Abstract

By using the fast-Fourier-transform method and by solving the rate equations, we derived a model of a generalized self-filtering unstable resonator of CO2 lasers with transverse flow. Near- and far-field distributions of various configurations are calculated. The focal length of one mirror is shown as an important parameter to match the resonator with special requirements of application. The theoretical results are in good agreement with experimental data.

© 1992 Optical Society of America

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  1. A. E. Siegman, “Unstable optical resonators for laser applications,” Proc. IEEE 53, 277–287 (1965).
    [CrossRef]
  2. E. A. Sziklas, A. E. Siegman, “Mode calculations in unstable resonators with flowing saturable gain. 2: Fast Fourier transform method,” Appl. Opt. 14, 1874–1889 (1975).
    [CrossRef] [PubMed]
  3. P. C. Gobbi, G. C. Reali, “A novel unstable resonator configuration with a self filtering aperture,” Opt. Commun. 52, 195–198 (1984).
    [CrossRef]
  4. P. G. Gobbi, S. Morosi, G. C. Reali, A. S. Zarkasi, “Novel unstable resonator configuration with a self-filtering aperture: experimental characterization of the Nd:YAG loaded cavity,” Appl. Opt. 24, 26–33 (1985).
    [CrossRef] [PubMed]
  5. R. Barbini, A. Ghigo, K. N. Iger, A. Palucci, S. Ribezzo, “Injection-locked single mode high-power low-divergence TEA CO2-laser using SFUR configuration,” Opt. Commun. 60, 239–243 (1986).
    [CrossRef]
  6. P. Di Lazzaro, T. Hermsen, T. Letardi, C. E. Zheng, “Self-filtering unstable resonator: an approximate analytical model with comparison to computed and XeCl laser experimental results,” Opt. Commun. 61, 393–396 (1987).
    [CrossRef]
  7. A. J. Ivanchenko, W. W. Krasheninikov, A. G. Ponomarenko, W. B. Shuljatev, “Self-filtering resonator in a cw-CO2-laser,” Kvantovaya Elektron. (Moscow)16, 305–307 (1989).
  8. R. Bhatnagar, S. K. Dixit, B. Singh, S. V. Nakhe, “Performance of a copper vapor laser with self-filtered unstable resonator,” Opt. Commun. 74, 93–96 (1989).
    [CrossRef]
  9. P. G. Gobbi, G. C. Reali, “Numerical optimization study of a self-filtering unstable resonator for high power solid state and gas lasers,” in Conference on Lasers and Electro-Optics; Vol. 12 of 1985 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1985), paper WM37.
  10. P. Di Lazzaro, T. W. Hermsen, C. Zheng, “A generalization of the self-filtering unstable resonator,” IEEE J. Quantum Electron. 24, 1543–1547 (1988).
    [CrossRef]
  11. A. Luches, V. Nassisi, M. R. Perrone, E. Radiotis, “High mode volume self-filtering unstable resonator applied to a short pulse XeCl laser,” Opt. Commun. 71, 97–102 (1989).
    [CrossRef]
  12. A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
  13. O. Svelto, Principles of Lasers (Plenum, New York, 1989), App. B.
  14. P. G. Gobbi, G. C. Reali, “Numerical study of a self-filtering resonator,” in Southwest Conference Optics ’85, R. S. McDowell, S. C. Stotlar, eds., Proc. Soc. Photo-Opt. Instrum. Eng.540, 119–123 (1985).
  15. P. G. Gobbi, G. C. Reali, “Mode analysis of a self-filtering unstable resonator with a Gaussian transmission aperture,” Opt. Commun. 57, 355–359 (1986).
    [CrossRef]
  16. W. Triebel, E. Ose, G. Michel, A. Petrich, “Experimental and theoretical investigation of a transverse-flow CO2-laser,” in 7th International Symposium on Gas Flow and Chemical Lasers, D. Schuoecker, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1031, 41–47 (1988).

