1. Introduction

In recent years, carrier-envelope phase control of few-cycle mode-locked lasers has become an enabling technology, paving the way for a wide range of applications like optical clocks, frequency metrology or nonlinear optical processes. While the mode-locking mechanism produces pulses with identical amplitude and pulse duration, each output pulse from a free-running mode-locked laser is affected by a carrier-envelope phase shift (CEPS) per round trip. The carrier wave propagates with phase velocity whereas the envelope of the pulse propagates with group velocity. The CEPS is determined by the resulting slip between the envelope and the carrier oscillation. Thus, it becomes particularly significant for femtosecond pulses in the few-cycle regime. For phase-sensitive applications like extreme nonlinear processes, high harmonic generation or attosecond pulse generation (see e.g [1–3].), it is crucial to stabilize and control the carrier-envelope phase (CEP) accurately. Most commonly, this is achieved by feedback from modulating the pump power [4–6] which changes the intracavity power. Therefore, it is important to understand the dependence of the carrier-envelope phase slip on the intracavity energy so that the operation and design of phase-stabilized lasers can be optimized and noise sources identified.

The CEPS was first measured for solitonlike pulses in a laser cavity operating in the net negative dispersion regime [7]. A decrease of the CEPS for increasing pulse energies was explained by a power-dependent wavelength shift, which, together with the negative group delay dispersion, dominated over the accumulated soliton phase [7]. A detailed analysis of the CEPS [8] based on the perturbed Nonlinear Schrödinger equation for the fundamental soliton revealed that the pulse energy modulates the group delay due to self-steepening. The resulting timing shift contributes twice as much to the CEPS and is of opposite sign as the nonlinear phase due to self-phase modulation. Later measurements of the CEPS using continuum generation in microstructure fiber confirmed that the self-steepening mechanism prevailed [9–11]. For dispersion-managed solitons [12], which best describe the pulse dynamics in few-cycle Ti:sapphire lasers, the Kerr-induced phase shift was derived to be reduced compared to classical solitons [8]. Under strong dispersion management, an analytical and numerical evaluation of the phase slip for dispersion-managed solitons presented that contributions from the shock-term and the phase slip can nearly cancel each other [13]. Analytical and experimental studies showed that the intensity-related spectral shift could be reduced by minimizing the carrier frequency shift [14]. For a broader spectrum, which was obtained for smaller magnitudes of net group delay dispersion (GDD), the coupling between the negative GDD (values as high as −400 fs²) and intensity fluctuations decreased [14]. In addition, the Raman effect was identified as a possible mechanism contributing to spectral shifts [14]. For octave-spanning Ti:sapphire lasers, measurements of the pump power-dependent carrier envelope offset frequency agreed sufficiently well with previous results based on soliton perturbation theory so that it was suggested that spectral shifts could be neglected for broadband intracavity spectra [15]. All these results indicate that the CEP dynamics depend strongly on the laser configuration. Therefore, our objective is to perform a thorough study of the CEP dynamics of prismless sub-two-cycle laser systems that generate octave-spanning spectra directly from the laser cavity and operate close to zero net dispersion. Specifically in this context, the contributions due to self-steepening and the Raman effect are elucidated.

In previous work, we presented a one-dimensional model for octave-spanning Ti:sapphire lasers which captured the detailed pulse formation in these lasers [16,17]. This allowed us to quantitatively predict the output spectrum and pulse shape of the generated sub-two-cycle pulses in good agreement with experimental results [18].

In this paper, our model [16,19] is extended to evaluate the carrier-envelope phase dynamics. We first study how self-steepening affects the pulse shaping and output spectrum of octave-spanning lasers. Then, we identify different contributions towards the CEPS amongst which we find the timing shift due to self-steepening the most significant effect.

2. Laser modeling

The pulse evolution in Kerr-lens mode-locked Ti:sapphire lasers is well described by dispersion-managed mode-locking (DMM) [12]: In systems where the group delay dispersion periodically changes its sign, with small net GDD per round trip when compared to the absolute GDD in each section, dispersion-managed solitary pulses are formed. Due to the encountered GDD and nonlinearities, i.e. predominantly self-phase modulation, the pulses undergo temporal and spectral broadening and recompression during each cavity round trip.

