Abstract

Surface plasmon-coupled emission microscopy (SPCEM) was proposed as a high sensitivity technique that makes use of a thin layer of metal deposited on glass slides to efficiently excite fluorophores and to collect the emission light. However, the theoretical aspect of SPCEM imaging has not been well studied. We propose a model for SPCEM and show, through theoretical analysis and empirical results, that the point spread function of SPCEM is irregular and has an annular-like structure, significantly different from the familiar point spread function of the conventional wide-field microscopy. This result is due to the highly polarized and anisotropic emission caused by the metal layer.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Axelrod, "Total internal reflection microscopy in cell biology," Traffic 2, 764-774 (2001).
    [CrossRef] [PubMed]
  2. S. E. Sund, and D. Axelrod, "Actin dynamics at the living cell submembrane imaged by total internal reflection fluorescence photobleaching," Biophys. J. 79, 1655-1669 (2000).
    [CrossRef] [PubMed]
  3. J. A. Steyer, H. Horstmann, and W. Almers, "Transport, docking and exocytosis of single secretory granules in live chromaffin cells," Nature 388, 474-478 (1997).
    [CrossRef] [PubMed]
  4. E. L. Schmid, A. Tairi, R. Hovius, and H. Vogel, "Screening ligands for membrane protein receptors by total internal reflection fluorescence: The 5-HT3 serotonin receptor," Anal. Chem. 70, 1331-1338 (1998).
    [CrossRef] [PubMed]
  5. J. Borejdo, Z. Gryzyncski, N. Calander, P. Muthu, and I. Gryzyncski, "Application of surface plasmon coupled emission to study of muscle," Biophys. J. 91, 2626-2635 (2006).
    [CrossRef] [PubMed]
  6. J. R. Lakowicz, "Directional surface plasmon-coupled emission: a new method for high sensitivity detection," Biochem. and Biophys. Res. Comm. 307, 435-439 (2003).
    [CrossRef]
  7. J. R. Lakowicz, "Radiative decay engineering 3. surface plasmon-coupled directional emission," Anal. Biochem. 324, 153-169 (2004).
    [CrossRef]
  8. I. Gryzyncski, J. Malicka, Z. Gryzyncski, and J. R. Lakowicz, "Radiative decay engineering 4. experimental studies of surface-plasmon coupled directional emission," Anal. Biochem. 324, 170-1822004.
    [CrossRef]
  9. J. Malicka, I. Gryzyncski, Z. Gryzyncski, and J. R. Lakowicz, "DNA hybridization using surface plasmoncoupled emssion," Anal. Chem. 75, 6629-6633 (2003).
    [CrossRef] [PubMed]
  10. J. Borejdo, N. Calander, Z. Gryzyncski, and I. Gryzyncski, "Fluorescence correlation spectroscopy in surface plasmon coupled emission microscope," Opt. Express 14, 7878-7888 (2006).
    [CrossRef] [PubMed]
  11. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, (Springer-Verlag, 1986)
  12. B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. II. structure of the image field in an aplanatic system," Proc. Roy. Soc. (London) A 253, 358-379 (1959).
    [CrossRef]
  13. E. Wolf, "Electromagnetic diffraction in optical systems. I. an integral representation of the image field. Proc. Roy. Soc. (London) A 253, 349-357 (1959).
    [CrossRef]
  14. P. Torok, "Propagation of electromagnetic dipole waves through dielectric interfaces," Opt. Lett. 25, 1463-1465 (2000).
    [CrossRef]
  15. H. F. Arnoldus and J. T. Foley. "Transmission of dipole radiation through interfaces and the phenomenon of anti-critical angles," J. Opt. Soc. Am. A 21, 1109-1117 (2004).
    [CrossRef]
  16. E. H. Hellen and D. Axelrod, "Fluorescence emission at dielectric and metal-film interfaces," J. Opt. Soc. Am. B 4, 337-349 (1987).
    [CrossRef]
  17. P. Torok, P. Varga, Z. Laczik, and G. R. Booker. "Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation," J. Opt. Soc. Am. A 12, 325-332 (1995).
    [CrossRef]
  18. P. Torok, P. D. Higdon, and T. Wilson, "Theory for confocal and conventional microscopes imaging small dielectric scatterers," J. Mod. Opt. 45, 1681-1698 (1998).
    [CrossRef]
  19. P. Torok and C. J. R. Sheppard, "The role of pinhole size in high-aperture two- and three-photon microscopy," in Confocal and Two-Photon Microscopy: Foundations, Applications, and Advances, Alberto Diaspro, ed. (Wiley- Liss, Inc., New York, 2002), pp. 127-151.
  20. J. Enderlein, and M . Böhmer, "Influence of interface-dipole interactions on the efficiency of fluorescence light collection near surfaces," Opt. Lett. 28, 941-943 (2003).
    [CrossRef] [PubMed]
  21. C. J. R. Sheppard and P. Torok, "An electromagnetic theory of imaging in fluorescence microscopy, and imaging in polarization fluorescence microscopy," Bioimaging 5, 205-218 (1997).
    [CrossRef]
  22. E. D. Palik, Handbook of Optical Constants of Solids, (Academic Press, 1985).
  23. E. Chung, D. Kim, and P. T. So, "Extended resolution wide-field optical imaging: objective-launched standingwave total internal reflection fluorescence microscopy," Opt. Lett. 31, 945-947 (2006).
    [CrossRef] [PubMed]
  24. M. A. A. Neil, R. Juskaitis, and T. Wilson, "Method of obtaining optical sectioning by using structured light in a conventional microscope," Opt. Lett. 22, 1905-1907 (1997).
    [CrossRef]
  25. J. Enderlein, and T. Ruckstuhl, "The efficiency of surface-plasmon coupled emission for sensitive fluorescence detection," Opt. Express 13, 8855-8865 (2005).
    [CrossRef] [PubMed]

