## Carrier-envelope phase dynamics and noise analysis in octave-spanning Ti:sapphire lasers

Optics Express, Vol. 14, Issue 6, pp. 2497-2511 (2006)

http://dx.doi.org/10.1364/OE.14.002497

Acrobat PDF (617 KB)

### Abstract

We investigate the carrier-envelope phase dynamics of octave-spanning Ti:sapphire lasers and perform a complete noise analysis of the carrier-envelope phase stabilization. We model the effect of the laser dynamics on the residual carrier-envelope phase noise by deriving a transfer function representation of the octave-spanning frequency comb. The modelled phase noise and the experimental results show excellent agreement. This greatly enhances our capability of predicting the dependence of the residual carrier-envelope phase noise on the feedback loop filter, the carrier-envelope frequency control mechanism and the pump laser used.

© 2006 Optical Society of America

## 1. Introduction

1. T. Udem, J. Reichert, R. Holzwarth, and T.W. Hänsch, “Accurate measurement of large optical frequency differences with a mode-locked laser,” Opt. Lett. **24**, 881–883 (1999). [CrossRef]

2. J. Ye, J. L. Hall, and S. A. Diddams, “Precision phase control of an ultrawide-bandwidth femtosecond laser: a network of ultrastable frequency marks across the visible spectrum,” Opt. Lett. **25**, 1675–1677 (2000). [CrossRef]

3. D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers and Direct Optical Frequency Synthesis,” Science **288**, 635–639 (2000). [CrossRef] [PubMed]

4. R. Holzwarth, T. Udem, T.W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical Frequency Synthesizer for Precision Spectroscopy,” Phys. Rev. Lett. **85**, 2264–2268 (2000). [CrossRef] [PubMed]

5. J. von Zanthier, T. Becker, M. Eichenseer, A.Y. Nevsky, C. Schwedes, E. Peik, H. Walther, R. Holzwarth, J. Reichert, T. Udem, T.W. Hänsch, P.V. Pokasov, M. N. Skvortsov, and S. N. Bagayev, “Absolute frequency measurement of the In^{+} clock transition with a mode-locked laser,” Opt. Lett. **25**, 1729–1731 (2000). [CrossRef]

6. M. Niering, R. Holzwarth, J. Reichert, P. Pokasov, T. Udem, M. Weitz, T.W. Hänsch, P. Lemonde, G. Santarelli, M. Abgrall, P. Laurent, C. Salomon, and A. Clairon, “Measurement of the Hydrogen 1S-2S Transition Frequency by Phase Coherent Comparison with a Microwave Cesium Fountain Clock,” Phys. Rev. Lett. **84**, 5496–5499 (2000). [CrossRef] [PubMed]

7. T. Udem, S. A. Diddams, K. R. Vogel, C.W. Oates, E. A. Curtis, W. D. Lee, W. M. Itano, R. E. Drullinger, J. C. Bergquist, and L. Hollberg, “Absolute Frequency Measurements of the Hg^{+} and Ca Optical Clock Transitions with a Femtosecond Laser,” Phys. Rev. Lett. **86**, 4996–4999 (2001). [CrossRef] [PubMed]

8. A. Baltus̆ka, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T.W. Hänsch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature **421**, 611–615 (2003). [CrossRef] [PubMed]

*f*

_{CE}, of modelocked lasers is a prerequisite for many applications in both time and frequency domain. For octave-spanning lasers which do not use intracavity prisms for dispersion compensation [9

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

10. 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**, 5163–5169 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-13-5163. [CrossRef] [PubMed]

*f*

_{CE}to intracavity power, i.e., by controlling the intracavity pulse energy via modulation of the pump power [11

11. L. Xu, C. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T.W. Hänsch, “Route to phase control of ultrashort light pulses,” Opt. Lett. **21**, 2008–2010 (1996). [CrossRef] [PubMed]

12. S. T. Cundiff, J. Ye, and J. L. Hall, “Optical frequency synthesis based on mode-locked lasers,” Rev. Sci. Instrum. **72**, 3749–3771 (2001). [CrossRef]

