Abstract

Transformation circle theory is simpler than other methods for analyzing laser resonators. In our analysis only σ circles and simple mathematical knowledge are used to analyze the stability and calculate the parameters of the laser resonator, which further simplifies the transformation theory. The results agree well with the well-known matrix theory. Two- and three-mirror (including a thermal lens) laser resonators are used as examples to present the stability formula and the Gaussian beam dimensions at the mirrors. Furthermore, we apply the commonly used example in which the laser medium is close to the cavity mirror.

© 2007 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  3. H. Weber, Laser Resonators (Huazhong Institute of Technology, 1983) (in Chinese).
  4. S. M. Wang and D. Zhao, Matrix Optics (CHEP-Springer, 2000).
  5. M. Zhao and B. Lu, "Gaussian beam transformation through resonator mirror," Laser J. 2, 19-22 (1989) (in Chinese).
  6. M. Spurr and M. Dunn, "Euclidean light: high-school geometry to solve problems in Gaussian beam optics," Opt. Photonics News 13(8), 40-44 (2002).
    [CrossRef]
  7. F. Song, G. Y. Zhang, J. J. Xu, and C. B. Zhang, "Analysis of laser diode pumped solid-state laser resonator by the transform circle approach," Chin. Phys. Lett. 17, 203-205 (2000).
    [CrossRef]
  8. F. Song, C. B. Zhang, X. Ding, J. J. Xu, G. Y. Zhang, M. Leigh, and N. Peyghambarian, "Determination of thermal focal length and pumping radius in gain medium in laser-diode-pumped Nd:YVO4 lasers," Appl. Phys. Lett. 81, 2145-2147 (2002).
    [CrossRef]
  9. G. Y. Zhang, "Solutions and characters of the thermal-insensitive resonator," Acta. Phys. Sin. 40, 407-413 (1991) (in Chinese).
  10. G. Y. Zhang, "Graphic design method of optical resonator (III)," Lasers 4, 41-46 (1977) (in Chinese).
  11. G. Y. Zhang and S. G. Guo, Graphic Analysis and Design Method of Optical Resonator (National Defence Industry Press, 2003), pp. 4-18 (in Chinese).
  12. C. B. Zhang, F. Song, F. Z. Meng, X. Ding, G. Y. Zhang, and M. R. Shang, "Measurement of the laser diode end-pumped solid laser's thermal focal length by using the curve of output," Acta. Phys. Sin. 51, 1517-1520 (2002) (in Chinese).
  13. G. Y. Zhang, C. B. Zhang, X. Ding, F. Song, S. G. Guo, and F. Z. Meng, "Control on dynamical stablity of solid laser resonator," Acta. Phys. Sin. 51, 253-258 (2002) (in Chinese).
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  16. "Propagation characteristics of laser beams," www.mellesgriot.com/pdf/CatalogX/X_36_6-9.pdf.
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    [CrossRef]

2005 (1)

2002 (4)

M. Spurr and M. Dunn, "Euclidean light: high-school geometry to solve problems in Gaussian beam optics," Opt. Photonics News 13(8), 40-44 (2002).
[CrossRef]

F. Song, C. B. Zhang, X. Ding, J. J. Xu, G. Y. Zhang, M. Leigh, and N. Peyghambarian, "Determination of thermal focal length and pumping radius in gain medium in laser-diode-pumped Nd:YVO4 lasers," Appl. Phys. Lett. 81, 2145-2147 (2002).
[CrossRef]

C. B. Zhang, F. Song, F. Z. Meng, X. Ding, G. Y. Zhang, and M. R. Shang, "Measurement of the laser diode end-pumped solid laser's thermal focal length by using the curve of output," Acta. Phys. Sin. 51, 1517-1520 (2002) (in Chinese).

G. Y. Zhang, C. B. Zhang, X. Ding, F. Song, S. G. Guo, and F. Z. Meng, "Control on dynamical stablity of solid laser resonator," Acta. Phys. Sin. 51, 253-258 (2002) (in Chinese).

2000 (1)

F. Song, G. Y. Zhang, J. J. Xu, and C. B. Zhang, "Analysis of laser diode pumped solid-state laser resonator by the transform circle approach," Chin. Phys. Lett. 17, 203-205 (2000).
[CrossRef]

1996 (1)

1991 (2)

G. Y. Zhang, "A dynamic analysis on solid laser resonator," Acta. Phys. Sin. 40, 1065-1073 (1991) (in Chinese).

G. Y. Zhang, "Solutions and characters of the thermal-insensitive resonator," Acta. Phys. Sin. 40, 407-413 (1991) (in Chinese).

