Abstract

In many photonics and fiber-optics applications, the irradiance distribution in the very near field (z/ D < 0.25) behind a circular or annular aperture is of interest. We present the results of detailed calculations of the irradiance distribution throughout the entire space behind an annular aperture. Included as a special case of the annular aperture is the circular aperture and the opaque circular disk. A log–log plot over many orders of magnitude in axial distance provides particular insight. The behavior throughout the Fresnel and Fraunhofer region is well known; however, we pay particular attention to the behavior in the near field. A variety of subtle effects in the near field are presented and discussed.

© 2002 Optical Society of America

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References

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  1. J. M. Stone, Radiation and Optics (McGraw-Hill, New York, 1963), Chap. 10, p. 204.
  2. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 4, pp. 69–74.
  3. A. Sommerfeld, Optics: Lectures on Theoretical Physics (Academic, P, New York, 1954), Vol. 4, Chap. 5, p. 215.
  4. J. E. Harvey, J. L. Forgham, “The Spot of Arago: new relevance for an old phenomenon,” Am. J. Phys. 52, 243–247 (1984).
    [CrossRef]
  5. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980), Chap. 8, pp. 370–375.
  6. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), p. 735.

1984 (1)

J. E. Harvey, J. L. Forgham, “The Spot of Arago: new relevance for an old phenomenon,” Am. J. Phys. 52, 243–247 (1984).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980), Chap. 8, pp. 370–375.

Forgham, J. L.

J. E. Harvey, J. L. Forgham, “The Spot of Arago: new relevance for an old phenomenon,” Am. J. Phys. 52, 243–247 (1984).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 4, pp. 69–74.

Harvey, J. E.

J. E. Harvey, J. L. Forgham, “The Spot of Arago: new relevance for an old phenomenon,” Am. J. Phys. 52, 243–247 (1984).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), p. 735.

Sommerfeld, A.

A. Sommerfeld, Optics: Lectures on Theoretical Physics (Academic, P, New York, 1954), Vol. 4, Chap. 5, p. 215.

Stone, J. M.

J. M. Stone, Radiation and Optics (McGraw-Hill, New York, 1963), Chap. 10, p. 204.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980), Chap. 8, pp. 370–375.

Am. J. Phys. (1)

J. E. Harvey, J. L. Forgham, “The Spot of Arago: new relevance for an old phenomenon,” Am. J. Phys. 52, 243–247 (1984).
[CrossRef]

Other (5)

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1980), Chap. 8, pp. 370–375.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), p. 735.

J. M. Stone, Radiation and Optics (McGraw-Hill, New York, 1963), Chap. 10, p. 204.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 4, pp. 69–74.

A. Sommerfeld, Optics: Lectures on Theoretical Physics (Academic, P, New York, 1954), Vol. 4, Chap. 5, p. 215.

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Figures (8)

Fig. 1
Fig. 1

Axial irradiance distribution throughout the Fresnel region that is due to diffraction from (a) a circular aperture, (b) an annular aperture, and (c) a circular obscuration.

Fig. 2
Fig. 2

Geometric relationship between the diffracting aperture and the observation space.

Fig. 3
Fig. 3

Axial irradiance distribution illustrated throughout the whole space behind a circular aperture illuminated with a unit amplitude plane wave.

Fig. 4
Fig. 4

Axial irradiance distribution with Fresnel numbers indicated for a limited range of axial positions throughout the Fresnel region.

Fig. 5
Fig. 5

Axial irradiance distribution illustrated throughout the whole space behind an annular aperture (ε = 0.5) illuminated with a unit amplitude plane wave.

Fig. 6
Fig. 6

Curves indicate the envelope of the oscillations in the axial irradiance distribution in the near field behind a circular and an annular aperture. The axial irradiance distribution behind a circular obscuration (of diameter equal to that of the circle and annulus) is also illustrated.

Fig. 7
Fig. 7

(a) Axial irradiance distribution in the very near field behind a circular aperture. (b) Axial irradiance distribution in the very near field behind an annular aperture (ε = 0.5) and a circular obscuration of the same diameter.

Fig. 8
Fig. 8

Axial irradiance distribution in the very near field behind an annular aperture with obscuration ratio (a) ε = 0.20, (b) ε = 0.35, (c) ε = 0.65, and (d) ε = 0.80.

Equations (15)

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E200; z  E02-2 cosk D28z=4E0 sin2k D216z.
z  k2x12+y12max.
U2x2, y2=Aiλ--U1x1, y1×expikll cosn, ldx1dy1.
U20, 0; z=-ikAr1=d/2D/2expikllzl r1dr1,
l2=z2+r12, r1dr1=ldl.
U20, 0; z=-ikAzl=z2+d2/41/2z2+D2/41/2expiklldl.
a=1+D2/4z2, b=1+d2/4z2,
U20, 0; z=-ikAzl=zbza l-1 expikldl.
u=l-1,dv=expikldl,du=-l-2, v=expikl/ik,
U20, 0; z=-Az expikllzbza-Azl=zbza l-2×expikldl.
U20, 0; z=-Az expiklln=0n!iklnzbza.
E20, 0; z=|U20, 0; z|2=E0z2expikzaza-expikzbzb2,
E20, 0; z=|U20, 0; z|2=E0a+bab-2ab coskza-b,
z3  π4λx2-x12+y2-y12max2.
N=D24λz.

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