Abstract

A rigorous theory of radiation from dipoles embedded inside an arbitrary multilayer system is presented. In particular, we derive explicit expressions for the angular distribution of the electromagnetic field and the intensity radiated by the dipole into the surrounding media. Under the assumptions of mutual incoherence of the dipole radiation the calculations are extended to a layer of radiating dipoles. Special configurations corresponding to (i) a single dipole near a dielectric interface, (ii) a dipole layer surrounded by semi-infinite dielectric media, and (iii) a dipole layer placed on top of a waveguide layer are discussed in detail. This theoretical analysis has important consequences for the optimization of optical chemical sensors and biosensors that are based on fluorescence emission.

© 2000 Optical Society of America

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References

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  1. S. Rabbany, B. Donner, F. Ligler, “Optical immunosensors,” Crit. Rev. Biomed. Eng. 22, 307–346 (1994).
    [PubMed]
  2. R. Chance, A. Prock, R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).
  3. G. Ford, W. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. 113, 195–287 (1984).
    [CrossRef]
  4. H. Stuart, D. Hall, “Enhanced dipole–dipole interaction between elementary radiators near a surface,” Phys. Rev. Lett. 80, 5663–5666 (1998).
    [CrossRef]
  5. R. Gruhlke, W. Holland, D. Hall, “Optical emission from coupled surface plasmons,” Opt. Lett. 12, 364–366 (1987).
    [CrossRef] [PubMed]
  6. W. Holland, D. Hall, “Waveguide mode enhancement of molecular fluorescence,” Opt. Lett. 10, 414–416 (1985).
    [CrossRef] [PubMed]
  7. E.-H. Lee, R. Benner, J. Fenn, R. Chang, “Angular distribution of fluorescence from liquids and monodispersed spheres by evanescent wave excitation,” Appl. Opt. 18, 862–868 (1979).
    [CrossRef]
  8. V. Ratner, “Calculation of the angular distribution and waveguide capture efficiency of the light emitted by a fluorophore situated at or adsorbed to the waveguide side wall,” Sens. Actuators B 17, 113–119 (1994).
    [CrossRef]
  9. J. Enderlein, T. Ruckstuhl, S. Seeger, “Highly efficient optical detection of surface-generated fluorescence,” Appl. Opt. 38, 724–732 (1999).
    [CrossRef]
  10. L. Landau, E. Lifschitz, Electrodynamics of Continuous Media (Pergamon, Oxford, UK, 1960).
  11. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1959).
  12. C. Brinker, G. Scherer, Solgel Science (Academic, New York, 1990).
  13. C. McDonagh, B. MacCraith, A. McEvoy, “Tailoring of sol-gel films for optical sensing of oxygen in gas and aqueous phase,” Anal. Chem. 70, 45–50 (1998).
    [CrossRef] [PubMed]
  14. Although only the modes propagating in the medium n2 at angles θ ∈ 〈θc02, θc12〉 are considered, the statement is valid for modes propagating in the entire range of angles θ ∈ 〈-90°, 90°〉.
  15. J.-F. Gouin, A. Doyle, B. MacCraith, “Fluorescence capture by planar waveguide a platform for optical sensors,” Electron. Lett. 34, 1685–1687 (1998).
    [CrossRef]

1999 (1)

1998 (3)

C. McDonagh, B. MacCraith, A. McEvoy, “Tailoring of sol-gel films for optical sensing of oxygen in gas and aqueous phase,” Anal. Chem. 70, 45–50 (1998).
[CrossRef] [PubMed]

J.-F. Gouin, A. Doyle, B. MacCraith, “Fluorescence capture by planar waveguide a platform for optical sensors,” Electron. Lett. 34, 1685–1687 (1998).
[CrossRef]

H. Stuart, D. Hall, “Enhanced dipole–dipole interaction between elementary radiators near a surface,” Phys. Rev. Lett. 80, 5663–5666 (1998).
[CrossRef]

1994 (2)

S. Rabbany, B. Donner, F. Ligler, “Optical immunosensors,” Crit. Rev. Biomed. Eng. 22, 307–346 (1994).
[PubMed]

V. Ratner, “Calculation of the angular distribution and waveguide capture efficiency of the light emitted by a fluorophore situated at or adsorbed to the waveguide side wall,” Sens. Actuators B 17, 113–119 (1994).
[CrossRef]

1987 (1)

1985 (1)

1984 (1)

G. Ford, W. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. 113, 195–287 (1984).
[CrossRef]

1979 (1)

1978 (1)

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

Benner, R.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1959).

Brinker, C.

C. Brinker, G. Scherer, Solgel Science (Academic, New York, 1990).

Chance, R.

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

Chang, R.

Donner, B.

