Abstract

We consider a simple analytical model for the electric near field of a semi-infinite conical probe and apply it to study the incident angle dependence of the field for the case of side illumination by both the plane wave and the Gaussian beam. The electric near field is shown to peak when approaching the grazing incidence. In some cases, a peak can also occur at an incident angle somewhat below 90°. The results obtained are in qualitative agreement with those for a thin semi-infinite wire and previously published results for the finite-size conical probes.

© 2007 Optical Society of America

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    [Crossref]
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  18. Our numerical simulations show that this statement is also valid for ν>0.
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    [Crossref]
  29. O. J. F. Martin, C. Girard, and A. Dereux, “Generalized field propagator for Electromagnetic Scattering and Light Confinement,” Phys. Rev. Lett. 74, 526–529 (1995).
    [Crossref] [PubMed]
  30. N. I. Petrov, “Focusing of beams into subwavelength area in an inhomogeneous medium,” Opt. Express 9, 658–673 (2001).
    [Crossref] [PubMed]
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    [Crossref]
  32. C. Durkan and I. V. Shvets, “Polarization effects in Reflection-Mode Scanning Near-Field Optical Microscopy,” J. Appl. Phys. 83, 1837–1843 (1998).
    [Crossref]
  33. O. J. F. Martin and C. Girard, “Controlling and Tuning Strong Optical Field Gradients at a Local Probe Microscope Tip Apex,” Appl. Phys. Lett. 70, 705 (1997).
    [Crossref]
  34. M. S. Anderson, “Locally enhanced Raman Spectroscopy with an Atomic Force Microscope,” Appl. Phys. Lett. 76, 3130–3132 (2000).
    [Crossref]
  35. M. S. Anderson and W. T. Pike, “A Raman-Atomic Force Microscope for Apertureless-Near-Field Spectroscopy and Optical Trapping,” Rev. Sci. Instrum. 73, 1198–1203 (2002).
    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
  38. K. C. Neuman and S. M. Block, “Optical Trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004).
    [Crossref]
  39. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal Field Modes probed by Single Molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
    [Crossref] [PubMed]
  40. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized beam,” Phys. Rev. Lett. 91, 233901 (2003).
    [Crossref] [PubMed]
  41. D. Mehtani, N. Lee, R. D. Hartschuh, A. Kisliuk, M. D. Foster, A. P. Sokolov, and J. F. Maguire, “Nano-Raman Spectroscopy with side-illumination optics,” J. Raman Spectrosc. 36, 1068–1075 (2005).
    [Crossref]
  42. C. C. Neacsu, J. Dreyer, N. Behr, and M. B. Raschke, “Scanning-probe Raman Spectroscopy with single-molecule sensitivity,” Phys. Rev. B 73, 193406 (2006).
    [Crossref]
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    [Crossref]

2007 (2)

R. Ossikovski, Q. Nguen, and G. Picardi, “Simple model for the polarization effects in tip-enhanced Raman Spectroscopy,” Phys. Rev. B 75, 045412 (2007).
[Crossref]

Q. Nguyen, R. Ossikovski, and J. Schreiber, “Contrast enhancement on Crystalline Silicon in polarized reflection mode tip-enhanced Raman Spectroscopy,” Opt. Commun. 274, 231–235 (2007).
[Crossref]

2006 (6)

R. Esteban, R. Vogelgesang, and K. Kern, “Simulation of Optical Near and Far Fields of Dielectric Apertureless Scanning Probe,” Nanotechnology 17, 475–482 (2006).
[Crossref]

A. Bouhelier, “Field-enhanced scanning near-field optical microscopy,” Microsc. Res. Tech. 69, 563–579 (2006).
[Crossref] [PubMed]

R. M. Roth, N. C. Panoiu, M. M. Adams, R. M. Osgood, C. C. Neacsu, and M. B. Raschke, “Resonant-Plasmon Field enhancement from asymmetrically illuminated Conical Metallic-Probe Tips,” Opt. Express 14, 2921–2931 (2006).
[Crossref] [PubMed]

A. V. Goncharenko, J. K. Wang, and Y. C. Chang, “Electric Near-Field Enhancement of a Sharp Semi-Infinite Conical Probe: Material and Cone Angle Dependence,” Phys. Rev. B 74, 235442 (2006).
[Crossref]

P. C. Chaumet, “Fully vectorial highly nonparaxial beam close to the waist,” J. Opt. Soc. Am. A 23, 3197–3202 (2006).
[Crossref]

C. C. Neacsu, J. Dreyer, N. Behr, and M. B. Raschke, “Scanning-probe Raman Spectroscopy with single-molecule sensitivity,” Phys. Rev. B 73, 193406 (2006).
[Crossref]

2005 (3)

D. Mehtani, N. Lee, R. D. Hartschuh, A. Kisliuk, M. D. Foster, A. P. Sokolov, and J. F. Maguire, “Nano-Raman Spectroscopy with side-illumination optics,” J. Raman Spectrosc. 36, 1068–1075 (2005).
[Crossref]

K. O. Greulich, “Single molecule studies of DNA and RNA,” ChemPhysChem 6, 2458–2471 (2005).
[Crossref] [PubMed]

R. Cicchetti and A. Faraone, “On the optical behavior of the Electromagnetic Field excited by a Semi-Infinite Traveling-Wave Current,” IEEE Trans. Antennas Propag. 53, 4015–4025 (2005).
[Crossref]

2004 (1)

K. C. Neuman and S. M. Block, “Optical Trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004).
[Crossref]

2003 (6)

D. Richards, “Near-Field Microscopy: throwing light on the Nanoworld,” Phil. Trans. R. Soc. Lond. A 361, 2843–2857 (2003).
[Crossref]

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

N. I. Petrov, “Evanescent and propagating fields of a strongly focused beam,” J. Opt. Soc. Am. 20, 2385–2389 (2003).
[Crossref]

W. X. Sun and Z. X. Shen, “Optimizing the Near Field around Silver Tips,” J. Opt. Soc. Am. A 20, 2254–2259 (2003).
[Crossref]

