Abstract

We demonstrate detailed simulations and experiments of near-field phase-response in a single silver nanoparticle. The plasmon-photon interaction is directly observed in the vicinity of silver nanoparticles through a near-field scanning optical microscope (NSOM). Our results manifest the correlation of phase-response and size-dependent optical enhancement. Detailed interference behaviors between optical excitation and plasmon mediated re-radiation are revealed on a single particle basis. This observation facilitates nano-applications in controlling the spatial distribution of surface plasmon (SP) modes by means of nanostructures.

©2008 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Direct near-field optical imaging of plasmonic resonances in metal nanoparticle pairs

Hsing-Ying Lin, Chen-Han Huang, Chih-Han Chang, Yun-Chiang Lan, and Hsiang-Chen Chui
Opt. Express 18(1) 165-172 (2010)

Near-field spectroscopy of surface plasmons in flat gold nanoparticles

M. Achermann, K. L. Shuford, G. C. Schatz, D. H. Dahanayaka, L. A. Bumm, and V. I. Klimov
Opt. Lett. 32(15) 2254-2256 (2007)

Super-resolution and frequency-dependent efficiency of near-field optical disks with silver nanoparticles

Ming-Yaw Ng and Wei-Chih Liu
Opt. Express 13(23) 9422-9430 (2005)

More Recommended Articles

References

  • View by:
  • Article Order
  • |
  • Year
  • |
  • Author
  • |
  • Publication
  1. U. Kreibig, B. Schmitz, and H. D. Breuer, “Separation of plasmon-polariton modes of small metal particles,” Phys. Rev. B 36, 5027 (1987).
    [Crossref]
  2. S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275, 1102–1106 (1997).
    [Crossref] [PubMed]
  3. J. J. Mock, M. Barbic, D. R. Smith, D. A. Schultz, and S. Schultz, “Shape effects in plasmon resonance of individual colloidal silver nanoparticles,” J. Chem. Phys. 116, 6755–6759 (2002).
    [Crossref]
  4. R. P. Van Duyne, J. C. Hulteen, and D. A. Treichel, “Atomic force microscopy and surface-enhanced Raman spectroscopy. I. Ag island films and Ag film over polymer nanosphere surfaces supported on glass,” J. Chem. Phys. 99, 2101–2115 (1993).
    [Crossref]
  5. K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Ultrasensitive chemical analysis by Raman spectroscopy,” Chem. Rev. 99, 2957–2976 (1999).
    [Crossref]
  6. M. Moskovits, “Surface-enhanced spectroscopy,” Rev. Mod. Phys. 57, 783–826 (1985).
    [Crossref]
  7. V. P. Drachev, M. D. Thoreson, V. Nashine, E. N. Khaliullin, D. Ben-Amotz, V. J. Davisson, and V. M. Shalaev, “Adaptive silver films for surface-enhanced Raman spectroscopy of biomolecules,” J. Raman Spectrosc. 36, 648–656 (2005).
    [Crossref]
  8. Z. B. Wang et al., “Energy flow around a small particle investigated by classical Mie theory,” Phys. Rev. B 70, 035418 (2004).
    [Crossref]
  9. C. Bohren and D. Huffiman, Absorption and Scattering of Light by Small Particles (John Wiley &Sons, New York, 1982).
  10. S. R. Emory, W. E. Haskins, and S. Nie, “Direct observation of size-dependent optical enhancement in single metal nanoparticles,” J. Am. Chem. Soc. 120, 8009–8010 (1998).
    [Crossref]
  11. J. Prikulis et al., “Phase-sensitive near-field imaging of metal nanoparticles,” J. Appl. Phys. 92, 6211–6214 (2002).
    [Crossref]
  12. A. A. Mikhailovsky, M. A. Petruska, M. I. Stockman, and V. I. Klimov, “Broadband near-field interference spectroscopy of metal nanoparticles using a femtosecond white-light continuum,” Opt. Lett. 28, 1686–1688 (2003).
    [Crossref] [PubMed]
  13. B.-H. Choi, H.-H. Lee, S. Jin, S. Chun, and S.-H. Kim, “Characterization of the optical properties of silver nanoparticle films,” Nanotechnology 18, 075706 (2007).
    [Crossref] [PubMed]
  14. M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Coherent control of femtosecond energy localization in nanosystems,” Phys. Rev. Lett. 88, 067402 (2002).
    [Crossref] [PubMed]
  15. J. F. Wolf, P. E. Hillner, R. Bilewicz, P. Kölsch, and J. P. Rabe, “Novel scanning near-field optical microscope (SNOM)/scanning confocal optical microscope based on normal force distance regulation and bent etched fiber tips,” Rev. Sci. Instrum. 70, 2751 (1999).
    [Crossref]