S. Rabbany, B. Donner, F. Ligler, “Optical immunosensors,” Crit. Rev. Biomed. Eng. 22, 307–346 (1994).
[PubMed]

Doyle, A.

J.-F. Gouin, A. Doyle, B. MacCraith, “Fluorescence capture by planar waveguide a platform for optical sensors,” Electron. Lett. 34, 1685–1687 (1998).
[CrossRef]

Enderlein, J.

Fenn, J.

Ford, G.

G. Ford, W. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. 113, 195–287 (1984).
[CrossRef]

Gouin, J.-F.

J.-F. Gouin, A. Doyle, B. MacCraith, “Fluorescence capture by planar waveguide a platform for optical sensors,” Electron. Lett. 34, 1685–1687 (1998).
[CrossRef]

Gruhlke, R.

Hall, D.

Holland, W.

Landau, L.

L. Landau, E. Lifschitz, Electrodynamics of Continuous Media (Pergamon, Oxford, UK, 1960).

Lee, E.-H.

Lifschitz, E.

L. Landau, E. Lifschitz, Electrodynamics of Continuous Media (Pergamon, Oxford, UK, 1960).

Ligler, F.

S. Rabbany, B. Donner, F. Ligler, “Optical immunosensors,” Crit. Rev. Biomed. Eng. 22, 307–346 (1994).
[PubMed]

MacCraith, B.

J.-F. Gouin, A. Doyle, B. MacCraith, “Fluorescence capture by planar waveguide a platform for optical sensors,” Electron. Lett. 34, 1685–1687 (1998).
[CrossRef]

C. McDonagh, B. MacCraith, A. McEvoy, “Tailoring of sol-gel films for optical sensing of oxygen in gas and aqueous phase,” Anal. Chem. 70, 45–50 (1998).
[CrossRef] [PubMed]

McDonagh, C.

C. McDonagh, B. MacCraith, A. McEvoy, “Tailoring of sol-gel films for optical sensing of oxygen in gas and aqueous phase,” Anal. Chem. 70, 45–50 (1998).
[CrossRef] [PubMed]

McEvoy, A.

C. McDonagh, B. MacCraith, A. McEvoy, “Tailoring of sol-gel films for optical sensing of oxygen in gas and aqueous phase,” Anal. Chem. 70, 45–50 (1998).
[CrossRef] [PubMed]

Prock, A.

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

Rabbany, S.

S. Rabbany, B. Donner, F. Ligler, “Optical immunosensors,” Crit. Rev. Biomed. Eng. 22, 307–346 (1994).
[PubMed]

Ratner, V.

V. Ratner, “Calculation of the angular distribution and waveguide capture efficiency of the light emitted by a fluorophore situated at or adsorbed to the waveguide side wall,” Sens. Actuators B 17, 113–119 (1994).
[CrossRef]

Ruckstuhl, T.

Scherer, G.

C. Brinker, G. Scherer, Solgel Science (Academic, New York, 1990).

Seeger, S.

Silbey, R.

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

Stuart, H.

H. Stuart, D. Hall, “Enhanced dipole–dipole interaction between elementary radiators near a surface,” Phys. Rev. Lett. 80, 5663–5666 (1998).
[CrossRef]

Weber, W.

G. Ford, W. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. 113, 195–287 (1984).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1959).

Adv. Chem. Phys. (1)

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

Anal. Chem. (1)

C. McDonagh, B. MacCraith, A. McEvoy, “Tailoring of sol-gel films for optical sensing of oxygen in gas and aqueous phase,” Anal. Chem. 70, 45–50 (1998).
[CrossRef] [PubMed]

Appl. Opt. (2)

Crit. Rev. Biomed. Eng. (1)

S. Rabbany, B. Donner, F. Ligler, “Optical immunosensors,” Crit. Rev. Biomed. Eng. 22, 307–346 (1994).
[PubMed]

Electron. Lett. (1)

J.-F. Gouin, A. Doyle, B. MacCraith, “Fluorescence capture by planar waveguide a platform for optical sensors,” Electron. Lett. 34, 1685–1687 (1998).
[CrossRef]

Opt. Lett. (2)

Phys. Rep. (1)

G. Ford, W. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. 113, 195–287 (1984).
[CrossRef]

Phys. Rev. Lett. (1)

H. Stuart, D. Hall, “Enhanced dipole–dipole interaction between elementary radiators near a surface,” Phys. Rev. Lett. 80, 5663–5666 (1998).
[CrossRef]

Sens. Actuators B (1)

V. Ratner, “Calculation of the angular distribution and waveguide capture efficiency of the light emitted by a fluorophore situated at or adsorbed to the waveguide side wall,” Sens. Actuators B 17, 113–119 (1994).
[CrossRef]

Other (4)

Although only the modes propagating in the medium n2 at angles θ ∈ 〈θc02, θc12〉 are considered, the statement is valid for modes propagating in the entire range of angles θ ∈ 〈-90°, 90°〉.

