Abstract

We investigate the inelastic scattering on spherical particles that contain one concentric inclusion in the case of input and output resonances, using a geometrical optics method. The excitation of resonances is included in geometrical optics by use of the concept of tunneled rays. To get a quantitative description of optical tunneling on spherical surfaces, we derive appropriate Fresnel-type reflection and transmission coefficients for the tunneled rays. We calculate the inelastic scattering cross section in the case of input and output resonances and investigate the influence of the distribution of the active material in the particle as well as the influence of the inclusion on inelastic scattering.

© 2003 Optical Society of America

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References

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  1. H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976).
    [CrossRef]
  2. H. Chew, M. Sculley, M. Kerker, P. J. McNulty, D. D. Cooke, “Raman and fluorescent scattering by molecules embedded in small particles: results for coherent optical processes,” J. Opt. Soc. Am. 68, 1686–1689 (1979).
    [CrossRef]
  3. M. Kerker, S. D. Druger, “Raman and fluorescent scattering by molecules embedded in spheres with radii up to several multiples of the wavelength,” Appl. Opt. 18, 1172–1179 (1979).
    [CrossRef] [PubMed]
  4. M. Kerker, P. J. McNulty, M. Sculley, H. Chew, D. D. Cooke, “Raman and fluorescent scattering by molecules embedded in small particles: results for incoherent optical processes,” J. Opt. Soc. Am. 68, 1676–1686 (1979).
    [CrossRef]
  5. H. Chew, M. Kerker, P. J. McNulty, “Raman and fluorescent scattering by molecules embedded in concentric spheres,” J. Opt. Soc. Am. 66, 440–444 (1976).
    [CrossRef]
  6. F. Borghese, P. Denti, R. Saija, O. I. Sindoni, “Optical properties of spheres containing a spherical eccentric inclusion,” J. Opt. Soc. Am. 9, 1327–1335 (1992).
    [CrossRef]
  7. F. Borghese, P. Denti, R. Saija, “Optical properties of spheres containing several inclusions,” Appl. Opt. 33, 484–493 (1994).
    [CrossRef] [PubMed]
  8. K. A. Fuller, “Morphology-dependent resonances in eccentrically stratified spheres,” Opt. Lett. 19, 1272–1274 (1994).
    [CrossRef] [PubMed]
  9. J. A. Roumeliotis, N. B. Kakogiannos, J. D. Kanellopoulos, “Scattering from a sphere of small radius embedded into a dielectric one,” IEEE Trans. Microwave Theory Tech. 43, 155–168 (1995).
    [CrossRef]
  10. N. C. Skaropoulos, M. P. Ioannidou, D. P. Chrissoulidis, “Indirect mode-matching solution to scattering from a dielectric sphere with an eccentric inclusion,” J. Opt. Soc. Am. A 11, 1859–1866 (1994).
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  13. N. Velesco, T. Kaiser, G. Schweiger, “Geometrical optics calculation of the internal field of a large spherical particle by using geometrical optics approximation,” Appl. Opt. 36, 8724–8728 (1997).
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  14. N. Velesco, G. Schweiger, “Geometrical optics calculation of inelastic scattering on large particles,” Appl. Opt. 38, 1046–1052 (1999).
    [CrossRef]
  15. S. C. Hill, R. E. Brenner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber, R. K. Chang, eds. (World Scientific, Singapore, 1988), pp. 3–61.
  16. G. Roll, G. Schweiger, “Geometrical optics model of Mie resonances,” J. Opt. Soc. Am. A 17, 1301–1311 (2000).
    [CrossRef]
  17. H. M. Nussenzveig, W. J. Wiscombe, “Diffraction as tunneling,” Phys. Rev. Lett. 59, 1667–1670 (1987).
    [CrossRef] [PubMed]
  18. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  19. G. Roll, T. Kaiser, S. Lange, G. Schweiger, “Ray interpretation of multipole fields in spherical dielectric cavities,” J. Opt. Soc. Am. A 15, 2879–2891 (1998).
    [CrossRef]
  20. E. A. Hovenac, J. A. Lock, “Assessing the contributions of surface waves and complex rays to far-field Mie scattering by use of the Debye series,” J. Opt. Soc. Am. A 9, 781–795 (1992).
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2000 (1)

1999 (1)

1998 (1)

1997 (1)

1995 (1)

J. A. Roumeliotis, N. B. Kakogiannos, J. D. Kanellopoulos, “Scattering from a sphere of small radius embedded into a dielectric one,” IEEE Trans. Microwave Theory Tech. 43, 155–168 (1995).
[CrossRef]

1994 (3)

1992 (4)

1987 (1)

H. M. Nussenzveig, W. J. Wiscombe, “Diffraction as tunneling,” Phys. Rev. Lett. 59, 1667–1670 (1987).
[CrossRef] [PubMed]

1979 (3)

1976 (2)

H. Chew, M. Kerker, P. J. McNulty, “Raman and fluorescent scattering by molecules embedded in concentric spheres,” J. Opt. Soc. Am. 66, 440–444 (1976).
[CrossRef]

H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976).
[CrossRef]

Alexander, D. R.

