Abstract

The focusing characteristics of long-distance flying optics were studied systemically for TEMmn Gaussian beams. The results show that the ABCD law of parameter q can be extended to Gaussian modes of any order when waist radius w in the imaginary part of parameter q is replaced by Rayleigh range ZR of a certain resonator in the equation. The difference between the real focal length and the geometric focal length, defined as Δf, was calculated for laser applications. A novel self-adaptive optical system was demonstrated for precisely controlling the focusing characteristics of long-distance flying optics. Theoretical analyses and experimental results were consistent.

© 2006 Optical Society of America

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References

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  1. Z. G. Cheng, J. B. Jiang, X. Q. Li, G. L. Xu, J. A. Xia, W. M. Steen, and G. Dearden, "Focus characteristics of long distance flying optics," Sci. China A 43, 980-987 (2000).
    [CrossRef]
  2. Y. J. Li and E. Wolf, "Focal shift in focused truncated Gaussian beam," Opt. Commun. 42, 151-156 (1982).
    [CrossRef]
  3. C. J. R. Sheppard and P. Torok, "Dependence of focal shift on Fresnel number and angular aperture," Opt. Lett. 23, 1903-1904 (1998).
    [CrossRef]
  4. M. P. Givens, "Focal shift in diffracted converging spherical waves," Opt. Commun. 41, 145-148 (1982).
    [CrossRef]
  5. Y. J. Li, "Propagation of focal shift through axisymmetrical optical system," Opt. Commun. 95, 13-17 (1993).
    [CrossRef]
  6. B. K. Zhou, Y. Z. Gao, J. H. Chen, and T. R. Chen, Principles of the Laser (Defense Industry Press, Beijing, 1980; in Chinese).
  7. H. Kogelnik, "Imaging of optics modes-resonators with internal lenses," Bell Syst. Tech. J. 44, 455-494 (1965).
  8. R. Ifflander and H. Weber, "Focusing of multimode laser beams with variable beam parameters," Opt. Acta 33, 1083-1090 (1986).
    [CrossRef]
  9. Q. Z. Zhao, Z. G. Cheng, J. B. Jiang, C. Q. Liu, X. L. Chai, and H. J. Huang, "Controlling the focusing characteristics of flying optics through an adaptive beam expander," Chin. J. Lasers A 29, 339-342 (2002).
  10. Y. N. Zhang, G. Ku, Z. H. Cheng, Z. M. Du, and F. Li, "The focus variable mirror with varying pressure in the flexible focus variable system," Chin. J. Lasers A 25, 998-1002 (1998).

2002 (1)

Q. Z. Zhao, Z. G. Cheng, J. B. Jiang, C. Q. Liu, X. L. Chai, and H. J. Huang, "Controlling the focusing characteristics of flying optics through an adaptive beam expander," Chin. J. Lasers A 29, 339-342 (2002).

2000 (1)

Z. G. Cheng, J. B. Jiang, X. Q. Li, G. L. Xu, J. A. Xia, W. M. Steen, and G. Dearden, "Focus characteristics of long distance flying optics," Sci. China A 43, 980-987 (2000).
[CrossRef]

1998 (2)

C. J. R. Sheppard and P. Torok, "Dependence of focal shift on Fresnel number and angular aperture," Opt. Lett. 23, 1903-1904 (1998).
[CrossRef]

Y. N. Zhang, G. Ku, Z. H. Cheng, Z. M. Du, and F. Li, "The focus variable mirror with varying pressure in the flexible focus variable system," Chin. J. Lasers A 25, 998-1002 (1998).

1993 (1)

Y. J. Li, "Propagation of focal shift through axisymmetrical optical system," Opt. Commun. 95, 13-17 (1993).
[CrossRef]

1986 (1)

R. Ifflander and H. Weber, "Focusing of multimode laser beams with variable beam parameters," Opt. Acta 33, 1083-1090 (1986).
[CrossRef]

1982 (2)

M. P. Givens, "Focal shift in diffracted converging spherical waves," Opt. Commun. 41, 145-148 (1982).
[CrossRef]

Y. J. Li and E. Wolf, "Focal shift in focused truncated Gaussian beam," Opt. Commun. 42, 151-156 (1982).
[CrossRef]

1965 (1)

H. Kogelnik, "Imaging of optics modes-resonators with internal lenses," Bell Syst. Tech. J. 44, 455-494 (1965).

Chai, X. L.

