Abstract

A model is proposed that describes the essential optical process in the recently observed resonant light scattering from a microsphere resonator that is strongly coupled to the substrate. The experimentally observed field patterns across the resonance can be reproduced quite well by a numerical calculation taking into account only a few vector spherical waves that are converted from nonpropagating to propagating waves at the substrate surface. Explicit consideration of the multiple-reflection effect is not necessary to reproduce the experimental results. Comparison of the experiment and the calculation suggests the splitting of degenerate resonance modes that have different azimuthal mode numbers within a single broad resonance line. These results are discussed on the basis of the strongly coupled nature of the system.

© 2000 Optical Society of America

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References

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  27. Mode numbers were deduced from the comparison between the observed resonance peaks and the calculated distributions of s=1 and s=2 resonant modes for an isolated sphere (Fig. 3 in Ref. 24). Because the measurement was carried out over more than three spectral ranges, the mode number can be obtained uniquely, along with geometrical parameters (radius and refractive index of the sphere), by adjusting them so that the particular spacing patterns between the adjacent modes are reproduced.
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  32. We are interested in the near field of the WGM’s because the distance between the sphere and the substrate is smaller than the wavelength (actually zero in this case). The near field is strongly enhanced in the resonant condition, just as the field inside the sphere.
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  38. T. P. Burghardt, N. L. Thompson, “Effect of planar dielectric interfaces on fluorescence emission and detection,” Biophys. J. 46, 729–737 (1984).
    [CrossRef] [PubMed]
  39. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
    [CrossRef]
  40. Finer details that do not match with the observed pattern (Fig. 3) are probably due to our complete negligence of the nonresonant contribution, which tends to smear out the interference.

1999 (4)

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, M. Kuwata-Gonokami, “Tight-binding photonic molecule modes of resonant bispheres,” Phys. Rev. Lett. 82, 4623–4626 (1999).
[CrossRef]

M. L. Gorodetsky, V. S. Ilchenko, “Optical microsphere resonator: optimal coupling to high-Q whispering-gallery modes,” J. Opt. Soc. Am. B 16, 147–154 (1999).
[CrossRef]

H. Ishikawa, H. Tamaru, K. Miyano, “Observation of a modulation effect caused by a microsphere resonator strongly coupled to a dielectric substrate,” Opt. Lett. 24, 643–645 (1999).
[CrossRef]

A. Shinya, M. Fukui, “Finite-difference time-domain analysis of the interaction of Gaussian evanescent light with a single dielectric sphere or ordered dielectric spheres,” Opt. Rev. 6, 215–223 (1999).
[CrossRef]

1998 (5)

1997 (6)

1996 (1)

1995 (4)

1994 (1)

M. L. Gorodetsky, V. S. Ilchenko, “High-Q optical whispering-gallery microresonators: precession approach for spherical mode analysis and emission patterns with prism couplers,” Opt. Commun. 113, 133–143 (1994).
[CrossRef]

1993 (1)

Y. Yamamoto, R. E. Slusher, “Optical process in microcavities,” Phys. Today 46, 66–74 (1993).
[CrossRef]

1992 (2)

S. Schiller, I. I. Yu, M. M. Fejer, R. L. Byer, “Fused silica monolithic total-internal-reflection resonator,” Opt. Lett. 17, 378–380 (1992).
[CrossRef] [PubMed]

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

1991 (1)

1990 (1)

D. C. Prieve, N. A. Frej, “Total internal reflection microscopy: a qualitative tool for the measurement of colloidal forces,” Langmuir 6, 396–403 (1990).
[CrossRef]

1989 (1)

J. P. Barton, D. R. Alexander, S. A. Shaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

1987 (1)

T. Takemori, M. Inoue, K. Ohtaka, “Optical response of a sphere coupled to a metal substrate,” J. Phys. Soc. Jpn. 56, 1587–1602 (1987).
[CrossRef]

1986 (1)

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica A 137, 209–242 (1986).
[CrossRef]

1984 (2)

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

T. P. Burghardt, N. L. Thompson, “Effect of planar dielectric interfaces on fluorescence emission and detection,” Biophys. J. 46, 729–737 (1984).
[CrossRef] [PubMed]

1979 (1)

1959 (1)

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Alexander, D. R.

J. P. Barton, D. R. Alexander, S. A. Shaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

Arnold, S.

