## 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

Carrier-envelope phase control of femtosecond frequency combs has enabled major progress in frequency metrology, high-resolution laser spectroscopy, optical clocks and in high-field physics [1, 2, 3, 4, 5, 6, 7, 8]. Active control of the carrier-envelope frequency, *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, 10] this control can be achieved by utilizing the response of *f*
_{CE} to intracavity power, i.e., by controlling the intracavity pulse energy via modulation of the pump power [11, 12].

Detailed studies of the intensity-related *f*
_{CE} dynamics have been performed in Ti:sapphire mode-locked lasers which employ external fiber broadening for the generation of octave-spanning spectra and *f*
_{CE} control [13]. A connection between the dependence of the carrier-envelope
frequency, the laser repetition rate and the center frequency of the pulse spectrum on intracavity pulse energy was found. The experiments showed that the carrier-envelope phase control coefficient *C _{fPp}* = Δ

*f*

_{CE}=Δ

*P*

_{p}, where Δ

*P*

_{p}is a change in pump power, could change signs, consistent with the shifting of the spectrum with pump power also changing signs. In order to obtain optimum conditions for

*f*

_{CE}control, the dependence of

*C*

_{fPp}on intracavity power and, therefore, on pump power is of prime importance.

In octave-spanning lasers the carrier-envelope phase dynamics is much simpler. This is due to the fact that the center frequency of the pulse has no observable change with pump power, because the pulse spectrum fills up all the available bandwidth and is in fact limited by the bandwidth of the output couplers available. Then a change of the carrier-envelope frequency due to changes of the center frequency of the pulse is absent and the carrier-envelope frequency responds only to changes in the intracavity pulse energy and the concomitant changes in phase and group velocity related to the nonlinear refractive index. This response has a simple linear behavior, provided care is taken to work in a power region where the laser is operating in a unique single pulse regime, i.e., no continuous-wave (cw) component or multiple pulses are present.

In this paper, we present a full characterization of the intensity-related carrier-envelope phase dynamics in octave-spanning frequency combs (OSFCs) and its impact on carrier-envelope phase control. We begin by describing, in Section 2, the observation of the intensity-dependent shift of the carrier-envelope frequency and its connection with mode-locking theory. Then,motivated by our observations of different carrier-envelope phase jitter in two OSFCs pumped by different lasers, we present in Section 3 a complete noise analysis of a carrier-envelope phase stabilized system, which shows that the laser dynamics related to pulse energy and gain plays an important role in the overall control loop and needs to be taken into account if ultimately low residual carrier-envelope phase jitter is desired.

In order to include the gain dynamics in the noise analysis, a transfer function representation of the OSFC stabilization is given, which enables a quantitative noise analysis and eventually optimization of the overall system. It is shown that there is a major difference in the laser dynamics of the continuous-wave running laser and of the mode-locked laser, and that this dynamics has impact in the noise analysis. Measurement of the transfer function of the OSFC confirms the global behavior of the theoretical predictions and leads to excellent agreement between the computed and measured carrier-envelope phase dynamics and noise characteristics.

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

In this chapter we summarize the linear and nonlinear effects in the laser cavity that may lead to a carrier-envelope phase shift per round trip Δϕ_{CE} and, therefore, contribute to the carrier-envelope
frequency via

Thus, in the frequency domain, the pulse train emitted by a mode-locked laser corresponds to a comb of frequencies equidistantly spaced by the repetition frequency frep and offset from zero by the carrier-envelope frequency *f*
_{CE}, i.e., *fn* = *n*
*f*
_{rep} + *f*
_{CE}, with integer *n* [12]. If we assume that the laser operates at carrier frequency *f*_{c}
, then the complex carrier wave of the pulse is given by

where *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

and for the subsequent carrier-envelope frequency

In a dispersive medium, group and phase velocities depend on the carrier frequency. Therefore, if the carrier frequency shifts as a function of the intracavity pulse energy, the linear carrier-envelope frequency becomes energy and pump power dependent as found in Ref. [13].

In a mode-locked laser there are also nonlinear processes at work that may directly lead to an energy-depended carrier-envelope frequency. There are many effects that may contribute to such a shift. Here we re-derive briefly the effects due to the intensity-dependent refractive index as discussed by Haus and Ippen [14] for the case of a laser with strong soliton-like pulse shaping which can be evaluated analytically using soliton perturbation theory. We then argue that the same analysis holds for the general case where steady-state pulse formation is different from conventional soliton pulse shaping.