2007 (1)

B.-H. Choi, H.-H. Lee, S. Jin, S. Chun, and S.-H. Kim, “Characterization of the optical properties of silver nanoparticle films,” Nanotechnology 18, 075706 (2007).
[Crossref] [PubMed]

2005 (1)

V. P. Drachev, M. D. Thoreson, V. Nashine, E. N. Khaliullin, D. Ben-Amotz, V. J. Davisson, and V. M. Shalaev, “Adaptive silver films for surface-enhanced Raman spectroscopy of biomolecules,” J. Raman Spectrosc. 36, 648–656 (2005).
[Crossref]

2004 (1)

Z. B. Wang et al., “Energy flow around a small particle investigated by classical Mie theory,” Phys. Rev. B 70, 035418 (2004).
[Crossref]

2003 (1)

2002 (3)

J. Prikulis et al., “Phase-sensitive near-field imaging of metal nanoparticles,” J. Appl. Phys. 92, 6211–6214 (2002).
[Crossref]

M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Coherent control of femtosecond energy localization in nanosystems,” Phys. Rev. Lett. 88, 067402 (2002).
[Crossref] [PubMed]

J. J. Mock, M. Barbic, D. R. Smith, D. A. Schultz, and S. Schultz, “Shape effects in plasmon resonance of individual colloidal silver nanoparticles,” J. Chem. Phys. 116, 6755–6759 (2002).
[Crossref]

1999 (2)

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Ultrasensitive chemical analysis by Raman spectroscopy,” Chem. Rev. 99, 2957–2976 (1999).
[Crossref]

J. F. Wolf, P. E. Hillner, R. Bilewicz, P. Kölsch, and J. P. Rabe, “Novel scanning near-field optical microscope (SNOM)/scanning confocal optical microscope based on normal force distance regulation and bent etched fiber tips,” Rev. Sci. Instrum. 70, 2751 (1999).
[Crossref]

1998 (1)

S. R. Emory, W. E. Haskins, and S. Nie, “Direct observation of size-dependent optical enhancement in single metal nanoparticles,” J. Am. Chem. Soc. 120, 8009–8010 (1998).
[Crossref]

1997 (1)

S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275, 1102–1106 (1997).
[Crossref] [PubMed]

1993 (1)

R. P. Van Duyne, J. C. Hulteen, and D. A. Treichel, “Atomic force microscopy and surface-enhanced Raman spectroscopy. I. Ag island films and Ag film over polymer nanosphere surfaces supported on glass,” J. Chem. Phys. 99, 2101–2115 (1993).
[Crossref]

1987 (1)

U. Kreibig, B. Schmitz, and H. D. Breuer, “Separation of plasmon-polariton modes of small metal particles,” Phys. Rev. B 36, 5027 (1987).
[Crossref]

1985 (1)

M. Moskovits, “Surface-enhanced spectroscopy,” Rev. Mod. Phys. 57, 783–826 (1985).
[Crossref]

Barbic, M.

J. J. Mock, M. Barbic, D. R. Smith, D. A. Schultz, and S. Schultz, “Shape effects in plasmon resonance of individual colloidal silver nanoparticles,” J. Chem. Phys. 116, 6755–6759 (2002).
[Crossref]

Ben-Amotz, D.

