Abstract

Three-dimensional phase distribution near the focal plane is demonstrated by cross-section profiles of cophasal surfaces. The phase distribution in the focal plane and the phase anomaly in the focus are discussed in greater detail. Numerical results obtained are arranged systematically with Fresnel numbers sampled between 0.5 and 100, a sampling that covers conventional and unconventional cases.

© 1985 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 8.8.
  2. A. Rubinowicz, “On the anomalous propagation of phase in the focus,” Phys. Rev. 54, 931–936 (1938).
    [CrossRef]
  3. E. H. Linfoot, E. Wolf, “Phase distribution near focus in an aberration-free diffraction image,” Proc. Phys. Soc. London 69, 823–832 (1956).
    [CrossRef]
  4. R. W. Boyd, “Intuitive explanation of the phase anomaly of focused light beams,” J. Opt. Soc. Am. 70, 877–880 (1980).
    [CrossRef]
  5. J. J. Stamnes, “Focusing of two-dimensional waves,” J. Opt. Soc. Am. 71, 15–31 (1981).
    [CrossRef]
  6. H. J. Erkkila, M. E. Rogers, “Diffracted field in the focal volume of a converging wave,” J. Opt. Soc. Am. 71, 904–905 (1981).
    [CrossRef]
  7. J. J. Stamnes, B. Spjelkavik, “Focusing of small angular aperture in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
    [CrossRef]
  8. E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
    [CrossRef]
  9. Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
    [CrossRef]
  10. Y. Li, “Dependence of the focal shift on Fresnel number and f number,” J. Opt. Soc. Am. 72, 770–774 (1982).
    [CrossRef]
  11. Y. Li, “Encircled energy for systems of different Fresnel numbers,” Optik 64, 207–218 (1983).
  12. 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]
  13. G. W. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, London, 1962), Sec. 16.5.
  14. G. W. Farnell, “Measured phase distribution in the image space of a microwave lens,” Can. J. Phys. 36, 935–943 (1958).
    [CrossRef]
  15. Y. Li, “The anomalous propagation of phase in the focus: a reexamination,” Opt. Commun. 53, 359–363 (1985).
    [CrossRef]

1985 (1)

Y. Li, “The anomalous propagation of phase in the focus: a reexamination,” Opt. Commun. 53, 359–363 (1985).
[CrossRef]

1984 (1)

1983 (1)

Y. Li, “Encircled energy for systems of different Fresnel numbers,” Optik 64, 207–218 (1983).

1982 (1)

1981 (5)

J. J. Stamnes, “Focusing of two-dimensional waves,” J. Opt. Soc. Am. 71, 15–31 (1981).
[CrossRef]

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

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

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

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

1980 (1)

1958 (1)

G. W. Farnell, “Measured phase distribution in the image space of a microwave lens,” Can. J. Phys. 36, 935–943 (1958).
[CrossRef]

1956 (1)

E. H. Linfoot, E. Wolf, “Phase distribution near focus in an aberration-free diffraction image,” Proc. Phys. Soc. London 69, 823–832 (1956).
[CrossRef]

1938 (1)

A. Rubinowicz, “On the anomalous propagation of phase in the focus,” Phys. Rev. 54, 931–936 (1938).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 8.8.

Boyd, R. W.

Erkkila, H. J.

Farnell, G. W.

G. W. Farnell, “Measured phase distribution in the image space of a microwave lens,” Can. J. Phys. 36, 935–943 (1958).
[CrossRef]

Li, Y.

Y. Li, “The anomalous propagation of phase in the focus: a reexamination,” Opt. Commun. 53, 359–363 (1985).
[CrossRef]

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, “Encircled energy for systems of different Fresnel numbers,” Optik 64, 207–218 (1983).

Y. Li, “Dependence of the focal shift on Fresnel number and f number,” J. Opt. Soc. Am. 72, 770–774 (1982).
[CrossRef]

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

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

Linfoot, E. H.

E. H. Linfoot, E. Wolf, “Phase distribution near focus in an aberration-free diffraction image,” Proc. Phys. Soc. London 69, 823–832 (1956).
[CrossRef]

Rogers, M. E.

Rubinowicz, A.

A. Rubinowicz, “On the anomalous propagation of phase in the focus,” Phys. Rev. 54, 931–936 (1938).
[CrossRef]

Spjelkavik, B.

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

Stamnes, J. J.

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

J. J. Stamnes, “Focusing of two-dimensional waves,” J. Opt. Soc. Am. 71, 15–31 (1981).
[CrossRef]

Watson, G. W.

G. W. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, London, 1962), Sec. 16.5.

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]

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

E. H. Linfoot, E. Wolf, “Phase distribution near focus in an aberration-free diffraction image,” Proc. Phys. Soc. London 69, 823–832 (1956).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 8.8.

Can. J. Phys. (1)

G. W. Farnell, “Measured phase distribution in the image space of a microwave lens,” Can. J. Phys. 36, 935–943 (1958).
[CrossRef]

J. Opt. Soc. Am. (4)

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

Opt. Commun. (4)

Y. Li, “The anomalous propagation of phase in the focus: a reexamination,” Opt. Commun. 53, 359–363 (1985).
[CrossRef]

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

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

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

Optik (1)

Y. Li, “Encircled energy for systems of different Fresnel numbers,” Optik 64, 207–218 (1983).

Phys. Rev. (1)

A. Rubinowicz, “On the anomalous propagation of phase in the focus,” Phys. Rev. 54, 931–936 (1938).
[CrossRef]

Proc. Phys. Soc. London (1)

E. H. Linfoot, E. Wolf, “Phase distribution near focus in an aberration-free diffraction image,” Proc. Phys. Soc. London 69, 823–832 (1956).
[CrossRef]

Other (2)

G. W. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, London, 1962), Sec. 16.5.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 8.8.