We start from the description of a mode-locked laser by a master equation of the form

where we have already factored out the carrier wave [15]. Here, *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*. ${D}_{2}=\frac{{d}^{2}k}{d{f}_{\text{c}}^{2}}\frac{{L}_{\mathrm{cav}}}{{8\pi}^{2}}$ is the group-velocity dispersion (GVD) parameter for the cavity. The Kerr coefficient is δ = (2π=λ_{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]. Nevertheless we want to understand this propagation equation as an effective equation of motion for the laser, where some of the parameters need to be determined self-consistently [15].

Let us assume that the laser operates in the negative GVD regime, where a conventional soliton-like pulse forms, and that it is stabilized by the effective saturable absorber action against the filtering effects. Then the steady-state pulse solution is close to a fundamental soliton, i.e., a symmetric sech-shaped pulse that acquires an energy-dependent nonlinear phase shift per round trip due to the nonlinear index

see Ref. [14]. The nonlinear phase shift per round trip is

A more careful treatment of the influence of the Kerr effect on the pulse propagation, especially for few-cycle pulses, needs to take the self-steepening of the pulse into account, i.e., the variation of the index during an optical cycle, by adding to the master equation the term [17]

We emphasize that this term is a consequence of the Kerr effect and is not related to soliton propagation. It can be viewed as a perturbation to the master equation (5). For pulses with τ much longer than an optical cycle, this self-steepening term is unimportant in pulse shaping, because it is on the order of 1/ω_{c}τ « 1. However, this term is always of importance when the phase shifts acquired by the pulse during propagation are considered. Haus and Ippen found, by using soliton perturbation theory based on the eigensolutions of the unperturbed linearized Schrödinger equation, analytic expressions for the changes in phase and group velocity. Obviously, the nonlinear phase shift per round trip of the soliton adds an additional phase shift to the pulse in each round trip.

If the term in (8) is applied to a real and symmetric waveform, it generates an odd waveform. An odd waveform added as a perturbation to the symmetric waveform of the steady-state pulse leads, to first order, to a temporal shift of the steady-state pulse. For a soliton-like steady-state solution this timing shift can be evaluated with soliton perturbation theory, i.e., using the basis functions of the linearized operator, and results in a timing shift [14, 18]

In total, the compound effect of self-phase modulation, self-steepening, and linear dispersion on the pulse results in a carrier-envelope frequency of

$$\phantom{\rule{1em}{0ex}}=-\frac{{f}_{\text{R}}}{4\pi}{\mathit{\delta A}}_{0}^{2}+2\frac{{f}_{\text{R}}}{4\pi}{\mathit{\delta A}}_{0}^{2}+{f}_{\text{c}}\left(1-\frac{{v}_{g}\left({f}_{\text{c}}\right)}{{v}_{p}\left({f}_{\text{c}}\right)}\right).$$

As the above expression shows, the term arising from the group delay change due to self-steepening is twice as large and of opposite sign compared with the one due to self-phase modulation. In total we obtain

We emphasize that soliton perturbation theory was only used in this derivation for analytical evaluation of timing shifts. If the pulse shaping in the laser is not governed by conventional soliton formation but rather by dispersion-managed soliton dynamics [16] or a saturable absorber, the fundamental physics stays the same. If the steady-state solution has a real and symmetric component, the self-steepening term converts this component via the derivative into a real and odd term, which is to first order a timing shift in the autonomous dynamics of the free running mode-locked laser. Another mechanism that leads to a timing shift is, for example, the action of a slow saturable absorber, which absorbs only the front of the pulse. So care needs to be taken to include all relevant effects when a given laser system is analyzed.

The derivation above shows that the group velocity change due to self-steepening of the pulse leads to a change in sign of the energy-dependent contribution at fixed center wavelength of the pulse. We checked this prediction by observing the carrier-envelope frequency shift in a 200MHz repetition rate octave-spanning laser. We identified which of the peaks in the RF spectrum corresponded to the carrier-envelope frequency by inserting BaF_{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).

