Abstract

The theoretical basis for resolution enhancement in standing-wave total internal reflection microscopy (SW-TIRM) is examined. This technique relies on the formation of an excitation field containing super-diffraction-limited spatial-frequency components. Although the fluorescence generated at the object planes contains high-frequency information of the object distribution, this information is lost at the image plane, where the detection optics acts as a low-pass filter. From the perspective of point-spread-function (PSF) engineering, one can show that if this excitation field is translatable experimentally, the high-frequency information can be extracted from a set of images where the excitation fields have different displacement vectors. We have developed algorithms to combine this image set to generate a composite image with an effective PSF that is equal to the product of the excitation field and the Fraunhofer PSF. This approach can easily be extended to incorporate nonlinear excitation modalities into SW-TIRM for further resolution improvement. We theoretically examine high-resolution imaging based on the addition of two-photon, pump–probe, and stimulated-emission depletion methods to SW-TIRM and show that resolution better than 1/20 of the emission wavelength may be achievable.

© 2001 Optical Society of America

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    [CrossRef]
  2. G. Binning, H. Rohrer, “Scanning tunneling microscopy,” IBM J. Res. Dev. 30, 355 (1986).
  3. E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, R. L. Kostalak, “Breaking the diffraction barrier: optical microscopy on a nanometric scale,” Science 251, 1469–1470 (1991).
    [CrossRef]
  4. R. C. Dunn, G. R. Holtom, L. Mets, X. S. Xie, “Near-field fluorescence imaging and fluorescence lifetime measurement of light harvesting complexes in intact photosynthetic membranes,” J. Phys. Chem. 98, 3094–3098 (1994).
    [CrossRef]
  5. V. Subramaniam, A. K. Kirsch, T. M. Jovin, “Cell biological applications of scanning near-field optical microscopy (SNOM),” Cell Mol. Biol. 44, 689–700 (1998).
    [PubMed]
  6. A. K. Kirsch, V. Subramaniam, G. Striker, C. Schnetter, D. J. Arndt-Jovin, T. M. Jovin, “Continuous wave two-photon scanning near-field optical microscopy,” Biophys. J. 75, 1513–1521 (1998).
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  13. P. E. Haenninen, S. W. Hell, J. Salo, E. Soini, “Two-photon excitation 4Pi confocal microscope: enhanced axial resolution microscope for biological research,” Appl. Phys. Lett. 66, 1698 (1995).
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  14. M. Schrader, M. Kozubek, S. W. Hell, T. Wilson, “Opti-cal transfer functions of 4Pi confocal microscopes: theory and experiment,” Opt. Lett. 22, 436–438 (1997).
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  16. W. J. Denk, D. W. Piston, W. W. Webb, “Two-photon molecular excitation laser-scanning microscopy,” in Handbook of Biological Confocal Microscopy, J. B. Pawley, ed. (Plenum, New York, 1995), pp. 445–458.
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    [CrossRef]
  18. C. Y. Dong, P. T. So, T. French, E. Gratton, “Fluorescence lifetime imaging by asynchronous pump-probe microscopy,” Biophys. J. 69, 2234–2242 (1995).
    [CrossRef] [PubMed]
  19. C. Y. Dong, P. T. C. So, Ch. Buehler, E. Gratton, “Spatial resolution in scanning pump-probe fluorescence microscopy,” Optik 106, 7–14 (1997).
  20. Ch. Buehler, C. Y. Dong, P. T. C. So, T. French, E. Gratton, “Time-resolved polarization imaging by pump-probe (stimulated emission) fluorescence microscopy,” Biophys. J. 79, 536–549 (2000).
    [CrossRef] [PubMed]
  21. S. W. Hell, J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Opt. Lett. 19, 780–782 (1994).
    [CrossRef] [PubMed]
  22. T. A. Klar, S. W. Hell, “Subdiffraction resolution in far-field fluorescence microscopy,” Opt. Lett. 24, 954–956 (1999).
    [CrossRef]
  23. M. Dyba, T. A. Klar, S. Jakobs, S. W. Hell, “Ultrafast dynamics microscopy,” Appl. Phys. Lett. 77, 597–599 (2000).
    [CrossRef]
  24. G. Cragg, P. T. C. So, “Standing-wave total internal reflection microscopy,” Opt. Lett. 25, 46–48 (2000).
    [CrossRef]
  25. J. T. Frohn, H. F. Knapp, A. Stemmer, “True optical resolution beyond the Rayleigh limit achieved by standing wave illumination,” Proc. Natl. Acad. Sci. USA 97, 7232–7236 (2000).
    [CrossRef] [PubMed]
  26. B. Bailey, D. L. Farkas, D. L. Taylor, F. Lanni, “Enhancement of axial resolution in fluorescence microscopy by standing-wave excitation,” Nature 366, 44–48 (1993).
    [CrossRef] [PubMed]
  27. V. Krishnamurthi, B. Bailey, F. Lanni, “Imaging processing in 3-D standing-wave fluroescence microscopy,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. Kino, T. Wilson, eds., Proc. SPIE2655, 18–25 (1996).
    [CrossRef]
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  29. P. R. Bevington, D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1991).
  30. C. Xu, J. Guild, W. W. Webb, W. Denk, “Determination of absolute two-photon excitation cross sections by in situ second-order autocorrelation,” Opt. Lett. 20, 2372–2374 (1995).
    [CrossRef] [PubMed]
  31. C. Xu, W. Zipfel, J. B. Shear, R. M. Williams, W. W. Webb, “Multiphoton fluorescence excitation: new spectral windows for biological nonlinear microscopy,” Proc. Natl. Acad. Sci. USA 93, 10763–10768 (1996).
    [CrossRef] [PubMed]
  32. P. T. C. So, C. Y. Dong, K. M. Berland, T. French, E. Gratton, “Time-resolved stimulated-emission and absorption microscopy,” in Topics in Fluorescence V, J. R. Lakowicz, ed., (Plenum, New York, 1998), 427–469.
  33. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. R. Flannery, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK1992).