1990 (1)

1989 (2)

S. Nemoto, "Waist shift of a Gaussian beam by a dielectric plate," Appl. Opt. 28, 1643-1647 (1989).
[CrossRef] [PubMed]

M. Zhao and B. Lu, "Gaussian beam transformation through resonator mirror," Laser J. 2, 19-22 (1989) (in Chinese).

1988 (1)

1985 (1)

1977 (1)

G. Y. Zhang, "Graphic design method of optical resonator (III)," Lasers 4, 41-46 (1977) (in Chinese).

1976 (1)

A. E. Siegman, "A canonical formulation for analyzing multielement unstable resonators," IEEE J. Quantum Electron. 12, 35-40 (1976).
[CrossRef]

1969 (1)

P. Baues, "The connection of geometric optics with the propagation of Gaussian beams and the theory of optical resonators," Opto-Electronics 1, 103-118 (1969).
[CrossRef]

1967 (1)

1966 (1)

1965 (1)

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

Alda, J.

J. Alda, "Laser and gaussian beam propagation and transformation," in Encyclopaedia of Optical Engineering (Marcel Dekker, 2003), pp. 999-1013.

Arnaud, J.

Baues, P.

P. Baues, "The connection of geometric optics with the propagation of Gaussian beams and the theory of optical resonators," Opto-Electronics 1, 103-118 (1969).
[CrossRef]

Deschamps, G. A.

G. A. Deschamps and P. E. Mast, in Proceedings of the Symposium on Quasi-optics, J. Fox, ed. (Polytechnic, 1964).

Ding, X.

F. Song, C. B. Zhang, X. Ding, J. J. Xu, G. Y. Zhang, M. Leigh, and N. Peyghambarian, "Determination of thermal focal length and pumping radius in gain medium in laser-diode-pumped Nd:YVO4 lasers," Appl. Phys. Lett. 81, 2145-2147 (2002).
[CrossRef]

C. B. Zhang, F. Song, F. Z. Meng, X. Ding, G. Y. Zhang, and M. R. Shang, "Measurement of the laser diode end-pumped solid laser's thermal focal length by using the curve of output," Acta. Phys. Sin. 51, 1517-1520 (2002) (in Chinese).

G. Y. Zhang, C. B. Zhang, X. Ding, F. Song, S. G. Guo, and F. Z. Meng, "Control on dynamical stablity of solid laser resonator," Acta. Phys. Sin. 51, 253-258 (2002) (in Chinese).

Dunn, M.

M. Spurr and M. Dunn, "Euclidean light: high-school geometry to solve problems in Gaussian beam optics," Opt. Photonics News 13(8), 40-44 (2002).
[CrossRef]

Guo, S. G.

G. Y. Zhang, C. B. Zhang, X. Ding, F. Song, S. G. Guo, and F. Z. Meng, "Control on dynamical stablity of solid laser resonator," Acta. Phys. Sin. 51, 253-258 (2002) (in Chinese).

G. Y. Zhang and S. G. Guo, Graphic Analysis and Design Method of Optical Resonator (National Defence Industry Press, 2003), pp. 4-18 (in Chinese).

Kogelink, H.

Kogelnik, H.

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

Laures, P.

Leigh, M.

F. Song, C. B. Zhang, X. Ding, J. J. Xu, G. Y. Zhang, M. Leigh, and N. Peyghambarian, "Determination of thermal focal length and pumping radius in gain medium in laser-diode-pumped Nd:YVO4 lasers," Appl. Phys. Lett. 81, 2145-2147 (2002).
[CrossRef]

Li, C.

Li, T.

Lu, B.

M. Zhao and B. Lu, "Gaussian beam transformation through resonator mirror," Laser J. 2, 19-22 (1989) (in Chinese).

Mast, P. E.

G. A. Deschamps and P. E. Mast, in Proceedings of the Symposium on Quasi-optics, J. Fox, ed. (Polytechnic, 1964).

Meng, F. Z.

G. Y. Zhang, C. B. Zhang, X. Ding, F. Song, S. G. Guo, and F. Z. Meng, "Control on dynamical stablity of solid laser resonator," Acta. Phys. Sin. 51, 253-258 (2002) (in Chinese).

C. B. Zhang, F. Song, F. Z. Meng, X. Ding, G. Y. Zhang, and M. R. Shang, "Measurement of the laser diode end-pumped solid laser's thermal focal length by using the curve of output," Acta. Phys. Sin. 51, 1517-1520 (2002) (in Chinese).

