Abstract

A simple model of the electromagnetic field coupled to matter as a system of coupled quantum harmonic oscillators (QHOs) with a photon number conserving Hamiltonian is specialized to a translationally invariant optical fiber and then equipped with a QHO model for a one-photon fluorophore source. In the narrowband fluorescence limit, this fluorophore model becomes the same as the classical model for a current loop antenna.

© 2007 Optical Society of America

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  1. R. W. C. Vance and F. Ladouceur, "One-photon electrodynamics in optical fiber with fluorophore systems. I. A one-photon correspondence principle for electromagnetic field propagation in matter," J. Opt. Soc. Am. B 24, 928-941 (2007).
    [CrossRef]
  2. R. W. C. Vance and F. Ladouceur, "One-photon electrodynamics in optical fiber with fluorophore systems. II. One-polariton propagation in matter and fibers from the one-photon correspondence principle," J. Opt. Soc. Am. B 24, 942-958 (2007).
    [CrossRef]
  3. I. Bialynicki-Birula, "On the wave function of the photon," Acta Phys. Pol. A 86, 97-116 (1994).
  4. I. Bialynicki-Birula, "The photon wave function," in Coherence and Quantum Optics VII, J.H.Eberly, L.Mandel, and E.Wolf, eds. (Plenum, 1996), pp. 313-322.
  5. I. Bialynicki-Birula, "Photon wave function," Prog. Opt. 36, 245-294 (1996).
    [CrossRef]
  6. B. Huttner and S. M. Barnett, "Quantization of the electromagnetic field in dielectrics," Phys. Rev. A 46, 4306-4322 (1992).
    [CrossRef] [PubMed]
  7. J. J. Hopfield, "Theory of the contribution of excitons to the complex dielectric constant of crystals," Phys. Rev. 112, 1555-1567 (1958).
    [CrossRef]
  8. P. M. Delaney, M. R. Harris, and R. G. King, "Fiber-optic laser scanning confocal microscope suitable for fluorescence imaging," Appl. Opt. 33, 573-577 (1994).
    [CrossRef] [PubMed]
  9. C. H. Bennett and G. Brassard, "Quantum cryptography: public key distribution and coin tossing," in Proceedings of the IEEE International Conference on Computers Systems and Signal Processing (IEEE, 1984), pp. 175-179.
  10. D. R. Sandison, R. M. Williams, K. S. Wells, J. Strickler, and W. W. Webb, "Quantitative fluorescence confocal laser scanning microscopy (CLSM)," in Handbook of Biological Confocal Microscopy, J.P.Pawley, ed. (Plenum, 1995), pp. 39-53.
  11. A. Draaijer, R. Sanders, and H. C. Gerritsen, "Fluorescence lifetime imaging, a new tool in confocal microscopy," in Handbook of Biological Confocal Microscopy, J.P.Pawley, ed. (Plenum, 1995), pp. 491-505.
  12. R. P. Feynman, "Angular momentum," in The Feynman Lectures on Physics (Addison-Wesley, 1989), Vol. 3, pp. 18/1-18/22.
  13. G. Weber, "Rotational Brownian motion and polarization of the fluorescence of solutions," Adv. Protein Chem. 8, 415-459 (1953).
    [CrossRef] [PubMed]
  14. G. Weber, "Dependence of the polarization of the fluorescence on the concentration," Trans. Faraday Soc. 50, 552-555 (1954).
    [CrossRef]
  15. G. Weber, "Fluorescence-polarization spectrum and electronic-energy transfer in proteins," Biochem. J. 75, 345-352 (1960).
    [PubMed]
  16. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983).
  17. R. E. Collin, "The receiving antenna," in Antenna Theory: Part I, R.E.Collin and F.J.Zucker, eds. (McGraw-Hill, 1969), pp. 93-137.
  18. R. Y. Tsien and A. Waggoner, "Fluorophores for confocal microscopy," in Handbook of Biological Confocal Microscopy, J.P.Pawley, ed. (Plenum, 1995), pp. 267-279.
  19. M. Boroditsky, R. Vrijen, T. Krauss, R. Coccioli, R. Bhat, and E. Yablonovitch, "Spontaneous emission extraction and Purcell enhancement from thin-film 2-D photonic crystals," J. Lightwave Technol. 17, 2096-2112 (1999).
    [CrossRef]
  20. P. M. Delaney, Optiscan Imaging Limited, 15 Normanby Road, Notting Hill, Victoria 3168, Australia (private communication, 1994) suggested the calculation of powers and inner products at any simply connected S∞ in the confocal system, not just at the fiber tip. The soundness of this procedure was then proven from the Lorentz lemma by one of the authors (R.W.C. Vance).
  21. M. O. Scully and M. Suhail Zubairy, Quantum Optics (Cambridge U. Press, 1997).
  22. D. Marcuse, Engineering Quantum Electrodynamics (Harcourt, Brace, 1970).
  23. G. Dubost and A. Bellossi, "Modèle physique de la membrane cellulaire dans les spectres infrarouge, visible et ultraviolet," Rev. Sci. Tech. Déf. 64, 113-127 (2004).

