Abstract

Imaging, rather than focusing, in systems of finite Fresnel number is considered. Incoherent systems, coherent systems, and scanning systems are discussed. Scanning systems exhibit spatially invariant imaging, while for conventional systems, coherent imaging is necessarily space variant. The optical transfer functions for incoherent systems are presented. Focusing with a circular pupil is compared with Gaussian beam propagation.

© 1986 Optical Society of America

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References

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  1. J. H. Erkkila, M. E. Rogers, “Diffracted fields in the focal volume of a converging wave,” J. Opt. Soc. Am. 71, 904–905 (1981).
    [CrossRef]
  2. J. J. Stamnes, S. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt Commun. 40, 81–85 (1981).
    [CrossRef]
  3. Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt Soc. Am. A 1, 801–808 (1984).
    [CrossRef]
  4. H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
    [CrossRef]
  5. Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
    [CrossRef]
  6. T. Wilson, C. J. R. Sheppard, “Imaging with finite values of Fresnel number,” J. Opt. Soc. Am. 72, 1639–1641 (1982).
    [CrossRef]
  7. Rayleigh, “On pinhole photography,” Phil. Mag. 31, 87–99 (1891); also in Scientific Papers III (Cambridge U. Press, 1902), pp. 429–440.
  8. O. E. Myers, “Studies of transmission zone plates,” Am. J. Phys. 19, 359–365 (1951).
    [CrossRef]
  9. J. Zemanek, “Beam behavior within the near-field of a vibrating piston,” J. Acoust. Soc. Am. 49, 181–191 (1971).
    [CrossRef]
  10. J. C. Lockwood, J. G. Willette, “High-speed method for computing the exact solution for the pressure variations in the near-field of a baffled piston,” J. Acoust. Soc. Am. 53, 735–741 (1973).
    [CrossRef]
  11. H. T. O’Neill, “Theory of focusing radiators,” J. Acoust. Soc. Am. 21, 516–526 (1949).
    [CrossRef]
  12. B. G. Lucas, T. G. Muir, “The field of a focusing source,” J. Acoust. Soc. Am. 72, 1289 (1982).
    [CrossRef]
  13. P. A. Stockseth, “Properties of a defocused optical system,” J. Opt. Soc. Am. 59, 1314–1321 (1969).
    [CrossRef]
  14. A. Arimoto, “Intensity distribution of aberration-free diffraction patterns due to circular apertures in large F-number optical systems,” Opt. Acta 23, 245–250 (1976).
    [CrossRef]
  15. M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970).
  16. H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
    [CrossRef]
  17. W. T. Cathey, “Angular spectrum and coherent imaging,” J. Opt. Soc. Am. 64, 1503–1506 (1974).
    [CrossRef]

1984 (1)

Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt Soc. Am. A 1, 801–808 (1984).
[CrossRef]

1982 (2)

T. Wilson, C. J. R. Sheppard, “Imaging with finite values of Fresnel number,” J. Opt. Soc. Am. 72, 1639–1641 (1982).
[CrossRef]

B. G. Lucas, T. G. Muir, “The field of a focusing source,” J. Acoust. Soc. Am. 72, 1289 (1982).
[CrossRef]

1981 (3)

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

J. H. Erkkila, M. E. Rogers, “Diffracted fields in the focal volume of a converging wave,” J. Opt. Soc. Am. 71, 904–905 (1981).
[CrossRef]

J. J. Stamnes, S. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt Commun. 40, 81–85 (1981).
[CrossRef]

1976 (1)

A. Arimoto, “Intensity distribution of aberration-free diffraction patterns due to circular apertures in large F-number optical systems,” Opt. Acta 23, 245–250 (1976).
[CrossRef]

1974 (1)

1973 (1)

J. C. Lockwood, J. G. Willette, “High-speed method for computing the exact solution for the pressure variations in the near-field of a baffled piston,” J. Acoust. Soc. Am. 53, 735–741 (1973).
[CrossRef]

1971 (1)

