Abstract

We treat the problem of a coherent spherical wave passing through a circular aperture and subsequently focused by a lens. Using the Huygens–Fresnel integral representation, an expression is derived for the axial intensity beyond the lens. It predicts that the large-F-number focal shift may be reversed or eliminated, depending on the aperture–lens separation.

© 1987 Optical Society of America

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References

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  1. The exposition of the Lommel analysis used here is from M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).
  2. M. E. Hufford, H. T. Davis, “The diffraction of light by a circular opening and the Lommel wave theory,” Phys. Rev. 33, 589–597 (1929).
    [CrossRef]
  3. G. W. Farnell, “Calculated intensity and phase distribution in the image space of a microwave lens,” Can. J. Phys. 35, 777–784 (1957).
    [CrossRef]
  4. G. W. Farnell, “Measured phase distribution in the image space of a microwave lens,” Can. J. Phys. 36, 935–943 (1958).
    [CrossRef]
  5. D. A. Holmes, J. E. Korka, P. V. Avizonas, “Parametric study of apertured focused Gaussian beams,” Appl. Opt. 3, 565–574 (1972).
    [CrossRef]
  6. U. Farrukh, “Diffraction and focusing of truncated Gaussian beams,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1974).
  7. 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]
  8. M. A. Gusinow, M. E. Riley, M. A. Palmer, “Focusing in a large F-number optical system,” Opt. Quantum Electron. 9, 465–471 (1977).
    [CrossRef]
  9. E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
    [CrossRef]
  10. Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
    [CrossRef]
  11. Y. Li, “Dependence of the focal shift on Fresnel number and F number,”J. Opt. Soc. Am. 72, 770–774 (1982).
    [CrossRef]
  12. Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
    [CrossRef]
  13. Y. Li, H. Platzer, “An experimental investigation of diffraction patterns in low-fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
    [CrossRef]
  14. 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]
  15. R. M. Simonds, “Analysis of the focal shift of converging spherical waves using the convolution method,” J. Opt. Soc. Am. A 2, 830–832 (1985).
    [CrossRef]
  16. Y. Li, “Phase distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 2, 1677–1686 (1985).
    [CrossRef]
  17. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  18. I. S. Gradshteyn, I. M. Ryghik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 717.
  19. R. G. Wenzel, J. M. Telle, J. L. Carlsten, “Fresnel diffraction in an optical system containing lenses,” J. Opt. Soc. Am. A 3, 838–842 (1986).
    [CrossRef]
  20. R. G. Wenzel, “Fresnel diffraction in an optical system with focusing,” in Proceedings of the International Conference on Lasers ’85, Las Vegas, Nev., Dec. 2–6, 1985 (Society of Optical and Quantum Electronics, McLean, Va., 1986).
  21. B. Newnam, “Laser-induced damage phenomena in dielectric films, solids, and inorganic liquids,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1972).

1986 (1)

1985 (2)

1984 (1)

1983 (1)

Y. Li, H. Platzer, “An experimental investigation of diffraction patterns in low-fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
[CrossRef]

1982 (2)

Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
[CrossRef]

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

1981 (2)

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]

1977 (1)

M. A. Gusinow, M. E. Riley, M. A. Palmer, “Focusing in a large F-number optical system,” Opt. Quantum Electron. 9, 465–471 (1977).
[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]

1972 (1)

D. A. Holmes, J. E. Korka, P. V. Avizonas, “Parametric study of apertured focused Gaussian beams,” Appl. Opt. 3, 565–574 (1972).
[CrossRef]

1958 (1)

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

1957 (1)

G. W. Farnell, “Calculated intensity and phase distribution in the image space of a microwave lens,” Can. J. Phys. 35, 777–784 (1957).
[CrossRef]

1929 (1)

M. E. Hufford, H. T. Davis, “The diffraction of light by a circular opening and the Lommel wave theory,” Phys. Rev. 33, 589–597 (1929).
[CrossRef]

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]

Avizonas, P. V.

D. A. Holmes, J. E. Korka, P. V. Avizonas, “Parametric study of apertured focused Gaussian beams,” Appl. Opt. 3, 565–574 (1972).
[CrossRef]

Born, M.

The exposition of the Lommel analysis used here is from M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

Carlsten, J. L.

Davis, H. T.

M. E. Hufford, H. T. Davis, “The diffraction of light by a circular opening and the Lommel wave theory,” Phys. Rev. 33, 589–597 (1929).
[CrossRef]

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]

G. W. Farnell, “Calculated intensity and phase distribution in the image space of a microwave lens,” Can. J. Phys. 35, 777–784 (1957).
[CrossRef]

Farrukh, U.

U. Farrukh, “Diffraction and focusing of truncated Gaussian beams,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1974).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryghik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 717.

Gusinow, M. A.

M. A. Gusinow, M. E. Riley, M. A. Palmer, “Focusing in a large F-number optical system,” Opt. Quantum Electron. 9, 465–471 (1977).
[CrossRef]

Holmes, D. A.

