Abstract

The evolution of the main parameters (spot size, radius of curvature, and far-field divergence) of the beam provided by a high-gain, short-pulse Xe–Cl laser equipped with a plane-parallel Gaussian resonator has been investigated both theoretically and experimentally as a function of the number of round trips. It is shown that a theoretical analysis, formerly developed to study the growth of coherence of a Schell–Gauss model beam propagating through a periodic sequence of Gaussian apertures, provides a satisfactory description of the temporal evolution of the Xe–Cl laser beam. A good agreement between theoretical and experimental data has been found.

© 1995 Optical Society of America

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References

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  1. M. R. Perrone, A. Piegari, S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1427 (1993).
    [CrossRef]
  2. A. E. Siegman, Lasers(University Science, Mill Valley, Calif., 1986), and references cited therein.
  3. A. Caprara, G. C. Reali, “Time-resolved M2of nanosecond pulses from a Q-switched variable-reflectivity-mirror Nd:YAG laser,” Opt. Lett. 17, 414–416 (1992).
    [CrossRef] [PubMed]
  4. J. J. Chang, “Time-resolved beam-quality characterization of copper vapor lasers with unstable resonators,” J. Opt. Soc. Am. A 11, 2255–2265 (1994).
  5. W. Coutts, M. D. Ainsworth, J. A. Piper, “Observation of the temporal evolution of transverse coherence in copper vapor lasers,” Opt. Commun. 87, 245–248 (1992).
    [CrossRef]
  6. T. Omatsu, K. Kuroda, “Time-resolved measurements of spatial coherence of a copper vapor laser using a reversal shearing interferometer,” Opt. Commun. 87, 278–286 (1992).
    [CrossRef]
  7. T. Omatsu, K. Kuroda, T. Shimura, M. Chiharu, M. Ito, I. Ogura, “Time-resolved measurements of beam divergence of a copper vapor laser using a saturable absorber,” Opt. Commun. 85, 343–349 (1991).
    [CrossRef]
  8. G. Cincotti, P. DeSantis, G. Guattari, C. Palma, “Propagation of partially coherent beams in a periodic sequence of lenses and Gaussian apertures,” Pure Appl. Opt. 3, 561–571 (1994).
    [CrossRef]
  9. F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
    [CrossRef]
  10. A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
    [CrossRef]
  11. B. Lu, B. Zhang, B. Cai, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams behaving like Gaussian Schell-model beams,” Opt. Commun. 101, 49–52 (1993).
    [CrossRef]
  12. S. Scaglione, D. Flori, I. Soymie, A. Piegari, “Laser optical coating produced by ion beam assisted deposition,” Thin Solid Films 214, 188–193 (1992).
    [CrossRef]
  13. Q. He, J. Turunen, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 67, 245–250 (1988).
    [CrossRef]

1994 (2)

J. J. Chang, “Time-resolved beam-quality characterization of copper vapor lasers with unstable resonators,” J. Opt. Soc. Am. A 11, 2255–2265 (1994).

G. Cincotti, P. DeSantis, G. Guattari, C. Palma, “Propagation of partially coherent beams in a periodic sequence of lenses and Gaussian apertures,” Pure Appl. Opt. 3, 561–571 (1994).
[CrossRef]

1993 (2)

B. Lu, B. Zhang, B. Cai, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams behaving like Gaussian Schell-model beams,” Opt. Commun. 101, 49–52 (1993).
[CrossRef]

M. R. Perrone, A. Piegari, S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1427 (1993).
[CrossRef]

1992 (4)

A. Caprara, G. C. Reali, “Time-resolved M2of nanosecond pulses from a Q-switched variable-reflectivity-mirror Nd:YAG laser,” Opt. Lett. 17, 414–416 (1992).
[CrossRef] [PubMed]

S. Scaglione, D. Flori, I. Soymie, A. Piegari, “Laser optical coating produced by ion beam assisted deposition,” Thin Solid Films 214, 188–193 (1992).
[CrossRef]

W. Coutts, M. D. Ainsworth, J. A. Piper, “Observation of the temporal evolution of transverse coherence in copper vapor lasers,” Opt. Commun. 87, 245–248 (1992).
[CrossRef]

T. Omatsu, K. Kuroda, “Time-resolved measurements of spatial coherence of a copper vapor laser using a reversal shearing interferometer,” Opt. Commun. 87, 278–286 (1992).
[CrossRef]

1991 (1)

T. Omatsu, K. Kuroda, T. Shimura, M. Chiharu, M. Ito, I. Ogura, “Time-resolved measurements of beam divergence of a copper vapor laser using a saturable absorber,” Opt. Commun. 85, 343–349 (1991).
[CrossRef]

1988 (1)

Q. He, J. Turunen, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 67, 245–250 (1988).
[CrossRef]

1982 (1)

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

1980 (1)

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

Ainsworth, M. D.

