Abstract

Fluorescent and Raman scattering by molecules embedded in dielectric particles is strongly dependent on the morphology and optical properties of the particle and the distribution of active molecules within the particle. In this paper, the formalism is derived for the case where the scattering molecules are embedded in an infinite dielectric cylinder. Analytical results for the scattered fields are given for arbitrary angles of incidence. The general results, which involve an integral and a sum, are rather lengthy. Accordingly, the saddle-point method has been used to carry out the integration approximately. Numerical results are given for perpendicular incidence and for observation in the plane perpendicular to the cylinder axis, for single dipoles variously located within the cylinder, and for a uniform distribution of isotropic incoherent dipoles. The angular distribution and polarization of the scattered irradiance depends sensitively upon cylinder radius and refractive index, so that this effect must be considered if inelastic scattering signals are to be used as a diagnostic tool.

© 1980 Optical Society of America

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Corrections

H. Chew, D. D. Cooke, and M. Kerker, "Raman and fluorescent scattering by molecules embedded in dielectric cylinders: erratum," Appl. Opt. 19, 1741-1741 (1980)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-19-11-1741

References

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  1. E. S. Etz, G. J. Rosasco, J. J. Blaha, in Environmental Pollutants, T. Toribara et al., Eds. (Plenum, New York, 1978), pp. 413ff.
    [Crossref]
  2. B. H. Mayall, B. L. Gledhill, Eds., Proceedings of the Sixth Conference on Automated Cytology, J. Histochem. Cytochem.27, 1–641 (1979).
  3. H. Chew, P. J. McNulty, M. Kerker, Phys. Rev. A 13, 396 (1976).
    [Crossref]
  4. H. Chew, M. Kerker, P. J. McNulty, J. Opt. Soc. Am. 66, 440 (1976).
    [Crossref]
  5. M. Kerker, P. J. McNulty, M. Sculley, H. Chew, D. D. Cooke, J. Opt. Soc. Am. 68, 1676 (1978).
    [Crossref]
  6. H. Chew, M. Sculley, M. Kerker, P. J. McNulty, D. D. Cooke, J. Opt. Soc. Am. 68, 1686 (1978).
    [Crossref]
  7. M. Kerker, S. D. Druger, Appl. Opt. 18, 1180 (1979).
    [Crossref] [PubMed]
  8. E.-H. Lee, R. E. Benner, J. B. Fenn, R. K. Chang, Appl. Opt. 17, 1980 (1978).
    [Crossref]
  9. J. P. Kratohvil, M.-P. Lee, M. Kerker, Appl. Opt. 17, 1978 (1978).
    [Crossref] [PubMed]
  10. P. J. McNulty, S. D. Druger, M. Kerker, H. W. Chew, Appl. Opt. 18, 1484 (1979).
    [Crossref] [PubMed]
  11. C. T. Tai, Dyadic Green’s Functions in Electromagnetic Theory (Intext Publishers, Columbus, 1971). Our notation is the same as in this book, with minor modifications: Our η and ζ correspond to Tai’s ζ and η, respectively, and our wave numbers are denoted by k and k′ instead of k1 and k2. We also use lower-case instead of capital letters for the arguments of the dyadic Green’s functions.
  12. We are anticipating the use of the boundary conditions at ρ = a and have accordingly assumed that ρ > ρ′.
  13. M. Kerker, Scattering of Light and other Electromagnetic Radiation (Academic, New York, 1969).
  14. The details may be found in Ref. 11.

1979 (2)

1978 (4)

1976 (2)

H. Chew, M. Kerker, P. J. McNulty, J. Opt. Soc. Am. 66, 440 (1976).
[Crossref]

H. Chew, P. J. McNulty, M. Kerker, Phys. Rev. A 13, 396 (1976).
[Crossref]

Benner, R. E.

Blaha, J. J.

E. S. Etz, G. J. Rosasco, J. J. Blaha, in Environmental Pollutants, T. Toribara et al., Eds. (Plenum, New York, 1978), pp. 413ff.
[Crossref]

Chang, R. K.

