Abstract

A new and simple formula valid within the framework of the Debye theory is derived for determining the structure of focused fields in diffraction-limited systems. It is first applied to study the field behavior in the focal region, and the results are compared with those of the classic theory of Lommel. The field distribution in the intermediate zone between the focal region and the far zone is then studied, and the changes of the field with increasing distance from the geometrical focus are examined. An estimate is obtained for the distance from focus at which the field behaves as a cutoff portion of a uniform spherical wave.

© 1995 Optical Society of America

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References

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  1. For comprehensive reviews, see E. Wolf, “The diffraction theory of aberrations,” Rep. Progr. Phys. (The Physical Society, London) 14, 95–120 (1951), Secs. 3.1 and 3.2, and J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Sec. 1.5.
    [CrossRef]
  2. G. B. Airy, “On the diffraction of an object-glass with circular aperture,” Trans. Cambridge Philos. Soc. 5, 283–291 (1835).
  3. E. Lommel, “Die Beugungserscheinungen einer kreisrunden Oeffnung und eines kreisnunden Schirmchens,” Abh. Bayer Akad. Math. Naturwiss. 15, 229–328 (1885). An account of Lommel’s theory in English is given in Ref. 17 below, Sec. 8.8.
  4. M. Berek, “Über Kohärenz und Konsonanz des Lichtes,” Z. Phys. 40, 420–450 (1927).
    [CrossRef]
  5. P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Physik 30, 755–776 (1909). An account of Debye’s theory in English is given in A. Sommerfeld, Optics (Academic, New York, 1954), Sec. 45.
    [CrossRef]
  6. J. Picht, Optische Abbildung (Vieweg, Braunschweig, 1931).
    [CrossRef]
  7. F. Zernike, B. R. A. Nijboer, “Théorie de la diffraction des aberrations,” in La Théorie des Images Optiques, P. Fleury, A. Maréchal, C. Anglade, eds. (Editions de la Revue d’Optique, Paris, 1949), p. 227.
  8. E. H. Linfoot, E. Wolf, “Diffraction images in systems with an annular aperture,” Proc. Phys. Soc. B 66, 145–149 (1953).
    [CrossRef]
  9. E. H. Linfoot, E. Wolf, “Phase distribution near focus in an aberration-free diffraction image,” Proc. Phys. Soc. B 69, 823–832 (1956).
    [CrossRef]
  10. A. G. Van Nie, “Rigorous calculation of the electromagnetic field for wave beams,” Philips Res. Rep. 19, 378–394 (1964).
  11. H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Technol. J. 44, 455–494 (1965).
  12. E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
    [CrossRef]
  13. Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
    [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. W. Wang, E. Wolf, “Far-zone behavior of focused fields in systems with different Fresnel numbers,” Opt. Commun. (to be published).
  16. For more general symmetry properties of focused fields, see E. Collett, E. Wolf, “Symmetry properties of focused fields,” Opt. Lett. 5, 264–266 (1980).
    [CrossRef] [PubMed]
  17. See, for example, M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).
  18. Strictly speaking, this behavior does not apply to field points on, or sufficiently close to, the zaxis or to points located on the geometrical shadow boundary or in the immediate vicinity of it. However, these two regions of anomalous behavior of the Debye integral become negligibly small when the distance from the aperture is sufficiently large [see Sherman, Chew, “Aperture and far-field distributions expressed by the Debye integral representation of focused fields,” J. Opt. Soc. Am. 72, 1076–1083 (1982)].
    [CrossRef]

1984 (1)

1982 (1)

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]

1980 (1)

1965 (1)

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Technol. J. 44, 455–494 (1965).

1964 (1)

A. G. Van Nie, “Rigorous calculation of the electromagnetic field for wave beams,” Philips Res. Rep. 19, 378–394 (1964).

1956 (1)

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

1953 (1)

E. H. Linfoot, E. Wolf, “Diffraction images in systems with an annular aperture,” Proc. Phys. Soc. B 66, 145–149 (1953).
[CrossRef]

1927 (1)

M. Berek, “Über Kohärenz und Konsonanz des Lichtes,” Z. Phys. 40, 420–450 (1927).
[CrossRef]

1909 (1)

P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Physik 30, 755–776 (1909). An account of Debye’s theory in English is given in A. Sommerfeld, Optics (Academic, New York, 1954), Sec. 45.
[CrossRef]

1885 (1)

E. Lommel, “Die Beugungserscheinungen einer kreisrunden Oeffnung und eines kreisnunden Schirmchens,” Abh. Bayer Akad. Math. Naturwiss. 15, 229–328 (1885). An account of Lommel’s theory in English is given in Ref. 17 below, Sec. 8.8.

