Abstract

The axial resolution of fluorescence microscopes can be considerably improved by superposing two illumination beams and by adding coherently the two wavefronts emitted by the luminescent sample. This solution has been implemented in 4Pi microscopes. Theoretical and experimental results have shown that a considerable improvement of the axial resolution can be obtained with these microscopes. However, the lateral resolution remains limited by diffraction. We propose a configuration of a 4Pi microscope in which the lateral displacement of the source modifies the collection efficiency function (CEF). Numerical calculations based on an approximate scalar theory and on exact vector-wave-optics results of the field distribution of the electromagnetic field in image space show that the lateral extent of the CEF can be reduced by a factor greater than 2 with respect to the diffraction limit. We show that, with this solution, the resolution in the transverse plane of 4Pi type B and 4Pi type C microscopes can be improved significantly.

© 2006 Optical Society of America

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References

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  1. M. Nagorni and S. W. Hell, "Coherent use of opposing lenses for axial resolution increase in fluorescence microscopy. I. Comparative study of concepts," J. Opt. Soc. Am. A 18, 36-48 (2001).
    [CrossRef]
  2. P.-F. Lenne, E. Etienne, and H. Rigneault, "Subwavelength patterns and high detection efficiency in fluorescence correlation spectroscopy using photonic structures," Appl. Phys. Lett. 80, 4106-4108 (2002).
    [CrossRef]
  3. S. Hell, "Double-confocal scanning microscope," European patent application EP0491289 (June 24, 1992).
  4. C. J. R. Sheppard and Y. Gong, "Improvement in axial resolution by interference confocal microscopy," Optik (Stuttgart) 87, 129-132 (1991).
  5. S. Hell and E. H. K. Stelzer, "Properties of a 4Pi confocal fluorescence microscope," J. Opt. Soc. Am. A 9, 2159-2166 (1992).
    [CrossRef]
  6. M. Gu and C. J. R. Sheppard, "Three-dimensional transfer functions in 4Pi confocal microscopes," J. Opt. Soc. Am. A 11, 1619-1627 (1994).
    [CrossRef]
  7. S. W. Hell, S. Lindek, C. Cremer, and E. H. K. Stelzer, "Measurement of the 4Pi-confocal point-spread function proves 75 nm axial resolution," Appl. Phys. Lett. 64, 1335-1337 (1994).
    [CrossRef]
  8. D. E. Koppel, D. Axelrod, J. Schlessinger, E. L. Elson, and W. W. Webb, "Dynamics of fluorescence marker concentration as a probe of mobility," Biophys. J. 16, 1315-1329 (1976).
    [CrossRef] [PubMed]
  9. C. M. Blanca, J. Bewersdorf, and S. W. Hell, "Single sharp spot in fluorescence microscopy of two opposing lenses," Appl. Phys. Lett. 79, 2321-2323 (2001).
    [CrossRef]
  10. M. Martinez-Corraz, A. Pons, and M. T. Caballero, "Axial apodization in 4Pi-confocal microscopy by annular binary filters," J. Opt. Soc. Am. A 19, 1532-1536 (2002).
    [CrossRef]
  11. M. Martinez-Corraz, M. T. Caballero, A. Pons, and P. Andrés, "Sidelobe decline in single-photon 4Pi microscopy by Toraldo rings," Micron 34, 319-325 (2003).
    [CrossRef]
  12. H. Gugel, J. Bewersdorf, S. Jakobs, J. Engelhardt, R. Storz, and S. W. Hell, "Cooperative 4Pi excitation and detection yields sevenfold sharper optical sections in live-cell microscopy," Biophys. J. 87, 4146-4152 (2004).
    [CrossRef] [PubMed]
  13. B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system," Proc. R. Soc. London Ser. A 235, 358-379 (1959).
  14. J. Enderlein, "Theoretical study of detection of a dipole emitter through an objective with high numerical aperture," Opt. Lett. 25, 634-636 (2000).
    [CrossRef]
  15. M. Bohmer and J. Enderlein, "Orientation imaging of single molecules by wide-field epifluorescence microscopy," J. Opt. Soc. Am. B 20, 554-559 (2003).
    [CrossRef]
  16. S. H. Wiersma, P. Török, T. D. Visser, and P. Varga, "Comparison of different theories for focusing through a plane interface," J. Opt. Soc. Am. A 14, 1482-1490 (1997).
    [CrossRef]
  17. P. Torok, "Propagation of electromagnetic dipole waves through dielectric interfaces," Opt. Lett. 25, 1463-1465 (2000).
    [CrossRef]
  18. O. Haeberlé, "Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: Conventional microscopy," Opt. Commun. 216, 55-63 (2003).
    [CrossRef]
  19. M. Dyba and S. W. Hell, "Focal spots of size lambda/23 open up far-field fluorescence microscopy at 33 nm axial resolution," Phys. Rev. Lett. 88, 163901 (2002).
    [CrossRef] [PubMed]