Previously, we presented details on the parameter choice for a one-dimensional laser model based on DMM and the Nonlinear Schrödinger Equation [16,17,19]. The Ti:sapphire crystal is implemented with gain saturation g(z, t), self-phase modulation (self-phase modulation coefficient δ = 4∙10⁻⁷ (Wmm)⁻¹), fast saturable absorber q(z, t) to model the Kerr-lens mode-locking process and second as well as higher order dispersive terms (expressed in terms of a material phase Ф) as shown in Eq. (1). Details on the modeling of the physical phenomena can be found in [16]. For this work, the additional nonlinear term of self-steepening is incorporated, the 4th summand in Eq. (1). This equation describes the evolution of the pulse envelope A(t), with |A(t)|² in units of power, in a retarded time frame t along the axis of propagation z in the crystal. With the split-step Fourier method this expression can be solved numerically.

Figure 1 illustrates the experimental setup for a prismless octave-spanning, sub-two-cycle laser ring cavity with two outputs [18]. The 1f-2f output is coupled out through one of the DCMs, which spectrally selectively enhances the 1f and 2f frequency components (corresponding to 1160 nm and 580 nm). This separate output coupling allows phase-stabilization of the laser independently from the main output through a 1f-2f interferometer feedback loop that controls the pump power [18].

 

Fig. 1 Experimental setup of the phase-stabilized 500 MHz ring laser [18]. The 1f-2f output is used for self-referencing in the 1f-2f interferometer to phase-stabilize the laser.

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3. Impact of self-steepening on octave-spanning Ti:sapphire lasers

3.1 Results for the output spectrum

The devised model captured the octave-spanning output spectrum and the generated sub-two-cycle pulses accurately, allowing for a direct comparison with experimental results. This enabled us to derive a realistic pulse shape, energy and duration for the output pulses, which were in good accordance with the measured results [16].

Figures 2(a) and 2(b) show the two output spectra: the main laser output and the 1f-2f output from the experimental setup and simulation for comparable output power. The numerical data is plotted for the cases with and without self-steepening (ss) in the Ti:sapphire crystal. On average, both spectra agree very well with the measured data. The self-steepening process slightly enhances the shorter wavelengths while the longer wavelengths are attenuated (difference of ~-6 dB between plots with and without ss), i.e. leads to a blue-shift of the spectrum. Although the numerical evaluation with ss results in a slight underestimation for the long wavelengths and gives an upper estimate for the short wavelengths, overall, a better agreement is obtained. The positions of local maxima and minima are barely affected by self-steepening, indicating that these are determined by dispersion and mirror characteristics.

 

Fig. 2 Power spectral density (PSD) of simulated spectrum with self-steepening (w ss) and without self-steepening (w/o ss) compared to the measured output spectrum. The wavelengths corresponding to the 1f- and 2f frequencies are marked. Each curve is normalized to its maximum amplitude.

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3.2 Pulse dynamics in the Ti:sapphire crystal

In Fig. 3 , the impact of self-steepening in the time domain is illustrated. The pulse envelope in units of absolute instantaneous power for the geometric middle and end of the Ti:sapphire crystal is shown in steady state for simulation results with and without self-steepening. The full-width at half maximum (FWHM) pulse duration is slightly different in the middle of the crystal at 11 fs (no ss) and 11.8 fs (with ss). However, stronger temporal broadening is induced in the presence of self-steepening: the pulse with ss has a ~2.4 fs wider FWHM (without ss the pulse FWHM duration is 45.5 fs compared to 47.9 fs with ss at the end of the Ti:sapphire crystal). Due to this additional temporal broadening the peak instantaneous pulse power is slightly reduced and the nonlinear phase shift from self-phase modulation, accumulated in the crystal, decreases. To make a valid comparison, the simulation was optimized to produce similar pulse peak power, pulse durations and self-phase modulation contributions in both cases and the simulation parameters were adjusted accordingly.

 

Fig. 3 Pulse in the geometric middle and at the end of the Ti:sapphire crystal for simulation with and without self-steepening (no ss).

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In general, for clean Gaussian pulses in a medium without dispersion the peak shifts towards the trailing part of the pulse, which results in a steeper trailing edge [20]. For dispersion-managed pulses, a redistribution of the power from the leading to the trailing edge of the pulse is observed in Fig. 3. This shift is more pronounced for longer propagation lengths in the crystal and higher pulse energies. Thus, although the pulse duration within the Ti:sapphire crystal does not generally fall significantly below three optical cycles, it is important to consider the asymmetric impact of self-steepening on the pulse characteristics and the output spectrum.

