Abstract

An extensive one-dimensional laser model based on dispersion managed mode locking is presented that accurately describes the pulse dynamics of octave-spanning titanium:sapphire lasers generating sub-two-cycle pulses. By including detailed characteristics for the intracavity elements (mirrors and output coupler), it is demonstrated that the spectral output and temporal pulse shape of these lasers can be predicted quantitatively in very good agreement with experimental results.

© 2009 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  4. S. Rausch, T. Binhammer, A. Harth, J. Kim, R. Ell, F. X. Kärtner, and U. Morgner, “Controlled waveforms on the single-cycle scale from a femtosecond oscillator,” Opt. Express 16, 9739-9745 (2008).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  25. A. Benedick, J. Birge, R. Ell, O. D. Mücke, M. Sander, and F. X. Kärtner, “Octave spanning 1 GHz Ti:sapphire oscillator for HeNe CH4-based frequency combs and clocks,” in Proceedings of the European Conference on Lasers and Electro-Optics (IEEE, 2007) paper CF3-1-MON.
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    [CrossRef]
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    [CrossRef]
  29. Schott Optical Glass Catalog (Schott Glass Technologies Inc., 1992).
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    [CrossRef]
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    [CrossRef]
  32. J. R. Birge and F. X. Kärtner, “Efficient analytic computation of dispersion from multilayer structures,” Appl. Opt. 45, 1478-1483 (2006).
    [CrossRef]
  33. J. R. Birge, H. Crespo, M. Sander, and F. X. Kärtner, “Non-intrusive sub-two-cycle carrier-envelope stabilized pulses using engineered chirped mirrors,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies (2008), paper CTuC3.
  34. G. Lenz, B. J. Eggleton, C. R. Giles, C. K. Madsen, and R. E. Slusher, “Dispersive properties of optical filters for WDM systems,” IEEE J. Quantum Electron. 34, 1390-1402 (1998).
    [CrossRef]
  35. R. Ell, J. R. Birge, M. Araghchini, and F. X. Kärtner, “Carrier-envelope phase control by a composite plate,” Opt. Express 14, 5829-5837 (2006).
    [CrossRef]

2008

S. Rausch, T. Binhammer, A. Harth, J. Kim, R. Ell, F. X. Kärtner, and U. Morgner, “Controlled waveforms on the single-cycle scale from a femtosecond oscillator,” Opt. Express 16, 9739-9745 (2008).
[CrossRef]

H. M. Crespo, J. R. Birge, E. L. Falcão-Filho, M. Y. Sander, A. Benedick, and F. X. Kärtner, “Nonintrusive phase stabilization of sub-two-cycle pulses from a prismless octave-spanning Ti:sapphire laser,” Opt. Lett. 33, 833-835 (2008).
[CrossRef]

H. M. Crespo, J. R. Birge, M. Y. Sander, E. L. Falcão-Filho, A. Benedick, and F. X. Kärtner, “Phase stabilization of sub-two-cycle pulses from prismless octave-spanning Ti:sapphire lasers,” J. Opt. Soc. Am. B 25, B147-B154 (2008).
[CrossRef]

M. Y. Sander, H. M. Crespo, J. R. Birge, and F. X. Kärtner, “Modeling of octave-spanning sub-two-cycle titanium:sapphire lasers: simulation and experiment,” in Conference on Ultrafast Phenomena (European Physical Society and Optical Society of America, 2008), paper THUIIIc.

J. R. Birge, H. Crespo, M. Sander, and F. X. Kärtner, “Non-intrusive sub-two-cycle carrier-envelope stabilized pulses using engineered chirped mirrors,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies (2008), paper CTuC3.

2007

J. R. Birge and F. X. Kärtner, “Efficient optimization of multilayer coatings for ultrafast optics using analytic gradients of dispersion,” Appl. Opt. 46, 2656-2662 (2007).
[CrossRef]

A. Benedick, J. Birge, R. Ell, O. D. Mücke, M. Sander, and F. X. Kärtner, “Octave spanning 1 GHz Ti:sapphire oscillator for HeNe CH4-based frequency combs and clocks,” in Proceedings of the European Conference on Lasers and Electro-Optics (IEEE, 2007) paper CF3-1-MON.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007).