L. Landau, E. Lifschitz, Electrodynamics of Continuous Media (Pergamon, Oxford, UK, 1960).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1959).

C. Brinker, G. Scherer, Solgel Science (Academic, New York, 1990).

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Figures (7)

Fig. 1
Fig. 1

Dipole, enclosed inside a multilayer system, that is radiating into the solid angle dΩ in the direction characterized by the wave vector k = [q, k z ] or, equivalently, by angles θ and ϕ.

Fig. 2
Fig. 2

Schematic drawing of the radiating dipole placed within a multilayer system. The dipole, depicted by the bold arrow, is placed at the interface of the jth and the (j + 1)th layer. Media with refractive indices n 0 and n N+1 are semi-infinite toward -∞ and +∞, respectively. The interface between the medium n 0 and the first layer is at z = 0.

Fig. 3
Fig. 3

(a) Decomposition of the incident-plane and the reflected-plane electromagnetic waves into s- and p-polarized components that are characterized by the unit vectors κs± and κp±, respectively. (b) Decomposition of the field in the jth layer into the plane waves that propagate in the positive and the negative z directions. The amplitudes of the s- and the p-polarized components are denoted by Eq,s,j± and Eq,p,j±, respectively.

Fig. 4
Fig. 4

Distributions of the intensity radiated by a randomly oriented dipole placed near the interface of media with refractive indices of n 0 = 1.0 and n 3 = 1.515. The solid and the dashed–dotted curves correspond to the dipole placed inside the lower-refractive-index medium at distances of t 2 = 0 and t 2 = 0.2λ, respectively, from the interface. θc03 = arcsin(n 0/n 3) = 41.3° denotes the critical angle of the n 0n 3 interface. The values of I(θ) for the angle ranges θ ∈ 〈0°, 90°〉 and θ ∈ 〈90°, 180°〉 correspond to the radiation into the media with refractive indices n 3 and n 0, respectively.

Fig. 5
Fig. 5

Distributions of the normalized intensity I(θ)/T 1 radiated by a layer of randomly oriented dipoles that are evenly distributed across a layer with a refractive index of n 1 = 1.43 and a thickness T 1. This layer is surrounded by media with refractive indices of n 0 = 1.0 and n 2 = 1.515. The solid, the dashed, and the dashed–dotted curves correspond to thicknesses of T 1 = 1.5λ, 0.3λ, 0.01λ, respectively. Angles θc02 = 41.3° and θc12 = 70.7° represent the critical angles of the n 0n 2 and the n 1n 2 interfaces, respectively. The values of I(θ)/T 1 for the angle ranges θ ∈ 〈0°, 90°〉 and θ ∈ 〈90°, 180°〉 correspond to the radiation into the media with refractive indices n 2 and n 0, respectively.

Fig. 6
Fig. 6

(a) Solid curve corresponds to the distribution of the intensity radiated by a layer of randomly oriented dipoles that are evenly distributed across a layer with a refractive index of n 1 = 1.43 and a thickness of T 1 = 0.3λ that is placed on top of a waveguide layer (n 2 = 1.6, T 2 = 2λ). The waveguide system is surrounded by media with refractive indices of n 0 = 1.0 and n 3 = 1.515. The values of I(θ) for the angle ranges θ ∈ 〈0°, 90°〉 and θ ∈ 〈90°, 180°〉 correspond to the radiation into the surrounding media with refractive indices of n 3 = 1.515 and n 0 = 1.0, respectively. For comparison, the dashed curve shows the intensity distribution when the waveguide layer is not present. (b) Magnitude of the intensity propagated along the waveguide layer as a function of angle θ2 inside the layer. The angles θc03 = 41.3° and θc13 = 70.7° represent the critical angles of the n 0n 3 and the n 1n 3 interfaces, respectively. (c) Detail of (b) plotted in a log10 scale.

Fig. 7
Fig. 7

Configuration of an optical sensor with a dipole layer that is coated on top of a glass slide. The fluorescence should (preferably) be detected at the edge of the glass slide rather than at its sides. D, detector.

