Abstract

The carrier-envelope phase dynamics of few-cycle octave-spanning Ti:sapphire lasers are analyzed based on a numerical one-dimensional dispersion-managed laser model. The dominant contribution to the carrier-envelope phase shift with respect to intracavity energy arises from the asymmetric impact of self-steepening on pulse formation and laser output. We show that this term is larger by a factor of four than the energy-dependent round trip phase and is thus more significant than in the corresponding result for conventional soliton lasers. Frequency shifts due to the Raman effect are studied and found to be of minor impact for octave-spanning lasers.

©2010 Optical Society of America

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 [13].), 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 [46] 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 [911]. 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.

A(z,t)z​ ​ =[g(z,t)q(z,t)  ​ jδ{|A(z,t)|2+jωC​ ​ ​ t(|A(z,t)|2)}jϕ(jt)​ ​]A(z,t)​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​
The free space path of the cavity includes specially designed double-chirped mirror (DCM) pairs and wedges of BaF2 and fused silica for broadband dispersion compensation [18]. This propagation path is modelled with transfer functions for the DCMs, characterized by their reflectivity and measured dispersive phase, including higher order dispersion contributions, and material dispersion of second order and higher for the wedges and air path. Thus, it is important that the detailed mirror and output coupler characteristics were incorporated to accurately reflect the laser dynamics.

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].

 figure: Fig. 1

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.

 figure: Fig. 2

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.

 figure: Fig. 3

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 TR. 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 ωc. Small deviations from the round trip time are expressed by the timing shift τ. Based on this chosen formalism, ФCE denotes the carrier-envelope phase shift CEPS per round trip.

E(t)=nA(tnTRnτ)  ​ejωc(tnTRnτ)ejnΦCE​ ​

4.2 Contributions to the CEPS

The CEPS ФCE itself is determined by two different contributions: a linear term ФCE, Lin due to the difference between the phase vp and group velocity vg in the optical materials, and a term ФCE, 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 ωc.

ΦCE=ΦCE,NL  ​ +ΦCE,Lin=  ​Φ+ωcτ+ΦCE==Φ+ωcτ+ωcL(1vg1vp)=Φ+ωcτ+ωciLi(kiω|ωCkiωc)
To study the impact of intracavity energy changes on the CEPS, the expression of interest is the derivative of the CEPS ФCE with respect to the intracavity pulse energy W:
ΦCEW  ​=ΦW  ​ +ωcτW+τωcWωciLi2kiω2|ωcωcW
This derivative consists of four contributions to the carrier envelope phase shift: The change of the round trip phase Ф and the timing shift τ w.r.t. energy and two terms depending on shifts of the carrier frequency ωc. The last term in Eq. (4) is proportional to the net group delay dispersion, the second order dispersion 2kiω2|ωc from the i different materials in the cavity summed over their respective optical path lengths Li. Since our octave-spanning lasers operate close to zero net dispersion, the contribution of this last term is expected to be small.

4.3 Discussion of carrier frequency

Eq. (3) and Eq. (4) include the carrier frequency ωc 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 ωc = 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:

ΦW=0.06  rad/nJ and τW=0.1  fs/nJ
This leads to the following expression for the carrier envelope phase shift over energy:
ΦCEW  ​=ΦW  ​ +ωcτW=0.06  rad/nJ +0.23  rad/nJ =0.17  rad/nJ
The timing shift term of 0.23 rad/nJ is four times larger than the phase shift contribution of −0.06 rad/nJ and both effects partially compensate each other. A comparison of this result with that for conventional soliton lasers, derived by Haus and Ippen [8], shows that for octave-spanning lasers, whose dynamics are closer to dispersion-managed solitons, the timing shift induced CEPS contribution dominates over the phase shift, roughly by a factor of four. We thus quantitatively characterized the reduced Kerr-induced phase previously suggested [8,13] for octave-spanning dispersion-managed lasers. Because of pulse breathing, the peak intensity varies within the Ti:sapphire crystal, so that smaller nonlinear phase shifts are accumulated per round trip.

 figure: Fig. 4

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 ΔωRaman, 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].

