Abstract

In many laser applications it is important to analyze the effect of the aberrations of optical systems which 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, peak intensity, and tolerance conditions for uniform and Gaussian beams in an optical system with primary spherical aberration are obtained on the basis of diffraction theory. The results presented are very useful for designing focusing systems for use with laser beams.

© 1982 Optical Society of America

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References

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  1. D. A. Holmes, J. E. Korka, P. V. Avizonis, Appl. Opt. 11, 565 (1972).
    [CrossRef] [PubMed]
  2. A. Arimoto, J. Opt. Soc. Am. 64, 850 (1974).
    [CrossRef]
  3. D. D. Lowenthal, Appl. Opt. 13, 2126 (1974).
    [CrossRef] [PubMed]
  4. E. Skear, J. Opt. Soc. Am. 65, 1520 (1975).
    [CrossRef]
  5. J. T. Hunt, P. A. Renard, R. G. Nelson, Appl. Opt. 15, 1458 (1976).
    [CrossRef] [PubMed]
  6. A. Yoshida, T. Asakura, Opt. Commun. 14, 211 (1975).
    [CrossRef]
  7. A. Yoshida, T. Asakura, Opt. Commun. 25, 133 (1978).
    [CrossRef]
  8. A. Yoshida, T. Asakura, Opt. Laser Technol. 11, 49 (1979).
    [CrossRef]
  9. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), Chap. 9.
  10. W. B. King, J. Opt. Soc. Am. 58, 655 (1968).
    [CrossRef]

1979 (1)

A. Yoshida, T. Asakura, Opt. Laser Technol. 11, 49 (1979).
[CrossRef]

1978 (1)

A. Yoshida, T. Asakura, Opt. Commun. 25, 133 (1978).
[CrossRef]

1976 (1)

1975 (2)

A. Yoshida, T. Asakura, Opt. Commun. 14, 211 (1975).
[CrossRef]

E. Skear, J. Opt. Soc. Am. 65, 1520 (1975).
[CrossRef]

1974 (2)

1972 (1)

1968 (1)

Arimoto, A.

Asakura, T.

A. Yoshida, T. Asakura, Opt. Laser Technol. 11, 49 (1979).
[CrossRef]

A. Yoshida, T. Asakura, Opt. Commun. 25, 133 (1978).
[CrossRef]

A. Yoshida, T. Asakura, Opt. Commun. 14, 211 (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.

King, W. B.

Korka, J. E.

Lowenthal, D. D.

Nelson, R. G.

Renard, P. A.

Skear, E.

Wolf, E.

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

Yoshida, A.

A. Yoshida, T. Asakura, Opt. Laser Technol. 11, 49 (1979).
[CrossRef]

A. Yoshida, T. Asakura, Opt. Commun. 25, 133 (1978).
[CrossRef]

A. Yoshida, T. Asakura, Opt. Commun. 14, 211 (1975).
[CrossRef]

Appl. Opt. (3)

J. Opt. Soc. Am. (3)

Opt. Commun. (2)

A. Yoshida, T. Asakura, Opt. Commun. 14, 211 (1975).
[CrossRef]

A. Yoshida, T. Asakura, Opt. Commun. 25, 133 (1978).
[CrossRef]

Opt. Laser Technol. (1)

A. Yoshida, T. Asakura, Opt. Laser Technol. 11, 49 (1979).
[CrossRef]

Other (1)

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

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Figures (4)

Fig. 1
Fig. 1

Beam optical system.

Fig. 2
Fig. 2

Positions of diffraction focus (xF,zF) for Gaussian beams near the Gaussian focus in the diffraction field for various values of ρ0 from 0 to 0.6 for f/a = 3, S = −5λ, and ω = 0.2.

Fig. 3
Fig. 3

Normalized peak intensity Is: (a) for uniform beams as a function of the aberration parameter |S|b4/λ for ρ0/b = 0, 0.2, 0.4, and 0.6; and (b) for Gaussian beams as a function of the aberration parameter |S|ω4/λ for ρ0/ω = 0, 0.4, 0.8, and 1.2.