2007 (2)

2004 (1)

G. Dubost and A. Bellossi, "Modèle physique de la membrane cellulaire dans les spectres infrarouge, visible et ultraviolet," Rev. Sci. Tech. Déf. 64, 113-127 (2004).

1999 (1)

1996 (1)

I. Bialynicki-Birula, "Photon wave function," Prog. Opt. 36, 245-294 (1996).
[CrossRef]

1994 (2)

1992 (1)

B. Huttner and S. M. Barnett, "Quantization of the electromagnetic field in dielectrics," Phys. Rev. A 46, 4306-4322 (1992).
[CrossRef] [PubMed]

1960 (1)

G. Weber, "Fluorescence-polarization spectrum and electronic-energy transfer in proteins," Biochem. J. 75, 345-352 (1960).
[PubMed]

1958 (1)

J. J. Hopfield, "Theory of the contribution of excitons to the complex dielectric constant of crystals," Phys. Rev. 112, 1555-1567 (1958).
[CrossRef]

1954 (1)

G. Weber, "Dependence of the polarization of the fluorescence on the concentration," Trans. Faraday Soc. 50, 552-555 (1954).
[CrossRef]

1953 (1)

G. Weber, "Rotational Brownian motion and polarization of the fluorescence of solutions," Adv. Protein Chem. 8, 415-459 (1953).
[CrossRef] [PubMed]

Barnett, S. M.

B. Huttner and S. M. Barnett, "Quantization of the electromagnetic field in dielectrics," Phys. Rev. A 46, 4306-4322 (1992).
[CrossRef] [PubMed]

Bellossi, A.

G. Dubost and A. Bellossi, "Modèle physique de la membrane cellulaire dans les spectres infrarouge, visible et ultraviolet," Rev. Sci. Tech. Déf. 64, 113-127 (2004).

Bennett, C. H.

C. H. Bennett and G. Brassard, "Quantum cryptography: public key distribution and coin tossing," in Proceedings of the IEEE International Conference on Computers Systems and Signal Processing (IEEE, 1984), pp. 175-179.

Bhat, R.

Bialynicki-Birula, I.

I. Bialynicki-Birula, "Photon wave function," Prog. Opt. 36, 245-294 (1996).
[CrossRef]

I. Bialynicki-Birula, "On the wave function of the photon," Acta Phys. Pol. A 86, 97-116 (1994).

I. Bialynicki-Birula, "The photon wave function," in Coherence and Quantum Optics VII, J.H.Eberly, L.Mandel, and E.Wolf, eds. (Plenum, 1996), pp. 313-322.

Boroditsky, M.

Brassard, G.

C. H. Bennett and G. Brassard, "Quantum cryptography: public key distribution and coin tossing," in Proceedings of the IEEE International Conference on Computers Systems and Signal Processing (IEEE, 1984), pp. 175-179.

Coccioli, R.

Collin, R. E.

R. E. Collin, "The receiving antenna," in Antenna Theory: Part I, R.E.Collin and F.J.Zucker, eds. (McGraw-Hill, 1969), pp. 93-137.

Delaney, P. M.

