Abstract

Axial irradiance distribution arising from the diffraction of a uniform, converging, spherical wave at a circular aperture is studied on the basis of scalar boundary-diffraction wave theory. The combined effects of Fresnel number and angular aperture on the focal shift are evaluated, and the validity of the results is checked against the Kirchhoff boundary conditions.

© 2003 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principal of Optics, 7th ed. (Pergamon, New York, 1999) Chap. 8.
  2. J. J. Stamnes, Waves in the Focal Region (Adam Hilger, Bristol, UK, 1986) Chap. 12.
  3. 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]
  4. Y. Li, “Dependence of the focal shift on Fresnel number and f number,” J. Opt. Soc. Am. 72, 770–774 (1982).
    [CrossRef]
  5. H. H. Hopkins, “The Airy disc formula for systems of higher relative aperture,” Proc. Phys. Soc. London 55, 116–128 (1943).
    [CrossRef]
  6. B. Richard, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
    [CrossRef]
  7. H. H. Hopkins, “Diffraction theory of laser read-out systems for optical video discs,” J. Opt. Soc. Am. 69, 4–24 (1979).
    [CrossRef]
  8. W. Hsu, R. Barakat, “Stratton–Chu vectorial diffraction of electromagnetic fields by aperture with application to small-Fresnel-number systems,” J. Opt. Soc. Am. A 11, 623–629 (1994).
    [CrossRef]
  9. E. W. Marchand, E. Wolf, “Consistent formulation of Kirchhof’s diffraction theory,” J. Opt. Soc. Am. 56, 1712–1722 (1966).
    [CrossRef]
  10. E. W. Marchand, E. Wolf, “Diffraction at small aperture in black screen,” J. Opt. Soc. Am. 59, 79–90 (1969).
    [CrossRef]
  11. W. Wang, E. Wolf, “Far-zone behavior of focused fields in systems with different Fresnel number,” Opt. Commun. 119, 453–459 (1995).
    [CrossRef]
  12. K. Miyamoto, E. Wolf, “Generalization of the Maggi–Rubinowiz theory of the boundary diffraction wave—Part I,” J. Opt. Soc. Am. 52, 615–625 (1962).
    [CrossRef]
  13. K. Miyamoto, E. Wolf, “Generalization of the Maggi-Rubinowiz theory of the boundary diffraction wave—Part II,” J. Opt. Soc. Am. 52, 626–637 (1962).
    [CrossRef]
  14. A. Rubinowicz, “Die Beugungswelle in der Kirchhoffschen Theorie der Beugungserscheinungen,” Ann. Phys. (Leipzig) 53, 257–278 (1917). Reprints in Selected Papers on Scalar Wave Diffraction, K. E. Oughstun, ed., Proc. SPIE51, 165–186 (1992).
    [CrossRef]
  15. C. J. R. Sheppard, P. Törö, “Dependence of focal shift on Fresnel number and angular aperture,” Opt. Lett. 23, 1803–1804 (1998).
    [CrossRef]
  16. M. J. Ehrlich, S. Silver, G. Held, “Studies of the diffraction of electromagnetic waves by circular aperture and complementary obstacles: the near-zone field,” J. Appl. Phys. 26, 336–345 (1955).
    [CrossRef]
  17. J. S. Asvestas, “Diffraction by a black screen,” J. Opt. Soc. Am. 65, 155–158 (1975).
    [CrossRef]
  18. Y. Li, E. Wolf, “Focal shift in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
    [CrossRef]
  19. Y. Li, “Focal shift formulae,” Optik (Stuttgart) 69, 41–42 (1984).

1998 (1)

1995 (1)

W. Wang, E. Wolf, “Far-zone behavior of focused fields in systems with different Fresnel number,” Opt. Commun. 119, 453–459 (1995).
[CrossRef]

1994 (1)

1984 (2)

1982 (1)

1981 (1)

Y. Li, E. Wolf, “Focal shift in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

1979 (1)

1975 (1)

1969 (1)

1966 (1)

1962 (2)

1959 (1)

B. Richard, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
[CrossRef]

1955 (1)

M. J. Ehrlich, S. Silver, G. Held, “Studies of the diffraction of electromagnetic waves by circular aperture and complementary obstacles: the near-zone field,” J. Appl. Phys. 26, 336–345 (1955).
[CrossRef]

1943 (1)

H. H. Hopkins, “The Airy disc formula for systems of higher relative aperture,” Proc. Phys. Soc. London 55, 116–128 (1943).
[CrossRef]

1917 (1)

A. Rubinowicz, “Die Beugungswelle in der Kirchhoffschen Theorie der Beugungserscheinungen,” Ann. Phys. (Leipzig) 53, 257–278 (1917). Reprints in Selected Papers on Scalar Wave Diffraction, K. E. Oughstun, ed., Proc. SPIE51, 165–186 (1992).
[CrossRef]

Asvestas, J. S.

Barakat, R.

Born, M.

M. Born, E. Wolf, Principal of Optics, 7th ed. (Pergamon, New York, 1999) Chap. 8.

Ehrlich, M. J.

M. J. Ehrlich, S. Silver, G. Held, “Studies of the diffraction of electromagnetic waves by circular aperture and complementary obstacles: the near-zone field,” J. Appl. Phys. 26, 336–345 (1955).
[CrossRef]

Held, G.

M. J. Ehrlich, S. Silver, G. Held, “Studies of the diffraction of electromagnetic waves by circular aperture and complementary obstacles: the near-zone field,” J. Appl. Phys. 26, 336–345 (1955).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, “Diffraction theory of laser read-out systems for optical video discs,” J. Opt. Soc. Am. 69, 4–24 (1979).
[CrossRef]

H. H. Hopkins, “The Airy disc formula for systems of higher relative aperture,” Proc. Phys. Soc. London 55, 116–128 (1943).
[CrossRef]

Hsu, W.