4. Analysis of the carrier envelope phase dynamics

In the following, the characteristics of the mode-locked laser pulse train and the carrier envelope phase dynamics are evaluated. The CEP is an important parameter for both time and frequency domain applications. Control of the CEP is usually achieved by amplitude modulation of the Ti:sapphire pump laser. The resulting modulation of the intracavity power changes the strength of the nonlinear processes in the crystal and thus alters the group velocity, which induces a CEPS.

4.1 Mode-locked pulse train description

The pulse train E(t) of a mode-locked laser in Eq. (2) is described by a pulse with envelope A(t), as determined from our numerical model, which repeats with round trip time T_R. Per definition, A(t) can be complex and can thus be separated into a real amplitude and a constant phase for each round trip. This envelope is multiplied with a carrier wave of radial frequency ω_c. Small deviations from the round trip time are expressed by the timing shift τ. Based on this chosen formalism, Ф_CE denotes the carrier-envelope phase shift CEPS per round trip.

4.2 Contributions to the CEPS

The CEPS Ф_CE itself is determined by two different contributions: a linear term Ф_{CE, Lin} due to the difference between the phase v_p and group velocity v_g in the optical materials, and a term Ф_{CE, NL} arising predominantly from nonlinear effects in the cavity. The latter can be split into a round trip phase Ф and a timing shift dependent on the carrier frequency ω_c.

4.3 Discussion of carrier frequency

Eq. (3) and Eq. (4) include the carrier frequency ω_c and frequency shifts as a function of energy. For different carrier frequency choices for ultrafast pulses the reader is referred to literature [2,5,21,22]. Each carrier frequency implies a unique pulse envelope description and consequently influences the values for the phase and timing shift per round trip. In addition, the group velocity, phase velocity and the dispersion are determined by the carrier frequency.

Here, a fixed carrier is chosen at the maximum of the gain spectrum with ω_c = 2.356·10¹⁵ rad/s, commonly utilized in simulations. Consistent with this choice, timing shifts and phase shifts per round trip are extracted from the numerical evaluation. By definition, no frequency shifts occur for this chosen representation of the carrier frequency so that Eq. (4) is reduced to the first two terms. However, for the lasers considered, the spectrum fills up the whole gain bandwidth and is limited by the output coupler and mirror bandwidth so that the center frequency is clamped and only very small frequency shifts can occur. If one chose a carrier frequency as the weighted average of the spectrum (the center of gravity), the frequency shifts would only make a small contribution of the order 1·10⁻¹² rad/(s·nJ), while at the same time, the phase and timing shift contributions would differ from the case of a fixed carrier frequency. Thus, the resulting CEPS, which determines the carrier-envelope offset frequency and which is characteristic for any specific laser configuration, does not depend on the formal description chosen for the carrier frequency.

4.4 Numerical evaluation of the CEPS

For steady-state pulses with energy W the numerical evaluation converges towards unique values for the round trip phase shift Φ and the nonlinear timing shift τ. According to our definition, Ф captures the incremental round trip phase change in steady state and as such includes effects from second and higher order dispersion and self-phase modulation in the crystal. This phase shift increases with higher energies, as the self-phase modulation gets stronger, from about -π/2 to −3π/2 (actual values from −0.6π to −1.4π) for typical pulse energies between 24 nJ to 69 nJ as shown in Fig. 4a . The asymmetry in the spectrum from self-steepening translates into a timing shift that varies from 3.2 fs to 7.7 fs (with fluctuations within a range of 0.2 fs, partly limited by the numerical resolution) see Fig. 4b. This timing shift can also include possible delays due to KLM saturable absorber action. However since a fast saturable absorber model is assumed, no significant contributions were noted. For both parameters a linear relationship w.r.t pulse energy is found, which is characterized by slopes of different magnitude and opposite sign:

 

Fig. 4 (a) Round trip phase shift Φ and (b) nonlinear timing shift τ dependency on intracavity pulse energy W. The linear relationship is described by the respective slopes.

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4.5 Influence of Raman shift on CEP dynamics

In our simulations, the Raman effect is not incorporated as the pulse dynamics are most significantly determined by the interaction of dispersion and self-phase modulation in the crystal. As the vibrational resonance lines in Ti:sapphire are of narrow bandwidth and have a small Raman gain [23,24], any anticipated Raman shift is expected to be small. Additional mechanisms that can induce small frequency shifts like frequency dependant reabsorption, upconversion and two-photon processes in the crystal are assumed to have a negligible impact on the octave-spanning intracavity spectrum [25], as there has been no indication that they play any role for our lasers. Therefore we will study the red-shift of the spectrum due to the Raman effect only and its impact on the CEP dynamics.

