Abstract

An appropriate form for the expansion of an aberration function for an optical system of high numerical aperture is considered. The effects on the defocus signal of a confocal imaging system of aberrations, high aperture, finite Fresnel number, system configuration, and surface tilt are discussed.

© 1988 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), p. 461.
  2. C. J. R. Sheppard, “Imaging in High-Aperture Optical Systems,” J. Opt. Soc. Am. A 14, 1353 (1987).
  3. N. N. Lebedev, Special Functions and Their Applications (Dover, New York, 1972), p. 198.
  4. C. J. R. Sheppard, A. Choudhury, “Imaging in the Scanning Microscope,” Opt. Acta 24, 1051 (1977).
    [CrossRef]
  5. C. J. R. Sheppard, T. Wilson, “Depth of Field in the Scanning Microscope,” Opt. Lett. 3, 115 (1978).
    [CrossRef] [PubMed]
  6. I. J. Cox, D. K. Hamilton, C. J. R. Sheppard, “Observation of Optical Signatures of Materials,” Appl. Phys. Lett. 41, 604 (1982).
    [CrossRef]
  7. T. R. Corle, C.-H. Chou, G. S. Kino, “Depth Response of Confocal Optical Microscopes,” Opt. Lett. 11, 770 (1986).
    [CrossRef] [PubMed]
  8. D. K. Hamilton, C. J. R. Sheppard, “Interferometric Measurements of the Complex Amplitude of the Defocus Signal V(z) in the Confocal Scanning Optical Microscope,” J. Appl. Phys. 60, 2708 (1986).
    [CrossRef]
  9. H. J. Matthews, D. K. Hamilton, C. J. R. Sheppard, “Aberration Measurement by Confocal Interferometry,” to be published.
  10. C. J. R. Sheppard, T. Wilson, “Effects of High Angles of Convergence on V(z) in the Scanning Acoustic Microscope,” Appl. Phys. Lett. 38, 858 (1981).
    [CrossRef]
  11. A. Atalar, “An Angular Spectrum Approach to Contrast in Reflection Acoustic Microscopy,” J. Appl. Phys. 49, 5130 (1978).
    [CrossRef]
  12. C. J. R. Sheppard, “Imaging in Optical Systems of Finite Fresnel Number,” J. Opt. Soc. Am. A 3, 1428 (1986).
    [CrossRef]
  13. J. Stamnes, Waves in Focal Regions (Adam Hilger, London, 1987).
  14. C.-H. Chou, G. S. Kino, “The Evaluation of V(z) in a Type II Reflection Microscope,” unpublished.
  15. K. K. Liang, G. S. Kino, B. T. Khuri-Yakub, “Material Characterisation by the Inversion of V(z),” IEEE Trans. Sonics Ultrason. SU-32, 213 (1985).
    [CrossRef]
  16. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), Eq. (7.438.39).
  17. C. J. R. Sheppard, D. K. Hamilton, I. J. Cox, “Optical Microscopy with Extended Depth of Field,” Proc. R. Soc. London Ser. A 387, 171 (1983).
    [CrossRef]

1987 (1)

C. J. R. Sheppard, “Imaging in High-Aperture Optical Systems,” J. Opt. Soc. Am. A 14, 1353 (1987).

1986 (3)

T. R. Corle, C.-H. Chou, G. S. Kino, “Depth Response of Confocal Optical Microscopes,” Opt. Lett. 11, 770 (1986).
[CrossRef] [PubMed]

D. K. Hamilton, C. J. R. Sheppard, “Interferometric Measurements of the Complex Amplitude of the Defocus Signal V(z) in the Confocal Scanning Optical Microscope,” J. Appl. Phys. 60, 2708 (1986).
[CrossRef]

C. J. R. Sheppard, “Imaging in Optical Systems of Finite Fresnel Number,” J. Opt. Soc. Am. A 3, 1428 (1986).
[CrossRef]

1985 (1)

K. K. Liang, G. S. Kino, B. T. Khuri-Yakub, “Material Characterisation by the Inversion of V(z),” IEEE Trans. Sonics Ultrason. SU-32, 213 (1985).
[CrossRef]

1983 (1)

C. J. R. Sheppard, D. K. Hamilton, I. J. Cox, “Optical Microscopy with Extended Depth of Field,” Proc. R. Soc. London Ser. A 387, 171 (1983).
[CrossRef]

1982 (1)

I. J. Cox, D. K. Hamilton, C. J. R. Sheppard, “Observation of Optical Signatures of Materials,” Appl. Phys. Lett. 41, 604 (1982).
[CrossRef]

