Abstract

We show an anomalous behavior in a diffractive lens in which the spot size at the focus reaches a minimum at a numerical aperture of 0.5 and then increases significantly at higher values. Theoretical and experimental results are presented, along with a comparison with refractive aplanatic lenses, in which the anomaly does not appear to exist.

© 2000 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 64.
  2. P. Debye, Ann. Physik 30, 755 (1909).
    [CrossRef]
  3. H. H. Hopkins, Proc. Phys. Soc. London 55, 116 (1943).
    [CrossRef]
  4. E. Wolf, Proc. R. Soc. London Ser. A 253, 349 (1959).
    [CrossRef]
  5. B. Richards and E. Wolf, Proc. R. Soc. London Ser. A 253, 358 (1959).
    [CrossRef]
  6. C. J. R. Sheppard, A. Choudhury, and J. Gannaway, Microwaves Opt. Acoust. 1, 129 (1977).
    [CrossRef]
  7. A. Yoshida and T. Askura, Optik 40, 322 (1974).
  8. R. Barakat, Appl. Opt. 26, 3790 (1987).
    [CrossRef] [PubMed]
  9. C. J. R. Sheppard and T. Wilson, Proc. R. Soc. London Ser. A 379, 145 (1982).
    [CrossRef]
  10. J. Kedmi and A. A. Friesem, J. Opt. Soc. Am. A 3, 2011 (1986).
    [CrossRef]
  11. Y. Amitai and A. A. Friesem, Opt. Lett. 13, 833 (1988).
    [CrossRef]
  12. E. Hasman, N. Davidson, Y. Danziger, and A. A. Friesem, Fiber Integr. Opt. 16, 1 (1997).
    [CrossRef]
  13. A. Carswell, Phys. Rev. Lett. 15, 647 (1965).
    [CrossRef]
  14. M. Schmitz and O. Bryngdahl, Opt. Commun. 153, 118 (1998).
    [CrossRef]

1998 (1)

M. Schmitz and O. Bryngdahl, Opt. Commun. 153, 118 (1998).
[CrossRef]

1997 (1)

E. Hasman, N. Davidson, Y. Danziger, and A. A. Friesem, Fiber Integr. Opt. 16, 1 (1997).
[CrossRef]

1988 (1)

Y. Amitai and A. A. Friesem, Opt. Lett. 13, 833 (1988).
[CrossRef]

1987 (1)

1986 (1)

1982 (1)

C. J. R. Sheppard and T. Wilson, Proc. R. Soc. London Ser. A 379, 145 (1982).
[CrossRef]

1977 (1)

C. J. R. Sheppard, A. Choudhury, and J. Gannaway, Microwaves Opt. Acoust. 1, 129 (1977).
[CrossRef]

1974 (1)

A. Yoshida and T. Askura, Optik 40, 322 (1974).

1965 (1)

A. Carswell, Phys. Rev. Lett. 15, 647 (1965).
[CrossRef]

1959 (2)

E. Wolf, Proc. R. Soc. London Ser. A 253, 349 (1959).
[CrossRef]

B. Richards and E. Wolf, Proc. R. Soc. London Ser. A 253, 358 (1959).
[CrossRef]

1943 (1)

H. H. Hopkins, Proc. Phys. Soc. London 55, 116 (1943).
[CrossRef]

1909 (1)

P. Debye, Ann. Physik 30, 755 (1909).
[CrossRef]

Amitai, Y.

Y. Amitai and A. A. Friesem, Opt. Lett. 13, 833 (1988).
[CrossRef]

Askura, T.

A. Yoshida and T. Askura, Optik 40, 322 (1974).

Barakat, R.

Bryngdahl, O.

M. Schmitz and O. Bryngdahl, Opt. Commun. 153, 118 (1998).
[CrossRef]

Carswell, A.

A. Carswell, Phys. Rev. Lett. 15, 647 (1965).
[CrossRef]

Choudhury, A.

C. J. R. Sheppard, A. Choudhury, and J. Gannaway, Microwaves Opt. Acoust. 1, 129 (1977).
[CrossRef]

Danziger, Y.

E. Hasman, N. Davidson, Y. Danziger, and A. A. Friesem, Fiber Integr. Opt. 16, 1 (1997).
[CrossRef]

Davidson, N.

E. Hasman, N. Davidson, Y. Danziger, and A. A. Friesem, Fiber Integr. Opt. 16, 1 (1997).
[CrossRef]

Debye, P.