ΔωRaman=iπ161τsechgRδωRG(ωR,iτsech)LAW
The Raman shift described by Eq. (7) is given by the gain gR of each individual Raman line ωR,i with linewidth δωR [23]. The function G(ωR,i τ sech) represents a convolution integral for the impact of each Raman line on the perturbed pulse in terms of the normalized pulse bandwidth of the Raman resonance frequency ωR,i and the sech pulse duration τsech = τFWHM /1.76. It decreases linearly for small τsech, reaches a relative minimum for ωRτsech = 1.2 at −0.97 (τsech ~10 fs) and then exponentially approaches zero for increasing ωRτsech [23,24]. The Raman frequency shift is also proportional to the pulse energy W, crystal length L and the inverse of the beam cross-section A so that the net shift is equal to the sum of the contributions of all individual lines.

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 τFilter and τGain due to the mirror and gain filtering interaction. This leads to the following differential equation:

TΔω=ΔωFilter+ΔωGain+ΔωRaman=ΔωτFilterΔωτGain+ΔωRamanwith 1τGain=43ΩG2τsech2
Applying results for the balancing action of a parabolic filter [23], the gain filtering in Eq. (8) is described by a parabolic filter of bandwidth ΩG = 240·1012 rad/s. The DCMs and the output coupler are approximated by a rectangular filtering function (for details on the transfer function see [16]). The resulting frequency shift caused by a rectangular filter is derived in the Appendix, Eq. (14), and its impact captured by τFilter. Thus, the spectrum, Raman shifted by ΔωRaman, experiences a pull towards the center frequency of the filter. Once steady state is reached, the overall frequency shift settles at the value ΔωTotal as given by Eq. (9).

The numerical values for the time constants for different pulse durations τsech and the corresponding FWHM spectral bandwidth ΔfFWHM 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.

ΔωTotal=11τFilter+1τGainΔωRaman=i11τFilter+1τGainπ161τsechgRδωR,iG(ωRτsech)LAW
To derive the frequency shift ΔωTotal from Eq. (9) the following assumptions are made: Firstly, to our knowledge, exact values for the Raman parameters in Ti:sapphire have not been determined. Thus, the Ti:sapphire Raman spectrum is modeled as consisting of two pronounced Raman lines of α-Al2O3 (corundum) at 418 cm−1 and 645 cm−1. The other five lines with smaller amplitude, which are more pronounced for different crystal orientation and light polarization, are neglected [26,27]. Secondly, the Raman shift is estimated based on data for the Raman gain gR = 6.9 10−9 cm/W in diamond with a linewidth δωR = 0.38·1012 rad/s [23,24] for both Raman lines. Thirdly, to accurately evaluate filtering effects from the overlap between the spectrum and the filter functions, the maximum spectral bandwidth needs to be appropriately represented. Therefore, Table 2 lists the corresponding sech pulse values τsech = τFWHM /1.76 = 3.57 fs and τsech = 2.38 fs, for the narrowest and broadest FWHM intracavity spectrum value of 50 and 75 THz. This results in maximum frequency shifts with energy of around −1.468∙1012 rad/(s·nJ) as shown in Table 2. For an assumed net dispersion per round trip of ~7 fs2, each Raman line has a contribution between 0.005 rad/nJ to 0.017 rad/nJ. This value is mainly given by the term including the net cavity dispersion, the fourth summand in Eq. (4). The third term, the product of the timing shift and the frequency shifts, only slightly reduce the CEPS. The total CEPS from the two lines thus varies between 0.03 rad/nJ and 0.01 rad/nJ depending on the spectral bandwidth and the net intracavity dispersion. Since the gain parameters for quartz reduces these values by a factor of one third, this calculation provides an upper limit estimate for a Raman induced CEPS of 0.02 rad/nJ (average of the two calculated values). However, overall, even with a small additional term from Raman shifts (at most the same order as half the round trip phase shift) for the CEPS dynamics, the dominant contribution to the phase shift still originates from the nonlinear timing shifts and thus the self-steepening process.

Tables Icon

Table 1. Time constants for relaxation behavior from filtering by mirrors and output coupler τFilter and from gain filtering τGain for a given pulse duration τsech.

Tables Icon

Table 2. Raman shift ΔωRaman, value of convolution function G(ωRτsech), resulting frequency shift ΔωTotal and CEPS contribution for each Raman line ωR for different pulse parameters τsech.

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 fCE,Soliton was derived as ΔfCE,SolitonΔP=Ln24λAeffτsech [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, Aeff ≈589 μm2 the beam cross section and τsech = τFWHM /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 τsech ≈(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) ΔfCE,Soliton/ΔPis 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.