1989

R. Bhatnagar, S. K. Dixit, B. Singh, S. V. Nakhe, “Performance of a copper vapor laser with self-filtered unstable resonator,” Opt. Commun. 74, 93–96 (1989).
[CrossRef]

A. Luches, V. Nassisi, M. R. Perrone, E. Radiotis, “High mode volume self-filtering unstable resonator applied to a short pulse XeCl laser,” Opt. Commun. 71, 97–102 (1989).
[CrossRef]

1988

P. Di Lazzaro, T. W. Hermsen, C. Zheng, “A generalization of the self-filtering unstable resonator,” IEEE J. Quantum Electron. 24, 1543–1547 (1988).
[CrossRef]

1987

P. Di Lazzaro, T. Hermsen, T. Letardi, C. E. Zheng, “Self-filtering unstable resonator: an approximate analytical model with comparison to computed and XeCl laser experimental results,” Opt. Commun. 61, 393–396 (1987).
[CrossRef]

1986

R. Barbini, A. Ghigo, K. N. Iger, A. Palucci, S. Ribezzo, “Injection-locked single mode high-power low-divergence TEA CO2-laser using SFUR configuration,” Opt. Commun. 60, 239–243 (1986).
[CrossRef]

P. G. Gobbi, G. C. Reali, “Mode analysis of a self-filtering unstable resonator with a Gaussian transmission aperture,” Opt. Commun. 57, 355–359 (1986).
[CrossRef]

1985

1984

P. C. Gobbi, G. C. Reali, “A novel unstable resonator configuration with a self filtering aperture,” Opt. Commun. 52, 195–198 (1984).
[CrossRef]

1975

1965

A. E. Siegman, “Unstable optical resonators for laser applications,” Proc. IEEE 53, 277–287 (1965).
[CrossRef]

1961

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).

Barbini, R.

R. Barbini, A. Ghigo, K. N. Iger, A. Palucci, S. Ribezzo, “Injection-locked single mode high-power low-divergence TEA CO2-laser using SFUR configuration,” Opt. Commun. 60, 239–243 (1986).
[CrossRef]

Bhatnagar, R.

R. Bhatnagar, S. K. Dixit, B. Singh, S. V. Nakhe, “Performance of a copper vapor laser with self-filtered unstable resonator,” Opt. Commun. 74, 93–96 (1989).
[CrossRef]

Di Lazzaro, P.

P. Di Lazzaro, T. W. Hermsen, C. Zheng, “A generalization of the self-filtering unstable resonator,” IEEE J. Quantum Electron. 24, 1543–1547 (1988).
[CrossRef]

P. Di Lazzaro, T. Hermsen, T. Letardi, C. E. Zheng, “Self-filtering unstable resonator: an approximate analytical model with comparison to computed and XeCl laser experimental results,” Opt. Commun. 61, 393–396 (1987).
[CrossRef]

Dixit, S. K.

R. Bhatnagar, S. K. Dixit, B. Singh, S. V. Nakhe, “Performance of a copper vapor laser with self-filtered unstable resonator,” Opt. Commun. 74, 93–96 (1989).
[CrossRef]

Fox, A. G.

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).

Ghigo, A.

R. Barbini, A. Ghigo, K. N. Iger, A. Palucci, S. Ribezzo, “Injection-locked single mode high-power low-divergence TEA CO2-laser using SFUR configuration,” Opt. Commun. 60, 239–243 (1986).
[CrossRef]

Gobbi, P. C.

P. C. Gobbi, G. C. Reali, “A novel unstable resonator configuration with a self filtering aperture,” Opt. Commun. 52, 195–198 (1984).
[CrossRef]

Gobbi, P. G.

P. G. Gobbi, G. C. Reali, “Mode analysis of a self-filtering unstable resonator with a Gaussian transmission aperture,” Opt. Commun. 57, 355–359 (1986).
[CrossRef]

P. G. Gobbi, S. Morosi, G. C. Reali, A. S. Zarkasi, “Novel unstable resonator configuration with a self-filtering aperture: experimental characterization of the Nd:YAG loaded cavity,” Appl. Opt. 24, 26–33 (1985).
[CrossRef] [PubMed]

P. G. Gobbi, G. C. Reali, “Numerical study of a self-filtering resonator,” in Southwest Conference Optics ’85, R. S. McDowell, S. C. Stotlar, eds., Proc. Soc. Photo-Opt. Instrum. Eng.540, 119–123 (1985).

P. G. Gobbi, G. C. Reali, “Numerical optimization study of a self-filtering unstable resonator for high power solid state and gas lasers,” in Conference on Lasers and Electro-Optics; Vol. 12 of 1985 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1985), paper WM37.

Hermsen, T.

P. Di Lazzaro, T. Hermsen, T. Letardi, C. E. Zheng, “Self-filtering unstable resonator: an approximate analytical model with comparison to computed and XeCl laser experimental results,” Opt. Commun. 61, 393–396 (1987).
[CrossRef]

Hermsen, T. W.