2006 (3)

2005 (1)

2004 (3)

H. F. Arnoldus and J. T. Foley. "Transmission of dipole radiation through interfaces and the phenomenon of anti-critical angles," J. Opt. Soc. Am. A 21, 1109-1117 (2004).
[CrossRef]

J. R. Lakowicz, "Radiative decay engineering 3. surface plasmon-coupled directional emission," Anal. Biochem. 324, 153-169 (2004).
[CrossRef]

I. Gryzyncski, J. Malicka, Z. Gryzyncski, and J. R. Lakowicz, "Radiative decay engineering 4. experimental studies of surface-plasmon coupled directional emission," Anal. Biochem. 324, 170-1822004.
[CrossRef]

2003 (3)

J. Malicka, I. Gryzyncski, Z. Gryzyncski, and J. R. Lakowicz, "DNA hybridization using surface plasmoncoupled emssion," Anal. Chem. 75, 6629-6633 (2003).
[CrossRef] [PubMed]

J. R. Lakowicz, "Directional surface plasmon-coupled emission: a new method for high sensitivity detection," Biochem. and Biophys. Res. Comm. 307, 435-439 (2003).
[CrossRef]

J. Enderlein, and M . Böhmer, "Influence of interface-dipole interactions on the efficiency of fluorescence light collection near surfaces," Opt. Lett. 28, 941-943 (2003).
[CrossRef] [PubMed]

2001 (1)

D. Axelrod, "Total internal reflection microscopy in cell biology," Traffic 2, 764-774 (2001).
[CrossRef] [PubMed]

2000 (2)

S. E. Sund, and D. Axelrod, "Actin dynamics at the living cell submembrane imaged by total internal reflection fluorescence photobleaching," Biophys. J. 79, 1655-1669 (2000).
[CrossRef] [PubMed]

P. Torok, "Propagation of electromagnetic dipole waves through dielectric interfaces," Opt. Lett. 25, 1463-1465 (2000).
[CrossRef]

1998 (2)

E. L. Schmid, A. Tairi, R. Hovius, and H. Vogel, "Screening ligands for membrane protein receptors by total internal reflection fluorescence: The 5-HT3 serotonin receptor," Anal. Chem. 70, 1331-1338 (1998).
[CrossRef] [PubMed]

P. Torok, P. D. Higdon, and T. Wilson, "Theory for confocal and conventional microscopes imaging small dielectric scatterers," J. Mod. Opt. 45, 1681-1698 (1998).
[CrossRef]