Chew, H.

Chew, H. W.

Cooke, D. D.

Druger, S. D.

Etz, E. S.

E. S. Etz, G. J. Rosasco, J. J. Blaha, in Environmental Pollutants, T. Toribara et al., Eds. (Plenum, New York, 1978), pp. 413ff.
[Crossref]

Fenn, J. B.

Kerker, M.

Kratohvil, J. P.

Lee, E.-H.

Lee, M.-P.

McNulty, P. J.

Rosasco, G. J.

E. S. Etz, G. J. Rosasco, J. J. Blaha, in Environmental Pollutants, T. Toribara et al., Eds. (Plenum, New York, 1978), pp. 413ff.
[Crossref]

Sculley, M.

Tai, C. T.

C. T. Tai, Dyadic Green’s Functions in Electromagnetic Theory (Intext Publishers, Columbus, 1971). Our notation is the same as in this book, with minor modifications: Our η and ζ correspond to Tai’s ζ and η, respectively, and our wave numbers are denoted by k and k′ instead of k1 and k2. We also use lower-case instead of capital letters for the arguments of the dyadic Green’s functions.

Appl. Opt. (4)

J. Opt. Soc. Am. (3)

Phys. Rev. A (1)

H. Chew, P. J. McNulty, M. Kerker, Phys. Rev. A 13, 396 (1976).
[Crossref]

Other (6)

E. S. Etz, G. J. Rosasco, J. J. Blaha, in Environmental Pollutants, T. Toribara et al., Eds. (Plenum, New York, 1978), pp. 413ff.
[Crossref]

B. H. Mayall, B. L. Gledhill, Eds., Proceedings of the Sixth Conference on Automated Cytology, J. Histochem. Cytochem.27, 1–641 (1979).

C. T. Tai, Dyadic Green’s Functions in Electromagnetic Theory (Intext Publishers, Columbus, 1971). Our notation is the same as in this book, with minor modifications: Our η and ζ correspond to Tai’s ζ and η, respectively, and our wave numbers are denoted by k and k′ instead of k1 and k2. We also use lower-case instead of capital letters for the arguments of the dyadic Green’s functions.

We are anticipating the use of the boundary conditions at ρ = a and have accordingly assumed that ρ > ρ′.

M. Kerker, Scattering of Light and other Electromagnetic Radiation (Academic, New York, 1969).

The details may be found in Ref. 11.

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Figures (12)

Fig. 1
Fig. 1

The scattering geometry. The cylinder axis passes through the origin and is along the z axis. The incident plane wave is along the x axis, and the scattering plane is the x-y plane. The fluorescing molecule is located at a point r′ inside the cylinder, whose radius is a. The scattering angle, designated as θsc in Sec. III of the text and as theta on the following figure coordinates, corresponds to ϕ in this figure.

Fig. 2
Fig. 2

Scattered irradiance with electric vector parallel to cylinder axis. I1 vs θsc for α = 5.0, m = 1.5 for a single dipole located at radial distances along the positive-y axis 0.0001a, 0.25a, 0.50a, and 0.75a.

Fig. 3
Fig. 3

Same as Fig. 2 for scattered irradiance with electric vector perpendicular to cylinder axis I2 and dipole locations along the positive-x axis.

Fig. 4
Fig. 4

Same as Fig. 2 for I1 and dipole locations along the negative-x axis.

Fig. 5
Fig. 5

Same as Fig. 2 for I2 and dipole locations along the negative-y axis.

Fig. 6
Fig. 6

I1 vs θsc for cylinder uniformly filled with dipoles and various values of α as indicated, m = 1.2.

Fig. 7
Fig. 7

Same as Fig. 6 for I2.

Fig. 8
Fig. 8

Same as Fig. 6 for m = 1.5.

Fig. 9
Fig. 9

Same as Fig. 6 for m = 1.5 and I2.