2006

2005

2004

2003

2002

2001

2000

H. A. Haus, “Mode-locking of lasers,” IEEE J. Quantum Electron. 6, 1173-1185 (2000).
[CrossRef]

1999

1998

V. P. Kalosha, M. Müller, J. Herrmann, and S. Gatz, “Spatiotemporal model of femtosecond pulse generation in Kerr-lens mode-locked solid-state lasers,” J. Opt. Soc. Am. B 15, 535-550 (1998).
[CrossRef]

F. X. Kärtner, J. Aus der Au, and U. Keller, “Mode-locking with slow and fast saturable absorbers-what's the difference?” IEEE J. Sel. Top. Quantum Electron. 4, 159-168 (1998).
[CrossRef]

R. Muijlwijk, “Update of the Edlen's formulae for the refractive index of air,” Metrologia 25, 189 (1998).
[CrossRef]

G. Lenz, B. J. Eggleton, C. R. Giles, C. K. Madsen, and R. E. Slusher, “Dispersive properties of optical filters for WDM systems,” IEEE J. Quantum Electron. 34, 1390-1402 (1998).
[CrossRef]

1996

1995

1994

1993

1992

Schott Optical Glass Catalog (Schott Glass Technologies Inc., 1992).

1991

1976

H. Haus, “Parameter ranges for CW passive mode locking,” IEEE J. Quantum Electron. 12, 169-176 (1976).
[CrossRef]

Ablowitz, M. J.

Q. Quraishi, S. T. Cundiff, B. Ilan, and M. J. Ablowitz, “Dynamics of nonlinear and dispersion managed solitons,” Phys. Rev. Lett. 94, 243904 (2005).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007).

Angelow, G.

Araghchini, M.

Aus der Au, J.

F. X. Kärtner, J. Aus der Au, and U. Keller, “Mode-locking with slow and fast saturable absorbers-what's the difference?” IEEE J. Sel. Top. Quantum Electron. 4, 159-168 (1998).
[CrossRef]

Bartels, A.

Benedick, A.

Binhammer, T.

Birge, J.

A. Benedick, J. Birge, R. Ell, O. D. Mücke, M. Sander, and F. X. Kärtner, “Octave spanning 1 GHz Ti:sapphire oscillator for HeNe CH4-based frequency combs and clocks,” in Proceedings of the European Conference on Lasers and Electro-Optics (IEEE, 2007) paper CF3-1-MON.

Birge, J. R.

H. M. Crespo, J. R. Birge, E. L. Falcão-Filho, M. Y. Sander, A. Benedick, and F. X. Kärtner, “Nonintrusive phase stabilization of sub-two-cycle pulses from a prismless octave-spanning Ti:sapphire laser,” Opt. Lett. 33, 833-835 (2008).
[CrossRef]

H. M. Crespo, J. R. Birge, M. Y. Sander, E. L. Falcão-Filho, A. Benedick, and F. X. Kärtner, “Phase stabilization of sub-two-cycle pulses from prismless octave-spanning Ti:sapphire lasers,” J. Opt. Soc. Am. B 25, B147-B154 (2008).
[CrossRef]

M. Y. Sander, H. M. Crespo, J. R. Birge, and F. X. Kärtner, “Modeling of octave-spanning sub-two-cycle titanium:sapphire lasers: simulation and experiment,” in Conference on Ultrafast Phenomena (European Physical Society and Optical Society of America, 2008), paper THUIIIc.

J. R. Birge, H. Crespo, M. Sander, and F. X. Kärtner, “Non-intrusive sub-two-cycle carrier-envelope stabilized pulses using engineered chirped mirrors,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies (2008), paper CTuC3.

