Abstract

The wave-front spacing in the focal region of an aplanatic focusing system is investigated by use of the vector theory of electromagnetic diffraction for monochromatic, linearly polarized incident light. It is shown that, in systems of high numerical aperture, the wave-front spacing near the focus is significantly larger than the wavelength of the incident light and that the wave-front spacing changes significantly within a few wavelengths of the focus and can be less than a wavelength.

© 2005 Optical Society of America

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References

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  1. By wave-front spacing we mean the smallest distance between surfaces of constant phase on which the values differ by 2?. For monochromatic plane-wave fields this distance equals the wavelength, and this is also approximately the case for focused waves of low angular aperture.
  2. C. M. J. Mecca, Y. Li, and E. Wolf, Opt. Commun. 182, 265 (2000).
    [CrossRef]
  3. Y. Li, C. Mecca, and E. Wolf, J. Mod. Opt. 50, 199 (2003).
    [CrossRef]
  4. S. Stalling, G. ’T Hooft, and J. Braat, J. Mod. Opt. 51, 775 (2004).
  5. Y. Li, E. Wolf, G. Gbur, and T. D. Visser, J. Mod. Opt. 51, 779 (2004).
    [CrossRef]
  6. E. H. Linfoot and E. Wolf, Proc. Phys. Soc. London Sect. B 69, 823 (1956).
    [CrossRef]
  7. F. R. Tolmon and J. G. Wood, J. Sci. Instrum. 33, 236 (1956).
    [CrossRef]
  8. J. W. Gates, J. Sci. Instrum. 33, 507 (1956).
    [CrossRef]
  9. C. F. Bruce and B. S. Thornton, J. Sci. Instrum. 34, 203 (1956).
    [CrossRef]
  10. C. J. R. Sheppard and T. Wilson, Appl. Phys. Lett. 38, 858 (1981).
    [CrossRef]
  11. D. K. Hamilton and C. J. R. Sheppard, J. Appl. Phys. 60, 2708 (1986).
    [CrossRef]
  12. C. J. R. Sheppard and H. J. Matthews, J. Opt. Soc. Am. A 4, 1354 (1987).
    [CrossRef]
  13. J. F. Bliegen, Appl. Opt. 28, 1972 (1989).
    [CrossRef]
  14. G. S. Kino and S. C. Chim, Appl. Opt. 29, 3775 (1990).
    [CrossRef] [PubMed]
  15. T. Wilson, R. Jusaitis, N. P. Rea, and D. K. Hamilton, Opt. Commun. 119, 1 (1994).
    [CrossRef]
  16. F. C. Chang and G. S. Kino, Appl. Opt. 37, 3471 (1998).
    [CrossRef]
  17. A. Dubois, J. Selb, L. Vabre, and A. Boccara, Appl. Opt. 39, 2326 (2000).
    [CrossRef]
  18. B. Richards and E. Wolf, Proc. R. Soc. London Ser. A 253, 358 (1959).
    [CrossRef]
  19. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), pp. 664–665. ?(z) is sometimes referred to as the Gouy phase of the Gaussian beam.

2004 (2)

S. Stalling, G. ’T Hooft, and J. Braat, J. Mod. Opt. 51, 775 (2004).

Y. Li, E. Wolf, G. Gbur, and T. D. Visser, J. Mod. Opt. 51, 779 (2004).
[CrossRef]

2003 (1)

Y. Li, C. Mecca, and E. Wolf, J. Mod. Opt. 50, 199 (2003).
[CrossRef]

2000 (2)

C. M. J. Mecca, Y. Li, and E. Wolf, Opt. Commun. 182, 265 (2000).
[CrossRef]

A. Dubois, J. Selb, L. Vabre, and A. Boccara, Appl. Opt. 39, 2326 (2000).
[CrossRef]

1998 (1)

1994 (1)

T. Wilson, R. Jusaitis, N. P. Rea, and D. K. Hamilton, Opt. Commun. 119, 1 (1994).
[CrossRef]

1990 (1)

1989 (1)

1987 (1)

1986 (1)

D. K. Hamilton and C. J. R. Sheppard, J. Appl. Phys. 60, 2708 (1986).
[CrossRef]

1981 (1)

C. J. R. Sheppard and T. Wilson, Appl. Phys. Lett. 38, 858 (1981).
[CrossRef]

1959 (1)

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

1956 (4)

E. H. Linfoot and E. Wolf, Proc. Phys. Soc. London Sect. B 69, 823 (1956).
[CrossRef]

F. R. Tolmon and J. G. Wood, J. Sci. Instrum. 33, 236 (1956).
[CrossRef]

J. W. Gates, J. Sci. Instrum. 33, 507 (1956).
[CrossRef]

C. F. Bruce and B. S. Thornton, J. Sci. Instrum. 34, 203 (1956).
[CrossRef]

’T Hooft, G.

