Abstract

Near-field photodetection optical microscopy (NPOM) is a scanning probe technique that has been developed to perform nanometer-scale optical intensity mapping and spectroscopy. In NPOM a nanometer-scale photodiode detector absorbs power directly as it is scanned in the near field of an illuminated sample surface. A model of photodetection in the near and intermediate fields is presented. A brief review of far-field absorption is given for comparison. Far-field absorption measurements measure the imaginary part of the polarizability to first order. In contrast, photodetection in the near field measures the real part of the polarizability. Other aspects of near-field photodetection are also examined, including contrast mechanisms and lateral resolution. NPOM measurements performed on isolated 300-nm spheres show good agreement with the theory.

© 2001 Optical Society of America

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References

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  1. E. Betzig, J. K. Trautman, “Near-field optics. Microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
    [CrossRef] [PubMed]
  2. H. Heinzelmann, D. W. Pohl, “Scanning near-field optical microscopy,” Appl. Phys. A: Solids Surf. 59, 89–107 (1994).
    [CrossRef]
  3. F. Zenhausen, Y. Martin, H. K. Wickramasinghe, “Scanning interferometric apertureless microscopy: optical imaging at 10 angstrom resolution,” Science 269, 1083–1085 (1995).
    [CrossRef]
  4. E. J. Sánchez, L. Novotny, X. Sunney Xie, “Near-field fluorescence microscopy based on two-photon excitation with metal tips,” Phys. Rev. Lett. 82, 4014–4017 (1999).
    [CrossRef]
  5. D. R. Busath, R. C. Davis, C. C. Williams, “Near-field photodetection optical microscopy (NPOM): a novel probe for optical characterization on a subwavelength spatial scale,” in Scanning Probe Microscopies II, C. C. Williams, ed., Proc. SPIE1855, 75–85 (1993).
    [CrossRef]
  6. H. U. Danzebrink, U. C. Fischer, “The concept of an optoelectronic probe for near field microscopy,” in Near Field Optics, D. W. Pohl, D. Courjon, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1993), pp. 303–308.
  7. D. R. Busath, “Near-field photodetection probe for near-field optical microscopy,” Ph.D. thesis (University of Utah, Salt Lake City, Utah, 1994).
  8. H.-U. Danzebrink, “Optoelectronic detector probes for scanning near-field optical microscopy,” J. Microsc. (Oxford) 167, 276–280 (1994).
    [CrossRef]
  9. R. C. Davis, C. C. Williams, P. Neuzil, “Micromachined submicrometer photodiode for scanning probe microscopy,” Appl. Phys. Lett. 66, 2309–2311 (1995).
    [CrossRef]
  10. H.-U. Danzebrink, G. Wilkening, O. Ohlsson, “Near-field optoelectronic detector probes based on standard scanning force cantilevers,” Appl. Phys. Lett. 67, 1981–1983 (1995).
    [CrossRef]
  11. S. Akamine, H. Kuwano, H. Yamada, “Scanning near-field optical microscope using an atomic force microscope cantilever with integrated photodiode,” Appl. Phys. Lett. 68, 579–581 (1996).
    [CrossRef]
  12. R. C. Davis, C. C. Williams, P. Neuzil, “Optical intensity mapping on the nanometer scale by near-field photodetection optical microscopy,” Opt. Lett. 21, 447–449 (1996).
    [CrossRef] [PubMed]
  13. R. C. Davis, C. C. Williams, “Nanometer-scale absorption spectroscopy by near-field photodetection optical microscopy,” Appl. Phys. Lett. 69, 1179–1181 (1996).
    [CrossRef]
  14. R. C. Davis, “Nanometer scale optical intensity mapping and absorption spectroscopy by near-field photodetection optical microscopy,” Ph.D. dissertation (University of Utah, Salt Lake City, Utah, 1996).
  15. C. Girard, D. Courjon, “Model for scanning tunneling optical microscopy: a microscopic self consistent approach,” Phys. Rev. B 42, 9340–9349 (1990).
    [CrossRef]
  16. D. Van Labeke, D. Barchiesi, “Theoretical problems in scanning near field optical microscopy,” in Near Field Optics, D. W. Pohl, D. Courjon, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1993), pp. 157–178.
  17. L. Novotny, D. W. Pohl, P. Regli, “Light propagation through nanometer-sized structures: the two-dimensional aperture scanning optical microscope,” J. Opt. Soc. Am. A 11, 1768–1779 (1994).
    [CrossRef]
  18. L. Novotny, “Allowed and forbidden light in near-field optics. II. Interacting dipolar particles,” J. Opt. Soc. Am. A 14, 105–113 (1997).
    [CrossRef]
  19. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  20. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  21. D. R. Pettit, T. W. Peterson, “Coherent detection of scattered light from submicron aerosols,” Aerosol. Sci. Technol. 2, 351–368 (1983).
    [CrossRef]
  22. D. R. Pettit, “Coherent extinction and self-homodyning by small particles,” Appl. Opt. 26, 5136–5142 (1987).
    [CrossRef] [PubMed]
  23. R. G. Newton, “Optical theorem and beyond,” Am. J. Phys. 44, 639–642 (1976).
    [CrossRef]
  24. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1998).

