Abstract

The theory of morphology-dependent resonances of a spherical particle is developed in analogy with the theory of quantum-mechanical shape resonances. Exact analytic formulas for predicting the widths of the resonances for both real and complex indices of refraction are developed.

© 1993 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. C. Hill, R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles,P. W. Barber, R. K. Chang, eds. (World Scientific, Singapore, 1988).
  2. H. M. Nussenzveig, “Tunneling effects in diffractive scattering and resonances,” Comments At. Mol. Phys. 23, 175–187 (1989).
  3. L. G. Guimarães, H. M. Nussenzveig, “Theory of Mie resonances and ripple fluctuations,” Opt. Commun. 89, 363–369 (1992).
    [CrossRef]
  4. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Interscience, New York, 1983).
  5. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  6. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  7. C. T. Tai, Dyadic Green’s Functions in Electromagnetic Theory (Intext, Scranton, Pa., 1971).
  8. P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects,” Phys. Rev. 127, 1837–1843 (1962).
    [CrossRef]
  9. M. Abramowitz, I. A. Stegun, eds. Handbook of Mathematical Functions (Dover, New York, 1965).
  10. J. L. Dehmer, “Shape resonances in molecular fields,” in Resonances in Electron-Molecule Scattering, van der Waals Complexes, and Reactive Chemical Dynamics,D. G. Truhlar, ed. (American Chemical Society, Washington, D.C., 1984).
    [CrossRef]
  11. J. P. Toennies, W. Weiz, G. Wolf, “Observation of orbiting resonances in H2—rare gas scattering,” J. Chem. Phys. 64, 5305–5307 (1976).
    [CrossRef]
  12. R. L. Hightower, C. B. Richardson, “Resonant Mie scattering from a layered sphere,” Appl. Opt. 27, 4850–4855 (1985).
    [CrossRef]
  13. D. Q. Chowdhury, S. C. Hill, P. W. Barber, “Morphology-dependent resonances in radially inhomogeneous spheres,” J. Opt. Soc. Am. A 8, 1702–1705 (1991).
    [CrossRef]
  14. C. Kittel, Introduction to Solid State Physics, 5th ed. (Wiley, New York, 1976).
  15. B. A. Hunter, M. A. Box, B. Maier, “Resonance structure in weakly absorbing spheres,” J. Opt. Soc. Am. A 5, 1281–1286 (1988).
    [CrossRef]
  16. L. M. Folan, S. Arnold, S. D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985).
    [CrossRef]
  17. G. J. Rosasco, H. S. Bennett, “Internal field resonance structure: implications for optical absorption and scattering by microscopic particles,” J. Opt. Soc. Am. 68, 1242–1250 (1978).
    [CrossRef]
  18. H-B. Lin, A. L. Huston, J. D. Eversole, A. J. Campillo, “Cavity-mode identification of fluorescence and lasing in dye-doped microdroplets,” Appl. Opt. 31, 1982–1991 (1992).
    [CrossRef] [PubMed]
  19. C. C. Lam, P. T. Leung, K. Young, “Explicit asymptotic formulas for the positions, widths and strengths of resonances in Mie scattering,” J. Opt. Soc. Am. B 9, 1585–1592 (1992).
    [CrossRef]

1992 (3)

1991 (1)

1989 (1)

H. M. Nussenzveig, “Tunneling effects in diffractive scattering and resonances,” Comments At. Mol. Phys. 23, 175–187 (1989).

1988 (1)

1985 (2)

L. M. Folan, S. Arnold, S. D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985).
[CrossRef]

R. L. Hightower, C. B. Richardson, “Resonant Mie scattering from a layered sphere,” Appl. Opt. 27, 4850–4855 (1985).
[CrossRef]

1978 (1)

1976 (1)

J. P. Toennies, W. Weiz, G. Wolf, “Observation of orbiting resonances in H2—rare gas scattering,” J. Chem. Phys. 64, 5305–5307 (1976).
[CrossRef]

1962 (1)

P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects,” Phys. Rev. 127, 1837–1843 (1962).
[CrossRef]

Arnold, S.

L. M. Folan, S. Arnold, S. D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985).
[CrossRef]

Barber, P. W.

Benner, R. E.

S. C. Hill, R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles,P. W. Barber, R. K. Chang, eds. (World Scientific, Singapore, 1988).

Bennett, H. S.

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Interscience, New York, 1983).

Box, M. A.

Campillo, A. J.

Chowdhury, D. Q.

Dehmer, J. L.

