Abstract

We consider the electromagnetic diffraction occurring when light is focused by a lens without spherical aberration through a planar interface between materials of mismatched refractive indices, which focusing produces spherical aberration. By means of a rigorous vectorial electromagnetic treatment developed previously for this problem by Török et al. [ J. Opt. Soc. Am. A 12, 325 ( 1995)], the time-averaged electric energy density distributions in the region of the focused probe are numerically evaluated for air–glass and air–silicon interfaces as functions of lens numerical aperture and probe depth. Strehl intensity, lateral and axial sizes, and axial location of the probe are shown to be regular functions for low numerical apertures and probe depths but irregular functions for high numerical apertures and probe depths. An explanation to account for these occurrences is presented that also explains some previous experimental results of confocal microscopy.

© 1996 Optical Society of America

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References

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  1. P. Török, P. Varga, G. R. Booker, ‘Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: structure of the electromagnetic field. I,’ J. Opt. Soc. Am. A 12, 2136–2144 (1995).
    [CrossRef]
  2. P. Török, P. Varga, Z. Laczik, 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]
  3. In this paper we use the term Strehl intensity as the measure of the absolute maximum of the time-averaged electric energy density.This definition is slightly different from that given by, for example, M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970),Chap. 9, p. 462.
  4. For media with n1 > n2, the diffraction integrals of Ref. 2 can be used. For this case, when the solid semiangle of the illumination is greater than the critical angle, irregular (or evanescent) waves are generated.This problem is the subject of another paper (P. Török, C. J. R. Sheppard, P. Varga, ‘Study of evanescent waves for transmission near-field optical microscopy,’ J. Mod. Opt. 43, 1167–1183 (1996).
    [CrossRef]
  5. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortan, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 9, p. 340.
  6. This was also concluded from the paraxial approximation of our original diffraction integrals by Eqs. (51) and (52) of Ref. 2.
  7. J. J. Stamnes, Waves in Focal Regions, 1st ed. (Adam Hilger, Bristol, UK, 1986), Chap. 12, p. 356.
  8. P. Török, P. Varga, G. Németh, ‘Analytical solution of the diffraction integrals and interpretation of wave-front distortion when light is focused through a planar interface between materials of mismatched refractive indices,’ J. Opt. Soc. Am. A 12, 2660–2672 (1995).
    [CrossRef]
  9. E. Wolf, ‘Electromagnetic diffraction in optical systems I. An integral representation of the image field,’ Proc.R. Soc. London 253, 349–357 (1959).
    [CrossRef]
  10. Experimental evidence for such an irregular behavior was obtained by L. Majlof, P.-O. Forsgren, ‘Confocal microscopy: important considerations for accurate imaging,’ in Methods in Cell Biology, B. K. Kay, H. B. Peng, eds. (Academic, London, 1993), Vol. 38, pp. 79–95.Majlof and Forsgren used a biological confocal microscope to image embedded 0.5 μm beads and measured the probe depth dependence of the Strehl intensity and lateral probe size.
    [CrossRef]
  11. C. J. R. Sheppard, M. Gu, K. Brain, H. Zhou, “Influence of spherical aberration on axial imaging of confocal reflection microscopy,” Appl. Opt. 33, 616–624 (1994), calculated the axial response of a confocal reflection microscope when the object was a perfect reflector.When the unbalanced primary spherical aberration was considered as a function of the semiangular aperture at a constant probe depth, the full width at half-maximum and the maximum value of the axial intensity distribution exhibited an oscillatory behavior because of the presence of strong axial lobes; these calculated results were confirmed experimentally. The occurrence of sidelobes in intensity plots for a perfect reflector using confocal detection suggests that there are sidelobes in the energy density distributions in the probe, although a one-to-one correspondence between the results of Sheppard et al.and our results would not be expected.
    [CrossRef] [PubMed]
  12. M. Born, E. Wolf, Principles of Optics4th ed. (Pergamon, Oxford, 1970), Chap. 9, p. 472.

1996 (1)

For media with n1 > n2, the diffraction integrals of Ref. 2 can be used. For this case, when the solid semiangle of the illumination is greater than the critical angle, irregular (or evanescent) waves are generated.This problem is the subject of another paper (P. Török, C. J. R. Sheppard, P. Varga, ‘Study of evanescent waves for transmission near-field optical microscopy,’ J. Mod. Opt. 43, 1167–1183 (1996).
[CrossRef]

1995 (3)

1994 (1)

1959 (1)

E. Wolf, ‘Electromagnetic diffraction in optical systems I. An integral representation of the image field,’ Proc.R. Soc. London 253, 349–357 (1959).
[CrossRef]

Booker, G. R.

Born, M.

In this paper we use the term Strehl intensity as the measure of the absolute maximum of the time-averaged electric energy density.This definition is slightly different from that given by, for example, M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970),Chap. 9, p. 462.

M. Born, E. Wolf, Principles of Optics4th ed. (Pergamon, Oxford, 1970), Chap. 9, p. 472.

Brain, K.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortan, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 9, p. 340.

Forsgren, P.-O.

Experimental evidence for such an irregular behavior was obtained by L. Majlof, P.-O. Forsgren, ‘Confocal microscopy: important considerations for accurate imaging,’ in Methods in Cell Biology, B. K. Kay, H. B. Peng, eds. (Academic, London, 1993), Vol. 38, pp. 79–95.Majlof and Forsgren used a biological confocal microscope to image embedded 0.5 μm beads and measured the probe depth dependence of the Strehl intensity and lateral probe size.
[CrossRef]

Gu, M.