Borghese, F.

F. Borghese, P. Denti, R. Saija, “Optical properties of spheres containing several inclusions,” Appl. Opt. 33, 484–493 (1994).
[CrossRef] [PubMed]

F. Borghese, P. Denti, R. Saija, O. I. Sindoni, “Optical properties of spheres containing a spherical eccentric inclusion,” J. Opt. Soc. Am. 9, 1327–1335 (1992).
[CrossRef]

Brenner, R. E.

S. C. Hill, R. E. Brenner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber, R. K. Chang, eds. (World Scientific, Singapore, 1988), pp. 3–61.

Chew, H.

Chrissoulidis, D. P.

Cooke, D. D.

Denti, P.

F. Borghese, P. Denti, R. Saija, “Optical properties of spheres containing several inclusions,” Appl. Opt. 33, 484–493 (1994).
[CrossRef] [PubMed]

F. Borghese, P. Denti, R. Saija, O. I. Sindoni, “Optical properties of spheres containing a spherical eccentric inclusion,” J. Opt. Soc. Am. 9, 1327–1335 (1992).
[CrossRef]

Druger, S. D.

Fuller, K. A.

Hill, S. C.

S. C. Hill, R. E. Brenner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber, R. K. Chang, eds. (World Scientific, Singapore, 1988), pp. 3–61.

Hovenac, E. A.

Ioannidou, M. P.

Kaiser, T.

Kakogiannos, N. B.

J. A. Roumeliotis, N. B. Kakogiannos, J. D. Kanellopoulos, “Scattering from a sphere of small radius embedded into a dielectric one,” IEEE Trans. Microwave Theory Tech. 43, 155–168 (1995).
[CrossRef]

Kanellopoulos, J. D.

J. A. Roumeliotis, N. B. Kakogiannos, J. D. Kanellopoulos, “Scattering from a sphere of small radius embedded into a dielectric one,” IEEE Trans. Microwave Theory Tech. 43, 155–168 (1995).
[CrossRef]

Kerker, M.

Lange, S.

Lock, J. A.

McNulty, P. J.

Nussenzveig, H. M.

H. M. Nussenzveig, W. J. Wiscombe, “Diffraction as tunneling,” Phys. Rev. Lett. 59, 1667–1670 (1987).
[CrossRef] [PubMed]

Roll, G.

Roumeliotis, J. A.

J. A. Roumeliotis, N. B. Kakogiannos, J. D. Kanellopoulos, “Scattering from a sphere of small radius embedded into a dielectric one,” IEEE Trans. Microwave Theory Tech. 43, 155–168 (1995).
[CrossRef]

Saija, R.

F. Borghese, P. Denti, R. Saija, “Optical properties of spheres containing several inclusions,” Appl. Opt. 33, 484–493 (1994).
[CrossRef] [PubMed]

F. Borghese, P. Denti, R. Saija, O. I. Sindoni, “Optical properties of spheres containing a spherical eccentric inclusion,” J. Opt. Soc. Am. 9, 1327–1335 (1992).
[CrossRef]

Schweiger, G.

Sculley, M.

Sindoni, O. I.

F. Borghese, P. Denti, R. Saija, O. I. Sindoni, “Optical properties of spheres containing a spherical eccentric inclusion,” J. Opt. Soc. Am. 9, 1327–1335 (1992).
[CrossRef]

Skaropoulos, N. C.

van de Hulst, H. C.

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

Velesco, N.

Wiscombe, W. J.

H. M. Nussenzveig, W. J. Wiscombe, “Diffraction as tunneling,” Phys. Rev. Lett. 59, 1667–1670 (1987).
[CrossRef] [PubMed]

Zhang, J.

Appl. Opt. (6)

IEEE Trans. Microwave Theory Tech. (1)

J. A. Roumeliotis, N. B. Kakogiannos, J. D. Kanellopoulos, “Scattering from a sphere of small radius embedded into a dielectric one,” IEEE Trans. Microwave Theory Tech. 43, 155–168 (1995).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (4)

Opt. Lett. (1)

Phys. Rev. A (1)

H. Chew, P. J. McNulty, M. Kerker, “Model for Raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976).
[CrossRef]

Phys. Rev. Lett. (1)

H. M. Nussenzveig, W. J. Wiscombe, “Diffraction as tunneling,” Phys. Rev. Lett. 59, 1667–1670 (1987).
[CrossRef] [PubMed]

Other (2)

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

S. C. Hill, R. E. Brenner, “Morphology-dependent resonances,” in Optical Effects Associated with Small Particles, P. W. Barber, R. K. Chang, eds. (World Scientific, Singapore, 1988), pp. 3–61.

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Figures (11)

Fig. 1
Fig. 1

Tunneling of a ray into a particle.

Fig. 2
Fig. 2

Incident, reflected, and transmitted rays at the particle surface.

Fig. 3
Fig. 3

Reflection coefficient r˜22 for a sphere with size parameter 100 and refractive index n2=1.33 for the (a) TM case and (b) TE case, compared with the respective Fresnel coefficient for the plane surface.