W. Coutts, M. D. Ainsworth, J. A. Piper, “Observation of the temporal evolution of transverse coherence in copper vapor lasers,” Opt. Commun. 87, 245–248 (1992).
[CrossRef]

Cai, B.

B. Lu, B. Zhang, B. Cai, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams behaving like Gaussian Schell-model beams,” Opt. Commun. 101, 49–52 (1993).
[CrossRef]

Caprara, A.

Chang, J. J.

J. J. Chang, “Time-resolved beam-quality characterization of copper vapor lasers with unstable resonators,” J. Opt. Soc. Am. A 11, 2255–2265 (1994).

Chiharu, M.

T. Omatsu, K. Kuroda, T. Shimura, M. Chiharu, M. Ito, I. Ogura, “Time-resolved measurements of beam divergence of a copper vapor laser using a saturable absorber,” Opt. Commun. 85, 343–349 (1991).
[CrossRef]

Cincotti, G.

G. Cincotti, P. DeSantis, G. Guattari, C. Palma, “Propagation of partially coherent beams in a periodic sequence of lenses and Gaussian apertures,” Pure Appl. Opt. 3, 561–571 (1994).
[CrossRef]

Coutts, W.

W. Coutts, M. D. Ainsworth, J. A. Piper, “Observation of the temporal evolution of transverse coherence in copper vapor lasers,” Opt. Commun. 87, 245–248 (1992).
[CrossRef]

DeSantis, P.

G. Cincotti, P. DeSantis, G. Guattari, C. Palma, “Propagation of partially coherent beams in a periodic sequence of lenses and Gaussian apertures,” Pure Appl. Opt. 3, 561–571 (1994).
[CrossRef]

Flori, D.

S. Scaglione, D. Flori, I. Soymie, A. Piegari, “Laser optical coating produced by ion beam assisted deposition,” Thin Solid Films 214, 188–193 (1992).
[CrossRef]

Friberg, A. T.

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

Gori, F.

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

Guattari, G.

G. Cincotti, P. DeSantis, G. Guattari, C. Palma, “Propagation of partially coherent beams in a periodic sequence of lenses and Gaussian apertures,” Pure Appl. Opt. 3, 561–571 (1994).
[CrossRef]

He, Q.

Q. He, J. Turunen, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 67, 245–250 (1988).
[CrossRef]

Ito, M.

T. Omatsu, K. Kuroda, T. Shimura, M. Chiharu, M. Ito, I. Ogura, “Time-resolved measurements of beam divergence of a copper vapor laser using a saturable absorber,” Opt. Commun. 85, 343–349 (1991).
[CrossRef]

Kuroda, K.

T. Omatsu, K. Kuroda, “Time-resolved measurements of spatial coherence of a copper vapor laser using a reversal shearing interferometer,” Opt. Commun. 87, 278–286 (1992).
[CrossRef]

T. Omatsu, K. Kuroda, T. Shimura, M. Chiharu, M. Ito, I. Ogura, “Time-resolved measurements of beam divergence of a copper vapor laser using a saturable absorber,” Opt. Commun. 85, 343–349 (1991).
[CrossRef]

Lu, B.

B. Lu, B. Zhang, B. Cai, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams behaving like Gaussian Schell-model beams,” Opt. Commun. 101, 49–52 (1993).
[CrossRef]

Ogura, I.

T. Omatsu, K. Kuroda, T. Shimura, M. Chiharu, M. Ito, I. Ogura, “Time-resolved measurements of beam divergence of a copper vapor laser using a saturable absorber,” Opt. Commun. 85, 343–349 (1991).
[CrossRef]

Omatsu, T.

T. Omatsu, K. Kuroda, “Time-resolved measurements of spatial coherence of a copper vapor laser using a reversal shearing interferometer,” Opt. Commun. 87, 278–286 (1992).
[CrossRef]

T. Omatsu, K. Kuroda, T. Shimura, M. Chiharu, M. Ito, I. Ogura, “Time-resolved measurements of beam divergence of a copper vapor laser using a saturable absorber,” Opt. Commun. 85, 343–349 (1991).
[CrossRef]

Palma, C.