## 2. Carrier-envelope phase dynamics of mode-locked lasers

_{CE}and, therefore, contribute to the carrier-envelope frequency via

*f*

_{CE}, i.e.,

*fn*=

*n*

*f*

_{rep}+

*f*

_{CE}, with integer

*n*[12

12. S. T. Cundiff, J. Ye, and J. L. Hall, “Optical frequency synthesis based on mode-locked lasers,” Rev. Sci. Instrum. **72**, 3749–3771 (2001). [CrossRef]

*f*

_{c}, then the complex carrier wave of the pulse is given by

*v*

_{p}is the phase velocity of the carrier wave in the cavity. In the absence of nonlinearities the phase velocity is simply the ratio between frequency and wavenumber due to the linear refractive index of the media in the cavity, i.e.,

*v*

_{p}=

*v*

_{p}(

*f*

_{c}) = 2π

*f*

_{c}=

*k*(

*f*

_{c}). The envelope of a pulse that builds up in the cavity due to the mode-locking process will travel at the group velocity due to the presence of the linear media given by

*v*

_{g}=

*v*

_{g}(

*fc*) = 2π[d

*k*(

*fc*)= d

*fc*]

^{-1}. Therefore, after one round trip of the pulse over a distance 2

*L*

_{cav}, which takes the time

*T*

_{R}= 2

*L*

_{cav}=

*v*

_{g}, we obtain from Eq. (2) that the linear contribution to the carrier-envelope phase shift caused by the difference between phase and group velocities is

13. K.W. Holman, R. J. Jones, A. Marian, S. T. Cundiff, and J. Ye, “Detailed Studies and Control of Intensity- Related Dynamics of Femtosecond Frequency Combs From Mode-Locked Ti:Sapphire Lasers,” IEEE J. Sel. Top. Quantum Electron. **9**, 1018–1024 (2003). [CrossRef]

14. H. A. Haus and E. P. Ippen, “Group velocity of solitons,” Opt. Lett. **26**, 1654–1656 (2001). [CrossRef]

15. H. A. Haus, “Mode-Locking of Lasers,” IEEE J. Sel. Top. Quantum Electron. **6**, 1173–1185 (2000). [CrossRef]

*D*

_{irrev}is an operator that describes the irreversible dynamics occurring in a mode-locked laser such as gain, loss and saturable absorption.

*A*≡

*A*(

*T*,

*t*) is the slowly varying field envelope whose shape is investigated on two time scales: first, the global time

*T*which is coarse grained on the time scale of the round-trip time

*T*

_{R}, and second, the local time

*t*which resolves the resulting pulse shape.

*A*(

*T*,

*t*) is normalized such that |

*A*(

*T*,

*t*)|

^{2}is the instantaneous power and ∫d

*t*|

*A*(

*T*,

*t*)|

^{2}the pulse energy at time

*T*.

_{c})

*n*

_{2}

*L*/

*A*

_{eff}, where λ

_{c}is the carrier wavelength,

*n*

_{2}is the nonlinear index in cm/W,

*L*is the path length per round trip through the laser crystal, and

*A*

_{eff}is the effective mode cross-sectional area. Strictly speaking Eq. (5) only applies to a laser with small changes in pulse shape within one round trip. Obviously this is not the case for few-cycle laser pulses where the pulse formation is governed by dispersion-managed mode locking [16

16. 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]

15. H. A. Haus, “Mode-Locking of Lasers,” IEEE J. Sel. Top. Quantum Electron. **6**, 1173–1185 (2000). [CrossRef]

14. H. A. Haus and E. P. Ippen, “Group velocity of solitons,” Opt. Lett. **26**, 1654–1656 (2001). [CrossRef]

14. H. A. Haus and E. P. Ippen, “Group velocity of solitons,” Opt. Lett. **26**, 1654–1656 (2001). [CrossRef]

18. H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing: a linearized approach,” J. Opt. Soc. Am. B **7**, 386–392 (1990). [CrossRef]

16. 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]

_{2}material in the laser and observing which peak moved up in frequency (adding dispersion causes

*v*

_{g}=

*v*

_{p}to decrease, thus increasing the magnitude of the second term in Eq. (11)). We then varied the pump power and observed that the same peak also moved up in frequency, confirming the prediction by Eq. (11).