2000 (6)

P. T. C. So, C. Y. Dong, B. R. Masters, K. M. Berland, “Two-photon excitation fluorescence microscopy,” Annu. Rev. Biomed. Eng. 2, 399–429 (2000).
[CrossRef]

Ch. Buehler, C. Y. Dong, P. T. C. So, T. French, E. Gratton, “Time-resolved polarization imaging by pump-probe (stimulated emission) fluorescence microscopy,” Biophys. J. 79, 536–549 (2000).
[CrossRef] [PubMed]

M. Dyba, T. A. Klar, S. Jakobs, S. W. Hell, “Ultrafast dynamics microscopy,” Appl. Phys. Lett. 77, 597–599 (2000).
[CrossRef]

G. Cragg, P. T. C. So, “Standing-wave total internal reflection microscopy,” Opt. Lett. 25, 46–48 (2000).
[CrossRef]

J. T. Frohn, H. F. Knapp, A. Stemmer, “True optical resolution beyond the Rayleigh limit achieved by standing wave illumination,” Proc. Natl. Acad. Sci. USA 97, 7232–7236 (2000).
[CrossRef] [PubMed]

G. E. Cragg, P. T. C. So, “Standing wave total internal reflection microscopy—breaking the diffraction resolution limit,” Biophys. J. 78, 248a (2000).

1999 (1)

1998 (2)

V. Subramaniam, A. K. Kirsch, T. M. Jovin, “Cell biological applications of scanning near-field optical microscopy (SNOM),” Cell Mol. Biol. 44, 689–700 (1998).
[PubMed]

A. K. Kirsch, V. Subramaniam, G. Striker, C. Schnetter, D. J. Arndt-Jovin, T. M. Jovin, “Continuous wave two-photon scanning near-field optical microscopy,” Biophys. J. 75, 1513–1521 (1998).
[CrossRef] [PubMed]

1997 (2)

C. Y. Dong, P. T. C. So, Ch. Buehler, E. Gratton, “Spatial resolution in scanning pump-probe fluorescence microscopy,” Optik 106, 7–14 (1997).

M. Schrader, M. Kozubek, S. W. Hell, T. Wilson, “Opti-cal transfer functions of 4Pi confocal microscopes: theory and experiment,” Opt. Lett. 22, 436–438 (1997).
[CrossRef] [PubMed]

1996 (1)

C. Xu, W. Zipfel, J. B. Shear, R. M. Williams, W. W. Webb, “Multiphoton fluorescence excitation: new spectral windows for biological nonlinear microscopy,” Proc. Natl. Acad. Sci. USA 93, 10763–10768 (1996).
[CrossRef] [PubMed]

1995 (3)

C. Xu, J. Guild, W. W. Webb, W. Denk, “Determination of absolute two-photon excitation cross sections by in situ second-order autocorrelation,” Opt. Lett. 20, 2372–2374 (1995).
[CrossRef] [PubMed]

C. Y. Dong, P. T. So, T. French, E. Gratton, “Fluorescence lifetime imaging by asynchronous pump-probe microscopy,” Biophys. J. 69, 2234–2242 (1995).
[CrossRef] [PubMed]

P. E. Haenninen, S. W. Hell, J. Salo, E. Soini, “Two-photon excitation 4Pi confocal microscope: enhanced axial resolution microscope for biological research,” Appl. Phys. Lett. 66, 1698 (1995).
[CrossRef]

1994 (3)

S. Hell, W. S. Lindek, E. H. K. Stelzer, “Enhancing the axial resolution in far-field light microscopy: two-photon 4Pi confocal fluorescence microscopy,” J. Mod. Opt. 41, 675–681 (1994).
[CrossRef]

R. C. Dunn, G. R. Holtom, L. Mets, X. S. Xie, “Near-field fluorescence imaging and fluorescence lifetime measurement of light harvesting complexes in intact photosynthetic membranes,” J. Phys. Chem. 98, 3094–3098 (1994).
[CrossRef]

S. W. Hell, J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Opt. Lett. 19, 780–782 (1994).
[CrossRef] [PubMed]

1993 (1)

B. Bailey, D. L. Farkas, D. L. Taylor, F. Lanni, “Enhancement of axial resolution in fluorescence microscopy by standing-wave excitation,” Nature 366, 44–48 (1993).
[CrossRef] [PubMed]

1992 (1)

1991 (1)

E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, R. L. Kostalak, “Breaking the diffraction barrier: optical microscopy on a nanometric scale,” Science 251, 1469–1470 (1991).
[CrossRef]

1990 (1)

W. Denk, J. H. Strickler, W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

1986 (2)

G. Binning, C. F. Quante, C. Gerber, “Atomic force microscope,” Phys. Rev. Lett. 12, 930–933 (1986).
[CrossRef]

G. Binning, H. Rohrer, “Scanning tunneling microscopy,” IBM J. Res. Dev. 30, 355 (1986).

Arndt-Jovin, D. J.

A. K. Kirsch, V. Subramaniam, G. Striker, C. Schnetter, D. J. Arndt-Jovin, T. M. Jovin, “Continuous wave two-photon scanning near-field optical microscopy,” Biophys. J. 75, 1513–1521 (1998).
[CrossRef] [PubMed]

Bailey, B.

B. Bailey, D. L. Farkas, D. L. Taylor, F. Lanni, “Enhancement of axial resolution in fluorescence microscopy by standing-wave excitation,” Nature 366, 44–48 (1993).
[CrossRef] [PubMed]

V. Krishnamurthi, B. Bailey, F. Lanni, “Imaging processing in 3-D standing-wave fluroescence microscopy,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. Kino, T. Wilson, eds., Proc. SPIE2655, 18–25 (1996).
[CrossRef]

Berland, K. M.

P. T. C. So, C. Y. Dong, B. R. Masters, K. M. Berland, “Two-photon excitation fluorescence microscopy,” Annu. Rev. Biomed. Eng. 2, 399–429 (2000).
[CrossRef]

P. T. C. So, C. Y. Dong, K. M. Berland, T. French, E. Gratton, “Time-resolved stimulated-emission and absorption microscopy,” in Topics in Fluorescence V, J. R. Lakowicz, ed., (Plenum, New York, 1998), 427–469.

Betzig, E.

E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, R. L. Kostalak, “Breaking the diffraction barrier: optical microscopy on a nanometric scale,” Science 251, 1469–1470 (1991).
[CrossRef]

Bevington, P. R.

P. R. Bevington, D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1991).

Binning, G.

G. Binning, C. F. Quante, C. Gerber, “Atomic force microscope,” Phys. Rev. Lett. 12, 930–933 (1986).
[CrossRef]

G. Binning, H. Rohrer, “Scanning tunneling microscopy,” IBM J. Res. Dev. 30, 355 (1986).

Buehler, Ch.