1997 (3)

C. J. R. Sheppard and P. Torok, "An electromagnetic theory of imaging in fluorescence microscopy, and imaging in polarization fluorescence microscopy," Bioimaging 5, 205-218 (1997).
[CrossRef]

M. A. A. Neil, R. Juskaitis, and T. Wilson, "Method of obtaining optical sectioning by using structured light in a conventional microscope," Opt. Lett. 22, 1905-1907 (1997).
[CrossRef]

J. A. Steyer, H. Horstmann, and W. Almers, "Transport, docking and exocytosis of single secretory granules in live chromaffin cells," Nature 388, 474-478 (1997).
[CrossRef] [PubMed]

1995 (1)

1987 (1)

Almers, W.

J. A. Steyer, H. Horstmann, and W. Almers, "Transport, docking and exocytosis of single secretory granules in live chromaffin cells," Nature 388, 474-478 (1997).
[CrossRef] [PubMed]

Arnoldus, H. F.

Axelrod, D.

D. Axelrod, "Total internal reflection microscopy in cell biology," Traffic 2, 764-774 (2001).
[CrossRef] [PubMed]

S. E. Sund, and D. Axelrod, "Actin dynamics at the living cell submembrane imaged by total internal reflection fluorescence photobleaching," Biophys. J. 79, 1655-1669 (2000).
[CrossRef] [PubMed]

E. H. Hellen and D. Axelrod, "Fluorescence emission at dielectric and metal-film interfaces," J. Opt. Soc. Am. B 4, 337-349 (1987).
[CrossRef]

Böhmer, M

Booker, G. R.

Borejdo, J.

J. Borejdo, N. Calander, Z. Gryzyncski, and I. Gryzyncski, "Fluorescence correlation spectroscopy in surface plasmon coupled emission microscope," Opt. Express 14, 7878-7888 (2006).
[CrossRef] [PubMed]

J. Borejdo, Z. Gryzyncski, N. Calander, P. Muthu, and I. Gryzyncski, "Application of surface plasmon coupled emission to study of muscle," Biophys. J. 91, 2626-2635 (2006).
[CrossRef] [PubMed]

Calander, N.

J. Borejdo, Z. Gryzyncski, N. Calander, P. Muthu, and I. Gryzyncski, "Application of surface plasmon coupled emission to study of muscle," Biophys. J. 91, 2626-2635 (2006).
[CrossRef] [PubMed]

J. Borejdo, N. Calander, Z. Gryzyncski, and I. Gryzyncski, "Fluorescence correlation spectroscopy in surface plasmon coupled emission microscope," Opt. Express 14, 7878-7888 (2006).
[CrossRef] [PubMed]

Chung, E.

Enderlein, J.

Foley, J. T.

Gryzyncski, I.

J. Borejdo, N. Calander, Z. Gryzyncski, and I. Gryzyncski, "Fluorescence correlation spectroscopy in surface plasmon coupled emission microscope," Opt. Express 14, 7878-7888 (2006).
[CrossRef] [PubMed]

J. Borejdo, Z. Gryzyncski, N. Calander, P. Muthu, and I. Gryzyncski, "Application of surface plasmon coupled emission to study of muscle," Biophys. J. 91, 2626-2635 (2006).
[CrossRef] [PubMed]

I. Gryzyncski, J. Malicka, Z. Gryzyncski, and J. R. Lakowicz, "Radiative decay engineering 4. experimental studies of surface-plasmon coupled directional emission," Anal. Biochem. 324, 170-1822004.
[CrossRef]

J. Malicka, I. Gryzyncski, Z. Gryzyncski, and J. R. Lakowicz, "DNA hybridization using surface plasmoncoupled emssion," Anal. Chem. 75, 6629-6633 (2003).
[CrossRef] [PubMed]

Gryzyncski, Z.