*f*

_{CE}to eventually shift in the opposite direction. This is a consequence of the fact that the Kerr lens mode-locking (KLM) action does not increase indefinitely, i.e., there is an upper value for the pulse energy above which a further increase in pump power will either contribute to cw breakthrough or to multiple pulses. Therefore, care must be taken to operate at an optimum pump power level which is significantly below this threshold value for single pulse instabilities. From the data shown in Fig. 1, the pump power to carrier-envelope frequency conversion coefficient for the 200 MHz lasers is

*C*

_{fPp}= 11MHz/W. Fig. 1 also shows the relative change of the intracavity pulse energy as a function of the same variation in pump power. The appearance of a cw component is also explicitly indicated in this measurement by the abrupt change in the observed slope. The shallow slope of the change in the average power in pulsed operation as compared to the change in power in cw operation is an indication of the strength of the saturable absorption and the bandwidth limitation of the laser. From this data, we can infer a relatively weak response in the change of the pulse energy in mode-locked operation and therefore a correspondingly weak response in the

*f*

_{CE}change, which in fact will be confirmed in the transfer function analysis discussed in the next section.

*f*

_{CE}in Fig. 1 with the theoretically derived result from soliton perturbation theory. It turns out that the measurement and theory agree very well, despite the fact that the laser dynamics differ from the ideal conventional soliton operation regime.

*P*

_{intra}is the intracavity power in mode-locked operation, can now be determined in terms of known cavity parameters under the assumption of a fixed pulse width

*L*= 4mm is the path length per round trip through the Ti:sapphire crystal,

*n*

_{2}= 3 × 10

^{-20}m

^{2}/W is the nonlinear index of refraction for Ti:sapphire, λ

_{c}= 800nm is the carrier wavelength,

*A*

_{eff}= π

*w*

_{0}= 16μm) is the mode cross sectional area, τ=τ

_{FWHM}=1.76 with a pulse width of τ

_{FWHM}= 5fs, and P

_{intra}= 12W is the intracavity power. This expression gives, for a 5% change in intracavity power, a corresponding change in

*f*

_{CE}of 9.6 MHz, which agrees well with the results shown in Fig. 1.

## 3. Noise analysis of carrier-envelope frequency stabilized lasers

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

10. 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**, 5163–5169 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-13-5163. [CrossRef] [PubMed]

_{4}pump laser (Verdi-V10, Coherent) and the other a multi-longitudinal-mode (mlm) Nd:YVO

_{4}pump laser (Millennia Xs, Spectra-Physics). Fig. 2 depicts the measured relative intensity noise (RIN) for the mlm pump laser and slm pump laser. The mlm pump laser shows significantly higher RIN in the high-frequency range, whereas the slm pump laser has higher RIN at very low frequencies. Recently, S.Witte

*et al*. [19

19. S. Witte, R. T. Zinkstok, W. Hogervorst, and K. S. E. Eikema, “Control and precise measurement of carrier-envelope phase dynamics,” Appl. Phys. B **78**, 5–12 (2004). [CrossRef]

*et al*., the high-frequency carrier-envelope phase noise is found to be larger for the mlm pump laser. The carrier-envelope phase fluctuations at lower frequencies, which are smaller for the mlm pump laser, are strongly suppressed by the large proportional-integral (PI) control loop gain and therefore do not contribute significantly to the residual carrier-envelope phase noise. The residual carrier-envelope phase fluctuations of the OSFC pumped by the mlm pump laser amount to 0.257 rad, compared to only 0.117 rad if the slm laser is used. First of all it is surprising that the mlm pumped system is only a factor of 2.2 worse than the slm pumped system despite the fact that the high-frequency noise of the mlm pump is worse. As we will see this is so because the feedback gain is large below 100 kHz. Obviously the system pumped by the mlm pump laser could easily do equally well if the feedback-loop bandwidth could be extended by one order of magnitude. The reasons why the high-frequency noise of the mlm pump can not be further suppressed will be elaborated further in the following feedback analysis.

*f*

_{CE}-stabilized laser is a phase-lock loop (PLL) [20

20. F. M. Gardner, *Phaselock Techniques* (John Wiley, 3rd edition, Hoboken, NJ, 2005). [CrossRef]

*f*

_{CE}is determined by the cavity parameters and alignment, equivalent to the center frequency of oscillation of the VCO in a PLL. A voltage applied to the acousto-optic modulator (AOM) driver changes this frequency by an amount proportional to the equivalent VCO gain of the system. The model depicted in Fig. 4 includes all the electronic components used in the stabilization (phase detector, AOM and loop filter), whose transfer characteristics are easy to measure and to describe by an analytic model. Assuming an instantaneous response of the carrier-envelope frequency to the pump power via a constant

*C*

_{fPp}, one is not able to reproduce the measured carrier-envelope phase noise spectrum. Therefore, the impact of the frequency response of the OSFC system must be taken into account in the analysis, which was done by considering the transfer function between the intracavity laser power (or pulse energy) and the pump power via the laser gain dynamics.