1981 (1)

C. J. R. Sheppard, T. Wilson, “Effects of High Angles of Convergence on V(z) in the Scanning Acoustic Microscope,” Appl. Phys. Lett. 38, 858 (1981).
[CrossRef]

1978 (2)

A. Atalar, “An Angular Spectrum Approach to Contrast in Reflection Acoustic Microscopy,” J. Appl. Phys. 49, 5130 (1978).
[CrossRef]

C. J. R. Sheppard, T. Wilson, “Depth of Field in the Scanning Microscope,” Opt. Lett. 3, 115 (1978).
[CrossRef] [PubMed]

1977 (1)

C. J. R. Sheppard, A. Choudhury, “Imaging in the Scanning Microscope,” Opt. Acta 24, 1051 (1977).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), Eq. (7.438.39).

Atalar, A.

A. Atalar, “An Angular Spectrum Approach to Contrast in Reflection Acoustic Microscopy,” J. Appl. Phys. 49, 5130 (1978).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), p. 461.

Chou, C.-H.

T. R. Corle, C.-H. Chou, G. S. Kino, “Depth Response of Confocal Optical Microscopes,” Opt. Lett. 11, 770 (1986).
[CrossRef] [PubMed]

C.-H. Chou, G. S. Kino, “The Evaluation of V(z) in a Type II Reflection Microscope,” unpublished.

Choudhury, A.

C. J. R. Sheppard, A. Choudhury, “Imaging in the Scanning Microscope,” Opt. Acta 24, 1051 (1977).
[CrossRef]

Corle, T. R.

Cox, I. J.

C. J. R. Sheppard, D. K. Hamilton, I. J. Cox, “Optical Microscopy with Extended Depth of Field,” Proc. R. Soc. London Ser. A 387, 171 (1983).
[CrossRef]

I. J. Cox, D. K. Hamilton, C. J. R. Sheppard, “Observation of Optical Signatures of Materials,” Appl. Phys. Lett. 41, 604 (1982).
[CrossRef]

Hamilton, D. K.

D. K. Hamilton, C. J. R. Sheppard, “Interferometric Measurements of the Complex Amplitude of the Defocus Signal V(z) in the Confocal Scanning Optical Microscope,” J. Appl. Phys. 60, 2708 (1986).
[CrossRef]

C. J. R. Sheppard, D. K. Hamilton, I. J. Cox, “Optical Microscopy with Extended Depth of Field,” Proc. R. Soc. London Ser. A 387, 171 (1983).
[CrossRef]

I. J. Cox, D. K. Hamilton, C. J. R. Sheppard, “Observation of Optical Signatures of Materials,” Appl. Phys. Lett. 41, 604 (1982).
[CrossRef]

H. J. Matthews, D. K. Hamilton, C. J. R. Sheppard, “Aberration Measurement by Confocal Interferometry,” to be published.

Khuri-Yakub, B. T.

K. K. Liang, G. S. Kino, B. T. Khuri-Yakub, “Material Characterisation by the Inversion of V(z),” IEEE Trans. Sonics Ultrason. SU-32, 213 (1985).
[CrossRef]

Kino, G. S.

T. R. Corle, C.-H. Chou, G. S. Kino, “Depth Response of Confocal Optical Microscopes,” Opt. Lett. 11, 770 (1986).
[CrossRef] [PubMed]

K. K. Liang, G. S. Kino, B. T. Khuri-Yakub, “Material Characterisation by the Inversion of V(z),” IEEE Trans. Sonics Ultrason. SU-32, 213 (1985).
[CrossRef]

C.-H. Chou, G. S. Kino, “The Evaluation of V(z) in a Type II Reflection Microscope,” unpublished.

Lebedev, N. N.

N. N. Lebedev, Special Functions and Their Applications (Dover, New York, 1972), p. 198.

Liang, K. K.

K. K. Liang, G. S. Kino, B. T. Khuri-Yakub, “Material Characterisation by the Inversion of V(z),” IEEE Trans. Sonics Ultrason. SU-32, 213 (1985).
[CrossRef]

Matthews, H. J.

H. J. Matthews, D. K. Hamilton, C. J. R. Sheppard, “Aberration Measurement by Confocal Interferometry,” to be published.

Sheppard, C. J. R.