1999 (1)

E. J. Sánchez, L. Novotny, X. Sunney Xie, “Near-field fluorescence microscopy based on two-photon excitation with metal tips,” Phys. Rev. Lett. 82, 4014–4017 (1999).
[CrossRef]

1997 (1)

1996 (3)

S. Akamine, H. Kuwano, H. Yamada, “Scanning near-field optical microscope using an atomic force microscope cantilever with integrated photodiode,” Appl. Phys. Lett. 68, 579–581 (1996).
[CrossRef]

R. C. Davis, C. C. Williams, P. Neuzil, “Optical intensity mapping on the nanometer scale by near-field photodetection optical microscopy,” Opt. Lett. 21, 447–449 (1996).
[CrossRef] [PubMed]

R. C. Davis, C. C. Williams, “Nanometer-scale absorption spectroscopy by near-field photodetection optical microscopy,” Appl. Phys. Lett. 69, 1179–1181 (1996).
[CrossRef]

1995 (3)

R. C. Davis, C. C. Williams, P. Neuzil, “Micromachined submicrometer photodiode for scanning probe microscopy,” Appl. Phys. Lett. 66, 2309–2311 (1995).
[CrossRef]

H.-U. Danzebrink, G. Wilkening, O. Ohlsson, “Near-field optoelectronic detector probes based on standard scanning force cantilevers,” Appl. Phys. Lett. 67, 1981–1983 (1995).
[CrossRef]

F. Zenhausen, Y. Martin, H. K. Wickramasinghe, “Scanning interferometric apertureless microscopy: optical imaging at 10 angstrom resolution,” Science 269, 1083–1085 (1995).
[CrossRef]

1994 (3)

L. Novotny, D. W. Pohl, P. Regli, “Light propagation through nanometer-sized structures: the two-dimensional aperture scanning optical microscope,” J. Opt. Soc. Am. A 11, 1768–1779 (1994).
[CrossRef]

H.-U. Danzebrink, “Optoelectronic detector probes for scanning near-field optical microscopy,” J. Microsc. (Oxford) 167, 276–280 (1994).
[CrossRef]

H. Heinzelmann, D. W. Pohl, “Scanning near-field optical microscopy,” Appl. Phys. A: Solids Surf. 59, 89–107 (1994).
[CrossRef]

1992 (1)

E. Betzig, J. K. Trautman, “Near-field optics. Microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
[CrossRef] [PubMed]

1990 (1)

C. Girard, D. Courjon, “Model for scanning tunneling optical microscopy: a microscopic self consistent approach,” Phys. Rev. B 42, 9340–9349 (1990).
[CrossRef]

1987 (1)

1983 (1)

D. R. Pettit, T. W. Peterson, “Coherent detection of scattered light from submicron aerosols,” Aerosol. Sci. Technol. 2, 351–368 (1983).
[CrossRef]

1976 (1)

R. G. Newton, “Optical theorem and beyond,” Am. J. Phys. 44, 639–642 (1976).
[CrossRef]

Akamine, S.

S. Akamine, H. Kuwano, H. Yamada, “Scanning near-field optical microscope using an atomic force microscope cantilever with integrated photodiode,” Appl. Phys. Lett. 68, 579–581 (1996).
[CrossRef]

Barchiesi, D.

D. Van Labeke, D. Barchiesi, “Theoretical problems in scanning near field optical microscopy,” in Near Field Optics, D. W. Pohl, D. Courjon, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1993), pp. 157–178.

Betzig, E.

E. Betzig, J. K. Trautman, “Near-field optics. Microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
[CrossRef] [PubMed]

Bohren, C. F.

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

Busath, D. R.