ΔfCEΔP=12πΔΦCEΔWCΔfCE,SolitonΔP  with C2.62.6Ln24λAeffτsech=30.3  MHz/W
Classical soliton perturbation theory yields that the timing shift magnitude is twice as large as the nonlinear phase shift compared to the factor of four between the timing shift and round trip phase in Eq. (6) established from our numerical evaluation. However, our total CEPS incorporates the possible impact of Raman shifts. In addition, since our round trip phase definition includes not only contributions from self-phase modulation but also from second and higher order dispersive effects, which cannot easily be singled out, the determined factor of 2.6 demonstrates the effect of a reduced nonlinear phase contribution in these lasers. Given the rough estimates for τsech and Aeff and that the pulse dynamics in octave-spanning lasers are not governed by classical soliton behavior, Eq. (10) can be seen as a adequate approximation for the carrier-envelope phase dynamics in dispersion-managed laser systems.

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 gfilter(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 Δωf and is characterized by a bandwidth of Ωf.

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

f¯ω(t)=jA0τ2tanh(tτsech)  sech(tτsech)
The frequency shift Δωfilter due to the filter action gfilter(t) on the pulse envelope A(t) is defined as:
Δωfilter=Ref¯ω*Δa(t)dt=Ref¯ω*gfilter(t)A(t)dt
With Parseval’s theorem for the inner product rule this expression goes over into:
f¯ω*Δa(t)dt=f¯ω*gfilter(t)A(t)dt=F{f¯ω*}F{gfilter(t)}F{A(t)}dω
Evaluating this integral in the frequency domain and using the following Fourier transform pairs leads to the expression for Δωfilter in Eq. (14):
sech(t)π  sech(π2ω)tanh(t)  sech(t)jπω  sech(π2ω)
Δωfilter=Ref¯ω*Δa(t)dt=RejA0τ2tanh(tτsech)  sech(tτsech)gfilter(t)A(t)dt==RejA0τsech2tanh(tτsech)  sech(tτsech)F1{  rect(ωΔωΔωfΩf)}A0  sech(tτsech)dt=ReF{f¯ω*}F{gfilter(t)}F{A(t)}dω==RejA0τsech2(jτsech2)πω  sech(π2τsechω)  rect(ωΔωΔωfΩf)...​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ...(A0)πτsech  sech(π2τsechω)dω=​ ​ ​ ​ ​ ​ ​ =Ωf2+Δω+ΔωfΩf2+Δω+Δωfπ2τsechωsech2(π2τsechω)dω==π2τsech[4π2τsech2  log[cosh(π2τsechω)]+2ωπτsech  tanh(π2τsechω)]|Ωf2+Δω+ΔωfΩf2+Δω+Δωf==4τsech{ log[cosh(π2τsech(Ωf2+Δω+Δωf))]+  log [cosh(π2τsech(Ωf2+Δω+Δωf))]}+...+2π[(Ωf2+Δω+Δωf)  tanh(π2τsech(Ωf2+Δω+Δωf))]...2π[(Ωf2+Δω+Δωf)  tanh(π2τsech(Ωf2+Δω+Δωf))]
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 τsech as parameter. As the argument π2τsech(Ωf2+Δω+Δωf) 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 Δωfilter, depending on the offset Δω between the center of the spectrum and the carrier frequency with the soliton pulse width τsech = τFWHM/1.76 as parameter. Typical values for the filter consist of Ωf = 1.4·1015 rad/s (span from 600 nm to 1100 nm) and a negligible offset Δωf 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:

ΔωfilterΔωf(Ωf2+Δωf+ΔωΩf2+Δωf+Δωπ2τsechωsech2(π2τsechω)dω|Δω=0)Δω=π2τsech(Ωf2+Δωf+Δω)sech2(π2τsech(Ωf2+Δωf+Δω))|Δω=0Δω+......+π2τsech(Ωf2+Δωf+Δω)sech2(π2τsech(Ωf2+Δωf+Δω))|Δω=0Δω=π2τsech[(Ωf2+Δωf)sech2(π2τsech(Ωf2+Δωf))(Ωf2+Δωf)sech2(π2τsech(Ωf2+Δωf))]Δω
Δωfilter=forΔωf=0π2τsechΩfsech2(τsechπ2Ωf2)Δω=1τfilterΔω
The impact of the filter on the spectrum in Eq. (15) can be expressed in terms of a relaxation constant 1filter. The linear approximation for a rectangular filter with a bandwidth of Ωf = 1.4·1015 rad/s and a pulse with τsech = 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).

 figure: Fig. 5

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

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 figure: Fig. 6

Fig. 6 Frequency shifts Δωfilter induced by a rectangular filter of bandwidth of Ωf = 1.4 ·1015 rad/s on a pulse with τsech = 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.