D. A. Holmes, J. E. Korka, P. V. Avizonas, “Parametric study of apertured focused Gaussian beams,” Appl. Opt. 3, 565–574 (1972).
[CrossRef]

Hufford, M. E.

M. E. Hufford, H. T. Davis, “The diffraction of light by a circular opening and the Lommel wave theory,” Phys. Rev. 33, 589–597 (1929).
[CrossRef]

Korka, J. E.

D. A. Holmes, J. E. Korka, P. V. Avizonas, “Parametric study of apertured focused Gaussian beams,” Appl. Opt. 3, 565–574 (1972).
[CrossRef]

Li, Y.

Y. Li, “Phase distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 2, 1677–1686 (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, H. Platzer, “An experimental investigation of diffraction patterns in low-fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
[CrossRef]

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

Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (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]

Newnam, B.

B. Newnam, “Laser-induced damage phenomena in dielectric films, solids, and inorganic liquids,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1972).

Palmer, M. A.

M. A. Gusinow, M. E. Riley, M. A. Palmer, “Focusing in a large F-number optical system,” Opt. Quantum Electron. 9, 465–471 (1977).
[CrossRef]

Platzer, H.

Y. Li, H. Platzer, “An experimental investigation of diffraction patterns in low-fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
[CrossRef]

Riley, M. E.

M. A. Gusinow, M. E. Riley, M. A. Palmer, “Focusing in a large F-number optical system,” Opt. Quantum Electron. 9, 465–471 (1977).
[CrossRef]

Ryghik, I. M.

I. S. Gradshteyn, I. M. Ryghik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 717.

Simonds, R. M.

Telle, J. M.

Wenzel, R. G.

R. G. Wenzel, J. M. Telle, J. L. Carlsten, “Fresnel diffraction in an optical system containing lenses,” J. Opt. Soc. Am. A 3, 838–842 (1986).
[CrossRef]

R. G. Wenzel, “Fresnel diffraction in an optical system with focusing,” in Proceedings of the International Conference on Lasers ’85, Las Vegas, Nev., Dec. 2–6, 1985 (Society of Optical and Quantum Electronics, McLean, Va., 1986).

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 shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
[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]

The exposition of the Lommel analysis used here is from M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

Appl. Opt. (1)

D. A. Holmes, J. E. Korka, P. V. Avizonas, “Parametric study of apertured focused Gaussian beams,” Appl. Opt. 3, 565–574 (1972).
[CrossRef]

Can. J. Phys. (2)

G. W. Farnell, “Calculated intensity and phase distribution in the image space of a microwave lens,” Can. J. Phys. 35, 777–784 (1957).
[CrossRef]

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. (1)

J. Opt. Soc. Am. A (4)

Opt. Acta (2)

Y. Li, H. Platzer, “An experimental investigation of diffraction patterns in low-fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
[CrossRef]

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. (3)

Y. Li, E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (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]

Opt. Quantum Electron. (1)

M. A. Gusinow, M. E. Riley, M. A. Palmer, “Focusing in a large F-number optical system,” Opt. Quantum Electron. 9, 465–471 (1977).
[CrossRef]

Phys. Rev. (1)

M. E. Hufford, H. T. Davis, “The diffraction of light by a circular opening and the Lommel wave theory,” Phys. Rev. 33, 589–597 (1929).
[CrossRef]

Other (6)

The exposition of the Lommel analysis used here is from M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

U. Farrukh, “Diffraction and focusing of truncated Gaussian beams,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1974).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

I. S. Gradshteyn, I. M. Ryghik, Table of Integrals, Series, and Products (Academic, New York, 1980), p. 717.

R. G. Wenzel, “Fresnel diffraction in an optical system with focusing,” in Proceedings of the International Conference on Lasers ’85, Las Vegas, Nev., Dec. 2–6, 1985 (Society of Optical and Quantum Electronics, McLean, Va., 1986).

B. Newnam, “Laser-induced damage phenomena in dielectric films, solids, and inorganic liquids,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1972).

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Figures (7)

Fig. 1
Fig. 1

The field variables in the Huygens–Fresnel integral formulation of the diffraction problem.

Fig. 2
Fig. 2

The radially symmetric problem treated here, defining the variables used.

Fig. 3
Fig. 3

Fresnel number F versus distance z for z0 = 0, f, 2f. For these plots, a = 2 mm, λ = 0.5 μm, f = 50 cm, and R0 + z0 = 300 cm.

Fig. 4
Fig. 4

Experimental radial intensity profiles in the neighborhood of focus.

Fig. 5
Fig. 5

Axial intensity in the neighborhood of the focus of a 100-cm lens for 0.5-μm light. The ordinates are intensity in units of the intensity at the geometrical focus. The aperture radius for A is 20 mm, and that for B is 1 mm. The abscissa of A extends 2.5 mm on either side of focus, whereas that of B extends 1000 mm on either side of focus.

Fig. 6
Fig. 6

Calculated axial intensity profiles for z0 = 0, f, 2f. For these plots, a = 1.5 mm, λ = 0.5 μm, f = 100 cm, and R0 = ∞.

