Abstract

Nonparaxial effects on optical imaging are considered. It is shown that the dominant effect is on the axial variation in the complex amplitude, affecting most strongly depth of focus, interference microscopy, and the imaging of objects with appreciable depth. A so-called pseudoparaxial approximation is introduced, which gives reasonable prediction of these effects up to quite large angular apertures.

© 1987 Optical Society of America

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References

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  1. B. Richards, “Diffraction in systems of high relative aperture,” in Astronomical Optics and Related Subjects, Z. Kopal, ed. (North-Holland, Amsterdam, 1955), pp. 352–359.
  2. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
    [CrossRef]
  3. A. Boivin, J. Dow, E. Wolf, “Energy flow in the neighborhood of the focus of a coherent beam,”J. Opt. Soc. Am. 57, 1171–1175 (1967).
    [CrossRef]
  4. C. J. R. Sheppard, T. Wilson, “The image of a single point in microscopes of large numerical aperture,” Proc. R. Soc. London Ser. A 379, 145–158 (1982).
    [CrossRef]
  5. C. J. R. Sheppard, “Electromagnetic field in the focal region of wide-angular annular lens and mirror systems,” Microwaves Opt. Acoust. 2, 163–166 (1978).
    [CrossRef]
  6. C. J. R. Sheppard, J. N. Gannaway, A. Choudhury, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” Microwaves Opt. Acoust. 1, 129–132 (1977)
    [CrossRef]
  7. R. Barakat, D. Lev, “Transfer function and total illuminance of high NA systems obeying the sine condition,”J. Opt. Soc. Am. 53, 324–332 (1963).
    [CrossRef]
  8. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).
  9. C. J. R. Sheppard, T. Wilson, “Effects of high angles of convergence on V(z) in the scanning acoustic microscope,” Appl. Phys. Lett. 38, 858–859 (1981).
    [CrossRef]
  10. C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,”J. Opt. Soc. Am. 54, 240–244 (1964).
    [CrossRef]
  11. C. J. R. Sheppard, A. Choudhury, “Imaging in the scanning microscope,” Opt. Acta 24, 1051–1073 (1977).
    [CrossRef]
  12. D. K. Hamilton, C. J. R. Sheppard, “A confocal interference microscope,” Opt. Acta 29, 1573–1577 (1982).
    [CrossRef]
  13. C. J. R. Sheppard, J. M. Heaton, “Images of surface steps in coherent illumination,” Optik 68, 267–280 (1984).
  14. C. J. R. Sheppard, J. M. Heaton, “Confocal images of straight edges and surface steps,” Optik 68, 371–380 (1984).

1984 (2)

C. J. R. Sheppard, J. M. Heaton, “Images of surface steps in coherent illumination,” Optik 68, 267–280 (1984).

C. J. R. Sheppard, J. M. Heaton, “Confocal images of straight edges and surface steps,” Optik 68, 371–380 (1984).

1982 (2)

D. K. Hamilton, C. J. R. Sheppard, “A confocal interference microscope,” Opt. Acta 29, 1573–1577 (1982).
[CrossRef]

C. J. R. Sheppard, T. Wilson, “The image of a single point in microscopes of large numerical aperture,” Proc. R. Soc. London Ser. A 379, 145–158 (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, 858–859 (1981).
[CrossRef]

1978 (1)

C. J. R. Sheppard, “Electromagnetic field in the focal region of wide-angular annular lens and mirror systems,” Microwaves Opt. Acoust. 2, 163–166 (1978).
[CrossRef]

1977 (2)

C. J. R. Sheppard, J. N. Gannaway, A. Choudhury, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” Microwaves Opt. Acoust. 1, 129–132 (1977)
[CrossRef]

C. J. R. Sheppard, A. Choudhury, “Imaging in the scanning microscope,” Opt. Acta 24, 1051–1073 (1977).
[CrossRef]

1967 (1)

A. Boivin, J. Dow, E. Wolf, “Energy flow in the neighborhood of the focus of a coherent beam,”J. Opt. Soc. Am. 57, 1171–1175 (1967).
[CrossRef]

1964 (1)

C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,”J. Opt. Soc. Am. 54, 240–244 (1964).
[CrossRef]

1963 (1)

R. Barakat, D. Lev, “Transfer function and total illuminance of high NA systems obeying the sine condition,”J. Opt. Soc. Am. 53, 324–332 (1963).
[CrossRef]

1959 (1)

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Barakat, R.