D. R. Busath, R. C. Davis, C. C. Williams, “Near-field photodetection optical microscopy (NPOM): a novel probe for optical characterization on a subwavelength spatial scale,” in Scanning Probe Microscopies II, C. C. Williams, ed., Proc. SPIE1855, 75–85 (1993).
[CrossRef]

D. R. Busath, “Near-field photodetection probe for near-field optical microscopy,” Ph.D. thesis (University of Utah, Salt Lake City, Utah, 1994).

Courjon, D.

C. Girard, D. Courjon, “Model for scanning tunneling optical microscopy: a microscopic self consistent approach,” Phys. Rev. B 42, 9340–9349 (1990).
[CrossRef]

Danzebrink, H. U.

H. U. Danzebrink, U. C. Fischer, “The concept of an optoelectronic probe for near field microscopy,” in Near Field Optics, D. W. Pohl, D. Courjon, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1993), pp. 303–308.

Danzebrink, H.-U.

H.-U. Danzebrink, G. Wilkening, O. Ohlsson, “Near-field optoelectronic detector probes based on standard scanning force cantilevers,” Appl. Phys. Lett. 67, 1981–1983 (1995).
[CrossRef]

H.-U. Danzebrink, “Optoelectronic detector probes for scanning near-field optical microscopy,” J. Microsc. (Oxford) 167, 276–280 (1994).
[CrossRef]

Davis, R. C.

R. C. Davis, C. C. Williams, “Nanometer-scale absorption spectroscopy by near-field photodetection optical microscopy,” Appl. Phys. Lett. 69, 1179–1181 (1996).
[CrossRef]

R. C. Davis, C. C. Williams, P. Neuzil, “Optical intensity mapping on the nanometer scale by near-field photodetection optical microscopy,” Opt. Lett. 21, 447–449 (1996).
[CrossRef] [PubMed]

R. C. Davis, C. C. Williams, P. Neuzil, “Micromachined submicrometer photodiode for scanning probe microscopy,” Appl. Phys. Lett. 66, 2309–2311 (1995).
[CrossRef]

D. R. Busath, R. C. Davis, C. C. Williams, “Near-field photodetection optical microscopy (NPOM): a novel probe for optical characterization on a subwavelength spatial scale,” in Scanning Probe Microscopies II, C. C. Williams, ed., Proc. SPIE1855, 75–85 (1993).
[CrossRef]

R. C. Davis, “Nanometer scale optical intensity mapping and absorption spectroscopy by near-field photodetection optical microscopy,” Ph.D. dissertation (University of Utah, Salt Lake City, Utah, 1996).

Fischer, U. C.

H. U. Danzebrink, U. C. Fischer, “The concept of an optoelectronic probe for near field microscopy,” in Near Field Optics, D. W. Pohl, D. Courjon, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1993), pp. 303–308.

Girard, C.

C. Girard, D. Courjon, “Model for scanning tunneling optical microscopy: a microscopic self consistent approach,” Phys. Rev. B 42, 9340–9349 (1990).
[CrossRef]

Heinzelmann, H.

H. Heinzelmann, D. W. Pohl, “Scanning near-field optical microscopy,” Appl. Phys. A: Solids Surf. 59, 89–107 (1994).
[CrossRef]

Huffman, D. R.

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

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1998).

Kuwano, H.

S. Akamine, H. Kuwano, H. Yamada, “Scanning near-field optical microscope using an atomic force microscope cantilever with integrated photodiode,” Appl. Phys. Lett. 68, 579–581 (1996).
[CrossRef]

Martin, Y.

F. Zenhausen, Y. Martin, H. K. Wickramasinghe, “Scanning interferometric apertureless microscopy: optical imaging at 10 angstrom resolution,” Science 269, 1083–1085 (1995).
[CrossRef]

Neuzil, P.

R. C. Davis, C. C. Williams, P. Neuzil, “Optical intensity mapping on the nanometer scale by near-field photodetection optical microscopy,” Opt. Lett. 21, 447–449 (1996).
[CrossRef] [PubMed]

R. C. Davis, C. C. Williams, P. Neuzil, “Micromachined submicrometer photodiode for scanning probe microscopy,” Appl. Phys. Lett. 66, 2309–2311 (1995).
[CrossRef]

Newton, R. G.

R. G. Newton, “Optical theorem and beyond,” Am. J. Phys. 44, 639–642 (1976).
[CrossRef]

Novotny, L.

Ohlsson, O.

H.-U. Danzebrink, G. Wilkening, O. Ohlsson, “Near-field optoelectronic detector probes based on standard scanning force cantilevers,” Appl. Phys. Lett. 67, 1981–1983 (1995).
[CrossRef]

Peterson, T. W.

