Abstract

The problem of molecular fluorescence in the vicinity of a gradient-index medium is studied theoretically through classical modeling of a radiating dipole. A previously developed formulation involving the Green dyadic for an inhomogeneous medium is applied to the present problem. Normalized lifetimes for the admolecules are calculated and compared with those for a homogeneous medium. The results are illustrated by numerical examples assuming certain simple forms for the index profile.

© 2000 Optical Society of America

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References

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  1. See A. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1949), p. 236; originally published in Ann. Phys. (Leipzig) 28, 665 (1909).
  2. R. R. Chance, A. Prock, R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978), and references therein.
  3. D. G. Deppe, C. Lei, “Spontaneous emission from a dipole in a semiconductor microcavity,” J. Appl. Phys. 70, 3443–3448 (1991).
    [CrossRef]
  4. W. L. Blacke, P. T. Leung, “Molecular fluorescence at a rough surface: the orientation effects,” Phys. Rev. B 56, 12625–12631 (1997), and references therein.
    [CrossRef]
  5. P. T. Leung, T. F. George, “Molecular fluorescence spectroscopy in the vicinity of a microstructure,” J. Chim. Phys. (France) 92, 226–247 (1995), and references therein.
  6. E. W. Marchand, Gradient Index Optics (Academic, New York, 1978).
  7. R. Jacobsson, “Optical properties of a class of inhomogeneous thin films,” Opt. Acta 10, 309–323 (1963); also in Physics of Thin Films (Academic, New York, 1975), Vol. 8, pp. 51–98.
    [CrossRef]
  8. G. Eichmann, “Quasi-geometric optics of media with inhomogeneous index of refraction,” J. Opt. Soc. Am. 61, 161–168 (1971).
    [CrossRef]
  9. C. C. Constantinou, “Path-integral analysis of tapered, graded-index waveguides,” J. Opt. Soc. Am. A 8, 1240–1244 (1991).
    [CrossRef]
  10. R. Srivastava, C. K. Kao, R. V. Ramaswamy, “WKB analysis of planar surface waveguides with truncated index profiles,” J. Lightwave Technol. LT-5, 1605–1608 (1987).
    [CrossRef]
  11. R. L. Hartman, “Green dyadic calculations for inhomogeneous optical media,” J. Opt. Soc. Am. A (to be published).
  12. For modeling molecular fluorescence with nonlocal dielectric response from the substrate surface, see, e.g., P. T. Leung, “Decay of molecules at spherical surfaces: nonlocal effects,” Phys. Rev. B 42, 7622–7625 (1990); P. T. Leung, M. H. Hider, “Nonlocal electrodynamic modeling of frequency shifts for molecules at rough metal surfaces,” J. Chem. Phys. 98, 5019–5022 (1993).
    [CrossRef]
  13. R. L. Hartman, S. M. Cohen, P. T. Leung, “A note on the Green dyadic calculation of the decay rates for admolecules at multiple planar interfaces,” J. Chem. Phys. 110, 2189–2194 (1999).
    [CrossRef]

1999 (1)

R. L. Hartman, S. M. Cohen, P. T. Leung, “A note on the Green dyadic calculation of the decay rates for admolecules at multiple planar interfaces,” J. Chem. Phys. 110, 2189–2194 (1999).
[CrossRef]

1997 (1)

W. L. Blacke, P. T. Leung, “Molecular fluorescence at a rough surface: the orientation effects,” Phys. Rev. B 56, 12625–12631 (1997), and references therein.
[CrossRef]

1995 (1)

P. T. Leung, T. F. George, “Molecular fluorescence spectroscopy in the vicinity of a microstructure,” J. Chim. Phys. (France) 92, 226–247 (1995), and references therein.

