Abstract

We determined theoretically that the nonlinear dynamics of a Gaussian beam is configuration dependent in a general cavity. This prediction was confirmed by numerical simulation in a Kerr-lens mode-locked cavity for which the self-focusing effect is considered the nonlinear source in both the spatial and the temporal domains. Period doubling, tripling, and quadrupling can occur in these configurations with the products of generalized cavity G parameters equal to 1/2, 1/4 (or 3/4), and (2±2)/4, respectively. The dynamic behavior of the cavity beam will become irregular if the nonlinear effect is further increased.

© 2000 Optical Society of America

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References

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  1. D. E. Spence, P. N. Kean, and W. Sibbett, “60-fs pulse generation from a self-mode-locked Ti:sapphire laser,” Opt. Lett. 16, 42–44 (1991).
    [CrossRef] [PubMed]
  2. M. Piche, “Beam reshaping and self-mode-locking in nonlinear laser resonators,” Opt. Commun. 86, 156–160 (1991).
    [CrossRef]
  3. G. W. Pearson, C. Radzewicz, and J. S. Krasinski, “Analysis of self-focusing mode-locking lasers with additional highly nonlinear self-focusing elements,” Opt. Commun. 94, 221–226 (1992).
    [CrossRef]
  4. J. Herrmann, “Theory of Kerr-lens mode locking: role of self-focusing and radially varying gain,” J. Opt. Soc. Am. B 11, 498–512 (1994).
    [CrossRef]
  5. G. Cerullo, S. De Silvestri, V. Magni, and L. Pallaro, “Resonators for Kerr-lens mode-locked femtosecond Ti:sapphire lasers,” Opt. Lett. 19, 807–809 (1994).
    [CrossRef] [PubMed]
  6. K.-H. Lin and W.-F. Hsieh, “An analytical design of asymmetrical Kerr lens mode-locking laser cavities,” J. Opt. Soc. Am. B 11, 737–741 (1994).
    [CrossRef]
  7. J. L. A. Chilla and O. E. Martinez, “Spatial-temporal analysis of the self-mode-locked Ti:sapphire laser,” J. Opt. Soc. Am. B 10, 638–643 (1993).
    [CrossRef]
  8. K.-H. Lin and W.-F. Hsieh, “Analytical spatio-temporal design of Kerr lens mode-locked laser resonators,” J. Opt. Soc. Am. B 13, 1786–1793 (1996).
    [CrossRef]
  9. D. Cote and H. M. van Driel, “Period doubling of a femtosecond Ti:sapphire laser by total mode locking,” Opt. Lett. 23, 715–717 (1998).
    [CrossRef]
  10. S. R. Bolton, R. A. Jenks, C. N. Elkinton, and G. Sucha, “Pulse-resolved measurements of subharmonic oscillations in a Kerr-lens mode-locked Ti:sapphire laser,” J. Opt. Soc. Am. B 16, 339–344 (1999).
    [CrossRef]
  11. V. L. Kalashnikov, I. G. Poloyko, and V. P. Mikhailov, “Regular, quasi-periodic, and chaotic behavior in continuous-wave solid-state Kerr-lens mode-locked lasers,” J. Opt. Soc. Am. B 14, 2691–2695 (1997).
    [CrossRef]
  12. Q. Xing, L. Chai, W. Zhang, and C. Wang, “Regular, period-doubling, quasi-periodic, and chaotic behavior in a self-mode-locked Ti:sapphire laser,” Opt. Commun. 162, 71–74 (1999).
    [CrossRef]
  13. M.-D. Wei, W.-F. Hsieh, and C. C. Sung, “Dynamics of an optical resonator determined by its iterative map of beam parameters,” Opt. Commun. 146, 201–207 (1998).
    [CrossRef]
  14. J. M. Greene, “A method for determining a stochastic transition,” J. Math. Phys. 20, 1183–1201 (1979).
    [CrossRef]
  15. A. E. Seigman, Lasers (University Science, Mill Valley, Calif., 1986), Chaps. 20 and 21.
  16. H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965); H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1556 (1966).
    [CrossRef] [PubMed]
  17. H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode-locking,” IEEE J. Quantum Electron. 28, 2086–2096 (1992).
    [CrossRef]
  18. M.-D. Wei, W.-F. Hsieh, and C. C. Sung, “The preferable resonators for Kerr-lens mode-locking determined by stability factors of their iterative maps,” Opt. Commun. 155, 406–412 (1998).
    [CrossRef]
  19. L. M. Sanchez and A. A. Hnilo, “Optical cavities as iterative maps in the complex plane,” Opt. Commun. 166, 229–238 (1999).
    [CrossRef]
  20. K.-H. Lin, Y. Lai, and W.-F. Hsieh, “Simple analytic method of cavity design for astigmatism compensated Kerr lens mode-locked ring lasers and its application,” J. Opt. Soc. Am. B 12, 468–475 (1995).
    [CrossRef]
  21. M. Mechendale, T. R. Nelson, F. G. Omenetto, and W. A. Schroeder, “Thermal effects in laser pumped Kerr-lens modelocked Ti:sapphire lasers,” Opt. Commun. 136, 150–159 (1997).
    [CrossRef]
  22. B. E. Lemoff and C. P. J. Barty, “Generation of high-peak-power 20 fs pulses from a regeneratively initiated, self-mode-locked Ti:sapphire laser,” Opt. Lett. 17, 1367–1369 (1992).
    [CrossRef]
  23. R. S. MacKay, Renormalisation in Area-Preserving Maps (World Scientific, Singapore, 1993), Chap. 1.
  24. A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics (Springer-Verlag, New York, 1992), Chap. 3.
  25. J.-M. Shieh, F. Ganikhanov, K.-H. Lin, W.-F. Hsieh, and C.-L. Pan, “Completely self-starting picosecond and femtosecond Kerr-lens mode-locked Ti:sapphire laser,” J. Opt. Soc. Am. B 12, 945–949 (1995).
    [CrossRef]
  26. F. Salin and J. Squier, “Gain guiding in solid-state lasers,” Opt. Lett. 17, 1352–1354 (1992).
    [CrossRef] [PubMed]