2004 (1)

H. Gugel, J. Bewersdorf, S. Jakobs, J. Engelhardt, R. Storz, and S. W. Hell, "Cooperative 4Pi excitation and detection yields sevenfold sharper optical sections in live-cell microscopy," Biophys. J. 87, 4146-4152 (2004).
[CrossRef] [PubMed]

2003 (3)

M. Martinez-Corraz, M. T. Caballero, A. Pons, and P. Andrés, "Sidelobe decline in single-photon 4Pi microscopy by Toraldo rings," Micron 34, 319-325 (2003).
[CrossRef]

M. Bohmer and J. Enderlein, "Orientation imaging of single molecules by wide-field epifluorescence microscopy," J. Opt. Soc. Am. B 20, 554-559 (2003).
[CrossRef]

O. Haeberlé, "Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: Conventional microscopy," Opt. Commun. 216, 55-63 (2003).
[CrossRef]

2002 (3)

M. Dyba and S. W. Hell, "Focal spots of size lambda/23 open up far-field fluorescence microscopy at 33 nm axial resolution," Phys. Rev. Lett. 88, 163901 (2002).
[CrossRef] [PubMed]

P.-F. Lenne, E. Etienne, and H. Rigneault, "Subwavelength patterns and high detection efficiency in fluorescence correlation spectroscopy using photonic structures," Appl. Phys. Lett. 80, 4106-4108 (2002).
[CrossRef]

M. Martinez-Corraz, A. Pons, and M. T. Caballero, "Axial apodization in 4Pi-confocal microscopy by annular binary filters," J. Opt. Soc. Am. A 19, 1532-1536 (2002).
[CrossRef]

2001 (2)

C. M. Blanca, J. Bewersdorf, and S. W. Hell, "Single sharp spot in fluorescence microscopy of two opposing lenses," Appl. Phys. Lett. 79, 2321-2323 (2001).
[CrossRef]

M. Nagorni and S. W. Hell, "Coherent use of opposing lenses for axial resolution increase in fluorescence microscopy. I. Comparative study of concepts," J. Opt. Soc. Am. A 18, 36-48 (2001).
[CrossRef]

2000 (2)

1997 (1)

1994 (2)

M. Gu and C. J. R. Sheppard, "Three-dimensional transfer functions in 4Pi confocal microscopes," J. Opt. Soc. Am. A 11, 1619-1627 (1994).
[CrossRef]

S. W. Hell, S. Lindek, C. Cremer, and E. H. K. Stelzer, "Measurement of the 4Pi-confocal point-spread function proves 75 nm axial resolution," Appl. Phys. Lett. 64, 1335-1337 (1994).
[CrossRef]

1992 (2)

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

S. Hell, "Double-confocal scanning microscope," European patent application EP0491289 (June 24, 1992).

1991 (1)

C. J. R. Sheppard and Y. Gong, "Improvement in axial resolution by interference confocal microscopy," Optik (Stuttgart) 87, 129-132 (1991).

1976 (1)

D. E. Koppel, D. Axelrod, J. Schlessinger, E. L. Elson, and W. W. Webb, "Dynamics of fluorescence marker concentration as a probe of mobility," Biophys. J. 16, 1315-1329 (1976).
[CrossRef] [PubMed]

1959 (1)

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system," Proc. R. Soc. London Ser. A 235, 358-379 (1959).

Andrés, P.