J. Borejdo, N. Calander, Z. Gryzyncski, and I. Gryzyncski, "Fluorescence correlation spectroscopy in surface plasmon coupled emission microscope," Opt. Express 14, 7878-7888 (2006).
[CrossRef] [PubMed]

J. Borejdo, Z. Gryzyncski, N. Calander, P. Muthu, and I. Gryzyncski, "Application of surface plasmon coupled emission to study of muscle," Biophys. J. 91, 2626-2635 (2006).
[CrossRef] [PubMed]

I. Gryzyncski, J. Malicka, Z. Gryzyncski, and J. R. Lakowicz, "Radiative decay engineering 4. experimental studies of surface-plasmon coupled directional emission," Anal. Biochem. 324, 170-1822004.
[CrossRef]

J. Malicka, I. Gryzyncski, Z. Gryzyncski, and J. R. Lakowicz, "DNA hybridization using surface plasmoncoupled emssion," Anal. Chem. 75, 6629-6633 (2003).
[CrossRef] [PubMed]

Hellen, E. H.

Higdon, P. D.

P. Torok, P. D. Higdon, and T. Wilson, "Theory for confocal and conventional microscopes imaging small dielectric scatterers," J. Mod. Opt. 45, 1681-1698 (1998).
[CrossRef]

Horstmann, H.

J. A. Steyer, H. Horstmann, and W. Almers, "Transport, docking and exocytosis of single secretory granules in live chromaffin cells," Nature 388, 474-478 (1997).
[CrossRef] [PubMed]

Hovius, R.

E. L. Schmid, A. Tairi, R. Hovius, and H. Vogel, "Screening ligands for membrane protein receptors by total internal reflection fluorescence: The 5-HT3 serotonin receptor," Anal. Chem. 70, 1331-1338 (1998).
[CrossRef] [PubMed]

Juskaitis, R.

Kim, D.

Laczik, Z.

Lakowicz, J. R.

J. R. Lakowicz, "Radiative decay engineering 3. surface plasmon-coupled directional emission," Anal. Biochem. 324, 153-169 (2004).
[CrossRef]

I. Gryzyncski, J. Malicka, Z. Gryzyncski, and J. R. Lakowicz, "Radiative decay engineering 4. experimental studies of surface-plasmon coupled directional emission," Anal. Biochem. 324, 170-1822004.
[CrossRef]

J. Malicka, I. Gryzyncski, Z. Gryzyncski, and J. R. Lakowicz, "DNA hybridization using surface plasmoncoupled emssion," Anal. Chem. 75, 6629-6633 (2003).
[CrossRef] [PubMed]

J. R. Lakowicz, "Directional surface plasmon-coupled emission: a new method for high sensitivity detection," Biochem. and Biophys. Res. Comm. 307, 435-439 (2003).
[CrossRef]

Malicka, J.

I. Gryzyncski, J. Malicka, Z. Gryzyncski, and J. R. Lakowicz, "Radiative decay engineering 4. experimental studies of surface-plasmon coupled directional emission," Anal. Biochem. 324, 170-1822004.
[CrossRef]

J. Malicka, I. Gryzyncski, Z. Gryzyncski, and J. R. Lakowicz, "DNA hybridization using surface plasmoncoupled emssion," Anal. Chem. 75, 6629-6633 (2003).
[CrossRef] [PubMed]

Muthu, P.

J. Borejdo, Z. Gryzyncski, N. Calander, P. Muthu, and I. Gryzyncski, "Application of surface plasmon coupled emission to study of muscle," Biophys. J. 91, 2626-2635 (2006).
[CrossRef] [PubMed]

Neil, M. A. A.

Ruckstuhl, T.

Schmid, E. L.

E. L. Schmid, A. Tairi, R. Hovius, and H. Vogel, "Screening ligands for membrane protein receptors by total internal reflection fluorescence: The 5-HT3 serotonin receptor," Anal. Chem. 70, 1331-1338 (1998).
[CrossRef] [PubMed]

Sheppard, C. J. R.

C. J. R. Sheppard and P. Torok, "An electromagnetic theory of imaging in fluorescence microscopy, and imaging in polarization fluorescence microscopy," Bioimaging 5, 205-218 (1997).
[CrossRef]

So, P. T.

Steyer, J. A.

J. A. Steyer, H. Horstmann, and W. Almers, "Transport, docking and exocytosis of single secretory granules in live chromaffin cells," Nature 388, 474-478 (1997).
[CrossRef] [PubMed]

Sund, S. E.

