Abstract

We find that soft-aperture Kerr-lens mode locking (SAKLM) in a Ti:sapphire laser always takes place around certain discrete cavity configurations accompanied by peculiar beam patterns. By correlating the superposition of proper transverse modes to the observed beam patterns and fitting the beat frequencies to the resonance equation, we discovered that the patterns for SAKLM are formed by phase locking of the fundamental mode with specific degenerate transverse modes that occur near different degenerate cavity configurations. As predicted by the stability analysis, period-3 and period-2 SAKLM does occur around the 1/3- and 1/4-degenerate cavity configurations, which cannot be fully explained in terms of total mode locking of TEM00 and TEM01 modes.

© 2001 Optical Society of America

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References

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  1. D. E. Spence, P. N. Kean, and W. Sibbett, “60-fsec pulse generation from a self-mode-locked Ti:sapphire laser,” Opt. Lett. 16, 42–44 (1991).
    [CrossRef] [PubMed]
  2. J. Hermann, “Theory of Kerr-lens mode locking: role of self-focusing and radially varying gain,” J. Opt. Soc. Am. B 11, 498–512 (1994).
    [CrossRef]
  3. 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]
  4. K.-H. Lin and W.-F. Hsieh, “Analytical design of symmetrical Kerr-lens mode-locking laser cavities,” J. Opt. Soc. Am. B 11, 737–741 (1994).
    [CrossRef]
  5. K.-H. Lin, Y. Lai, and W.-F. Hsieh, “Simple analytical method of cavity design for astigmatism-compensated Kerr-lens mode-locked ring lasers and its applications,” J. Opt. Soc. Am. B 12, 468–475 (1995).
    [CrossRef]
  6. J. L. A. Chilla and O. E. Martinez, “Spatial-temporal analysis of the self-mode-locked Ti:sapphire,” J. Opt. Soc. Am. B 10, 638–643 (1993).
    [CrossRef]
  7. K.-H. Lin and W.-F. Hsieh, “Analytical spatial-temporal design of Kerr lens mode-locked laser resonators,” J. Opt. Soc. Am. B 13, 1786–1793 (1996).
    [CrossRef]
  8. G. Cerullo, S. D. Silvestri, and V. Magni, “Self-starting Kerr-lens mode locking of a Ti:sapphire laser,” Opt. Lett. 19, 1040–1042 (1994).
    [CrossRef] [PubMed]
  9. M. Lai, “Self-starting, self-mode-locked Ti:sapphire laser,” Opt. Lett. 19, 722–724 (1994).
    [CrossRef] [PubMed]
  10. D.-G. Juang, Y.-C. Chen, S.-H. Hsu, K.-H. Lin, and W.-F. Hsieh, “Differential gain and buildup dynamics of self-starting Kerr lens mode-locked Ti:sapphire laser without an internal aperture,” J. Opt. Soc. Am. B 14, 2116–2121 (1997).
    [CrossRef]
  11. J.-G. Lai, K.-H. Lin, D.-G. Juang, and W.-F. Hsieh, “Construction of a wavelength-tunable self-starting Kerr lens mode locked Ti:sapphire laser system,” Chin. J. Phys. (Taipei) 34, 111–121 (1996).
  12. 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]
  13. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chaps. 20 and 21.
  14. 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]
  15. M.-D. Wei and W.-F. Hsieh, “Cavity-configuration-dependent nonlinear dynamics in Kerr-lens mode-locked lasers,” J. Opt. Soc. Am. B 17, 1335–1342 (2000).
    [CrossRef]
  16. M.-D. Wei and W.-F. Hsieh, “Bifurcation of fundamental Gaussian modes in Kerr-lens mode-locked lasers,” Opt. Commun. 168, 161–166 (1999).
    [CrossRef]
  17. H.-H. Wu, C.-C. Sheu, T.-W. Chen, M.-D. Wei, and W.-F. Hsieh, “Observation of power drop and low threshold due to beam waist shrinkage around critical configurations in an end-pumped Nd:YVO4 laser,” Opt. Commun. 165, 225–229 (1999).
    [CrossRef]
  18. H.-H. Wu and W.-F. Hsieh, “Observation of multipass transverse modes in an axially pumped solid state laser with different fractional degenerate resonator configurations,” J. Opt. Soc. Am. B 18, 7–12 (2001).
    [CrossRef]
  19. S. R. Bolton, R. A. Jenks, C. N. Elkinton, and G. Sucha, “Pulse-resolved measurements of subharmonic oscillations in a Kerr-lens mode-locked laser,” J. Opt. Soc. Am. B 16, 339–343 (1999).
    [CrossRef]
  20. 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]