M. Idemen, “Confluent Tip Singularity of the Electromagnetic Field at the Apex of a Material Cone,” Wave Motion 38, 251–277 (2003).
[Crossref]

R. Bachelot, F. H’Dhili, D. Barchiesi, G. Lerondel, R. Fikri, P. Royer, N. Landraud, J. Peretti, F. Chaput, G. Lampel, J. P. Boilot, and K. Lahlil, “Apertureless near-field optical microscopy: a study of the local tip field enhancement using photosensitive Azobenzene-containing films,” J. Appl. Phys. 94, 2060–2072 (2003).
[Crossref]

2002 (3)

S. Takahashi and A. V. Zayats, “Near-Field Second-Harmonic Generation at a Metal Tip Apex,” Appl. Phys. Lett. 80, 3479–3481 (2002).
[Crossref]

C. G. Chen, P. T. Konkola, J. Ferrera, R. K. Heilmann, and M. L. Schattenburg, “Analysis of Vector Gaussian Beam Propagation and the Validity of Paraxial and Spherical Approximations,” J. Opt. Soc. Am. A 19, 404–412 (2002).
[Crossref]

M. S. Anderson and W. T. Pike, “A Raman-Atomic Force Microscope for Apertureless-Near-Field Spectroscopy and Optical Trapping,” Rev. Sci. Instrum. 73, 1198–1203 (2002).
[Crossref]

2001 (2)

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal Field Modes probed by Single Molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref] [PubMed]

N. I. Petrov, “Focusing of beams into subwavelength area in an inhomogeneous medium,” Opt. Express 9, 658–673 (2001).
[Crossref] [PubMed]

2000 (1)

M. S. Anderson, “Locally enhanced Raman Spectroscopy with an Atomic Force Microscope,” Appl. Phys. Lett. 76, 3130–3132 (2000).
[Crossref]

1998 (1)

C. Durkan and I. V. Shvets, “Polarization effects in Reflection-Mode Scanning Near-Field Optical Microscopy,” J. Appl. Phys. 83, 1837–1843 (1998).
[Crossref]

1997 (4)

O. J. F. Martin and C. Girard, “Controlling and Tuning Strong Optical Field Gradients at a Local Probe Microscope Tip Apex,” Appl. Phys. Lett. 70, 705 (1997).
[Crossref]

L. Novotny, R. X. Bean, and X. S. Xie, “Theory of Nanometric Optical Tweezers,” Phys. Rev. Lett. 79, 645–648 (1997).
[Crossref]

O. J. F. Martin and C. Girard, “Controlling and tuning strong optical field gradients at a local probe microscope tip apex,” Appl. Phys. Lett. 70, 705–707 (1997).
[Crossref]

M. J. Hagmann, “Intensification of Optical Electric Fields caused by the Interaction with a Metal Tip in Photofield Emission and Laser-Assisted Scanning Tunneling Microscopy,” J. Vac. Sci. Technol. B 15, 597–601 (1997).
[Crossref]

1995 (1)

O. J. F. Martin, C. Girard, and A. Dereux, “Generalized field propagator for Electromagnetic Scattering and Light Confinement,” Phys. Rev. Lett. 74, 526–529 (1995).
[Crossref] [PubMed]

1994 (1)

1985 (1)

J. Van Bladel, “Field singularities at the tip of a Dielectric Cone,” IEEE Trans. Antennas Propag. 33, 893–895 (1985).
[Crossref]

1978 (1)

D.C. Chang, S.W. Lee, and L. Rispin, “Simple formula for current on a cylindrical receiving antenna,” IEEE Trans. Antennas Propag. 26, 683–690 (1978).
[Crossref]

1973 (1)

J. C. Wiesner and T. E. Everhart, “Point-Cathode Electron Sources — Electron Optics of the Initial Diode Region,” J. Appl. Phys. 44, 2140–2148 (1973).
[Crossref]

1953 (1)

W. P. Dyke, J. K. Trolan, W. W. Dolan, and G. Barnes, “The Field Emitter: Fabrication, Electron Microscopy, and Electric Field Calculations,” J. Appl. Phys. 24, 570–576 (1953).
[Crossref]

1949 (1)

R. N. Hall, “The application of non-integral Legendre functions to potential problems,” J. Appl. Phys. 20, 925–931 (1949).
[Crossref]

Adams, M. M.

Anderson, M. S.

M. S. Anderson and W. T. Pike, “A Raman-Atomic Force Microscope for Apertureless-Near-Field Spectroscopy and Optical Trapping,” Rev. Sci. Instrum. 73, 1198–1203 (2002).
[Crossref]

M. S. Anderson, “Locally enhanced Raman Spectroscopy with an Atomic Force Microscope,” Appl. Phys. Lett. 76, 3130–3132 (2000).
[Crossref]

Bachelot, R.

R. Bachelot, F. H’Dhili, D. Barchiesi, G. Lerondel, R. Fikri, P. Royer, N. Landraud, J. Peretti, F. Chaput, G. Lampel, J. P. Boilot, and K. Lahlil, “Apertureless near-field optical microscopy: a study of the local tip field enhancement using photosensitive Azobenzene-containing films,” J. Appl. Phys. 94, 2060–2072 (2003).
[Crossref]

Balanis, C.A.

C.A. Balanis, Antenna Theory. Analysis and Design (Wiley, New York, 1997).

Barchiesi, D.

R. Bachelot, F. H’Dhili, D. Barchiesi, G. Lerondel, R. Fikri, P. Royer, N. Landraud, J. Peretti, F. Chaput, G. Lampel, J. P. Boilot, and K. Lahlil, “Apertureless near-field optical microscopy: a study of the local tip field enhancement using photosensitive Azobenzene-containing films,” J. Appl. Phys. 94, 2060–2072 (2003).
[Crossref]

Barnes, G.

W. P. Dyke, J. K. Trolan, W. W. Dolan, and G. Barnes, “The Field Emitter: Fabrication, Electron Microscopy, and Electric Field Calculations,” J. Appl. Phys. 24, 570–576 (1953).
[Crossref]

Bateman, H.