### 3.1. Transfer function representation for the pulse energy versus pump power dynamics

21. F. X. Kärtner, L. R. Brovelli, D. Kopf, M. Kamp, I. Calasso, and U. Keller, “Control of solid state laser dynamics by semiconductor devices,” Opt. Eng. **34**, 2024–2036 (1995). [CrossRef]

22. F. X. Kärtner, I. D. Jung, and U. Keller, ”Soliton Mode-Locking with Saturable Absorbers,” IEEE J. Sel. Top.Quantum Electron. **2**, 540–556 (1996). [CrossRef]

*T*_{R}= cavity round-trip time,- τ
_{L}= upper state lifetime, *l*= total non-saturable loss,*q*(*E*) ≡*q*_{ml}(*E*) = effective energy-dependent saturable absorber and filter loss (for more details, e.g., see Ref. [22]),22. F. X. Kärtner, I. D. Jung, and U. Keller, ”Soliton Mode-Locking with Saturable Absorbers,” IEEE J. Sel. Top.Quantum Electron.

**2**, 540–556 (1996). [CrossRef]*g*_{0}= small-signal gain, which is proportional to pump power,*E*_{sat}= saturation energy of the gain medium.

_{stim}is the stimulated lifetime given by τ

_{L}(1+τ

_{L}

*E*

_{s}=

*T*

_{R}

*E*

_{sat})

^{-1}. By taking the Laplace transform of the above equations, it is straightforward to derive a

*pump power*to

*pulse energy*transfer function, by writing

*g*

_{0}as

*K*

_{g0}

*P*

_{p}. Defining the pump parameter

*r*= 1+τ

_{L}

*E*

_{s}=

*T*

_{R}

*E*

_{sat}, which indicates how many times the laser operates above threshold, we arrive at

_{p}=

*T*

_{R}=

*l*is the photon decay time due to the linear cavity losses in cw operation. So far we have considered the mode-locked case, but a similar relation can be obtained for the intracavity power in cw operation simply by setting the saturable absorber terms in Eq. (18) to zero. However, the laser parameters, like mode cross section in the gain medium likely change values when the laser changes from cw operation to mode-locked operation, but since we have no way of measuring them in mode-locked operation, we have estimated them based on our knowledge of laser parameters in cw operation and used those to examine the effect of the inclusion of the saturable absorber into the transfer function. Fig. 6 shows the pump power to intracavity power transfer functions, in amplitude and phase, for cw and mode-locked operation for different values of the term

*q*/∂

*E*> 0) intracavity energy fluctuations. In cw operation its absence usually leads to pronounced relaxation oscillations, see Fig. 6. As we can see, the stronger the effective inverse saturable absorption, the more damped become the relaxation oscillations in the laser, and the weaker becomes the response at all frequencies. This result has to be expected, because the stronger the inverse saturable absorption the more clamped become the pulse energy and the average power.

*q*(

*E*) = 0. The parameter

*r*was measured to be 3.22, while the intracavity loss,

*l*, which determines the value of τ

_{p}, was determined by matching the relaxation oscillation frequency in the model with the measured result. The value obtained was

*l*=0.22.

*K*

_{g0}was then calculated using the relationship

*g*

_{0}=

*K*

_{g0}

*P*

_{p}=

*rl*= 0.15. As can be seen in Fig. 8, the model describes well the gain dynamics in cw operation. For the mode-locked case, we included the effect of the saturable absorber and modified the other parameters in such a way as to match the measured transfer function as close as possible, especially for frequencies beyond 10kHz, where the impact on the final noise calculation is most pronounced. This was achieved by setting