S. Stalling, G. ’T Hooft, and J. Braat, J. Mod. Opt. 51, 775 (2004).

Bliegen, J. F.

Boccara, A.

Braat, J.

S. Stalling, G. ’T Hooft, and J. Braat, J. Mod. Opt. 51, 775 (2004).

Bruce, C. F.

C. F. Bruce and B. S. Thornton, J. Sci. Instrum. 34, 203 (1956).
[CrossRef]

Chang, F. C.

Chim, S. C.

Dubois, A.

Gates, J. W.

J. W. Gates, J. Sci. Instrum. 33, 507 (1956).
[CrossRef]

Gbur, G.

Y. Li, E. Wolf, G. Gbur, and T. D. Visser, J. Mod. Opt. 51, 779 (2004).
[CrossRef]

Hamilton, D. K.

T. Wilson, R. Jusaitis, N. P. Rea, and D. K. Hamilton, Opt. Commun. 119, 1 (1994).
[CrossRef]

D. K. Hamilton and C. J. R. Sheppard, J. Appl. Phys. 60, 2708 (1986).
[CrossRef]

Jusaitis, R.

T. Wilson, R. Jusaitis, N. P. Rea, and D. K. Hamilton, Opt. Commun. 119, 1 (1994).
[CrossRef]

Kino, G. S.

Li, Y.

Y. Li, E. Wolf, G. Gbur, and T. D. Visser, J. Mod. Opt. 51, 779 (2004).
[CrossRef]

Y. Li, C. Mecca, and E. Wolf, J. Mod. Opt. 50, 199 (2003).
[CrossRef]

C. M. J. Mecca, Y. Li, and E. Wolf, Opt. Commun. 182, 265 (2000).
[CrossRef]

Linfoot, E. H.

E. H. Linfoot and E. Wolf, Proc. Phys. Soc. London Sect. B 69, 823 (1956).
[CrossRef]

Matthews, H. J.

Mecca, C.

Y. Li, C. Mecca, and E. Wolf, J. Mod. Opt. 50, 199 (2003).
[CrossRef]

Mecca, C. M. J.

C. M. J. Mecca, Y. Li, and E. Wolf, Opt. Commun. 182, 265 (2000).
[CrossRef]

Rea, N. P.

T. Wilson, R. Jusaitis, N. P. Rea, and D. K. Hamilton, Opt. Commun. 119, 1 (1994).
[CrossRef]

Richards, B.

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

Selb, J.

Sheppard, C. J. R.

C. J. R. Sheppard and H. J. Matthews, J. Opt. Soc. Am. A 4, 1354 (1987).
[CrossRef]

D. K. Hamilton and C. J. R. Sheppard, J. Appl. Phys. 60, 2708 (1986).
[CrossRef]

C. J. R. Sheppard and T. Wilson, Appl. Phys. Lett. 38, 858 (1981).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), pp. 664–665. ?(z) is sometimes referred to as the Gouy phase of the Gaussian beam.

Stalling, S.

S. Stalling, G. ’T Hooft, and J. Braat, J. Mod. Opt. 51, 775 (2004).

Thornton, B. S.

C. F. Bruce and B. S. Thornton, J. Sci. Instrum. 34, 203 (1956).
[CrossRef]

Tolmon, F. R.

F. R. Tolmon and J. G. Wood, J. Sci. Instrum. 33, 236 (1956).
[CrossRef]

Vabre, L.

Visser, T. D.

Y. Li, E. Wolf, G. Gbur, and T. D. Visser, J. Mod. Opt. 51, 779 (2004).
[CrossRef]

Wilson, T.

T. Wilson, R. Jusaitis, N. P. Rea, and D. K. Hamilton, Opt. Commun. 119, 1 (1994).
[CrossRef]

C. J. R. Sheppard and T. Wilson, Appl. Phys. Lett. 38, 858 (1981).
[CrossRef]

Wolf, E.