P. M. Delaney, M. R. Harris, and R. G. King, "Fiber-optic laser scanning confocal microscope suitable for fluorescence imaging," Appl. Opt. 33, 573-577 (1994).
[CrossRef] [PubMed]

P. M. Delaney, Optiscan Imaging Limited, 15 Normanby Road, Notting Hill, Victoria 3168, Australia (private communication, 1994) suggested the calculation of powers and inner products at any simply connected S∞ in the confocal system, not just at the fiber tip. The soundness of this procedure was then proven from the Lorentz lemma by one of the authors (R.W.C. Vance).

Draaijer, A.

A. Draaijer, R. Sanders, and H. C. Gerritsen, "Fluorescence lifetime imaging, a new tool in confocal microscopy," in Handbook of Biological Confocal Microscopy, J.P.Pawley, ed. (Plenum, 1995), pp. 491-505.

Dubost, G.

G. Dubost and A. Bellossi, "Modèle physique de la membrane cellulaire dans les spectres infrarouge, visible et ultraviolet," Rev. Sci. Tech. Déf. 64, 113-127 (2004).

Feynman, R. P.

R. P. Feynman, "Angular momentum," in The Feynman Lectures on Physics (Addison-Wesley, 1989), Vol. 3, pp. 18/1-18/22.

Gerritsen, H. C.

A. Draaijer, R. Sanders, and H. C. Gerritsen, "Fluorescence lifetime imaging, a new tool in confocal microscopy," in Handbook of Biological Confocal Microscopy, J.P.Pawley, ed. (Plenum, 1995), pp. 491-505.

Harris, M. R.

Hopfield, J. J.

J. J. Hopfield, "Theory of the contribution of excitons to the complex dielectric constant of crystals," Phys. Rev. 112, 1555-1567 (1958).
[CrossRef]

Huttner, B.

B. Huttner and S. M. Barnett, "Quantization of the electromagnetic field in dielectrics," Phys. Rev. A 46, 4306-4322 (1992).
[CrossRef] [PubMed]

King, R. G.

Krauss, T.

Ladouceur, F.

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983).

Marcuse, D.

D. Marcuse, Engineering Quantum Electrodynamics (Harcourt, Brace, 1970).

Sanders, R.

A. Draaijer, R. Sanders, and H. C. Gerritsen, "Fluorescence lifetime imaging, a new tool in confocal microscopy," in Handbook of Biological Confocal Microscopy, J.P.Pawley, ed. (Plenum, 1995), pp. 491-505.

Sandison, D. R.

D. R. Sandison, R. M. Williams, K. S. Wells, J. Strickler, and W. W. Webb, "Quantitative fluorescence confocal laser scanning microscopy (CLSM)," in Handbook of Biological Confocal Microscopy, J.P.Pawley, ed. (Plenum, 1995), pp. 39-53.

Scully, M. O.

M. O. Scully and M. Suhail Zubairy, Quantum Optics (Cambridge U. Press, 1997).

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983).

Strickler, J.

D. R. Sandison, R. M. Williams, K. S. Wells, J. Strickler, and W. W. Webb, "Quantitative fluorescence confocal laser scanning microscopy (CLSM)," in Handbook of Biological Confocal Microscopy, J.P.Pawley, ed. (Plenum, 1995), pp. 39-53.

Tsien, R. Y.

R. Y. Tsien and A. Waggoner, "Fluorophores for confocal microscopy," in Handbook of Biological Confocal Microscopy, J.P.Pawley, ed. (Plenum, 1995), pp. 267-279.

Vance, R. W. C.

Vrijen, R.

Waggoner, A.

R. Y. Tsien and A. Waggoner, "Fluorophores for confocal microscopy," in Handbook of Biological Confocal Microscopy, J.P.Pawley, ed. (Plenum, 1995), pp. 267-279.

Webb, W. W.

D. R. Sandison, R. M. Williams, K. S. Wells, J. Strickler, and W. W. Webb, "Quantitative fluorescence confocal laser scanning microscopy (CLSM)," in Handbook of Biological Confocal Microscopy, J.P.Pawley, ed. (Plenum, 1995), pp. 39-53.

Weber, G.