Ch. Buehler, C. Y. Dong, P. T. C. So, T. French, E. Gratton, “Time-resolved polarization imaging by pump-probe (stimulated emission) fluorescence microscopy,” Biophys. J. 79, 536–549 (2000).
[CrossRef] [PubMed]

C. Y. Dong, P. T. C. So, Ch. Buehler, E. Gratton, “Spatial resolution in scanning pump-probe fluorescence microscopy,” Optik 106, 7–14 (1997).

Cragg, G.

Cragg, G. E.

G. E. Cragg, P. T. C. So, “Standing wave total internal reflection microscopy—breaking the diffraction resolution limit,” Biophys. J. 78, 248a (2000).

Denk, W.

Denk, W. J.

W. J. Denk, D. W. Piston, W. W. Webb, “Two-photon molecular excitation laser-scanning microscopy,” in Handbook of Biological Confocal Microscopy, J. B. Pawley, ed. (Plenum, New York, 1995), pp. 445–458.

Dong, C. Y.

P. T. C. So, C. Y. Dong, B. R. Masters, K. M. Berland, “Two-photon excitation fluorescence microscopy,” Annu. Rev. Biomed. Eng. 2, 399–429 (2000).
[CrossRef]

Ch. Buehler, C. Y. Dong, P. T. C. So, T. French, E. Gratton, “Time-resolved polarization imaging by pump-probe (stimulated emission) fluorescence microscopy,” Biophys. J. 79, 536–549 (2000).
[CrossRef] [PubMed]

C. Y. Dong, P. T. C. So, Ch. Buehler, E. Gratton, “Spatial resolution in scanning pump-probe fluorescence microscopy,” Optik 106, 7–14 (1997).

C. Y. Dong, P. T. So, T. French, E. Gratton, “Fluorescence lifetime imaging by asynchronous pump-probe microscopy,” Biophys. J. 69, 2234–2242 (1995).
[CrossRef] [PubMed]

P. T. C. So, C. Y. Dong, K. M. Berland, T. French, E. Gratton, “Time-resolved stimulated-emission and absorption microscopy,” in Topics in Fluorescence V, J. R. Lakowicz, ed., (Plenum, New York, 1998), 427–469.

Dunn, R. C.

R. C. Dunn, G. R. Holtom, L. Mets, X. S. Xie, “Near-field fluorescence imaging and fluorescence lifetime measurement of light harvesting complexes in intact photosynthetic membranes,” J. Phys. Chem. 98, 3094–3098 (1994).
[CrossRef]

Dyba, M.

M. Dyba, T. A. Klar, S. Jakobs, S. W. Hell, “Ultrafast dynamics microscopy,” Appl. Phys. Lett. 77, 597–599 (2000).
[CrossRef]

Farkas, D. L.

B. Bailey, D. L. Farkas, D. L. Taylor, F. Lanni, “Enhancement of axial resolution in fluorescence microscopy by standing-wave excitation,” Nature 366, 44–48 (1993).
[CrossRef] [PubMed]

Flannery, B. R.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. R. Flannery, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK1992).

French, T.

Ch. Buehler, C. Y. Dong, P. T. C. So, T. French, E. Gratton, “Time-resolved polarization imaging by pump-probe (stimulated emission) fluorescence microscopy,” Biophys. J. 79, 536–549 (2000).
[CrossRef] [PubMed]

C. Y. Dong, P. T. So, T. French, E. Gratton, “Fluorescence lifetime imaging by asynchronous pump-probe microscopy,” Biophys. J. 69, 2234–2242 (1995).
[CrossRef] [PubMed]

P. T. C. So, C. Y. Dong, K. M. Berland, T. French, E. Gratton, “Time-resolved stimulated-emission and absorption microscopy,” in Topics in Fluorescence V, J. R. Lakowicz, ed., (Plenum, New York, 1998), 427–469.

Frohn, J. T.

J. T. Frohn, H. F. Knapp, A. Stemmer, “True optical resolution beyond the Rayleigh limit achieved by standing wave illumination,” Proc. Natl. Acad. Sci. USA 97, 7232–7236 (2000).
[CrossRef] [PubMed]

Gerber, C.

G. Binning, C. F. Quante, C. Gerber, “Atomic force microscope,” Phys. Rev. Lett. 12, 930–933 (1986).
[CrossRef]

Gratton, E.

Ch. Buehler, C. Y. Dong, P. T. C. So, T. French, E. Gratton, “Time-resolved polarization imaging by pump-probe (stimulated emission) fluorescence microscopy,” Biophys. J. 79, 536–549 (2000).
[CrossRef] [PubMed]

C. Y. Dong, P. T. C. So, Ch. Buehler, E. Gratton, “Spatial resolution in scanning pump-probe fluorescence microscopy,” Optik 106, 7–14 (1997).

C. Y. Dong, P. T. So, T. French, E. Gratton, “Fluorescence lifetime imaging by asynchronous pump-probe microscopy,” Biophys. J. 69, 2234–2242 (1995).
[CrossRef] [PubMed]

P. T. C. So, C. Y. Dong, K. M. Berland, T. French, E. Gratton, “Time-resolved stimulated-emission and absorption microscopy,” in Topics in Fluorescence V, J. R. Lakowicz, ed., (Plenum, New York, 1998), 427–469.

Guild, J.

Haenninen, P. E.

P. E. Haenninen, S. W. Hell, J. Salo, E. Soini, “Two-photon excitation 4Pi confocal microscope: enhanced axial resolution microscope for biological research,” Appl. Phys. Lett. 66, 1698 (1995).
[CrossRef]

Harris, T. D.

E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, R. L. Kostalak, “Breaking the diffraction barrier: optical microscopy on a nanometric scale,” Science 251, 1469–1470 (1991).
[CrossRef]

Hell, S.

S. Hell, W. S. Lindek, E. H. K. Stelzer, “Enhancing the axial resolution in far-field light microscopy: two-photon 4Pi confocal fluorescence microscopy,” J. Mod. Opt. 41, 675–681 (1994).
[CrossRef]

S. Hell, E. H. K. Stelzer, “Properties of a 4Pi confocal fluorescence microscope,” J. Opt. Soc. Am. A 9, 2159–2166 (1992).
[CrossRef]

Hell, S. W.