Ataka, T.

Barton, J. P.

H.-B. Lin, J. D. Eversole, A. J. Campillo, J. P. Barton, “Excitation localization principle for spherical microcavities,” Opt. Lett. 23, 1921–1923 (1998).
[CrossRef]

J. P. Barton, D. R. Alexander, S. A. Shaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

Birks, T. A.

Bobbert, P. A.

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica A 137, 209–242 (1986).
[CrossRef]

Borghese, F.

Burghardt, T. P.

T. P. Burghardt, N. L. Thompson, “Effect of planar dielectric interfaces on fluorescence emission and detection,” Biophys. J. 46, 729–737 (1984).
[CrossRef] [PubMed]

Byer, R. L.

Campillo, A. J.

Cheung, G.

Chew, H.

Chiba, N.

Chu, S. T.

B. E. Little, S. T. Chu, H. A. Haus, “Track changing by use of the phase response of microspheres and resonators,” Opt. Lett. 23, 894–896 (1998).
[CrossRef]

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

Connolly, J.

Denti, P.

Dubreuil, N.

Duvreuil, N.

Edamatsu, K.

Eversole, J. D.

Fan, S.

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, H. A. Haus, “Channel drop filters in photonic crystals,” Opt. Express. 3, 4–11 (1998).
[CrossRef] [PubMed]

Fejer, M. M.

Ford, G. W.

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

Foresi, J.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

Frej, N. A.

D. C. Prieve, N. A. Frej, “Total internal reflection microscopy: a qualitative tool for the measurement of colloidal forces,” Langmuir 6, 396–403 (1990).
[CrossRef]

Fucile, E.

Fujimura, T.

Fukui, M.

A. Shinya, M. Fukui, “Finite-difference time-domain analysis of the interaction of Gaussian evanescent light with a single dielectric sphere or ordered dielectric spheres,” Opt. Rev. 6, 215–223 (1999).
[CrossRef]

Gorodetsky, M. L.

M. L. Gorodetsky, V. S. Ilchenko, “Optical microsphere resonator: optimal coupling to high-Q whispering-gallery modes,” J. Opt. Soc. Am. B 16, 147–154 (1999).
[CrossRef]

M. L. Gorodetsky, V. S. Ilchenko, “High-Q optical whispering-gallery microresonators: precession approach for spherical mode analysis and emission patterns with prism couplers,” Opt. Commun. 113, 133–143 (1994).
[CrossRef]

Goto, K.

Griffel, G.

Guimarães, L. G.

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

Hagness, S. C.

Hare, J.

Haroche, S.

Haus, H. A.

B. E. Little, S. T. Chu, H. A. Haus, “Track changing by use of the phase response of microspheres and resonators,” Opt. Lett. 23, 894–896 (1998).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, H. A. Haus, “Channel drop filters in photonic crystals,” Opt. Express. 3, 4–11 (1998).
[CrossRef] [PubMed]

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

Hecht, E.

E. Hecht, Optics (Addison-Wesley, Reading, Mass., 1998).

Ho, S. T.

Ilchenko, V. S.

M. L. Gorodetsky, V. S. Ilchenko, “Optical microsphere resonator: optimal coupling to high-Q whispering-gallery modes,” J. Opt. Soc. Am. B 16, 147–154 (1999).
[CrossRef]

M. L. Gorodetsky, V. S. Ilchenko, “High-Q optical whispering-gallery microresonators: precession approach for spherical mode analysis and emission patterns with prism couplers,” Opt. Commun. 113, 133–143 (1994).
[CrossRef]

Imada, A.

Inoue, M.

T. Takemori, M. Inoue, K. Ohtaka, “Optical response of a sphere coupled to a metal substrate,” J. Phys. Soc. Jpn. 56, 1587–1602 (1987).
[CrossRef]

Ishikawa, H.

Itho, T.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

Jacques, F.

Jimba, Y.

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, M. Kuwata-Gonokami, “Tight-binding photonic molecule modes of resonant bispheres,” Phys. Rev. Lett. 82, 4623–4626 (1999).
[CrossRef]

Joannopoulos, J. D.

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, H. A. Haus, “Channel drop filters in photonic crystals,” Opt. Express. 3, 4–11 (1998).
[CrossRef] [PubMed]

Kaiser, T.