G. Weber, "Fluorescence-polarization spectrum and electronic-energy transfer in proteins," Biochem. J. 75, 345-352 (1960).
[PubMed]

G. Weber, "Dependence of the polarization of the fluorescence on the concentration," Trans. Faraday Soc. 50, 552-555 (1954).
[CrossRef]

G. Weber, "Rotational Brownian motion and polarization of the fluorescence of solutions," Adv. Protein Chem. 8, 415-459 (1953).
[CrossRef] [PubMed]

Wells, K. S.

D. R. Sandison, R. M. Williams, K. S. Wells, J. Strickler, and W. W. Webb, "Quantitative fluorescence confocal laser scanning microscopy (CLSM)," in Handbook of Biological Confocal Microscopy, J.P.Pawley, ed. (Plenum, 1995), pp. 39-53.

Williams, R. M.

D. R. Sandison, R. M. Williams, K. S. Wells, J. Strickler, and W. W. Webb, "Quantitative fluorescence confocal laser scanning microscopy (CLSM)," in Handbook of Biological Confocal Microscopy, J.P.Pawley, ed. (Plenum, 1995), pp. 39-53.

Yablonovitch, E.

Zubairy, M. Suhail

M. O. Scully and M. Suhail Zubairy, Quantum Optics (Cambridge U. Press, 1997).

Acta Phys. Pol. A (1)

I. Bialynicki-Birula, "On the wave function of the photon," Acta Phys. Pol. A 86, 97-116 (1994).

Adv. Protein Chem. (1)

G. Weber, "Rotational Brownian motion and polarization of the fluorescence of solutions," Adv. Protein Chem. 8, 415-459 (1953).
[CrossRef] [PubMed]

Appl. Opt. (1)

Biochem. J. (1)

G. Weber, "Fluorescence-polarization spectrum and electronic-energy transfer in proteins," Biochem. J. 75, 345-352 (1960).
[PubMed]

J. Lightwave Technol. (1)

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

Phys. Rev. (1)

J. J. Hopfield, "Theory of the contribution of excitons to the complex dielectric constant of crystals," Phys. Rev. 112, 1555-1567 (1958).
[CrossRef]

Phys. Rev. A (1)

B. Huttner and S. M. Barnett, "Quantization of the electromagnetic field in dielectrics," Phys. Rev. A 46, 4306-4322 (1992).
[CrossRef] [PubMed]

Prog. Opt. (1)

I. Bialynicki-Birula, "Photon wave function," Prog. Opt. 36, 245-294 (1996).
[CrossRef]

Rev. Sci. Tech. Déf. (1)

G. Dubost and A. Bellossi, "Modèle physique de la membrane cellulaire dans les spectres infrarouge, visible et ultraviolet," Rev. Sci. Tech. Déf. 64, 113-127 (2004).

Trans. Faraday Soc. (1)

G. Weber, "Dependence of the polarization of the fluorescence on the concentration," Trans. Faraday Soc. 50, 552-555 (1954).
[CrossRef]

Other (11)

I. Bialynicki-Birula, "The photon wave function," in Coherence and Quantum Optics VII, J.H.Eberly, L.Mandel, and E.Wolf, eds. (Plenum, 1996), pp. 313-322.

P. M. Delaney, Optiscan Imaging Limited, 15 Normanby Road, Notting Hill, Victoria 3168, Australia (private communication, 1994) suggested the calculation of powers and inner products at any simply connected S∞ in the confocal system, not just at the fiber tip. The soundness of this procedure was then proven from the Lorentz lemma by one of the authors (R.W.C. Vance).

M. O. Scully and M. Suhail Zubairy, Quantum Optics (Cambridge U. Press, 1997).

D. Marcuse, Engineering Quantum Electrodynamics (Harcourt, Brace, 1970).

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983).

R. E. Collin, "The receiving antenna," in Antenna Theory: Part I, R.E.Collin and F.J.Zucker, eds. (McGraw-Hill, 1969), pp. 93-137.

R. Y. Tsien and A. Waggoner, "Fluorophores for confocal microscopy," in Handbook of Biological Confocal Microscopy, J.P.Pawley, ed. (Plenum, 1995), pp. 267-279.

C. H. Bennett and G. Brassard, "Quantum cryptography: public key distribution and coin tossing," in Proceedings of the IEEE International Conference on Computers Systems and Signal Processing (IEEE, 1984), pp. 175-179.