Fig. 4
Fig. 4

Tolerance regions: (a) for uniform beams as to parameters ρ0/b and |S|b4/λ; and (b) for Gaussian beams as to parameters ρ0/ω and |S|ω4/λ.

Equations (45)

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u = k ( a / f ) 2 z , υ = k ( a / f ) ( x 2 + y 2 ) 1 / 2 ,
I ( u , υ , ψ ) = | 0 2 π 0 δ P ( r , ϕ ) exp { i [ k Φ B ( r , ϕ ) υ r cos ( ϕ ψ ) ( 1 / 2 ) u ( r 2 + ρ 0 2 + 2 ρ 0 r cos ϕ ) ] } rdrd ϕ | 2 ,
Φ S ( ρ , θ ) = S ρ 4 ,
Φ B ( r , ϕ ) = S ρ 0 4 + 4 S ρ 0 2 r 2 + 4 S ρ 0 3 r cos ϕ + Sr 4 + 4 S ρ 0 r 3 cos ϕ + 2 S ρ 0 2 r 2 cos 2 ϕ ,
υ r cos ϕ ( 1 / 2 ) u ( r 2 + ρ 0 2 + 2 ρ 0 r cos ϕ ) .
Φ ( r , ϕ ) = D r 2 + K r cos ϕ + Sr 4 + 4 S ρ 0 r 3 cos ϕ + 2 S ρ 0 2 r 2 cos 2 ϕ .
D = 4 S ρ 0 2 ( 1 / 2 ) ( u / k ) ,
K = 4 S ρ 0 2 υ / k ρ 0 ( u / k ) .
i = I / I 0 = | 0 2 π 0 δ P ( r , ϕ ) exp [ ik Φ ( r , ϕ ) ] rdrd ϕ | 2 | 0 2 π 0 δ P ( r , ϕ ) rdrd ϕ | 2 .
i = 1 ( 2 π / λ ) 2 [ Φ ¯ 2 ( Φ ¯ ) 2 ] ,
Φ ¯ n = 0 2 π 0 δ Φ n ( r , ϕ ) rdrd ϕ 0 2 π 0 δ P ( r , ϕ ) rdrd ϕ .
i = 1 ( 2 π / λ ) 2 E ,
E = Φ ¯ 2 ( Φ ¯ ) 2 .
P ( r , ϕ ) = 1 .
i = 1 ( 2 π / λ ) 2 E u ,
E u = ( 1 / 2 ) b 4 D 2 + ( 4 / 45 ) b 8 S 2 + ( 1 / 4 ) b 2 K 2 + ( 1 / 8 ) b 6 C 2 + ( 1 / 6 ) b 4 A 2 + ( 1 / 6 ) b 6 D S + ( 1 / 3 ) b 4 K C .
C = 4 S ρ 0 and A = 2 S ρ 0 2 .
E u / D = ( 1 / 6 ) b 4 D + ( 1 / 6 ) b 6 S = 0 ,
E u / K = ( 1 / 2 ) b 2 K + ( 1 / 3 ) b 4 C = 0 .
u = 2 ( 2 π / λ ) S ( 4 ρ 0 2 + b 2 ) ,
υ = ( 2 π / λ ) S [ 4 ρ 0 3 + ( 2 / 3 ) b 2 ρ 0 ] .
z F = 2 ( f / a ) 2 S ( 4 ρ 0 2 + b 2 ) ,
x F = ( f / a ) S [ 4 ρ 0 3 + ( 2 / 3 ) b 2 ρ 0 ] .
P ( r , ϕ ) = exp [ ( a / w ) 2 r 2 ] = exp [ ( r 2 / ω 2 ) ] ,
i = 1 ( 2 π / λ ) 2 Eg ,
Eg = ω 4 D 2 + 20 ω 8 S 2 + ( 1 / 2 ) ω 2 K 2 + 3 ω 6 C 2 + ω 4 A 2 + 8 ω 6 D S + 2 ω 4 K C .