References and links

1. P. Christov, M. M. Murnane, and H. C. Kapteyn, ““High-Harmonic Generation of Attosecond Pulses in the “Single-Cycle” Regime,” Phys. Rev. Lett. 78(7), 1251–1254 (1997). [CrossRef]  

2. T. Brabec and F. Krausz, “Intense few-cycle laser fields: frontiers of nonlinear optics,” Rev. Mod. Phys. 72(2), 545–591 (2000). [CrossRef]  

3. A. Baltuška, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. 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(6923), 611–615 (2003). [CrossRef]   [PubMed]  

4. A. Poppe, R. Holzwarth, A. Apolonski, G. Tempea, Ch. Spielmann, T. W. Hänsch, and F. Krausz, “Few-cycle optical waveform synthesis,” Appl. Phys. B 72, 373–376 (2001).

5. S. T. Cundiff, “Phase stabilization of ultrashort optical pulses,” J. Phys. D Appl. Phys. 35(8), 201 (2002). [CrossRef]  

6. O. D. Mücke, R. Ell, A. Winter, J. W. 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(13), 5163–5169 (2005). [CrossRef]   [PubMed]  

7. L. Xu, Ch. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T. W. Hänsch, “Route to phase control of ultrashort light pulses,” Opt. Lett. 21(24), 2008–2010 (1996). [CrossRef]   [PubMed]  

8. H. A. Haus and E. P. Ippen, “Group velocity of solitons,” Opt. Lett. 26(21), 1654–1656 (2001). [CrossRef]  

9. T. M. Fortier, J. Ye, S. T. Cundiff, and R. S. Windeler, “Nonlinear phase noise generated in air-silica microstructure fiber and its effect on carrier-envelope phase,” Opt. Lett. 27(6), 445–447 (2002). [CrossRef]  

10. F. W. Helbing, G. Steinmeyer, U. Keller, R. S. Windeler, J. Stenger, and H. R. Telle, “Carrier-envelope offset dynamics of mode-locked lasers,” Opt. Lett. 27(3), 194–196 (2002). [CrossRef]  

11. 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(1), 5–12 (2004). [CrossRef]  

12. 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(11), 1999–2004 (1999). [CrossRef]  

13. M. J. Ablowitz, B. Ilan, and S. T. Cundiff, “Carrier-envelope phase slip of ultrashort dispersion-managed solitons,” Opt. Lett. 29(15), 1808–1810 (2004). [CrossRef]   [PubMed]  

14. 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(4), 1018–1024 (2003). [CrossRef]  

15. L. Matos, O. D. Mücke, J. Chen, and F. X. Kärtner, “Carrier-envelope phase dynamics and noise analysis in octave-spanning Ti:sapphire lasers,” Opt. Express 14(6), 2497–2511 (2006). [CrossRef]   [PubMed]  

16. M. Y. Sander, J. Birge, A. Benedick, H. M. Crespo, and F. X. Kärtner, “Dynamics of dispersion managed octave-spanning titanium:sapphire lasers,” J. Opt. Soc. Am. B 26(4), 743–749 (2009). [CrossRef]  

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

18. H. M. Crespo, J. R. Birge, M. Y. Sander, E. L. Falcão-Filho, A. Benedick, and F. X. Kärtner, “Phase stabilization of sub-two-cycle pulses from prismless octave-spanning Ti:sapphire lasers,” J. Opt. Soc. Am. B 25(7), B147–B154 (2008). [CrossRef]  

19. M. Y. Sander, and F. X. Kärtner, “Carrier-envelope phase dynamics of octave-spanning Titanium:sapphire lasers,” Conference on Lasers and Electro Optics (CLEO), Baltimore, MD, June 1–5, 2009, CMN4.