Equations (38)

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Etotr=1k03  d3kEk expik·r.
Etotρ, z=1k02  d2qEq expiq·ρ,
Eq=1k0  dkzEq,kzexpikzz.
q=0, q.
Ejr=1k02x=s,p  d2qEq,x,j+ expiq·ρ×expikzjz-zj-1+Eq,x,j- expiq·ρ×exp-ikzjz-zj-1,
Eq,x,j±=Eq,x,j±κq,x,j±,
κq,s,j±=1, 0, 0,  κq,p,j±=1kj0, ±kzj, -q
kzj=kj2-q21/2,  kzj0,
Aq,x,jEq,x,j+, Eq,x,j-T,
Ts,j=11kzjk0-kzjk0,  Tp,j=kzjkj-kzjkj-kjk0-kjk0,  Pj=expikzjtj00exp-ikzjtj,
Aq,x,j=Pj-1Tx,j-1Tx,j+1·Aq,x,j+1,  j=0, 1, , N.
ρx,j=Eq,x,j-Eq,x,j+,  τx,j=Eq,x,j+1+Eq,x,j+.
ρs,j=kzj-kzj+1kzj+kzj+1,  ρp,j=kj+12kzj-kj2kzj+1kj+12kzj+kj2kzj+1,  τs,j=2kzjkzj+kzj+1,  τp,j=2kjkj+1kzjkj+12kzj+kj2kzj+1.
Mx,j=Tx,j-1Tx,j+1,
Mx,j=1τx,j1ρx,jρx,j1.
Aq,x,0=Lx,j·Aq,x,j+1,  j=0, , N,
Lx,jMx,0P1-1Mx,1P2-1  Mx,j.
Lx,jαx,j+αx,j-βx,j+βx,j-.
jr, t=-iωμ exp-iωtδr.
Eμr=1k03  d3kEkμ expik·r,
Ekμ=-12π30nj2μ+k×k×μk2-kj2,
Eμρ, z=-12π30nj21k03  d2q expiq·ρ× dkzμ+k×k×μkz2-kzj2expikzz,
Eμρ, z=-i8π20nj21k03  d2q expiq·ρ×exp±ikzjzkj±×kj±×μkzj,
kj±=q, ±kzj.
Eμρ, z>0=i8π2k00  d2q1kzj×μ·κq,s,j+κq,s,j++μ·κq,p,j+κq,p,j+×expiq·ρexp+ikzjz.
Eq,x,jμ±Eq,x,jμ±κq,x,j±,  Eq,x,jμ±=i8π20k0kzjμ·κq,x,j±,  x=s, p,
Aq,x,0=0, Eq,x,0-T,  Aq,x,N+1=Eq,x,N+1+, 0T.
Aq,x,0=Mx,0P1-1Mx,1  Mx,j-1·Aq,x,j,  Aq,x,j+1=Pj+1-1Mx,j+1Pj+2-1  Mx,N·Aq,x,N+1.
Aq,x,0=Mx,0P1-1Mx,1  Mx,j-1×Aq,x,j+Pj-1·0Eq,x,jμ-,  Aq,x,j+1+Eq,x,j+1μ+0=Pj+1-1Mx,j+1Pj+2-1  Mx,N·Aq,x,N+1.
Aq,x,0=Lx,N·Aq,x,N+1+Lx,j-1Pj-1·-Eq,x,jμ++Eq,x,jμ-.
Eq,x,N+1+=Ux,j-1+ exp-ikzjtjEq,x,jμ++Ux,j-1- exp+ikzjtjEq,x,jμ-,  Eq,x,0-=Vx,j-1+ exp-ikzjtjEq,x,jμ++Vx,j-1- exp+ikzjtjEq,x,jμ-,
Ux,j±=±αx,j±αx,N+,  Vx,j±=±βx,N+αx,j±αx,N+-βx,j±.
Ek,x,j±=kzjkjEq,x,j±.
2η0nN+1 Ik,N+1+μ=x=s,p|Ux,j-1+|2|Eq,x,jμ+|2exp+2kzjtj+|Ux,j-1-|2|Eq,x,jμ-|2exp-2kzjtj+2Ux,j-1+Ux,j-1-*Eq,x,jμ+Eq,x,jμ-*×exp-2ikzjtjkz,N+1kN+12.
1/k02|k|=kN+1kz>0 Ik,N+1+μd3k= IN+1+θ, ϕ; μdΩN+1,
IN+1+θ, ϕ; μ=nN+12Ik,N+1+μ,
2η0nN+1 Ik,N+1+μ=x=s,p|Ux,j-1+|2|Ek,x,jμ+|2 exp+T˜jT sinchT˜j+|Ux,j-1-|2|Ek,x,jμ-|2 exp-T˜jT sinchT˜j+2Ux,j-1+Ux,j-1-*Ek,x,jμ+Ek,x,jμ-*exp-iT˜j×T sincT˜jkz,N+1kN+12,
Ik,N+1+=13Ik,N+1+μ=μ1, 0, 0+Ik,N+1+μ=μ0, 1, 0+Ik,N+1+μ=μ0, 0, 1.

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