J. Zemanek, “Beam behavior within the near-field of a vibrating piston,” J. Acoust. Soc. Am. 49, 181–191 (1971).
[CrossRef]

1969 (1)

1966 (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

1955 (1)

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
[CrossRef]

1951 (1)

O. E. Myers, “Studies of transmission zone plates,” Am. J. Phys. 19, 359–365 (1951).
[CrossRef]

1949 (1)

H. T. O’Neill, “Theory of focusing radiators,” J. Acoust. Soc. Am. 21, 516–526 (1949).
[CrossRef]

1891 (1)

Rayleigh, “On pinhole photography,” Phil. Mag. 31, 87–99 (1891); also in Scientific Papers III (Cambridge U. Press, 1902), pp. 429–440.

Arimoto, A.

A. Arimoto, “Intensity distribution of aberration-free diffraction patterns due to circular apertures in large F-number optical systems,” Opt. Acta 23, 245–250 (1976).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970).

Cathey, W. T.

Erkkila, J. H.

Hopkins, H. H.

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
[CrossRef]

Kogelnik, H.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Li, T.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Li, Y.

Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt Soc. Am. A 1, 801–808 (1984).
[CrossRef]

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

Lockwood, J. C.

J. C. Lockwood, J. G. Willette, “High-speed method for computing the exact solution for the pressure variations in the near-field of a baffled piston,” J. Acoust. Soc. Am. 53, 735–741 (1973).
[CrossRef]

Lucas, B. G.

B. G. Lucas, T. G. Muir, “The field of a focusing source,” J. Acoust. Soc. Am. 72, 1289 (1982).
[CrossRef]

Muir, T. G.

B. G. Lucas, T. G. Muir, “The field of a focusing source,” J. Acoust. Soc. Am. 72, 1289 (1982).
[CrossRef]

Myers, O. E.

O. E. Myers, “Studies of transmission zone plates,” Am. J. Phys. 19, 359–365 (1951).
[CrossRef]

O’Neill, H. T.

H. T. O’Neill, “Theory of focusing radiators,” J. Acoust. Soc. Am. 21, 516–526 (1949).
[CrossRef]

Rayleigh,

Rayleigh, “On pinhole photography,” Phil. Mag. 31, 87–99 (1891); also in Scientific Papers III (Cambridge U. Press, 1902), pp. 429–440.

Rogers, M. E.

Sheppard, C. J. R.

Spjelkavik, S.

J. J. Stamnes, S. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt Commun. 40, 81–85 (1981).
[CrossRef]

Stamnes, J. J.

J. J. Stamnes, S. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt Commun. 40, 81–85 (1981).
[CrossRef]

Stockseth, P. A.

Willette, J. G.

J. C. Lockwood, J. G. Willette, “High-speed method for computing the exact solution for the pressure variations in the near-field of a baffled piston,” J. Acoust. Soc. Am. 53, 735–741 (1973).
[CrossRef]

Wilson, T.

Wolf, E.

Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt Soc. Am. A 1, 801–808 (1984).
[CrossRef]

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970).

Zemanek, J.

J. Zemanek, “Beam behavior within the near-field of a vibrating piston,” J. Acoust. Soc. Am. 49, 181–191 (1971).
[CrossRef]

Am. J. Phys. (1)

O. E. Myers, “Studies of transmission zone plates,” Am. J. Phys. 19, 359–365 (1951).
[CrossRef]

J. Acoust. Soc. Am. (4)

J. Zemanek, “Beam behavior within the near-field of a vibrating piston,” J. Acoust. Soc. Am. 49, 181–191 (1971).
[CrossRef]

J. C. Lockwood, J. G. Willette, “High-speed method for computing the exact solution for the pressure variations in the near-field of a baffled piston,” J. Acoust. Soc. Am. 53, 735–741 (1973).
[CrossRef]

H. T. O’Neill, “Theory of focusing radiators,” J. Acoust. Soc. Am. 21, 516–526 (1949).
[CrossRef]

B. G. Lucas, T. G. Muir, “The field of a focusing source,” J. Acoust. Soc. Am. 72, 1289 (1982).
[CrossRef]

J. Opt Soc. Am. A (1)

Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt Soc. Am. A 1, 801–808 (1984).
[CrossRef]

