Abstract

We present analytical expressions of quadratic and cubic phases in a three-element resonator for Kerr-lens mode locking. Variation of quadratic and cubic phases with wedge angles of Ti:sapphire and apex angles of a prismatic output coupler are calculated. The results provide a theoretical basis for the design of a three-element cavity.

© 2001 Optical Society of America

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References

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  1. M. Ramaswamy-Paye, J. G. Fujimoto, “Compact dispersion-compensating geometry for Kerr-lens mode-locked femtosecond lasers,” Opt. Lett. 19, 1756–1758 (1994).
    [CrossRef] [PubMed]
  2. B. E. Bouma, J. G. Fujimoto, “Compact Kerr-lens mode-locked resonators,” Opt. Lett. 21, 134–136 (1996).
    [CrossRef] [PubMed]
  3. R. Zhang, D. Pang, J. Sun, Q. Wang, S. Zhang, G. Wen, “Analytical expressions of group-delay dispersion and cubic phase for four-prism sequence used at other than Brewster’s angle,” Opt. Laser Technol. 31, 373–379 (1999).
    [CrossRef]

1999

R. Zhang, D. Pang, J. Sun, Q. Wang, S. Zhang, G. Wen, “Analytical expressions of group-delay dispersion and cubic phase for four-prism sequence used at other than Brewster’s angle,” Opt. Laser Technol. 31, 373–379 (1999).
[CrossRef]

1996

1994

Bouma, B. E.

Fujimoto, J. G.

Pang, D.

R. Zhang, D. Pang, J. Sun, Q. Wang, S. Zhang, G. Wen, “Analytical expressions of group-delay dispersion and cubic phase for four-prism sequence used at other than Brewster’s angle,” Opt. Laser Technol. 31, 373–379 (1999).
[CrossRef]

Ramaswamy-Paye, M.

Sun, J.

R. Zhang, D. Pang, J. Sun, Q. Wang, S. Zhang, G. Wen, “Analytical expressions of group-delay dispersion and cubic phase for four-prism sequence used at other than Brewster’s angle,” Opt. Laser Technol. 31, 373–379 (1999).
[CrossRef]

Wang, Q.

R. Zhang, D. Pang, J. Sun, Q. Wang, S. Zhang, G. Wen, “Analytical expressions of group-delay dispersion and cubic phase for four-prism sequence used at other than Brewster’s angle,” Opt. Laser Technol. 31, 373–379 (1999).
[CrossRef]

Wen, G.

R. Zhang, D. Pang, J. Sun, Q. Wang, S. Zhang, G. Wen, “Analytical expressions of group-delay dispersion and cubic phase for four-prism sequence used at other than Brewster’s angle,” Opt. Laser Technol. 31, 373–379 (1999).
[CrossRef]

Zhang, R.

R. Zhang, D. Pang, J. Sun, Q. Wang, S. Zhang, G. Wen, “Analytical expressions of group-delay dispersion and cubic phase for four-prism sequence used at other than Brewster’s angle,” Opt. Laser Technol. 31, 373–379 (1999).
[CrossRef]

Zhang, S.

R. Zhang, D. Pang, J. Sun, Q. Wang, S. Zhang, G. Wen, “Analytical expressions of group-delay dispersion and cubic phase for four-prism sequence used at other than Brewster’s angle,” Opt. Laser Technol. 31, 373–379 (1999).
[CrossRef]

Opt. Laser Technol.

R. Zhang, D. Pang, J. Sun, Q. Wang, S. Zhang, G. Wen, “Analytical expressions of group-delay dispersion and cubic phase for four-prism sequence used at other than Brewster’s angle,” Opt. Laser Technol. 31, 373–379 (1999).
[CrossRef]

Opt. Lett.

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Figures (7)

Fig. 1
Fig. 1

Schematic of a three-element cavity.

Fig. 2
Fig. 2

Two rays with different wavelengths in a three-element cavity.

Fig. 3
Fig. 3

Light traces in a POC in one arm of a three-element cavity.

Fig. 4
Fig. 4

(a) Quadratic phase and (b) cubic phase in the cavity when the Ti:sapphire and the POC angles change. The cavity length is 300 mm. The POC was made with fused silica.

Fig. 5
Fig. 5

Cavity length with group-delay dispersion compensated in a three-element cavity: (a) Brewster angle POC and a varied Ti:sapphire wedge angle and (b) Brewster angle Ti:sapphire and a varied POC apex angle.

Fig. 6
Fig. 6

Cubic phase with group-delay dispersion compensated in a three-element cavity: (a) Brewster angle POC and a varied Ti:sapphire wedge angle and (b) Brewster angle Ti:sapphire and a varied POC apex angle.