P. Debye, Ann. Physik 30, 755 (1909).
[CrossRef]

Friesem, A. A.

E. Hasman, N. Davidson, Y. Danziger, and A. A. Friesem, Fiber Integr. Opt. 16, 1 (1997).
[CrossRef]

Y. Amitai and A. A. Friesem, Opt. Lett. 13, 833 (1988).
[CrossRef]

J. Kedmi and A. A. Friesem, J. Opt. Soc. Am. A 3, 2011 (1986).
[CrossRef]

Gannaway, J.

C. J. R. Sheppard, A. Choudhury, and J. Gannaway, Microwaves Opt. Acoust. 1, 129 (1977).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 64.

Hasman, E.

E. Hasman, N. Davidson, Y. Danziger, and A. A. Friesem, Fiber Integr. Opt. 16, 1 (1997).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, Proc. Phys. Soc. London 55, 116 (1943).
[CrossRef]

Kedmi, J.

Richards, B.

B. Richards and E. Wolf, Proc. R. Soc. London Ser. A 253, 358 (1959).
[CrossRef]

Schmitz, M.

M. Schmitz and O. Bryngdahl, Opt. Commun. 153, 118 (1998).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard and T. Wilson, Proc. R. Soc. London Ser. A 379, 145 (1982).
[CrossRef]

C. J. R. Sheppard, A. Choudhury, and J. Gannaway, Microwaves Opt. Acoust. 1, 129 (1977).
[CrossRef]

Wilson, T.

C. J. R. Sheppard and T. Wilson, Proc. R. Soc. London Ser. A 379, 145 (1982).
[CrossRef]

Wolf, E.

B. Richards and E. Wolf, Proc. R. Soc. London Ser. A 253, 358 (1959).
[CrossRef]

E. Wolf, Proc. R. Soc. London Ser. A 253, 349 (1959).
[CrossRef]

Yoshida, A.

A. Yoshida and T. Askura, Optik 40, 322 (1974).

Ann. Physik (1)

P. Debye, Ann. Physik 30, 755 (1909).
[CrossRef]

Appl. Opt. (1)

Fiber Integr. Opt. (1)

E. Hasman, N. Davidson, Y. Danziger, and A. A. Friesem, Fiber Integr. Opt. 16, 1 (1997).
[CrossRef]

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

Microwaves Opt. Acoust. (1)

C. J. R. Sheppard, A. Choudhury, and J. Gannaway, Microwaves Opt. Acoust. 1, 129 (1977).
[CrossRef]

Opt. Commun. (1)

M. Schmitz and O. Bryngdahl, Opt. Commun. 153, 118 (1998).
[CrossRef]

Opt. Lett. (1)

Y. Amitai and A. A. Friesem, Opt. Lett. 13, 833 (1988).
[CrossRef]

Optik (1)

A. Yoshida and T. Askura, Optik 40, 322 (1974).

Phys. Rev. Lett. (1)

A. Carswell, Phys. Rev. Lett. 15, 647 (1965).
[CrossRef]

Proc. Phys. Soc. London (1)

H. H. Hopkins, Proc. Phys. Soc. London 55, 116 (1943).
[CrossRef]

Proc. R. Soc. London Ser. A (3)

E. Wolf, Proc. R. Soc. London Ser. A 253, 349 (1959).
[CrossRef]

B. Richards and E. Wolf, Proc. R. Soc. London Ser. A 253, 358 (1959).
[CrossRef]

C. J. R. Sheppard and T. Wilson, Proc. R. Soc. London Ser. A 379, 145 (1982).
[CrossRef]

Other (1)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 64.

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Figures (4)

Fig. 1
Fig. 1

Calculated spot size as a function of NA for diffractive (solid curves) and aplanatic (dashed curves) lenses. The spot sizes are based on first-zero and 84% encircled-energy criteria.

Fig. 2
Fig. 2

Detected energy at the focal plane as a function of knife-edge displacement for different NA’s and corresponding apertures L.

Fig. 3
Fig. 3

Intensity distributions at the focus for (a) NA=0.29, (b) NA=0.62, (c) NA=0.78.

Fig. 4
Fig. 4

Experimental spot size as a function of NA for a diffractive cylindrical lens. Asterisks, 72.5% encircled-energy criterion; circles, FWHM criterion; solid curve, the scalar approximation.

Equations (2)

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UˆP=-iλΩAωaˆsx,sy×expiksxx+syy+szzdsxdsysz,
aˆsx,sy=1-sx21+szxˆ;sxsy1+szyˆ;-sxzˆ.

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