J. L. Dehmer, “Shape resonances in molecular fields,” in Resonances in Electron-Molecule Scattering, van der Waals Complexes, and Reactive Chemical Dynamics,D. G. Truhlar, ed. (American Chemical Society, Washington, D.C., 1984).
[CrossRef]

Druger, S. D.

L. M. Folan, S. Arnold, S. D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985).
[CrossRef]

Eversole, J. D.

Folan, L. M.

L. M. Folan, S. Arnold, S. D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985).
[CrossRef]

Guimarães, L. G.

L. G. Guimarães, H. M. Nussenzveig, “Theory of Mie resonances and ripple fluctuations,” Opt. Commun. 89, 363–369 (1992).
[CrossRef]

Hightower, R. L.

Hill, S. C.

D. Q. Chowdhury, S. C. Hill, P. W. Barber, “Morphology-dependent resonances in radially inhomogeneous spheres,” J. Opt. Soc. Am. A 8, 1702–1705 (1991).
[CrossRef]

S. C. Hill, R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles,P. W. Barber, R. K. Chang, eds. (World Scientific, Singapore, 1988).

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Interscience, New York, 1983).

Hunter, B. A.

Huston, A. L.

Kerker, M.

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

Kittel, C.

C. Kittel, Introduction to Solid State Physics, 5th ed. (Wiley, New York, 1976).

Lam, C. C.

Leung, P. T.

Lin, H-B.

Maier, B.

Nussenzveig, H. M.

L. G. Guimarães, H. M. Nussenzveig, “Theory of Mie resonances and ripple fluctuations,” Opt. Commun. 89, 363–369 (1992).
[CrossRef]

H. M. Nussenzveig, “Tunneling effects in diffractive scattering and resonances,” Comments At. Mol. Phys. 23, 175–187 (1989).

Richardson, C. B.

Rosasco, G. J.

Tai, C. T.

C. T. Tai, Dyadic Green’s Functions in Electromagnetic Theory (Intext, Scranton, Pa., 1971).

Toennies, J. P.

J. P. Toennies, W. Weiz, G. Wolf, “Observation of orbiting resonances in H2—rare gas scattering,” J. Chem. Phys. 64, 5305–5307 (1976).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Weiz, W.

J. P. Toennies, W. Weiz, G. Wolf, “Observation of orbiting resonances in H2—rare gas scattering,” J. Chem. Phys. 64, 5305–5307 (1976).
[CrossRef]

Wolf, G.

J. P. Toennies, W. Weiz, G. Wolf, “Observation of orbiting resonances in H2—rare gas scattering,” J. Chem. Phys. 64, 5305–5307 (1976).
[CrossRef]

Wyatt, P. J.

P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects,” Phys. Rev. 127, 1837–1843 (1962).
[CrossRef]

Young, K.

Appl. Opt. (2)

Chem. Phys. Lett. (1)

L. M. Folan, S. Arnold, S. D. Druger, “Enhanced energy transfer within a microparticle,” Chem. Phys. Lett. 118, 322–327 (1985).
[CrossRef]

Comments At. Mol. Phys. (1)

H. M. Nussenzveig, “Tunneling effects in diffractive scattering and resonances,” Comments At. Mol. Phys. 23, 175–187 (1989).

J. Chem. Phys. (1)

J. P. Toennies, W. Weiz, G. Wolf, “Observation of orbiting resonances in H2—rare gas scattering,” J. Chem. Phys. 64, 5305–5307 (1976).
[CrossRef]

J. Opt. Soc. Am. (1)

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

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

Opt. Commun. (1)

L. G. Guimarães, H. M. Nussenzveig, “Theory of Mie resonances and ripple fluctuations,” Opt. Commun. 89, 363–369 (1992).
[CrossRef]

Phys. Rev. (1)

P. J. Wyatt, “Scattering of electromagnetic plane waves from inhomogeneous spherically symmetric objects,” Phys. Rev. 127, 1837–1843 (1962).
[CrossRef]

Other (8)

M. Abramowitz, I. A. Stegun, eds. Handbook of Mathematical Functions (Dover, New York, 1965).

J. L. Dehmer, “Shape resonances in molecular fields,” in Resonances in Electron-Molecule Scattering, van der Waals Complexes, and Reactive Chemical Dynamics,D. G. Truhlar, ed. (American Chemical Society, Washington, D.C., 1984).
[CrossRef]

S. C. Hill, R. E. Benner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles,P. W. Barber, R. K. Chang, eds. (World Scientific, Singapore, 1988).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Interscience, New York, 1983).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

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

C. T. Tai, Dyadic Green’s Functions in Electromagnetic Theory (Intext, Scranton, Pa., 1971).

C. Kittel, Introduction to Solid State Physics, 5th ed. (Wiley, New York, 1976).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Effective potential associated with a spherical dielectric particle.