Fig. 1(a) shows the carrier-envelope frequency shift as a function of pump power. As Eq. (5) predicts, the carrier-envelope frequency shift follows linearly the pump power over the range where the intracavity laser power (or pulse energy) depends linearly on the pump power. This is the case as long the laser operates in a unique single pulse regime. For higher pump powers, cw background radiation breaks through and the theory on which Eq. (11) is based no longer holds because the intracavity power is now divided between two components, the pulse and the cw solution, see Fig. 1(b) and (c). The observed turning point is not inherent to the pulse dynamics itself, but is due to the appearance of a cw component in the spectrum, as is easy to see in Fig. 1. After the cw component is present, any increase in pump power enhances the cw component and most likely decreases the pulse energy, causing *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.

Now, we can compare quantitatively the measured shift of *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.

The conversion coefficient

where *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

where *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${w}_{0}^{2}$ (*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.

Despite this surprisingly good agreement, one has to be aware that the spot size and other parameters are rough estimates, which may easily change depending on cavity alignment. Also the pulse width is not constant in the crystal but rather stretching and compressing by more than a factor of 2. Nevertheless, the experimental observations in Fig. 1 agree well with the above theoretical estimate obtained from conventional soliton formation.

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

The design and overall setup of carrier-envelope phase stabilized OSFC has been extensively discussed in Refs. [9, 10]. We have built two of these OSFCs at 200MHz repetition rate with slightly different loop filter designs and pumped by different pump lasers. One is using a single-longitudinal-mode (slm) Nd:YVO_{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] also characterized the influence of RIN of these pump lasers on the residual carrier-envelope phase noise for a 10-fs Ti:sapphire laser employing chirped mirrors for intracavity dispersion compensation and external spectral broadening in a microstructure fiber, however, no rigorous noise analysis has been performed so far. The purpose of this section is to elucidate the impact of the RIN of different pump lasers on the finally achievable carrier-envelope phase noise and how the feedback mechanism and the design of the feedback loop employed impacts residual carrier-envelope phase noise.

Fig. 3 shows the corresponding spectrally-resolved and integrated carrier-envelope phase error measured for the two 200MHz OSFCs which are directly carrier-envelope phase locked. In
agreement with the data measured by S. Witte *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.

From a control systems point of view, the *f*
_{CE}-stabilized laser is a phase-lock loop (PLL) [20], where the voltage-controlled oscillator (VCO) is the carrier-envelope frequency controlled OSFC, which is the block indicated by the dashed frame in Fig. 4. When the laser is turned on, the carrier-envelope frequency *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

The starting point for derivation of the transfer function are laser rate equations for pulse energy and gain, which can be derived from the master equation (5) by proper elimination of the remaining degrees of freedom in the mode-locked laser as has been derived for example in the case of soliton lasers mode locked by slow saturable absorbers [21, 22]. One can write

where we have used:

*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]),*g*_{0}= small-signal gain, which is proportional to pump power,*E*_{sat}= saturation energy of the gain medium.

Many assumptions have been made when using these equations to describe the energy and gain dynamics. For example, possible frequency shifts and back action of the background radiation onto the energy and gain dynamics, i.e., the details of the pulse shaping mechanism, are neglected. We have denoted the time coordinate as capital *T* to emphasize the fact that this dynamics occurs on the time scale of many cavity round trips. This is possible for solidstate laser gain media because the interaction cross section is small and therefore the gain saturates with average power rather than with a single passage of the pulse through the gain medium [21]. Note that the most important term in the rate equations is the mode locking related energy-dependent loss *q*
_{ml}(*E*), which comprises the loss during saturation of the absorber as well as additional losses due to the bandwidth limitations of the system, which increase with additional spectral broadening or increasing intracavity pulse energy [22]. A typical characteristic dependence of *q*
_{ml} on intracavity pulse energy *E* is shown in Fig. 5. To
derive a transfer function for the laser, we linearize Eqs. (14) and (15) around the steadystate operating point, denoted by the subscript s: *E* = *E*
_{s} +Δ*E*, *g* = *g*s +Δ*g*, *g*
_{0} = *g*
_{0s} +Δ*g*
_{0}, $q\left(E\right)=q\left({E}_{\text{s}}+\Delta E\right)\approx q\left({E}_{\text{s}}\right)+\frac{\partial q}{\partial E}{|}_{E={E}_{\text{s}}}\Delta E\equiv {q}_{\text{s}}+\frac{\partial q}{\partial {E}_{\text{s}}}\Delta E$ to get the set of linearized equations

where τ_{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

where τ_{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 $\frac{\partial q}{\partial {E}_{\text{s}}}{P}_{\text{s}}$. It is obvious that the mode locking of the laser drastically changes the transfer characteristic between pump power and intracavity power due to the pulse operation, which introduces the term $\frac{\partial q}{\partial {E}_{\text{s}}}{P}_{\text{s}}$ into the rate equation (16). Depending on its sign this term enhances (for ∂q/∂E < 0 it may lead to Q-switching when large enough) or strongly
damps (for ∂*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.