G. Cincotti, P. DeSantis, G. Guattari, C. Palma, “Propagation of partially coherent beams in a periodic sequence of lenses and Gaussian apertures,” Pure Appl. Opt. 3, 561–571 (1994).
[CrossRef]

Perrone, M. R.

M. R. Perrone, A. Piegari, S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1427 (1993).
[CrossRef]

Piegari, A.

M. R. Perrone, A. Piegari, S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1427 (1993).
[CrossRef]

S. Scaglione, D. Flori, I. Soymie, A. Piegari, “Laser optical coating produced by ion beam assisted deposition,” Thin Solid Films 214, 188–193 (1992).
[CrossRef]

Piper, J. A.

W. Coutts, M. D. Ainsworth, J. A. Piper, “Observation of the temporal evolution of transverse coherence in copper vapor lasers,” Opt. Commun. 87, 245–248 (1992).
[CrossRef]

Reali, G. C.

Scaglione, S.

M. R. Perrone, A. Piegari, S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1427 (1993).
[CrossRef]

S. Scaglione, D. Flori, I. Soymie, A. Piegari, “Laser optical coating produced by ion beam assisted deposition,” Thin Solid Films 214, 188–193 (1992).
[CrossRef]

Shimura, T.

T. Omatsu, K. Kuroda, T. Shimura, M. Chiharu, M. Ito, I. Ogura, “Time-resolved measurements of beam divergence of a copper vapor laser using a saturable absorber,” Opt. Commun. 85, 343–349 (1991).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers(University Science, Mill Valley, Calif., 1986), and references cited therein.

Soymie, I.

S. Scaglione, D. Flori, I. Soymie, A. Piegari, “Laser optical coating produced by ion beam assisted deposition,” Thin Solid Films 214, 188–193 (1992).
[CrossRef]

Sudol, R. J.

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

Turunen, J.

Q. He, J. Turunen, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 67, 245–250 (1988).
[CrossRef]

Zhang, B.

B. Lu, B. Zhang, B. Cai, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams behaving like Gaussian Schell-model beams,” Opt. Commun. 101, 49–52 (1993).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. R. Perrone, A. Piegari, S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1427 (1993).
[CrossRef]

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

J. J. Chang, “Time-resolved beam-quality characterization of copper vapor lasers with unstable resonators,” J. Opt. Soc. Am. A 11, 2255–2265 (1994).

Opt. Commun. (7)

W. Coutts, M. D. Ainsworth, J. A. Piper, “Observation of the temporal evolution of transverse coherence in copper vapor lasers,” Opt. Commun. 87, 245–248 (1992).
[CrossRef]

T. Omatsu, K. Kuroda, “Time-resolved measurements of spatial coherence of a copper vapor laser using a reversal shearing interferometer,” Opt. Commun. 87, 278–286 (1992).
[CrossRef]

T. Omatsu, K. Kuroda, T. Shimura, M. Chiharu, M. Ito, I. Ogura, “Time-resolved measurements of beam divergence of a copper vapor laser using a saturable absorber,” Opt. Commun. 85, 343–349 (1991).
[CrossRef]

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

B. Lu, B. Zhang, B. Cai, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams behaving like Gaussian Schell-model beams,” Opt. Commun. 101, 49–52 (1993).
[CrossRef]

Q. He, J. Turunen, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 67, 245–250 (1988).
[CrossRef]

Opt. Lett. (1)

Pure Appl. Opt. (1)

G. Cincotti, P. DeSantis, G. Guattari, C. Palma, “Propagation of partially coherent beams in a periodic sequence of lenses and Gaussian apertures,” Pure Appl. Opt. 3, 561–571 (1994).
[CrossRef]

Thin Solid Films (1)

S. Scaglione, D. Flori, I. Soymie, A. Piegari, “Laser optical coating produced by ion beam assisted deposition,” Thin Solid Films 214, 188–193 (1992).
[CrossRef]

Other (1)

A. E. Siegman, Lasers(University Science, Mill Valley, Calif., 1986), and references cited therein.

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Figures (8)

Fig. 1
Fig. 1

Schematic experimental layout: M1, aluminized mirror; M2, Gaussian reflectivity mirror; B, beam mirror; L, resonator length; La, laser active length; Pn, pinhole; Pd, photodiode.