J. Opt. Soc. Am. (4)

Opt Commun. (1)

J. J. Stamnes, S. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt Commun. 40, 81–85 (1981).
[CrossRef]

Opt. Acta (1)

A. Arimoto, “Intensity distribution of aberration-free diffraction patterns due to circular apertures in large F-number optical systems,” Opt. Acta 23, 245–250 (1976).
[CrossRef]

Opt. Commun. (1)

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

Phil. Mag. (1)

Rayleigh, “On pinhole photography,” Phil. Mag. 31, 87–99 (1891); also in Scientific Papers III (Cambridge U. Press, 1902), pp. 429–440.

Proc. IEEE (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

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

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
[CrossRef]

Other (1)

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970).

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Figures (4)

Fig. 1
Fig. 1

Gaussian beam geometry.

Fig. 2
Fig. 2

Optical transfer function for an incoherent system of finite Fresnel number, as a function of defocus.

Fig. 3
Fig. 3

Geometry of focusing of a wave to geometrical focus F.

Fig. 4
Fig. 4

Geometry in a plane perpendicular to the optic axis.

Equations (47)

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f = ( f + z ) [ 1 + { π W 0 2 λ ( f + z 0 ) } 2 ] ,
z = π W 0 4 λ 2 ( f + z 0 ) .
a 2 = W 0 2 [ 1 + { λ ( f + z 0 ) π W 0 2 } 2 ] ,
N = a 2 / λ F ,
z 0 f = ( W 0 a ) 2 = 1 1 + π 2 N 2 .
W 1 2 = W 0 2 ( 1 + π 2 N 2 ) π 2 N 2 ,
W 1 a = 1 π N ,
W 1 λ = 1 π ( f a ) .
W 0 λ = N ( f / a ) ( 1 + π 2 N 2 ) 1 / 2 .
υ = π N ( r / a ) ,
U ( υ ) / U ( 0 ) = exp ( υ 2 / 4 ) exp ( j υ 2 / 4 π N ) ,
υ = 2 π N r / a 1 + z / f ,
u = 2 π N z / f 1 + z / f ,
sin α = sin α 0 1 ( 1 + z / f ) .
U ( υ ) = 2 J 1 ( υ ) υ exp ( j υ 2 4 π N ) .
U ( υ x s , υ y s ) = + U ( υ x , υ y ) t ( υ x υ x s , υ y υ y s ) d υ x d υ y .
c ( m , n ) = + u ( υ x , υ y ) exp [ j ( υ x m + υ y n ) ] d υ x d υ y ,
c ( m , 0 ) = 0 2 J 1 ( υ ) υ exp ( j υ 2 4 π N ) J 0 ( m υ ) υ d υ .
U ( υ , u ) = B ( u ) exp j Φ ( u , υ ) 0 1 J 0 ( υ p ) exp ( j u ρ 2 / 2 ) ρ d ρ ,
exp j Φ ( u , υ ) = F ( u ) exp ( j υ 2 4 π N 2 u )
c ( m ) = 0 U ( υ , u ) J 0 ( υ m ) υ d υ = B ( u ) F ( u ) 0 0 1 J 0 ( υ ρ ) J 0 ( υ m ) exp ( j u ρ 2 / 2 )