Nemoto, S.

Peyghambarian, N.

F. Song, C. B. Zhang, X. Ding, J. J. Xu, G. Y. Zhang, M. Leigh, and N. Peyghambarian, "Determination of thermal focal length and pumping radius in gain medium in laser-diode-pumped Nd:YVO4 lasers," Appl. Phys. Lett. 81, 2145-2147 (2002).
[CrossRef]

Shang, M. R.

C. B. Zhang, F. Song, F. Z. Meng, X. Ding, G. Y. Zhang, and M. R. Shang, "Measurement of the laser diode end-pumped solid laser's thermal focal length by using the curve of output," Acta. Phys. Sin. 51, 1517-1520 (2002) (in Chinese).

Shi, X.

Siegman, A. E.

A. E. Siegman, "A canonical formulation for analyzing multielement unstable resonators," IEEE J. Quantum Electron. 12, 35-40 (1976).
[CrossRef]

Song, F.

C. B. Zhang, F. Song, F. Z. Meng, X. Ding, G. Y. Zhang, and M. R. Shang, "Measurement of the laser diode end-pumped solid laser's thermal focal length by using the curve of output," Acta. Phys. Sin. 51, 1517-1520 (2002) (in Chinese).

G. Y. Zhang, C. B. Zhang, X. Ding, F. Song, S. G. Guo, and F. Z. Meng, "Control on dynamical stablity of solid laser resonator," Acta. Phys. Sin. 51, 253-258 (2002) (in Chinese).

F. Song, C. B. Zhang, X. Ding, J. J. Xu, G. Y. Zhang, M. Leigh, and N. Peyghambarian, "Determination of thermal focal length and pumping radius in gain medium in laser-diode-pumped Nd:YVO4 lasers," Appl. Phys. Lett. 81, 2145-2147 (2002).
[CrossRef]

F. Song, G. Y. Zhang, J. J. Xu, and C. B. Zhang, "Analysis of laser diode pumped solid-state laser resonator by the transform circle approach," Chin. Phys. Lett. 17, 203-205 (2000).
[CrossRef]

Spurr, M.

M. Spurr and M. Dunn, "Euclidean light: high-school geometry to solve problems in Gaussian beam optics," Opt. Photonics News 13(8), 40-44 (2002).
[CrossRef]

Visser, T. D.

Wang, S. M.

S. M. Wang and D. Zhao, Matrix Optics (CHEP-Springer, 2000).

Weber, H.

H. Weber, Laser Resonators (Huazhong Institute of Technology, 1983) (in Chinese).

Wiersma, S. H.

Xu, J. J.

F. Song, C. B. Zhang, X. Ding, J. J. Xu, G. Y. Zhang, M. Leigh, and N. Peyghambarian, "Determination of thermal focal length and pumping radius in gain medium in laser-diode-pumped Nd:YVO4 lasers," Appl. Phys. Lett. 81, 2145-2147 (2002).
[CrossRef]

F. Song, G. Y. Zhang, J. J. Xu, and C. B. Zhang, "Analysis of laser diode pumped solid-state laser resonator by the transform circle approach," Chin. Phys. Lett. 17, 203-205 (2000).
[CrossRef]

Zhang, C. B.

F. Song, C. B. Zhang, X. Ding, J. J. Xu, G. Y. Zhang, M. Leigh, and N. Peyghambarian, "Determination of thermal focal length and pumping radius in gain medium in laser-diode-pumped Nd:YVO4 lasers," Appl. Phys. Lett. 81, 2145-2147 (2002).
[CrossRef]

C. B. Zhang, F. Song, F. Z. Meng, X. Ding, G. Y. Zhang, and M. R. Shang, "Measurement of the laser diode end-pumped solid laser's thermal focal length by using the curve of output," Acta. Phys. Sin. 51, 1517-1520 (2002) (in Chinese).

G. Y. Zhang, C. B. Zhang, X. Ding, F. Song, S. G. Guo, and F. Z. Meng, "Control on dynamical stablity of solid laser resonator," Acta. Phys. Sin. 51, 253-258 (2002) (in Chinese).

F. Song, G. Y. Zhang, J. J. Xu, and C. B. Zhang, "Analysis of laser diode pumped solid-state laser resonator by the transform circle approach," Chin. Phys. Lett. 17, 203-205 (2000).
[CrossRef]

Zhang, G. Y.

G. Y. Zhang, C. B. Zhang, X. Ding, F. Song, S. G. Guo, and F. Z. Meng, "Control on dynamical stablity of solid laser resonator," Acta. Phys. Sin. 51, 253-258 (2002) (in Chinese).