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Figures (6)

Fig. 1
Fig. 1

Illustration of the notation.

Fig. 2
Fig. 2

Phase distribution in the focal plane in systems with different Fresnel numbers N. OR1, OR2, and OR3 are the radii of the first three dark rings in the Airy pattern.

Fig. 3
Fig. 3

Profiles of cophasal surfaces in the immediate neighborhood of the focal plane in an F/3.5 system with different Fresnel numbers N. OR1 and OR2 are radii of the first and the second dark rings in the Airy pattern.

Fig. 4
Fig. 4

Phase anomaly along geometrical rays through the focus in an F/3.5 system with different Fresnel numbers N. The angle θ denotes the inclination of the ray to the axis.

Fig. 5
Fig. 5

Difference between phase anomaly δN evaluated at the points z = −0.2f and z = +0.2f in an F/3.5 system.

Fig. 6
Fig. 6

Rapidity of the change of δN in the focus.

Equations (32)

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ϕ N ( u N , υ N ) = 1 1 u N / 2 π N [ 4 F 2 u N + 1 4 π N υ N 2 ] α ( u N , υ N ) π 2 , ( mod 2 π ) .
u N = 2 π N z / f 1 + z / f ,
u N = 2 π N r / a 1 + z / f [ r = ( x 2 + y 2 ) 1 / 2 ] ,
sin α = S ( C 2 + S 2 ) 1 / 2 , cos α = C ( C 2 + S 2 ) 1 / 2 ,
C i S = 2 0 1 J 0 ( υ N ρ ) exp ( i 2 u N ρ 2 ) ρ d ρ .
ϕ N ( 0 , 0 ) = π 2 ± m π , ( m = 0 , 1 , 2 , ) .
ϕ N ( 0 , 0 ) = π 2
ϕ N ( 0 , υ ) = 1 4 π N υ 2 α ( 0 , υ ) π 2 , ( mod 2 π ) .
υ = 2 π N r a .
C = Besinc ( υ ) = 2 J 1 ( υ ) / υ and S = 0 .
sin α = 0 and cos α = Besinc ( υ ) / | Besinc ( υ ) | .
0 r < R 1 , R 2 p < r < R 2 p + 1 ,
R 2 p 1 < r < R 2 p , ( p = 1 , 2 , 3 , ) .
ϕ N ( 0 , υ ) = 1 4 π N υ 2 π 2 ( mod 2 π )
ϕ N ( 0 , υ ) = 1 4 π N υ 2 + π 2 ( mod 2 π )
1 1 u N / 2 π N ( 4 F 2 u N + 1 4 π N u N 2 ) 1 2 u N + β ( u N , υ N ) = constant ,
sin β = U 2 ( U 1 2 + U 2 2 ) 1 / 2 , cos β = U 1 ( U 1 2 + U 2 2 ) 1 / 2 ,
sin β = V 0 cos Θ [ ( V 0 cos Θ ) 2 + ( V 1 + sin Θ ) 2 ] 1 / 2 , cos β = V 1 + sin Θ [ ( V 0 cos Θ ) 2 + ( V 1 + sin Θ ) 2 ] 1 / 2 , Θ = 1 2 ( u N + υ N 2 u N ) .
G ( z , r ; ϕ N ) = 0 ,
κ = | G z G r ( G z r + G r z ) ( G r ) 2 G z z ( G z ) 2 G r r ( G z 2 + G r 2 ) 3 / 2 | z = 0 r = 0 ,
κ = 1 f ( 1 1 / 16 F 2 ) .
= 1 κ f
υ N u N = 2 F tan θ .
ϕ N ( u N , θ ) = 4 F 2 u N 1 u N / 2 π N ( 1 + u N 4 π N tan 2 θ ) α ( u N , θ ) π 2 ( mod 2 π ) .
D ( P ) = exp [ i ϕ ( u N , θ ) ] R / f , 0 θ tan 1 ( 2 F ) ,
ϕ ( u N , θ ) = k R when u N 0 = + k R when u N 0 } .
R = ( z 2 + r 2 ) 1 / 2 = 1 k 4 F 2 | u N | 1 u N / 2 π N sec θ
δ N ( u N , θ ) = ϕ N ( u N , θ ) ϕ ( u N , θ ) ,
= 4 F 2 u N 1 u N / 2 π N [ ( 1 sec θ ) + u N 4 π N tan 2 θ ] α ( u N , θ ) π 2 , ( mod 2 π ) , 0 tan θ ( 2 F ) 1 ,
Δ = δ N ( u N , θ ) z = 0.2 f δ N ( u N , θ ) z = + 0.2 f ,
δ N ( 0 , θ ) = ( δ N u N d u N d z ) z = 0 u N = 0 = π N 2 [ 1 + 16 F 2 ( sec θ 1 ) ] , 0 tan θ ( 2 F ) 1 .
δ N ( 0 , θ ) = π N 2 [ 1 + 8 ( F tan θ ) 2 ] , 0 ( F tan θ ) 0.5 .

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