Abstract

The position of the most focused point of a diffracted Gaussian beam through a finite aperture lens is investigated experimentally and numerically. It depends on the definition of the point, but it will be of most practical interest to define the point as the position where the maximum energy is received by a finite aperture detector. In considering the problem, we also discuss the effect of the curvature of the equiphase surface of the incident beam on the diffraction field.

© 1987 Optical Society of America

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References

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  1. N. Saga, T. 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]
  2. K. Tanaka, N. Saga, K. Hauchi, “Focusing of a Gaussian Beam Through a Finite Aperture Lens,” Appl. Opt. 24, 1098 (1985).
    [CrossRef] [PubMed]
  3. J. J. Stamnes, H. Heier, S. Ljunggren, “Encircled Energy for Systems with Centrally Obscured Circular Pupils,” Appl. Opt. 21, 1628 (1982).
    [CrossRef] [PubMed]
  4. K. Tanaka, N. Saga, H. Mizokami, “Field Spread of a Diffracted Gaussian Beam Through a Circular Aperture,” Appl. Opt. 24, 1102 (1985).
    [CrossRef] [PubMed]

1985 (2)

1982 (1)

1981 (1)

Fukumitsu, O.

Hauchi, K.

Heier, H.

Ljunggren, S.

Mizokami, H.

Saga, N.

Stamnes, J. J.

Tanaka, K.

Tanaka, T.

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Figures (5)

Fig. 1
Fig. 1

Incident beam, finite aperture lens, and detector.

Fig. 2
Fig. 2

Received energy normalized by the total diffracted energy. The incident beam has the smallest spot size ws = 0.608 mm at zs = −2000 mm. The radii of the lens and the detector are a and b, respectively: (a) f = 30 cm, a = 0.317 mm; (b) f = 30 cm, a = 0.609 mm; (c) f = 30 cm, a = 1.305 mm; (d) f = 50 cm, a = 0.317 mm; (e) f = 50 cm, a = 0.609 mm; (f) f = 50 cm, a = 1.305 mm.

Fig. 3
Fig. 3

Incident beam with large curvature of the equiphase surface.

Fig. 4
Fig. 4

Intensity distributions of the diffraction field with large curvature of the equiphase surface of the incident field through a finite aperture. The parameter d is the optical path difference between the center and the edge of the aperture normalized by the wavelength. The incident beam has the smallest spot size ws = 0.106 mm at zs = −1308 mm: (a) d = 0.57, a = 0.9745 mm; (b) d = 0.62, a = 1.0215 mm; (c) d = 0.75, a = 1.1120 mm; (d) d = 0.81, a = 1.1625 mm; (e) d = 0.89, a = 1.2140 mm; (f) d = 0.96, a = 1.2615 mm.

Fig. 5
Fig. 5

Numerical results for various optical path difference d normalized by the wavelength for the same incidence conditions as in Fig. 4: (a) d = 0.57; (b) d = 0.62; (c) d = 0.75; (d) d = 0.81; (e) d = 0.89; (f) d = 0.96; (g) d = 1.03; (h) d = 1.12; (i) d = 1.20.

Equations (12)

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ψ 00 ( r , z ) = κ π exp [ - i k ( z - z s ) - 1 2 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 .
P ( 0 ) = 0 a 0 2 π ψ 00 2 r d r d θ = 1 - exp ( - κ 0 2 a 2 ) ,
U 1 ( x , y , z ) = i λ z exp [ - i k z - i k ( x 2 + y 2 ) 2 z ] 0 a 0 2 π U 0 ( x 0 , y 0 , 0 ) × exp [ i k ( x x 0 + y y 0 ) z - i k ( x 0 2 + y 0 2 ) 2 z ] d x 0 d y 0 ,
U 1 ( r , θ , z ) = i λ z exp ( - i k z - i k r 2 2 z ) 0 a 0 2 π U 0 ( r 0 , θ 0 , 0 ) × exp [ i k r r 0 cos ( θ - θ 0 ) z - i k r 0 2 2 z ] r 0 d r 0 d θ 0 .
U ( r , z ) = i λ z exp ( - i k z - i k r 2 2 z ) 0 a 0 2 π κ 0 π exp [ i k z s - 1 2 κ 0 2 σ 0 2 r 0 2 + i tan - 1 ξ 0 + i k r 0 2 2 f + i k r r 0 cos ( θ - θ 0 ) z - i k r 0 2 2 z ] r 0 d r 0 d θ 0 .
J n ( z ) = 1 2 π 0 2 π exp [ i ( n θ - z sin θ ) ] d θ ,
U ( r , z ) = i 2 π κ 0 λ z exp [ - i k ( z - z s ) + i tan - 1 ξ 0 - i k r 2 2 z ] × 0 a J 0 ( k r r 0 z ) exp [ - 1 2 ( κ 0 2 σ 0 2 + i k z - i k f ) r 0 2 ] r 0 d r 0 .
P ( z ) = 0 b 0 2 π U ( r , z ) 2 r d r d θ .
V ( r , z ) = i 2 π κ 0 λ z exp [ - i k ( z - z s ) + i tan - 1 ξ 0 - i k r 2 2 z ] × 0 a J 0 ( k r r 0 z ) exp ( - 1 2 κ 0 2 σ 0 2 r 0 2 ) r 0 d r 0 .
N a = k a 2 z ,             β = κ 0 a ,             p = 2 z s λ w s 2 .
I ( r , z ) = V ( r , z ) 2 = N a 2 β 2 π a 2 { [ 0 1 J 0 ( N a R R 0 ) exp ( - 1 2 β 2 R 0 2 ) × cos ( p 4 π β 2 R 0 2 ) R 0 d R 0 ] 2 + [ 0 1 J 0 ( N a R R 0 ) exp ( - 1 2 β 2 R 0 2 ) × sin ( p 4 π β 2 R 0 2 ) R 0 d R 0 ] 2 } ,

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