Fig. 7
Fig. 7

Calculated axial intensity (in units of the intensity at the aperture) for plane-wave diffraction in the experimental configuration used by Newnam, i.e., a = 0.56 mm, λ = 0.6943 μm, z0 = 109.5 cm, f = 20.7 cm, and R0 = 154.5 cm. The Kirchhoff and Huygens–Fresnel plots were indistinguishable. The region near the aperture image was deleted because the oscillation frequency became too great for the calculational grid.

Tables (1)

Tables Icon

Table 1 Values of q for Which F = ±1 from Eq. (20)a

Equations (31)

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U ( x , y , z ) = e i k z i λ z d x d y ( U ( x , y ) × exp { i k 2 z [ ( x - x ) 2 + ( y - y ) 2 ] } ) ,
U ( r 0 ) = 2 π e i k z 0 i λ z 0 0 a d r 1 { r 1 U ( r 1 ) × exp [ i k 2 z 0 ( r 0 2 + r 1 2 ) ] J 0 ( k r 0 r 1 z 0 ) } ,
U ( r 1 ) = exp ( i k r 1 2 2 R 0 ) .
U ( r 0 ) = 2 π e i k z 0 i λ z 0 exp ( i k r 0 2 2 z 0 ) × 0 a d r 1 { r 1 exp [ i k r 1 2 2 ( 1 R 0 + 1 z 0 ) ] J 0 ( k r 0 r 1 z 0 ) } .
U ( r 0 ) = U ( r 0 ) exp ( - i k r 0 2 2 f ) ,
U ( z ) = 2 π e i k z i λ z 0 d r 0 [ r 0 U ( r 0 ) exp ( i k r 0 2 2 z ) ] .
U ( z ) = - 4 π 2 exp [ i k ( z 0 + z ) ] λ 2 z 0 z 0 d r 0 0 a d r 1 ( r 0 r 1 J 0 ( k r 0 r 1 z 0 ) × exp { i k 2 [ r 0 2 ( 1 z 0 + 1 z - 1 f ) + r 1 2 ( 1 R 0 + 1 z 0 ) ] } ) .
0 x ν + 1 exp ( ± i α x 2 ) J ν ( β x ) d x = β ν ( 2 α ) ν + 1 exp [ ± i ( ν + 1 2 π - β 2 4 α ) ] .
U ( z ) = - 2 π i f exp [ i k ( z 0 + z ) ] λ ( z 0 f + z f - z 0 z ) 0 a d r 1 { r 1 exp [ i π F ( z ) r 1 2 a 2 ] } ,
F ( z ) = a 2 λ { 1 / [ z 0 - z f / ( z - f ) ] + 1 / R 0 } .
U ( z ) = R 0 f exp [ i k ( z 0 + z ) ] ( R 0 + z 0 ) ( z - f ) - z f { exp [ i π F ( z ) ] - 1 } .
G ( z ) = ( R 0 f ) / [ ( R 0 + z 0 ) ( z - f ) - z f ]
U ( z ) = G ( z ) exp [ i k ( z 0 + z ) ] ( exp [ i π F ( z ) ] - 1 ) .
I ( z ) = U ( z ) 2 = { 2 G ( z ) sin [ π 2 F ( z ) ] } 2 .
q = z - f ( R 0 + z 0 ) R 0 + z 0 - f ,
G ( q ) = - 1 q R 0 f ( R 0 + z 0 - f )
F ( q ) = - a 2 q λ R 0 { ( R 0 + z 0 - f ) 2 R 0 f 2 - [ R 0 ( z 0 - f ) + ( z 0 - f ) 2 ] q } .
I ( q = 0 ) = lim q 0 { 2 G ( q ) sin [ π 2 F ( q ) ] } 2 = [ π a 2 λ R 0 f ( R 0 + z 0 - f ) ] 2 ,
q ( F ) = λ f 2 F 1 + ( z 0 - f ) / R 0 { 1 λ ( z 0 - f ) F - a 2 [ 1 + ( z 0 - f ) / R 0 ] } .
q ( F ) = - λ f 2 F λ f F + a 2 .
F ( z 0 = f ) = - a 2 q λ f 2 .
I ( z 0 = f ) = [ π a 2 λ f sin ( u / 4 ) u / 4 ] 2 ,
G ( q ) = - ( f / q )
F ( q ) = - a 2 λ f q / f 1 + q / f ,
G ( F ) = 1 + a 2 / λ f F .
I ( F ) = [ π a 2 λ f ( 1 + λ f F a 2 ) sin ( π F / 2 ) π F / 2 ] 2 .
I ( z ) = [ 2 G ( z ) sin ½ θ ( z ) ] 2 ,
θ = 2 π λ { a 2 2 ( 1 z 0 + 1 R 0 ) - z z 0 f ( z 0 f - z 0 z + z f ) - z z 0 f [ ( z 0 f - z 0 z + z f ) 2 - a 2 f 2 ] 1 / 2 } .
a f z 0 f - z 0 z + z f ,
q = - λ f 2 F / a 2 ,
ν = - λ f 2 F ν / q a 2

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