Fig. 7
Fig. 7

Comparison of the quadratic phase and the cubic phase of a three-element cavity and a four-prism sequence: (a) fused silica and (b) Schott SF10.

Equations (41)

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1CO cos β-1CO cos β=2R,
CC sin β+OCtan Δθ=OC+CC sin βtan Δθ,
CO tan Δθ=CO tan Δθ.
COCO=ΔθΔθ.
OC=R21cos β1-ΔθΔθ,
OC=R21cos βΔθΔθ-1.
sinθλ+α=nλsin α, sinθλ+α=nλsin α,
Δθ=sin α1-n2 sin2 α1/2dndλ Δλ,
Δθ=sin α1-n2 sin2 α1/2dndλ Δλ,
ΔθΔθ=sin α/1-n2 sin2 α1/2dndλsin α/1-n2 sin2 α1/2dndλ
Pλ=2nA1B1+B1C+CO+OD1+nD1C1.
Pλ=P1λ+P2λ,
P1λ=2nA1B1+B1C+CO,
P2λ=2nD1C1+OD1.
P2λ=2OB=2BG+GO=2EF sin θ+OF cos θ,
dP2dλ=dP2dθdθdndndλ,
d2P2dλ2=dP2dθdθdnd2ndλ2+dP2dθd2θdn2+d2P2dθ2dθdn2×dndλ2,
d3P2dλ3=dP2dθdθdnd3ndλ3+3d2P2dθ2dθdn2+dP2dθd2θdn2dndλd2ndλ2+d3P2dθ3dθdn3+3 d2P2dθ2dθdnd2θdn2+dP2dθd3θdn3dndλ3.
12dP2dθ=EF cos θ-OF sin θ=FH-FG=EB,
EB=ED1 sin θ=ED1 sinπ2-α-θ=ED1 cosα+θ,ED1=C1D1sin α.
cosα+θ=1-sin2α+θ1/2=1-n2 sin2 α1/2.
EB=C1D11-n2 sin2 α1/2sin α.
dP2dθ=2C1D11-n2 sin2 α1/2sin α=2C1D11dθ/dn,d2P2dθ2=2-EF sin θ-OF cos θ=2-BG-OG=-2OB.
d2P2dθ2=-2nC1D1-2D1O,
d3P2dθ3=2-EF cos θ+OF sin θ=-dP2dθ=-2C1D11dθ/dn.
θ=sin-1n sin α-α.
dθdn=sin α-2-n2-1/2,
d2θdn2=nsin α-2-n2-3/2=ndθdn3,
d3θdn3=sin α-2-n2-3/2+3n2sin α-2-n2-5/2=dθdn3+3n2dθdn5.
d2P2dλ2=2C1D1d2ndλ2-2D1O sin2 α1-n2 sin2 αdndλ2.
d3P2dλ3=2C1D1d3ndλ3-6D1O sin2 α1-n2 sin2 αdndλd2ndλ2-6D1On sin4 α1-n2 sin2 α2dndλ3.
P1λ=2nA1B1+B1O.
d2P1λdλ=2A1B1d2ndλ2-2B1O sin2 α1-n2 sin2 αdndλ2,
d3P1λdλ3=2A1B1d3ndλ3-6B1O sin2 α1-n2 sin2 αdndλd2ndλ2 -6B1On sin4 α1-n2 sin2 α2dndλ3.
P1λ=2nA1B1+B1C+CO, P1λ=2nA1B1+B1C+CO.
d2P1dλ2=d2P1dλ2, d3P1dλ3=d3P1dλ3.
d2Pdλ2=d2P1dλ2+d2P2dλ2=2A1B1d2ndλ2-B1O sin2 α1-n2 sin2 αdndλ2+C1D1d2ndλ2-D1O sin2 α1-n2 sin2 αdndλ2,
d3Pdλ3=d3P1dλ3+d3P2dλ3=2A1B1d3ndλ3-3B1O sin2 α1-n2 sin2 αdndλd2ndλ2-3B1On sin4 α1-n2 sin2 α2dndλ3+C1D1d3ndλ3-3D1O sin2 α1-n2 sin2 αdndλd2ndλ2-3D1On sin4 α1-n2 sin2 α2dndλ3.
d2Pdλ2=2A1B1d2ndλ2-B1Odndλ2+C1D1d2ndλ2-D1Odndλ2,
d3Pdλ3=2A1B1d3ndλ3-3B1O dndλd2ndλ2-3B1On dndλ3+C1D1d3ndλ3-3D1O dndλd2ndλ2-3D1Ondndλ3.
d2ϕdω2=λ32πc2d2Pdλ2, d3ϕdω3=-λ42π2c33 d2Pdλ2+λ d3Pdλ3.

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