Abstract

A simple modification of the known Maréchal intensity formula is proposed. It enables one to evaluate the near-the-diffraction-focus intensity distribution for well-corrected optical systems with small Fresnel numbers of the focusing geometry. As an example, the focal-shift effect in perfect systems is briefly reexamined.

© 1983 Optical Society of America

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References

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  1. A. Maréchal, Rev. Opt. 2, 257–277 (1947).
  2. D. D. Lowenthal, Appl. Opt. 13, 2126–2133 (1974); A. Yoshida, Appl. Opt. 21, 1812–1816 (1982).
    [CrossRef] [PubMed]
  3. S. Szapiel, Opt. Lett. 2, 124–126 (1978).
    [CrossRef] [PubMed]
  4. S. Szapiel, J. Opt. Soc. Am. 72, 947–956 (1982); V. N. Mahajan, J. Opt. Soc. Am. 72, 1258–1266 (1982).
    [CrossRef]
  5. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), Chap. 9.
  6. A. Boivin, Théorie et Calcul des Figures de Diffraction de Révolution (Laval U. Press, Quebec, Canada, 1964), pp. 42–64.
  7. Y. Li, E. Wolf, Opt. Commun. 39, 211–215 (1981).
    [CrossRef]
  8. Y. Li, E. Wolf, Opt. Commun. 42, 151–156 (1982).
    [CrossRef]
  9. Y. Li, J. Opt. Soc. Am. 72, 770–774 (1982).
    [CrossRef]
  10. For a detailed bibliography concerning the focal shift, see Ref. 9; also see Ref. 6, pp. 69–80.
  11. See, for example, Ref. 5, p. 183, Eq. (10).
  12. The coefficient W20 represents the relevant wave-front deviation at the pupil edge, expressed in wavelengths.
  13. J. D. Zook, T. C. Lee, Appl. Opt. 11, 2140–2145 (1972).
    [CrossRef] [PubMed]
  14. D. A. Holmes, J. E. Korka, P. V. Avizonis, Appl. Opt. 11, 565–574 (1972).
    [CrossRef] [PubMed]
  15. The same values were plotted versus the relative axial distance z/R in Ref. 14, p. 567, Fig. 2.
  16. A second solution of Eq. (20) does not satisfy the condition δ2iT/δD2 < 0 that is necessary for the existing maximum value of the normalized intensity iT
  17. Specifically, in the case of uniform illumination, the focal-shift parameters obtained from expressions (18), (21), and (22) are in a good agreement with the exact numerical results that have been calculated in Ref. 9 by using the Young–Rubinowicz diffraction theory. When F = π, for instance, from Eq. (22) we obtain (ΔR/R)opt = −0.385, whereas Li9 gives in this case (N = 1; f-number → ∞) the relative focal shift Δf/f = −0.398.
  18. The necessary integrations are performed in Refs. 2–4. See, for example, Ref. 4, Eq. (A2).
  19. C. S. Williams, Appl. Opt. 12, 872–876 (1973), specifically, see Eq. (21) on p. 874.
    [CrossRef] [PubMed]
  20. S. Szapiel, K. Patorski, Opt. Acta 26, 439–446 (1979).
    [CrossRef]
  21. A small quantitive influence of the f-number is observed only in the low-Fresnel-number and low-f-number systems. See Ref. 9 for details.
  22. See, for example, G. Wade, ed., Acoustic Imaging (Plenum, New York, 1976), Chap. 11; S. Cornbleet, Microwave Optics (Academic, New York, 1976), Chap. 3; also see Refs. 13 and 14.

1982 (3)

1981 (1)

Y. Li, E. Wolf, Opt. Commun. 39, 211–215 (1981).
[CrossRef]

1979 (1)

S. Szapiel, K. Patorski, Opt. Acta 26, 439–446 (1979).
[CrossRef]

1978 (1)

1974 (1)

1973 (1)

1972 (2)

1947 (1)

A. Maréchal, Rev. Opt. 2, 257–277 (1947).

Avizonis, P. V.

Boivin, A.

A. Boivin, Théorie et Calcul des Figures de Diffraction de Révolution (Laval U. Press, Quebec, Canada, 1964), pp. 42–64.

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), Chap. 9.

Holmes, D. A.

Korka, J. E.

Lee, T. C.

Li, Y.

Y. Li, E. Wolf, Opt. Commun. 42, 151–156 (1982).
[CrossRef]

Y. Li, J. Opt. Soc. Am. 72, 770–774 (1982).
[CrossRef]

Y. Li, E. Wolf, Opt. Commun. 39, 211–215 (1981).
[CrossRef]

Lowenthal, D. D.

Maréchal, A.

A. Maréchal, Rev. Opt. 2, 257–277 (1947).

Patorski, K.

S. Szapiel, K. Patorski, Opt. Acta 26, 439–446 (1979).
[CrossRef]

Szapiel, S.

Williams, C. S.

Wolf, E.

Y. Li, E. Wolf, Opt. Commun. 42, 151–156 (1982).
[CrossRef]

Y. Li, E. Wolf, Opt. Commun. 39, 211–215 (1981).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), Chap. 9.

Zook, J. D.

