Abstract

The diffraction field through a finite aperture lens is obtained by using the Kirchhoff-Huygens formula. The most focused point of the diffraction field differs depending on the definition of it. The positions of the maximum axial intensity, the minimum field spread, and the maximum encircled energy are calculated and compared. They vary depending on the incidence conditions, aperture radius, and the focal length.

© 1985 Optical Society of America

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References

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  1. R. G. Schell, G. Tyras, “Irradiance from Truncated-Gaussian Field,” J. Opt. Soc. Am. 61, 31 (1971).
    [CrossRef]
  2. K. Tanaka, M. Shibukawa, O. Fukumitsu, “Diffraction of a Wave Beam by an Aperture,” IEEE Trans. Microwave Theory Tech. MTT-20, 749 (1972).
    [CrossRef]
  3. K. Tanaka, O. Fukumitsu, “Study on the Mode Expansion for the Diffraction Field of a Wave Beam by an Aperture,” Trans. IECE Jpn. 57-B207 (1974).
  4. D. A. Holmes, J. E. Korka, P. V. Avizonis, “Parametric Study of Apertured Focused Gaussian Beams,” Appl. Opt. 11, 565 (1972).
    [CrossRef] [PubMed]
  5. Y. Li, E. Wolf, “Focal Shift in Focused Truncated Gaussian Beams,” Opt. Commun. 42, 151 (1982).
    [CrossRef]
  6. N. Saga, K. Tanaka, O. Fukumitsu, “Diffraction of a Gaussian Beam Through a Finite Aperture Lens and the Resulting Heterodyne Efficiency,” Appl. Opt. 20, 2827 (1981).
    [CrossRef] [PubMed]
  7. V. N. Mahajan, “Axial Irradiance and Optimum Focusing of Laser Beams,” Appl. Opt. 22, 3042 (1983).
    [CrossRef] [PubMed]
  8. P. Belland, J. P. Crenn, “Changes in the Characteristics of a Gaussian Beam Weakly Diffracted by a Circular Aperture,” Appl. Opt. 21, 522 (1982).
    [CrossRef] [PubMed]
  9. G. P. Agrawal, M. Lax, “Fraunhofer Diffraction in the Beam Approximation from Two Longitudinally Separated Slits,”J. Opt. Soc. Am. 72, 164 (1982).
    [CrossRef]

1983 (1)

1982 (3)

1981 (1)

1974 (1)

K. Tanaka, O. Fukumitsu, “Study on the Mode Expansion for the Diffraction Field of a Wave Beam by an Aperture,” Trans. IECE Jpn. 57-B207 (1974).

1972 (2)

K. Tanaka, M. Shibukawa, O. Fukumitsu, “Diffraction of a Wave Beam by an Aperture,” IEEE Trans. Microwave Theory Tech. MTT-20, 749 (1972).
[CrossRef]

D. A. Holmes, J. E. Korka, P. V. Avizonis, “Parametric Study of Apertured Focused Gaussian Beams,” Appl. Opt. 11, 565 (1972).
[CrossRef] [PubMed]

1971 (1)

Agrawal, G. P.

Avizonis, P. V.

Belland, P.

Crenn, J. P.

Fukumitsu, O.

N. Saga, K. Tanaka, O. Fukumitsu, “Diffraction of a Gaussian Beam Through a Finite Aperture Lens and the Resulting Heterodyne Efficiency,” Appl. Opt. 20, 2827 (1981).
[CrossRef] [PubMed]

K. Tanaka, O. Fukumitsu, “Study on the Mode Expansion for the Diffraction Field of a Wave Beam by an Aperture,” Trans. IECE Jpn. 57-B207 (1974).

K. Tanaka, M. Shibukawa, O. Fukumitsu, “Diffraction of a Wave Beam by an Aperture,” IEEE Trans. Microwave Theory Tech. MTT-20, 749 (1972).
[CrossRef]

Holmes, D. A.

Korka, J. E.

Lax, M.

Li, Y.

Y. Li, E. Wolf, “Focal Shift in Focused Truncated Gaussian Beams,” Opt. Commun. 42, 151 (1982).
[CrossRef]

Mahajan, V. N.

Saga, N.

Schell, R. G.

Shibukawa, M.

K. Tanaka, M. Shibukawa, O. Fukumitsu, “Diffraction of a Wave Beam by an Aperture,” IEEE Trans. Microwave Theory Tech. MTT-20, 749 (1972).
[CrossRef]

Tanaka, K.