Q. Z. Zhao, Z. G. Cheng, J. B. Jiang, C. Q. Liu, X. L. Chai, and H. J. Huang, "Controlling the focusing characteristics of flying optics through an adaptive beam expander," Chin. J. Lasers A 29, 339-342 (2002).

Chen, J. H.

B. K. Zhou, Y. Z. Gao, J. H. Chen, and T. R. Chen, Principles of the Laser (Defense Industry Press, Beijing, 1980; in Chinese).

Chen, T. R.

B. K. Zhou, Y. Z. Gao, J. H. Chen, and T. R. Chen, Principles of the Laser (Defense Industry Press, Beijing, 1980; in Chinese).

Cheng, Z. G.

Q. Z. Zhao, Z. G. Cheng, J. B. Jiang, C. Q. Liu, X. L. Chai, and H. J. Huang, "Controlling the focusing characteristics of flying optics through an adaptive beam expander," Chin. J. Lasers A 29, 339-342 (2002).

Z. G. Cheng, J. B. Jiang, X. Q. Li, G. L. Xu, J. A. Xia, W. M. Steen, and G. Dearden, "Focus characteristics of long distance flying optics," Sci. China A 43, 980-987 (2000).
[CrossRef]

Cheng, Z. H.

Y. N. Zhang, G. Ku, Z. H. Cheng, Z. M. Du, and F. Li, "The focus variable mirror with varying pressure in the flexible focus variable system," Chin. J. Lasers A 25, 998-1002 (1998).

Dearden, G.

Z. G. Cheng, J. B. Jiang, X. Q. Li, G. L. Xu, J. A. Xia, W. M. Steen, and G. Dearden, "Focus characteristics of long distance flying optics," Sci. China A 43, 980-987 (2000).
[CrossRef]

Du, Z. M.

Y. N. Zhang, G. Ku, Z. H. Cheng, Z. M. Du, and F. Li, "The focus variable mirror with varying pressure in the flexible focus variable system," Chin. J. Lasers A 25, 998-1002 (1998).

Gao, Y. Z.

B. K. Zhou, Y. Z. Gao, J. H. Chen, and T. R. Chen, Principles of the Laser (Defense Industry Press, Beijing, 1980; in Chinese).

Givens, M. P.

M. P. Givens, "Focal shift in diffracted converging spherical waves," Opt. Commun. 41, 145-148 (1982).
[CrossRef]

Huang, H. J.

Q. Z. Zhao, Z. G. Cheng, J. B. Jiang, C. Q. Liu, X. L. Chai, and H. J. Huang, "Controlling the focusing characteristics of flying optics through an adaptive beam expander," Chin. J. Lasers A 29, 339-342 (2002).

Ifflander, R.

R. Ifflander and H. Weber, "Focusing of multimode laser beams with variable beam parameters," Opt. Acta 33, 1083-1090 (1986).
[CrossRef]

Jiang, J. B.

Q. Z. Zhao, Z. G. Cheng, J. B. Jiang, C. Q. Liu, X. L. Chai, and H. J. Huang, "Controlling the focusing characteristics of flying optics through an adaptive beam expander," Chin. J. Lasers A 29, 339-342 (2002).

Z. G. Cheng, J. B. Jiang, X. Q. Li, G. L. Xu, J. A. Xia, W. M. Steen, and G. Dearden, "Focus characteristics of long distance flying optics," Sci. China A 43, 980-987 (2000).
[CrossRef]

Kogelnik, H.

H. Kogelnik, "Imaging of optics modes-resonators with internal lenses," Bell Syst. Tech. J. 44, 455-494 (1965).

Ku, G.

Y. N. Zhang, G. Ku, Z. H. Cheng, Z. M. Du, and F. Li, "The focus variable mirror with varying pressure in the flexible focus variable system," Chin. J. Lasers A 25, 998-1002 (1998).

Li, F.

Y. N. Zhang, G. Ku, Z. H. Cheng, Z. M. Du, and F. Li, "The focus variable mirror with varying pressure in the flexible focus variable system," Chin. J. Lasers A 25, 998-1002 (1998).