D. R. Sandison, R. M. Williams, K. S. Wells, J. Strickler, and W. W. Webb, "Quantitative fluorescence confocal laser scanning microscopy (CLSM)," in Handbook of Biological Confocal Microscopy, J.P.Pawley, ed. (Plenum, 1995), pp. 39-53.

A. Draaijer, R. Sanders, and H. C. Gerritsen, "Fluorescence lifetime imaging, a new tool in confocal microscopy," in Handbook of Biological Confocal Microscopy, J.P.Pawley, ed. (Plenum, 1995), pp. 491-505.

R. P. Feynman, "Angular momentum," in The Feynman Lectures on Physics (Addison-Wesley, 1989), Vol. 3, pp. 18/1-18/22.

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Figures (6)

Fig. 1
Fig. 1

Jablonski diagram for fluoresceinlike fluorophore.

Fig. 2
Fig. 2

Quickening fluorophore relaxation with an idealized optical resonator.

Fig. 3
Fig. 3

Generalized Smith chart visualization of frequency pulling and relaxation quickening in the idealized resonator.

Fig. 4
Fig. 4

Fluorophore driving a two-moded axisymmetric optical fiber in wavenumber (momentum) space; (inset) analogous picture for fluorophore in a homogeneous medium.

Fig. 5
Fig. 5

Coupling from currents into fiber calculation after Chap. 21 of Snyder and Love[16] in the presence of a lossless lens system.

Fig. 6
Fig. 6

Lossy matter model (left) and mechanical thermodynamic equivalent (right).

Equations (74)