Fig. 10
Fig. 10

I1 and I2 vs α for cylinder uniformly filled with dipoles and m = 1.2, θsc as indicated.

Fig. 11
Fig. 11

I1 vs α for cylinder uniformly filled with dipoles and m = 1.5, θsc as indicated.

Fig. 12
Fig. 12

Same as Fig. 11 for I2.

Equations (50)

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E dip ( r ) = 4 π k 2 G ˜ 0 ( r , r ) × P ,
M n o e η ( h ) = [ ( - sin n ϕ cos n ϕ ) n J n ( η ρ ) ρ ρ ^ - ( cos n ϕ sin n ϕ ) η J n ( η ρ ) ϕ ^ ] × exp ( i h z ) , η 2 = k 2 - h 2 ,
N n o e η ( h ) = [ i h η J n ( η ρ ) ( cos n ϕ sin n ϕ ) ρ ^ + i h n ρ J n ( η ρ ) ( - sin n ϕ cos n ϕ ) ϕ ^ + η 2 J n ( η ρ ) ( cos n ϕ sin n ϕ ) k ^ ] exp ( i h z ) k .
M n o e ζ ( 1 ) ( h ) = M n o e η ( h ) with η replaced by ζ , J n by H n ( 1 ) ,
N n o e ζ ( 1 ) ( h ) = N n o e η ( h ) with η replaced by ζ , J n by H n ( 1 ) and k by k .
E d i p ( r ) = 4 π k 2 G ˜ 0 ( r , r ) × P = i k 2 2 - d h e , 0 n = 0 2 - δ n 0 η 2 [ α n o e M n o e η ( 1 ) ( h ) + β n o e ( h ) N n o e η ( 1 ) ( h ) ] ,
M n o e η ( 1 ) and N n o e η ( 1 ) are equal to M n o e η and N n o e η
α n o e ( h ) = P · M n o e η ( - h ) ,
β n o e ( h ) = P · N n o e η ( - h ) ,
E i ( r ) = i k 2 - d h e , o n = 0 2 - δ n 0 η 2 [ a n o e ( h ) M n o e η ( h ) + b n o e ( h ) N n o e η ( h ) ] ,
E s ( r ) = i k 2 - d h e , o n = 0 2 - δ n 0 ζ 2 [ A n o e ( h ) M n o e ζ ( 1 ) ( h ) + B n o e ( h ) N n o e ζ ( 1 ) ( h ) ] .
ρ ^ × [ E i ( r ) + E dip ( r ) ] = ρ ^ × E s ( r ) ,
ρ ^ × [ H i ( r ) + H dip ( r ) ] = ρ ^ × H s ( r ) ,
B = - i c ω × E = - i k × E ,
H s ( r ) = - i k × E s = k 2 2 - d h e , 0 n = 0 2 - δ n 0 ζ 2 × [ A n o e ( h ) N n o e ζ ( 1 ) ( h ) + B n o e ( h ) M n o e ζ ( 1 ) ( h ) ] ,
H dip ( r ) = - i μ k × E dip ( r ) = k k 2 μ - d h e , 0 n = 0 2 - δ n 0 η 2 [ α n o e ( h ) N n o e η ( h ) + B n o e h ) M n o e ζ ( h ) ] ,
A e n ( h ) = 2 i π η 2 a Δ μ { [ k 2 J n ( η a ) H n ( 1 ) ( ζ a ) μ η k - k J n ( η a ) H n ( 1 ) ( ζ a ) ζ ] α e n ( h ) + i h n k J n ( η a ) H n ( 1 ) ( ζ a ) μ k a ( 1 η 2 - 1 ζ 2 ) β o n ( h ) } ,