V. P. Drachev, M. D. Thoreson, V. Nashine, E. N. Khaliullin, D. Ben-Amotz, V. J. Davisson, and V. M. Shalaev, “Adaptive silver films for surface-enhanced Raman spectroscopy of biomolecules,” J. Raman Spectrosc. 36, 648–656 (2005).
[Crossref]

Bergman, D. J.

M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Coherent control of femtosecond energy localization in nanosystems,” Phys. Rev. Lett. 88, 067402 (2002).
[Crossref] [PubMed]

Bilewicz, R.

J. F. Wolf, P. E. Hillner, R. Bilewicz, P. Kölsch, and J. P. Rabe, “Novel scanning near-field optical microscope (SNOM)/scanning confocal optical microscope based on normal force distance regulation and bent etched fiber tips,” Rev. Sci. Instrum. 70, 2751 (1999).
[Crossref]

Bohren, C.

C. Bohren and D. Huffiman, Absorption and Scattering of Light by Small Particles (John Wiley &Sons, New York, 1982).

Breuer, H. D.

U. Kreibig, B. Schmitz, and H. D. Breuer, “Separation of plasmon-polariton modes of small metal particles,” Phys. Rev. B 36, 5027 (1987).
[Crossref]

Choi, B.-H.

B.-H. Choi, H.-H. Lee, S. Jin, S. Chun, and S.-H. Kim, “Characterization of the optical properties of silver nanoparticle films,” Nanotechnology 18, 075706 (2007).
[Crossref] [PubMed]

Chun, S.

B.-H. Choi, H.-H. Lee, S. Jin, S. Chun, and S.-H. Kim, “Characterization of the optical properties of silver nanoparticle films,” Nanotechnology 18, 075706 (2007).
[Crossref] [PubMed]

Dasari, R. R.

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Ultrasensitive chemical analysis by Raman spectroscopy,” Chem. Rev. 99, 2957–2976 (1999).
[Crossref]

Davisson, V. J.

V. P. Drachev, M. D. Thoreson, V. Nashine, E. N. Khaliullin, D. Ben-Amotz, V. J. Davisson, and V. M. Shalaev, “Adaptive silver films for surface-enhanced Raman spectroscopy of biomolecules,” J. Raman Spectrosc. 36, 648–656 (2005).
[Crossref]

Drachev, V. P.

V. P. Drachev, M. D. Thoreson, V. Nashine, E. N. Khaliullin, D. Ben-Amotz, V. J. Davisson, and V. M. Shalaev, “Adaptive silver films for surface-enhanced Raman spectroscopy of biomolecules,” J. Raman Spectrosc. 36, 648–656 (2005).
[Crossref]

Emory, S. R.

S. R. Emory, W. E. Haskins, and S. Nie, “Direct observation of size-dependent optical enhancement in single metal nanoparticles,” J. Am. Chem. Soc. 120, 8009–8010 (1998).
[Crossref]

S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275, 1102–1106 (1997).
[Crossref] [PubMed]

Faleev, S. V.

M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Coherent control of femtosecond energy localization in nanosystems,” Phys. Rev. Lett. 88, 067402 (2002).
[Crossref] [PubMed]

Feld, M. S.

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Ultrasensitive chemical analysis by Raman spectroscopy,” Chem. Rev. 99, 2957–2976 (1999).
[Crossref]

Haskins, W. E.

S. R. Emory, W. E. Haskins, and S. Nie, “Direct observation of size-dependent optical enhancement in single metal nanoparticles,” J. Am. Chem. Soc. 120, 8009–8010 (1998).
[Crossref]

Hillner, P. E.

J. F. Wolf, P. E. Hillner, R. Bilewicz, P. Kölsch, and J. P. Rabe, “Novel scanning near-field optical microscope (SNOM)/scanning confocal optical microscope based on normal force distance regulation and bent etched fiber tips,” Rev. Sci. Instrum. 70, 2751 (1999).
[Crossref]

Huffiman, D.

C. Bohren and D. Huffiman, Absorption and Scattering of Light by Small Particles (John Wiley &Sons, New York, 1982).

Hulteen, J. C.