J. R. Birge and F. X. Kärtner, “Efficient optimization of multilayer coatings for ultrafast optics using analytic gradients of dispersion,” Appl. Opt. 46, 2656-2662 (2007).
[CrossRef]

J. R. Birge and F. X. Kärtner, “Efficient analytic computation of dispersion from multilayer structures,” Appl. Opt. 45, 1478-1483 (2006).
[CrossRef]

R. Ell, J. R. Birge, M. Araghchini, and F. X. Kärtner, “Carrier-envelope phase control by a composite plate,” Opt. Express 14, 5829-5837 (2006).
[CrossRef]

J. R. Birge, R. Ell, and F. X. Kärtner, “Two-dimensional spectral shearing interferometry for few-cycle pulse characterization,” Opt. Lett. 31, 2063-2065 (2006).
[CrossRef]

O. D. Mücke, R. Ell, A. Winter, J. Kim, J. R. Birge, L. Matos, and F. X. Kärtner, “Self-referenced 200 MHz octave-spanning Ti:sapphire laser with 50 attosecond carrier-envelope phase jitter,” Opt. Express 13, 5136-5169 (2005).
[CrossRef]

Boiko, A.

Brabec, T.

Bravec, T.

Chen, Y.

Cho, S. H.

Christov, I. P.

Crespo, H.

J. R. Birge, H. Crespo, M. Sander, and F. X. Kärtner, “Non-intrusive sub-two-cycle carrier-envelope stabilized pulses using engineered chirped mirrors,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies (2008), paper CTuC3.

Crespo, H. M.

M. Y. Sander, H. M. Crespo, J. R. Birge, and F. X. Kärtner, “Modeling of octave-spanning sub-two-cycle titanium:sapphire lasers: simulation and experiment,” in Conference on Ultrafast Phenomena (European Physical Society and Optical Society of America, 2008), paper THUIIIc.

H. M. Crespo, J. R. Birge, E. L. Falcão-Filho, M. Y. Sander, A. Benedick, and F. X. Kärtner, “Nonintrusive phase stabilization of sub-two-cycle pulses from a prismless octave-spanning Ti:sapphire laser,” Opt. Lett. 33, 833-835 (2008).
[CrossRef]

H. M. Crespo, J. R. Birge, M. Y. Sander, E. L. Falcão-Filho, A. Benedick, and F. X. Kärtner, “Phase stabilization of sub-two-cycle pulses from prismless octave-spanning Ti:sapphire lasers,” J. Opt. Soc. Am. B 25, B147-B154 (2008).
[CrossRef]

M. V. Tognetti, M. N. Miranda, and H. M. Crespo, “Dispersion-managed mode-locking dynamics in a Ti:sapphire laser,” Phys. Rev. A 74, 033809 (2006).
[CrossRef]

Cundiff, S. T.

Q. Quraishi, S. T. Cundiff, B. Ilan, and M. J. Ablowitz, “Dynamics of nonlinear and dispersion managed solitons,” Phys. Rev. Lett. 94, 243904 (2005).
[CrossRef]

Diddams, S. A.

Eggleton, B. J.

G. Lenz, B. J. Eggleton, C. R. Giles, C. K. Madsen, and R. E. Slusher, “Dispersive properties of optical filters for WDM systems,” IEEE J. Quantum Electron. 34, 1390-1402 (1998).
[CrossRef]

Ell, R.

Falcão-Filho, E. L.

Fortier, T. M.

Fujimoto, F. G.

Fujimoto, J. G.

Gatz, S.

Giles, C. R.

G. Lenz, B. J. Eggleton, C. R. Giles, C. K. Madsen, and R. E. Slusher, “Dispersive properties of optical filters for WDM systems,” IEEE J. Quantum Electron. 34, 1390-1402 (1998).
[CrossRef]

Harth, A.

Haus, H.

H. Haus, “Parameter ranges for CW passive mode locking,” IEEE J. Quantum Electron. 12, 169-176 (1976).
[CrossRef]

Haus, H. A.

Herrmann, J.

Huang, C.-P.

Ilan, B.

Q. Quraishi, S. T. Cundiff, B. Ilan, and M. J. Ablowitz, “Dynamics of nonlinear and dispersion managed solitons,” Phys. Rev. Lett. 94, 243904 (2005).
[CrossRef]

Ippen, E. P.