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]

C. Liu, T. Kaiser, S. Lange, G. Schweiger, “Structural resonances in a dielectric sphere illuminated by an evanescent wave,” Opt. Commun. 117, 521–531 (1995).
[CrossRef]

Kerker, M.

Knight, J. C.

Koda, T.

Kuwata-Gonokami, M.

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, M. Kuwata-Gonokami, “Tight-binding photonic molecule modes of resonant bispheres,” Phys. Rev. Lett. 82, 4623–4626 (1999).
[CrossRef]

Laine, J.-P.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

Lange, S.

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]

C. Liu, T. Kaiser, S. Lange, G. Schweiger, “Structural resonances in a dielectric sphere illuminated by an evanescent wave,” Opt. Commun. 117, 521–531 (1995).
[CrossRef]

Lefèvre, V.

Lefèvre-Seguin, V.

Leventhal, D. K.

Lin, H.-B.

Little, B. E.

B. E. Little, S. T. Chu, H. A. Haus, “Track changing by use of the phase response of microspheres and resonators,” Opt. Lett. 23, 894–896 (1998).
[CrossRef]

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

Liu, C.

C. Liu, T. Kaiser, S. Lange, G. Schweiger, “Structural resonances in a dielectric sphere illuminated by an evanescent wave,” Opt. Commun. 117, 521–531 (1995).
[CrossRef]

Lock, J. A.

Miyano, K.

Miyazaki, H.

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, M. Kuwata-Gonokami, “Tight-binding photonic molecule modes of resonant bispheres,” Phys. Rev. Lett. 82, 4623–4626 (1999).
[CrossRef]

Morris, N.

Mukaiyama, T.

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, M. Kuwata-Gonokami, “Tight-binding photonic molecule modes of resonant bispheres,” Phys. Rev. Lett. 82, 4623–4626 (1999).
[CrossRef]

Muramatsu, H.

Nussenzveig, H. M.

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

Ohtaka, K.

T. Takemori, M. Inoue, K. Ohtaka, “Optical response of a sphere coupled to a metal substrate,” J. Phys. Soc. Jpn. 56, 1587–1602 (1987).
[CrossRef]

Prieve, D. C.

D. C. Prieve, N. A. Frej, “Total internal reflection microscopy: a qualitative tool for the measurement of colloidal forces,” Langmuir 6, 396–403 (1990).
[CrossRef]

Rafizadeh, D.

Raimond, J. M.

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Roll, G.

Saija, R.

Sandoghdar, V.

Schiller, S.

Schweiger, G.

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]

C. Liu, T. Kaiser, S. Lange, G. Schweiger, “Structural resonances in a dielectric sphere illuminated by an evanescent wave,” Opt. Commun. 117, 521–531 (1995).
[CrossRef]

Serpengüzel, A.

Shaub, S. A.

J. P. Barton, D. R. Alexander, S. A. Shaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

Shimada, R.

Shinya, A.

A. Shinya, M. Fukui, “Finite-difference time-domain analysis of the interaction of Gaussian evanescent light with a single dielectric sphere or ordered dielectric spheres,” Opt. Rev. 6, 215–223 (1999).
[CrossRef]

Sindoni, O. I.

Slusher, R. E.

Y. Yamamoto, R. E. Slusher, “Optical process in microcavities,” Phys. Today 46, 66–74 (1993).
[CrossRef]

Stair, K. A.

Taflove, A.

Takeda, K.

T. Mukaiyama, K. Takeda, H. Miyazaki, Y. Jimba, M. Kuwata-Gonokami, “Tight-binding photonic molecule modes of resonant bispheres,” Phys. Rev. Lett. 82, 4623–4626 (1999).
[CrossRef]

Takemori, T.

T. Takemori, M. Inoue, K. Ohtaka, “Optical response of a sphere coupled to a metal substrate,” J. Phys. Soc. Jpn. 56, 1587–1602 (1987).
[CrossRef]

Tamaru, H.

Taskent, D.

Thompson, N. L.

T. P. Burghardt, N. L. Thompson, “Effect of planar dielectric interfaces on fluorescence emission and detection,” Biophys. J. 46, 729–737 (1984).
[CrossRef] [PubMed]

Tiberio, R. C.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Villeneuve, P. R.

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, H. A. Haus, “Channel drop filters in photonic crystals,” Opt. Express. 3, 4–11 (1998).
[CrossRef] [PubMed]

Vlieger, J.