Holtom, G. R.

R. C. Dunn, G. R. Holtom, L. Mets, X. S. Xie, “Near-field fluorescence imaging and fluorescence lifetime measurement of light harvesting complexes in intact photosynthetic membranes,” J. Phys. Chem. 98, 3094–3098 (1994).
[CrossRef]

Jakobs, S.

M. Dyba, T. A. Klar, S. Jakobs, S. W. Hell, “Ultrafast dynamics microscopy,” Appl. Phys. Lett. 77, 597–599 (2000).
[CrossRef]

Jovin, T. M.

A. K. Kirsch, V. Subramaniam, G. Striker, C. Schnetter, D. J. Arndt-Jovin, T. M. Jovin, “Continuous wave two-photon scanning near-field optical microscopy,” Biophys. J. 75, 1513–1521 (1998).
[CrossRef] [PubMed]

V. Subramaniam, A. K. Kirsch, T. M. Jovin, “Cell biological applications of scanning near-field optical microscopy (SNOM),” Cell Mol. Biol. 44, 689–700 (1998).
[PubMed]

Kirsch, A. K.

A. K. Kirsch, V. Subramaniam, G. Striker, C. Schnetter, D. J. Arndt-Jovin, T. M. Jovin, “Continuous wave two-photon scanning near-field optical microscopy,” Biophys. J. 75, 1513–1521 (1998).
[CrossRef] [PubMed]

V. Subramaniam, A. K. Kirsch, T. M. Jovin, “Cell biological applications of scanning near-field optical microscopy (SNOM),” Cell Mol. Biol. 44, 689–700 (1998).
[PubMed]

Klar, T. A.

M. Dyba, T. A. Klar, S. Jakobs, S. W. Hell, “Ultrafast dynamics microscopy,” Appl. Phys. Lett. 77, 597–599 (2000).
[CrossRef]

T. A. Klar, S. W. Hell, “Subdiffraction resolution in far-field fluorescence microscopy,” Opt. Lett. 24, 954–956 (1999).
[CrossRef]

Knapp, H. F.

J. T. Frohn, H. F. Knapp, A. Stemmer, “True optical resolution beyond the Rayleigh limit achieved by standing wave illumination,” Proc. Natl. Acad. Sci. USA 97, 7232–7236 (2000).
[CrossRef] [PubMed]

Kostalak, R. L.

E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, R. L. Kostalak, “Breaking the diffraction barrier: optical microscopy on a nanometric scale,” Science 251, 1469–1470 (1991).
[CrossRef]

Kozubek, M.

Krishnamurthi, V.

V. Krishnamurthi, B. Bailey, F. Lanni, “Imaging processing in 3-D standing-wave fluroescence microscopy,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. Kino, T. Wilson, eds., Proc. SPIE2655, 18–25 (1996).
[CrossRef]

Lanni, F.

B. Bailey, D. L. Farkas, D. L. Taylor, F. Lanni, “Enhancement of axial resolution in fluorescence microscopy by standing-wave excitation,” Nature 366, 44–48 (1993).
[CrossRef] [PubMed]

V. Krishnamurthi, B. Bailey, F. Lanni, “Imaging processing in 3-D standing-wave fluroescence microscopy,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. Kino, T. Wilson, eds., Proc. SPIE2655, 18–25 (1996).
[CrossRef]

Lindek, W. S.

S. Hell, W. S. Lindek, E. H. K. Stelzer, “Enhancing the axial resolution in far-field light microscopy: two-photon 4Pi confocal fluorescence microscopy,” J. Mod. Opt. 41, 675–681 (1994).
[CrossRef]

Masters, B. R.

P. T. C. So, C. Y. Dong, B. R. Masters, K. M. Berland, “Two-photon excitation fluorescence microscopy,” Annu. Rev. Biomed. Eng. 2, 399–429 (2000).
[CrossRef]

B. R. Masters, Selected Papers on Confocal Microscopy (SPIE Press, Bellingham, 1996).

Mets, L.

R. C. Dunn, G. R. Holtom, L. Mets, X. S. Xie, “Near-field fluorescence imaging and fluorescence lifetime measurement of light harvesting complexes in intact photosynthetic membranes,” J. Phys. Chem. 98, 3094–3098 (1994).
[CrossRef]

Piston, D. W.

W. J. Denk, D. W. Piston, W. W. Webb, “Two-photon molecular excitation laser-scanning microscopy,” in Handbook of Biological Confocal Microscopy, J. B. Pawley, ed. (Plenum, New York, 1995), pp. 445–458.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. R. Flannery, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK1992).

Quante, C. F.

G. Binning, C. F. Quante, C. Gerber, “Atomic force microscope,” Phys. Rev. Lett. 12, 930–933 (1986).
[CrossRef]

Robinson, D. K.

P. R. Bevington, D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1991).

Rohrer, H.

G. Binning, H. Rohrer, “Scanning tunneling microscopy,” IBM J. Res. Dev. 30, 355 (1986).

Salo, J.

P. E. Haenninen, S. W. Hell, J. Salo, E. Soini, “Two-photon excitation 4Pi confocal microscope: enhanced axial resolution microscope for biological research,” Appl. Phys. Lett. 66, 1698 (1995).
[CrossRef]

Schnetter, C.

A. K. Kirsch, V. Subramaniam, G. Striker, C. Schnetter, D. J. Arndt-Jovin, T. M. Jovin, “Continuous wave two-photon scanning near-field optical microscopy,” Biophys. J. 75, 1513–1521 (1998).
[CrossRef] [PubMed]

Schrader, M.

Shear, J. B.

C. Xu, W. Zipfel, J. B. Shear, R. M. Williams, W. W. Webb, “Multiphoton fluorescence excitation: new spectral windows for biological nonlinear microscopy,” Proc. Natl. Acad. Sci. USA 93, 10763–10768 (1996).
[CrossRef] [PubMed]

Sheppard, C. J. R.

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984).

So, P. T.

C. Y. Dong, P. T. So, T. French, E. Gratton, “Fluorescence lifetime imaging by asynchronous pump-probe microscopy,” Biophys. J. 69, 2234–2242 (1995).
[CrossRef] [PubMed]

So, P. T. C.