Jirauschek, C.

Kalosha, V. P.

Kapteyn, H. C.

Kärtner, F.

Kärtner, F. X.

M. Y. Sander, H. M. Crespo, J. R. Birge, and F. X. Kärtner, “Modeling of octave-spanning sub-two-cycle titanium:sapphire lasers: simulation and experiment,” in Conference on Ultrafast Phenomena (European Physical Society and Optical Society of America, 2008), paper THUIIIc.

H. M. Crespo, J. R. Birge, E. L. Falcão-Filho, M. Y. Sander, A. Benedick, and F. X. Kärtner, “Nonintrusive phase stabilization of sub-two-cycle pulses from a prismless octave-spanning Ti:sapphire laser,” Opt. Lett. 33, 833-835 (2008).
[CrossRef]

H. M. Crespo, J. R. Birge, M. Y. Sander, E. L. Falcão-Filho, A. Benedick, and F. X. Kärtner, “Phase stabilization of sub-two-cycle pulses from prismless octave-spanning Ti:sapphire lasers,” J. Opt. Soc. Am. B 25, B147-B154 (2008).
[CrossRef]

J. R. Birge, H. Crespo, M. Sander, and F. X. Kärtner, “Non-intrusive sub-two-cycle carrier-envelope stabilized pulses using engineered chirped mirrors,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies (2008), paper CTuC3.

S. Rausch, T. Binhammer, A. Harth, J. Kim, R. Ell, F. X. Kärtner, and U. Morgner, “Controlled waveforms on the single-cycle scale from a femtosecond oscillator,” Opt. Express 16, 9739-9745 (2008).
[CrossRef]

J. R. Birge and F. X. Kärtner, “Efficient optimization of multilayer coatings for ultrafast optics using analytic gradients of dispersion,” Appl. Opt. 46, 2656-2662 (2007).
[CrossRef]

A. Benedick, J. Birge, R. Ell, O. D. Mücke, M. Sander, and F. X. Kärtner, “Octave spanning 1 GHz Ti:sapphire oscillator for HeNe CH4-based frequency combs and clocks,” in Proceedings of the European Conference on Lasers and Electro-Optics (IEEE, 2007) paper CF3-1-MON.

J. R. Birge, R. Ell, and F. X. Kärtner, “Two-dimensional spectral shearing interferometry for few-cycle pulse characterization,” Opt. Lett. 31, 2063-2065 (2006).
[CrossRef]

J. R. Birge and F. X. Kärtner, “Efficient analytic computation of dispersion from multilayer structures,” Appl. Opt. 45, 1478-1483 (2006).
[CrossRef]

R. Ell, J. R. Birge, M. Araghchini, and F. X. Kärtner, “Carrier-envelope phase control by a composite plate,” Opt. Express 14, 5829-5837 (2006).
[CrossRef]

O. D. Mücke, R. Ell, A. Winter, J. Kim, J. R. Birge, L. Matos, and F. X. Kärtner, “Self-referenced 200 MHz octave-spanning Ti:sapphire laser with 50 attosecond carrier-envelope phase jitter,” Opt. Express 13, 5136-5169 (2005).
[CrossRef]

L. Matos, D. Kleppner, O. Kuzucu, T. R. Schibli, J. Kim, E. P. Ippen, and F. X. Kärtner, “Direct frequency comb generation from an octave-spanning, prismless Ti:sapphire laser,” Opt. Lett. 29, 1683-1685 (2004).
[CrossRef]

R. Ell, U. Morgner, F. X. Kärtner, F. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, T. Schudi, M. J. Lederer, A. Boiko, and B. Luther-Davies, “Generation of 5-fs pulses and octave-spanning spectra directly from a Ti:sapphire laser,” Opt. Lett. 26, 373-375 (2001).
[CrossRef]

Y. Chen, F. X. Kärtner, U. Morgner, S. H. Cho, H. A. Haus, E. P. Ippen, and J. G. Fujimoto, “Dispersion-managed mode locking,” J. Opt. Soc. Am. B 16, 1999-2004 (1999).
[CrossRef]

F. X. Kärtner, J. Aus der Au, and U. Keller, “Mode-locking with slow and fast saturable absorbers-what's the difference?” IEEE J. Sel. Top. Quantum Electron. 4, 159-168 (1998).
[CrossRef]

Kean, P. N.