H. Bateman and A. Erdelyi, Higher Transcendental Functions, (McGraw-Hill, 1985) Vol. 1.

Bean, R. X.

L. Novotny, R. X. Bean, and X. S. Xie, “Theory of Nanometric Optical Tweezers,” Phys. Rev. Lett. 79, 645–648 (1997).
[Crossref]

Behr, N.

C. C. Neacsu, J. Dreyer, N. Behr, and M. B. Raschke, “Scanning-probe Raman Spectroscopy with single-molecule sensitivity,” Phys. Rev. B 73, 193406 (2006).
[Crossref]

Beversluis, M. R.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal Field Modes probed by Single Molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref] [PubMed]

Block, S. M.

K. C. Neuman and S. M. Block, “Optical Trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004).
[Crossref]

Boilot, J. P.

R. Bachelot, F. H’Dhili, D. Barchiesi, G. Lerondel, R. Fikri, P. Royer, N. Landraud, J. Peretti, F. Chaput, G. Lampel, J. P. Boilot, and K. Lahlil, “Apertureless near-field optical microscopy: a study of the local tip field enhancement using photosensitive Azobenzene-containing films,” J. Appl. Phys. 94, 2060–2072 (2003).
[Crossref]

Bouhelier, A.

A. Bouhelier, “Field-enhanced scanning near-field optical microscopy,” Microsc. Res. Tech. 69, 563–579 (2006).
[Crossref] [PubMed]

Bowman, J. J.

J. J. Bowman, J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, eds., (Hemisphere, New York, 1987).

J. J. Bowman, J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, eds., (Hemisphere, New York, 1987).

Brown, T. G.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal Field Modes probed by Single Molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref] [PubMed]

Chang, D.C.

D.C. Chang, S.W. Lee, and L. Rispin, “Simple formula for current on a cylindrical receiving antenna,” IEEE Trans. Antennas Propag. 26, 683–690 (1978).
[Crossref]

Chang, Y. C.

A. V. Goncharenko, J. K. Wang, and Y. C. Chang, “Electric Near-Field Enhancement of a Sharp Semi-Infinite Conical Probe: Material and Cone Angle Dependence,” Phys. Rev. B 74, 235442 (2006).
[Crossref]

Chaput, F.

R. Bachelot, F. H’Dhili, D. Barchiesi, G. Lerondel, R. Fikri, P. Royer, N. Landraud, J. Peretti, F. Chaput, G. Lampel, J. P. Boilot, and K. Lahlil, “Apertureless near-field optical microscopy: a study of the local tip field enhancement using photosensitive Azobenzene-containing films,” J. Appl. Phys. 94, 2060–2072 (2003).
[Crossref]

Chaumet, P. C.

Chen, C. G.

Cicchetti, R.

R. Cicchetti and A. Faraone, “On the optical behavior of the Electromagnetic Field excited by a Semi-Infinite Traveling-Wave Current,” IEEE Trans. Antennas Propag. 53, 4015–4025 (2005).
[Crossref]

Dereux, A.

O. J. F. Martin, C. Girard, and A. Dereux, “Generalized field propagator for Electromagnetic Scattering and Light Confinement,” Phys. Rev. Lett. 74, 526–529 (1995).
[Crossref] [PubMed]

Dolan, W. W.

W. P. Dyke, J. K. Trolan, W. W. Dolan, and G. Barnes, “The Field Emitter: Fabrication, Electron Microscopy, and Electric Field Calculations,” J. Appl. Phys. 24, 570–576 (1953).
[Crossref]

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

Dreyer, J.

C. C. Neacsu, J. Dreyer, N. Behr, and M. B. Raschke, “Scanning-probe Raman Spectroscopy with single-molecule sensitivity,” Phys. Rev. B 73, 193406 (2006).
[Crossref]

Durkan, C.

C. Durkan and I. V. Shvets, “Polarization effects in Reflection-Mode Scanning Near-Field Optical Microscopy,” J. Appl. Phys. 83, 1837–1843 (1998).
[Crossref]

Dyke, W. P.

W. P. Dyke, J. K. Trolan, W. W. Dolan, and G. Barnes, “The Field Emitter: Fabrication, Electron Microscopy, and Electric Field Calculations,” J. Appl. Phys. 24, 570–576 (1953).
[Crossref]

Erdelyi, A.

H. Bateman and A. Erdelyi, Higher Transcendental Functions, (McGraw-Hill, 1985) Vol. 1.

Esteban, R.

R. Esteban, R. Vogelgesang, and K. Kern, “Simulation of Optical Near and Far Fields of Dielectric Apertureless Scanning Probe,” Nanotechnology 17, 475–482 (2006).
[Crossref]

Everhart, T. E.

J. C. Wiesner and T. E. Everhart, “Point-Cathode Electron Sources — Electron Optics of the Initial Diode Region,” J. Appl. Phys. 44, 2140–2148 (1973).
[Crossref]

Faraone, A.

R. Cicchetti and A. Faraone, “On the optical behavior of the Electromagnetic Field excited by a Semi-Infinite Traveling-Wave Current,” IEEE Trans. Antennas Propag. 53, 4015–4025 (2005).
[Crossref]

Felsen, L. B.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Wiley, 1994) and references therein.
[Crossref]

Ferrera, J.

Fikri, R.

R. Bachelot, F. H’Dhili, D. Barchiesi, G. Lerondel, R. Fikri, P. Royer, N. Landraud, J. Peretti, F. Chaput, G. Lampel, J. P. Boilot, and K. Lahlil, “Apertureless near-field optical microscopy: a study of the local tip field enhancement using photosensitive Azobenzene-containing films,” J. Appl. Phys. 94, 2060–2072 (2003).
[Crossref]

Foster, M. D.

D. Mehtani, N. Lee, R. D. Hartschuh, A. Kisliuk, M. D. Foster, A. P. Sokolov, and J. F. Maguire, “Nano-Raman Spectroscopy with side-illumination optics,” J. Raman Spectrosc. 36, 1068–1075 (2005).
[Crossref]

Girard, C.

