Abstract

In many laser applications optical systems are employed to focus off-axis beams that only partially fill the system aperture. In this paper analytical expressions that give the position of the diffraction focus, normalized peak intensity, and tolerance conditions for uniform and Gaussian off-axis beams in an optical system which suffers from astigmatism together with longitudinal focal shift aberration are obtained on the basis of diffraction theory. The results are useful for designing focusing systems for use with laser beams.

© 1990 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), Chap. 9.
  2. D. A. Holmes, J. E. Korka, P. V. Avizonis, “Parametric Study of Apertured Focused Gaussian Beams,” Appl. Opt. 11, 565–574 (1972).
    [CrossRef] [PubMed]
  3. D. D. Lowenthal, “Marechal Intensity Criteria Modified for Gaussian Beams,” Appl. Opt. 13, 2126–2133 (1974).
    [CrossRef] [PubMed]
  4. J. T. Hunt, P. A. Renard, R. G. Nelson, “Focusing Properties of an Aberrated Laser Beam,” Appl. Opt. 15, 1458–1464 (1976).
    [CrossRef] [PubMed]
  5. A. Yoshida, T. Asakura, “Effect of Aberrations on Off-Axis Gaussian Beams,” Opt. Commun. 14, 211–214 (1975).
    [CrossRef]
  6. A. Yoshida, T. Asakura, “Diffraction Patterns of Off-Axis Gaussian Beams in the Presence of Third-Order Spherical Aberration in the Optical System,” Opt. Commun. 19, 387–392 (1976).
    [CrossRef]
  7. A. Yoshida, T. Asakura, “Diffraction Patterns of Off-Axis Gaussian Beams in the Optical System with Astigmatism and Coma,” Opt. Commun. 25, 133–136 (1978).
    [CrossRef]
  8. A. Yoshida, “Spherical Aberration in Beam Optical Systems,” Appl. Opt. 21, 1812–1816 (1982).
    [CrossRef] [PubMed]

1982

1978

A. Yoshida, T. Asakura, “Diffraction Patterns of Off-Axis Gaussian Beams in the Optical System with Astigmatism and Coma,” Opt. Commun. 25, 133–136 (1978).
[CrossRef]

1976

A. Yoshida, T. Asakura, “Diffraction Patterns of Off-Axis Gaussian Beams in the Presence of Third-Order Spherical Aberration in the Optical System,” Opt. Commun. 19, 387–392 (1976).
[CrossRef]

J. T. Hunt, P. A. Renard, R. G. Nelson, “Focusing Properties of an Aberrated Laser Beam,” Appl. Opt. 15, 1458–1464 (1976).
[CrossRef] [PubMed]

1975

A. Yoshida, T. Asakura, “Effect of Aberrations on Off-Axis Gaussian Beams,” Opt. Commun. 14, 211–214 (1975).
[CrossRef]

1974

1972

Asakura, T.

A. Yoshida, T. Asakura, “Diffraction Patterns of Off-Axis Gaussian Beams in the Optical System with Astigmatism and Coma,” Opt. Commun. 25, 133–136 (1978).
[CrossRef]

A. Yoshida, T. Asakura, “Diffraction Patterns of Off-Axis Gaussian Beams in the Presence of Third-Order Spherical Aberration in the Optical System,” Opt. Commun. 19, 387–392 (1976).
[CrossRef]

A. Yoshida, T. Asakura, “Effect of Aberrations on Off-Axis Gaussian Beams,” Opt. Commun. 14, 211–214 (1975).
[CrossRef]

Avizonis, P. V.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), Chap. 9.

Holmes, D. A.

Hunt, J. T.

Korka, J. E.

Lowenthal, D. D.

Nelson, R. G.

Renard, P. A.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), Chap. 9.

Yoshida, A.