S. E. Sund, and D. Axelrod, "Actin dynamics at the living cell submembrane imaged by total internal reflection fluorescence photobleaching," Biophys. J. 79, 1655-1669 (2000).
[CrossRef] [PubMed]

Tairi, A.

E. L. Schmid, A. Tairi, R. Hovius, and H. Vogel, "Screening ligands for membrane protein receptors by total internal reflection fluorescence: The 5-HT3 serotonin receptor," Anal. Chem. 70, 1331-1338 (1998).
[CrossRef] [PubMed]

Torok, P.

P. Torok, "Propagation of electromagnetic dipole waves through dielectric interfaces," Opt. Lett. 25, 1463-1465 (2000).
[CrossRef]

P. Torok, P. D. Higdon, and T. Wilson, "Theory for confocal and conventional microscopes imaging small dielectric scatterers," J. Mod. Opt. 45, 1681-1698 (1998).
[CrossRef]

C. J. R. Sheppard and P. Torok, "An electromagnetic theory of imaging in fluorescence microscopy, and imaging in polarization fluorescence microscopy," Bioimaging 5, 205-218 (1997).
[CrossRef]

P. Torok, P. Varga, Z. Laczik, and G. R. Booker. "Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation," J. Opt. Soc. Am. A 12, 325-332 (1995).
[CrossRef]

Varga, P.

Vogel, H.

E. L. Schmid, A. Tairi, R. Hovius, and H. Vogel, "Screening ligands for membrane protein receptors by total internal reflection fluorescence: The 5-HT3 serotonin receptor," Anal. Chem. 70, 1331-1338 (1998).
[CrossRef] [PubMed]

Wilson, T.

P. Torok, P. D. Higdon, and T. Wilson, "Theory for confocal and conventional microscopes imaging small dielectric scatterers," J. Mod. Opt. 45, 1681-1698 (1998).
[CrossRef]

M. A. A. Neil, R. Juskaitis, and T. Wilson, "Method of obtaining optical sectioning by using structured light in a conventional microscope," Opt. Lett. 22, 1905-1907 (1997).
[CrossRef]

Anal. Biochem. (2)

J. R. Lakowicz, "Radiative decay engineering 3. surface plasmon-coupled directional emission," Anal. Biochem. 324, 153-169 (2004).
[CrossRef]

I. Gryzyncski, J. Malicka, Z. Gryzyncski, and J. R. Lakowicz, "Radiative decay engineering 4. experimental studies of surface-plasmon coupled directional emission," Anal. Biochem. 324, 170-1822004.
[CrossRef]

Anal. Chem. (2)

J. Malicka, I. Gryzyncski, Z. Gryzyncski, and J. R. Lakowicz, "DNA hybridization using surface plasmoncoupled emssion," Anal. Chem. 75, 6629-6633 (2003).
[CrossRef] [PubMed]

E. L. Schmid, A. Tairi, R. Hovius, and H. Vogel, "Screening ligands for membrane protein receptors by total internal reflection fluorescence: The 5-HT3 serotonin receptor," Anal. Chem. 70, 1331-1338 (1998).
[CrossRef] [PubMed]

Biochem. and Biophys. Res. Comm. (1)

J. R. Lakowicz, "Directional surface plasmon-coupled emission: a new method for high sensitivity detection," Biochem. and Biophys. Res. Comm. 307, 435-439 (2003).
[CrossRef]

Bioimaging (1)

C. J. R. Sheppard and P. Torok, "An electromagnetic theory of imaging in fluorescence microscopy, and imaging in polarization fluorescence microscopy," Bioimaging 5, 205-218 (1997).
[CrossRef]

Biophys. J. (2)

J. Borejdo, Z. Gryzyncski, N. Calander, P. Muthu, and I. Gryzyncski, "Application of surface plasmon coupled emission to study of muscle," Biophys. J. 91, 2626-2635 (2006).
[CrossRef] [PubMed]

S. E. Sund, and D. Axelrod, "Actin dynamics at the living cell submembrane imaged by total internal reflection fluorescence photobleaching," Biophys. J. 79, 1655-1669 (2000).
[CrossRef] [PubMed]