R. P. Van Duyne, J. C. Hulteen, and D. A. Treichel, “Atomic force microscopy and surface-enhanced Raman spectroscopy. I. Ag island films and Ag film over polymer nanosphere surfaces supported on glass,” J. Chem. Phys. 99, 2101–2115 (1993).
[Crossref]

Itzkan, I.

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Ultrasensitive chemical analysis by Raman spectroscopy,” Chem. Rev. 99, 2957–2976 (1999).
[Crossref]

Jin, S.

B.-H. Choi, H.-H. Lee, S. Jin, S. Chun, and S.-H. Kim, “Characterization of the optical properties of silver nanoparticle films,” Nanotechnology 18, 075706 (2007).
[Crossref] [PubMed]

Khaliullin, E. N.

V. P. Drachev, M. D. Thoreson, V. Nashine, E. N. Khaliullin, D. Ben-Amotz, V. J. Davisson, and V. M. Shalaev, “Adaptive silver films for surface-enhanced Raman spectroscopy of biomolecules,” J. Raman Spectrosc. 36, 648–656 (2005).
[Crossref]

Kim, S.-H.

B.-H. Choi, H.-H. Lee, S. Jin, S. Chun, and S.-H. Kim, “Characterization of the optical properties of silver nanoparticle films,” Nanotechnology 18, 075706 (2007).
[Crossref] [PubMed]

Klimov, V. I.

Kneipp, H.

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Ultrasensitive chemical analysis by Raman spectroscopy,” Chem. Rev. 99, 2957–2976 (1999).
[Crossref]

Kneipp, K.

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Ultrasensitive chemical analysis by Raman spectroscopy,” Chem. Rev. 99, 2957–2976 (1999).
[Crossref]

Kölsch, P.

J. F. Wolf, P. E. Hillner, R. Bilewicz, P. Kölsch, and J. P. Rabe, “Novel scanning near-field optical microscope (SNOM)/scanning confocal optical microscope based on normal force distance regulation and bent etched fiber tips,” Rev. Sci. Instrum. 70, 2751 (1999).
[Crossref]

Kreibig, U.

U. Kreibig, B. Schmitz, and H. D. Breuer, “Separation of plasmon-polariton modes of small metal particles,” Phys. Rev. B 36, 5027 (1987).
[Crossref]

Lee, H.-H.

B.-H. Choi, H.-H. Lee, S. Jin, S. Chun, and S.-H. Kim, “Characterization of the optical properties of silver nanoparticle films,” Nanotechnology 18, 075706 (2007).
[Crossref] [PubMed]

Mikhailovsky, A. A.

Mock, J. J.

J. J. Mock, M. Barbic, D. R. Smith, D. A. Schultz, and S. Schultz, “Shape effects in plasmon resonance of individual colloidal silver nanoparticles,” J. Chem. Phys. 116, 6755–6759 (2002).
[Crossref]

Moskovits, M.

M. Moskovits, “Surface-enhanced spectroscopy,” Rev. Mod. Phys. 57, 783–826 (1985).
[Crossref]

Nashine, V.

V. P. Drachev, M. D. Thoreson, V. Nashine, E. N. Khaliullin, D. Ben-Amotz, V. J. Davisson, and V. M. Shalaev, “Adaptive silver films for surface-enhanced Raman spectroscopy of biomolecules,” J. Raman Spectrosc. 36, 648–656 (2005).
[Crossref]

Nie, S.

S. R. Emory, W. E. Haskins, and S. Nie, “Direct observation of size-dependent optical enhancement in single metal nanoparticles,” J. Am. Chem. Soc. 120, 8009–8010 (1998).
[Crossref]

S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275, 1102–1106 (1997).
[Crossref] [PubMed]

Petruska, M. A.

Prikulis, J.

J. Prikulis et al., “Phase-sensitive near-field imaging of metal nanoparticles,” J. Appl. Phys. 92, 6211–6214 (2002).
[Crossref]

Rabe, J. P.