1991 (2)

D. G. Deppe, C. Lei, “Spontaneous emission from a dipole in a semiconductor microcavity,” J. Appl. Phys. 70, 3443–3448 (1991).
[CrossRef]

C. C. Constantinou, “Path-integral analysis of tapered, graded-index waveguides,” J. Opt. Soc. Am. A 8, 1240–1244 (1991).
[CrossRef]

1990 (1)

For modeling molecular fluorescence with nonlocal dielectric response from the substrate surface, see, e.g., P. T. Leung, “Decay of molecules at spherical surfaces: nonlocal effects,” Phys. Rev. B 42, 7622–7625 (1990); P. T. Leung, M. H. Hider, “Nonlocal electrodynamic modeling of frequency shifts for molecules at rough metal surfaces,” J. Chem. Phys. 98, 5019–5022 (1993).
[CrossRef]

1987 (1)

R. Srivastava, C. K. Kao, R. V. Ramaswamy, “WKB analysis of planar surface waveguides with truncated index profiles,” J. Lightwave Technol. LT-5, 1605–1608 (1987).
[CrossRef]

1978 (1)

R. R. Chance, A. Prock, R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978), and references therein.

1971 (1)

1963 (1)

R. Jacobsson, “Optical properties of a class of inhomogeneous thin films,” Opt. Acta 10, 309–323 (1963); also in Physics of Thin Films (Academic, New York, 1975), Vol. 8, pp. 51–98.
[CrossRef]

Blacke, W. L.

W. L. Blacke, P. T. Leung, “Molecular fluorescence at a rough surface: the orientation effects,” Phys. Rev. B 56, 12625–12631 (1997), and references therein.
[CrossRef]

Chance, R. R.

R. R. Chance, A. Prock, R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978), and references therein.

Cohen, S. M.

R. L. Hartman, S. M. Cohen, P. T. Leung, “A note on the Green dyadic calculation of the decay rates for admolecules at multiple planar interfaces,” J. Chem. Phys. 110, 2189–2194 (1999).
[CrossRef]

Constantinou, C. C.

Deppe, D. G.

D. G. Deppe, C. Lei, “Spontaneous emission from a dipole in a semiconductor microcavity,” J. Appl. Phys. 70, 3443–3448 (1991).
[CrossRef]

Eichmann, G.

George, T. F.

P. T. Leung, T. F. George, “Molecular fluorescence spectroscopy in the vicinity of a microstructure,” J. Chim. Phys. (France) 92, 226–247 (1995), and references therein.

Hartman, R. L.

R. L. Hartman, S. M. Cohen, P. T. Leung, “A note on the Green dyadic calculation of the decay rates for admolecules at multiple planar interfaces,” J. Chem. Phys. 110, 2189–2194 (1999).
[CrossRef]

R. L. Hartman, “Green dyadic calculations for inhomogeneous optical media,” J. Opt. Soc. Am. A (to be published).

Jacobsson, R.

R. Jacobsson, “Optical properties of a class of inhomogeneous thin films,” Opt. Acta 10, 309–323 (1963); also in Physics of Thin Films (Academic, New York, 1975), Vol. 8, pp. 51–98.
[CrossRef]

Kao, C. K.

R. Srivastava, C. K. Kao, R. V. Ramaswamy, “WKB analysis of planar surface waveguides with truncated index profiles,” J. Lightwave Technol. LT-5, 1605–1608 (1987).
[CrossRef]

Lei, C.

D. G. Deppe, C. Lei, “Spontaneous emission from a dipole in a semiconductor microcavity,” J. Appl. Phys. 70, 3443–3448 (1991).
[CrossRef]

Leung, P. T.

R. L. Hartman, S. M. Cohen, P. T. Leung, “A note on the Green dyadic calculation of the decay rates for admolecules at multiple planar interfaces,” J. Chem. Phys. 110, 2189–2194 (1999).
[CrossRef]

W. L. Blacke, P. T. Leung, “Molecular fluorescence at a rough surface: the orientation effects,” Phys. Rev. B 56, 12625–12631 (1997), and references therein.
[CrossRef]

P. T. Leung, T. F. George, “Molecular fluorescence spectroscopy in the vicinity of a microstructure,” J. Chim. Phys. (France) 92, 226–247 (1995), and references therein.