A o n ( h ) = 2 i π η 2 a Δ μ { [ k 2 J n ( η a ) H n ( 1 ) ( ζ a ) μ η k - k J n ( η a ) H n ( 1 ) ( ζ a ) ζ ] α o n ( h ) - i h n k J n ( η a ) H n ( 1 ) ( ζ a ) μ k a ( 1 η 2 - 1 ζ 2 ) β e n ( h ) } ,
B e n ( h ) = 2 i π η 2 a Δ μ { [ J n ( η a ) H n ( 1 ) ( ζ a ) η - J n ( η a ) H n ( 1 ) ( ζ a ) μ ζ ] k β e n ( h ) + i h n J n ( η a ) H n ( 1 ) ( ζ a ) a ( 1 η 2 - 1 ζ 2 ) α o n ( h ) } ,
B o n ( h ) = 2 i π η 2 a Δ μ { [ J n ( η a ) H n ( 1 ) ( ζ a ) η - J n ( η a ) H n ( 1 ) ( ζ a ) μ ζ ] k β o n ( h ) + i h n J n ( η a ) H n ( 1 ) ( ζ a ) a ( 1 η 2 - 1 ζ 2 ) α e n ( h ) } ,
Δ = j n 2 ( η a ) [ H n ( 1 ) ( ζ a ) ] 2 h 2 n 2 μ k a 2 ( 1 η 2 - 1 ζ 2 ) 2 + [ J n ( η a ) H n ( 1 ) ( ζ a ) μ ζ - H n ( 1 ) ( ζ a ) J n ( η a ) η ] × [ k 2 J n ( η a ) H n ( 1 ) ( η a ) μ k η - k J n ( η a ) H n ( 1 ) ( ζ a ) ζ ] .
α n o e and β n o e
E s ( r ) r k 2 - d h exp [ i ( ζ ρ + h z ) ] ( 2 π i ζ ρ ) 1 / 2 e , o n = 0 i - n ( 2 - δ n o ) × ( cos n ϕ sin n ϕ ) [ A n o e ( h ) ζ ϕ ^ + i B n o e ( h ) k ( - h ζ ρ ^ + k ^ ) ] .
- G ( h ) ( 2 π ζ ρ i ) 1 / 2 exp [ i ( ζ ρ + h z ) ] d h = G ( k cos θ ) exp ( i k r ) i r .
E s ( r ) k exp ( i k r ) i r sin θ e , o n = 0 i - n ( cos n ϕ sin n ϕ ) ( 2 - δ n o ) × [ A n o e ( k cos θ ) ϕ ^ - i B n o e ( k cos θ ) θ ^ ] = k exp ( i k r ) i r sin θ n = 0 i - n ( 2 - δ n o ) × { [ A e n ( k cos θ ) ϕ ^ - i B e n ( k cos θ ) θ ^ ] cos n ϕ + [ A o n ( k cos θ ) ϕ ^ - i B o n ( k cos θ ) θ ^ ] sin n ϕ } ,
A n o e ( h ) ,             B n o e ( h )
M n o e η ( - h ) ,             N n o e η ( - h ) .
- exp ( - i h z ) d z = 2 π δ ( h ) .
E s ( r ) 2 π k exp ( i k ρ ) ( 2 π i ρ ) 1 / 2 n = 0 i - n ( 2 - δ n o ) { [ A e n ( 0 ) ϕ ^ + i B e n ( 0 ) k ^ ] cos n ϕ + [ A o n ( 0 ) ϕ ^ + i B o n ( 0 ) k ^ ] sin n ϕ } ,
A e n ( 0 ) = 2 i α e n ( 0 ) π k a [ ( k / k ) J n ( k a ) H n ( 1 ) ( k a ) - μ J n ( k a ) H n ( 1 ) ( k a ) ] ,
A o n ( 0 ) = 2 i α o n ( 0 ) π k a [ ( k / k ) J n ( k a ) H n ( 1 ) ( k a ) - μ J n ( k a ) H n ( 1 ) ( k a ) ] ,
B e n ( 0 ) = 2 i β e n ( 0 ) π k a [ μ J n ( k a ) H n ( 1 ) ( k a ) - ( k / k ) J n ( k a ) H n ( 1 ) ( k a ) ] .
B o n ( 0 ) = 2 i β o n ( 0 ) π k a [ μ J n ( k a ) H n ( 1 ) ( k a ) - ( k / k ) J n ( k a ) H n ( 1 ) ( k a ) ] .