Fig. 2
Fig. 2

Experimental intensity profile of the Gaussian mirror (circles) and Gaussian profile (solid curve).

Fig. 3
Fig. 3

Temporal evolution of the intracavity laser pulse at different radial distances r from the optical axis: (a) r = 0.0 mm, (b) r = 1.5 mm, (c) r = 2.2 mm. a.u., arbitrary units.

Fig. 4
Fig. 4

Normalized intensity profiles of the intracavity laser beam at 40 cm from the Gaussian mirror for different round trips. Circles, experimental results; solid curves, Gaussian profiles obtained with the theoretical spot-size values given in Table 1.

Fig. 5
Fig. 5

Evolution of the intracavity beam spot size versus z. Circles, experimental results; curves, numerical profiles obtained with the propagation formulas for the Schell–Gauss beams.

Fig. 6
Fig. 6

Normalized intensity profiles of the intracavity laser beam at 190 cm from the Gaussian mirror for different round trips. Circles, experimental results; solid curves, Gaussian profiles obtained with the corresponding numerical values used for spot sizes.

Fig. 7
Fig. 7

Normalized-intensity far-field profile versus far-field angle for different round trips.

Fig. 8
Fig. 8

Spot-size and divergence beam evolution versus round-trip number. Solid curve, divergence; dashed curve, spot size.

Tables (3)

Tables Icon

Table 1 Numerical Spot Size and Far-Field Divergence Values versus Cavity Round-Trip Number

Tables Icon

Table 2 Influence of Ri and i on Beam Spot Size at 40 cm from the Gaussian Mirror

Tables Icon

Table 3 Influence of Ri and ui on Far-Field Beam Divergence

Equations (21)

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R ( ρ ) = exp ( ρ 2 / 2 ) ,
F = π 2 / λ L .
g = 1 L / R ,
W ( ρ 1 , ρ 2 , z = 0 ) = A exp ( 2 ρ 1 2 2 ) δ ( ρ 1 ρ 2 ) ,
W ( r 1 , r 2 , z = 2 L ) = i 2 λ L W ( ρ 1 , ρ 2 , z = 0 ) × exp { i k 4 L [ ( r 1 ρ 1 ) 2 ( r 2 ρ 2 ) 2 ] } d 2 ρ 1 d 2 ρ 2 = i A 2 λ L exp [ i k 4 L ( r 1 2 r 2 2 ) ] exp [ ( r 1 r 2 ) 2 2 σ μ 2 ] ,
W ( r 1 , r 2 ) = W ( r 1 , r 2 , z = L ) exp ( r 1 2 + r 2 2 2 ) ,
W N ( r 1 , r 2 ) = A exp ( r 1 2 + r 2 2 w N 2 ) exp [ ( r 1 r 2 ) 2 2 σ μ N 2 ] × exp [ i π λ R N ( r 1 2 + r 2 2 ) ] .
I N ( r ) = W N ( r , r ) .
( w norm ) N = w N λ L / π ,
α N = ( σ μ ) N / w N ,
M N 2 = ( 1 + 1 / α N 2 ) 1 / 2 .
c N = 1 + L / R N .
1 ( w norm ) N + 1 2 = 1 ( w norm ) N 2 1 c N 2 + ( M N 2 ) 2 / ( w norm ) N 4 + 1 F ,
( M N + 1 2 ) 2 = 1 + [ ( M N 2 ) 2 1 ] { 1 + ( w norm ) N 2 / F [ c N 2 + M N 4 / ( w norm ) N 4 ] } ,
c N + 1 = 2 c N c N 2 + ( M N 2 ) 2 / ( w norm ) N 4 ,
θ N w N ( ζ ) ζ λ M N 2 π w 0 N ,
w 0 N 2 = w N 2 1 + ( π w N 2 / λ M N 2 R N ) 2 .
θ N = λ M N 2 π w N [ 1 + ( π w N 2 λ M N 2 R N ) 2 ] 1 / 2 .
θ N = 1000 [ ( λ π L ) ( M N 2 ) 2 + ( w norm ) N 4 ( c N 1 ) 2 ( w norm ) N 2 ] 1 / 2 ,
g = 1 L / R = 1 , F = π 2 / λ L = 11 ,
σ μ o = w i ( M o 2 ) 2 1 = 0.03 mm 100 λ .

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