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i F ± ( r , t ) t = ± c × F ± ( r , t ) + all atoms m ( C ± , m ( r m r ) ψ m ( t ) ) + K ± ( r 0 r ) ψ 0 ( t ) ,
i ψ m ( t ) t = ω m ψ m ( t ) + ( ( L B B C + , m ( r ) ) r F + ( r , t ) + ( L B B C , m ( r ) ) r F ( r , l ) ) r = r m ,
i ψ 0 ( t ) t = ω 0 ψ 0 ( t ) + ( ( L B B K + ( r ) ) r F + ( r , t ) + ( L B B K ( r ) ) r F ( r , t ) ) r = r 0 ,
K ± ( r ) = 1 8 π 3 all k exp ( i k r ) K ± ( k ) f ± ( k ) d 3 k .
ψ 0 ( 0 ) = 1 , ψ m ( 0 ) = 0 , F ± ( r , 0 ) = 0 .
i F ± ( r , t ) t + i G ± ( r , t ) t = ± c × F ± ( r , t ) i J ± ( r , t ) ,
J ± ( r , t ) = i K ± ( r 0 r ) ψ 0 ( t ) ,
i ψ 0 ( t ) t = ω 0 ψ 0 ( t ) + ( ( L B B K + ( r ) ) r F + ( r , t ) + ( L B B K ( r ) ) r F ( r , t ) ) r = r 0 ,
× H ̆ ( r , t ) = J ̆ E ( r , t ) + D ̆ ( r , t ) t
J ̆ E ( r , t ) = ε 0 2 ( J + ( r , t ) + J ( r , t ) ) = i K E ( r 0 r ) ψ 0 ( t ) × E ̆ ( r , t ) = J ̆ M ( r , t ) + B ̆ ( r , t ) t
J ̆ M ( r , t ) = i μ 0 2 ( J + ( r , t ) J ( r , t ) ) = i K M ( r 0 r ) ψ 0 ( t )
i ψ 0 ( t ) t = ω 0 ψ 0 ( t ) + ( ( L B B K E ( r ) ) r E ̆ ( r , t ) + ( L B B K M ( r ) ) r H ̆ ( r , t ) ) r = r 0
K E ( r ) = ε 0 2 ( K + ( r ) + K ( r ) )
K M ( r ) = i μ 0 2 ( K + ( r ) K ( r ) )
K ± ( r ) = J ̆ ± ( r , t ) = J ̆ M ( r , t ) = J ̆ E ( r , t ) = 0 .
ψ 0 ( t ) 2 t = 2 all r Re ( J + ( r , t ) * ( L B B F + ( r , t ) ) + J ( r , t ) * ( L B B F ( r , t ) ) ) d 3 r ,
all r ρ ( r , t ) t d 3 r = all r ( S ( r , t ) W ( r , t ) t ) d 3 r ,
ρ ( r , t ) = 2 Re [ ( L B B F + ( r , t ) ) ( F + ( r , t ) * + G + ( r , t ) * ) + ( L B B F ( r , t ) ) ( F ( r , t ) * + G ( r , t ) * ) ] ,
W ( r , t ) = 2 Re [ J + ( r , t ) * ( L B B F + ( r , t ) ) + J ( r , t ) * ( L B B F ( r , t ) ) ] ,
S ( r , t ) = 2 c Im [ F + ( r , t ) * × ( L B B F + ( r , t ) ) F ( r , t ) * × ( L B B F ( r , t ) ) ] .
ρ ( r , t ) t = S ( r , t ) W ( r , t ) t ,
ψ 0 ( t ) 2 t = all r ρ ( r , t ) t d 3 r = 2 Re [ ( F + + G + ) , F + B B + ( F + G ) , F B B ] ,
G ± ( r ) = ( ϵ r + μ r 2 1 ) F ± ( r ) + ϵ r μ r 2 F ( r )
F M , ± ( r , t ) = 1 2 [ ( ε r + μ r ) F ± ( r , t ) + ( ε r μ r ) F ( r , t ) ] = ε 2 E ̆ ( r , t ) ± i μ 2 H ̆ ( r , t )
J M , ± ( r , t ) = 1 2 ε r μ r [ ( ε r + μ r ) J ± ( r , t ) + ( μ r ε r ) J ( r , t ) ]
K M , ± ( r ) = 1 2 ε r μ r [ ( ε r + μ r ) K ± ( r ) + ( μ r ε r ) K ( r ) ]
i F M , ± ( r , t ) t = ± c ϵ r μ r × F M , ± ( r , t ) i J M , ± ( r , t ) ,
i ψ 0 ( t ) t = ω 0 ψ 0 ( t ) + all r ( F M , + ( r , t ) K M , + ( r r 0 ) + F M , ( r , t ) K M , ( r r 0 ) ) d 3 r .