P. Di Lazzaro, T. W. Hermsen, C. Zheng, “A generalization of the self-filtering unstable resonator,” IEEE J. Quantum Electron. 24, 1543–1547 (1988).
[CrossRef]

Iger, K. N.

R. Barbini, A. Ghigo, K. N. Iger, A. Palucci, S. Ribezzo, “Injection-locked single mode high-power low-divergence TEA CO2-laser using SFUR configuration,” Opt. Commun. 60, 239–243 (1986).
[CrossRef]

Ivanchenko, A. J.

A. J. Ivanchenko, W. W. Krasheninikov, A. G. Ponomarenko, W. B. Shuljatev, “Self-filtering resonator in a cw-CO2-laser,” Kvantovaya Elektron. (Moscow)16, 305–307 (1989).

Krasheninikov, W. W.

A. J. Ivanchenko, W. W. Krasheninikov, A. G. Ponomarenko, W. B. Shuljatev, “Self-filtering resonator in a cw-CO2-laser,” Kvantovaya Elektron. (Moscow)16, 305–307 (1989).

Letardi, T.

P. Di Lazzaro, T. Hermsen, T. Letardi, C. E. Zheng, “Self-filtering unstable resonator: an approximate analytical model with comparison to computed and XeCl laser experimental results,” Opt. Commun. 61, 393–396 (1987).
[CrossRef]

Li, T.

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).

Luches, A.

A. Luches, V. Nassisi, M. R. Perrone, E. Radiotis, “High mode volume self-filtering unstable resonator applied to a short pulse XeCl laser,” Opt. Commun. 71, 97–102 (1989).
[CrossRef]

Michel, G.

W. Triebel, E. Ose, G. Michel, A. Petrich, “Experimental and theoretical investigation of a transverse-flow CO2-laser,” in 7th International Symposium on Gas Flow and Chemical Lasers, D. Schuoecker, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1031, 41–47 (1988).

Morosi, S.

Nakhe, S. V.

R. Bhatnagar, S. K. Dixit, B. Singh, S. V. Nakhe, “Performance of a copper vapor laser with self-filtered unstable resonator,” Opt. Commun. 74, 93–96 (1989).
[CrossRef]

Nassisi, V.

A. Luches, V. Nassisi, M. R. Perrone, E. Radiotis, “High mode volume self-filtering unstable resonator applied to a short pulse XeCl laser,” Opt. Commun. 71, 97–102 (1989).
[CrossRef]

Ose, E.

W. Triebel, E. Ose, G. Michel, A. Petrich, “Experimental and theoretical investigation of a transverse-flow CO2-laser,” in 7th International Symposium on Gas Flow and Chemical Lasers, D. Schuoecker, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1031, 41–47 (1988).

Palucci, A.

R. Barbini, A. Ghigo, K. N. Iger, A. Palucci, S. Ribezzo, “Injection-locked single mode high-power low-divergence TEA CO2-laser using SFUR configuration,” Opt. Commun. 60, 239–243 (1986).
[CrossRef]

Perrone, M. R.

A. Luches, V. Nassisi, M. R. Perrone, E. Radiotis, “High mode volume self-filtering unstable resonator applied to a short pulse XeCl laser,” Opt. Commun. 71, 97–102 (1989).
[CrossRef]

Petrich, A.

W. Triebel, E. Ose, G. Michel, A. Petrich, “Experimental and theoretical investigation of a transverse-flow CO2-laser,” in 7th International Symposium on Gas Flow and Chemical Lasers, D. Schuoecker, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1031, 41–47 (1988).

Ponomarenko, A. G.

A. J. Ivanchenko, W. W. Krasheninikov, A. G. Ponomarenko, W. B. Shuljatev, “Self-filtering resonator in a cw-CO2-laser,” Kvantovaya Elektron. (Moscow)16, 305–307 (1989).

Radiotis, E.

A. Luches, V. Nassisi, M. R. Perrone, E. Radiotis, “High mode volume self-filtering unstable resonator applied to a short pulse XeCl laser,” Opt. Commun. 71, 97–102 (1989).
[CrossRef]

Reali, G. C.