To estimate the magnitude of the frequency shift due to the Raman effect Δω_Raman, we follow the approach by Haus et al. (see [23]): Eq. (7) was obtained based on soliton perturbation theory for the Nonlinear Schrödinger equation, which included effects of self-phase modulation and second order dispersion. This formalism is applicable to materials with a weak Raman effect as is the case for Ti:sapphire [25].

The steady state frequency shift Δω that builds up in the cavity per round trip is not only determined by the value from Eq. (7) but is also influenced by the filtering from the mirrors and the limited gain bandwidth. The resulting Raman frequency offset undergoes a relaxation behavior during each round trip that can be described by two time constants τ_Filter and τ_Gain due to the mirror and gain filtering interaction. This leads to the following differential equation:

The numerical values for the time constants for different pulse durations τ_sech and the corresponding FWHM spectral bandwidth Δf_FWHM in the transform-limited case are listed in Table 1 . As the inverse of the gain time constant is at least one order of magnitude larger than the filtering term, it is the dominating term as expected from its narrower filtering bandwidth.

Table 1. Time constants for relaxation behavior from filtering by mirrors and output coupler τ_Filter and from gain filtering τ_Gain for a given pulse duration τ_sech.

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Table 2. Raman shift Δω_Raman, value of convolution function G(ω_Rτ_sech), resulting frequency shift Δω_Total and CEPS contribution for each Raman line ω_R for different pulse parameters τ_sech.

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We saw earlier that the contribution of the Raman term to the CEPS scaled with the sign and magnitude of the intracavity net dispersion. For laser systems with net negative dispersion, the operating point can theoretically be chosen such that the Raman CEPS contribution is of opposite sign and of equal value as the contribution from the phase shift and self-steepening. In this case, the CEPS cannot be impacted by pump power modulation but other mechanisms have to be applied, like slightly tilting one of the cavity mirrors.

4.6 Comparison with soliton theory

Having quantitatively determined a value for the CEPS with energy for dispersion-managed solitary pulses, we compare these results to classical soliton theory and translate the CEPS into measurable laser characteristics. A coefficient for the on power P dependent carrier envelope offset frequency f_CE,Soliton was derived as $\frac{\Delta f_{CE,Soliton}}{\Delta P}=\frac{L{n}_2}{4\lambda A_{eff}{\tau }_{sech}}$ [8,14] with soliton perturbation theory. For the studied laser configuration, L = 2.15 mm is the optical path length of the Ti:sapphire crystal per round trip, n₂ = 3∙10⁻²⁰ m²/W its nonlinear index of refraction, λ = 800 nm the carrier wavelength, A_eff ≈589 μm² the beam cross section and τ_sech = τ_FWHM /1.76 the temporal width for a corresponding soliton sech-shaped pulse. With Eq. (10) we can convert the derived numerical result for the CEPS over energy of (0.17 + 0.02) rad/nJ = 0.19 rad/nJ into a carrier-envelope offset frequency change with intracavity power of (27.1 + 3.2) MHz/W = 30.3 MHz/W. For a soliton pulse parameter of τ_sech ≈(2.38 fs + 3.56 fs)/2 = 2.97 fs (average value of the corresponding temporal width of a soliton, calculated from the minimum and maximum spectral bandwidth in the Ti:sapphire crystal due to pulse breathing) $\Delta f_{CE,Soliton}/\Delta P$is calculated to be 11.5 MHz/W according to Eq. (10). This result then allows us to determine the factor C ≈2.6, which relates the CEPS of dispersion managed solitons to the nonlinear phase shift of a classical soliton.

5. Conclusion

The presented one-dimensional numerical analysis, which includes self-steepening for the Ti:sapphire crystal and higher-order dispersion of the intracavity elements, accurately describes the laser dynamics of octave-spanning lasers that generate sub-two-cycle pulses.