*q*

_{s}=

*l*,

*l*= 0.17,

*r*= 3.5 and

*K*

_{g0}= 0.23. Fig. 8 shows that the approximations made in the model do not fully describe the system, i.e., we neglect the interaction with the continuum, which is an infinite-dimensional system. In the measurements, a significant change in the amplitude response from low to high frequencies, in the 1–100 kHz range, is observed in mode-locked operation, which is an indication of additional slow processes occurring in the mode-locked laser that are absent in the cw laser. When pushing the laser to octave-spanning operation, i.e., for the shortest pulse and widest spectrum, one is always pushing the laser towards its stability boundary resulting in modes that approach zero damping time, i.e., dynamics with time constants of many round trips. These modes, which are neglected in the analysis, are most likely responsible for the deviations in the low-frequency range of the laser when mode locked. Nevertheless, the model gives good qualitative and quantitative description of the laser dynamics and the transfer function mimics the global behavior of the measured transfer function while still being simple. It confirms our observations on the strength of the saturable absorption in such systems, explicit in the measurements shown in Fig. 1. As will be shown in the next section, the inclusion of this transfer function in the noise analysis is essential in deriving the correct noise behavior of the system.

### 3.2. Determination of the carrier-envelope phase error

_{P}(

*s*), which is the RIN multiplied by the square of the pump power, is converted to the carrier-envelope phase noise spectral density

*S*

_{ϕ}(

*s*) in the laser and is partially suppressed in the feedback loop. The feedback path consists of the phase detector, the loop filter and the AOM, with transfer functions denoted by

*H*

_{PD}(

*s*),

*H*

_{LF}(

*s*) and

*H*

_{AOM}(

*s*), respectively. Table 1 shows the corresponding analytic expressions. The loop filter consists of a simple PI controller with time constants τ

_{1}, τ

_{2}, and τ

_{3}. The fused-silica AOM used in our experiments has a 5mm aperture, and the unfocused pump beam of 2.3mm diameter is placed as close as possible to the piezo-electric transducer. Then, to first order and up to a small drop in amplitude for higher frequencies, the AOM is equivalent to a delay line with a propagation delay given by the time it takes for the acoustic wave to travel from the piezo-electric transducer to the optical beam. We set this delay to 1.73μs to match the measured transfer function. The limitations of the bandwidth of the AOM are not taken into account here because the measurements show that it is approximately flat out to 1 MHz. The calculated and measured transfer functions for the loop filter and the AOM are shown in Fig. 10. We consider the phase detector (Analog Devices AD9901) transfer function to be flat based on its datasheet.

**RIN**measurement in Fig. 2 as the input noise source, we calculate

*S*ϕ(

*s*) using the transfer functions in Table 1 to derive the closed loop transfer function that describes the conversion of pump noise

*S*

_{P}(

*s*) to carrier-envelope phase noise

*S*ϕ(

*s*),

*S*ϕ(

*s*) is obtained by multiplying the intensity noise spectrum of the pump laser,

*S*P(

*s*), with the magnitude squared of

*H*

_{CL}given by expression (19)

*H*

_{Tisa}(

*s*) are those which closely match the measured and calculated OSFC transfer function in mode-locked operation in the range between 10 kHz and 1 MHz. When only the pump noise is used as a noise source, we found that the calculated and measured

*S*ϕ(

*s*) showed good agreement up to 200 kHz, beyond which point the measurement showed enhanced noise. This was determined to be due to electronic noise at the output of the phase detector, which was then included as an additional noise source in the analysis to get a good fit beyond 200 kHz.We added this white noise source

*S*

_{PD}in the loop, which converts to a final contribution to

*S*ϕ(

*s*) by the closed-loop transfer function

*H*

_{AOM}(

*s*) and

*H*

_{Tisa}(

*s*) contribute to a decrease in the phase margin. However, now that the different phase contributions to the feedback loop are well understood, the noise suppression at high frequencies could be improved by custom design of the control electronics. For example, by adding a lead-lag compensator [23], the limitation arising from the added phases could be reduced. Furthermore, the contribution to the open-loop phase from the time delay of the AOM could be further suppressed by replacing the AOM by an electro-optic modulator (EOM) or using an AOM/EOM combination. The intensity noise in the frequency range 1 kHz-10MHz amounts to 0.06%, which could easily be reduced by an EOM (e.g., made of rubidium titanyl phosphate (RTP)) with a ~1V modulation amplitude. It remains to be seen if an AOM/EOM combination indeed can further suppress the phase error.