For modeling molecular fluorescence with nonlocal dielectric response from the substrate surface, see, e.g., P. T. Leung, “Decay of molecules at spherical surfaces: nonlocal effects,” Phys. Rev. B 42, 7622–7625 (1990); P. T. Leung, M. H. Hider, “Nonlocal electrodynamic modeling of frequency shifts for molecules at rough metal surfaces,” J. Chem. Phys. 98, 5019–5022 (1993).
[CrossRef]

Marchand, E. W.

E. W. Marchand, Gradient Index Optics (Academic, New York, 1978).

Prock, A.

R. R. Chance, A. Prock, R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978), and references therein.

Ramaswamy, R. V.

R. Srivastava, C. K. Kao, R. V. Ramaswamy, “WKB analysis of planar surface waveguides with truncated index profiles,” J. Lightwave Technol. LT-5, 1605–1608 (1987).
[CrossRef]

Silbey, R.

R. R. Chance, A. Prock, R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978), and references therein.

Sommerfeld, A.

See A. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1949), p. 236; originally published in Ann. Phys. (Leipzig) 28, 665 (1909).

Srivastava, R.

R. Srivastava, C. K. Kao, R. V. Ramaswamy, “WKB analysis of planar surface waveguides with truncated index profiles,” J. Lightwave Technol. LT-5, 1605–1608 (1987).
[CrossRef]

Adv. Chem. Phys. (1)

R. R. Chance, A. Prock, R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978), and references therein.

J. Appl. Phys. (1)

D. G. Deppe, C. Lei, “Spontaneous emission from a dipole in a semiconductor microcavity,” J. Appl. Phys. 70, 3443–3448 (1991).
[CrossRef]

J. Chem. Phys. (1)

R. L. Hartman, S. M. Cohen, P. T. Leung, “A note on the Green dyadic calculation of the decay rates for admolecules at multiple planar interfaces,” J. Chem. Phys. 110, 2189–2194 (1999).
[CrossRef]

J. Chim. Phys. (France) (1)

P. T. Leung, T. F. George, “Molecular fluorescence spectroscopy in the vicinity of a microstructure,” J. Chim. Phys. (France) 92, 226–247 (1995), and references therein.

J. Lightwave Technol. (1)

R. Srivastava, C. K. Kao, R. V. Ramaswamy, “WKB analysis of planar surface waveguides with truncated index profiles,” J. Lightwave Technol. LT-5, 1605–1608 (1987).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Acta (1)

R. Jacobsson, “Optical properties of a class of inhomogeneous thin films,” Opt. Acta 10, 309–323 (1963); also in Physics of Thin Films (Academic, New York, 1975), Vol. 8, pp. 51–98.
[CrossRef]

Phys. Rev. B (2)

W. L. Blacke, P. T. Leung, “Molecular fluorescence at a rough surface: the orientation effects,” Phys. Rev. B 56, 12625–12631 (1997), and references therein.
[CrossRef]

For modeling molecular fluorescence with nonlocal dielectric response from the substrate surface, see, e.g., P. T. Leung, “Decay of molecules at spherical surfaces: nonlocal effects,” Phys. Rev. B 42, 7622–7625 (1990); P. T. Leung, M. H. Hider, “Nonlocal electrodynamic modeling of frequency shifts for molecules at rough metal surfaces,” J. Chem. Phys. 98, 5019–5022 (1993).
[CrossRef]

Other (3)

R. L. Hartman, “Green dyadic calculations for inhomogeneous optical media,” J. Opt. Soc. Am. A (to be published).

E. W. Marchand, Gradient Index Optics (Academic, New York, 1978).

See A. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1949), p. 236; originally published in Ann. Phys. (Leipzig) 28, 665 (1909).

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Figures (3)

Fig. 1
Fig. 1

Schematic illustration of the index profiles for the gradient-index medium considered in the modeling. The index for the medium that contains the molecule (z<0) is fixed at 1.5, whereas the index for the film varies linearly with z for 0z4d in the form n+ik=0.06+iN2z4d-1+4.11, N=0, 1, 2, 3, 4, 5, for the cases (a)–(f). Thus case (a) corresponds to the homogeneous index value used in the computation of Fig. 2 of Ref. 2. The film thickness is fixed at 4d. For z>4d the index is fixed at 0.06+4.11i.