The carrier envelope phase dynamics in dispersion-managed octave-spanning Ti:sapphire lasers are dominated by nonlinear timing shifts due to self-steepening, which attenuates the longer wavelengths in the spectrum and redistributes power to the trailing edge of the pulse. For the studied 500 MHz dispersion-managed laser system this contribution is about four times larger than term from the round trip phase shift. The Raman frequency shift is found to play a minor role due to clamping of the octave-spanning spectrum by intracavity filtering from the gain and cavity mirrors. A quantitative value for the carrier envelope phase shift with energy of 0.19 rad/nJ was derived together with an analytical expression based on a modified result from soliton perturbation theory as a first estimate for the CEPS. In conclusion, we presented a thorough study of the carrier envelope phase dynamics and demonstrated that our numerical model, which can be adapted to different laser cavity configurations, is a powerful tool for the design and optimization of octave-spanning lasers.

Appendix

Derivation of frequency shift for a rectangular filter function

For current octave-spanning Ti:sapphire laser systems the overall filtering behavior g_filter(t) exerted by the DCMs and the output coupler on the pulse is better matched by a rectangular filter function in the frequency domain than by a parabolic filter. Thus, the impact of a rectangular filter function is evaluated with soliton perturbation theory. The rectangular filter function is centered relative to the carrier frequency of the intracavity spectrum at Δω_f and is characterized by a bandwidth of Ω_f.

The adjoint f_ω(t) for the frequency-perturbation function in soliton perturbation theory is given by [23]:

(11)$${\underset{¯}{f}}_{\omega }(t)=\frac{j}{A_0{\tau }^2}\tanh \left(\frac{t}{{\tau }_{\text{sech}}}\right)\text{\hspace{0.17em} sech}\left(\frac{t}{{\tau }_{\text{sech}}}\right)$$

The frequency shift Δω_filter due to the filter action g_filter(t) on the pulse envelope A(t) is defined as:

(12)$$\Delta {\omega }_{filter}=\mathrm{Re}{\displaystyle \int {\underset{¯}{f}}_{\omega }^{*}\Delta a(t)dt}=\mathrm{Re}{\displaystyle \int {\underset{¯}{f}}_{\omega }^{*}{g}_{filter}(t)A(t)dt}$$

With Parseval’s theorem for the inner product rule this expression goes over into:

(13)$${\displaystyle \int {\underset{¯}{f}}_{\omega }^{*}\Delta a(t)dt}={\displaystyle \int {\underset{¯}{f}}_{\omega }^{*}{g}_{filter}(t)A(t)dt}={\displaystyle \int F\left\{{\underset{¯}{f}}_{\omega }^{*}\right\}F\{g_{filter}(t)\}F\{A(t)\}d\omega }$$

Evaluating this integral in the frequency domain and using the following Fourier transform pairs leads to the expression for Δω_filter in Eq. (14):

$$\begin{array}{l}\text{sech}(t)\to \pi \text{\hspace{0.17em} sech}\left(\frac{\pi }{2}\omega \right)\\ \tanh (t)\text{\hspace{0.17em} sech}(t)\to -j\pi \omega \text{\hspace{0.17em} sech}\left(\frac{\pi }{2}\omega \right)\end{array}$$