The model is verified by measuring the transfer function of the laser in cw and mode-locked operation. The measurement is performed by using a network analyzer as shown in Fig. 7. Care has to be taken to assure that no cw component is present in mode-locked operation during the measurements. The results are shown in Fig. 8. Also shown in the same plots are the amplitude and phase of the transfer function given by Eq. (18). For the cw case, *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 $\frac{\partial q}{\partial {E}_{\text{s}}}{P}_{\text{s}}=\frac{150r}{{\tau}_{L}}$, *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

To calculate the carrier-envelope phase noise spectrum of the OSFC, a linear noise analysis is performed. The block diagram in Fig. 9 shows the closed-loop system. The input noise source, characterized by the power spectral density of the pump noise S_{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.

In the noise analysis we consider the case of the OSFC pumped by the multi-longitudinal-mode pump laser because of the increased high-frequency noise in such systems as discussed above. Given the **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*),

Then, *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)

The calculated and measured carrier-envelope phase noise are shown in Fig. 11. Already given in the previous section, the values for the parameters used in *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

where the magnitude of the noise source SPD was estimated based on the measurement. Improved noise performance could be obtained by increasing the closed-loop bandwidth, which is currently about 100 kHz. This bandwidth is dictated by the phase margin of the feedback loop, which according to the Nyquist theorem may run unstable when the gain is larger than 1 while the phase approaches 180° [23]. From a control systems perspective, the VCO has integrating behavior, i.e., it integrates a frequency deviation into a phase deviation, thus causing the open-loop transfer function to start off with a -90° phase, as can be seen in Fig. 12. When the bandwidth of the gain medium is approached, additional phase accumulates from the gain dynamics and the time delay from the AOM (see Fig. 12), which renders the system unstable if the gain setting is not properly reduced. Of course, for a reasonably stable system and optimum operation of the feedback loop, large enough gain and phase margins are necessary. This imposes a limitation to the maximum loop gain since the jointly added phase from *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

We have performed a complete characterization and analysis of the intensity-related carrier-envelope phase dynamics in OSFCs and our studies confirm the validity of the theoretical treatment by Haus and Ippen of soliton-like propagation in the few-cycle pulse regime. The inclusion of the self-steepening term in the expression describing the dependence of *f*
_{CE} on intensity correctly describes our experimental observations. The intensity dependence of *f*
_{CE}
in OSFCs is found to be significantly simpler than in systems which employ external fiber broadening, where the effect of intensity-dependent spectral shifts of the comb spectrum was found to be of importance [13]. Here, there is a simple linear behavior which is universal, provided the correct pump power level is used to prevent the appearance of pulse instabilities such as cw breakthrough. The pulse energy in the octave-spanning regime is strongly clamped by the KLM action and the combination of SPM and bandwidth limitation, making the intensity-related carrier-envelope frequency shift much smaller than one would expect from the cw laser model.

A complete noise analysis of a carrier-envelope phase stabilized OSFCs was performed. It was found that the inclusion of the pump power to intracavity power transfer function of the laser, that was derived from the linearized rate equations for the system, was essential to obtain correct predictions for the carrier-envelope phase noise behavior. Most importantly the finite response time of the gain introduces a phase delay that adds a phase to the closed-loop transfer function of the system which, together with the phase delay due to the AOM, decreases the phase margin and limits the loop bandwidth to ~100 kHz. This limited bandwidth is insufficient to fully suppress the intensity-dependent high-frequency carrier-envelope phase noise in the system. Because these different contributions are now well understood, the stabilization electronics can be further optimized to suppress the noise in OSFCs pumped by multi-longitudinal-mode lasers to the level of those pumped by single-longitudinal-mode lasers.

## Acknowledgements

This research has been supported by ONR N00014-02-1-0717 and AFOSR FA9550-04-1-0011. O. D. Mücke acknowledges support from the Alexander von Humboldt Foundation. We thank E. P. Ippen for useful comments on the manuscript.

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