1999 (3)

Q. Xing, L. Chai, W. Zhang, and C. Wang, “Regular, period-doubling, quasi-periodic, and chaotic behavior in a self-mode-locked Ti:sapphire laser,” Opt. Commun. 162, 71–74 (1999).
[CrossRef]

L. M. Sanchez and A. A. Hnilo, “Optical cavities as iterative maps in the complex plane,” Opt. Commun. 166, 229–238 (1999).
[CrossRef]

S. R. Bolton, R. A. Jenks, C. N. Elkinton, and G. Sucha, “Pulse-resolved measurements of subharmonic oscillations in a Kerr-lens mode-locked Ti:sapphire laser,” J. Opt. Soc. Am. B 16, 339–344 (1999).
[CrossRef]

1998 (3)

D. Cote and H. M. van Driel, “Period doubling of a femtosecond Ti:sapphire laser by total mode locking,” Opt. Lett. 23, 715–717 (1998).
[CrossRef]

M.-D. Wei, W.-F. Hsieh, and C. C. Sung, “Dynamics of an optical resonator determined by its iterative map of beam parameters,” Opt. Commun. 146, 201–207 (1998).
[CrossRef]

M.-D. Wei, W.-F. Hsieh, and C. C. Sung, “The preferable resonators for Kerr-lens mode-locking determined by stability factors of their iterative maps,” Opt. Commun. 155, 406–412 (1998).
[CrossRef]

1997 (2)

V. L. Kalashnikov, I. G. Poloyko, and V. P. Mikhailov, “Regular, quasi-periodic, and chaotic behavior in continuous-wave solid-state Kerr-lens mode-locked lasers,” J. Opt. Soc. Am. B 14, 2691–2695 (1997).
[CrossRef]