F. Song, C. B. Zhang, X. Ding, J. J. Xu, G. Y. Zhang, M. Leigh, and N. Peyghambarian, "Determination of thermal focal length and pumping radius in gain medium in laser-diode-pumped Nd:YVO4 lasers," Appl. Phys. Lett. 81, 2145-2147 (2002).
[CrossRef]

C. B. Zhang, F. Song, F. Z. Meng, X. Ding, G. Y. Zhang, and M. R. Shang, "Measurement of the laser diode end-pumped solid laser's thermal focal length by using the curve of output," Acta. Phys. Sin. 51, 1517-1520 (2002) (in Chinese).

F. Song, G. Y. Zhang, J. J. Xu, and C. B. Zhang, "Analysis of laser diode pumped solid-state laser resonator by the transform circle approach," Chin. Phys. Lett. 17, 203-205 (2000).
[CrossRef]

G. Y. Zhang, "Solutions and characters of the thermal-insensitive resonator," Acta. Phys. Sin. 40, 407-413 (1991) (in Chinese).

G. Y. Zhang, "A dynamic analysis on solid laser resonator," Acta. Phys. Sin. 40, 1065-1073 (1991) (in Chinese).

G. Y. Zhang, "Graphic design method of optical resonator (III)," Lasers 4, 41-46 (1977) (in Chinese).

G. Y. Zhang and S. G. Guo, Graphic Analysis and Design Method of Optical Resonator (National Defence Industry Press, 2003), pp. 4-18 (in Chinese).

Zhao, D.

S. M. Wang and D. Zhao, Matrix Optics (CHEP-Springer, 2000).

Zhao, M.

M. Zhao and B. Lu, "Gaussian beam transformation through resonator mirror," Laser J. 2, 19-22 (1989) (in Chinese).

Acta. Phys. Sin. (4)

G. Y. Zhang, "Solutions and characters of the thermal-insensitive resonator," Acta. Phys. Sin. 40, 407-413 (1991) (in Chinese).

C. B. Zhang, F. Song, F. Z. Meng, X. Ding, G. Y. Zhang, and M. R. Shang, "Measurement of the laser diode end-pumped solid laser's thermal focal length by using the curve of output," Acta. Phys. Sin. 51, 1517-1520 (2002) (in Chinese).

G. Y. Zhang, C. B. Zhang, X. Ding, F. Song, S. G. Guo, and F. Z. Meng, "Control on dynamical stablity of solid laser resonator," Acta. Phys. Sin. 51, 253-258 (2002) (in Chinese).

G. Y. Zhang, "A dynamic analysis on solid laser resonator," Acta. Phys. Sin. 40, 1065-1073 (1991) (in Chinese).

Appl. Opt. (6)

Appl. Phys. Lett. (1)

F. Song, C. B. Zhang, X. Ding, J. J. Xu, G. Y. Zhang, M. Leigh, and N. Peyghambarian, "Determination of thermal focal length and pumping radius in gain medium in laser-diode-pumped Nd:YVO4 lasers," Appl. Phys. Lett. 81, 2145-2147 (2002).
[CrossRef]

Bell. Syst. Tech. J. (1)

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

Chin. Opt. Lett. (1)

Chin. Phys. Lett. (1)

F. Song, G. Y. Zhang, J. J. Xu, and C. B. Zhang, "Analysis of laser diode pumped solid-state laser resonator by the transform circle approach," Chin. Phys. Lett. 17, 203-205 (2000).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. E. Siegman, "A canonical formulation for analyzing multielement unstable resonators," IEEE J. Quantum Electron. 12, 35-40 (1976).
[CrossRef]

J. Opt. Soc. Am. A (1)

Laser J. (1)

M. Zhao and B. Lu, "Gaussian beam transformation through resonator mirror," Laser J. 2, 19-22 (1989) (in Chinese).

Lasers (1)

G. Y. Zhang, "Graphic design method of optical resonator (III)," Lasers 4, 41-46 (1977) (in Chinese).

Opt. Photonics News (1)

M. Spurr and M. Dunn, "Euclidean light: high-school geometry to solve problems in Gaussian beam optics," Opt. Photonics News 13(8), 40-44 (2002).
[CrossRef]

Opto-Electronics (1)

P. Baues, "The connection of geometric optics with the propagation of Gaussian beams and the theory of optical resonators," Opto-Electronics 1, 103-118 (1969).
[CrossRef]

Other (6)

H. Weber, Laser Resonators (Huazhong Institute of Technology, 1983) (in Chinese).

S. M. Wang and D. Zhao, Matrix Optics (CHEP-Springer, 2000).

G. Y. Zhang and S. G. Guo, Graphic Analysis and Design Method of Optical Resonator (National Defence Industry Press, 2003), pp. 4-18 (in Chinese).