Fig. 2
Fig. 2

Radial wave functions for the three TE, n = 40 resonances.

Fig. 3
Fig. 3

Behavior of the TE wave function in the vicinity of a resonance: behavior for a size parameter value slightly above resonance (top); on resonance (middle); below resonance (bottom).

Fig. 4
Fig. 4

Effective potential function for a layered sphere with a positive dielectric core covered by a negative dielectric layer.

Fig. 5
Fig. 5

Effective potential associated with a negative dielectric particle, (a) k2 > 0, (b) k2 < 0.

Equations (67)

Equations on this page are rendered with MathJax. Learn more.

× × E k 2 m 2 ( r ) E = 0 .
M n , m ( r , θ , ϕ ) = exp ( i m ϕ ) k r S n ( r ) X n , m ( θ ) , N n , m ( r , θ , ϕ ) = exp ( i m ϕ ) k 2 m 2 ( r ) [ 1 r T n ( r ) r Y n , m ( θ ) + 1 r 2 T n ( r ) Z n , m ( θ ) ] ,
X n , m ( θ ) = i π n , m ( θ ) ê θ τ n , m ( θ ) ê ϕ , Y n , m ( θ ) = τ n , m ( θ ) ê θ i π n , m ( θ ) ê ϕ , Z n , m ( θ ) = n ( n + 1 ) P n m ( cos θ ) ê r ,
π n , m ( θ ) = m sin θ P n m ( cos θ ) , τ n , m ( θ ) = θ P n m ( cos θ ) .
d 2 S n ( r ) d r 2 + [ k 2 m 2 ( r ) n ( n + 1 ) r 2 ] S n ( r ) = 0 ,
d 2 T n ( r ) d r 2 2 m ( r ) d m ( r ) d r d T n ( r ) d r + [ k 2 m 2 ( r ) n ( n + 1 ) r 2 ] T n ( r ) = 0 .
ψ n ( mkr ) = mkr j n ( mkr ) ,
χ n ( mkr ) = mkr n n ( mkr ) ,
S n ( r ) = B n [ χ n ( k r ) + β n ψ n ( k r ) ] ,
T n ( r ) = A n [ χ n ( k r ) + α n ψ n ( k r ) ] ,
U n ( r ) = 1 k [ S n ( r ) / S n ( r ) ] ,
V n ( r ) = 1 k m 2 ( r ) [ T n ( r ) / T n ( r ) ] ,
D n ( x ) = ψ n ( x ) / ψ n ( x ) ,
G n ( x ) = χ n ( x ) / χ n ( x ) .
U n ( a ) = χ n ( k a ) + β n ψ n ( k a ) χ n ( k a ) + β n ψ n ( k a ) ,
V n ( a ) = χ n ( k a ) + α n ψ n ( k a ) χ n ( k a ) + α n ψ n ( k a ) ,
β n = χ n ( k a ) ψ n ( k a ) [ G n ( k a ) U n ( a ) D n ( k a ) U n ( a ) ] ,
α n = χ n ( k a ) ψ n ( k a ) [ G n ( k a ) V n ( a ) D n ( k a ) V n ( a ) ] .
S n ( r ) = T n ( r ) = ψ n ( mkr ) .
U n ( r ) = m D n ( mkr ) ,
V n ( r ) = ( 1 / m ) D n ( mkr ) .
β n = χ n ( x ) ψ n ( x ) [ G n ( x ) m D n ( m x ) D n ( x ) m D n ( m x ) ] ,
α n = χ n ( x ) ψ n ( x ) [ m G n ( x ) D n ( m x ) m D n ( x ) D n ( m x ) ] ,
b n = 1 1 i β n ,
a n = 1 1 i α n .
d 2 ψ ( r ) d r 2 + [ V ( r ) + n ( n + 1 ) r 2 ] ψ ( r ) = E ψ ( r ) ,
V ( r ) = k 2 [ 1 m 2 ( r ) ]