Keller, U.

F. X. Kärtner, J. Aus der Au, and U. Keller, “Mode-locking with slow and fast saturable absorbers-what's the difference?” IEEE J. Sel. Top. Quantum Electron. 4, 159-168 (1998).
[CrossRef]

Kelly, S. M. J.

Kim, J.

Kleppner, D.

Krausz, F.

Kuzucu, O.

Lederer, M. J.

Lenz, G.

G. Lenz, B. J. Eggleton, C. R. Giles, C. K. Madsen, and R. E. Slusher, “Dispersive properties of optical filters for WDM systems,” IEEE J. Quantum Electron. 34, 1390-1402 (1998).
[CrossRef]

Luther-Davies, B.

Madsen, C. K.

G. Lenz, B. J. Eggleton, C. R. Giles, C. K. Madsen, and R. E. Slusher, “Dispersive properties of optical filters for WDM systems,” IEEE J. Quantum Electron. 34, 1390-1402 (1998).
[CrossRef]

Matos, L.

Miranda, M. N.

M. V. Tognetti, M. N. Miranda, and H. M. Crespo, “Dispersion-managed mode-locking dynamics in a Ti:sapphire laser,” Phys. Rev. A 74, 033809 (2006).
[CrossRef]

Moores, J. D.

Morgner, U.

Mücke, O. D.

A. Benedick, J. Birge, R. Ell, O. D. Mücke, M. Sander, and F. X. Kärtner, “Octave spanning 1 GHz Ti:sapphire oscillator for HeNe CH4-based frequency combs and clocks,” in Proceedings of the European Conference on Lasers and Electro-Optics (IEEE, 2007) paper CF3-1-MON.

O. D. Mücke, R. Ell, A. Winter, J. Kim, J. R. Birge, L. Matos, and F. X. Kärtner, “Self-referenced 200 MHz octave-spanning Ti:sapphire laser with 50 attosecond carrier-envelope phase jitter,” Opt. Express 13, 5136-5169 (2005).
[CrossRef]

Muijlwijk, R.

R. Muijlwijk, “Update of the Edlen's formulae for the refractive index of air,” Metrologia 25, 189 (1998).
[CrossRef]

Müller, M.

Murnane, M. M.

Nelson, L. E.

Quraishi, Q.

Q. Quraishi, S. T. Cundiff, B. Ilan, and M. J. Ablowitz, “Dynamics of nonlinear and dispersion managed solitons,” Phys. Rev. Lett. 94, 243904 (2005).
[CrossRef]

Rausch, S.

Sander, M.

J. R. Birge, H. Crespo, M. Sander, and F. X. Kärtner, “Non-intrusive sub-two-cycle carrier-envelope stabilized pulses using engineered chirped mirrors,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies (2008), paper CTuC3.

A. Benedick, J. Birge, R. Ell, O. D. Mücke, M. Sander, and F. X. Kärtner, “Octave spanning 1 GHz Ti:sapphire oscillator for HeNe CH4-based frequency combs and clocks,” in Proceedings of the European Conference on Lasers and Electro-Optics (IEEE, 2007) paper CF3-1-MON.

Sander, M. Y.

Scheuer, V.

Schibli, T. R.

Schudi, T.

Sibbett, W.

Slusher, R. E.

G. Lenz, B. J. Eggleton, C. R. Giles, C. K. Madsen, and R. E. Slusher, “Dispersive properties of optical filters for WDM systems,” IEEE J. Quantum Electron. 34, 1390-1402 (1998).
[CrossRef]

Spence, D. E.

Spielmann, Ch.

Stoev, V. D.

Tognetti, M. V.

M. V. Tognetti, M. N. Miranda, and H. M. Crespo, “Dispersion-managed mode-locking dynamics in a Ti:sapphire laser,” Phys. Rev. A 74, 033809 (2006).
[CrossRef]

Winter, A.