G. A. Deschamps and P. E. Mast, in Proceedings of the Symposium on Quasi-optics, J. Fox, ed. (Polytechnic, 1964).

"Propagation characteristics of laser beams," www.mellesgriot.com/pdf/CatalogX/X_36_6-9.pdf.

J. Alda, "Laser and gaussian beam propagation and transformation," in Encyclopaedia of Optical Engineering (Marcel Dekker, 2003), pp. 999-1013.

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Figures (6)

Fig. 1
Fig. 1

Characteristics of the light mode in a two-mirror resonator.

Fig. 2
Fig. 2

Changes of R and ω with the propagation of the Gaussian beam.

Fig. 3
Fig. 3

Two kinds of stabilized resonator.

Fig. 4
Fig. 4

Propagation circles in a resonator containing a thin lens.

Fig. 5
Fig. 5

Plot of ω i 1 / f .

Fig. 6
Fig. 6

Thermal focal length in the laser resonator when the gain medium is close to the all-reflective mirror.

Equations (24)

Equations on this page are rendered with MathJax. Learn more.

b 0 = L ( R 1 L ) ( R 2 L ) ( R 1 + R 2 L ) R 1 + R 2 2 L .
ω 0 = b 0 λ / π ,
R 2 < R 1 < L < R 1 + R 2  or   0 < L < R 2 < R 1 ( if   R 2 < R 1 ) ,
R 1 < R 2 < L < R 1 + R 2   or   0 < L < R 1 < R 2 ( if   R 1 < R 2 ) .
0 < ( 1 L R 1 ) ( 1 L R 2 ) < 1.
1 L 1 + 1 L 1 = 1 f ,
1 K 1 + 1 K 1 = 1 f ,
K 1 = L 1 R 1 , K 1 = L 1 + R 1 ,
L 1 = f L 1 L 1 f ,
R 1 = f 2 R 1 ( L 1 R 1 f ) ( L 1 f ) .
0 ( 1 L R 1 ) ( 1 L R 2 ) 1 ,
0 [ f ( L 1 R 1 ) ( L 1 f ) L 2 ( L 1 R 1 f ) ( L 1 f ) ] f 2 R 1 × [ ( L 1 f ) ( R 2 L 2 ) + f L 1 ] R 2 ( L 1 f ) 1.
ω 02 = b 02 λ π = [ ( λ π ) 2 ( L 2 L 1 ) ( R 1 L 2 + L 1 ) ( R 2 L 2 + L 1 ) ( R 1 + R 2 L 2 + L 1 ) ( R 1 + R 2 2 L 2 + L 1 ) 2 ] 1 / 4 .
ω 01 = b 01 λ π = [ ( λ π ) 2 ( L 1 L 2 ) ( R 2 L 1 + L 2 ) ( R 1 L 1 + L 2 ) ( R 2 + R 1 L 1 + L 2 ) ( R 2 + R 1 2 L 1 + 2 L 2 ) 2 ] 1 / 4 ,
L 2 = f L 2 L 2 f ,
R 2 = f ( L 2 R 2 ) L 2 ( R 2 + f ) f L 2 L 2 f .
L f 2 = L 2 1 2 R 2 R 2 2 R 1 2 + ( 2 L 2 2 L 1 R 1 R 2 ) 2 8 L 2 8 L 1 4 R 1 4 R 2 .
ω f = ω 02 1 + ( λ L f 2 π ω 02 2 ) 2 .
ω 1 = ω 01 1 + ( λ L 01 π ω 01 2 ) 2 ,
ω 2 = ω 02 1 + ( λ L 02 π ω 02 2 ) 2 ,
L 02 = 1 2 R 2 + R 2 2 R 1 2 + ( 2 L 2 2 L 1 R 1 R 2 ) 2 8 L 2 8 L 1 4 R 1 4 R 2 .
L 01 = 1 2 R 1 + R 1 2 R 2 2 + ( 2 L 1 2 L 2 R 2 R 1 ) 2 8 L 1 8 L 2 4 R 2 4 R 1 .
0 < f 2 R 1 f L 2 ( R 1 + f ) f 2 R 1 ( 1 L 2 R 2 ) < 1.
0 < ( 1 L 2 R 1 ) ( 1 L 2 R 2 ) < 1.

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