P. G. Gobbi, G. C. Reali, “Mode analysis of a self-filtering unstable resonator with a Gaussian transmission aperture,” Opt. Commun. 57, 355–359 (1986).
[CrossRef]

P. G. Gobbi, S. Morosi, G. C. Reali, A. S. Zarkasi, “Novel unstable resonator configuration with a self-filtering aperture: experimental characterization of the Nd:YAG loaded cavity,” Appl. Opt. 24, 26–33 (1985).
[CrossRef] [PubMed]

P. C. Gobbi, G. C. Reali, “A novel unstable resonator configuration with a self filtering aperture,” Opt. Commun. 52, 195–198 (1984).
[CrossRef]

P. G. Gobbi, G. C. Reali, “Numerical study of a self-filtering resonator,” in Southwest Conference Optics ’85, R. S. McDowell, S. C. Stotlar, eds., Proc. Soc. Photo-Opt. Instrum. Eng.540, 119–123 (1985).

P. G. Gobbi, G. C. Reali, “Numerical optimization study of a self-filtering unstable resonator for high power solid state and gas lasers,” in Conference on Lasers and Electro-Optics; Vol. 12 of 1985 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1985), paper WM37.

Ribezzo, S.

R. Barbini, A. Ghigo, K. N. Iger, A. Palucci, S. Ribezzo, “Injection-locked single mode high-power low-divergence TEA CO2-laser using SFUR configuration,” Opt. Commun. 60, 239–243 (1986).
[CrossRef]

Shuljatev, W. B.

A. J. Ivanchenko, W. W. Krasheninikov, A. G. Ponomarenko, W. B. Shuljatev, “Self-filtering resonator in a cw-CO2-laser,” Kvantovaya Elektron. (Moscow)16, 305–307 (1989).

Siegman, A. E.

Singh, B.

R. Bhatnagar, S. K. Dixit, B. Singh, S. V. Nakhe, “Performance of a copper vapor laser with self-filtered unstable resonator,” Opt. Commun. 74, 93–96 (1989).
[CrossRef]

Svelto, O.

O. Svelto, Principles of Lasers (Plenum, New York, 1989), App. B.

Sziklas, E. A.

Triebel, W.

W. Triebel, E. Ose, G. Michel, A. Petrich, “Experimental and theoretical investigation of a transverse-flow CO2-laser,” in 7th International Symposium on Gas Flow and Chemical Lasers, D. Schuoecker, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1031, 41–47 (1988).

Zarkasi, A. S.

Zheng, C.

P. Di Lazzaro, T. W. Hermsen, C. Zheng, “A generalization of the self-filtering unstable resonator,” IEEE J. Quantum Electron. 24, 1543–1547 (1988).
[CrossRef]

Zheng, C. E.

P. Di Lazzaro, T. Hermsen, T. Letardi, C. E. Zheng, “Self-filtering unstable resonator: an approximate analytical model with comparison to computed and XeCl laser experimental results,” Opt. Commun. 61, 393–396 (1987).
[CrossRef]

Appl. Opt.

Bell Syst. Tech. J.

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).

IEEE J. Quantum Electron.

P. Di Lazzaro, T. W. Hermsen, C. Zheng, “A generalization of the self-filtering unstable resonator,” IEEE J. Quantum Electron. 24, 1543–1547 (1988).
[CrossRef]

Opt. Commun.

A. Luches, V. Nassisi, M. R. Perrone, E. Radiotis, “High mode volume self-filtering unstable resonator applied to a short pulse XeCl laser,” Opt. Commun. 71, 97–102 (1989).
[CrossRef]

R. Barbini, A. Ghigo, K. N. Iger, A. Palucci, S. Ribezzo, “Injection-locked single mode high-power low-divergence TEA CO2-laser using SFUR configuration,” Opt. Commun. 60, 239–243 (1986).
[CrossRef]

P. Di Lazzaro, T. Hermsen, T. Letardi, C. E. Zheng, “Self-filtering unstable resonator: an approximate analytical model with comparison to computed and XeCl laser experimental results,” Opt. Commun. 61, 393–396 (1987).
[CrossRef]

R. Bhatnagar, S. K. Dixit, B. Singh, S. V. Nakhe, “Performance of a copper vapor laser with self-filtered unstable resonator,” Opt. Commun. 74, 93–96 (1989).
[CrossRef]

P. G. Gobbi, G. C. Reali, “Mode analysis of a self-filtering unstable resonator with a Gaussian transmission aperture,” Opt. Commun. 57, 355–359 (1986).
[CrossRef]

P. C. Gobbi, G. C. Reali, “A novel unstable resonator configuration with a self filtering aperture,” Opt. Commun. 52, 195–198 (1984).
[CrossRef]

Proc. IEEE

A. E. Siegman, “Unstable optical resonators for laser applications,” Proc. IEEE 53, 277–287 (1965).
[CrossRef]

Other

W. Triebel, E. Ose, G. Michel, A. Petrich, “Experimental and theoretical investigation of a transverse-flow CO2-laser,” in 7th International Symposium on Gas Flow and Chemical Lasers, D. Schuoecker, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1031, 41–47 (1988).