D. R. Pettit, T. W. Peterson, “Coherent detection of scattered light from submicron aerosols,” Aerosol. Sci. Technol. 2, 351–368 (1983).
[CrossRef]

Pettit, D. R.

D. R. Pettit, “Coherent extinction and self-homodyning by small particles,” Appl. Opt. 26, 5136–5142 (1987).
[CrossRef] [PubMed]

D. R. Pettit, T. W. Peterson, “Coherent detection of scattered light from submicron aerosols,” Aerosol. Sci. Technol. 2, 351–368 (1983).
[CrossRef]

Pohl, D. W.

Regli, P.

Sánchez, E. J.

E. J. Sánchez, L. Novotny, X. Sunney Xie, “Near-field fluorescence microscopy based on two-photon excitation with metal tips,” Phys. Rev. Lett. 82, 4014–4017 (1999).
[CrossRef]

Sunney Xie, X.

E. J. Sánchez, L. Novotny, X. Sunney Xie, “Near-field fluorescence microscopy based on two-photon excitation with metal tips,” Phys. Rev. Lett. 82, 4014–4017 (1999).
[CrossRef]

Trautman, J. K.

E. Betzig, J. K. Trautman, “Near-field optics. Microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
[CrossRef] [PubMed]

van de Hulst, H. C.

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

Van Labeke, D.

D. Van Labeke, D. Barchiesi, “Theoretical problems in scanning near field optical microscopy,” in Near Field Optics, D. W. Pohl, D. Courjon, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1993), pp. 157–178.

Wickramasinghe, H. K.

F. Zenhausen, Y. Martin, H. K. Wickramasinghe, “Scanning interferometric apertureless microscopy: optical imaging at 10 angstrom resolution,” Science 269, 1083–1085 (1995).
[CrossRef]

Wilkening, G.

H.-U. Danzebrink, G. Wilkening, O. Ohlsson, “Near-field optoelectronic detector probes based on standard scanning force cantilevers,” Appl. Phys. Lett. 67, 1981–1983 (1995).
[CrossRef]

Williams, C. C.

R. C. Davis, C. C. Williams, “Nanometer-scale absorption spectroscopy by near-field photodetection optical microscopy,” Appl. Phys. Lett. 69, 1179–1181 (1996).
[CrossRef]

R. C. Davis, C. C. Williams, P. Neuzil, “Optical intensity mapping on the nanometer scale by near-field photodetection optical microscopy,” Opt. Lett. 21, 447–449 (1996).
[CrossRef] [PubMed]

R. C. Davis, C. C. Williams, P. Neuzil, “Micromachined submicrometer photodiode for scanning probe microscopy,” Appl. Phys. Lett. 66, 2309–2311 (1995).
[CrossRef]

D. R. Busath, R. C. Davis, C. C. Williams, “Near-field photodetection optical microscopy (NPOM): a novel probe for optical characterization on a subwavelength spatial scale,” in Scanning Probe Microscopies II, C. C. Williams, ed., Proc. SPIE1855, 75–85 (1993).
[CrossRef]

Yamada, H.

S. Akamine, H. Kuwano, H. Yamada, “Scanning near-field optical microscope using an atomic force microscope cantilever with integrated photodiode,” Appl. Phys. Lett. 68, 579–581 (1996).
[CrossRef]

Zenhausen, F.

F. Zenhausen, Y. Martin, H. K. Wickramasinghe, “Scanning interferometric apertureless microscopy: optical imaging at 10 angstrom resolution,” Science 269, 1083–1085 (1995).
[CrossRef]

Aerosol. Sci. Technol. (1)

D. R. Pettit, T. W. Peterson, “Coherent detection of scattered light from submicron aerosols,” Aerosol. Sci. Technol. 2, 351–368 (1983).
[CrossRef]

Am. J. Phys. (1)

R. G. Newton, “Optical theorem and beyond,” Am. J. Phys. 44, 639–642 (1976).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. A: Solids Surf. (1)

H. Heinzelmann, D. W. Pohl, “Scanning near-field optical microscopy,” Appl. Phys. A: Solids Surf. 59, 89–107 (1994).
[CrossRef]

Appl. Phys. Lett. (4)

R. C. Davis, C. C. Williams, P. Neuzil, “Micromachined submicrometer photodiode for scanning probe microscopy,” Appl. Phys. Lett. 66, 2309–2311 (1995).
[CrossRef]