Eg / D = 2 ω 4 D + 8 ω 6 S = 0 ,
Eg / K = ω 2 K + 2 ω 4 C = 0 .
u = 8 ( 2 π / λ ) S ( ρ 0 2 + ω 2 ) ,
υ = 4 ( 2 π / λ ) S ρ 0 3 .
z F = 8 ( f / a ) 2 S ( ρ 0 2 + ω 2 ) ,
x F = 4 ( f / a ) S ρ 0 3 .
I S = 1 ( π 2 / 45 ) ( Sb 4 / λ ) 2 [ 1 + 40 ( ρ 0 / b ) 2 + 120 ( ρ 0 / b ) 4 ] .
I S = 1 16 π 2 ( S ω 4 / λ ) 2 [ 1 + 4 ( ρ 0 / ω ) 2 + ( ρ 0 / ω ) 4 ] .
I S 0.8 ,
| S | b 4 / λ ( 3 / π ) [ 1 + 40 ( ρ 0 / b ) 2 + 120 ( ρ 0 / b ) 4 ] ( 1 / 2 ) .
( ρ 0 / b ) 2 ( 1 / 2 ) [ 7 / 90 + ( 3 / 10 π 2 ) ( λ / Sb 4 ) 2 ] 1 / 2 ( 1 / 6 ) .
| S | ω 4 / λ ( 1 / 4 5 π ) [ 1 + 4 ( ρ 0 / ω ) 2 + ( ρ 0 / ω ) 4 ] ( 1 / 2 ) .
( ρ 0 / ω ) 2 ( 1 / 2 ) [ 12 + ( 1 / 20 π 2 ) ( λ / S ω 4 ) 2 ] 1 / 2 2 .
Φ ¯ = 1 0 2 π 0 b rdrd ϕ 0 2 π 0 b ( D r 2 + K r cos ϕ + Sr 4 + 4 S ρ 0 r 3 cos ϕ + 2 S ρ 0 2 r 2 cos 2 ϕ ) rdrd ϕ = ( 1 / π b 2 ) [ ( π / 2 ) D b 4 + ( π / 3 ) S b 6 ] = ( D / 2 ) b 2 + ( 1 / 3 ) S b 4 .
( Φ ¯ ) 2 = ( 1 / 4 ) b 4 D 2 + ( 1 / 9 ) b 8 S 2 + ( 1 / 3 ) b 6 D S .
Φ ¯ 2 = 1 0 2 π 0 b rdrd ϕ 0 2 π 0 b ( D r 2 + K r cos ϕ + Sr 4 + 4 S ρ 0 r 3 cos ϕ + 2 S ρ 0 2 r 2 cos 2 ϕ ) 2 rdrd ϕ = ( 1 / 3 ) b 4 D 2 + ( 1 / 5 ) b 8 S 2 + ( 1 / 4 ) b 2 K 2 + ( 1 / 8 ) b 6 C 2 + ( 1 / 6 ) b 4 A 2 ,
Φ ¯ = 1 0 2 π 0 b exp ( r 2 / ω 2 ) rdrd ϕ 0 2 π 0 b exp ( r 2 / ω 2 ) × ( D r 2 + K r cos ϕ + Sr 4 + 4 S ρ 0 r 3 cos ϕ ) + 2 S ρ 0 2 r 2 cos 2 ϕ ) rdrd ϕ = ( 1 / π ω 2 ) [ 2 π D ( ω 4 / 2 ) + 2 π S ω 6 ] = D ω 2 + 2 S ω 4 ,
( Φ ¯ ) 2 = ω 4 D 2 + 4 ω 8 S 2 + 4 ω 6 D S .
Φ ¯ 2 = 1 0 2 π 0 b exp ( r 2 / ω 2 ) rdrd ϕ 0 2 π 0 b exp ( r 2 / ω 2 ) × ( D r 2 + K r cos ϕ + Sr 4 + 4 S ρ 0 r 3 cos ϕ ) + 2 S ρ 0 2 r 2 cos 2 ϕ ) 2 rdrd ϕ = 2 ω 4 D 2 + 24 ω 8 S 2 + ( 1 / 2 ) ω 2 K 2 + 3 ω 6 C 2 + ω 4 A 2 + 12 ω 6 D S + 2 ω 4 K C ,

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