P. T. C. So, C. Y. Dong, B. R. Masters, K. M. Berland, “Two-photon excitation fluorescence microscopy,” Annu. Rev. Biomed. Eng. 2, 399–429 (2000).
[CrossRef]

Ch. Buehler, C. Y. Dong, P. T. C. So, T. French, E. Gratton, “Time-resolved polarization imaging by pump-probe (stimulated emission) fluorescence microscopy,” Biophys. J. 79, 536–549 (2000).
[CrossRef] [PubMed]

G. E. Cragg, P. T. C. So, “Standing wave total internal reflection microscopy—breaking the diffraction resolution limit,” Biophys. J. 78, 248a (2000).

G. Cragg, P. T. C. So, “Standing-wave total internal reflection microscopy,” Opt. Lett. 25, 46–48 (2000).
[CrossRef]

C. Y. Dong, P. T. C. So, Ch. Buehler, E. Gratton, “Spatial resolution in scanning pump-probe fluorescence microscopy,” Optik 106, 7–14 (1997).

P. T. C. So, C. Y. Dong, K. M. Berland, T. French, E. Gratton, “Time-resolved stimulated-emission and absorption microscopy,” in Topics in Fluorescence V, J. R. Lakowicz, ed., (Plenum, New York, 1998), 427–469.

Soini, E.

P. E. Haenninen, S. W. Hell, J. Salo, E. Soini, “Two-photon excitation 4Pi confocal microscope: enhanced axial resolution microscope for biological research,” Appl. Phys. Lett. 66, 1698 (1995).
[CrossRef]

Stelzer, E. H. K.

S. Hell, W. S. Lindek, E. H. K. Stelzer, “Enhancing the axial resolution in far-field light microscopy: two-photon 4Pi confocal fluorescence microscopy,” J. Mod. Opt. 41, 675–681 (1994).
[CrossRef]

S. Hell, E. H. K. Stelzer, “Properties of a 4Pi confocal fluorescence microscope,” J. Opt. Soc. Am. A 9, 2159–2166 (1992).
[CrossRef]

Stemmer, A.

J. T. Frohn, H. F. Knapp, A. Stemmer, “True optical resolution beyond the Rayleigh limit achieved by standing wave illumination,” Proc. Natl. Acad. Sci. USA 97, 7232–7236 (2000).
[CrossRef] [PubMed]

Strickler, J. H.

W. Denk, J. H. Strickler, W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

Striker, G.

A. K. Kirsch, V. Subramaniam, G. Striker, C. Schnetter, D. J. Arndt-Jovin, T. M. Jovin, “Continuous wave two-photon scanning near-field optical microscopy,” Biophys. J. 75, 1513–1521 (1998).
[CrossRef] [PubMed]

Subramaniam, V.

A. K. Kirsch, V. Subramaniam, G. Striker, C. Schnetter, D. J. Arndt-Jovin, T. M. Jovin, “Continuous wave two-photon scanning near-field optical microscopy,” Biophys. J. 75, 1513–1521 (1998).
[CrossRef] [PubMed]

V. Subramaniam, A. K. Kirsch, T. M. Jovin, “Cell biological applications of scanning near-field optical microscopy (SNOM),” Cell Mol. Biol. 44, 689–700 (1998).
[PubMed]

Taylor, D. L.

B. Bailey, D. L. Farkas, D. L. Taylor, F. Lanni, “Enhancement of axial resolution in fluorescence microscopy by standing-wave excitation,” Nature 366, 44–48 (1993).
[CrossRef] [PubMed]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. R. Flannery, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK1992).

Trautman, J. K.

E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, R. L. Kostalak, “Breaking the diffraction barrier: optical microscopy on a nanometric scale,” Science 251, 1469–1470 (1991).
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. R. Flannery, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK1992).

Webb, W. W.

C. Xu, W. Zipfel, J. B. Shear, R. M. Williams, W. W. Webb, “Multiphoton fluorescence excitation: new spectral windows for biological nonlinear microscopy,” Proc. Natl. Acad. Sci. USA 93, 10763–10768 (1996).
[CrossRef] [PubMed]

C. Xu, J. Guild, W. W. Webb, W. Denk, “Determination of absolute two-photon excitation cross sections by in situ second-order autocorrelation,” Opt. Lett. 20, 2372–2374 (1995).
[CrossRef] [PubMed]

W. Denk, J. H. Strickler, W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

W. J. Denk, D. W. Piston, W. W. Webb, “Two-photon molecular excitation laser-scanning microscopy,” in Handbook of Biological Confocal Microscopy, J. B. Pawley, ed. (Plenum, New York, 1995), pp. 445–458.

Weiner, J. S.

E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, R. L. Kostalak, “Breaking the diffraction barrier: optical microscopy on a nanometric scale,” Science 251, 1469–1470 (1991).
[CrossRef]

Wichmann, J.

Williams, R. M.

C. Xu, W. Zipfel, J. B. Shear, R. M. Williams, W. W. Webb, “Multiphoton fluorescence excitation: new spectral windows for biological nonlinear microscopy,” Proc. Natl. Acad. Sci. USA 93, 10763–10768 (1996).
[CrossRef] [PubMed]

Wilson, T.

M. Schrader, M. Kozubek, S. W. Hell, T. Wilson, “Opti-cal transfer functions of 4Pi confocal microscopes: theory and experiment,” Opt. Lett. 22, 436–438 (1997).
[CrossRef] [PubMed]

T. Wilson, Confocal Microscopy (Academic, London, 1990).

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984).

Xie, X. S.

R. C. Dunn, G. R. Holtom, L. Mets, X. S. Xie, “Near-field fluorescence imaging and fluorescence lifetime measurement of light harvesting complexes in intact photosynthetic membranes,” J. Phys. Chem. 98, 3094–3098 (1994).
[CrossRef]

Xu, C.

C. Xu, W. Zipfel, J. B. Shear, R. M. Williams, W. W. Webb, “Multiphoton fluorescence excitation: new spectral windows for biological nonlinear microscopy,” Proc. Natl. Acad. Sci. USA 93, 10763–10768 (1996).
[CrossRef] [PubMed]

C. Xu, J. Guild, W. W. Webb, W. Denk, “Determination of absolute two-photon excitation cross sections by in situ second-order autocorrelation,” Opt. Lett. 20, 2372–2374 (1995).
[CrossRef] [PubMed]

Zipfel, W.