C. J. R. Sheppard, “Imaging in High-Aperture Optical Systems,” J. Opt. Soc. Am. A 14, 1353 (1987).

D. K. Hamilton, C. J. R. Sheppard, “Interferometric Measurements of the Complex Amplitude of the Defocus Signal V(z) in the Confocal Scanning Optical Microscope,” J. Appl. Phys. 60, 2708 (1986).
[CrossRef]

C. J. R. Sheppard, “Imaging in Optical Systems of Finite Fresnel Number,” J. Opt. Soc. Am. A 3, 1428 (1986).
[CrossRef]

C. J. R. Sheppard, D. K. Hamilton, I. J. Cox, “Optical Microscopy with Extended Depth of Field,” Proc. R. Soc. London Ser. A 387, 171 (1983).
[CrossRef]

I. J. Cox, D. K. Hamilton, C. J. R. Sheppard, “Observation of Optical Signatures of Materials,” Appl. Phys. Lett. 41, 604 (1982).
[CrossRef]

C. J. R. Sheppard, T. Wilson, “Effects of High Angles of Convergence on V(z) in the Scanning Acoustic Microscope,” Appl. Phys. Lett. 38, 858 (1981).
[CrossRef]

C. J. R. Sheppard, T. Wilson, “Depth of Field in the Scanning Microscope,” Opt. Lett. 3, 115 (1978).
[CrossRef] [PubMed]

C. J. R. Sheppard, A. Choudhury, “Imaging in the Scanning Microscope,” Opt. Acta 24, 1051 (1977).
[CrossRef]

H. J. Matthews, D. K. Hamilton, C. J. R. Sheppard, “Aberration Measurement by Confocal Interferometry,” to be published.

Stamnes, J.

J. Stamnes, Waves in Focal Regions (Adam Hilger, London, 1987).

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), Eq. (7.438.39).

Wilson, T.

C. J. R. Sheppard, T. Wilson, “Effects of High Angles of Convergence on V(z) in the Scanning Acoustic Microscope,” Appl. Phys. Lett. 38, 858 (1981).
[CrossRef]

C. J. R. Sheppard, T. Wilson, “Depth of Field in the Scanning Microscope,” Opt. Lett. 3, 115 (1978).
[CrossRef] [PubMed]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), p. 461.

Appl. Phys. Lett. (2)

I. J. Cox, D. K. Hamilton, C. J. R. Sheppard, “Observation of Optical Signatures of Materials,” Appl. Phys. Lett. 41, 604 (1982).
[CrossRef]

C. J. R. Sheppard, T. Wilson, “Effects of High Angles of Convergence on V(z) in the Scanning Acoustic Microscope,” Appl. Phys. Lett. 38, 858 (1981).
[CrossRef]

IEEE Trans. Sonics Ultrason. (1)

K. K. Liang, G. S. Kino, B. T. Khuri-Yakub, “Material Characterisation by the Inversion of V(z),” IEEE Trans. Sonics Ultrason. SU-32, 213 (1985).
[CrossRef]

J. Appl. Phys. (2)

A. Atalar, “An Angular Spectrum Approach to Contrast in Reflection Acoustic Microscopy,” J. Appl. Phys. 49, 5130 (1978).
[CrossRef]

D. K. Hamilton, C. J. R. Sheppard, “Interferometric Measurements of the Complex Amplitude of the Defocus Signal V(z) in the Confocal Scanning Optical Microscope,” J. Appl. Phys. 60, 2708 (1986).
[CrossRef]

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

C. J. R. Sheppard, “Imaging in High-Aperture Optical Systems,” J. Opt. Soc. Am. A 14, 1353 (1987).

C. J. R. Sheppard, “Imaging in Optical Systems of Finite Fresnel Number,” J. Opt. Soc. Am. A 3, 1428 (1986).
[CrossRef]

Opt. Acta (1)

C. J. R. Sheppard, A. Choudhury, “Imaging in the Scanning Microscope,” Opt. Acta 24, 1051 (1977).
[CrossRef]

Opt. Lett. (2)

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

C. J. R. Sheppard, D. K. Hamilton, I. J. Cox, “Optical Microscopy with Extended Depth of Field,” Proc. R. Soc. London Ser. A 387, 171 (1983).
[CrossRef]

Other (6)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), p. 461.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), Eq. (7.438.39).

J. Stamnes, Waves in Focal Regions (Adam Hilger, London, 1987).

C.-H. Chou, G. S. Kino, “The Evaluation of V(z) in a Type II Reflection Microscope,” unpublished.

H. J. Matthews, D. K. Hamilton, C. J. R. Sheppard, “Aberration Measurement by Confocal Interferometry,” to be published.

N. N. Lebedev, Special Functions and Their Applications (Dover, New York, 1972), p. 198.

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Figures (4)

Fig. 1
Fig. 1

Geometry of a lens of high aperture.