O. J. F. Martin and C. Girard, “Controlling and Tuning Strong Optical Field Gradients at a Local Probe Microscope Tip Apex,” Appl. Phys. Lett. 70, 705 (1997).
[Crossref]

O. J. F. Martin and C. Girard, “Controlling and tuning strong optical field gradients at a local probe microscope tip apex,” Appl. Phys. Lett. 70, 705–707 (1997).
[Crossref]

O. J. F. Martin, C. Girard, and A. Dereux, “Generalized field propagator for Electromagnetic Scattering and Light Confinement,” Phys. Rev. Lett. 74, 526–529 (1995).
[Crossref] [PubMed]

Goncharenko, A. V.

A. V. Goncharenko, J. K. Wang, and Y. C. Chang, “Electric Near-Field Enhancement of a Sharp Semi-Infinite Conical Probe: Material and Cone Angle Dependence,” Phys. Rev. B 74, 235442 (2006).
[Crossref]

Greulich, K. O.

K. O. Greulich, “Single molecule studies of DNA and RNA,” ChemPhysChem 6, 2458–2471 (2005).
[Crossref] [PubMed]

H’Dhili, F.

R. Bachelot, F. H’Dhili, D. Barchiesi, G. Lerondel, R. Fikri, P. Royer, N. Landraud, J. Peretti, F. Chaput, G. Lampel, J. P. Boilot, and K. Lahlil, “Apertureless near-field optical microscopy: a study of the local tip field enhancement using photosensitive Azobenzene-containing films,” J. Appl. Phys. 94, 2060–2072 (2003).
[Crossref]

Hagmann, M. J.

M. J. Hagmann, “Intensification of Optical Electric Fields caused by the Interaction with a Metal Tip in Photofield Emission and Laser-Assisted Scanning Tunneling Microscopy,” J. Vac. Sci. Technol. B 15, 597–601 (1997).
[Crossref]

Hall, R. N.

R. N. Hall, “The application of non-integral Legendre functions to potential problems,” J. Appl. Phys. 20, 925–931 (1949).
[Crossref]

Hartschuh, R. D.

D. Mehtani, N. Lee, R. D. Hartschuh, A. Kisliuk, M. D. Foster, A. P. Sokolov, and J. F. Maguire, “Nano-Raman Spectroscopy with side-illumination optics,” J. Raman Spectrosc. 36, 1068–1075 (2005).
[Crossref]

Heilmann, R. K.

Hu, D.

D. Hu, M. Micic, N. Klymyshyn, Y. D. Suh, and H. P. Lu, “Correlated Topographic and Spectroscopic Imaging beyond Diffraction limit by Atomic Force Microscopy Metallic Tip-Enhanced Near-Field Fluorescence Lifetime Microscopy,” Rev. Sci. Instrum.74, 3347–3355 (2003).
[Crossref]

Idemen, M.

M. Idemen, “Confluent Tip Singularity of the Electromagnetic Field at the Apex of a Material Cone,” Wave Motion 38, 251–277 (2003).
[Crossref]

Kamel, A. H.

M. A. Salem, A. H. Kamel, and A. V. Osipov, “Electromagnetic fields in the presence of an infinite dielectric wedge,” Proc. R. Soc. A462, 2503–2522 (2006).
[Crossref]

Kern, K.

R. Esteban, R. Vogelgesang, and K. Kern, “Simulation of Optical Near and Far Fields of Dielectric Apertureless Scanning Probe,” Nanotechnology 17, 475–482 (2006).
[Crossref]

Kisliuk, A.

D. Mehtani, N. Lee, R. D. Hartschuh, A. Kisliuk, M. D. Foster, A. P. Sokolov, and J. F. Maguire, “Nano-Raman Spectroscopy with side-illumination optics,” J. Raman Spectrosc. 36, 1068–1075 (2005).
[Crossref]

Klymyshyn, N.

D. Hu, M. Micic, N. Klymyshyn, Y. D. Suh, and H. P. Lu, “Correlated Topographic and Spectroscopic Imaging beyond Diffraction limit by Atomic Force Microscopy Metallic Tip-Enhanced Near-Field Fluorescence Lifetime Microscopy,” Rev. Sci. Instrum.74, 3347–3355 (2003).
[Crossref]

Konkola, P. T.

Kriezis, Em. E.

Lahlil, K.

R. Bachelot, F. H’Dhili, D. Barchiesi, G. Lerondel, R. Fikri, P. Royer, N. Landraud, J. Peretti, F. Chaput, G. Lampel, J. P. Boilot, and K. Lahlil, “Apertureless near-field optical microscopy: a study of the local tip field enhancement using photosensitive Azobenzene-containing films,” J. Appl. Phys. 94, 2060–2072 (2003).
[Crossref]

Lampel, G.

R. Bachelot, F. H’Dhili, D. Barchiesi, G. Lerondel, R. Fikri, P. Royer, N. Landraud, J. Peretti, F. Chaput, G. Lampel, J. P. Boilot, and K. Lahlil, “Apertureless near-field optical microscopy: a study of the local tip field enhancement using photosensitive Azobenzene-containing films,” J. Appl. Phys. 94, 2060–2072 (2003).
[Crossref]

Landraud, N.

R. Bachelot, F. H’Dhili, D. Barchiesi, G. Lerondel, R. Fikri, P. Royer, N. Landraud, J. Peretti, F. Chaput, G. Lampel, J. P. Boilot, and K. Lahlil, “Apertureless near-field optical microscopy: a study of the local tip field enhancement using photosensitive Azobenzene-containing films,” J. Appl. Phys. 94, 2060–2072 (2003).
[Crossref]

Lee, N.

D. Mehtani, N. Lee, R. D. Hartschuh, A. Kisliuk, M. D. Foster, A. P. Sokolov, and J. F. Maguire, “Nano-Raman Spectroscopy with side-illumination optics,” J. Raman Spectrosc. 36, 1068–1075 (2005).
[Crossref]

Lee, S.W.