J. F. Wolf, P. E. Hillner, R. Bilewicz, P. Kölsch, and J. P. Rabe, “Novel scanning near-field optical microscope (SNOM)/scanning confocal optical microscope based on normal force distance regulation and bent etched fiber tips,” Rev. Sci. Instrum. 70, 2751 (1999).
[Crossref]

Schmitz, B.

U. Kreibig, B. Schmitz, and H. D. Breuer, “Separation of plasmon-polariton modes of small metal particles,” Phys. Rev. B 36, 5027 (1987).
[Crossref]

Schultz, D. A.

J. J. Mock, M. Barbic, D. R. Smith, D. A. Schultz, and S. Schultz, “Shape effects in plasmon resonance of individual colloidal silver nanoparticles,” J. Chem. Phys. 116, 6755–6759 (2002).
[Crossref]

Schultz, S.

J. J. Mock, M. Barbic, D. R. Smith, D. A. Schultz, and S. Schultz, “Shape effects in plasmon resonance of individual colloidal silver nanoparticles,” J. Chem. Phys. 116, 6755–6759 (2002).
[Crossref]

Shalaev, V. M.

V. P. Drachev, M. D. Thoreson, V. Nashine, E. N. Khaliullin, D. Ben-Amotz, V. J. Davisson, and V. M. Shalaev, “Adaptive silver films for surface-enhanced Raman spectroscopy of biomolecules,” J. Raman Spectrosc. 36, 648–656 (2005).
[Crossref]

Smith, D. R.

J. J. Mock, M. Barbic, D. R. Smith, D. A. Schultz, and S. Schultz, “Shape effects in plasmon resonance of individual colloidal silver nanoparticles,” J. Chem. Phys. 116, 6755–6759 (2002).
[Crossref]

Stockman, M. I.

Thoreson, M. D.

V. P. Drachev, M. D. Thoreson, V. Nashine, E. N. Khaliullin, D. Ben-Amotz, V. J. Davisson, and V. M. Shalaev, “Adaptive silver films for surface-enhanced Raman spectroscopy of biomolecules,” J. Raman Spectrosc. 36, 648–656 (2005).
[Crossref]

Treichel, D. A.

R. P. Van Duyne, J. C. Hulteen, and D. A. Treichel, “Atomic force microscopy and surface-enhanced Raman spectroscopy. I. Ag island films and Ag film over polymer nanosphere surfaces supported on glass,” J. Chem. Phys. 99, 2101–2115 (1993).
[Crossref]

Van Duyne, R. P.

R. P. Van Duyne, J. C. Hulteen, and D. A. Treichel, “Atomic force microscopy and surface-enhanced Raman spectroscopy. I. Ag island films and Ag film over polymer nanosphere surfaces supported on glass,” J. Chem. Phys. 99, 2101–2115 (1993).
[Crossref]

Wang, Z. B.

Z. B. Wang et al., “Energy flow around a small particle investigated by classical Mie theory,” Phys. Rev. B 70, 035418 (2004).
[Crossref]

Wolf, J. F.

J. F. Wolf, P. E. Hillner, R. Bilewicz, P. Kölsch, and J. P. Rabe, “Novel scanning near-field optical microscope (SNOM)/scanning confocal optical microscope based on normal force distance regulation and bent etched fiber tips,” Rev. Sci. Instrum. 70, 2751 (1999).
[Crossref]

Chem. Rev. (1)

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, “Ultrasensitive chemical analysis by Raman spectroscopy,” Chem. Rev. 99, 2957–2976 (1999).
[Crossref]

J. Am. Chem. Soc. (1)

S. R. Emory, W. E. Haskins, and S. Nie, “Direct observation of size-dependent optical enhancement in single metal nanoparticles,” J. Am. Chem. Soc. 120, 8009–8010 (1998).
[Crossref]

J. Appl. Phys. (1)

J. Prikulis et al., “Phase-sensitive near-field imaging of metal nanoparticles,” J. Appl. Phys. 92, 6211–6214 (2002).
[Crossref]