Appl. Opt. (4)

J. Opt. Soc. Am. (2)

Opt. Acta (1)

S. Szapiel, K. Patorski, Opt. Acta 26, 439–446 (1979).
[CrossRef]

Opt. Commun. (2)

Y. Li, E. Wolf, Opt. Commun. 39, 211–215 (1981).
[CrossRef]

Y. Li, E. Wolf, Opt. Commun. 42, 151–156 (1982).
[CrossRef]

Opt. Lett. (1)

Rev. Opt. (1)

A. Maréchal, Rev. Opt. 2, 257–277 (1947).

Other (11)

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), Chap. 9.

A. Boivin, Théorie et Calcul des Figures de Diffraction de Révolution (Laval U. Press, Quebec, Canada, 1964), pp. 42–64.

For a detailed bibliography concerning the focal shift, see Ref. 9; also see Ref. 6, pp. 69–80.

See, for example, Ref. 5, p. 183, Eq. (10).

The coefficient W20 represents the relevant wave-front deviation at the pupil edge, expressed in wavelengths.

A small quantitive influence of the f-number is observed only in the low-Fresnel-number and low-f-number systems. See Ref. 9 for details.

See, for example, G. Wade, ed., Acoustic Imaging (Plenum, New York, 1976), Chap. 11; S. Cornbleet, Microwave Optics (Academic, New York, 1976), Chap. 3; also see Refs. 13 and 14.

The same values were plotted versus the relative axial distance z/R in Ref. 14, p. 567, Fig. 2.

A second solution of Eq. (20) does not satisfy the condition δ2iT/δD2 < 0 that is necessary for the existing maximum value of the normalized intensity iT

Specifically, in the case of uniform illumination, the focal-shift parameters obtained from expressions (18), (21), and (22) are in a good agreement with the exact numerical results that have been calculated in Ref. 9 by using the Young–Rubinowicz diffraction theory. When F = π, for instance, from Eq. (22) we obtain (ΔR/R)opt = −0.385, whereas Li9 gives in this case (N = 1; f-number → ∞) the relative focal shift Δf/f = −0.398.

The necessary integrations are performed in Refs. 2–4. See, for example, Ref. 4, Eq. (A2).

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Figures (1)

Fig. 1
Fig. 1

Plot of the axial intensity distributions in the neighborhood of the geometrical focus of a perfect, rotationally symmetric system with uniformly illuminated aperture. The normalized intensity iT is plotted versus the longitudinal wave defocusing D, expressed in wavelengths. The curves show the difference between the modified Maréchal result (solid lines) and an exact computer integration of the complete diffraction integral (dashed lines), for selected values of the factorized Fresnel number F. The appropriate exact values are in perfect agreement with results reported previously.15 Since |D| < λ, the corresponding wave-front deviations in the pupil are small enough for achieving a good accuracy of the Maréchal approximations. Also note that for F = 50 the focal-shift effect is negligible.

Equations (28)

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I ( W ) = ( λ z ) - 2 T ( r ) exp [ ikW ( r ) ] d r 2 ,
I ( 0 ) = ( λ R ) - 2 T ( r ) d r 2 ,
i T = I ( W ) / I ( 0 ) ,
i T = ( R / z ) 2 T ( r ) exp [ ikW ( r ) ] d r / T ( r ) d r 2 .
SD = T ( r ) exp [ ikW ( r ) ] d r / T ( r ) d r 2 .
SD [ 1 - ( k 2 / 2 ) E T ] 2 1 - k 2 E T ,
E T = [ T ( r ) W 2 ( r ) d r / T ( r ) d r ] - [ T ( r ) W ( r ) d r / T ( r ) d r ] 2 .
i T = ( R / z ) 2 SD ( R / z ) 2 [ 1 - ( k 2 / 2 ) E T ] 2 .
W 20 = [ Δ R a 2 / ( 2 z R ) ] + higher - order terms ,
W 20 D = ( z - R ) a 2 / ( 2 z R ) .
( R / z ) 2 = [ 1 - ( k D / F ) ] 2 ,
F = ( π a 2 ) / ( λ R )
i T = [ 1 - ( k D / F ) ] 2 SD ,
i T { [ 1 - ( k D / F ) ] [ 1 - ( k 2 / 2 ) E T ] } 2 .
i T SD [ 1 - ( k 2 / 2 ) E T ] 2 ,             ( F )
W ( r ) = D r 2 ,             T ( r ) = T ( r ) ,
E T = σ T D 2 ,
i T { [ 1 - ( k D / F ) ] [ 1 - ( k 2 / 2 ) σ T D 2 ] } 2 .
i T / D = 0.
3 k 2 σ T D 2 - 2 k F σ T D - 2 = 0.
D opt = [ F / ( 3 k ) ] ( 1 - A ) ,
( Δ R / R ) opt = ( z opt / R ) - 1 = ( 1 - A ) / ( 2 + A ) .
A = ( 1 + ) 1 / 2 = 1 + / 2 - 2 / 8 + ,
D opt - ( k F σ T ) - 1
( Δ R / R ) opt - ( 1 + F 2 σ T ) - 1 .
R / z opt max i T 1 + ( F 2 σ T ) - 1 .
Δ i T = max i T - 1 = ( F 2 σ T ) - 1
z opt = R / { 1 + [ ( 2 R ) / ( k w 2 ) ] 2 } ,

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