2001 (1)

2000 (1)

1999 (3)

M.-D. Wei and W.-F. Hsieh, “Bifurcation of fundamental Gaussian modes in Kerr-lens mode-locked lasers,” Opt. Commun. 168, 161–166 (1999).
[CrossRef]

H.-H. Wu, C.-C. Sheu, T.-W. Chen, M.-D. Wei, and W.-F. Hsieh, “Observation of power drop and low threshold due to beam waist shrinkage around critical configurations in an end-pumped Nd:YVO4 laser,” Opt. Commun. 165, 225–229 (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 laser,” J. Opt. Soc. Am. B 16, 339–343 (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, “The preferable resonators for Kerr-lens mode-locking determined by stability factors of their iterative maps,” Opt. Commun. 155, 406–412 (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]

1997 (1)

1996 (2)

J.-G. Lai, K.-H. Lin, D.-G. Juang, and W.-F. Hsieh, “Construction of a wavelength-tunable self-starting Kerr lens mode locked Ti:sapphire laser system,” Chin. J. Phys. (Taipei) 34, 111–121 (1996).

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

1995 (1)

1994 (5)

1993 (1)

1991 (1)

Bolton, S. R.

Cerullo, G.

Chen, T.-W.

H.-H. Wu, C.-C. Sheu, T.-W. Chen, M.-D. Wei, and W.-F. Hsieh, “Observation of power drop and low threshold due to beam waist shrinkage around critical configurations in an end-pumped Nd:YVO4 laser,” Opt. Commun. 165, 225–229 (1999).
[CrossRef]

Chen, Y.-C.

Chilla, J. L. A.

Cote, D.

De Silvestri, S.

Elkinton, C. N.

Hermann, J.

Hsieh, W.-F.

H.-H. Wu and W.-F. Hsieh, “Observation of multipass transverse modes in an axially pumped solid state laser with different fractional degenerate resonator configurations,” J. Opt. Soc. Am. B 18, 7–12 (2001).
[CrossRef]

M.-D. Wei and W.-F. Hsieh, “Cavity-configuration-dependent nonlinear dynamics in Kerr-lens mode-locked lasers,” J. Opt. Soc. Am. B 17, 1335–1342 (2000).
[CrossRef]

M.-D. Wei and W.-F. Hsieh, “Bifurcation of fundamental Gaussian modes in Kerr-lens mode-locked lasers,” Opt. Commun. 168, 161–166 (1999).
[CrossRef]

H.-H. Wu, C.-C. Sheu, T.-W. Chen, M.-D. Wei, and W.-F. Hsieh, “Observation of power drop and low threshold due to beam waist shrinkage around critical configurations in an end-pumped Nd:YVO4 laser,” Opt. Commun. 165, 225–229 (1999).
[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]

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]

D.-G. Juang, Y.-C. Chen, S.-H. Hsu, K.-H. Lin, and W.-F. Hsieh, “Differential gain and buildup dynamics of self-starting Kerr lens mode-locked Ti:sapphire laser without an internal aperture,” J. Opt. Soc. Am. B 14, 2116–2121 (1997).
[CrossRef]

J.-G. Lai, K.-H. Lin, D.-G. Juang, and W.-F. Hsieh, “Construction of a wavelength-tunable self-starting Kerr lens mode locked Ti:sapphire laser system,” Chin. J. Phys. (Taipei) 34, 111–121 (1996).