J. Chem. Phys. (2)

J. J. Mock, M. Barbic, D. R. Smith, D. A. Schultz, and S. Schultz, “Shape effects in plasmon resonance of individual colloidal silver nanoparticles,” J. Chem. Phys. 116, 6755–6759 (2002).
[Crossref]

R. P. Van Duyne, J. C. Hulteen, and D. A. Treichel, “Atomic force microscopy and surface-enhanced Raman spectroscopy. I. Ag island films and Ag film over polymer nanosphere surfaces supported on glass,” J. Chem. Phys. 99, 2101–2115 (1993).
[Crossref]

J. Raman Spectrosc. (1)

V. P. Drachev, M. D. Thoreson, V. Nashine, E. N. Khaliullin, D. Ben-Amotz, V. J. Davisson, and V. M. Shalaev, “Adaptive silver films for surface-enhanced Raman spectroscopy of biomolecules,” J. Raman Spectrosc. 36, 648–656 (2005).
[Crossref]

Nanotechnology (1)

B.-H. Choi, H.-H. Lee, S. Jin, S. Chun, and S.-H. Kim, “Characterization of the optical properties of silver nanoparticle films,” Nanotechnology 18, 075706 (2007).
[Crossref] [PubMed]

Opt. Lett. (1)

Phys. Rev. B (2)

Z. B. Wang et al., “Energy flow around a small particle investigated by classical Mie theory,” Phys. Rev. B 70, 035418 (2004).
[Crossref]

U. Kreibig, B. Schmitz, and H. D. Breuer, “Separation of plasmon-polariton modes of small metal particles,” Phys. Rev. B 36, 5027 (1987).
[Crossref]

Phys. Rev. Lett. (1)

M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Coherent control of femtosecond energy localization in nanosystems,” Phys. Rev. Lett. 88, 067402 (2002).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

M. Moskovits, “Surface-enhanced spectroscopy,” Rev. Mod. Phys. 57, 783–826 (1985).
[Crossref]

Rev. Sci. Instrum. (1)

J. F. Wolf, P. E. Hillner, R. Bilewicz, P. Kölsch, and J. P. Rabe, “Novel scanning near-field optical microscope (SNOM)/scanning confocal optical microscope based on normal force distance regulation and bent etched fiber tips,” Rev. Sci. Instrum. 70, 2751 (1999).
[Crossref]

Science (1)

S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275, 1102–1106 (1997).
[Crossref] [PubMed]

Other (1)

C. Bohren and D. Huffiman, Absorption and Scattering of Light by Small Particles (John Wiley &Sons, New York, 1982).

Cited By

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

Alert me when this article is cited.


Figures (4)

Fig. 1.
Fig. 1.

(a). NSOM setup. (b) The absorption spectrum of Ag nanoparticles. The spectral positions of the NSOM excitations are shown with RGB lines. Inset: SEM image of Ag nanoparticles.

Download Full Size | PPT Slide | PDF

Fig. 2.
Fig. 2.

Simulation results of the far-field interference from Etip and Es with phase difference (a) ϕ=0, (b) ϕ=π/2, and (c) ϕ=π. (d) Their cross-sectional views are combined in (d), showing the dramatic difference between on- and off-resonance.

Download Full Size | PPT Slide | PDF

Fig. 3.
Fig. 3.

Measured NSOM images of a 50-nm nanoparticle with (a) λ=633 nm, (b) λ=532 nm, and (c) λ=488 nm. Their cross-sectional views are combined in (d).

Download Full Size | PPT Slide | PDF

Fig. 4.
Fig. 4.

Particle size spectra for RGB excitations. The tendency of shorter wavelength resonant with smaller silver nanoparticles is evident.

Download Full Size | PPT Slide | PDF

Equations (4)

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

E tip ( r ) ∝ exp ⁡ [ i ( k r − ω t ) ] [ A k r + B exp ⁡ ( − r η ) ] ,
P = α E tip ,
α = ∣ α ∣ exp ⁡ ( i ϕ ) ,
E s ( r 1 ) ≈ [ Z 0 k 2 ( 4 π i k r 1 ) ] d P d t [ exp ⁡ ( − i k r 1 ) ] ,

Metrics