F M , ± ( r , t ) = 1 8 π 3 k = 0 [ ψ ± ( k c ϵ r μ r t ) κ ± ( k c ϵ r μ r ) { k : k = ω ϵ r μ r c } exp ( i k r ) K M , ± ( k ) * i ( k ) ± i j ( k ) 2 d 2 k ] d k ;
i ψ ± ( ω , t ) t = ω ψ ± ( ω , t ) + κ ± ( ω ) * ψ 0 ( t ) ,
i ψ 0 ( t ) t = ω 0 ψ 0 ( t ) + ϵ r μ r c ω = 0 k = ω ϵ r μ r c ( K M , + ( k ) 2 ψ + ( ω , t ) κ + ( ω ) + K M , ( k ) 2 ψ ( ω , t ) κ ( ω ) ) d 2 k d ω .
κ ± ( ω ) = ϵ r μ r c k = ω ϵ r μ r c K M , ± ( k ) 2 d 2 k ,
ψ ± ( ω , t ) = i κ ± ( ω ) 0 t e i ω ( u t ) ψ 0 ( u ) d u ,
d ψ 0 ( t ) d t = i ω 0 ψ 0 ( t ) 0 t R ( t v ) ψ 0 ( v ) d v ,
R ( t ) = 0 e i ω t ( κ + ( ω ) 2 + κ ( ω ) 2 ) d u ,
ψ 0 ( t ) exp ( i ω 0 t t 2 τ ) , τ 1 2 π ( κ + 2 + κ 2 ) ω = ω 0 .
τ = c 3 4 π 2 ω 0 2 ϵ r μ r ( ϵ r + μ r ) ( K + 2 + K 2 ) = c 4 π 2 k fluor 2 ϵ r μ r ϵ r + μ r 1 K + 2 + K 2 ,
i ψ 0 ( t ) t = ω 0 ψ 0 ( t ) + d + , F + ( r 0 , t ) + d , F ( r 0 , t ) .
i f ( t ) t = [ ω 0 ω i A ( ω ) τ ] f ( t ) τ τ ̃ + i τ ( ω ω 0 ) = A ( ω ) ,
ω 2 π Δ r ω r + δ 2 τ ω 0 2 π Δ r + δ 2 τ .
F ± ( r , z , t ) = e i ω t ( j = 1 N [ a j L ± j ( r , ω ) exp ( i β j ( ω ) z ) ] + j = 1 Q = 0 a ( Q ) L R , ± j ( r , Q , ω ) exp ( i β j ( Q , ω ) z ) d Q ) ,
c z A ( F + × F ¯ + * F × F ¯ * ) z d A = A ( F ¯ + * J + + F + J ¯ + * + F ¯ * J + F J ¯ * ) d A ,
c z A ( F + × F ¯ + F × F ¯ ) z d A = A ( F ¯ + J + F + J ¯ + + F ¯ J F J ¯ ) d A ,
d a j ( z , ω ) d z = 1 c A ( L + , j * ( r , ω ) J + ( r , ω ) + L , j * ( r , ω ) J ( r , ω ) ) exp ( i β j ( ω ) z ) d A A ( L + , j ( r , ω ) × L + , j * ( r , ω ) L , j ( r , ω ) × L , j * ( r , ω ) ) z d A ,
a j ( ω ) = 1 c all r ( L + , j * ( r , ω ) J + ( r , ω ) + L , j * ( r , ω ) J ( r , ω ) ) exp ( i β j ( ω ) z ) d 3 r A ( L + , j ( r , ω ) × L + , j * ( r , ω ) L , j ( r , ω ) × L , j * ( r , ω ) ) z d A ,
J ± ( r , ω ) = i K ± ( r 0 r ) 2 π 0 exp ( i ω t ) ψ 0 ( t ) d t i K ± ( r 0 r ) 2 π ( i ( ω 0 ω ) + 1 2 τ ) ,
i all k ( L ̃ + , j ( k , ω ) 2 + L ̃ , j ( k , ω ) 2 ) β j ( ω ) k 2 + β j ( ω ) 2 d k x d k .
F ± , j ( r , t ) = b j ( ω 0 ) L ± , j ( r , ω 0 ) 1 2 π 0 exp ( i ω t ) exp ( i β j ( ω ) z ) 0 exp ( i ω u ) ψ 0 ( u ) d u d ω ,
b j ( ω ) = i c all r ( L + , j * ( r , ω ) K + ( r 0 r ) + L , j * ( r , ω ) K ( r 0 r ) ) exp ( i β j ( ω ) z ) d 3 r A ( L + , j ( r , ω ) × L + , j * ( r , ω ) L , j ( r , ω ) × L , j * ( r , ω ) ) z d A = 2 π c all k exp ( i k r 0 ) ( L ̃ + , j * ( k , ω ) K + ( k ) + L ̃ , j * ( k , ω ) K ( k ) ) δ ( k z β j ( ω ) ) d 3 k all k ( L ̃ + , j ( k , ω ) 2 + L ̃ , j ( k , ω ) 2 ) β j ( ω ) k 2 + β j ( ω ) 2 d k x d k y = 2 π c all k j exp ( i k r 0 ) ( L ̃ + , j * ( k , ω ) K + ( k + β j ( ω ) z ) + L ̃ , j * ( k , ω ) K ( k + β j ( ω ) z ) ) d k x d k y all k ( L ̃ + , j ( k , ω ) 2 + L ̃ , j ( k , ω ) 2 ) β j ( ω ) k 2 + β j ( ω ) 2 d k x d k y .