M. Martinez-Corraz, M. T. Caballero, A. Pons, and P. Andrés, "Sidelobe decline in single-photon 4Pi microscopy by Toraldo rings," Micron 34, 319-325 (2003).
[CrossRef]

Axelrod, D.

D. E. Koppel, D. Axelrod, J. Schlessinger, E. L. Elson, and W. W. Webb, "Dynamics of fluorescence marker concentration as a probe of mobility," Biophys. J. 16, 1315-1329 (1976).
[CrossRef] [PubMed]

Bewersdorf, J.

H. Gugel, J. Bewersdorf, S. Jakobs, J. Engelhardt, R. Storz, and S. W. Hell, "Cooperative 4Pi excitation and detection yields sevenfold sharper optical sections in live-cell microscopy," Biophys. J. 87, 4146-4152 (2004).
[CrossRef] [PubMed]

C. M. Blanca, J. Bewersdorf, and S. W. Hell, "Single sharp spot in fluorescence microscopy of two opposing lenses," Appl. Phys. Lett. 79, 2321-2323 (2001).
[CrossRef]

Blanca, C. M.

C. M. Blanca, J. Bewersdorf, and S. W. Hell, "Single sharp spot in fluorescence microscopy of two opposing lenses," Appl. Phys. Lett. 79, 2321-2323 (2001).
[CrossRef]

Bohmer, M.

Caballero, M. T.

M. Martinez-Corraz, M. T. Caballero, A. Pons, and P. Andrés, "Sidelobe decline in single-photon 4Pi microscopy by Toraldo rings," Micron 34, 319-325 (2003).
[CrossRef]

M. Martinez-Corraz, A. Pons, and M. T. Caballero, "Axial apodization in 4Pi-confocal microscopy by annular binary filters," J. Opt. Soc. Am. A 19, 1532-1536 (2002).
[CrossRef]

Cremer, C.

S. W. Hell, S. Lindek, C. Cremer, and E. H. K. Stelzer, "Measurement of the 4Pi-confocal point-spread function proves 75 nm axial resolution," Appl. Phys. Lett. 64, 1335-1337 (1994).
[CrossRef]

Dyba, M.

M. Dyba and S. W. Hell, "Focal spots of size lambda/23 open up far-field fluorescence microscopy at 33 nm axial resolution," Phys. Rev. Lett. 88, 163901 (2002).
[CrossRef] [PubMed]

Elson, E. L.

D. E. Koppel, D. Axelrod, J. Schlessinger, E. L. Elson, and W. W. Webb, "Dynamics of fluorescence marker concentration as a probe of mobility," Biophys. J. 16, 1315-1329 (1976).
[CrossRef] [PubMed]

Enderlein, J.

Engelhardt, J.

H. Gugel, J. Bewersdorf, S. Jakobs, J. Engelhardt, R. Storz, and S. W. Hell, "Cooperative 4Pi excitation and detection yields sevenfold sharper optical sections in live-cell microscopy," Biophys. J. 87, 4146-4152 (2004).
[CrossRef] [PubMed]

Etienne, E.

P.-F. Lenne, E. Etienne, and H. Rigneault, "Subwavelength patterns and high detection efficiency in fluorescence correlation spectroscopy using photonic structures," Appl. Phys. Lett. 80, 4106-4108 (2002).
[CrossRef]

Gong, Y.

C. J. R. Sheppard and Y. Gong, "Improvement in axial resolution by interference confocal microscopy," Optik (Stuttgart) 87, 129-132 (1991).

Gu, M.

Gugel, H.

H. Gugel, J. Bewersdorf, S. Jakobs, J. Engelhardt, R. Storz, and S. W. Hell, "Cooperative 4Pi excitation and detection yields sevenfold sharper optical sections in live-cell microscopy," Biophys. J. 87, 4146-4152 (2004).
[CrossRef] [PubMed]

Haeberlé, O.

O. Haeberlé, "Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: Conventional microscopy," Opt. Commun. 216, 55-63 (2003).
[CrossRef]

Hell, S.

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

S. Hell, "Double-confocal scanning microscope," European patent application EP0491289 (June 24, 1992).

Hell, S. W.