J. Mod. Opt. (1)

P. Torok, P. D. Higdon, and T. Wilson, "Theory for confocal and conventional microscopes imaging small dielectric scatterers," J. Mod. Opt. 45, 1681-1698 (1998).
[CrossRef]

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

J. Opt. Soc. Am. B (1)

Nature (1)

J. A. Steyer, H. Horstmann, and W. Almers, "Transport, docking and exocytosis of single secretory granules in live chromaffin cells," Nature 388, 474-478 (1997).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (4)

Traffic (1)

D. Axelrod, "Total internal reflection microscopy in cell biology," Traffic 2, 764-774 (2001).
[CrossRef] [PubMed]

Other (5)

P. Torok and C. J. R. Sheppard, "The role of pinhole size in high-aperture two- and three-photon microscopy," in Confocal and Two-Photon Microscopy: Foundations, Applications, and Advances, Alberto Diaspro, ed. (Wiley- Liss, Inc., New York, 2002), pp. 127-151.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, (Springer-Verlag, 1986)

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. II. structure of the image field in an aplanatic system," Proc. Roy. Soc. (London) A 253, 358-379 (1959).
[CrossRef]

E. Wolf, "Electromagnetic diffraction in optical systems. I. an integral representation of the image field. Proc. Roy. Soc. (London) A 253, 349-357 (1959).
[CrossRef]

E. D. Palik, Handbook of Optical Constants of Solids, (Academic Press, 1985).

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 (8)

Fig. 1.
Fig. 1.

Ex citation of dipole by a p-polarized incident plane wave.

Fig. 2.
Fig. 2.

A schematic view of the SPCEM imaging process within a 4f optical system.

Fig. 3.
Fig. 3.

Axis convention used in the derivation of the field in medium 3.

Fig. 4.
Fig. 4.

Absolute of the electric field components on the image plane for a dipole which is oriented perpendicular to the metal interface. (a) Ex component (b) Ey component (c) Ez component

Fig. 5.
Fig. 5.

Absolute of the electric field components on the image plane for a dipole which is oriented parallel to the metal interface in the x direction. (a) Ex component (b) Ey component (c) Ez component

Fig. 6.
Fig. 6.

Ex perimental setup used to obtain the point spread function of SPCEM by imaging fluorescent beads of diameter below the diffraction limit.

Fig. 7.
Fig. 7.

A comparison of the theoretical and experimental point spread function. (a) Calculated point spread function of SPCEM (b) Actual point spread function of SPCEM obtained from experiments (c) Theoretical PSF after adding a linear polarizer between the objective and the tube lens (d) Ex perimental PSF after adding a linear polarizer to the relay optics

Fig. 8.
Fig. 8.

A comparison of the cross-sectional profile of the theoretical and experimental results. The calculated PSFs are shown in smooth lines and the experimental PSFs are shown as dashed lines. (a) Without linear polarizer (b) With linear polarizer

Tables (1)

Tables Icon

Table 1. Values used for numerical simulation.

Equations (36)