H.-U. Danzebrink, G. Wilkening, O. Ohlsson, “Near-field optoelectronic detector probes based on standard scanning force cantilevers,” Appl. Phys. Lett. 67, 1981–1983 (1995).
[CrossRef]

S. Akamine, H. Kuwano, H. Yamada, “Scanning near-field optical microscope using an atomic force microscope cantilever with integrated photodiode,” Appl. Phys. Lett. 68, 579–581 (1996).
[CrossRef]

R. C. Davis, C. C. Williams, “Nanometer-scale absorption spectroscopy by near-field photodetection optical microscopy,” Appl. Phys. Lett. 69, 1179–1181 (1996).
[CrossRef]

J. Microsc. (Oxford) (1)

H.-U. Danzebrink, “Optoelectronic detector probes for scanning near-field optical microscopy,” J. Microsc. (Oxford) 167, 276–280 (1994).
[CrossRef]

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

Opt. Lett. (1)

Phys. Rev. B (1)

C. Girard, D. Courjon, “Model for scanning tunneling optical microscopy: a microscopic self consistent approach,” Phys. Rev. B 42, 9340–9349 (1990).
[CrossRef]

Phys. Rev. Lett. (1)

E. J. Sánchez, L. Novotny, X. Sunney Xie, “Near-field fluorescence microscopy based on two-photon excitation with metal tips,” Phys. Rev. Lett. 82, 4014–4017 (1999).
[CrossRef]

Science (2)

E. Betzig, J. K. Trautman, “Near-field optics. Microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
[CrossRef] [PubMed]

F. Zenhausen, Y. Martin, H. K. Wickramasinghe, “Scanning interferometric apertureless microscopy: optical imaging at 10 angstrom resolution,” Science 269, 1083–1085 (1995).
[CrossRef]

Other (8)

D. R. Busath, R. C. Davis, C. C. Williams, “Near-field photodetection optical microscopy (NPOM): a novel probe for optical characterization on a subwavelength spatial scale,” in Scanning Probe Microscopies II, C. C. Williams, ed., Proc. SPIE1855, 75–85 (1993).
[CrossRef]

H. U. Danzebrink, U. C. Fischer, “The concept of an optoelectronic probe for near field microscopy,” in Near Field Optics, D. W. Pohl, D. Courjon, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1993), pp. 303–308.

D. R. Busath, “Near-field photodetection probe for near-field optical microscopy,” Ph.D. thesis (University of Utah, Salt Lake City, Utah, 1994).

D. Van Labeke, D. Barchiesi, “Theoretical problems in scanning near field optical microscopy,” in Near Field Optics, D. W. Pohl, D. Courjon, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1993), pp. 157–178.

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

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

R. C. Davis, “Nanometer scale optical intensity mapping and absorption spectroscopy by near-field photodetection optical microscopy,” Ph.D. dissertation (University of Utah, Salt Lake City, Utah, 1996).

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1998).

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Figures (10)

Fig. 1
Fig. 1

Near-field photodetection optical microscopy (NPOM). (a) An absorbing detector is brought into the near field of an illuminated sample surface. (b) The probe is rastor scanned and the photocurrent is recorded to generate a two-dimensional image of the optical intensity distribution.

Fig. 2
Fig. 2

Incident and scattered waves in the vicinity of an illuminated particle. Arrows indicate the direction of propagation of the waves.

Fig. 3
Fig. 3

Absorption and scattering by a particle.

Fig. 4
Fig. 4

Detector placed in the forward-scattering direction.

Fig. 5
Fig. 5

Detector and sample particle illuminated in the near field. Both are modeled as dipoles. Here the detector is directly above the sample particle.

Fig. 6
Fig. 6

Specific example of a polystyrene particle and a silicon probe.

Fig. 7
Fig. 7

The power absorbed by the detector for arbitrary r and θ is calculated.

Fig. 8
Fig. 8

The detector is scanned in the x direction above the sample particle. (a) The particle is illuminated below resonance. (b) The particle is illuminated above resonance.

Fig. 9
Fig. 9

Two particles, one at x=100 nm and one at x=-100 nm, illuminated below resonance. The detector is scanned in the x direction above two sample particles separated by 200 nm.

Fig. 10
Fig. 10

Probe in the intermediate field. (a) The probe is scanned at a constant height above the particle. (b) Calculated intensity profile for the scan path shown in (a). In this calculation the probe is in the intermediate field of the sample particle. (c) Experimentally measured photocurrent in good agreement with the calculated linescan.