M. Mechendale, T. R. Nelson, F. G. Omenetto, and W. A. Schroeder, “Thermal effects in laser pumped Kerr-lens modelocked Ti:sapphire lasers,” Opt. Commun. 136, 150–159 (1997).
[CrossRef]

1996 (1)

1995 (2)

1994 (3)

1993 (1)

1992 (4)

F. Salin and J. Squier, “Gain guiding in solid-state lasers,” Opt. Lett. 17, 1352–1354 (1992).
[CrossRef] [PubMed]

G. W. Pearson, C. Radzewicz, and J. S. Krasinski, “Analysis of self-focusing mode-locking lasers with additional highly nonlinear self-focusing elements,” Opt. Commun. 94, 221–226 (1992).
[CrossRef]

B. E. Lemoff and C. P. J. Barty, “Generation of high-peak-power 20 fs pulses from a regeneratively initiated, self-mode-locked Ti:sapphire laser,” Opt. Lett. 17, 1367–1369 (1992).
[CrossRef]

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode-locking,” IEEE J. Quantum Electron. 28, 2086–2096 (1992).
[CrossRef]

1991 (2)

M. Piche, “Beam reshaping and self-mode-locking in nonlinear laser resonators,” Opt. Commun. 86, 156–160 (1991).
[CrossRef]

D. E. Spence, P. N. Kean, and W. Sibbett, “60-fs pulse generation from a self-mode-locked Ti:sapphire laser,” Opt. Lett. 16, 42–44 (1991).
[CrossRef] [PubMed]

1979 (1)

J. M. Greene, “A method for determining a stochastic transition,” J. Math. Phys. 20, 1183–1201 (1979).
[CrossRef]

Barty, C. P. J.

Bolton, S. R.

Cerullo, G.

Chai, L.

Q. Xing, L. Chai, W. Zhang, and C. Wang, “Regular, period-doubling, quasi-periodic, and chaotic behavior in a self-mode-locked Ti:sapphire laser,” Opt. Commun. 162, 71–74 (1999).
[CrossRef]

Chilla, J. L. A.

Cote, D.

De Silvestri, S.

Elkinton, C. N.

Fujimoto, J. G.

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode-locking,” IEEE J. Quantum Electron. 28, 2086–2096 (1992).
[CrossRef]

Ganikhanov, F.

Greene, J. M.

J. M. Greene, “A method for determining a stochastic transition,” J. Math. Phys. 20, 1183–1201 (1979).
[CrossRef]

Haus, H. A.

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode-locking,” IEEE J. Quantum Electron. 28, 2086–2096 (1992).
[CrossRef]

Herrmann, J.

Hnilo, A. A.

L. M. Sanchez and A. A. Hnilo, “Optical cavities as iterative maps in the complex plane,” Opt. Commun. 166, 229–238 (1999).
[CrossRef]

Hsieh, W.-F.

Ippen, E. P.

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode-locking,” IEEE J. Quantum Electron. 28, 2086–2096 (1992).
[CrossRef]

Jenks, R. A.

Kalashnikov, V. L.

Kean, P. N.

Krasinski, J. S.

G. W. Pearson, C. Radzewicz, and J. S. Krasinski, “Analysis of self-focusing mode-locking lasers with additional highly nonlinear self-focusing elements,” Opt. Commun. 94, 221–226 (1992).
[CrossRef]

Lai, Y.

Lemoff, B. E.

Lin, K.-H.

Magni, V.

Martinez, O. E.

Mechendale, M.

M. Mechendale, T. R. Nelson, F. G. Omenetto, and W. A. Schroeder, “Thermal effects in laser pumped Kerr-lens modelocked Ti:sapphire lasers,” Opt. Commun. 136, 150–159 (1997).
[CrossRef]

Mikhailov, V. P.

Nelson, T. R.

M. Mechendale, T. R. Nelson, F. G. Omenetto, and W. A. Schroeder, “Thermal effects in laser pumped Kerr-lens modelocked Ti:sapphire lasers,” Opt. Commun. 136, 150–159 (1997).
[CrossRef]

Omenetto, F. G.