C. Xu, W. Zipfel, J. B. Shear, R. M. Williams, W. W. Webb, “Multiphoton fluorescence excitation: new spectral windows for biological nonlinear microscopy,” Proc. Natl. Acad. Sci. USA 93, 10763–10768 (1996).
[CrossRef] [PubMed]

Annu. Rev. Biomed. Eng. (1)

P. T. C. So, C. Y. Dong, B. R. Masters, K. M. Berland, “Two-photon excitation fluorescence microscopy,” Annu. Rev. Biomed. Eng. 2, 399–429 (2000).
[CrossRef]

Appl. Phys. Lett. (2)

P. E. Haenninen, S. W. Hell, J. Salo, E. Soini, “Two-photon excitation 4Pi confocal microscope: enhanced axial resolution microscope for biological research,” Appl. Phys. Lett. 66, 1698 (1995).
[CrossRef]

M. Dyba, T. A. Klar, S. Jakobs, S. W. Hell, “Ultrafast dynamics microscopy,” Appl. Phys. Lett. 77, 597–599 (2000).
[CrossRef]

Biophys. J. (4)

Ch. Buehler, C. Y. Dong, P. T. C. So, T. French, E. Gratton, “Time-resolved polarization imaging by pump-probe (stimulated emission) fluorescence microscopy,” Biophys. J. 79, 536–549 (2000).
[CrossRef] [PubMed]

G. E. Cragg, P. T. C. So, “Standing wave total internal reflection microscopy—breaking the diffraction resolution limit,” Biophys. J. 78, 248a (2000).

C. Y. Dong, P. T. So, T. French, E. Gratton, “Fluorescence lifetime imaging by asynchronous pump-probe microscopy,” Biophys. J. 69, 2234–2242 (1995).
[CrossRef] [PubMed]

A. K. Kirsch, V. Subramaniam, G. Striker, C. Schnetter, D. J. Arndt-Jovin, T. M. Jovin, “Continuous wave two-photon scanning near-field optical microscopy,” Biophys. J. 75, 1513–1521 (1998).
[CrossRef] [PubMed]

Cell Mol. Biol. (1)

V. Subramaniam, A. K. Kirsch, T. M. Jovin, “Cell biological applications of scanning near-field optical microscopy (SNOM),” Cell Mol. Biol. 44, 689–700 (1998).
[PubMed]

IBM J. Res. Dev. (1)

G. Binning, H. Rohrer, “Scanning tunneling microscopy,” IBM J. Res. Dev. 30, 355 (1986).

J. Mod. Opt. (1)

S. Hell, W. S. Lindek, E. H. K. Stelzer, “Enhancing the axial resolution in far-field light microscopy: two-photon 4Pi confocal fluorescence microscopy,” J. Mod. Opt. 41, 675–681 (1994).
[CrossRef]

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

J. Phys. Chem. (1)

R. C. Dunn, G. R. Holtom, L. Mets, X. S. Xie, “Near-field fluorescence imaging and fluorescence lifetime measurement of light harvesting complexes in intact photosynthetic membranes,” J. Phys. Chem. 98, 3094–3098 (1994).
[CrossRef]

Nature (1)

B. Bailey, D. L. Farkas, D. L. Taylor, F. Lanni, “Enhancement of axial resolution in fluorescence microscopy by standing-wave excitation,” Nature 366, 44–48 (1993).
[CrossRef] [PubMed]

Opt. Lett. (5)

Optik (1)

C. Y. Dong, P. T. C. So, Ch. Buehler, E. Gratton, “Spatial resolution in scanning pump-probe fluorescence microscopy,” Optik 106, 7–14 (1997).

Phys. Rev. Lett. (1)

G. Binning, C. F. Quante, C. Gerber, “Atomic force microscope,” Phys. Rev. Lett. 12, 930–933 (1986).
[CrossRef]

Proc. Natl. Acad. Sci. USA (2)

C. Xu, W. Zipfel, J. B. Shear, R. M. Williams, W. W. Webb, “Multiphoton fluorescence excitation: new spectral windows for biological nonlinear microscopy,” Proc. Natl. Acad. Sci. USA 93, 10763–10768 (1996).
[CrossRef] [PubMed]

J. T. Frohn, H. F. Knapp, A. Stemmer, “True optical resolution beyond the Rayleigh limit achieved by standing wave illumination,” Proc. Natl. Acad. Sci. USA 97, 7232–7236 (2000).
[CrossRef] [PubMed]

Science (2)

E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, R. L. Kostalak, “Breaking the diffraction barrier: optical microscopy on a nanometric scale,” Science 251, 1469–1470 (1991).
[CrossRef]

W. Denk, J. H. Strickler, W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef] [PubMed]

Other (9)

W. J. Denk, D. W. Piston, W. W. Webb, “Two-photon molecular excitation laser-scanning microscopy,” in Handbook of Biological Confocal Microscopy, J. B. Pawley, ed. (Plenum, New York, 1995), pp. 445–458.

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984).

T. Wilson, Confocal Microscopy (Academic, London, 1990).

J. B. Pawley, ed., Handbook of Confocal Microscopy (Plenum, New York, 1995).

B. R. Masters, Selected Papers on Confocal Microscopy (SPIE Press, Bellingham, 1996).

V. Krishnamurthi, B. Bailey, F. Lanni, “Imaging processing in 3-D standing-wave fluroescence microscopy,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. Kino, T. Wilson, eds., Proc. SPIE2655, 18–25 (1996).
[CrossRef]

P. R. Bevington, D. K. Robinson, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1991).

P. T. C. So, C. Y. Dong, K. M. Berland, T. French, E. Gratton, “Time-resolved stimulated-emission and absorption microscopy,” in Topics in Fluorescence V, J. R. Lakowicz, ed., (Plenum, New York, 1998), 427–469.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. R. Flannery, Numerical Recipes in C (Cambridge U. Press, Cambridge, UK1992).

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Figures (10)

Fig. 1
Fig. 1

(a) Franhoffer diffraction-limited PSF (dotted curves), the SW-TIRM PSF predicted by Eq. (10) (solid curves), and the composite image of a point source generated by numerical simulation (data points). (b) Franhoffer diffraction-limited MTF (dotted curves), and the SW-TIRM MTF (solid curves).

