed]

Other (2)

L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).

E. M. Lifshitz, L. D. Landau, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Elsevier Butterworth-Heinemann, 1984).

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Figures (2)

Fig. 1
Fig. 1

Three permittivities ϵ R ( ω ) , ϵ g ( ω ) , and [ ϵ g ( ω ) + ϵ R ( ω ) ] 2 as a function of the frequency ratio ω ω p .

Fig. 2
Fig. 2

Spontaneous emission rate as a function of the transition wavelength. The inset shows the geometry of the cavity.

Equations (22)

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u eff , e = ϵ 0 ϵ g ( r , ω 0 ) 4 E s ( r , t ) 2 ,
u eff , m = μ 0 μ g ( r , ω 0 ) 4 H s ( r , t ) 2 ,
A ( r , t ) = n / ( 2 ϵ 0 ω n ) [ f n ( r ) a n e i ω n t + H.c. ] ,
× μ 1 ( r , ω n ) × f n ( r ) = ( ω n / c ) 2 ϵ ( r , ω n ) f n ( r ) ,
[ ϵ ( r , ω n ) f n ( r ) ] = 0 .
E s ( r , t ) = n i ( 2 ω n ) / ϵ 0 f n ( r ) a n e i ( ω n ω 0 ) t ,
H s ( r , t ) = n 2 ϵ 0 ω n × f n ( r ) μ 0 μ ( r , ω n ) a n e i ( ω n ω 0 ) t .
U eff , em = n , n ω n ω n I n n e i ( ω n ω n ) t a n a n ,
I n n = V d r 1 2 { ϵ g ( r , ω 0 ) f n * ( r ) f n ( r ) + c 2 μ g ( r , ω 0 ) × f n * ( r ) × f n ( r ) ω n μ ( r , ω n ) ω n μ ( r , ω n ) } ,
I n n = V d r 1 2 { ϵ g ( r , ω 0 ) f n * ( r ) f n ( r ) + c 2 μ g ( r , ω 0 ) [ ω 0 μ ( r , ω 0 ) ] 2 × × f n * ( r ) × f n ( r ) } + O ( Δ ω ̃ n + Δ ω ̃ n ) ,
V d r [ f n * ( r ) ω n ϵ ( r , ω n ) f n ( r ) ] V d r [ c 2 × f n * ( r ) × f n ( r ) ω n μ ( r , ω n ) ] = 0 .
ω n ϵ ( r , ω n ) ω 0 ϵ ( r , ω 0 ) + ( ω n ω 0 ) ϵ g ( r , ω 0 ) ,
1 ω n μ ( r , ω n ) 1 ω 0 μ ( r , ω 0 ) ( ω n ω 0 ) μ g ( r , ω 0 ) [ ω 0 μ ( r , ω 0 ) ] 2 ,
I n n ( ω n ω n ) ω 0 = O ( Δ ω ̃ n 2 Δ ω ̃ n 2 ) 0 .
1 = V d r [ ϵ g ( r , ω 0 ) 2 f n ( r ) 2 + μ g ( r , ω 0 ) 2 c × f n ( r ) ω 0 μ ( r , ω 0 ) 2 ] .
V d r [ × f n * ( r ) ] [ × f n ( r ) ] = ω n 2 c 2 V d r ϵ ( r , ω n ) f n * ( r ) f n ( r ) .
I n n = V d r [ ϵ g ( r , ω 0 ) + ϵ ( r , ω 0 ) ] 2 f n * ( r ) f n ( r ) δ n n .
V d r ϵ ( r ) f n * ( r ) f n ( r ) = δ n n .
ϵ ( ω ) = ϵ b ω p 2 [ ω ( ω + i γ ) ] ,
ϵ R ( ω ) = ϵ b ω p 2 ( ω 2 + γ 2 ) ,
ϵ g ( ω ) = ϵ b + ω p 2 ( ω 2 γ 2 ) ( ω 2 + γ 2 ) 2 .
R sp ( ω ) = 2 ω c e d 2 3 ϵ 0 f EH 11 ( 0 ) 2 Δ ω c 2 ( ω ω c ) 2 + ( Δ ω c 2 ) 2 ,

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