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica A 137, 209–242 (1986).
[CrossRef]

Wang, D.-S.

Weber, W. H.

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

Wolf, E.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Yamamoto, Y.

Y. Yamamoto, R. E. Slusher, “Optical process in microcavities,” Phys. Today 46, 66–74 (1993).
[CrossRef]

Yu, I. I.

Zhang, J. P.

Zvyagin, A. V.

Appl. Opt. (1)

Biophys. J. (1)

T. P. Burghardt, N. L. Thompson, “Effect of planar dielectric interfaces on fluorescence emission and detection,” Biophys. J. 46, 729–737 (1984).
[CrossRef] [PubMed]

J. Appl. Phys. (1)

J. P. Barton, D. R. Alexander, S. A. Shaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: focal point positioning effects at resonance,” J. Appl. Phys. 65, 2900–2906 (1989).
[CrossRef]

J. Lightwave Technol. (1)

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[CrossRef]

J. Opt. Soc. Am. A (3)

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

J. Phys. Soc. Jpn. (1)

T. Takemori, M. Inoue, K. Ohtaka, “Optical response of a sphere coupled to a metal substrate,” J. Phys. Soc. Jpn. 56, 1587–1602 (1987).
[CrossRef]

Langmuir (1)

D. C. Prieve, N. A. Frej, “Total internal reflection microscopy: a qualitative tool for the measurement of colloidal forces,” Langmuir 6, 396–403 (1990).
[CrossRef]

Opt. Commun. (3)

C. Liu, T. Kaiser, S. Lange, G. Schweiger, “Structural resonances in a dielectric sphere illuminated by an evanescent wave,” Opt. Commun. 117, 521–531 (1995).
[CrossRef]

M. L. Gorodetsky, V. S. Ilchenko, “High-Q optical whispering-gallery microresonators: precession approach for spherical mode analysis and emission patterns with prism couplers,” Opt. Commun. 113, 133–143 (1994).
[CrossRef]

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

Opt. Express. (1)

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, H. A. Haus, “Channel drop filters in photonic crystals,” Opt. Express. 3, 4–11 (1998).
[CrossRef] [PubMed]

Opt. Lett. (12)

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Other (7)

Finer details that do not match with the observed pattern (Fig. 3) are probably due to our complete negligence of the nonresonant contribution, which tends to smear out the interference.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

We are interested in the near field of the WGM’s because the distance between the sphere and the substrate is smaller than the wavelength (actually zero in this case). The near field is strongly enhanced in the resonant condition, just as the field inside the sphere.

E. Hecht, Optics (Addison-Wesley, Reading, Mass., 1998).

Mode numbers were deduced from the comparison between the observed resonance peaks and the calculated distributions of s=1 and s=2 resonant modes for an isolated sphere (Fig. 3 in Ref. 24). Because the measurement was carried out over more than three spectral ranges, the mode number can be obtained uniquely, along with geometrical parameters (radius and refractive index of the sphere), by adjusting them so that the particular spacing patterns between the adjacent modes are reproduced.

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Figures (11)

Fig. 1
Fig. 1

Schematic diagram of the experimental setup (total reflection microscope). R, resonator (glass sphere); S, plane glass substrate; OIL, immersion oil; M, mirrors; P1, polarizer; P2, analyzer; L1, lens (f=60 cm); L2, relay lenses (×5); L3, objective (×5); O, objective (×40, N.A.=1.3); Σo, object plane (focal plane of the objective); Σt, transform plane; Σi, image plane.

Fig. 2
Fig. 2

Images taken by total reflection microscope: (a) cross section of totally reflected Gaussian beam (no sphere) and (b)–(d) modulation of totally reflected beam caused by the sphere. In each part the sphere is laterally displaced against the incident beam, which is held fixed.

Fig. 3
Fig. 3

Images taken at different wavelengths: (a) 563.25 nm, (b) 563.81 nm, (c) 577.92 nm, (d) 578.50 nm, (e) 593.55 nm, (f) 594.13 nm. Each pair is located on both sides of the first-order resonance peak (cf. Ref. 24).