Li, X. Q.

Z. G. Cheng, J. B. Jiang, X. Q. Li, G. L. Xu, J. A. Xia, W. M. Steen, and G. Dearden, "Focus characteristics of long distance flying optics," Sci. China A 43, 980-987 (2000).
[CrossRef]

Li, Y. J.

Y. J. Li, "Propagation of focal shift through axisymmetrical optical system," Opt. Commun. 95, 13-17 (1993).
[CrossRef]

Y. J. Li and E. Wolf, "Focal shift in focused truncated Gaussian beam," Opt. Commun. 42, 151-156 (1982).
[CrossRef]

Liu, C. Q.

Q. Z. Zhao, Z. G. Cheng, J. B. Jiang, C. Q. Liu, X. L. Chai, and H. J. Huang, "Controlling the focusing characteristics of flying optics through an adaptive beam expander," Chin. J. Lasers A 29, 339-342 (2002).

Sheppard, C. J. R.

C. J. R. Sheppard and P. Torok, "Dependence of focal shift on Fresnel number and angular aperture," Opt. Lett. 23, 1903-1904 (1998).
[CrossRef]

Steen, W. M.

Z. G. Cheng, J. B. Jiang, X. Q. Li, G. L. Xu, J. A. Xia, W. M. Steen, and G. Dearden, "Focus characteristics of long distance flying optics," Sci. China A 43, 980-987 (2000).
[CrossRef]

Torok, P.

C. J. R. Sheppard and P. Torok, "Dependence of focal shift on Fresnel number and angular aperture," Opt. Lett. 23, 1903-1904 (1998).
[CrossRef]

Weber, H.

R. Ifflander and H. Weber, "Focusing of multimode laser beams with variable beam parameters," Opt. Acta 33, 1083-1090 (1986).
[CrossRef]

Wolf, E.

Y. J. Li and E. Wolf, "Focal shift in focused truncated Gaussian beam," Opt. Commun. 42, 151-156 (1982).
[CrossRef]

Xia, J. A.

Z. G. Cheng, J. B. Jiang, X. Q. Li, G. L. Xu, J. A. Xia, W. M. Steen, and G. Dearden, "Focus characteristics of long distance flying optics," Sci. China A 43, 980-987 (2000).
[CrossRef]

Xu, G. L.

Z. G. Cheng, J. B. Jiang, X. Q. Li, G. L. Xu, J. A. Xia, W. M. Steen, and G. Dearden, "Focus characteristics of long distance flying optics," Sci. China A 43, 980-987 (2000).
[CrossRef]

Zhang, Y. N.

Y. N. Zhang, G. Ku, Z. H. Cheng, Z. M. Du, and F. Li, "The focus variable mirror with varying pressure in the flexible focus variable system," Chin. J. Lasers A 25, 998-1002 (1998).

Zhao, Q. Z.

Q. Z. Zhao, Z. G. Cheng, J. B. Jiang, C. Q. Liu, X. L. Chai, and H. J. Huang, "Controlling the focusing characteristics of flying optics through an adaptive beam expander," Chin. J. Lasers A 29, 339-342 (2002).

Zhou, B. K.

B. K. Zhou, Y. Z. Gao, J. H. Chen, and T. R. Chen, Principles of the Laser (Defense Industry Press, Beijing, 1980; in Chinese).

Bell Syst. Tech. J. (1)

H. Kogelnik, "Imaging of optics modes-resonators with internal lenses," Bell Syst. Tech. J. 44, 455-494 (1965).

Chin. J. Lasers A (2)

Q. Z. Zhao, Z. G. Cheng, J. B. Jiang, C. Q. Liu, X. L. Chai, and H. J. Huang, "Controlling the focusing characteristics of flying optics through an adaptive beam expander," Chin. J. Lasers A 29, 339-342 (2002).

Y. N. Zhang, G. Ku, Z. H. Cheng, Z. M. Du, and F. Li, "The focus variable mirror with varying pressure in the flexible focus variable system," Chin. J. Lasers A 25, 998-1002 (1998).