P. G. Gobbi, G. C. Reali, “Numerical optimization study of a self-filtering unstable resonator for high power solid state and gas lasers,” in Conference on Lasers and Electro-Optics; Vol. 12 of 1985 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1985), paper WM37.

A. J. Ivanchenko, W. W. Krasheninikov, A. G. Ponomarenko, W. B. Shuljatev, “Self-filtering resonator in a cw-CO2-laser,” Kvantovaya Elektron. (Moscow)16, 305–307 (1989).

O. Svelto, Principles of Lasers (Plenum, New York, 1989), App. B.

P. G. Gobbi, G. C. Reali, “Numerical study of a self-filtering resonator,” in Southwest Conference Optics ’85, R. S. McDowell, S. C. Stotlar, eds., Proc. Soc. Photo-Opt. Instrum. Eng.540, 119–123 (1985).

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Figures (10)

Fig. 1
Fig. 1

Resonator configuration with active medium (dashed boxes): M1, M2, end reflectors; Mf folding mirrors; S, scraper mirror; g1g4, gain; PM, power meter’s; D, detector (0.03 rad).

Fig. 2
Fig. 2

Measured small-signal gain distribution (solid curve) and the calculated pump rate (dotted curve) versus the normalized x coordinate. pA is the aperture radius in the case f1 = 10 m.

Fig. 3
Fig. 3

Near-field distribution without (dotted curve) and including (solid curve) the active medium, f1 = 10 m.

Fig. 4
Fig. 4

Near-field distributions. The parameter is the normalized aperture radius pA/pA.

Fig. 5
Fig. 5

Far-field distribution of the configuration f1 = 10 m.

Fig. 6
Fig. 6

Losses (solid curve) and mode volume (dotted curve) of the bare resonator.

Fig. 7
Fig. 7

θ1/e2 (solid curve) and θ0.9 (dotted curve) of the bare resonator.

Fig. 8
Fig. 8

θ1/e2 (solid curve) and θ0.9 (dotted curve) of the loaded resonator. The asterisks represent the calculated total power of the one-dimensional configuration.

Fig. 9
Fig. 9

Experimental intensity distribution obtained 7.2 m behind the resonator.

Fig. 10
Fig. 10

Numerical calculated intensity distribution obtained 7.2 m behind the resonator.

Equations (11)

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

T = T ( x , y ) = T ^ ( x , y ) exp [ i ϕ ( x , y ) ]
u ( x , y , z ) = T ( x , y ) u ( x , y , z ) .
u ˜ l ( f x , f y ) = u ˜ z 0 ( f x , f y ) * A ( f x , f y , l ) ,
A = { exp { i 2 π l [ 1 / λ 2 - ( f x 2 + f y 2 ) ] 1 / 2 } for f x 2 + f y 2 < 1 / λ 2 , exp { i 2 π l [ f x 2 + f y 2 - 1 / λ 2 ] 1 / 2 } for f x 2 + f y 2 > 1 / λ 2 .
g 2 = - L 1 g 1 L 2 - 2 g 1 ( L 1 + L 2 ) , p A = ( g 2 L 2 λ ) 1 / 2 ,
f a = - 2 g 2 2 g 2 - 1 L 2 + d ,
v g x N ( r , t ) = - σ I ( r , t ) N ( r , t ) - N ( r , t ) τ + W p ( r ) N g ,
d I ( r , t ) d t = c 0 2 σ h ν 0 V R R p N ( r , t ) d V - I ( r , t ) τ c .
N ( x ) = N g / v g 0 x W p ( x ) × exp { - [ σ I ( x ) + 1 / τ ) ( x - x ) ] / v g } d x .
N ( x i ) = N g W p ( x i ) σ I ( x i ) + 1 / τ ( 1 - exp { - [ σ I ( x i ) + 1 / τ ] Δ x / v g } ) + N ( x i - 1 ) exp { - [ σ I ( x i ) + 1 / τ ] Δ x / v g } ,
W p ( x i ) = g 0 ( x i ) - g 0 ( x i - 1 ) exp [ - Δ x / ( v g τ ) ] σ h ν 0 N g τ { 1 - exp [ - Δ x / ( v g τ ) ] } ,

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