Fig. 2
Fig. 2

(a) Simulated composite images for SW-TIRM imaging of two point objects separated by 20, 30, 40, 50, 60, 70, 80, 100, and 200 nm. (b) Simulated composite images for conventional fluorescence microscopic imaging of two point objects separated by 20, 30, 40, 50, 60, 70, 80, 100, and 200 nm. These images are normalized and are displaced from each other vertically for the clarity of the presentation.

Fig. 3
Fig. 3

The S/N distribution in reconstructed SW-TIRM images for different shift-vector choices: (a) {x}={0, 2π/3,-2π/3}, (b) {x}={0, π/2,-π/2}, (c) {x}={0, π/3,-π/3}, (d) {x}={0, π/4,-π/4}, (d) {x}={0, π/8,-π/8}, (e) {x}={0, π/16,-π/16}.

Fig. 4
Fig. 4

(a) Theoretical PSF of two-photon SW-TIRM at 900-nm excitation as predicted by Eq. (10) (solid curves) and simulated composite image of a point source imaged by two-photon SW-TIRM at 900 nm (data points). (b) MTF of two-photon SW-TIRM with 900-nm excitation (solid curves) and MTF of Fraunhofer diffraction PSF (dotted curves). (c) Theoretical PSF of two-photon SW-TIRM at 750-nm excitation as predicted by Eq. (10) (solid curves) and simulated composite image of a point source imaged by two-photon SW-TIRM at 750 nm (data points). (d) MTF of two-photon SW-TIRM with 750-nm excitation (solid curves) and MTF of Fraunhofer diffraction PSF (dotted curves). (e) S/N distribution in two-photon SW-TIRM images with excitation wavelength at 750 nm (upper curve) and at 900 nm (lower curve).

Fig. 5
Fig. 5

(a) Simulated composite images for two-photon SW-TIRM imaging at 900-nm excitation of two point objects separated by 20, 30, 40, 50, 60, 70, 80, 100, and 200 nm. (b) Simulated composite images of two-photon SW-TIRM imaging at 750-nm excitation of two point objects separated by 20, 30, 40, 50, 60, 70, 80, 100, and 200 nm. These images are normalized and are displaced from each other vertically for the clarity of the presentation.

Fig. 6
Fig. 6

(a) Theoretical PSF of pump-probe SW-TIRM in transient absorption mode as predicted by Eq. (10) (solid curves) and simulated composite image of a point source imaged by pump–probe SW-TIRM in transient absorption mode (data points). (b) MTF of pump–probe SW-TIRM in transient absorption mode (solid curves) and MTF of Fraunhofer diffraction PSF (dotted curves). (c) Theoretical PSF of pump–probe SW-TIRM in stimulated-emission mode as predicted by Eq. (10) (solid curves) and simulated composite image of a point source imaged by pump-probe SW-TIRM in stimulated-emission mode (data points). (d) MTF of pump-probe SW-TIRM in stimulated-emission mode (solid curves) and MTF of Fraunhofer diffraction PSF (dotted curve). (e) S/N distribution in pump–probe SW-TIRM images in stimulated-emission mode (upper curve) and in transient absorption mode (lower curve).

Fig. 7
Fig. 7

(a) Simulated composite images for pump–probe SW-TIRM imaging in transient absorption mode of two point objects separated by 20, 30, 40, 50, 60, 70, 80, 100, and 200 nm. (b) Simulated composite images of pump–probe SW-TIRM imaging in stimulated-emission mode of two point objects separated by 20, 30, 40, 50, 60, 70, 80, 100, and 200 nm. These images are normalized and are displaced from each other vertically for the clarity of the presentation.

Fig. 8
Fig. 8

(a) Fluorescence intensity distribution generated by evanescence fields in STED configuration. The parameters of this simulation are: σ01=σ23=5×10-7 μm2, τfluor=2 ns, τvibr=0.002 ns, Q=0.1 ns-1, hexc=2 W μm-2. A range of STED beam intensity is evaluated: hSTED=0 W μm-2 (square), hSTED=0.1 W μm-2 (circle), hSTED=1 W μm-2 (triangle), hSTED=10 W μm-2 (diamond), hSTED=50 W μm-2 (solid curve). Other parameters used in the simulation are specified in Section 3.B; in particular, the wavelength of the excitation beam is set at 450 nm, and the wavelength of the STED beam is set at 900 nm. (b) Fourier transform of fluorescence intensity distribution generated by evanescence fields in STED configuration with STED beam intensities of 50 W µm-2 (circles) and 0 W µm-2 (squares).

Fig. 9
Fig. 9

(a) Theoretical PSF of STED SW-TIRM imaged with use of 50 W µm-2 STED beam intensity as predicted by Eq. (10) (solid curves) and simulated composite image of a point source imaged by STED SW-TIRM with use of 50 W µm-2 STED beam (data points). (b) Simulated PSFs of STED SW-TIRM imaged with use of 50 W µm-2 STED beam intensity (solid curves), 10 W µm-2 STED beam intensity (circles), 1 W µm-2 STED beam intensity (squares), 0.1 W µm-2 STED beam intensity (triangles), and 0 W µm-2 STED beam intensity (diamonds). (c) MTFs of STED SW-TIRM PSFs with STED beam intensity set to 50 W µm-2, 10 W µm-2, 1 W µm-2, 0.1 W µm-2, and 0 W µm-2 (from right to left).

Fig. 10
Fig. 10

Simulated composite images for STED SW-TIRM imaging with 50 W µm-2 STED beam intensity of two point objects separated by 10, 20, 30, 40, 50, 60, 70, 80, 100, and 200 nm. These images are normalized and are displaced from each other vertically for the clarity of the presentation.