1835 (1)

G. B. Airy, “On the diffraction of an object-glass with circular aperture,” Trans. Cambridge Philos. Soc. 5, 283–291 (1835).

Airy, G. B.

G. B. Airy, “On the diffraction of an object-glass with circular aperture,” Trans. Cambridge Philos. Soc. 5, 283–291 (1835).

Berek, M.

M. Berek, “Über Kohärenz und Konsonanz des Lichtes,” Z. Phys. 40, 420–450 (1927).
[CrossRef]

Born, M.

See, for example, M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

Chew,

Collett, E.

Debye, P.

P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Physik 30, 755–776 (1909). An account of Debye’s theory in English is given in A. Sommerfeld, Optics (Academic, New York, 1954), Sec. 45.
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Technol. J. 44, 455–494 (1965).

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]

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

Linfoot, E. H.

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

E. H. Linfoot, E. Wolf, “Diffraction images in systems with an annular aperture,” Proc. Phys. Soc. B 66, 145–149 (1953).
[CrossRef]

Lommel, E.

E. Lommel, “Die Beugungserscheinungen einer kreisrunden Oeffnung und eines kreisnunden Schirmchens,” Abh. Bayer Akad. Math. Naturwiss. 15, 229–328 (1885). An account of Lommel’s theory in English is given in Ref. 17 below, Sec. 8.8.

Nijboer, B. R. A.

F. Zernike, B. R. A. Nijboer, “Théorie de la diffraction des aberrations,” in La Théorie des Images Optiques, P. Fleury, A. Maréchal, C. Anglade, eds. (Editions de la Revue d’Optique, Paris, 1949), p. 227.

Picht, J.

J. Picht, Optische Abbildung (Vieweg, Braunschweig, 1931).
[CrossRef]

Sherman,

Van Nie, A. G.

A. G. Van Nie, “Rigorous calculation of the electromagnetic field for wave beams,” Philips Res. Rep. 19, 378–394 (1964).

Wang, W.

W. Wang, E. Wolf, “Far-zone behavior of focused fields in systems with different Fresnel numbers,” Opt. Commun. (to be published).

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]

For more general symmetry properties of focused fields, see E. Collett, E. Wolf, “Symmetry properties of focused fields,” Opt. Lett. 5, 264–266 (1980).
[CrossRef] [PubMed]

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

E. H. Linfoot, E. Wolf, “Diffraction images in systems with an annular aperture,” Proc. Phys. Soc. B 66, 145–149 (1953).
[CrossRef]

W. Wang, E. Wolf, “Far-zone behavior of focused fields in systems with different Fresnel numbers,” Opt. Commun. (to be published).

For comprehensive reviews, see E. Wolf, “The diffraction theory of aberrations,” Rep. Progr. Phys. (The Physical Society, London) 14, 95–120 (1951), Secs. 3.1 and 3.2, and J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Sec. 1.5.
[CrossRef]

See, for example, M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

Zernike, F.

F. Zernike, B. R. A. Nijboer, “Théorie de la diffraction des aberrations,” in La Théorie des Images Optiques, P. Fleury, A. Maréchal, C. Anglade, eds. (Editions de la Revue d’Optique, Paris, 1949), p. 227.

Abh. Bayer Akad. Math. Naturwiss. (1)

E. Lommel, “Die Beugungserscheinungen einer kreisrunden Oeffnung und eines kreisnunden Schirmchens,” Abh. Bayer Akad. Math. Naturwiss. 15, 229–328 (1885). An account of Lommel’s theory in English is given in Ref. 17 below, Sec. 8.8.

Ann. Physik (1)

P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Physik 30, 755–776 (1909). An account of Debye’s theory in English is given in A. Sommerfeld, Optics (Academic, New York, 1954), Sec. 45.
[CrossRef]

Bell Syst. Technol. J. (1)

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Technol. J. 44, 455–494 (1965).

J. Opt. Soc. Am. (1)

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

Opt. Commun. (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]

Opt. Lett. (1)

Philips Res. Rep. (1)

A. G. Van Nie, “Rigorous calculation of the electromagnetic field for wave beams,” Philips Res. Rep. 19, 378–394 (1964).

Proc. Phys. Soc. B (2)

E. H. Linfoot, E. Wolf, “Diffraction images in systems with an annular aperture,” Proc. Phys. Soc. B 66, 145–149 (1953).
[CrossRef]

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

Trans. Cambridge Philos. Soc. (1)

G. B. Airy, “On the diffraction of an object-glass with circular aperture,” Trans. Cambridge Philos. Soc. 5, 283–291 (1835).