Fig. 4
Fig. 4

Intensity and phase distribution of a VSH (TE polarization, l=m=34). (a) Log-scale plot of {Re[Eϕ(r)]}2=|Eϕ(r)|2 cos[arg Eϕ(r)] in the equatorial plane (y=0). The size of the plotted region is 16 µm×16 µm. Parameters are arbitrarily chosen as a=2.5 µm, nsph=1.5, and λ=598.7205 nm(TE34,1 resonance). Solid and dotted lines indicate the location of the substrate surface and the plane of observation, respectively. (b) Log-scale plots of {Re[Eϕ(r)]}2 (lower curve) and |Eϕ(r)|2 (upper curve) along the z axis. The origin is at the center of the sphere. The arrow indicates the position of the observation plane used in the calculation. The straight line near the vertical axis (exponential decay of the evanescent component) is only a guide for the eye.

Fig. 5
Fig. 5

Schematic diagram of a microscope.

Fig. 6
Fig. 6

(a) Calculated intensity of a VSH at the substrate surface (TM polarization, l=m=34, in linear scale). (b) Calculated spatial variation of {Re(Ez)}2 of the same VSH at the substrate surface (magnified by four times). (c) Location and size of the sphere in the calculation. (d) Location and size of the sphere in the experiment (corresponding to Fig. 3). (e) Calculated intensity of the VSH at the focal plane of the microscope. (f)–(h) calculated spatial variations of the absolute square of each Cartesian component of the electric field of the VSH. The brightness is normalized within each figure, and the brightest spot in each figure respectively corresponds to (f) 44 (x component), (g) 5 (y component), and (h) 100 (z component) in arbitrary units.

Fig. 7
Fig. 7

Calculated VSH images to be observed with the microscope: (a) l=34,m=34, (b) l=34,m=32, (c) l=34,m=30.

Fig. 8
Fig. 8

(a)–(c) Calculated microscope image of superposition of two VSH’s and a totally reflected Gaussian beam. The wavelength is set at TM34,1 resonance (λ=578.3888 nm). The relative phases of the m=34 and m=32 modes against the Gaussian beam are respectively set as (a) (-π/6,-7π/12) and (b) (0,-π/12). (c), (d) Better calculated results: (c) At shorter wavelength, the amplitude of the m=32 mode is reduced by 33%; (d) at longer wavelength, the amplitude of the m=32 mode is enhanced by 33%. (e) Calculated image of superposition of the same two VSH’s and a totally reflected plane wave.

Fig. 9
Fig. 9

Illustration of the splitting of WGM’s having different azimuthal mode numbers.

Fig. 10
Fig. 10

(a) Experimentally observed TIRM images at the TM33,1 resonance wavelength for p-polarized excitation and s-polarized detection. (b), (c) Corresponding calculated images with (b) m=34 and m=32 and with (c) an additional m=30 mode with a phase of -π/3. The amplitudes of each mode are the same and the phase of the m=34,m=32, and m=30 modes are respectively set as 0, -π/3, and -π/3.

Fig. 11
Fig. 11

(a), (b) Experimentally observed TIRM images for a 5-µm sphere between the first-order resonances: (a) in the nonresonant excitation condition (λ=566.96 nm) and (b) at the center of the broad second-order resonance (λ=572.17 nm). (c) Experimentally observed TIRM images for a 3-µm sphere in the nonresonant excitation condition. (d), (e) Calculated images with the use of a tightly focused Gaussian beam as the scattered wave: (d) λ=566.96 nm, phase=23π/18; (e) λ=572.17 nm, phase=29π/18. (f), (g) Calculated images with the use of a VSH (second-order TM, l=m=30) as a scattered wave: (f) λ=566.96 nm, phase=3π/2; (g) λ=572.17 nm, phase=0.

Equations (57)