D.C. Chang, S.W. Lee, and L. Rispin, “Simple formula for current on a cylindrical receiving antenna,” IEEE Trans. Antennas Propag. 26, 683–690 (1978).
[Crossref]

Lerondel, G.

R. Bachelot, F. H’Dhili, D. Barchiesi, G. Lerondel, R. Fikri, P. Royer, N. Landraud, J. Peretti, F. Chaput, G. Lampel, J. P. Boilot, and K. Lahlil, “Apertureless near-field optical microscopy: a study of the local tip field enhancement using photosensitive Azobenzene-containing films,” J. Appl. Phys. 94, 2060–2072 (2003).
[Crossref]

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

Lu, H. P.

D. Hu, M. Micic, N. Klymyshyn, Y. D. Suh, and H. P. Lu, “Correlated Topographic and Spectroscopic Imaging beyond Diffraction limit by Atomic Force Microscopy Metallic Tip-Enhanced Near-Field Fluorescence Lifetime Microscopy,” Rev. Sci. Instrum.74, 3347–3355 (2003).
[Crossref]

Maguire, J. F.

D. Mehtani, N. Lee, R. D. Hartschuh, A. Kisliuk, M. D. Foster, A. P. Sokolov, and J. F. Maguire, “Nano-Raman Spectroscopy with side-illumination optics,” J. Raman Spectrosc. 36, 1068–1075 (2005).
[Crossref]

Marcuvitz, N.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Wiley, 1994) and references therein.
[Crossref]

Martin, O. J. F.

O. J. F. Martin and C. Girard, “Controlling and tuning strong optical field gradients at a local probe microscope tip apex,” Appl. Phys. Lett. 70, 705–707 (1997).
[Crossref]

O. J. F. Martin and C. Girard, “Controlling and Tuning Strong Optical Field Gradients at a Local Probe Microscope Tip Apex,” Appl. Phys. Lett. 70, 705 (1997).
[Crossref]

O. J. F. Martin, C. Girard, and A. Dereux, “Generalized field propagator for Electromagnetic Scattering and Light Confinement,” Phys. Rev. Lett. 74, 526–529 (1995).
[Crossref] [PubMed]

Mehtani, D.

D. Mehtani, N. Lee, R. D. Hartschuh, A. Kisliuk, M. D. Foster, A. P. Sokolov, and J. F. Maguire, “Nano-Raman Spectroscopy with side-illumination optics,” J. Raman Spectrosc. 36, 1068–1075 (2005).
[Crossref]

Micic, M.

D. Hu, M. Micic, N. Klymyshyn, Y. D. Suh, and H. P. Lu, “Correlated Topographic and Spectroscopic Imaging beyond Diffraction limit by Atomic Force Microscopy Metallic Tip-Enhanced Near-Field Fluorescence Lifetime Microscopy,” Rev. Sci. Instrum.74, 3347–3355 (2003).
[Crossref]

Neacsu, C. C.

Neuman, K. C.

K. C. Neuman and S. M. Block, “Optical Trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004).
[Crossref]

Nguen, Q.

R. Ossikovski, Q. Nguen, and G. Picardi, “Simple model for the polarization effects in tip-enhanced Raman Spectroscopy,” Phys. Rev. B 75, 045412 (2007).
[Crossref]

Nguyen, Q.

Q. Nguyen, R. Ossikovski, and J. Schreiber, “Contrast enhancement on Crystalline Silicon in polarized reflection mode tip-enhanced Raman Spectroscopy,” Opt. Commun. 274, 231–235 (2007).
[Crossref]

Novotny, L.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal Field Modes probed by Single Molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref] [PubMed]

L. Novotny, R. X. Bean, and X. S. Xie, “Theory of Nanometric Optical Tweezers,” Phys. Rev. Lett. 79, 645–648 (1997).
[Crossref]

Osgood, R. M.

Osipov, A. V.

M. A. Salem, A. H. Kamel, and A. V. Osipov, “Electromagnetic fields in the presence of an infinite dielectric wedge,” Proc. R. Soc. A462, 2503–2522 (2006).
[Crossref]

Ossikovski, R.

R. Ossikovski, Q. Nguen, and G. Picardi, “Simple model for the polarization effects in tip-enhanced Raman Spectroscopy,” Phys. Rev. B 75, 045412 (2007).
[Crossref]

Q. Nguyen, R. Ossikovski, and J. Schreiber, “Contrast enhancement on Crystalline Silicon in polarized reflection mode tip-enhanced Raman Spectroscopy,” Opt. Commun. 274, 231–235 (2007).
[Crossref]

Pandelakis, P. K.

Panoiu, N. C.

Papagiannakis, A. G.

Peretti, J.

R. Bachelot, F. H’Dhili, D. Barchiesi, G. Lerondel, R. Fikri, P. Royer, N. Landraud, J. Peretti, F. Chaput, G. Lampel, J. P. Boilot, and K. Lahlil, “Apertureless near-field optical microscopy: a study of the local tip field enhancement using photosensitive Azobenzene-containing films,” J. Appl. Phys. 94, 2060–2072 (2003).
[Crossref]

Petrov, N. I.

N. I. Petrov, “Evanescent and propagating fields of a strongly focused beam,” J. Opt. Soc. Am. 20, 2385–2389 (2003).
[Crossref]

N. I. Petrov, “Focusing of beams into subwavelength area in an inhomogeneous medium,” Opt. Express 9, 658–673 (2001).
[Crossref] [PubMed]

Picardi, G.

R. Ossikovski, Q. Nguen, and G. Picardi, “Simple model for the polarization effects in tip-enhanced Raman Spectroscopy,” Phys. Rev. B 75, 045412 (2007).
[Crossref]

Pike, W. T.

M. S. Anderson and W. T. Pike, “A Raman-Atomic Force Microscope for Apertureless-Near-Field Spectroscopy and Optical Trapping,” Rev. Sci. Instrum. 73, 1198–1203 (2002).
[Crossref]

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

Raschke, M. B.