× exp ( j υ 2 4 π N 2 u ) ρ d ρ υ d υ .
0 exp [ a 2 x 2 ] J 0 ( p x ) J 0 ( q x ) x d x = 1 2 a 2 exp ( p 2 + q 2 4 a 2 ) I 0 ( p q 2 a 2 )
c ( m ) = G ( u ) exp { j m 2 ( π N u 2 ) } × 0 1 exp ( j ρ 2 π N ) J 0 [ 2 ρ m ( π N u 2 ) ] ρ d ρ ,
m = m 1 + z / f
m π N = m ( π N u / 2 ) ,
c ( m ) = exp { j m 2 ( π 2 N 2 π N u / 2 ) } × 0 1 exp ( j ρ 2 π N ) J 0 ( 2 ρ m π N ) ρ d ρ .
c ( m ) = j exp { j m 2 π N ( 1 + z f ) } [ V 0 ( 2 π N , 2 π N m ) + j V 1 ( 2 π N , 2 π N m ) exp ( j π N ( m 2 + 1 ) ) ] m < 1 = exp { j m 2 π N ( 1 + z f ) } × [ U 1 ( 2 π N , 2 π N m ) j U 2 ( 2 π N , 2 π N m ) ] m 1.
V 0 ( 2 π N , 0 ) = 1 , V 1 ( 2 π N , 0 ) = 0 ,
c ( 1 ) = j 2 exp { j π N ( 1 + z f ) } [ e 2 π N J 0 ( 2 π N ) ] ,
U ( P ) = A j λ S σ s exp [ j k ( σ s ) ] d S ,
α s = ( z 0 z ) + [ ( x 1 x 0 ) 2 + ( y 1 y 0 ) 2 2 ( f + z 0 ) ] [ ( x 1 x ) 2 + ( y 1 y ) 2 2 ( f + z ) ] .
σ s = ( z 0 z ) + r 1 2 + r 0 2 2 ( f + z 0 ) r 1 2 + r 2 2 ( f + z ) r 1 { r 0 f + z 0 cos ( θ 1 θ 0 ) r f + z cos ( θ 1 θ ) } .
υ = k a r f + z , υ 0 = k a r 0 f + z 0 ,
u = k a 2 f + z , u 0 = k a 2 f + z 0 ,
ρ = r 1 / a ,
k ( σ s ) = 4 π 2 N 2 ( f a ) 2 ( 1 u 0 1 u ) + k 2 N ( υ 0 2 u 0 υ 2 u ) υ ρ cos ( θ 1 θ + β ) ,
tan β = υ 0 sin ( θ θ 0 ) υ 0 cos ( θ θ 0 ) υ ,
U ( P ) = 2 π A j λ ( u u 0 ) exp { j 4 π 2 N 2 ( f a ) 2 ( 1 u 0 1 u ) } × exp { j k 2 N ( υ 0 2 u 0 υ 2 u ) } 0 1 exp ( 1 / 2 j ρ 2 u ) J 0 ( υ ρ ) ρ d ρ .
U 0 = 2 π N ,
U ( P ) = 2 π A j λ ( u 2 π N ) × exp { j k f ( 1 2 π N u ) } exp { j k ( υ 0 2 4 π N υ 2 u ) } × 0 1 exp 1 2 j ρ 2 ( 2 π N u ) J 0 ( υ ρ ) ρ d ρ .
U ( P ) = 2 π A j λ exp { j k 4 π N ( υ 0 2 υ 2 ) } 0 1 J 0 ( υ ρ ) ρ d ρ
= π A j λ exp { j k 4 π N ( υ 0 2 υ 2 ) } [ 2 J 1 ( υ ) υ ] .
υ x = υ cos θ , υ x 0 = υ 0 cos θ 0 , υ x = υ x + υ x 0 ; υ y = υ sin θ , υ y 0 = υ 0 sin θ 0 , υ y = υ y + υ y 0 .
U ( P ) = π A j λ exp { j k 4 π N ( υ x 0 2 + υ y 0 2 υ x 2 υ y 2 ) } h ( υ x , υ y ) ,
U ( υ x 0 , υ y 0 ) = π A j λ + exp j k 4 π N ( υ x 0 2 + υ y 0 2 υ x 2 υ y 2 ) × t ( υ x , υ y ) h ( υ x υ x 0 , υ y υ y 0 ) d υ x d υ y .
U ( υ x 0 ) = exp { j k υ x 0 2 4 π N } + exp { j k υ x 0 2 4 π N } t ( υ x ) h ( υ x = υ x 0 ) d υ x .

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