Tables (1)

Tables Icon

Table 1 Characteristics of PSFs and MTFs of Fluorescence Microscopy (FM), SW-TIRM, Two-Photon SW-TIRM (2p SW-TIRM), Pump–Probe SW-TIRM (pp SW-TIRM), and STED SW-TIRM

Equations (65)

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P(r)=2J1(2πNA|r|/λe)2πNA|r|/λe2,
I(r)=[O(r)E(r)]P(r),
I(r,r)=[O(r)E(r-r)]P(r),
I(x, x)=[O(x)E(x-x)]P(x).
I(x)={x}f(x, x)I(x, x),
I(x)={x}f(x, x)-O(x)E(x-x)P(x-x)dx.
I(x)=-O(x){x}f(x, x)E(x-x)P(x-x)dx.
{x}f(x, x)E(x-x)=E(x-x),
I(x)=-O(x)E(x-x)P(x-x)dx.
I(x)=O(x)[E(x)P(x)].
E(x)=n=0an cos(nkx)+bn sin(nkx).
{x}n=0f(x, x){an cos[nk(x-x)]+bn sin[nk(x-x)]}=n=0an cos[nk(x-x)]+bn sin[nk(x-x)].
n=0an cos[nk(x-x)]+bn sin[nk(x-x)]=a0+[a1 cos(kx)+b1 sin(kx)]cos(kx)+[a1 sin(kx)-b1 cos(kx)]sin(kx)++[an cos(nkx)+bn sin(nkx)]cos(nkx)+[an sin(nkx)-bn cos(nkx)]sin(nkx)+ .
{x}n=0f(x, x){an cos[nk(x-x)]+bn sin[nk(x-x)]}=a0{x}f(x, x)+{x}f(x, x)[a1 cos(kx)-b1 sin(kx)]cos(kx)+{x}f(x, x)[a1 sin(kx)+b1 cos(kx)]sin(kx)++{x}f(x, x)[an cos(nkx)-bn sin(nkx)]cos(nkx)+{x}f(x, x)[an sin(nkx)+bn cos(nkx)]sin(nkx)+ .
{x}f(x, x)=1, {x}f(x, x)[a1 cos(kx)-b1 sin(kx)]=a1 cos(kx)+b1 sin(kx), {x}f(x, x)[a1 sin(kx)+b1 cos(kx)]=a1 sin(kx)-b1 cos(kx)  {x}f(x, x)[an cos(nkx)-bn sin(nkx)]=an cos(nkx)+bn sin(nkx),{x}f(x, x)[an sin(nkx)+bn cos(nkx)]=an sin(nkx)-bn cos(nkx)
f(x, x)=A0(x)+A1(x)cos(kx)+B1(x)sin(kx)++An(x)cos(nkx)+Bn(x)sin(nkx)+ .
11a1 cos(kx0)-b1 sin(kx0)a1 cos(kx2m)-b1 sin(kx2m)a1 sin(kx0)+b1 cos(kx0)a1 sin(kx2m)+b1 cos(kx2m)am cos(mkx0)-bm sin(mkx0)am cos(mkx2m)-bm sin(mkx2m)am sin(mkx0)+bm cos(mkx0)am sin(mkx2m)+bm cos(mkx2m)×A00A10B10Am,0Bm,0A01A11B11Am,1Bm,1A0,2mA1,2mB1,2,mAm,2mBm,2m=1a1b10-b1a10ambm-bmam.
S×A=E.
det(S)0,
S-1S=I,
A=S-1E.
S=11cos(kx0)cos(kx2m)sin(kx0)sin(kx2m)cos(mkx0)cos(mkx2m)sin(mkx0)sin(mkx2m),
A=S-1.
E(x)=1+cos(Kx).
K=4πn sin(θ)/λ,
f(x, x)=A0(x)+A1(x)cos Kx+A2(x)sin Kx.
S=111cos Kx0cos Kx1cos Kx2sin Kx0sin Kx1sin Kx2.
{x}=0, 2π3,-2π3
S=1111-1/2-1/203/2-3/2.
A=S-1=1/32/301/3-1/33/31/3-1/3-3/3.
O1(x)=1if|x|<1 nm0elsewhere,
O2(x, a)=1a<x<a+2 nm1-a-2 nm<x<-a0elsewhere.
N(x)={x}I(x, x)f(x, x)21/2.
S/N(x)=I(x)N(x).
O3(x)=O0,
E2p(x)=E(x)2=14[1+cos(Kx)]2=38+12 cos(Kx)+18 cos(2Kx).
f(x, x)=A0(x)+A1(x)cos Kx+A2(x)sin Kx+A3(x)cos 2Kx+A4(x)sin 2Kx.
{x}=0, 2π5K,-2π5K, 4π5K,-4π5K.
A=0.20.400.400.20.1240.38-0.3240.2350.20.124-0.38-0.324-0.2350.2-0.3240.2350.124-0.380.2-0.324-0.2350.1240.38.
Epu(x)=1+cos(Kx),
Epr(x)=1+cos(Kx+ϕ).
Epupr(x)=[1+cos(Kx)]2.
Epu(x)=1+cos(2Kx).
Epr(x)=1+cos(Kx+ϕ).
Epupr(x)=14[1+cos(2Kx)][1+cos(Kx)]=14+38 cos(Kx)+14 cos(2Kx)+18 cos(3Kx).
{x}=0, 2π7K,-2π7K, 4π7K,-4π7K, 6π7K,-6π7K,
f(x, x)=A0(x)+A1(x)cos Kx+A2(x)sin Kx+A3(x)cos 2Kx+A4(x)sin 2Kx+A5(x)cos 3Kx+A6(x)sin 3Kx,
A=0.1430.2860 0.2860 0.2860 0.1430.1780.223-0.0640.279-0.2570.1240.1430.178-0.223-0.064-0.279-0.257-0.1240.143-0.0640.279-0.257-0.1240.178-0.2230.143-0.064-0.279-0.2570.1240.1780.2230.143-0.2570.1240.178-0.223-0.0640.2790.143-0.257-0.1240.1780.223-0.064-0.279.
E(-x)=E(x).
{x}f(x, x)E(x-x)=E(x-x).
{x}=R,
f(x, x)=δx-x,
δx=0x01x=0.
{x}f(x, x)E(x-x)={x}Rδx-xE(x-x)=E(x-x).
I(x)={x}RI(x, x)δx-x.
{x}0xX.
f(x, x)=δx-nX-x,
{x}f(x, x)E(x-x)=0xXδx-nX-xE(x-x)=E(x-nX-x)=E(x-x).
dn0dt=hexc(x, t)σ01(n1-n0)+1τvibrn3,
dn1dt=hexc(x, t)σ01(n0-n1)-1τvibrn1,
dn2dt=1τvibrn1+hSTED(x, t)σ23(n3-n2)-1τfluor+Qn2,
dn3dt=hSTED(x, t)σ23(n2-n3)+1τfluor+Qn2-1τvibrn3.
hexc(x, t)=hexc(t)[1+cos(2Kx)].
hSTED(x, t)=hSTED(t)[1+cos(Kx+ϕ)].
hSTED(x, t)=hSTED(t)[1-cos(Kx)].

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