N. Saga, K. Tanaka, O. Fukumitsu, “Diffraction of a Gaussian Beam Through a Finite Aperture Lens and the Resulting Heterodyne Efficiency,” Appl. Opt. 20, 2827 (1981).
[CrossRef] [PubMed]

K. Tanaka, O. Fukumitsu, “Study on the Mode Expansion for the Diffraction Field of a Wave Beam by an Aperture,” Trans. IECE Jpn. 57-B207 (1974).

K. Tanaka, M. Shibukawa, O. Fukumitsu, “Diffraction of a Wave Beam by an Aperture,” IEEE Trans. Microwave Theory Tech. MTT-20, 749 (1972).
[CrossRef]

Tyras, G.

Wolf, E.

Y. Li, E. Wolf, “Focal Shift in Focused Truncated Gaussian Beams,” Opt. Commun. 42, 151 (1982).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Positions of the most focused point by a finite aperture lens. The solid and dotted lines represent the points of the maximum axial intensity and the minimum field spread, respectively. They are normalized by z0, the position of the beam waist transformed by an infinite aperture lens. w1 is the spot size of the incident beam at the lens: (a) beam waist incidence (ξ0 = 0); (b) convex equiphase surface (ξ0 = 1); (c) concave euqiphase surface (ξ0 = −1).

Fig. 2
Fig. 2

Encircled energy E(z) in a circle with radius b, normalized by total energy of the diffraction field. For comparison, the normalized axial intensity is shown (dotted line) also normalized by the intensity of the incident beam at the lens. The incident beam has its beam waist at the position of the lens: (a) P = k w s 2 / f = 30; (b) P = k w s 2 / f = 10.

Fig. 3
Fig. 3

Encircled energy in a circle with radius b at the positions of the maximum axial intensity zm and the minimum field spread z s : (a) P = k w s 2 / f = 100; (b) P = k w s 2 / f = 30.

Fig. 4
Fig. 4

Axial intensity distribution normalized by the maximum intensity Im. The abscissa z is also normalized by the position zm of the maximum axial intensity.

Fig. 5
Fig. 5

Axial intensity distribution of the diffraction field through a two aperture system, normalized by the intensity of the incident beam at the first aperture. d is the distance between two apertures, a and b are radii of the first and second apertures. The incident beam has its beam waist at the first aperture.

Equations (9)

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Ψ 00 ( r , z ) = κ π exp [ i k ( z z s ) ½ κ 2 σ 2 r 2 + i tan 1 ξ ] ,
ξ = 2 ( z z s ) k w s 2 , κ = 2 w s 1 + ξ 2 , σ 2 = 1 + i ξ .
U 00 ( r , z ) = 2 i π κ 0 λ z exp [ i k ( z z s ) + i tan 1 ξ 0 i k 2 z r 2 ] 0 a r 0 J 0 ( k r r 0 z ) exp { ½ [ κ 0 2 σ 0 2 + i k ( 1 z 1 f ) ] r 0 2 } d r 0 ,
I ( r , z ) = | U 00 ( r , z ) | 2 I 0 = P α 2 Z 2 { [ 0 1 r 0 J 0 ( P α R Z r 0 ) exp ( α 2 r 0 2 1 + ξ 0 2 ) cos ( s 1 r 0 2 ) d r 0 ] 2 + [ 0 1 r 0 J 0 ( P α R Z r 0 ) exp ( α 2 r 0 2 1 + ξ 0 2 ) sin ( s 1 r 0 2 ) d r 0 ] 2 } ,
P = k w s 2 f , α = a w s , Z = z f , R = r w s , s 1 = P α 2 ( 1 Z ) 2 Z + α 2 ξ 0 1 + ξ 0 2 .
I ( z ) = I ( 0 , z ) = P 2 α 4 4 Z 2 ( s 1 2 + s 2 2 ) [ 1 + exp ( 2 s 2 ) 2 exp ( s 2 ) cos s 1 ] .
2 ( s 1 s 1 2 + s 2 2 2 Z P α 2 ) ( cosh s 2 cos s 1 ) = sin s 1 ,
s 2 = α 2 1 + ξ 0 2 .
| U 00 ( r , z ) | = exp ( 1 ) | U 00 ( 0 , z ) | .

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