R. Barakat, D. Lev, “Transfer function and total illuminance of high NA systems obeying the sine condition,”J. Opt. Soc. Am. 53, 324–332 (1963).
[CrossRef]

Boivin, A.

A. Boivin, J. Dow, E. Wolf, “Energy flow in the neighborhood of the focus of a coherent beam,”J. Opt. Soc. Am. 57, 1171–1175 (1967).
[CrossRef]

Born, M.

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

Choudhury, A.

C. J. R. Sheppard, A. Choudhury, “Imaging in the scanning microscope,” Opt. Acta 24, 1051–1073 (1977).
[CrossRef]

C. J. R. Sheppard, J. N. Gannaway, A. Choudhury, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” Microwaves Opt. Acoust. 1, 129–132 (1977)
[CrossRef]

Dow, J.

A. Boivin, J. Dow, E. Wolf, “Energy flow in the neighborhood of the focus of a coherent beam,”J. Opt. Soc. Am. 57, 1171–1175 (1967).
[CrossRef]

Gannaway, J. N.

C. J. R. Sheppard, J. N. Gannaway, A. Choudhury, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” Microwaves Opt. Acoust. 1, 129–132 (1977)
[CrossRef]

Hamilton, D. K.

D. K. Hamilton, C. J. R. Sheppard, “A confocal interference microscope,” Opt. Acta 29, 1573–1577 (1982).
[CrossRef]

Heaton, J. M.

C. J. R. Sheppard, J. M. Heaton, “Images of surface steps in coherent illumination,” Optik 68, 267–280 (1984).

C. J. R. Sheppard, J. M. Heaton, “Confocal images of straight edges and surface steps,” Optik 68, 371–380 (1984).

Lev, D.

R. Barakat, D. Lev, “Transfer function and total illuminance of high NA systems obeying the sine condition,”J. Opt. Soc. Am. 53, 324–332 (1963).
[CrossRef]

McCutchen, C. W.

C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,”J. Opt. Soc. Am. 54, 240–244 (1964).
[CrossRef]

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

B. Richards, “Diffraction in systems of high relative aperture,” in Astronomical Optics and Related Subjects, Z. Kopal, ed. (North-Holland, Amsterdam, 1955), pp. 352–359.

Sheppard, C. J. R.

C. J. R. Sheppard, J. M. Heaton, “Confocal images of straight edges and surface steps,” Optik 68, 371–380 (1984).

C. J. R. Sheppard, J. M. Heaton, “Images of surface steps in coherent illumination,” Optik 68, 267–280 (1984).

D. K. Hamilton, C. J. R. Sheppard, “A confocal interference microscope,” Opt. Acta 29, 1573–1577 (1982).
[CrossRef]

C. J. R. Sheppard, T. Wilson, “The image of a single point in microscopes of large numerical aperture,” Proc. R. Soc. London Ser. A 379, 145–158 (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, 858–859 (1981).
[CrossRef]

C. J. R. Sheppard, “Electromagnetic field in the focal region of wide-angular annular lens and mirror systems,” Microwaves Opt. Acoust. 2, 163–166 (1978).
[CrossRef]

C. J. R. Sheppard, J. N. Gannaway, A. Choudhury, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” Microwaves Opt. Acoust. 1, 129–132 (1977)
[CrossRef]

C. J. R. Sheppard, A. Choudhury, “Imaging in the scanning microscope,” Opt. Acta 24, 1051–1073 (1977).
[CrossRef]

Wilson, T.

C. J. R. Sheppard, T. Wilson, “The image of a single point in microscopes of large numerical aperture,” Proc. R. Soc. London Ser. A 379, 145–158 (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, 858–859 (1981).
[CrossRef]

Wolf, E.