Li, Y.

Marchand, E. W.

Miyamoto, K.

Richard, B.

B. Richard, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
[CrossRef]

Rubinowicz, A.

A. Rubinowicz, “Die Beugungswelle in der Kirchhoffschen Theorie der Beugungserscheinungen,” Ann. Phys. (Leipzig) 53, 257–278 (1917). Reprints in Selected Papers on Scalar Wave Diffraction, K. E. Oughstun, ed., Proc. SPIE51, 165–186 (1992).
[CrossRef]

Sheppard, C. J. R.

Silver, S.

M. J. Ehrlich, S. Silver, G. Held, “Studies of the diffraction of electromagnetic waves by circular aperture and complementary obstacles: the near-zone field,” J. Appl. Phys. 26, 336–345 (1955).
[CrossRef]

Stamnes, J. J.

J. J. Stamnes, Waves in the Focal Region (Adam Hilger, Bristol, UK, 1986) Chap. 12.

Törö, P.

Wang, W.

W. Wang, E. Wolf, “Far-zone behavior of focused fields in systems with different Fresnel number,” Opt. Commun. 119, 453–459 (1995).
[CrossRef]

Wolf, E.

Ann. Phys. (Leipzig) (1)

A. Rubinowicz, “Die Beugungswelle in der Kirchhoffschen Theorie der Beugungserscheinungen,” Ann. Phys. (Leipzig) 53, 257–278 (1917). Reprints in Selected Papers on Scalar Wave Diffraction, K. E. Oughstun, ed., Proc. SPIE51, 165–186 (1992).
[CrossRef]

J. Appl. Phys. (1)

M. J. Ehrlich, S. Silver, G. Held, “Studies of the diffraction of electromagnetic waves by circular aperture and complementary obstacles: the near-zone field,” J. Appl. Phys. 26, 336–345 (1955).
[CrossRef]

J. Opt. Soc. Am. (7)

J. Opt. Soc. Am. A (2)

Opt. Commun. (2)

W. Wang, E. Wolf, “Far-zone behavior of focused fields in systems with different Fresnel number,” Opt. Commun. 119, 453–459 (1995).
[CrossRef]

Y. Li, E. Wolf, “Focal shift in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

Opt. Lett. (1)

Optik (Stuttgart) (1)

Y. Li, “Focal shift formulae,” Optik (Stuttgart) 69, 41–42 (1984).

Proc. Phys. Soc. London (1)

H. H. Hopkins, “The Airy disc formula for systems of higher relative aperture,” Proc. Phys. Soc. London 55, 116–128 (1943).
[CrossRef]

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

B. Richard, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
[CrossRef]

Other (2)

M. Born, E. Wolf, Principal of Optics, 7th ed. (Pergamon, New York, 1999) Chap. 8.

J. J. Stamnes, Waves in the Focal Region (Adam Hilger, Bristol, UK, 1986) Chap. 12.

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Figures (3)

Fig. 1
Fig. 1

Notation relating to the BDW theory when the observation point P is on the z axis.

Fig. 2
Fig. 2

Plots of normalized axial irradiance against the distance z in focal length f from the center of the aperture in systems of different parameters. The dashed curves are for reference only because the linear dimension of the aperture is smaller than the wavelength.

Fig. 3
Fig. 3

Dependence of the relative focal shift on the angular aperture α assuming the Fresnel number N from 0.1 to 0.7.

Tables (2)

Tables Icon

Table 1 Relative Focal Shift Δf/f in Small-Fresnel-Number Focusing Systems of Different Angular Aperture α

Tables Icon

Table 2 Upper Limit of the Angular Aperture αmax, Calculated From Equations (16) and (19)

Equations (32)

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N=a2/λf,
F=f/2a,
α=arcsin(a/f)=arcsin(1/2F)30°.
UK(P)=UG(P)+UB(P),
UG(P)=A exp[-ik(z0-a cot β)]z0-a cot β,
UB(P)=-A4πΓexp(-ikf)fexp(iks)s(s×r)lsr-srdl,
(s×r)lsr-sr=cot θ/2,inregionI,-cot(-θ/2),inregionII,
UB(P)=-A2 fsin β cot(θ/2)exp-ikf-asin β.
I(P)=UK(P)UK*(P).
I(P)=IFsin β/2sin α/2 sin ΘΘ2+(1+cos α)sin β4πN tan θ22,
Θ=πN1+cos α1-tan β/2tan α/2.
cos(2Θ-Θ0)=Θ1+Θ2(Θ22+Θ32)1/2,
Θ1=tan2θ2sin2θ21+tan βsin θ,
Θ2=2-tan βtan θ/2,
Θ3=πNtan α/2tan β1+cos β,
Θ0=arctan(Θ3/Θ2).
Δff=zm-z0f=-sin θmsin(α+θm),
a>λ.
N=a2λf=aλaf=aλsin α,
αm(1)=arcsin(N).
zm(15)λ,
zm-z0fλ-z0f,
Δffsin2 αN-cos α.
α<αmax=min(αm(1), αm(2))whenN<0.730°whenN0.7 .    (20a)(20b)
Θ=-πN2 Φ,
Φ=θα.
Θ10,
Θ2-2Φ,
Θ3πN(1+Φ),
Θ0-arctanπN2 Φ(1+Φ).
tan(πNΦ/2)πNΦ/2=1+Φ,
Δff=-Φm1+Φm.

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