A. Yoshida, “Spherical Aberration in Beam Optical Systems,” Appl. Opt. 21, 1812–1816 (1982).
[CrossRef] [PubMed]

A. Yoshida, T. Asakura, “Diffraction Patterns of Off-Axis Gaussian Beams in the Optical System with Astigmatism and Coma,” Opt. Commun. 25, 133–136 (1978).
[CrossRef]

A. Yoshida, T. Asakura, “Diffraction Patterns of Off-Axis Gaussian Beams in the Presence of Third-Order Spherical Aberration in the Optical System,” Opt. Commun. 19, 387–392 (1976).
[CrossRef]

A. Yoshida, T. Asakura, “Effect of Aberrations on Off-Axis Gaussian Beams,” Opt. Commun. 14, 211–214 (1975).
[CrossRef]

Appl. Opt.

Opt. Commun.

A. Yoshida, T. Asakura, “Effect of Aberrations on Off-Axis Gaussian Beams,” Opt. Commun. 14, 211–214 (1975).
[CrossRef]

A. Yoshida, T. Asakura, “Diffraction Patterns of Off-Axis Gaussian Beams in the Presence of Third-Order Spherical Aberration in the Optical System,” Opt. Commun. 19, 387–392 (1976).
[CrossRef]

A. Yoshida, T. Asakura, “Diffraction Patterns of Off-Axis Gaussian Beams in the Optical System with Astigmatism and Coma,” Opt. Commun. 25, 133–136 (1978).
[CrossRef]

Other

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), Chap. 9.

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Figures (3)

Fig. 1
Fig. 1

Off-axis beam optical system.

Fig. 2
Fig. 2

Positions of diffraction focus (xF,zF) for Gaussian beams of normalized spot radii ω = 0.2 and 0.4 for various values of ρ0 from 0 to 0.6 for three values of the coefficient D = 0, −2.5λ, −5λ, and A = 5λ, θ0 = 45°, f/a = 1.

Fig. 3
Fig. 3

Tolerance regions for (a) uniform beams for parameter ρ0/b and Ab2/λ and (b) Gaussian beams for parameters ρ0/ω and 2/λ.

Equations (25)