A. Boivin, J. Dow, E. Wolf, “Energy flow in the neighborhood of the focus of a coherent beam,”J. Opt. Soc. Am. 57, 1171–1175 (1967).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

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

Appl. Phys. Lett. (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, 858–859 (1981).
[CrossRef]

J. Opt. Soc. Am. (3)

C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image,”J. Opt. Soc. Am. 54, 240–244 (1964).
[CrossRef]

A. Boivin, J. Dow, E. Wolf, “Energy flow in the neighborhood of the focus of a coherent beam,”J. Opt. Soc. Am. 57, 1171–1175 (1967).
[CrossRef]

R. Barakat, D. Lev, “Transfer function and total illuminance of high NA systems obeying the sine condition,”J. Opt. Soc. Am. 53, 324–332 (1963).
[CrossRef]

Microwaves Opt. Acoust. (2)

C. J. R. Sheppard, “Electromagnetic field in the focal region of wide-angular annular lens and mirror systems,” Microwaves Opt. Acoust. 2, 163–166 (1978).
[CrossRef]

C. J. R. Sheppard, J. N. Gannaway, A. Choudhury, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” Microwaves Opt. Acoust. 1, 129–132 (1977)
[CrossRef]

Opt. Acta (2)

C. J. R. Sheppard, A. Choudhury, “Imaging in the scanning microscope,” Opt. Acta 24, 1051–1073 (1977).
[CrossRef]

D. K. Hamilton, C. J. R. Sheppard, “A confocal interference microscope,” Opt. Acta 29, 1573–1577 (1982).
[CrossRef]

Optik (2)

C. J. R. Sheppard, J. M. Heaton, “Images of surface steps in coherent illumination,” Optik 68, 267–280 (1984).

C. J. R. Sheppard, J. M. Heaton, “Confocal images of straight edges and surface steps,” Optik 68, 371–380 (1984).

Proc. R. Soc. London Ser. A (2)

C. J. R. Sheppard, T. Wilson, “The image of a single point in microscopes of large numerical aperture,” Proc. R. Soc. London Ser. A 379, 145–158 (1982).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Other (2)

B. Richards, “Diffraction in systems of high relative aperture,” in Astronomical Optics and Related Subjects, Z. Kopal, ed. (North-Holland, Amsterdam, 1955), pp. 352–359.

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

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Figures (8)

Fig. 1
Fig. 1

Geometry of the coordinate system.

Fig. 2
Fig. 2

Depth of field according to three different expressions as a function of numerical aperture. The exact expression (neglecting apodization effects) is ½sin2(α/2), whereas, according to the paraxial theory, depth of field is inversely proportional to sin2α. Assuming depth of field to be inversely proportional to α2 is a better approximation.

Fig. 3
Fig. 3

Geometry of a convergent beam.

Fig. 4
Fig. 4

Geometry of a confocal interference microscope.

Fig. 5
Fig. 5

Phase variation with defocus for a confocal system.

Fig. 6
Fig. 6

Phase gradient for small defocus in a confocal system. Curve 1 is from the exact high-angle theory; curves 2 and 3 are from paraxial approximations with two different definitions of the axial coordinate (see text).

Fig. 7
Fig. 7

Intensity at a phase step in a confocal system when focused at the center of the step, according to (a) paraxial theory, (b) exact high-angle theory, and (c) the pseudoparaxial theory.

Fig. 8
Fig. 8

As in Fig. 7 but focused on one side of the step.

Equations (29)