Fig. 2
Fig. 2

Geometry of a confocal system.

Fig. 3
Fig. 3

Defocus signal intensity I for confocal systems of various apertures: (a) aberration-free, A = 0; (b) with primary spherical aberration, A = π.

Fig. 4
Fig. 4

Defocus signal intensity I predicted by a small angle theory for a surface tilted at an angle γ: (a) true intensities; (b) normalized intensities.

Equations (26)

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U ( v , u , ψ ) = exp ( 1 4 i u csc 2 α 2 ) 0 1 0 2 π exp i [ k Φ ( ρ , ϕ ) - v ρ cos ( ϕ - ψ ) - 1 2 u ρ 2 ] d ϕ ρ d ρ .
v = k r sin α , u = 4 k z sin 2 α 2 , }
Φ ( ρ , ϕ ) = n = 0 m = 0 A n m R n m ( ρ ) cos m θ ,
U ( v , u , ψ ) = exp ( ¼ i u csc 2 α 2 ) 0 α 0 2 π × exp i [ k Φ - v sin θ sin α cos ( ϕ - ψ ) - ½ u sin 2 ( θ / 2 ) sin 2 ( α / 2 ) ] × cos 1 / 2 θ sin θ d ϕ d θ ,
Φ ( θ , Φ ) = n = 0 m = 0 B n m cos m ϕ P n m ( cos θ ) ,
U = 0 2 π 0 1 exp i [ k B n m cos m ϕ P n m ( 1 - 2 s 2 sin 2 α 2 ) ] × ( 1 - 2 s 2 sin 2 α 2 ) 1 / 2 × 4 s sin α 2 d s d ϕ ,
s = sin ( θ / 2 ) sin ( α / 2 ) .
u = 4 k B 10 sin 2 α 2 ,
v = - k B 11 sin α ,
Φ = B 20 P 2 0 ( 1 - 2 s 2 sin 2 α 2 )
= B 20 ( 1 - 6 s 2 sin 2 α 2 + 6 s 4 sin 4 α 2 ) .
P n m ( cos θ ) = sin m θ 2 m i = 0 n - m ( - ) i + m ( m + n + i ) ! ( n - m - i ) ! ( m + i ) ! ( i ) ! × sin 2 i θ 2 .
E ( u ) = exp ( 1 2 i u csc 2 α 2 ) 0 a 0 2 π E inc ( r , ϕ ) P ( r , ϕ ) × P ( r , π - ϕ ) R ( θ ) exp [ - i u sin 2 ( θ / 2 ) sin 2 ( α / 2 ) ] r d r ,
r = f sin θ d r = f cos θ d θ } ,
U ( u ) = exp ( ½ i u csc 2 α 2 ) 0 α 0 2 π P ( θ , ϕ ) P ( θ , π - ϕ ) R ( θ ) × exp [ - i u sin 2 ( θ / 2 ) sin 2 ( α / 2 ) ] × sin θ cos θ d ϕ d θ .
U ( u ) = exp ( ½ i u csc 2 α 2 ) 0 α 0 2 π P ( θ , ϕ ) P ( θ , π - ϕ ) R ( θ ) × exp [ - i u sin 2 ( θ / 2 ) sin 2 ( α / 2 ) ] sin θ d ϕ d θ .
t = sin 2 ( θ / 2 ) / sin 2 ( α / 2 )
U ( u ) = 2 π exp ( 1 2 i u csc 2 α 2 ) exp ( 2 i k B 0 ) × 0 1 exp i ( 2 A t 2 - u A t ) ( 1 - 2 t sin 2 α 2 ) d t ,
A = 6 k B 20 sin 4 ( α / 2 )
exp i ( a x 2 + 2 b c ) d x = π 2 a exp ( - i b 2 a ) F ( 2 a π ( a x + b ) ) ,
F ( p ) = 0 p exp ( i π 2 t 2 ) d t ,
= C ( p ) + i S ( p ) ,
u = u + 2 A cot 2 ( α / 2 )
U ( u ) = [ F ( A π ) ] - 1 exp { i A ( u 2 A ) [ 1 - ( u 2 A ) ] 2 } × { 1 2 ( 1 + u 2 A tan 2 α 2 ) { F [ A π ( 1 + u 2 A ) ] + F [ A π ( 1 - u 2 A ) ] } - 1 π A tan 2 α 2 sin u 2 exp i A 2 [ 1 - ( u 2 A ) 2 ] } .
u / 2 A = - cot 2 ( α / 2 ) ,
A = n π tan 2 ( α / 2 ) ,

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