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

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

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

Hsu, S.-H.

Jenks, R. A.

Juang, D.-G.

D.-G. Juang, Y.-C. Chen, S.-H. Hsu, K.-H. Lin, and W.-F. Hsieh, “Differential gain and buildup dynamics of self-starting Kerr lens mode-locked Ti:sapphire laser without an internal aperture,” J. Opt. Soc. Am. B 14, 2116–2121 (1997).
[CrossRef]

J.-G. Lai, K.-H. Lin, D.-G. Juang, and W.-F. Hsieh, “Construction of a wavelength-tunable self-starting Kerr lens mode locked Ti:sapphire laser system,” Chin. J. Phys. (Taipei) 34, 111–121 (1996).

Kean, P. N.

Lai, J.-G.

J.-G. Lai, K.-H. Lin, D.-G. Juang, and W.-F. Hsieh, “Construction of a wavelength-tunable self-starting Kerr lens mode locked Ti:sapphire laser system,” Chin. J. Phys. (Taipei) 34, 111–121 (1996).

Lai, M.

Lai, Y.

Lin, K.-H.

Magni, V.

Martinez, O. E.

Pallaro, L.

Sheu, C.-C.

H.-H. Wu, C.-C. Sheu, T.-W. Chen, M.-D. Wei, and W.-F. Hsieh, “Observation of power drop and low threshold due to beam waist shrinkage around critical configurations in an end-pumped Nd:YVO4 laser,” Opt. Commun. 165, 225–229 (1999).
[CrossRef]

Sibbett, W.

Silvestri, S. D.

Spence, D. E.

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.

Wei, M.-D.

M.-D. Wei and W.-F. Hsieh, “Cavity-configuration-dependent nonlinear dynamics in Kerr-lens mode-locked lasers,” J. Opt. Soc. Am. B 17, 1335–1342 (2000).
[CrossRef]

H.-H. Wu, C.-C. Sheu, T.-W. Chen, M.-D. Wei, and W.-F. Hsieh, “Observation of power drop and low threshold due to beam waist shrinkage around critical configurations in an end-pumped Nd:YVO4 laser,” Opt. Commun. 165, 225–229 (1999).
[CrossRef]

M.-D. Wei and W.-F. Hsieh, “Bifurcation of fundamental Gaussian modes in Kerr-lens mode-locked lasers,” Opt. Commun. 168, 161–166 (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]

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]

Wu, H.-H.

H.-H. Wu and W.-F. Hsieh, “Observation of multipass transverse modes in an axially pumped solid state laser with different fractional degenerate resonator configurations,” J. Opt. Soc. Am. B 18, 7–12 (2001).
[CrossRef]

H.-H. Wu, C.-C. Sheu, T.-W. Chen, M.-D. Wei, and W.-F. Hsieh, “Observation of power drop and low threshold due to beam waist shrinkage around critical configurations in an end-pumped Nd:YVO4 laser,” Opt. Commun. 165, 225–229 (1999).
[CrossRef]

Chin. J. Phys. (Taipei) (1)

J.-G. Lai, K.-H. Lin, D.-G. Juang, and W.-F. Hsieh, “Construction of a wavelength-tunable self-starting Kerr lens mode locked Ti:sapphire laser system,” Chin. J. Phys. (Taipei) 34, 111–121 (1996).