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(nsub sin α)/nair=x/x0,
Δα=nairnsub1x0 cos αΔx,
E(r)=l=1m=-ll(r>a)βTEp,s(l, m)hl(1)(k0r)Xl,m(θ, ϕ)+ik0βTMp,s(l, m)×[hl(1)(k0r)Xl,m(θ, ϕ)],
βTMp,s(l, m)=Bl exp(-γd)αTMp,s(l, m)(TMpolarization),
βTEp,s(l, m)=Cl exp(-γd)αTEp,s(l, m)(TEpolarization),
E(r)=1[l(l+1)]1/2exp(-γd)Blm=-llαTMp(l, m)×1krhl(1)(k0r)l(l+1)θimsin θrθϕYlm(θ, ϕ)+dd(k0r)hl(1)(k0r)0θimsin θrθϕYlm(θ, ϕ),
E(r, z)=--E(k, z)exp(ik·r)dk,
E(r, z)=--{E(+)(k)exp(ik·r)exp[ikz(z-zs)]+E(-)(k)exp(ik·r)×exp[-ikz(z-zs)]}dk,
=--{E(+)(k)exp[ik(+)·r]+E(-)(k)exp[ik(-)·r]}dk,
kz=kz(k)=(k02-|k|2)1/2(|k|k0)i(|k|2-k02)1/2(|k|>k0),
k(±)=k(±)(k)=k±kz(k)uz,
k(+)(k)·E(+)(k)=k(-)(k)·E(-)(k)=0.
E(r, z)=--E(+)(k)×exp(ik·r)exp[ikz(z-zs)dk].
E(+)(k, z)=1(2π)2--E(r, z)×exp(-ik·r)×exp[-ikz(z-zs)]dr.
E(r, z)=--E(+)(k)×exp(ik·r)exp[ikz(z-zs)]dk,
kz(k)=[(nsub k0)2-|k|2]1/2(|k|nsub k0)i[|k|2-(nsub k0)2]1/2(|k|>nsub k0)
E(+)(k)=T(k)·E(+)(k),
T(k)=tpupsubupair+tsussubusair,
tp=tp(k)=2nsub kz(k)nsub2 kz(k)+kz(k),
ts=ts(k)=2kz(k)kz(k)+kz(k).
E(r, z)=--T(k)·E(+)(k)×exp(ik·r)exp[ikz(z-zs)]dk.
E(r, zt)=--E(+)(k)exp(ik·r)×exp[ikz(zt-zs)]dk,
E(r, za)=--E(+)(k)exp(ik·r)×exp[ikz(za-zs)]dk,
E(r, zi)=--E(+)(k)exp(ik·r)×exp[ikz(zi-zs)]dk.
E(+)(k)=F(k)·Lo(k)·E(+)(k),
Lo(k)=uptupsub+ustussub,
F(k)=1(|k|N.A×k0)0(|k|>N.A.×k0),
E(+)(k)=Le(k)·E(+)(k),
Le(k)=upimgupt+usimgust,
E(+)(k)=P·E(+)(k),
P=uxux(ppolarization)uyuy(spolarization),
E(+)(k)=Le(k)·P·F(k)·Lo(k)·E(+)(k).
E(+)(k)=Le(k)·P·F(k)·Lo(k)×exp[ikz(zo-zs)]·T(k)·E(+)(k)
=F(k)Ux(k)·E(+)(k),
Ux(k)=exp[ikz(zo-zs)]ξp,s{tξ(k)[ux·uξt)(k)]2×uξimg(k)uξair(k)},
I(+)(r)=|k|<(N.A.×k0)×Ux(k)·E(+)(k)exp(ik·r)dk2.
Δλ-λFSR arg(rc)-arg(r0)2π,
ukair=ukair(k)=(kx, ky, kz)/k0,
upair=upair(k)=(kxkz, kykz,-|k|2)/(k0|k|),
usair=usair(k)=(-ky, kx, 0)/|k|,
uksub=uksub(k)=(kx, ky, kz)/(nsubk0),
upsub=upsub(k)=(kxkz, kykz,-|k|2)/(nsubk0|k|),
ussub=ussub(k)=(-ky, kx, 0)/|k|,
ukt=ukt(k)=(0, 0, 1),
upt=upt(k)=(kx, ky, 0)/|k|,
ust=ust(k)=(-ky, kx, 0)/|k|,
ukimg=ukimg(k)=(-kx,-ky, kz)/k0,
upimg=upimg(k)=(kxkz, kykz, |k|2)/(k0|k),
usimg=usimg(k)=(-ky, kx, 0)/|k|,
|ukair|=|upair|=|usair|=1,
|uksub|=|upsub|=|ussub|=1,
|ukt|=|upt|=|ust|=1,
|ukimg|=|upimg|=|usimg|=1,
ukair·upair=ukair·usair=upair·usair=0,
uksub·upsub=uksub·ussub=upsub·ussub=0,
ukt·upt=ukt·ust=upt·ust=0,
ukimg·upimg=ukimg·usimg=upimg·usimg=0.

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