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ϕ A ( ρ , θ ) = D ρ 2 + A ρ 2 cos 2 θ ,
ϕ A ( r , φ ) = ρ 0 2 ( D + A cos 2 θ 0 ) + D r 2 + [ 2 ( A + D ) ρ 0 cos θ 0 ] r cos φ + [ 2 D ρ 0 sin θ 0 ] r sin φ + A r 2 cos 2 φ .
u = k ( a / f ) 2 z             v = k ( a / f ) ( x 2 + y 2 ) 1 / 2 ,
I ( u , v , ψ ) = | 0 2 π 0 δ P ( r , φ ) exp { i [ k ϕ A ( r , φ ) - v r cos ( φ - ψ ) - ( 1 / 2 ) u ( r 2 + ρ 0 2 + 2 ρ 0 r cos φ ] } r d r d φ | 2 ,
ϕ ( r , φ ) = ϕ A ( r , φ ) - v r cos φ - ( 1 / 2 ) u ( r 2 + ρ 0 2 + 2 ρ 0 r cos φ ) ,
i = I I 0 = | 0 2 π 0 δ P ( r , φ ) exp [ i k ϕ ( r , φ ) ] r d r d φ | 2 | 0 2 π 0 δ P ( r , φ ) r d r d φ | 2 .
i = 1 - ( 2 π / λ ) 2 E = 1 - ( 2 π / λ ) 2 [ ϕ 2 ¯ - ( ϕ ¯ ) 2 ] ,
ϕ n ¯ = 0 2 π 0 δ P ( r , φ ) ϕ n ( r , φ ) r d r d φ / 0 2 π 0 δ P ( r , φ ) r d r d φ .
E = ( b 4 / 48 ) [ 4 D ( A + D ) + 3 A 2 - 2 ( A + 2 D ) ( u / k ) + u 2 / k 2 ] + b 2 ρ 0 2 [ D 2 + A ( A + 2 D ) cos 2 θ 0 ] + [ b 2 / ( 4 k 2 ) ] ( v 2 + u 2 ρ 0 2 + 2 u v ρ 0 cos θ 0 ) - ( b 2 ρ 0 / k ) × [ u ρ 0 ( D + A cos 2 θ 0 ) + v ( A + D ) cos θ 0 ] .
u = k [ ( 2 D + A ) b 2 + 24 D ρ 0 2 sin 2 θ 0 ] / ( b 2 + 12 ρ 0 2 sin 2 θ 0 ) ,
v = ρ 0 cos θ 0 [ 2 k ( A + D ) - u ] = A k ρ 0 cos θ 0 ( b 2 + 24 ρ 0 2 sin 2 θ 0 ) / ( b 2 + 12 ρ 0 2 sin 2 θ 0 ) .
z F = ( f / a ) 2 ( A + 2 D ) ,             x F = ( f / a ) A ρ 0 .
P ( r , φ ) = exp [ - ( a / ξ ) 2 r 2 ] = exp [ - ( r 2 / ω 2 ) ] ,
E = ( ω 4 / 2 ) [ 2 D ( A + D ) + A 2 - ( A + 2 D ) ( u / k ) + u 2 / ( 2 k 2 ) ] + ( ω 2 / 2 ) { 4 ρ 0 2 [ D 2 + A ( A + 2 D ) cos 2 θ 0 ] + ( 1 / k 2 ) ( u 2 ρ 0 2 + 2 u v ρ 0 cos θ 0 + v 2 ) - ( 4 ρ 0 / k ) × [ u ρ 0 ( D + A cos 2 θ 0 ) + v ( A + D ) cos θ 0 ] } .
u = k [ ( A + 2 D ) ω 2 + 4 D ρ 0 2 sin 2 θ 0 ] / ( ω 2 + 2 ρ 0 2 sin 2 θ 0 ) ,
v = ρ 0 cos θ 0 [ 2 k ( A + D ) - u ] = A k ρ 0 cos θ 0 ( ω 2 + 4 ρ 0 2 sin 2 θ 0 ) / ( ω 2 + 2 ρ 0 2 sin 2 θ 0 ) .
x M | y = 0 z = 0 = 2 ( f / a ) ( A + D ) ρ 0 cos θ 0 ,
I S = 1 - ( π 2 / 6 ) ( A b 2 / λ ) 2 [ 1 + 30 ( ρ 0 / b ) 2 sin 2 θ 0 + 216 ( ρ 0 / b ) 4 sin 4 θ 0 ] / [ 1 + 12 ( ρ 0 / b ) 2 sin 2 θ 0 ] 2 .
I S = 1 - ( π 2 / 6 ) ( A b 2 / λ ) 2 .
I S = 1 - π 2 ( A ω 2 / λ ) 2 [ 1 + 6 ( ρ 0 / ω ) 2 sin 2 θ 0 + 8 ( ρ 0 / ω ) 4 sin 4 θ 0 ] / [ 1 + 2 ( ρ 0 / ω ) 2 sin 2 θ 0 ] 2 .
I S = 1 - π 2 ( A ω 2 / λ ) 2 .
A b 2 / λ ( 1.2 / π ) [ 1 + 12 ( ρ 0 / b ) 2 sin 2 θ 0 ] × [ 1 + 30 ( ρ 0 / b ) 2 sin 2 θ 0 + 216 ( ρ 0 / b ) 4 sin 4 θ 0 ] - 1 / 2 .
A ( 1.2 / π ) λ .
A ω 2 / λ ( 1 / ( 5 π ) ) [ 1 + 2 ( ρ 0 / ω ) 2 sin 2 θ 0 ] × [ 1 + 6 ( ρ 0 / ω ) 2 sin 2 θ 0 + 8 ( ρ 0 / ω ) 4 sin 4 θ 0 ] - 1 / 2 .
A λ ω 2 / ( 5 π ) .

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