(14)$$\begin{array}{l}\Delta {\omega }_{filter}=\mathrm{Re}{\displaystyle \int {\underset{¯}{f}}_{\omega }^{*}\Delta a(t)dt}=\mathrm{Re}{\displaystyle \int -\frac{j}{A_0{\tau }^2}\tanh \left(\frac{t}{{\tau }_{\text{sech}}}\right)\text{\hspace{0.17em} sech}\left(\frac{t}{{\tau }_{\text{sech}}}\right)}{g}_{filter}(t)A(t)dt=\\ =\mathrm{Re}{\displaystyle \int -\frac{j}{A_0{\tau }_{\text{sech}}{}^2}\tanh \left(\frac{t}{{\tau }_{\text{sech}}}\right)\text{\hspace{0.17em} sech}\left(\frac{t}{{\tau }_{\text{sech}}}\right)}{F}^{-1}\left\{\text{\hspace{0.17em} rect}\left(\frac{\omega -\Delta \omega -\Delta {\omega }_f}{{\Omega }_f}\right)\right\}{A}_0\text{\hspace{0.17em} sech}\left(\frac{t}{{\tau }_{\text{sech}}}\right)dt\\ =\mathrm{Re}{\displaystyle \int F\left\{{\underset{¯}{f}}_{\omega }^{*}\right\}F\{g_{filter}(t)\}F\{A(t)\}d\omega }=\\ =\mathrm{Re}{\displaystyle \int -\frac{j}{A_0{\tau }_{\text{sech}}{}^2}(-j{\tau }_{\text{sech}}{}^2)\pi \omega \text{\hspace{0.17em} sech}\left(\frac{\pi }{2}{\tau }_{\text{sech}}\omega \right)}\text{\hspace{0.17em} rect}\left(\frac{\omega -\Delta \omega -\Delta {\omega }_f}{{\Omega }_f}\right)\cdot \mathrm{...}\\ \text{ }\mathrm{...}\cdot (A_0)\pi {\tau }_{\text{sech}}\text{\hspace{0.17em} sech}\left(\frac{\pi }{2}{\tau }_{\text{sech}}\omega \right)d\omega =\\ \text{ }={\displaystyle \underset{-\frac{{\Omega }_f}{2}+\Delta \omega +\Delta {\omega }_f}{\overset{\frac{{\Omega }_f}{2}+\Delta \omega +\Delta {\omega }_f}{\int }}-{\pi }^2{\tau }_{\text{sech}}\omega {\text{sech}}^2\left(\frac{\pi }{2}{\tau }_{\text{sech}}\omega \right)d\omega }=\\ ={-{\pi }^2{\tau }_{\text{sech}}\left[\frac{-4}{{\pi }^2{\tau }_{\text{sech}}{}^2}\text{\hspace{0.17em} log}\left[\cosh \left(\frac{\pi }{2}{\tau }_{\text{sech}}\omega \right)\right]+\frac{2\omega }{\pi {\tau }_{\text{sech}}}\text{\hspace{0.17em} tanh}\left(\frac{\pi }{2}{\tau }_{\text{sech}}\omega \right)\right]|}_{-\frac{{\Omega }_f}{2}+\Delta \omega +\Delta {\omega }_f}^{\frac{{\Omega }_f}{2}+\Delta \omega +\Delta {\omega }_f}=\\ =\frac{4}{{\tau }_{\text{sech}}}\left\{-\text{ log}\left[\cosh \left(\frac{\pi }{2}{\tau }_{\text{sech}}\left(-\frac{{\Omega }_f}{2}+\Delta \omega +\Delta {\omega }_f\right)\right)\right]+\text{\hspace{0.17em} log }\left[\cosh \left(\frac{\pi }{2}{\tau }_{\text{sech}}\left(\frac{{\Omega }_f}{2}+\Delta \omega +\Delta {\omega }_f\right)\right)\right]\right\}+\mathrm{...}\\ +2\pi \left[\left(-\frac{{\Omega }_f}{2}+\Delta \omega +\Delta {\omega }_f\right)\text{\hspace{0.17em} tanh}\left(\frac{\pi }{2}{\tau }_{\text{sech}}\left(-\frac{{\Omega }_f}{2}+\Delta \omega +\Delta {\omega }_f\right)\right)\right]-\mathrm{...}\\ -2\pi \left[\left(\frac{{\Omega }_f}{2}+\Delta \omega +\Delta {\omega }_f\right)\text{\hspace{0.17em} tanh}\left(\frac{\pi }{2}{\tau }_{\text{sech}}\left(\frac{{\Omega }_f}{2}+\Delta \omega +\Delta {\omega }_f\right)\right)\right]\end{array}$$

Instead of incorporating the frequency perturbation ω’ + Δω into the spectral description in Eq. (14), we use a variable substitution ω = ω’-Δω to include this offset only in the filter function which maintains the relative shift between the center of the spectrum and the filter. To evaluate this expression, a graphical approach is chosen with τ_sech as parameter. As the argument $\frac{\pi }{2}{\tau }_{\text{sech}}\left(-\frac{{\Omega }_f}{2}+\Delta \omega +\Delta {\omega }_f\right)$ of the hyperbolic functions is in general larger than 1, a Taylor series to obtain an analytical approximation cannot be applied.