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E x ( u , v ) = - 2 i ( I 0 + I 2 cos 2 χ ) exp ( ¼ i u cosec 2 α / 2 ) , E y ( u , v ) = - 2 i I 2 sin 2 χ exp ( ¼ i u cosec 2 α / 2 ) , E z ( u , v ) = - 2 I 1 cos ϕ exp ( ¼ i u cosec 2 α / 2 ) ,
I 0 ( u , v ) = 0 α cos 1 / 2 θ sin θ ( cos 2 θ 2 ) J 0 ( v sin θ sin α ) × exp ( - ½ i u sin 2 θ / 2 sin 2 α / 2 ) d θ , I 1 ( u , v ) = 0 α cos 1 / 2 θ sin θ J 1 ( v sin θ sin α ) × exp ( - ½ i u sin 2 θ / 2 sin 2 α / 2 ) d θ , I 2 ( u , v ) = 0 α cos 1 / 2 θ sin θ ( sin 2 θ z ) J 2 ( v sin θ sin α ) × exp ( - ½ i u sin 2 θ / 2 sin 2 α / 2 ) d θ ,
v = k r sin α
u = 4 k z sin 2 α 2 .
I 0 ( u , v ) = α 2 0 1 J 0 ( v ρ ) exp ( - ½ i u ρ 2 ) ρ d ρ .
A ( θ ) = sec 2 θ 2 .
U ( u , v ) = - 2 i exp ( ¼ i u cosec 2 α 2 ) × 0 α A ( θ ) J 0 ( v sin θ sin α ) exp ( - ½ i u sin 2 θ / 2 sin 2 α / 2 ) sin θ d θ .
s = sin θ / 2 sin α / 2 ,
U ( u , 0 ) = - 8 i sin 2 α 2 exp ( ¼ i u cosec 2 α 2 ) × 0 α A ( s ) exp ( - ½ i u s 2 ) s d s ;
U ( u , 0 ) = - 4 i sin 2 α 2 exp ( ¼ i u cot 2 α 2 ) ( sin u / 4 u / 4 )
= - 4 i sin 2 α 2 exp ( i k z cos 2 α 2 ) ( sin u / 4 u / 4 ) .
I ( 0 , u ) = 16 [ sin ( k r ) k r ] 2 .
I ( z ) = | 0 π / 2 [ S P o 1 ( θ ) P o 2 ( θ ) exp ( 2 i k z cos θ ) + R P r 1 ( θ ) P r 2 ( θ ) exp ( i ϕ ) ] sin θ cos θ d θ | 2 .
I ( z ) = 2 R S Re [ 0 π / 2 P o 1 ( θ ) P o 2 ( θ ) × exp ( 2 i k z cos θ - i ϕ ) sin θ cos θ d θ ] ,
R exp ( - i ϕ ) = 0 π / 2 P * r 1 ( θ ) P * r 2 ( θ ) sin θ cos θ d θ .
I ( z ) = 2 R S cos ( ψ - ϕ ) ,
S exp ( i ψ ) = S 0 π / 2 P s 1 ( ϕ ) P * s 2 ( ϕ ) × exp ( 2 i k z cos θ ) sin θ cos θ d θ .
ψ = u 2 cot 2 α 2 + tan - 1 { tan 2 α 2 [ 1 u / 2 - 1 tan ( u / 2 ) ] } ,
ψ = 2 k z { 1 - sin 2 α 2 [ 1 - tan 2 α 2 ] } .
ψ = 2 k z cos 2 α 2 ,
I ( z ) = 0 π / 2 S P o 1 ( θ ) P o 2 ( θ ) exp ( 2 i k z cos θ ) + R P r 1 ( θ ) P r 2 ( θ ) exp ( i ϕ ) 2 sin θ cos θ d θ .
I ( z ) = 2 S R Re [ e - i ϕ 0 π / 2 P o 1 ( θ ) P o 2 ( θ ) + P * r 1 ( θ ) P * r 2 ( θ ) × exp ( 2 i k z cos θ - i ϕ ) sin θ cos θ d θ ] .
m ˜ = sin θ / sin α .
c ( m ˜ , u ) = ( 1 - m ˜ 2 sin 2 α ) - 1 / 4 exp i u ( 1 - m ˜ 2 sin 2 α ) 4 sin 2 α / 2 , m ˜ < 1.
c ( m ˜ , u ) = exp ( i u 4 sin 2 α / 2 ) exp ( - i u m ˜ 2 2 ) ( sin 2 α 4 sin 2 α / 2 )
= exp ( i k z ) exp ( - i k z m ˜ 2 2 ) ,
c ( m ˜ , u ) = exp ( i u 4 sin 2 α / 2 ) exp ( - i u m ˜ 2 2 ) ,
2 sin α 2 = sin α ,
I ( ϕ ) = ¼ | U ( ϕ + ϕ 2 ) + U ( ϕ - ϕ 2 ) | 2 ,

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