Fig. 2
Fig. 2

Computed normalized lifetimes for an admolecule as a function of distance to the gradient-index film as described in Fig. 1. The molecular orientation is perpendicular to the film surface. Results with gradual variation corresponding to cases (a)–(f) in Fig. 1 are shown, with the labels (a), (b), and (f) indicated explicitly.

Fig. 3
Fig. 3

Same as in Fig. 2, except for a parallel-oriented molecule.

Equations (27)

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G(R, R)J(R)=i4πλ=0+dλn=0+ 2-δ(n, 0)λh(λ, z).
l=01[Ml,n,λ(h(λ, z))Ml,n,λ(-h(λ, z))Nl,n,λ(h(λ, z))Nl,n,λ(-h(λ, z))]Cl,n,λ(z)Fl,n,λ(z),
Cl,n,λ(z)cl,n,λ(z)cl,n,λ(z),Fl,n,λ(z)fl,n,λ(z)fl,n,λ(z)
dCdz=dhdz-iz-12hexp(-2ihz)2hexp(2ihz)2hiz-12hC+δ(z-zs)M(-h(zs))t-M(+h(zs))tJTcC+δ(z-zs)M(-h(zs))t-M(+h(zs))tJ,
C(zb)=0c(zb),C(zt)=c(zt)0,
dFdz=-iz+12h dhdz12hdhdz-1kdkdzexp(-2ihz)12hdhdz-1kdkdzexp(2ihz)iz-12h dhdzF+δ(z-zs)N(-h(zs))t-N(+h(zs))tJTfF+δ(z-zs)N(-h(zs))t-N(+h(zs))tJ,
F(zb)=0f(zb),F(zt)=f(zt)0.
bˆ=1+6πϵ0qns2p0ks3I(E0),
E0(R)=iωμG(R, R)J(R)dV(R),
J=-iωp0zˆ exp(-iωt)δ (R-zszˆ).
ΦcΦc(+,-)Πc(zb+, zb-)Φc(zt, zb),
ΦfΦf(+,-)Πf(zb+, zb-)Φf(zt, zb).
f0,0,λ(+)f0,0,λ(-)=(Φf)110(Φf)21-1-1N0,0,λ(-hs)tJ0,
f(-)=(Φf)21(Φf)11N(-hs)tJ=(Φf)21(Φf)11[-iωp0 exp(-iωt)][N(-hs)tzˆ]×δ(R-zszˆ).
zˆtE(zszˆ)=iω2μ4π[p0 exp(-iωt)]×λ=0+ (Φf)21(Φf)11[N(-hs)tzˆ]2λhs(λ)dλ.
N(-hs)tzˆ=λ2 exp(-ihszs)ks,
bˆ=1+3q2ks3Rλ=0+ (Φf)21(Φf)11λ3 exp(-2ihszs)hs(λ)dλ,
J=-iωp0xˆ exp(-iωt)δ (R-zszˆ).
xˆtG(zszˆ, zszˆ)J(zszˆ)
=ixˆt4πλ=0+ 2λhs(λ)×[c1,1,λM1,1,λ(-hs)+f0,1,λN0,1,λ(-hs)]dλ.
c1,1,λ(+)c1,1,λ(-)=(Φc)110(Φc)21-1-1M1,1,λ(-hs)t·J0,
c(-)=(Φc)21(Φc)11M(-hs)tJ=(Φc)21(Φc)11[-iωp0 exp(-iωt)]×[M(-hs)txˆ]δ(R-zzzˆ).
xˆtM(-hs)=λ exp(-ihszs)2,
xˆtN(-hs)=-ihsλ exp(-ihszs)2ks.
xˆtE(zszˆ)=iω2p0μ8πexp(-iωt)λ=0+ λ exp(-2ihszs)hs(λ)(Φc)21(Φc)11-hsks2 (Φf)21(Φf)11dλ.
bˆ=1+3q4ksRλ=0+ λ exp(-2ihszs)hs(λ)×(Φc)21(Φc)11-hsks2 (Φf)21(Φf)11dλ.
n+ik=0.06+iN2z4d-1+4.11,

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