M. Mechendale, T. R. Nelson, F. G. Omenetto, and W. A. Schroeder, “Thermal effects in laser pumped Kerr-lens modelocked Ti:sapphire lasers,” Opt. Commun. 136, 150–159 (1997).
[CrossRef]

Pallaro, L.

Pan, C.-L.

Pearson, G. W.

G. W. Pearson, C. Radzewicz, and J. S. Krasinski, “Analysis of self-focusing mode-locking lasers with additional highly nonlinear self-focusing elements,” Opt. Commun. 94, 221–226 (1992).
[CrossRef]

Piche, M.

M. Piche, “Beam reshaping and self-mode-locking in nonlinear laser resonators,” Opt. Commun. 86, 156–160 (1991).
[CrossRef]

Poloyko, I. G.

Radzewicz, C.

G. W. Pearson, C. Radzewicz, and J. S. Krasinski, “Analysis of self-focusing mode-locking lasers with additional highly nonlinear self-focusing elements,” Opt. Commun. 94, 221–226 (1992).
[CrossRef]

Salin, F.

Sanchez, L. M.

L. M. Sanchez and A. A. Hnilo, “Optical cavities as iterative maps in the complex plane,” Opt. Commun. 166, 229–238 (1999).
[CrossRef]

Schroeder, W. A.

M. Mechendale, T. R. Nelson, F. G. Omenetto, and W. A. Schroeder, “Thermal effects in laser pumped Kerr-lens modelocked Ti:sapphire lasers,” Opt. Commun. 136, 150–159 (1997).
[CrossRef]

Shieh, J.-M.

Sibbett, W.

Spence, D. E.

Squier, J.

Sucha, G.

Sung, C. C.

M.-D. Wei, W.-F. Hsieh, and C. C. Sung, “Dynamics of an optical resonator determined by its iterative map of beam parameters,” Opt. Commun. 146, 201–207 (1998).
[CrossRef]

M.-D. Wei, W.-F. Hsieh, and C. C. Sung, “The preferable resonators for Kerr-lens mode-locking determined by stability factors of their iterative maps,” Opt. Commun. 155, 406–412 (1998).
[CrossRef]

van Driel, H. M.

Wang, C.

Q. Xing, L. Chai, W. Zhang, and C. Wang, “Regular, period-doubling, quasi-periodic, and chaotic behavior in a self-mode-locked Ti:sapphire laser,” Opt. Commun. 162, 71–74 (1999).
[CrossRef]

Wei, M.-D.

M.-D. Wei, W.-F. Hsieh, and C. C. Sung, “Dynamics of an optical resonator determined by its iterative map of beam parameters,” Opt. Commun. 146, 201–207 (1998).
[CrossRef]

M.-D. Wei, W.-F. Hsieh, and C. C. Sung, “The preferable resonators for Kerr-lens mode-locking determined by stability factors of their iterative maps,” Opt. Commun. 155, 406–412 (1998).
[CrossRef]

Xing, Q.

Q. Xing, L. Chai, W. Zhang, and C. Wang, “Regular, period-doubling, quasi-periodic, and chaotic behavior in a self-mode-locked Ti:sapphire laser,” Opt. Commun. 162, 71–74 (1999).
[CrossRef]

Zhang, W.

Q. Xing, L. Chai, W. Zhang, and C. Wang, “Regular, period-doubling, quasi-periodic, and chaotic behavior in a self-mode-locked Ti:sapphire laser,” Opt. Commun. 162, 71–74 (1999).
[CrossRef]

IEEE J. Quantum Electron. (1)

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode-locking,” IEEE J. Quantum Electron. 28, 2086–2096 (1992).
[CrossRef]