Zhou, J.

Appl. Opt.

IEEE J. Quantum Electron.

H. Haus, “Parameter ranges for CW passive mode locking,” IEEE J. Quantum Electron. 12, 169-176 (1976).
[CrossRef]

G. Lenz, B. J. Eggleton, C. R. Giles, C. K. Madsen, and R. E. Slusher, “Dispersive properties of optical filters for WDM systems,” IEEE J. Quantum Electron. 34, 1390-1402 (1998).
[CrossRef]

H. A. Haus, “Mode-locking of lasers,” IEEE J. Quantum Electron. 6, 1173-1185 (2000).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

F. X. Kärtner, J. Aus der Au, and U. Keller, “Mode-locking with slow and fast saturable absorbers-what's the difference?” IEEE J. Sel. Top. Quantum Electron. 4, 159-168 (1998).
[CrossRef]

J. Opt. Soc. Am. B

Metrologia

R. Muijlwijk, “Update of the Edlen's formulae for the refractive index of air,” Metrologia 25, 189 (1998).
[CrossRef]

Opt. Express

Opt. Lett.

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

(a) Experimental setup of 500 MHz laser. (b) Schematic model for ring laser used in numerical analysis. The numbering in the ring laser denotes the cavity position used in Fig. 3. (c) Model of linear laser cavity.

Fig. 2
Fig. 2

(a) DCM reflectivity and output coupler reflectivity for the 500 MHz ring laser. (b) Designed and measured group delay for each type of DCM. The measured data is determined by white-light interferometry at normal incidence angle. (c) Amplitude and phase of linear passive cavity transfer function versus wavelength.

Fig. 3
Fig. 3

(a) Pulse breathing for the 500 MHz ring laser for one cavity round trip starting in the geometric middle of the TiSa crystal. (b) Spectral breathing of the power spectral density (PSD) for the 500 MHz ring laser. The position in the cavity labeling refers to Fig. 1b.

Fig. 4
Fig. 4

Comparison of the simulated and measured main output spectra: (a) linear 200 MHz laser, (b) 1 GHz ring laser. Each PSD curve is normalized to its maximum amplitude.

Fig. 5
Fig. 5

(a) Main output spectrum for the 500 MHz ring laser. (b) 1 f 2 f output spectrum for the 500 MHz ring laser. In both cases the measured spectrum is compared with the simulated output spectra using the designed and experimentally determined DCM data. Each PSD curve is normalized to its maximum amplitude in the center part of the spectrum.

Fig. 6
Fig. 6

Main output spectrum for different dispersion compensation with varying Ba F 2 insertion for the 500 MHz laser, normalized to its maximum amplitude. A decrease of 0.1 mm of Ba F 2 corresponds to a GD of 4 fs and a decrease in the spectral wings around 1160 nm of 12 dB .

Fig. 7
Fig. 7

Output pulse for 500 MHz laser after external compression through a delay line with 2 DCM pairs and F S Ba F 2 material. The measured pulse is retrieved from two-dimensional spectral shearing interferometry (2DSI) measurements and shows excellent agreement with the simulated output pulse.

Equations (6)

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A ( z , t ) z = [ g ( z , t ) q ( z , t ) j δ A ( z , t ) 2 j φ ( j t ) ] A ( z , t ) ,
with
q ( z , t ) = q 0 1 + A ( z , t ) 2 P Sat , g ( z , t ) = g 0 1 + A ( z , t ) 2 d t W Sat ( Δ ω L ) 2 ω 2 + ( Δ ω L ) 2 .
A 9 ( ω ) = S ( ω ) e j P ( ω ) A 2 ( ω ) ,
S ( ω ) = R 2 ( ω ) R 1 ( ω ) T OC ( ω ) R 2 ( ω ) R 1 ( ω ) ,
P ( ω ) = φ 2 ( ω ) + φ 1 ( ω ) + φ FS ( ω ) + φ Ba F 2 ( ω ) + φ 2 ( ω ) + φ 1 ( ω ) + φ Air ( ω ) .

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