J. Opt. Soc. Am. B (9)

M.-D. Wei and W.-F. Hsieh, “Cavity-configuration-dependent nonlinear dynamics in Kerr-lens mode-locked lasers,” J. Opt. Soc. Am. B 17, 1335–1342 (2000).
[CrossRef]

H.-H. Wu and W.-F. Hsieh, “Observation of multipass transverse modes in an axially pumped solid state laser with different fractional degenerate resonator configurations,” J. Opt. Soc. Am. B 18, 7–12 (2001).
[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 laser,” J. Opt. Soc. Am. B 16, 339–343 (1999).
[CrossRef]

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

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

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

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

D.-G. Juang, Y.-C. Chen, S.-H. Hsu, K.-H. Lin, and W.-F. Hsieh, “Differential gain and buildup dynamics of self-starting Kerr lens mode-locked Ti:sapphire laser without an internal aperture,” J. Opt. Soc. Am. B 14, 2116–2121 (1997).
[CrossRef]

Opt. Commun. (4)

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 and W.-F. Hsieh, “Bifurcation of fundamental Gaussian modes in Kerr-lens mode-locked lasers,” Opt. Commun. 168, 161–166 (1999).
[CrossRef]

H.-H. Wu, C.-C. Sheu, T.-W. Chen, M.-D. Wei, and W.-F. Hsieh, “Observation of power drop and low threshold due to beam waist shrinkage around critical configurations in an end-pumped Nd:YVO4 laser,” Opt. Commun. 165, 225–229 (1999).
[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]

Opt. Lett. (5)

Other (1)

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

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Figures (8)

Fig. 1
Fig. 1

Experimental setup of the Kerr-lens mode-locking Ti:sapphire laser. D1 and D2 are high-speed detectors, and B is beam splitter.

Fig. 2
Fig. 2

Measured output power (solid curve) and G1G2 for the sagittal (dash) and tangential (dotted) planes as a function of r2 in the LMS region. The thick parts of the solid curve label the mode-locking region at each degenerate configuration.

Fig. 3
Fig. 3

Measured (left column) and imitated patterns (right column) of the picosecond mode locking at the configuration of (a) G1G2=1/4 (sagittal), (b) G1G2=1/2 (sagittal), (c) G1G2=1/2 (tangential), and (d) G1G2=0.6545 (sagittal). The configurations as well as the amplitude and relative phases of transverse modes are listed atop each figure in the right column.

Fig. 4
Fig. 4

Typical measured autocorrelation trace (solid curve) and corresponding optical spectrum (right inset) at the state of femtosecond pulses. Note that the dashed curve represents the fitting result by use of the sech2 function.

Fig. 5
Fig. 5

Measured femtosecond mode locking (left column) and imitated patterns (right column) from the corresponding configurations of Fig. 3.

Fig. 6
Fig. 6

Pulse train of (a) period-3 and (b) period-2 mode locking.

Fig. 7
Fig. 7

Intensity distribution of the beam pattern for (a) measured period-3 mode locking, (b) the imitated one by the superposition of the TEM00 and TEM30, and (c) the imitated integration of three successive round-trip patterns of superposition of TEM00, TEM30, and TEM10 with its relative phase ϕβ equal to (d) 0, (e) 2π/3 and (f) 4π/3, respectively.

Fig. 8
Fig. 8

Beam pattern for (a) the measured period-2 mode locking, (b) the imitated one by the superposition of TEM00 and TEM04(π), and (c) the imitated integration of two successive round-trip patterns with the superposition of the TEM00, TEM04(π), and TEM02 with its relative phase ϕβ equal to 0 (right) and π (left) in (d).

Tables (2)

Tables Icon

Table 1 Chosen Transverse Eigenmodes and the Ratio of Radii in the Tangential and Sagittal Directions Used to Simulate the Imitated Beam Patterns of Picosecond ML in Fig. 3 with Their Results Correlated to the Measured Beam Patterns

Tables Icon

Table 2 Chosen Transverse Eignmodes and the Ratio of Radii in the Tangential and Sagittal Directions Used to Simulate the Imitated Beam Patterns of Femtosecond ML in Fig. 5 and Periods 2 and 3 with Their Results Correlated to the Measured Beam Patterns

Equations (4)

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a=(d2-feff)δfeff2-1,
b=a(d1+1)+(d2-2 feff)+1,
c=δfeff2,
d=(d1-feff)δfeff2-1.

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