Richards, D.

D. Richards, “Near-Field Microscopy: throwing light on the Nanoworld,” Phil. Trans. R. Soc. Lond. A 361, 2843–2857 (2003).
[Crossref]

Rispin, L.

D.C. Chang, S.W. Lee, and L. Rispin, “Simple formula for current on a cylindrical receiving antenna,” IEEE Trans. Antennas Propag. 26, 683–690 (1978).
[Crossref]

Roth, R. M.

Royer, P.

R. Bachelot, F. H’Dhili, D. Barchiesi, G. Lerondel, R. Fikri, P. Royer, N. Landraud, J. Peretti, F. Chaput, G. Lampel, J. P. Boilot, and K. Lahlil, “Apertureless near-field optical microscopy: a study of the local tip field enhancement using photosensitive Azobenzene-containing films,” J. Appl. Phys. 94, 2060–2072 (2003).
[Crossref]

Salem, M. A.

M. A. Salem, A. H. Kamel, and A. V. Osipov, “Electromagnetic fields in the presence of an infinite dielectric wedge,” Proc. R. Soc. A462, 2503–2522 (2006).
[Crossref]

Schattenburg, M. L.

Schreiber, J.

Q. Nguyen, R. Ossikovski, and J. Schreiber, “Contrast enhancement on Crystalline Silicon in polarized reflection mode tip-enhanced Raman Spectroscopy,” Opt. Commun. 274, 231–235 (2007).
[Crossref]

Senior, T. B. A.

J. J. Bowman, J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, eds., (Hemisphere, New York, 1987).

Shen, Z. X.

Shvets, I. V.

C. Durkan and I. V. Shvets, “Polarization effects in Reflection-Mode Scanning Near-Field Optical Microscopy,” J. Appl. Phys. 83, 1837–1843 (1998).
[Crossref]

Sokolov, A. P.

D. Mehtani, N. Lee, R. D. Hartschuh, A. Kisliuk, M. D. Foster, A. P. Sokolov, and J. F. Maguire, “Nano-Raman Spectroscopy with side-illumination optics,” J. Raman Spectrosc. 36, 1068–1075 (2005).
[Crossref]

Suh, Y. D.

D. Hu, M. Micic, N. Klymyshyn, Y. D. Suh, and H. P. Lu, “Correlated Topographic and Spectroscopic Imaging beyond Diffraction limit by Atomic Force Microscopy Metallic Tip-Enhanced Near-Field Fluorescence Lifetime Microscopy,” Rev. Sci. Instrum.74, 3347–3355 (2003).
[Crossref]

Sun, W. X.

Takahashi, S.

S. Takahashi and A. V. Zayats, “Near-Field Second-Harmonic Generation at a Metal Tip Apex,” Appl. Phys. Lett. 80, 3479–3481 (2002).
[Crossref]

Trolan, J. K.

W. P. Dyke, J. K. Trolan, W. W. Dolan, and G. Barnes, “The Field Emitter: Fabrication, Electron Microscopy, and Electric Field Calculations,” J. Appl. Phys. 24, 570–576 (1953).
[Crossref]

Uslenghi, P. L. E.

J. J. Bowman, J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, eds., (Hemisphere, New York, 1987).

Van Bladel, J.

J. Van Bladel, “Field singularities at the tip of a Dielectric Cone,” IEEE Trans. Antennas Propag. 33, 893–895 (1985).
[Crossref]

Vogelgesang, R.

R. Esteban, R. Vogelgesang, and K. Kern, “Simulation of Optical Near and Far Fields of Dielectric Apertureless Scanning Probe,” Nanotechnology 17, 475–482 (2006).
[Crossref]

Wang, J. K.

A. V. Goncharenko, J. K. Wang, and Y. C. Chang, “Electric Near-Field Enhancement of a Sharp Semi-Infinite Conical Probe: Material and Cone Angle Dependence,” Phys. Rev. B 74, 235442 (2006).
[Crossref]

Wiesner, J. C.

J. C. Wiesner and T. E. Everhart, “Point-Cathode Electron Sources — Electron Optics of the Initial Diode Region,” J. Appl. Phys. 44, 2140–2148 (1973).
[Crossref]

Xie, X. S.

L. Novotny, R. X. Bean, and X. S. Xie, “Theory of Nanometric Optical Tweezers,” Phys. Rev. Lett. 79, 645–648 (1997).
[Crossref]

Youngworth, K. S.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal Field Modes probed by Single Molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref] [PubMed]

Zayats, A. V.

S. Takahashi and A. V. Zayats, “Near-Field Second-Harmonic Generation at a Metal Tip Apex,” Appl. Phys. Lett. 80, 3479–3481 (2002).
[Crossref]

Appl. Phys. Lett. (4)

O. J. F. Martin and C. Girard, “Controlling and tuning strong optical field gradients at a local probe microscope tip apex,” Appl. Phys. Lett. 70, 705–707 (1997).
[Crossref]

S. Takahashi and A. V. Zayats, “Near-Field Second-Harmonic Generation at a Metal Tip Apex,” Appl. Phys. Lett. 80, 3479–3481 (2002).
[Crossref]

O. J. F. Martin and C. Girard, “Controlling and Tuning Strong Optical Field Gradients at a Local Probe Microscope Tip Apex,” Appl. Phys. Lett. 70, 705 (1997).
[Crossref]

M. S. Anderson, “Locally enhanced Raman Spectroscopy with an Atomic Force Microscope,” Appl. Phys. Lett. 76, 3130–3132 (2000).
[Crossref]

ChemPhysChem (1)

K. O. Greulich, “Single molecule studies of DNA and RNA,” ChemPhysChem 6, 2458–2471 (2005).
[Crossref] [PubMed]

IEEE Trans. Antennas Propag. (3)

D.C. Chang, S.W. Lee, and L. Rispin, “Simple formula for current on a cylindrical receiving antenna,” IEEE Trans. Antennas Propag. 26, 683–690 (1978).
[Crossref]