Laczik, Z.

Majlof, L.

Experimental evidence for such an irregular behavior was obtained by L. Majlof, P.-O. Forsgren, ‘Confocal microscopy: important considerations for accurate imaging,’ in Methods in Cell Biology, B. K. Kay, H. B. Peng, eds. (Academic, London, 1993), Vol. 38, pp. 79–95.Majlof and Forsgren used a biological confocal microscope to image embedded 0.5 μm beads and measured the probe depth dependence of the Strehl intensity and lateral probe size.
[CrossRef]

Németh, G.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortan, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 9, p. 340.

Sheppard, C. J. R.

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions, 1st ed. (Adam Hilger, Bristol, UK, 1986), Chap. 12, p. 356.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortan, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 9, p. 340.

Török, P.

Varga, P.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortan, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 9, p. 340.

Wolf, E.

E. Wolf, ‘Electromagnetic diffraction in optical systems I. An integral representation of the image field,’ Proc.R. Soc. London 253, 349–357 (1959).
[CrossRef]

M. Born, E. Wolf, Principles of Optics4th ed. (Pergamon, Oxford, 1970), Chap. 9, p. 472.

In this paper we use the term Strehl intensity as the measure of the absolute maximum of the time-averaged electric energy density.This definition is slightly different from that given by, for example, M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970),Chap. 9, p. 462.

Zhou, H.

Appl. Opt. (1)

J. Mod. Opt. (1)

For media with n1 > n2, the diffraction integrals of Ref. 2 can be used. For this case, when the solid semiangle of the illumination is greater than the critical angle, irregular (or evanescent) waves are generated.This problem is the subject of another paper (P. Török, C. J. R. Sheppard, P. Varga, ‘Study of evanescent waves for transmission near-field optical microscopy,’ J. Mod. Opt. 43, 1167–1183 (1996).
[CrossRef]

J. Opt. Soc. Am. A (3)

Proc.R. Soc. London (1)

E. Wolf, ‘Electromagnetic diffraction in optical systems I. An integral representation of the image field,’ Proc.R. Soc. London 253, 349–357 (1959).
[CrossRef]

Other (6)

Experimental evidence for such an irregular behavior was obtained by L. Majlof, P.-O. Forsgren, ‘Confocal microscopy: important considerations for accurate imaging,’ in Methods in Cell Biology, B. K. Kay, H. B. Peng, eds. (Academic, London, 1993), Vol. 38, pp. 79–95.Majlof and Forsgren used a biological confocal microscope to image embedded 0.5 μm beads and measured the probe depth dependence of the Strehl intensity and lateral probe size.
[CrossRef]

In this paper we use the term Strehl intensity as the measure of the absolute maximum of the time-averaged electric energy density.This definition is slightly different from that given by, for example, M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970),Chap. 9, p. 462.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in fortan, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 9, p. 340.

This was also concluded from the paraxial approximation of our original diffraction integrals by Eqs. (51) and (52) of Ref. 2.

J. J. Stamnes, Waves in Focal Regions, 1st ed. (Adam Hilger, Bristol, UK, 1986), Chap. 12, p. 356.

M. Born, E. Wolf, Principles of Optics4th ed. (Pergamon, Oxford, 1970), Chap. 9, p. 472.

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Figures (7)

Fig. 1
Fig. 1

Diagram showing light focused by a lens into two media separated by a planar interface.

Fig. 2
Fig. 2

Time-averaged electric energy density distribution along the optical (z) axis as a function of probe depth, calculated for air (n1 = 1.0), silcon (n2 = 3.5) and wavelength λ = 1.3 μm. The numerical aperture is 0.85 (in air). The energy levels are plotted on a linear gray-scale range.

Fig. 3
Fig. 3

Strehl intensity as a function of numerical aperture and probe depth. The projection shows contours of constant Strehl intensities. Calculations were for (a) air (n1 = 1.0), silicon (n2 = 3.5), and λ = 1.3 μm, and for (b) air (n1 = 1.0), glass (n2 = 1.5), and λ = 0.6328 μm.

Fig. 4
Fig. 4

Focus shift as a function of numerical aperture and probe depth. The projection shows contours of constant focus shift values. Calculations were for (a) air (n1 = 1.0), silicon (n2 = 3.5), and λ= 1.3 μm, and for (b) air (n1 = 1.0), glass (n2 = 1.5), and λ = 0.6328 μm.

Fig. 5
Fig. 5

(a), (c) Lateral (x) probe size and (b), (d) axial (z) probe size as a function of numerical aperture and probe depth. The figures show 50 equally spaced contours of constant size values. Calculated for (a), (b) air (n1 = 1.0), silicon (n2 = 3.5), and λ = 1.3 μm, and for (c), (d) air (n1 = 1.0), glass n2 = 1.5), and λ= 0.6328 μm. The contour keys are given in micrometers.

Fig. 6
Fig. 6

Lateral (x) probe size as a function of probe depth. Calculated for air (n1 = 1.0), glass (n2 = 1.5), numerical aperture 0.9, and wavelength λ = 0.6328 μm.

Fig. 7
Fig. 7

Reproduction of Fig. 5(b), with the addition of letters indicating the paraxial region (A), the high-aperture regular region (B), and the high-aperture irregular region (C).

Equations (2)

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w e ( r p , z , θ p ) = 1 16 π ( e · e * ) ,
| A 40 | 0.22 λ ,

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