Equations (58)

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

Ei=E0exp(ik  r-iωt),
Hi=0/μ0k×Ei.
Wa=-σS  dA.
S=Si+Sext+Ss.
Si=12Re(Ei×Hi*),
Sext=12Re[(Ei×Hs*)+(Es×Hi*)],
Ss=12Re(Es×Hs*).
Wa=-σS  dA,
Ws=σSs  dA,
Wext=-σSext  dA
Wext=Wa+Ws.
Wd=DS  dA=DSi  dA+DSext  dA+DSs  dA.
Wi=DSi  dA=SiD.
DSext  dAσSext  dA=-Wext.
WdWi-Wext.
Wext=12 0cE02k Im(α)+k4|α|26π,
Wabs=12 0cE02k Im(α),
Wsca=12 0cE02k4|α|26π,
qm E0exp(-iωt)-ω02x-γx˙=x¨.
x(ω)=q/mω02-ω2-iωγ E0exp(-iωt),
α(ω)=p0E0exp(-iωt)=q2/0mω02-ω2-iωγ=α0ω02ω02-ω2-iωγ,
α0=q20mω02.
Wext=12 0cE02k α0ω02ωγ(ω02-ω2)2+ω2γ2+k46πα02ω04(ω02-ω2)2+ω2γ2.
E(ω, r)=14π0exp(ik  r-iωt)1r3 [3rˆ(rˆ  p)-p]-ikr2 [3rˆ(rˆ  p)-p]-k2r [rˆ×(rˆ×p)],
H(ω, r)=-iω4πexp(ik  r-iωt)1r2-ikrp×rˆ.
UE(r)UH(r)=0|E|2μ0|H|21k2r2.
F=qE,
x=pq=0αEq,
x˙=-iω0αEq.
W=12Re(Fx˙*)=0c|E|22 k Im(α),
E1=E0xˆ+14π01r3 [3r^12(r^12  p2)-p2],
E2=E0xˆ+14π01r3 [3r^21(r^21  p1)-p1],
p1=α0E1,
p2=α0E2,
p1=p1xˆ,p2=p2xˆ,r^12=-zˆ,r^21=zˆ,
E1xˆ=E0-α2E24πr3xˆ,E2xˆ=E0-α1E14πr3xˆ,
E1+α24πr3 E2=E0,E2+α14πr3 E1=E0.
E1=E01-α24πr31-α1α216π2r6,E2=E01-α14πr31-α1α216π2r6.
E2=E01-αpolystyrene4πr31-αpolystyreneαsilicon16π2r6.
α=4π r-1r+2 α3,
αsilicon4π=0.8a3,αpolystyrene4π=0.3a3.
E2E0[1-0.3(a/r)3]=0.995E0.
E2E0xˆ+14π01r3 {3r^21[r^21  (α10E0xˆ)]-(α10E0xˆ)}E0xˆ+α1E04π1r3 [3r^21(r^21  xˆ)-xˆ],
|E2|2 =E02+Re(α1)E022π1r3 [3(rˆ  xˆ)2-1]+termsoforder|α|2r6.
W2=120cE22k Im(α2).
W212 0ck Im(α2)E02×1+Re(α1)2π1r3 (3 sin2 θ cos2 φ-1).
|E|2 =|Ei+Es|2 =|Ei|2 +2 Re(Ei  Es*)+|Es|2,
Re[α(w)]=α0ω02(ω02-ω2)(ω02-ω2)2+ω2γ2.
Re(α)α0.
Re(α)=0.
|E|2 =E021+Oα2r6.
Re(α)-α0ω02ω2.
Ei=E0exp[ik(z-ωt)]xˆ,
Es=14π0exp[ik(r-wt)]1r2 [3rˆ(rˆ  p)-p]-ikr2 [3rˆ(rˆ  p)-p]-k2r [rˆ×(rˆ×p)],
p=0ααE0xˆ.
|Etot|2 =|Ei|2+2 Re(Ei  Es*)+|Es|2.
2 Re(Ei  Es*)=Re|E0|2α*2πexp[ik(z-r)]1r3 [3(rˆ  xˆ)2-1]+ikr2 [3(rˆ  xˆ)2-1]-k2r [(rˆ  xˆ)2-1]=|E0|22πRe(α)cos[k(z-r)]-1r31-3x2r2+k2r1-x2r2+sin[k(z-r)]×kr21-3x2r2-|E0|22πIm(α)cos[k(z-r)]
×kr21-3x2r2-sin[k(z-r)]×-1r31-3x2r2+k2r1-x2r2.

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