R. Cicchetti and A. Faraone, “On the optical behavior of the Electromagnetic Field excited by a Semi-Infinite Traveling-Wave Current,” IEEE Trans. Antennas Propag. 53, 4015–4025 (2005).
[Crossref]

J. Van Bladel, “Field singularities at the tip of a Dielectric Cone,” IEEE Trans. Antennas Propag. 33, 893–895 (1985).
[Crossref]

J. Appl. Phys. (5)

R. N. Hall, “The application of non-integral Legendre functions to potential problems,” J. Appl. Phys. 20, 925–931 (1949).
[Crossref]

R. Bachelot, F. H’Dhili, D. Barchiesi, G. Lerondel, R. Fikri, P. Royer, N. Landraud, J. Peretti, F. Chaput, G. Lampel, J. P. Boilot, and K. Lahlil, “Apertureless near-field optical microscopy: a study of the local tip field enhancement using photosensitive Azobenzene-containing films,” J. Appl. Phys. 94, 2060–2072 (2003).
[Crossref]

W. P. Dyke, J. K. Trolan, W. W. Dolan, and G. Barnes, “The Field Emitter: Fabrication, Electron Microscopy, and Electric Field Calculations,” J. Appl. Phys. 24, 570–576 (1953).
[Crossref]

J. C. Wiesner and T. E. Everhart, “Point-Cathode Electron Sources — Electron Optics of the Initial Diode Region,” J. Appl. Phys. 44, 2140–2148 (1973).
[Crossref]

C. Durkan and I. V. Shvets, “Polarization effects in Reflection-Mode Scanning Near-Field Optical Microscopy,” J. Appl. Phys. 83, 1837–1843 (1998).
[Crossref]

J. Opt. Soc. Am. (1)

N. I. Petrov, “Evanescent and propagating fields of a strongly focused beam,” J. Opt. Soc. Am. 20, 2385–2389 (2003).
[Crossref]

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

J. Raman Spectrosc. (1)

D. Mehtani, N. Lee, R. D. Hartschuh, A. Kisliuk, M. D. Foster, A. P. Sokolov, and J. F. Maguire, “Nano-Raman Spectroscopy with side-illumination optics,” J. Raman Spectrosc. 36, 1068–1075 (2005).
[Crossref]

J. Vac. Sci. Technol. B (1)

M. J. Hagmann, “Intensification of Optical Electric Fields caused by the Interaction with a Metal Tip in Photofield Emission and Laser-Assisted Scanning Tunneling Microscopy,” J. Vac. Sci. Technol. B 15, 597–601 (1997).
[Crossref]

Microsc. Res. Tech. (1)

A. Bouhelier, “Field-enhanced scanning near-field optical microscopy,” Microsc. Res. Tech. 69, 563–579 (2006).
[Crossref] [PubMed]

Nanotechnology (1)

R. Esteban, R. Vogelgesang, and K. Kern, “Simulation of Optical Near and Far Fields of Dielectric Apertureless Scanning Probe,” Nanotechnology 17, 475–482 (2006).
[Crossref]

Opt. Commun. (1)

Q. Nguyen, R. Ossikovski, and J. Schreiber, “Contrast enhancement on Crystalline Silicon in polarized reflection mode tip-enhanced Raman Spectroscopy,” Opt. Commun. 274, 231–235 (2007).
[Crossref]

Opt. Express (2)

Phil. Trans. R. Soc. Lond. A (1)

D. Richards, “Near-Field Microscopy: throwing light on the Nanoworld,” Phil. Trans. R. Soc. Lond. A 361, 2843–2857 (2003).
[Crossref]

Phys. Rev. B (3)

A. V. Goncharenko, J. K. Wang, and Y. C. Chang, “Electric Near-Field Enhancement of a Sharp Semi-Infinite Conical Probe: Material and Cone Angle Dependence,” Phys. Rev. B 74, 235442 (2006).
[Crossref]

R. Ossikovski, Q. Nguen, and G. Picardi, “Simple model for the polarization effects in tip-enhanced Raman Spectroscopy,” Phys. Rev. B 75, 045412 (2007).
[Crossref]

C. C. Neacsu, J. Dreyer, N. Behr, and M. B. Raschke, “Scanning-probe Raman Spectroscopy with single-molecule sensitivity,” Phys. Rev. B 73, 193406 (2006).
[Crossref]

Phys. Rev. Lett. (4)

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal Field Modes probed by Single Molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref] [PubMed]

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

L. Novotny, R. X. Bean, and X. S. Xie, “Theory of Nanometric Optical Tweezers,” Phys. Rev. Lett. 79, 645–648 (1997).
[Crossref]

O. J. F. Martin, C. Girard, and A. Dereux, “Generalized field propagator for Electromagnetic Scattering and Light Confinement,” Phys. Rev. Lett. 74, 526–529 (1995).
[Crossref] [PubMed]

Rev. Sci. Instrum. (2)

M. S. Anderson and W. T. Pike, “A Raman-Atomic Force Microscope for Apertureless-Near-Field Spectroscopy and Optical Trapping,” Rev. Sci. Instrum. 73, 1198–1203 (2002).
[Crossref]

K. C. Neuman and S. M. Block, “Optical Trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004).
[Crossref]

Wave Motion (1)

M. Idemen, “Confluent Tip Singularity of the Electromagnetic Field at the Apex of a Material Cone,” Wave Motion 38, 251–277 (2003).
[Crossref]

Other (7)

H. Bateman and A. Erdelyi, Higher Transcendental Functions, (McGraw-Hill, 1985) Vol. 1.

Our numerical simulations show that this statement is also valid for ν>0.