J. Math. Phys. (1)

J. M. Greene, “A method for determining a stochastic transition,” J. Math. Phys. 20, 1183–1201 (1979).
[CrossRef]

J. Opt. Soc. Am. B (8)

J. L. A. Chilla and O. E. Martinez, “Spatial-temporal analysis of the self-mode-locked Ti:sapphire laser,” J. Opt. Soc. Am. B 10, 638–643 (1993).
[CrossRef]

J. Herrmann, “Theory of Kerr-lens mode locking: role of self-focusing and radially varying gain,” J. Opt. Soc. Am. B 11, 498–512 (1994).
[CrossRef]

K.-H. Lin and W.-F. Hsieh, “An analytical design of asymmetrical Kerr lens mode-locking laser cavities,” J. Opt. Soc. Am. B 11, 737–741 (1994).
[CrossRef]

K.-H. Lin, Y. Lai, and W.-F. Hsieh, “Simple analytic method of cavity design for astigmatism compensated Kerr lens mode-locked ring lasers and its application,” J. Opt. Soc. Am. B 12, 468–475 (1995).
[CrossRef]

J.-M. Shieh, F. Ganikhanov, K.-H. Lin, W.-F. Hsieh, and C.-L. Pan, “Completely self-starting picosecond and femtosecond Kerr-lens mode-locked Ti:sapphire laser,” J. Opt. Soc. Am. B 12, 945–949 (1995).
[CrossRef]

K.-H. Lin and W.-F. Hsieh, “Analytical spatio-temporal design of Kerr lens mode-locked laser resonators,” J. Opt. Soc. Am. B 13, 1786–1793 (1996).
[CrossRef]

V. L. Kalashnikov, I. G. Poloyko, and V. P. Mikhailov, “Regular, quasi-periodic, and chaotic behavior in continuous-wave solid-state Kerr-lens mode-locked lasers,” J. Opt. Soc. Am. B 14, 2691–2695 (1997).
[CrossRef]

S. R. Bolton, R. A. Jenks, C. N. Elkinton, and G. Sucha, “Pulse-resolved measurements of subharmonic oscillations in a Kerr-lens mode-locked Ti:sapphire laser,” J. Opt. Soc. Am. B 16, 339–344 (1999).
[CrossRef]

Opt. Commun. (7)

M.-D. Wei, W.-F. Hsieh, and C. C. Sung, “The preferable resonators for Kerr-lens mode-locking determined by stability factors of their iterative maps,” Opt. Commun. 155, 406–412 (1998).
[CrossRef]

L. M. Sanchez and A. A. Hnilo, “Optical cavities as iterative maps in the complex plane,” Opt. Commun. 166, 229–238 (1999).
[CrossRef]

M. Mechendale, T. R. Nelson, F. G. Omenetto, and W. A. Schroeder, “Thermal effects in laser pumped Kerr-lens modelocked Ti:sapphire lasers,” Opt. Commun. 136, 150–159 (1997).
[CrossRef]

M. Piche, “Beam reshaping and self-mode-locking in nonlinear laser resonators,” Opt. Commun. 86, 156–160 (1991).
[CrossRef]

G. W. Pearson, C. Radzewicz, and J. S. Krasinski, “Analysis of self-focusing mode-locking lasers with additional highly nonlinear self-focusing elements,” Opt. Commun. 94, 221–226 (1992).
[CrossRef]

Q. Xing, L. Chai, W. Zhang, and C. Wang, “Regular, period-doubling, quasi-periodic, and chaotic behavior in a self-mode-locked Ti:sapphire laser,” Opt. Commun. 162, 71–74 (1999).
[CrossRef]

M.-D. Wei, W.-F. Hsieh, and C. C. Sung, “Dynamics of an optical resonator determined by its iterative map of beam parameters,” Opt. Commun. 146, 201–207 (1998).
[CrossRef]

Opt. Lett. (5)

Other (4)

A. E. Seigman, Lasers (University Science, Mill Valley, Calif., 1986), Chaps. 20 and 21.

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965); H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1556 (1966).
[CrossRef] [PubMed]

R. S. MacKay, Renormalisation in Area-Preserving Maps (World Scientific, Singapore, 1993), Chap. 1.

A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics (Springer-Verlag, New York, 1992), Chap. 3.