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

E i = n 3 n 1 τ p , inc ( i sin 2 θ i ( n 1 n 3 ) 2 e x + sin θ i e z )
× exp ( ik 3 , inc x sin θ i ) exp ( k 3 , inc z sin 2 θ i ( n 1 n 3 ) 2 )
τ p , inc = t p 32 t p 21 exp ( ik 2 t cos θ 2 ) 1 + t p 32 t p 21 exp ( 2 ik 2 t cos θ 2 )
E 4 ( r , φ , z ) = ik 4 2 π Ω E ´ 4 sin θ 4 exp ( ik 4 r sin θ 4 cos ( ϕ φ ) )
× exp ( ik 4 z cos θ 4 ) exp ( i Φ ) d θ 4 d ϕ
E 1 ( r ) = i 2 π n 1 2 d 2 k 1 k 0 v 1 [ k 1 2 p ( p k 1 ) k 1 ] exp [ i k 1 ( r + d e z ) ]
k 1 = { k + k 0 v 1 e z for z > d k k 0 v 1 e z for z < d
e ρ = k k , e s = e z × e ρ , e z = e ρ × e s
e 1 , p = k 1 k 1 × e s , e 3 , p = k 3 k 3 × e s
k 3 = k + k 0 v 3 e z .
E 3 ( r ) = i k 0 2 π d 2 k 1 v 1 E 3 exp [ i k 3 ( r t e z ) ] exp [ ik 0 v 1 d ]
E 3 = [ E 3 , ρ E ´ 3 , s E ´ 3 , z ] = [ v 3 n 1 n 3 τ p ( v 1 p e ρ + αp e z ) τ s p e s α n 1 n 3 τ p ( v 1 p e ρ + αp e z ) ]
τ q = t q 12 t q 23 exp ( ik 2 t cos θ 2 ) 1 + t q 12 t q 23 exp ( 2 ik 2 t cos θ 2 ) , q = p , s .
Φ = k 0 v 1 d k 0 v 3 d ( n 3 n 1 ) k 0 v 3 t ( n 3 n 2 ) .
E 4 = [ E 4 , r E 4 , φ E 4 , z ] = A ( θ 3 , θ 4 ) ML 4 L 3 E 3
A ( θ 3 , θ 4 ) = cos θ 4 cos θ 3 ,
L n = [ cos θ n 0 sin θ n 0 1 0 sin θ n 0 cos θ n ] ,
M = [ cos ( φ ϕ ) sin ( φ ϕ ) 0 sin ( φ ϕ ) cos ( φ ϕ ) 0 0 0 1 ] ,
E 4 ( r , φ , z ) = [ E 4 , r E 4 , φ E 4 , z ] ,
E 4 , r = ik 4 2 { μ sin θ d cos ( ϕ d φ ) [ K 0 I + K 2 I ] μ cos θ d [ 2 i K 1 I ] }
E 4 , φ = ik 4 2 { μ sin θ d sin ( ϕ d φ ) [ K 0 I K 2 I ] }
E 4 , z = ik 4 2 { μ sin θ d cos ( ϕ d φ ) [ 2 i K 1 II ] μ cos θ d [ 2 i K 0 II ] }
K 0 I = 0 σ cos θ 4 cos θ 3 sin θ 4 ( τ s + τ p cos θ 1 cos θ 4 ) J 0 ( k 4 r sin θ 4 ) × exp ( ik 4 z cos θ 4 ) exp ( i Φ ) d θ 4
K 1 I = 0 σ cos θ 4 cos θ 3 sin θ 4 τ p sin θ 1 cos θ 4 J 1 ( k 4 r sin θ 4 ) × exp ( ik 4 z cos θ 4 ) exp ( i Φ ) d θ 4
K 2 I = 0 σ cos θ 4 cos θ 3 sin θ 4 ( τ s τ p cos θ 1 cos θ 4 ) J 2 ( k 4 r sin θ 4 ) × exp ( ik 4 z cos θ 4 ) exp ( i Φ ) d θ 4
K 1 II = 0 σ cos θ 4 cos θ 3 sin θ 4 τ p sin θ 1 cos θ 4 J 0 ( k 4 r sin θ 4 ) × exp ( ik 4 z cos θ 4 ) exp ( i Φ ) d θ 4
K 1 II = 0 σ cos θ 4 cos θ 3 sin θ 4 τ p cos θ 1 sin θ 4 J 1 ( k 4 r sin θ 4 ) × exp ( ik 4 z cos θ 4 ) exp ( i Φ ) d θ 4
NA of objective n 4 sin σ = mag ,
I ( E 4 , r 2 + E 4 , φ 2 + E 4 , z 2 ) P T
I 4 ( K 1 I 2 + K 0 II 2 ) P T .
I ( K 0 I 2 + K 2 I 2 + 4 cos 2 φ K 1 II 2 + 2 cos 2 φRe [ K 0 I K 2 I * ] ) P T .
I 1 0 π 0 2 π E i 2 I sin θ d d d .
I 2 0 π 0 2 π p E i 2 I sin θ d d d
E 4 = A ( θ 3 , θ 4 ) ML 4 R 1 PRL 3 E 3
R = [ cos ϕ sin ϕ 0 sin ϕ cosϕ 0 0 0 1 ]
P x = [ 1 0 0 0 0 0 0 0 1 ] , P y = [ 0 0 0 0 1 0 0 0 1 ]

Metrics