Y. Li, E. Wolf, G. Gbur, and T. D. Visser, J. Mod. Opt. 51, 779 (2004).
[CrossRef]

Y. Li, C. Mecca, and E. Wolf, J. Mod. Opt. 50, 199 (2003).
[CrossRef]

C. M. J. Mecca, Y. Li, and E. Wolf, Opt. Commun. 182, 265 (2000).
[CrossRef]

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

E. H. Linfoot and E. Wolf, Proc. Phys. Soc. London Sect. B 69, 823 (1956).
[CrossRef]

Wood, J. G.

F. R. Tolmon and J. G. Wood, J. Sci. Instrum. 33, 236 (1956).
[CrossRef]

Appl. Opt. (4)

Appl. Phys. Lett. (1)

C. J. R. Sheppard and T. Wilson, Appl. Phys. Lett. 38, 858 (1981).
[CrossRef]

J. Appl. Phys. (1)

D. K. Hamilton and C. J. R. Sheppard, J. Appl. Phys. 60, 2708 (1986).
[CrossRef]

J. Mod. Opt. (3)

Y. Li, C. Mecca, and E. Wolf, J. Mod. Opt. 50, 199 (2003).
[CrossRef]

S. Stalling, G. ’T Hooft, and J. Braat, J. Mod. Opt. 51, 775 (2004).

Y. Li, E. Wolf, G. Gbur, and T. D. Visser, J. Mod. Opt. 51, 779 (2004).
[CrossRef]

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

J. Sci. Instrum. (3)

F. R. Tolmon and J. G. Wood, J. Sci. Instrum. 33, 236 (1956).
[CrossRef]

J. W. Gates, J. Sci. Instrum. 33, 507 (1956).
[CrossRef]

C. F. Bruce and B. S. Thornton, J. Sci. Instrum. 34, 203 (1956).
[CrossRef]

Opt. Commun. (2)

C. M. J. Mecca, Y. Li, and E. Wolf, Opt. Commun. 182, 265 (2000).
[CrossRef]

T. Wilson, R. Jusaitis, N. P. Rea, and D. K. Hamilton, Opt. Commun. 119, 1 (1994).
[CrossRef]

Proc. Phys. Soc. London Sect. B (1)

E. H. Linfoot and E. Wolf, Proc. Phys. Soc. London Sect. B 69, 823 (1956).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

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

Other (2)

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), pp. 664–665. ?(z) is sometimes referred to as the Gouy phase of the Gaussian beam.

By wave-front spacing we mean the smallest distance between surfaces of constant phase on which the values differ by 2?. For monochromatic plane-wave fields this distance equals the wavelength, and this is also approximately the case for focused waves of low angular aperture.

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Figures (3)

Fig. 1
Fig. 1

Basic geometry. The coordinate system is centered at the focus. The vector r specifies the observation point. The focal length of the system is f, and α is the angular semi-aperture.

Fig. 2
Fig. 2

On-axis electric field E x ( z , 0 ) as a function of z (in units of λ) near the focus, for NA = 0.650 .

Fig. 3
Fig. 3

On-axis electric field E x ( z , 0 ) as a function of z (in units of λ) near the focus, for NA = 0.850 .

Tables (1)

Tables Icon

Table 1 Wave-front Spacing (in Units of λ), Defined by Eqs. (8, 9) in Systems with Different Numerical Apertures

Equations (13)

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e ( r ) = i k 2 π Ω a ( θ , ϕ ) exp ( i k s r ) sin θ d θ d ϕ ,
a x ( θ , ϕ ) = f cos θ [ cos θ + sin 2 ϕ ( 1 cos θ ) ] ,
a y ( θ , ϕ ) = f cos θ ( cos θ 1 ) sin ϕ cos ϕ ,
a z ( θ , ϕ ) = f cos θ sin θ cos ϕ ,
e x ( z ) = i A 0 α cos θ ( 1 + cos θ ) exp ( i k z cos θ ) sin θ d θ ,
= i A cos a 1 w ( 1 + w ) exp ( i k z w ) d w ,
e x ( z ) = exp ( i k z ) M ( k z ) ,
M ( k z ) = i A 0 1 cos α 1 ξ ( 2 ξ ) exp ( i k z ξ ) d ξ .
ψ ( z m ) = 2 π m , ( m = 0 , ± 1 , ± 2 , ) .
Δ z m = z m z m 1 , for m > 0 ,
Δ z m = z m z m + 1 , for m < 0 .
λ = λ [ 1 + 0.25 ( NA ) 2 + 0.1 ( NA ) 4 + 0.086 ( NA ) 6 ] .
z 1 = λ [ 1 + 1 2 π tan 1 ( z 1 z R ) ] .

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