H. Gugel, J. Bewersdorf, S. Jakobs, J. Engelhardt, R. Storz, and S. W. Hell, "Cooperative 4Pi excitation and detection yields sevenfold sharper optical sections in live-cell microscopy," Biophys. J. 87, 4146-4152 (2004).
[CrossRef] [PubMed]

M. Dyba and S. W. Hell, "Focal spots of size lambda/23 open up far-field fluorescence microscopy at 33 nm axial resolution," Phys. Rev. Lett. 88, 163901 (2002).
[CrossRef] [PubMed]

M. Nagorni and S. W. Hell, "Coherent use of opposing lenses for axial resolution increase in fluorescence microscopy. I. Comparative study of concepts," J. Opt. Soc. Am. A 18, 36-48 (2001).
[CrossRef]

C. M. Blanca, J. Bewersdorf, and S. W. Hell, "Single sharp spot in fluorescence microscopy of two opposing lenses," Appl. Phys. Lett. 79, 2321-2323 (2001).
[CrossRef]

S. W. Hell, S. Lindek, C. Cremer, and E. H. K. Stelzer, "Measurement of the 4Pi-confocal point-spread function proves 75 nm axial resolution," Appl. Phys. Lett. 64, 1335-1337 (1994).
[CrossRef]

Jakobs, S.

H. Gugel, J. Bewersdorf, S. Jakobs, J. Engelhardt, R. Storz, and S. W. Hell, "Cooperative 4Pi excitation and detection yields sevenfold sharper optical sections in live-cell microscopy," Biophys. J. 87, 4146-4152 (2004).
[CrossRef] [PubMed]

Koppel, D. E.

D. E. Koppel, D. Axelrod, J. Schlessinger, E. L. Elson, and W. W. Webb, "Dynamics of fluorescence marker concentration as a probe of mobility," Biophys. J. 16, 1315-1329 (1976).
[CrossRef] [PubMed]

Lenne, P.-F.

P.-F. Lenne, E. Etienne, and H. Rigneault, "Subwavelength patterns and high detection efficiency in fluorescence correlation spectroscopy using photonic structures," Appl. Phys. Lett. 80, 4106-4108 (2002).
[CrossRef]

Lindek, S.

S. W. Hell, S. Lindek, C. Cremer, and E. H. K. Stelzer, "Measurement of the 4Pi-confocal point-spread function proves 75 nm axial resolution," Appl. Phys. Lett. 64, 1335-1337 (1994).
[CrossRef]

Martinez-Corraz, M.

M. Martinez-Corraz, M. T. Caballero, A. Pons, and P. Andrés, "Sidelobe decline in single-photon 4Pi microscopy by Toraldo rings," Micron 34, 319-325 (2003).
[CrossRef]

M. Martinez-Corraz, A. Pons, and M. T. Caballero, "Axial apodization in 4Pi-confocal microscopy by annular binary filters," J. Opt. Soc. Am. A 19, 1532-1536 (2002).
[CrossRef]

Nagorni, M.

Pons, A.

M. Martinez-Corraz, M. T. Caballero, A. Pons, and P. Andrés, "Sidelobe decline in single-photon 4Pi microscopy by Toraldo rings," Micron 34, 319-325 (2003).
[CrossRef]

M. Martinez-Corraz, A. Pons, and M. T. Caballero, "Axial apodization in 4Pi-confocal microscopy by annular binary filters," J. Opt. Soc. Am. A 19, 1532-1536 (2002).
[CrossRef]

Richards, B.

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system," Proc. R. Soc. London Ser. A 235, 358-379 (1959).

Rigneault, H.

P.-F. Lenne, E. Etienne, and H. Rigneault, "Subwavelength patterns and high detection efficiency in fluorescence correlation spectroscopy using photonic structures," Appl. Phys. Lett. 80, 4106-4108 (2002).
[CrossRef]

Schlessinger, J.

D. E. Koppel, D. Axelrod, J. Schlessinger, E. L. Elson, and W. W. Webb, "Dynamics of fluorescence marker concentration as a probe of mobility," Biophys. J. 16, 1315-1329 (1976).
[CrossRef] [PubMed]

Sheppard, C. J. R.