## 4. Conclusion

## Acknowledgements

## References and links

1. | T. Udem, J. Reichert, R. Holzwarth, and T.W. Hänsch, “Accurate measurement of large optical frequency differences with a mode-locked laser,” Opt. Lett. |

2. | J. Ye, J. L. Hall, and S. A. Diddams, “Precision phase control of an ultrawide-bandwidth femtosecond laser: a network of ultrastable frequency marks across the visible spectrum,” Opt. Lett. |

3. | D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers and Direct Optical Frequency Synthesis,” Science |

4. | R. Holzwarth, T. Udem, T.W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical Frequency Synthesizer for Precision Spectroscopy,” Phys. Rev. Lett. |

5. | J. von Zanthier, T. Becker, M. Eichenseer, A.Y. Nevsky, C. Schwedes, E. Peik, H. Walther, R. Holzwarth, J. Reichert, T. Udem, T.W. Hänsch, P.V. Pokasov, M. N. Skvortsov, and S. N. Bagayev, “Absolute frequency measurement of the In |

6. | M. Niering, R. Holzwarth, J. Reichert, P. Pokasov, T. Udem, M. Weitz, T.W. Hänsch, P. Lemonde, G. Santarelli, M. Abgrall, P. Laurent, C. Salomon, and A. Clairon, “Measurement of the Hydrogen 1S-2S Transition Frequency by Phase Coherent Comparison with a Microwave Cesium Fountain Clock,” Phys. Rev. Lett. |

7. | T. Udem, S. A. Diddams, K. R. Vogel, C.W. Oates, E. A. Curtis, W. D. Lee, W. M. Itano, R. E. Drullinger, J. C. Bergquist, and L. Hollberg, “Absolute Frequency Measurements of the Hg |

8. | A. Baltus̆ka, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T.W. Hänsch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature |

9. | L. Matos, D. Kleppner, O. Kuzucu, T. R. Schibli, J. Kim, E. P. Ippen, and F. X. Kaertner, “Direct frequency comb generation from an octave-spanning, prismless Ti:sapphire laser,” Opt. Lett. |

10. | 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 |

11. | L. Xu, C. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T.W. Hänsch, “Route to phase control of ultrashort light pulses,” Opt. Lett. |

12. | S. T. Cundiff, J. Ye, and J. L. Hall, “Optical frequency synthesis based on mode-locked lasers,” Rev. Sci. Instrum. |

13. | K.W. Holman, R. J. Jones, A. Marian, S. T. Cundiff, and J. Ye, “Detailed Studies and Control of Intensity- Related Dynamics of Femtosecond Frequency Combs From Mode-Locked Ti:Sapphire Lasers,” IEEE J. Sel. Top. Quantum Electron. |

14. | H. A. Haus and E. P. Ippen, “Group velocity of solitons,” Opt. Lett. |

15. | H. A. Haus, “Mode-Locking of Lasers,” IEEE J. Sel. Top. Quantum Electron. |

16. | 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 |

17. | G. P. Agrawal, |

18. | H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing: a linearized approach,” J. Opt. Soc. Am. B |

19. | S. Witte, R. T. Zinkstok, W. Hogervorst, and K. S. E. Eikema, “Control and precise measurement of carrier-envelope phase dynamics,” Appl. Phys. B |

20. | F. M. Gardner, |

21. | F. X. Kärtner, L. R. Brovelli, D. Kopf, M. Kamp, I. Calasso, and U. Keller, “Control of solid state laser dynamics by semiconductor devices,” Opt. Eng. |

22. | F. X. Kärtner, I. D. Jung, and U. Keller, ”Soliton Mode-Locking with Saturable Absorbers,” IEEE J. Sel. Top.Quantum Electron. |

23. | G. F. Franklin, J. D. Powell, and A. Emami-Naeini, |

**OCIS Codes**

(120.3940) Instrumentation, measurement, and metrology : Metrology

(320.7090) Ultrafast optics : Ultrafast lasers

(320.7160) Ultrafast optics : Ultrafast technology

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: November 22, 2005

Revised Manuscript: March 13, 2006

Manuscript Accepted: March 14, 2006

Published: March 20, 2006

**Citation**

Lia Matos, Oliver D. Mücke, Jian Chen, and Franz X. Kärtner, "Carrier-envelope phase dynamics and noise analysis in octave-spanning Ti:sapphire lasers," Opt. Express **14**, 2497-2511 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-6-2497

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### References

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