Z. Phys. (1)

M. Berek, “Über Kohärenz und Konsonanz des Lichtes,” Z. Phys. 40, 420–450 (1927).
[CrossRef]

Other (5)

For comprehensive reviews, see E. Wolf, “The diffraction theory of aberrations,” Rep. Progr. Phys. (The Physical Society, London) 14, 95–120 (1951), Secs. 3.1 and 3.2, and J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Sec. 1.5.
[CrossRef]

J. Picht, Optische Abbildung (Vieweg, Braunschweig, 1931).
[CrossRef]

F. Zernike, B. R. A. Nijboer, “Théorie de la diffraction des aberrations,” in La Théorie des Images Optiques, P. Fleury, A. Maréchal, C. Anglade, eds. (Editions de la Revue d’Optique, Paris, 1949), p. 227.

W. Wang, E. Wolf, “Far-zone behavior of focused fields in systems with different Fresnel numbers,” Opt. Commun. (to be published).

See, for example, M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).

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Figures (6)

Fig. 1
Fig. 1

Illustration of the notation.

Fig. 2
Fig. 2

Normalized intensity I(ρ, z)/I0 (solid curves), (a) on the z axis and (b) in the geometrical focal plane z = 0, for several values of the semiangular aperture α of the focusing geometry. The normalized intensity IL(ρ, z)/I0 (dashed curves) obtained from Lommel’s theory is also shown for comparison. The dashed curves predicted from Lommel’s theory are independent of α.

Fig. 3
Fig. 3

Intensity contours in the geometrical focal region for two values of the semiangular aperture: (a) α = 30°, (b) α = 45°. The intensity at the geometrical focus is normalized to unity.

Fig. 4
Fig. 4

Distribution of (a) the normalized intensity I(r)/(|A|/r)2 and of (b) the phase on several selected spherical surfaces r = constant > 0 for three focusing systems with different semiangular apertures α.

Fig. 5
Fig. 5

Illustration of the distances r and R.

Fig. 6
Fig. 6

Evolution of the focused field.

Equations (31)

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N F a 2 λ f
U ( r ) = - i k 2 π A Ω exp ( i k s · r ) d Ω .
d Ω = d 2 s s z ,             [ s = ( s x , s y , 0 ) ] ,
U ( r ) = - i k 2 π A Ω exp ( i k s · ρ ) exp ( i k s z z ) d 2 s s z .
ρ = ( ρ cos ψ , ρ sin ψ , 0 ) , s = ( s cos ϕ , s sin ϕ , 0 ) .
U ( ρ , z ) = - i k 2 π A 0 sin α 1 s z exp ( i k s z z ) × 0 2 π exp [ i k ρ s cos ( ϕ - ψ ) ] d ϕ s d s = - i k A 0 sin α 1 s z exp ( i k s z z ) J 0 ( k ρ s ) s d s ,
s z 2 = 1 - s 2 , 2 s z d s z = - 2 s d s ,
U ( ρ , z ) = - i k A cos α 1 exp ( i k s z z ) J 0 ( k ρ 1 - s z 2 ) d s z .
U ( ρ , z ) = - i k A cos α 1 exp ( i k z p ) J 0 ( k ρ 1 - p 2 ) d p .
U ( ρ , - z ) = - U * ( ρ , z ) ,
U ( 0 , z ) = - i k A cos α 1 exp ( i k z p ) d p = - i k A 2 sin 2 α 2 sin ( k z sin 2 α 2 ) k z sin 2 α 2 × exp ( i k z cos 2 α 2 ) .
I ( 0 , z ) U ( 0 , z ) 2 = 4 k 2 A 2 sin 4 α 2 sin 2 ( k z sin 2 α 2 ) ( k z sin 2 α 2 ) 2 .
I L ( 0 , z ) = I 0 ( sin u / 4 u / 4 ) 2 ,
u = k z sin 2 α ,
I 0 = 1 4 k 2 A 2 sin 4 α .
d f λ 2 sin 2 ( α / 2 ) .
U ( ρ , 0 ) = - i k A cos α 1 J 0 ( k ρ 1 - p 2 ) d p .
U ( ρ , 0 ) = - i k A 0 sin α J 0 ( k ρ q ) q d q 1 - q 2 .
U ( ρ , 0 ) - i k A 0 sin α J 0 ( k ρ q ) q d q = - i k A sin 2 α J 1 ( k ρ sin α ) k ρ sin α .
I ( ρ , 0 ) U ( ρ , 0 ) 2 I 0 [ 2 J 1 ( v ) v ] 2 ,
v = k ρ sin α .
l f 0.61 λ sin α .
ρ = r sin θ ,             z = r cos θ ,
N r = k r sin 2 α .
N r = 2 π N F ( r f ) .
N r 2.5 × 10 2 ,
r 50 N F f r ϕ .
N r 2.5 × 10 3 ,
r 500 N F f r I .
R a 2 λ ,
r R N F f .

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