M. Gu and C. J. R. Sheppard, "Three-dimensional transfer functions in 4Pi confocal microscopes," J. Opt. Soc. Am. A 11, 1619-1627 (1994).
[CrossRef]

C. J. R. Sheppard and Y. Gong, "Improvement in axial resolution by interference confocal microscopy," Optik (Stuttgart) 87, 129-132 (1991).

Stelzer, E. H. K.

S. W. Hell, S. Lindek, C. Cremer, and E. H. K. Stelzer, "Measurement of the 4Pi-confocal point-spread function proves 75 nm axial resolution," Appl. Phys. Lett. 64, 1335-1337 (1994).
[CrossRef]

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

Storz, R.

H. Gugel, J. Bewersdorf, S. Jakobs, J. Engelhardt, R. Storz, and S. W. Hell, "Cooperative 4Pi excitation and detection yields sevenfold sharper optical sections in live-cell microscopy," Biophys. J. 87, 4146-4152 (2004).
[CrossRef] [PubMed]

Torok, P.

Török, P.

Varga, P.

Visser, T. D.

Webb, W. W.

D. E. Koppel, D. Axelrod, J. Schlessinger, E. L. Elson, and W. W. Webb, "Dynamics of fluorescence marker concentration as a probe of mobility," Biophys. J. 16, 1315-1329 (1976).
[CrossRef] [PubMed]

Wiersma, S. H.

Wolf, E.

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system," Proc. R. Soc. London Ser. A 235, 358-379 (1959).

Appl. Phys. Lett. (3)

P.-F. Lenne, E. Etienne, and H. Rigneault, "Subwavelength patterns and high detection efficiency in fluorescence correlation spectroscopy using photonic structures," Appl. Phys. Lett. 80, 4106-4108 (2002).
[CrossRef]

S. W. Hell, S. Lindek, C. Cremer, and E. H. K. Stelzer, "Measurement of the 4Pi-confocal point-spread function proves 75 nm axial resolution," Appl. Phys. Lett. 64, 1335-1337 (1994).
[CrossRef]

C. M. Blanca, J. Bewersdorf, and S. W. Hell, "Single sharp spot in fluorescence microscopy of two opposing lenses," Appl. Phys. Lett. 79, 2321-2323 (2001).
[CrossRef]

Biophys. J. (2)

H. Gugel, J. Bewersdorf, S. Jakobs, J. Engelhardt, R. Storz, and S. W. Hell, "Cooperative 4Pi excitation and detection yields sevenfold sharper optical sections in live-cell microscopy," Biophys. J. 87, 4146-4152 (2004).
[CrossRef] [PubMed]

D. E. Koppel, D. Axelrod, J. Schlessinger, E. L. Elson, and W. W. Webb, "Dynamics of fluorescence marker concentration as a probe of mobility," Biophys. J. 16, 1315-1329 (1976).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (5)

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

Micron (1)

M. Martinez-Corraz, M. T. Caballero, A. Pons, and P. Andrés, "Sidelobe decline in single-photon 4Pi microscopy by Toraldo rings," Micron 34, 319-325 (2003).
[CrossRef]

Opt. Commun. (1)

O. Haeberlé, "Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: Conventional microscopy," Opt. Commun. 216, 55-63 (2003).
[CrossRef]

Opt. Lett. (2)

Optik (Stuttgart) (1)

C. J. R. Sheppard and Y. Gong, "Improvement in axial resolution by interference confocal microscopy," Optik (Stuttgart) 87, 129-132 (1991).

Phys. Rev. Lett. (1)

M. Dyba and S. W. Hell, "Focal spots of size lambda/23 open up far-field fluorescence microscopy at 33 nm axial resolution," Phys. Rev. Lett. 88, 163901 (2002).
[CrossRef] [PubMed]

Proc. R. Soc. London Ser. A (1)

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system," Proc. R. Soc. London Ser. A 235, 358-379 (1959).

Other (1)

S. Hell, "Double-confocal scanning microscope," European patent application EP0491289 (June 24, 1992).

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Figures (9)

Fig. 1
Fig. 1

Schematic of the 4 Pi microscope. S, output signal; M, mirrors; BS, beam splitter; O 1 and O 2 , microscope objectives. The optical paths in the two arms of the interferometer are assumed to be equal. The 4 Pi arrangement is obtained from a 4Pi microscope[12] by adding an image inversion system in one arm of the interferometer.