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Figures (8)

Fig. 1
Fig. 1

Stability diagram for a general resonator. Besides the stable regions (0<G1G2<1), the specific configurations are shown as a, G1G2=1; b, G1G2=(2+2)/4; c, G1G2=3/4; d, G1G2=1/2; e, G1G2=1/4; f, G1G2=(2-2)/4.

Fig. 2
Fig. 2

Four-mirror KLM laser standing-wave resonator. A Kerr medium with length L is placed between curved mirrors M2 and M3 of high reflection. M1 is the output coupler and M4 are flat mirrors with high reflection.

Fig. 3
Fig. 3

Period-doubling bifurcation. As K>Kc=0.0008353, the solid line of period-2 represents the stable pair of solutions that corresponds to the self-consistent Gaussian beam of two round trips. Period-1 is unstable (shown as a dashed line).

Fig. 4
Fig. 4

Evolution of (a) spot size and (b) curvature for the period-2 solution. The two period-2 solutions, (w, 1/R)=(0.40 mm, 0) and (w, 1/R)=(1.33 mm, 0) with z=113.34 mm, x=42 mm, and K=0.002, are the self-consistent Gaussian beam solutions of two round trips. The variation of curvature between the two curved mirrors is too large to be shown in this figure.

Fig. 5
Fig. 5

Contours of the critical bifurcation parameter for period-2. The gray levels, from dark to light, correspond to Kc=0.02, 0.04, 0.06, 0.08, 0.1.

Fig. 6
Fig. 6

Evolution of (a) spot size and (b) curvature for the period-3 solution. The period-3 solutions are (w, 1/R)=(0.75 mm, 0), (w, 1/R)=(0.64 mm, 0.14 m-1), (w, 1/R)=(0.64 mm,-0.14 m-1) with z=114 mm, r1=49 mm, and K=0.15.

Fig. 7
Fig. 7

Spatial and temporal evolutions of period-3. (a) The evolution of the spatial parameter converges to a period-3 steady state, in which the fixed points are (w, 1/R)=q1(1.46 mm, 0.94 m-1), (w, 1/R)=q2(1.35 mm,-1.04 m-1), and (w, 1/R)=q3(2.71 mm,-0.01 m-1). (b) The pulse width also convergently evolves to σ1=86.72 fs, σ2=80.76 fs, and σ3=110.42 fs. Inset, extended plot of the last 10 iterations, the numbers are the subscripts of σ.

Fig. 8
Fig. 8

Evolution of the intensity fluctuation. The intensity fluctuation is normalized to the average intensity in this figure. The initial values are (a) w0=0.71 mm and (b) w0=0.705 mm, with z=114 mm, x=49 mm, and K=0.4.

Tables (1)

Tables Icon

Table 1 Configurations with Several z Values that Have Low-Order Resonances

Equations (15)

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u(r, ζ)=Uw(ζ)exp-i kr22q(ζ),
1q(ζ)=1R(ζ)-i λπw(ζ)2,
wn+1=fw(wn, Rn)=-πλ ImC+D(1/Rn-iλ/πw2)A+B(1/Rn-iλ/πw2)-1/2,
Rn+1=fR(wn, Rn)=ReC+D(1/Rn-iλ/πw2)A+B(1/Rn-iλ/πw2)-1,
Res=[2-Tr(MJ)]/4,
Res=sin2(θ/2).
Res=1-(2G1G2-1)2,
1qII=1RII-i λπwII2=1RI+LRI+L(1-K)λπwI22-iλπwI21+LRI2+L2(1-K)λπwI22,
K=8πn2Pλ2,
wn+1=fw(wn, Rn;K),
Rn+1=fR(wn, Rn;K).
Qn+1=FK(Qn),
1p=2ηc+i 2cσ2,
p2=At p1+BtCt p1+Dt,
1L-i λg0πwp21

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