L ± , j ( r , ω ) = exp ( ± i ϕ j ) ς j ( r , ω ) x ± i y 2 ,
b j ( ω ) = 2 π 3 c ς j ( r 0 , ω ) ( K + ( β j ( ω ) z ) e i ϕ j + K ( β j ( ω ) z ) e i ϕ j ) A ς j ( r , ω ) 2 d A .
max ϕ j [ 0 , 2 π ) K + ( β j ( ω ) z ) exp ( i ϕ j ) + K ( β j ( ω ) z ) exp ( i ϕ j ) = 2 G ( β j ( ω ) z ) K + 2 + K 2 = 1 2 π k fluor 2 c ϵ r μ r G ( β j ( ω ) z ) ( ϵ r + μ r ) τ ,
F ± , j ( r , t ) = x ± i y 2 k fluor π ϵ r μ r G ( β j ( ω ) z ) c ( ϵ r + μ r ) τ ς # j ( r 0 , ω ) ς # j ( r , ω ) × exp ( i β j ( ω ) z ( i ω + 1 2 τ ) ( t z v g ) ) U ( t z v g ) ,
v g = ( d β j d ω ω = ω 0 ) 1
all r ϵ r + μ r 2 ( F + , j ( r , t ) 2 + F , j ( r , t ) 2 ) d 3 r = ϵ r + μ r 2 2 π ϵ r μ r G ( β j ( ω ) z ) k fluor 2 c ( ϵ r + μ r ) τ ς # j ( r 0 , ω ) 2 τ v g = λ 2 4 π G ( β j ( ω ) z ) ϵ r μ r v g c ς # j ( r 0 , ω ) 2 ,
Prob ( coupling to j th mode ) = fluorophore effective antenna area × fluorophore gain along fiber axis × normalized mode intensity at fluorophore position × γ m .
i ψ c ( ω , t ) t = ω ψ c ( ω , t ) + C c ( ω ) ψ m ( t ) ,
i ψ m ( t ) t = ω m ψ m ( t ) + 0 ψ c ( u , t ) C c * ( u ) d u + f ( t ) ,
ψ c ( ω , t ) = ψ c ( ω , 0 ) exp ( i ω t ) i C c ( ω ) 0 t exp ( i ω ( t v ) ) ψ m ( v ) d v ,
i ψ m ( t ) t = ω 0 ψ m ( t ) + f ( t ) + ϕ c ( t ) i 0 t ψ m ( v ) R ( t v ) d v ,
R ( t ) = 0 exp ( i ω t ) C c ( ω ) 2 d ω ,
ϕ c ( t ) = 0 exp ( i ω t ) C c * ( ω ) ψ c ( ω , 0 ) d ω .
Ψ m ( s ) = ψ m ( 0 ) ( F ( s ) + Φ c ( s ) ) ( s + R ( s ) + i ω m ) .
f ( t ) = ( C + , m ( r ) r F + ( r , t ) + C , m ( r ) r F ( r , t ) ) r = r m ,
Ψ m ( s ) = ψ m ( 0 ) ( ( C + , m ( r ) r F + ( r , s ) + C , m ( r ) r F ( r , s ) ) r = r m + Φ c ( s ) ) ( s + R ( s ) + i ω m ) .
h m ( t ) = ω m 1 e i ω m t ,
h m ( t ) = L 1 ( H m ( s ) ) , H m ( s ) = 1 ω m ( s + R ( s ) + i ω m ) .
h m ( t ) = ω m 1 e t τ exp ( i ω m t ) .
ω m 1 e t τ sin ( ω m t ) t cos ( ω t + δ ) ,
τ 2 ( 1 + ( ω m 2 ω 2 ) τ 2 ) cos ( ω t + δ ) + 2 ω τ sin ( ω t + δ ) ( 1 + ( ω m + ω ) 2 τ 2 ) ( 1 + ( ω m ω ) 2 τ 2 ) ,
arg ( 1 + ( ω m 2 ω 2 ) τ 2 2 ω τ i ) ,
τ 2 exp ( i arg ( 1 + ( ω m 2 ω 2 ) τ 2 2 ω τ i ) ) 1 + ( ω m 2 ω 2 ) τ 2 2 ω τ i 1 + ( ω m 2 ω 2 ) τ 2 + 2 ω τ i 2 = τ 2 1 + ( ω m 2 ω 2 ) τ 2 + 2 ω τ i ,
ϵ ( ω ) = ( 1 + all matter kinds m ϵ r , m 1 + ( ω m 2 ω 2 ) τ m 2 + 2 ω τ m i ) ϵ 0 ,
μ ( ω ) = ( 1 + all matter kinds m μ r , m 1 + ( ω m 2 ω 2 ) τ m 2 + 2 ω τ m i ) μ 0 ,

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