Fig. 2
Fig. 2

Schematic representation of the sensor head of a system equivalent to a 4Pi microscope and of a system equivalent to a 4 Pi microscope. F is the common focus of the two microscope objectives O 1 and O 2 . O is a microscope objective identical to O 1 or to O 2 ( O 1 and O 2 are assumed to be identical). OA is the optical axis of the system. Dipole D is the dipole emitter. D moves in the vicinity of the focal plane FP. I D is the image of D through the 4Pi microscope. I D is the image of D through a 4 Pi microscope. In this representation, the beams emitted by D and by I D for the 4Pi microscope and by D and by I D for the 4 Pi microscope pass through O and are superimposed on the detector. (a) View in a plane containing the optical axis. (b) View in a plane orthogonal to the optical axis.

Fig. 3
Fig. 3

Optical system equivalent to a 4 Pi microscope. O is the microscope objective (the two microscope objectives O 1 and O 2 are assumed to be identical). S is the source that moves in the focal plane of O. S emits a spherical wave. A second spherical wave is emitted by I S . S and I S are symmetrical with respect to F. S and I S are the images, through the optical system ( O ; L ) , of S and I S , respectively.

Fig. 4
Fig. 4

Section of the CEF of the 4Pi and the 4 Pi microscopes in arbitrary units in the focal plane of the microscope objective. The source emits at λ = 525 nm . The diameter of the detector is ϕ = 100 μ m . M = 100 and NA = 1.45 . The intensity I of the pump beam at 488 nm , focused by O, is also represented.

Fig. 5
Fig. 5

Sections of the CEF of the 4 Pi microscope obtained with different microscope objectives. λ = 525 nm . NA = 1.45 , with m = 100 corresponding to an α Plan-FLUAR objective from Zeiss. NA = 1.25 and NA = 1 , with m = 100 corresponding to an N Plan-FLUAR microscope objective NA = 1.25 0.6 × 100 from Leica.

Fig. 6
Fig. 6

(a), (b) Comparison between the calculated CEF (arbitrary units) of (a) a 4Pi microscope and (b) a 4 Pi microscope. (c) Section of the CEF in the transverse plane. (d) Section of the CEF along the optical axis. The diameter ϕ of the pinhole is ϕ = 20 μ m . The wavelength of emission is λ 2 = 525 nm . In the numerical calculation, the microscope objectives O 1 and O 2 are oil immersion objectives ( NA = 1.4 , magnification m = 40 ).

Fig. 7
Fig. 7

Comparison between the calculated MDE (arbitrary units) of a 4Pi microscope and the calculated MDE of a 4 Pi microscope. The illumination beam is a Gaussian monochromatic beam (emission wavelength λ 1 = 488 nm ) linearly polarized along the x axis. (a) MDE of the 4Pi-C microscope (left) and MDE of a 4 Pi - C microscope (right) in the x y plane. (b) MDE of the 4Pi-C microscope (left) and MDE of a 4 Pi - C microscope (right) in the y z plane.

Fig. 8
Fig. 8

Comparison between the MDE (arbitrary units) of a 4Pi-C microscope and the MDE of a 4 Pi -C microscope. (a) Section of the MDE in the focal plane along the x axis. (b) Section of the MDE in the focal plane along the y axis. (c) Section of the MDE along the z axis.

Fig. 9
Fig. 9

CEF (arbitrary units) calculated with two different methods. In both calculations, λ = 525 nm , NA = 1.4 , m = 40 , ϕ = 20 μ m .

Equations (6)

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MDE ( r ) = CEF ( r ) I e ( r ) ,
I d ( x , y ) = [ J 1 ( u ) u ] 2 ,
I d ( x , y ) = [ J 1 ( u + ) u + + J 1 ( u ) u ] 2
u = k NA m ( x m d ) 2 + y 2 ,
u + = k NA m ( x + m d ) 2 + y 2 ,
Sig ( d ) = S PD I d ( x , y ) d x d y ,

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