Figure 5 displays the overall frequency shift Δω_filter, depending on the offset Δω between the center of the spectrum and the carrier frequency with the soliton pulse width τ_sech = τ_FWHM/1.76 as parameter. Typical values for the filter consist of Ω_f = 1.4·10¹⁵ rad/s (span from 600 nm to 1100 nm) and a negligible offset Δω_f for the transfer function including the output coupler and mirror impact. As expected, filters designed with a reasonable large bandwidth do not have any significant impact on the intracavity spectrum unless the spectral wings start overlapping with the cut-off regions close to the steep edges of the rectangular filter. In a first order approximation, we can therefore focus on the region around Δω = 0 and describe any possible shifts in terms of the derivative at Δω = 0:

$$\begin{array}{l}\Delta {\omega }_{filter}\approx \frac{\partial }{\partial \Delta {\omega }_f}\left({{\displaystyle \underset{-\frac{{\Omega }_f}{2}+\Delta {\omega }_{f+\Delta \omega }}{\overset{\frac{{\Omega }_f}{2}+\Delta {\omega }_f+\Delta \omega }{\int }}-{\pi }^2{\tau }_{\text{sech}}\omega {\text{sech}}^2\left(\frac{\pi }{2}{\tau }_{\text{sech}}\omega \right)d\omega }|}_{\Delta \omega =0}\right)\cdot \Delta \omega \\ =-{\pi }^2{\tau }_{\text{sech}}{\left(-\frac{{\Omega }_f}{2}+\Delta {\omega }_f+\Delta \omega \right){\text{sech}}^2\left(\frac{\pi }{2}{\tau }_{\text{sech}}\left(-\frac{{\Omega }_f}{2}+\Delta {\omega }_f+\Delta \omega \right)\right)|}_{\Delta \omega =0}\cdot \Delta \omega +\mathrm{...}\\ \mathrm{...}+{\pi }^2{\tau }_{\text{sech}}{\left(\frac{{\Omega }_f}{2}+\Delta {\omega }_f+\Delta \omega \right){\text{sech}}^2\left(\frac{\pi }{2}{\tau }_{\text{sech}}\left(\frac{{\Omega }_f}{2}+\Delta {\omega }_f+\Delta \omega \right)\right)|}_{\Delta \omega =0}\cdot \Delta \omega \\ =-{\pi }^2{\tau }_{\text{sech}}\left[\left(-\frac{{\Omega }_f}{2}+\Delta {\omega }_f\right){\text{sech}}^2\left(\frac{\pi }{2}{\tau }_{\text{sech}}\left(-\frac{{\Omega }_f}{2}+\Delta {\omega }_f\right)\right)-\left(\frac{{\Omega }_f}{2}+\Delta {\omega }_f\right){\text{sech}}^2\left(\frac{\pi }{2}{\tau }_{\text{sech}}\left(\frac{{\Omega }_f}{2}+\Delta {\omega }_f\right)\right)\right]\cdot \Delta \omega \end{array}$$

(15)$$\Delta {\omega }_{filter}\stackrel{for\Delta {\omega }_f=0}{=}-{\pi }^2{\tau }_{\text{sech}}{\Omega }_f{\text{sech}}^2\left({\tau }_{\text{sech}}\frac{\pi }{2}\frac{{\Omega }_f}{2}\right)\cdot \Delta \omega =-\frac{1}{{\tau }_{filter}}\cdot \Delta \omega$$

The impact of the filter on the spectrum in Eq. (15) can be expressed in terms of a relaxation constant 1/τ_filter. The linear approximation for a rectangular filter with a bandwidth of Ω_f = 1.4·10¹⁵ rad/s and a pulse with τ_sech = 3.5 fs is shown in Fig. 6 . For small frequency offsets the linear approximation follows the imposed frequency shift accurately. For frequency shifts with a magnitude larger than 0.4 ·10¹⁵ rad/s it is obvious that the slope increases more rapidly. However, for that regime, one can work with a different linear approximation. Overall, the impact of a rectangular filter distinguishes itself quite pronounced from that of a parabolic filter, which is described by the linear relationship in Eq. (8).

 

Fig. 5 Frequency shift Δω_filter due to a rectangular filter with bandwidth Ω_f = 1.4 · 10¹⁵ rad/s for a spectrum shifted by Δω from the carrier frequency for three values of soliton pulse width τ_sech.

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Fig. 6 Frequency shifts Δω_filter induced by a rectangular filter of bandwidth of Ω_f = 1.4 ·10¹⁵ rad/s on a pulse with τ_sech = 3.5 fs. The linear approximation according to Eq. (15) is plotted as a dashed line.

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Acknowledgements

The authors would like to thank Helder M. Crespo for contributing the experimental results shown in this paper. This research was supported in part by Defense Advanced Research Projects Agency (DARPA) grant HR0011-05-C-0155, and United States Air Force Office of Scientific Research (AFOSR) grant FA9550-07-1-0014.

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