- exp ( i h z ) f ( h ) d h = - exp ( i h z ) g ( h ) d h
- H n ( 1 ) ( ζ a ) ζ A o n - i h n H n ( 1 ) ( ζ a ) k a ζ 2 B e n + J n ( η a ) η a o n + i h n J n ( η a ) k a η 2 b e n = - H n ( 1 ) ( η a ) η α o n - i h n H n ( 1 ) ( η a ) k a η 2 β e n ,
i h n H n ( 1 ) ( ζ a ) k a ζ 2 A o n - H n ( 1 ) ( ζ a ) ζ 2 B e n - i h n J n ( η a ) μ k a η 2 a o n + k J n ( η a ) μ k η b e n = i h n H n ( 1 ) ( η a ) μ k a η 2 α o n - k H n ( 1 ) ( η a ) μ k η β e n ,
H n ( 1 ) ( ζ a ) A o n + 0 - J n ( η a ) a o n / μ + 0 = H n ( 1 ) ( η a ) α o n / μ ,
0 + k H n ( 1 ) ( ζ a ) B e n + 0 - k J n ( η a ) b e n = k H n ( 1 ) ( η a ) β e n ,
- H n ( 1 ) ( ζ a ) ζ A e n + i h n H n ( 1 ) ( ζ a ) k a ζ 2 B o n + J n ( η a ) η a e n - i h n J n ( η a ) k a η 2 b o n = - H n ( 1 ) ( η a ) η α e n + i h n H n ( 1 ) ( η a ) k a η 2 β o n ,
i h n H n ( 1 ) ( ζ a ) k a ζ 2 A e n + H n ( 1 ) ( ζ a ) ζ B o n - i h n J n ( η a ) μ k a η 2 a e n - k J n ( η a ) μ k η b o n = i h n H n ( 1 ) ( η a ) μ k a η 2 α e n + k H n ( 1 ) ( η a ) μ k η β o n ,
H n ( 1 ) ( ζ a ) A e n + 0 - J n ( η a ) a e n / μ + 0 = H n ( 1 ) ( η a ) a e n / μ ,
0 + k H n ( 1 ) ( ζ a ) B o n + 0 - k J n ( η a ) b o n = k H n ( 1 ) ( η a ) β o n .
c n I = - 2 h n l 2 k 0 3 ( m 2 - 1 ) H n ( 2 ) ( l a ) J n ( j a ) π m α 2 D ,
d n I = 2 i l 3 j [ l H n ( 2 ) ( l a ) J n ( j a ) - j J n ( j a ) H n ( 2 ) ( l a ) ] π a ,
c n II = 2 i l 3 j [ m 2 l J n ( j a ) H n ( 2 ) ( l a ) - j J n ( j a ) H n ( 2 ) ( l a ) ] π m a D ,
d n II = 2 h n l 2 k 0 3 ( m 2 - 1 ) J n ( j a ) H n ( 2 ) ( l a ) π α 2 D ,
D = h 2 n 2 m α 2 J n 2 ( j a ) [ H n ( 2 ) ( l a ) ] 2 ( m 2 - 1 ) 2 k 0 4 + j 2 l 2 × [ j J n ( j a ) H n ( 2 ) ( l a ) - l H n ( 2 ) ( l a ) J n ( j a ) ] × [ m l H n ( 2 ) ( l a ) J n ( j a ) - j m J n ( j a ) H n ( 2 ) ( l a ) ] .
c n I = d n II = 0 ,
d n I = 2 i π m a [ k 0 J n ( k a ) H n ( 2 ) ( k o a ) - k H n 2 ) ( k o a ) J n ( k a ) ] ,
d n II = 2 i π m a [ k J n ( k a ) H n ( 2 ) ( k o a ) - k o H n ( 2 ) ( k o a ) J n ( k a ) ] .

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