E = k 2 .
V n ( r ) = k 2 [ 1 m 2 ( r ) ] + n ( n + 1 ) r 2 .
p n 2 ( r ) = k 2 m 2 ( r ) n ( n + 1 ) r 2 .
V n ( r ) = { k 2 ( 1 m 2 ) + n ( n + 1 ) / r 2 r a n ( n + 1 ) / r 2 r > a .
r 1 = ( n + 1 / 2 ) / k m ,
r 2 = ( n + 1 / 2 ) / k ,
x B = ( n + 1 / 2 ) / m ,
x T = n + 1 / 2 .
G n ( x 0 ) = m D n ( m x 0 ) ,
m G n ( x 0 ) = D n ( m x 0 ) .
k U 2 = [ n ( n + 1 ) ] / m 2 a 2 .
Re [ b n ( x ) ] = 1 1 + β n 2 ( x ) .
β n ( x ) = β n ( x 0 ) ( x x 0 ) ,
w n ( x 0 ) = 2 β n ( x 0 ) .
β n ( x 0 ) = χ n ( x 0 ) ψ n ( x 0 ) [ m 2 D n ( m x 0 ) G n ( x 0 ) m D n ( m x 0 ) D n ( x 0 ) ] .
D n ( x ) = [ n ( n + 1 ) ] / x 2 1 D n 2 ( x ) ,
G n ( x ) = [ n ( n + 1 ) ] / x 2 1 G n 2 ( x ) ,
G n ( x ) D n ( x ) = [ ψ n ( x ) χ n ( x ) ] 1 .
w n ( x 0 ) = 2 ( m 2 1 ) χ n 2 ( x 0 ) .
w ¯ n ( x 0 ) = 2 α n ( x 0 ) .
α n ( x 0 ) = m χ n ( x 0 ) ψ n ( x 0 ) [ G n ( x 0 ) D n ( m x 0 ) m D n ( x 0 ) D n ( m x 0 ) ] .
w ¯ n ( x 0 ) = 2 / { ( m 2 1 ) χ n 2 ( x 0 ) [ n ( n + 1 ) m 2 x 0 2 + G n 2 ( x 0 ) ] } .
β n ( m ; x ) = β n ( m r , x 0 ) ( x x 0 ) + i γ n ( m ; x ) ,
γ n ( m ; x ) = m i × [ β n ( m ; x ) m ] m = m r .
Re [ b n ( x ) ] = [ 1 + γ n ] / { [ 1 + γ n ] 2 + [ β n ( x x 0 ) ] 2 } .
Re [ b n ( x 0 ) ] = 1 / ( 1 + γ n ) .
Re [ b n ( x 0 ± Δ x ) ] = 1 / 2 ( 1 + γ n ) ,
w n ( m i ; x 0 ) = w n ( 0 ; x 0 ) [ 1 + γ n ] ,
H n ( m i ; x 0 ) = w n ( 0 ; x 0 ) w n ( m i ; x 0 ) .
β n m = χ n ( x 0 ) ψ n ( x 0 ) [ D n ( m x 0 ) + m x 0 D n ( m x 0 ) m D n ( m x 0 ) D n ( x 0 ) ] .
β n m = x 0 χ 0 2 ( x 0 ) m { [ m 2 1 ] [ G n ( x 0 ) + G n ( x 0 ) / x 0 ] } .
γ n ( m ; x 0 ) = χ n 2 ( x 0 ) ( m r 2 1 ) m i m r [ 1 n ] x 0 ,
n = 1 ( m r 2 1 ) [ G n ( x 0 ) + G n ( x 0 ) / x 0 ] .
w n ( m i ; x 0 ) = w n ( 0 ; x 0 ) + 2 x 0 m i m r ( 1 n ) .
w n ( m i ; x 0 ) = 2 ( m r 2 1 ) χ n 2 ( x 0 ) + 2 x 0 m i m r .
α n m = χ n ( x 0 ) ψ n ( x 0 ) [ x 0 D n ( m x 0 ) G ( x 0 ) G ( x 0 ) D ( x 0 ) ] .
α n m = χ n 2 ( x 0 ) m [ x 0 + m 2 x 0 G n 2 ( x 0 ) n ( n + 1 ) m 2 x 0 ] .
w ¯ n ( m i ; x 0 ) = w ¯ n ( 0 ; x 0 ) + 2 x 0 m i m r ( 1 ¯ n ) ,
¯ n = G n ( x 0 ) G n ( x 0 ) / x 0 ( m 2 1 ) [ G n 2 ( x 0 ) + n ( n + 1 ) ( m 2 x 0 2 ) ] .
w ¯ n ( m i ; x 0 ) = 2 ( m r 2 1 ) χ n 2 ( x 0 ) [ n ( n + 1 ) ( m r 2 x 0 2 ) + G n 2 ( x 0 ) ] + 2 x 0 m i m r .

Metrics