M. A. Salem, A. H. Kamel, and A. V. Osipov, “Electromagnetic fields in the presence of an infinite dielectric wedge,” Proc. R. Soc. A462, 2503–2522 (2006).
[Crossref]

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Wiley, 1994) and references therein.
[Crossref]

J. J. Bowman, J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, eds., (Hemisphere, New York, 1987).

D. Hu, M. Micic, N. Klymyshyn, Y. D. Suh, and H. P. Lu, “Correlated Topographic and Spectroscopic Imaging beyond Diffraction limit by Atomic Force Microscopy Metallic Tip-Enhanced Near-Field Fluorescence Lifetime Microscopy,” Rev. Sci. Instrum.74, 3347–3355 (2003).
[Crossref]

C.A. Balanis, Antenna Theory. Analysis and Design (Wiley, New York, 1997).

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Figures (5)

Fig. 1.
Fig. 1.

Two schemes of illumination of a conical probe: side illumination (a) and total internal reflection illumination (b).

Fig. 2.
Fig. 2.

Equipotential surfaces for the “bowling” pin geometry.

Fig. 3.
Fig. 3.

Absolute value of the associated Legendre function P 1ν(cosθ 0) vs the incident angle θ i.

Fig. 4.
Fig. 4.

Dependence of the incident angle θi at which P 1 ν(cosθ 0) peaks vs the parameter ν.

Fig. 5.
Fig. 5.

Electric near field enhancement for a silver (solid lines) and silicon (dashed lines) conical probe at the point r=5 nm, θ=0° vs the beam waist radius.

Equations (22)

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

U = 2 i k m = 0 Λ m ν s exp ( i ν s π 2 ) j ν s ( k r ) P ν s m ( cos θ ) { ν s ( ν s + 1 ) 0 π α [ P ν s m ( cos α ) ] 2 sin α d α } 1
× [ m sin m ( φ φ 0 ) cos β ( sin θ 0 ) 1 P ν s m ( cos θ 0 ) + cos m ( φ φ 0 ) sin β θ 0 P ν s m ( cos θ 0 ) ]
V = 2 i η k m = 0 Λ m μ s exp ( i μ s π 2 ) j μ s ( k r ) P μ s m ( cos θ ) { μ s ( μ s + 1 ) 0 π α [ P μ s m ( cos α ) ] 2 sin α d α } 1
× [ m cos m ( φ φ 0 ) cos β θ 0 P μ s m ( cos θ 0 ) m sin m ( φ φ 0 ) sin β ( sin θ 0 ) 1 P μ s m ( cos θ 0 ) ] ,
E ( r , θ ) = r ̂ E r + θ ̂ E θ E 0 C ( k r ) ν 1 P ν 1 ( cos θ 0 ) cos β [ r ̂ + θ ̂ ν θ ] P ν ( cos θ )
ε 1 P ν + 1 ( cos α ) P ν ( cos α ) + ε 2 P ν + 1 ( cos α ) P ν ( cos α ) + ( ε 1 ε 2 ) cos α = 0 .
P ν 1 ( cos θ ) = 1 2 ν ( ν + 1 ) sin θ · 2 F 1 ( 2 + ν , 1 ν ; 2 ; 1 cos θ 2 )
E ( r , θ ) E 0 C ( k r ) ν 1 [ 1 + ν + 1 ν B ( r , r 0 ) ] P ν 1 ( cos θ 0 ) [ r ̂ + θ ̂ ν θ 1 B ( r , r 0 ) 1 + ν + 1 ν B ( r , r 0 ) ] P ν ( cos θ )
ψ 1 ( r , θ ) = ψ 0 ( k r ) ν [ 1 S ( r 0 r ) 2 ν + 1 ] P ν ( cos θ ) .
E i ( x ' , y ' , z ' ) E 0 exp [ ( x ' + y ' ) 2 w 0 2 ] exp ( i k z ' )
A i ( p , q ) = k 2 d x ' d y ' E i ( x ' , y ' , 0 ) exp [ i k ( p x ' + q y ' ) ] = 2 π f 2 exp ( p 2 + q 2 4 f 2 ) ,
E nf ( x ' , y ' , z ' ) = E 0 2 π f 2 π 2 π 2 exp ( 1 4 w 0 2 k 2 sin 2 θ ' ) exp ( i k z ' cos θ ' ) sin θ ' cos θ ' d θ '
0 2 π E ( α , θ 0 , θ ' , φ ' ) exp [ i k ( x ' sin θ ' cos φ ' + y ' sin θ ' sin φ ' ) ] d φ ' ,
E i ( x ' , z ' ) = E 0 k w 0 2 π π 2 π 2 exp ( 1 4 w 0 2 k 2 sin 2 θ ' ) exp [ i k ( x ' sin θ ' + z ' cos θ ' ) ] cos θ ' d θ ' .
E nf ( x ' , z ' ) = E 0 k w 0 2 π π 2 π 2 E ( α , θ 0 θ ' ) exp ( 1 4 w 0 2 k 2 sin 2 θ ' ) exp [ i k ( x ' sin θ ' + z ' cos θ ' ) ] cos θ ' d θ ' .
x ' = ( x s x ) cos θ i + ( z s z ) sin θ i , z ' = ( x s x ) sin θ i ( z s z ) cos θ i .
θ n = cos 1 ( 1 λ 2 l m n ) ,
E z ( r ) = i 4 π ω ε [ 0 I I ( z ' ) ( z ' 2 + k 2 ) G ( r r ' ) d z ' ] ,
E ρ ( r ) = i 4 π ω ε [ 0 I I ( z ' ) ρ ' z ' G ( r r ' ) d z ' ] ,
E ( r ' ) E 0 exp ( i k z z ' ) = E 0 [ cos ( k z ' cos θ i ) + i sin ( k z ' cos θ i ) ] = E 0 Φ ( z ' )
E z ( r ) = E 0 z ( r ) + K V d r ' [ G xz ( r , r ' ) E 0 x ( r ' ) + G yz ( r , r ' ) E 0 y ( r ' ) + G zz ( r , r ' ) E 0 z ( r ' ) ] = E 0 ( r ) sin θ 0
× [ 1 + K V d r ' Φ ( z ' ) G zz ( r , r ' ) ] + E 0 ( r ) cos θ 0 K V d r ' Φ ( z ' ) [ sin φ 0 G zx ( r , r ' ) + cos φ 0 G zy ( r , r ) ] ,

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