Fig. 4
Fig. 4

Phase of the reflection coefficient r˜22 for a sphere with size parameter 100 and refractive index n2=1.33 for the (a) TM case and (b) TE case, compared with the respective phase of the Fresnel coefficient for the plane surface.

Fig. 5
Fig. 5

Reflection coefficients r˜22 for a sphere with refractive index n2=1.33 and varying size parameters for the (a) TM case and (b) TE case, compared with the respective Fresnel coefficients for the plane surface.

Fig. 6
Fig. 6

Transmission coefficients t˜21 for a sphere with refractive index n2=1.33 and varying size parameters for the (a) TM case and (b) TE case, compared with the respective Fresnel coefficients for the plane surface.

Fig. 7
Fig. 7

Transmitted field for TE288 resonance, (a) calculated with geometrical optics by use of tunneled rays and (b) calculated with Mie theory.

Fig. 8
Fig. 8

Geometry for inelastic scattering.

Fig. 9
Fig. 9

Inelastic scattering cross section of a particle at resonance TE288 with one inclusion of radius r=0.5a, dependent on distribution of the active molecules for (a) input resonance and (b) output resonance. Refractive index of particle n2=1.333, refractive index of inclusion ninc=1.5, and perpendicular polarization of the incident wave.

Fig. 10
Fig. 10

Inelastic scattering cross section of a particle at resonance TE288 with one inclusion, dependent on the size of the inclusion for (a) input resonance and (b) output resonance when the active molecules are located in the inclusion. Refractive index of particle n2=1.333, refractive index of inclusion ni=1.5, and perpendicular polarization of the incident wave.

Fig. 11
Fig. 11

Inelastic scattering cross section of a particle at resonance TE288 with one inclusion, dependent on the size of the inclusion for (a) input resonance and (b) output resonance when the active molecules are located in the host particle. Refractive index of particle n2=1.333, refractive index of inclusion ni=1.5, and perpendicular polarization of the incident wave.

Equations (29)

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L1φ=utc1d=utc0n1d.
L2φ=utc2a sin τ=utc0n2a sin τ.
sin τ=dan1n2
d=sin τ an2/n1.
A exp[±iΦ(r)].
Φ(r)=Φr(kr)+Φϑ(ϑ)+Φφ(φ),
Φr(kr)=ϕr(kr)+iψr(kr),
ϕr(kr)=k2r2-Λ2-Λ arccosΛkrkrΛ0k<Λ,
ψr(kr)=0krΛΛ2-k2r2-Λ arccoshΛkrk<Λ,
Φϑ(ϑ)=ϕϑ(ϑ)+iψϑ(ϑ),
ϕϑ(ϑ)=Λ arccosΛ cos ϑΛ2-m2-m arccosm cot ϑΛ2-m2,
ψϑ(ϑ)=Λ arccoshΛ|cos ϑ|Λ2-m2-m arccoshm|cot ϑ|Λ2-m2,
Φφ(φ)=mφ,
rci=l+1/2k2.
d=rcik2k0n1=l+1/2k0n1.
hn(1)(kr)=Ir(kr)exp[+iΦr(kr)],
hn(2)(kr)=Ir(kr)exp[-iΦr(kr)],
Ir(kr)=1kr(k2r2-Λ2)1/2.
αhl(1)(k2a)Ir(k2a)+r22αhl(2)(k2a)Ir(k2a)
=t21hl(1)(k1a)Ir(k1a),
ddrrhl(1)(k2r)Ir(k2r)r=a+r22 ddrrhl(2)(k2r)Ir(k2r)r=a=t21α ddrrhl(1)(k1r)Ir(k1r)r=a,
α=k2k1 TMcase1TEcase.
r22=-hl(1)(k1a)Ir(k1a)ddrrhl(1)(k2r)Ir(k2r)r=a-α2hl(1)(k2a)Ir(k2a)ddrrhl(1)(k1r)Ir(k1r)r=ahl(1)(k1a)Ir(k1a)ddrrhl(2)(k2r)Ir(k2r)r=a-α2k2hl(2)(k2a)Ir(k2a)ddrn1rhl(1)(k1r)Ir(k1r)r=a,
t21=-αhl(1)(k2a)Ir(k2a)ddrrhl(2)(k2r)Ir(k2r)r=a-αhl(2)(k2a)Ir(k2a)ddrrhl(1)(k2r)Ir(k2r)r=ahl(1)(k1a)Ir(k1a)ddrrhl(2)(k2r)Ir(k2r)r=a-α2k2hl(2)(k2a)Ir(k2a)ddrn1rhl(1)(k1r)Ir(k1r)r=a.
τ=arcsinl+1/2k0n2a.
Eref=r22hl(2)(k2a)hl(1)(k2a)Einc.
Etr=t21hl(1)(k1a)hl(1)(k2a)Ir(k2a)Ir(k1a)Einc.
r˜22=r22hl(2)(k2a)hl(1)(k2a),
t˜21=t21hl(1)(k1a)hl(1)(k2a)Ir(k2a)Ir(k1a).

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