Opt. Acta (1)

R. Ifflander and H. Weber, "Focusing of multimode laser beams with variable beam parameters," Opt. Acta 33, 1083-1090 (1986).
[CrossRef]

Opt. Commun. (3)

Y. J. Li and E. Wolf, "Focal shift in focused truncated Gaussian beam," Opt. Commun. 42, 151-156 (1982).
[CrossRef]

M. P. Givens, "Focal shift in diffracted converging spherical waves," Opt. Commun. 41, 145-148 (1982).
[CrossRef]

Y. J. Li, "Propagation of focal shift through axisymmetrical optical system," Opt. Commun. 95, 13-17 (1993).
[CrossRef]

Opt. Lett. (1)

C. J. R. Sheppard and P. Torok, "Dependence of focal shift on Fresnel number and angular aperture," Opt. Lett. 23, 1903-1904 (1998).
[CrossRef]

Sci. China A (1)

Z. G. Cheng, J. B. Jiang, X. Q. Li, G. L. Xu, J. A. Xia, W. M. Steen, and G. Dearden, "Focus characteristics of long distance flying optics," Sci. China A 43, 980-987 (2000).
[CrossRef]

Other (1)

B. K. Zhou, Y. Z. Gao, J. H. Chen, and T. R. Chen, Principles of the Laser (Defense Industry Press, Beijing, 1980; in Chinese).

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Figures (8)

Fig. 1
Fig. 1

Parameters that influence the Gaussian beam's focusing characteristics.

Fig. 2
Fig. 2

Flying-optics laser processing system equipped with self-adaptive optics.

Fig. 3
Fig. 3

(Color online) Photograph of the self-adaptive optical system.

Fig. 4
Fig. 4

(Color online) Photograph of the deformable mirrors. Left, convex mirror; right, concave mirror.

Fig. 5
Fig. 5

Equivalent optical model of the flying-optics system in Fig. 2.

Fig. 6
Fig. 6

Simplified optical model of Fig. 5.

Fig. 7
Fig. 7

(Color online) Relationship between pressure and radius of curvature for a convex mirror.

Fig. 8
Fig. 8

(Color online) Relationship between pressure and radius of curvature for a concave mirror.

Tables (2)

Tables Icon

Table 1 Maximum Change of Focal Length for Typical Lasers ( f = 200 mm)

Tables Icon

Table 2 Experimental Results for the Self-Adaptive Optical System

Equations (22)

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w m n = M w 00 ,
θ m n = M θ 00 ,
Z R = π w m m 2 M 2 λ ,
w m n θ m n = M 2 w 00 θ 00 = M 2 ( λ / π ) ,
1 q 00 ( z ) = 1 R 00 ( z ) i λ π w 00 2 ( z ) ,
q 00 ( z ) = A q 00 ( z ) + B C q 00 ( z ) + D ,
[ A B C D ] = [ 1 f 0 1 ] [ 1 0 1 f 1 ] [ 1 z z 0 0 1 ] = [ 1 f f ( z z 0 ) f f ( z z 0 ) + f 1 f 1 ( z z 0 ) f ] ,
1 q 00 = i λ π w 00           2 .
1 q m n = i θ m n w m n ,
Z R = w m n / θ m n .
1 q m n = i Z R .
Δ f = f 2 ( z z 0 f ) / [ Z R       2 + ( z z 0 f ) 2 ] ,
w 2 = λ π M 2 f 2 Z R / [ Z R       2 + ( z z 0 f ) 2 ] ,
h = f 2 Z R / [ Z R       2 + ( z z 0 f ) 2 ] ,
Δ f max = f 2 / 2 Z R .
[ A B C D ] = [ 1 f 0 1 ] [ 1 0 1 f 1 ] [ 1 L 2 0 1 ] [ 1 0 1 f 2 1 ] [ 1 L 1 0 1 ] × [ 1 0 1 f 1 1 ] [ 1 L 0 1 ] .
[ A B C D ] = [ 1 f 0 1 ] [ a b c d ] [ 1 L 0 1 ] = [ a + c f ( a + c f ) L + ( b + d f ) c c L + d ] .
A Z R = D Z R ,
B + C Z R Z R = 0 ,
f = ( c L + d ) ( a L + b ) + a c Z R       2 ( c L + d ) 2 + c 2 Z R       2 ,
w 2 = 1 c L ( c L + 2 d ) + d 2 + c 2 Z R       2 w 2 ,
Z R = w m n θ m n = w 00 θ 00 .

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