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

21. T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002). [CrossRef]   [PubMed]  

22. T. R. Schibli, O. Kuzucu, J. Kim, E. P. Ippen, J. G. Fujimoto, F. X. Kaertner, V. Scheuer, and G. Angelow, “Toward single-cycle laser systems,” IEEE J. Sel. Top. Quantum Electron. 9(4), 990–1001 (2003). [CrossRef]  

23. H. A. Haus, I. Sorokina, and E. Sorokin, “Raman-induced redshift of ultrashort mode-locked laser pulses,” J. Opt. Soc. Am. B 15(1), 223–231 (1998). [CrossRef]  

24. I. T. Sorokina, E. Sorokin, E. Wintner, A. Cassanho, and H. P. Jenssen, “Raman induced pulse self-frequency shift in the sub-20 fs Kerr-lens mode-locked Cr:LiSGaF and Cr:LiSAF lasers,” OSA TOPS 19, 359–361 (1998).

25. V. L. Kalashnikov, E. Sorokin, and I. T. Sorokina, “Mechanisms of spectral shift in ultrashort-pulse laser oscillators,” J. Opt. Soc. Am. B 18(11), 1732–1741 (2001). [CrossRef]  

26. S. P. S. Porto and R. S. Krishnan, “Raman effect of Corundum,” J. Chem. Phys. 47(3), 1009–1012 (1967). [CrossRef]  

27. M. Kadleikova, J. Breza, and M. Vesely, “Raman spectra of synthetic sapphire,” Microelectron. J. 32(12), 955–958 (2001). [CrossRef]  

References

  • View by:

  1. P. Christov, M. M. Murnane, and H. C. Kapteyn, ““High-Harmonic Generation of Attosecond Pulses in the “Single-Cycle” Regime,” Phys. Rev. Lett. 78(7), 1251–1254 (1997).
    [Crossref]
  2. T. Brabec and F. Krausz, “Intense few-cycle laser fields: frontiers of nonlinear optics,” Rev. Mod. Phys. 72(2), 545–591 (2000).
    [Crossref]
  3. A. Baltuška, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. 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(6923), 611–615 (2003).
    [Crossref] [PubMed]
  4. A. Poppe, R. Holzwarth, A. Apolonski, G. Tempea, Ch. Spielmann, T. W. Hänsch, and F. Krausz, “Few-cycle optical waveform synthesis,” Appl. Phys. B 72, 373–376 (2001).
  5. S. T. Cundiff, “Phase stabilization of ultrashort optical pulses,” J. Phys. D Appl. Phys. 35(8), 201 (2002).
    [Crossref]
  6. O. D. Mücke, R. Ell, A. Winter, J. W. 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(13), 5163–5169 (2005).
    [Crossref] [PubMed]
  7. L. Xu, Ch. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T. W. Hänsch, “Route to phase control of ultrashort light pulses,” Opt. Lett. 21(24), 2008–2010 (1996).
    [Crossref] [PubMed]
  8. H. A. Haus and E. P. Ippen, “Group velocity of solitons,” Opt. Lett. 26(21), 1654–1656 (2001).
    [Crossref]
  9. T. M. Fortier, J. Ye, S. T. Cundiff, and R. S. Windeler, “Nonlinear phase noise generated in air-silica microstructure fiber and its effect on carrier-envelope phase,” Opt. Lett. 27(6), 445–447 (2002).
    [Crossref]
  10. F. W. Helbing, G. Steinmeyer, U. Keller, R. S. Windeler, J. Stenger, and H. R. Telle, “Carrier-envelope offset dynamics of mode-locked lasers,” Opt. Lett. 27(3), 194–196 (2002).
    [Crossref]
  11. 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(1), 5–12 (2004).
    [Crossref]
  12. 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(11), 1999–2004 (1999).
    [Crossref]
  13. M. J. Ablowitz, B. Ilan, and S. T. Cundiff, “Carrier-envelope phase slip of ultrashort dispersion-managed solitons,” Opt. Lett. 29(15), 1808–1810 (2004).
    [Crossref] [PubMed]
  14. 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(4), 1018–1024 (2003).
    [Crossref]
  15. L. Matos, O. D. Mücke, J. Chen, and F. X. Kärtner, “Carrier-envelope phase dynamics and noise analysis in octave-spanning Ti:sapphire lasers,” Opt. Express 14(6), 2497–2511 (2006).
    [Crossref] [PubMed]
  16. M. Y. Sander, J. Birge, A. Benedick, H. M. Crespo, and F. X. Kärtner, “Dynamics of dispersion managed octave-spanning titanium:sapphire lasers,” J. Opt. Soc. Am. B 26(4), 743–749 (2009).
    [Crossref]
  17. M. Y. Sander, H. M. Crespo, J. R. Birge, and F. X. Kärtner, “Modeling of octave-spanning sub-two-cycle titanium:sapphire lasers: Simulation and Experiment,” Conference on Ultrafast Phenomena, (European Physical Society and Optical Society of America, 2008), THUIIIc.
  18. H. M. Crespo, J. R. Birge, M. Y. Sander, E. L. Falcão-Filho, A. Benedick, and F. X. Kärtner, “Phase stabilization of sub-two-cycle pulses from prismless octave-spanning Ti:sapphire lasers,” J. Opt. Soc. Am. B 25(7), B147–B154 (2008).
    [Crossref]
  19. M. Y. Sander, and F. X. Kärtner, “Carrier-envelope phase dynamics of octave-spanning Titanium:sapphire lasers,” Conference on Lasers and Electro Optics (CLEO), Baltimore, MD, June 1–5, 2009, CMN4.
  20. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007).
  21. T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
    [Crossref] [PubMed]
  22. T. R. Schibli, O. Kuzucu, J. Kim, E. P. Ippen, J. G. Fujimoto, F. X. Kaertner, V. Scheuer, and G. Angelow, “Toward single-cycle laser systems,” IEEE J. Sel. Top. Quantum Electron. 9(4), 990–1001 (2003).
    [Crossref]
  23. H. A. Haus, I. Sorokina, and E. Sorokin, “Raman-induced redshift of ultrashort mode-locked laser pulses,” J. Opt. Soc. Am. B 15(1), 223–231 (1998).
    [Crossref]
  24. I. T. Sorokina, E. Sorokin, E. Wintner, A. Cassanho, and H. P. Jenssen, “Raman induced pulse self-frequency shift in the sub-20 fs Kerr-lens mode-locked Cr:LiSGaF and Cr:LiSAF lasers,” OSA TOPS 19, 359–361 (1998).
  25. V. L. Kalashnikov, E. Sorokin, and I. T. Sorokina, “Mechanisms of spectral shift in ultrashort-pulse laser oscillators,” J. Opt. Soc. Am. B 18(11), 1732–1741 (2001).
    [Crossref]
  26. S. P. S. Porto and R. S. Krishnan, “Raman effect of Corundum,” J. Chem. Phys. 47(3), 1009–1012 (1967).
    [Crossref]
  27. M. Kadleikova, J. Breza, and M. Vesely, “Raman spectra of synthetic sapphire,” Microelectron. J. 32(12), 955–958 (2001).
    [Crossref]

2009 (1)

2008 (1)

2006 (1)

2005 (1)

2004 (2)

M. J. Ablowitz, B. Ilan, and S. T. Cundiff, “Carrier-envelope phase slip of ultrashort dispersion-managed solitons,” Opt. Lett. 29(15), 1808–1810 (2004).
[Crossref] [PubMed]

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(1), 5–12 (2004).
[Crossref]

2003 (3)

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(4), 1018–1024 (2003).
[Crossref]

A. Baltuška, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. 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(6923), 611–615 (2003).
[Crossref] [PubMed]

T. R. Schibli, O. Kuzucu, J. Kim, E. P. Ippen, J. G. Fujimoto, F. X. Kaertner, V. Scheuer, and G. Angelow, “Toward single-cycle laser systems,” IEEE J. Sel. Top. Quantum Electron. 9(4), 990–1001 (2003).
[Crossref]

2002 (4)

2001 (4)

H. A. Haus and E. P. Ippen, “Group velocity of solitons,” Opt. Lett. 26(21), 1654–1656 (2001).
[Crossref]

A. Poppe, R. Holzwarth, A. Apolonski, G. Tempea, Ch. Spielmann, T. W. Hänsch, and F. Krausz, “Few-cycle optical waveform synthesis,” Appl. Phys. B 72, 373–376 (2001).

V. L. Kalashnikov, E. Sorokin, and I. T. Sorokina, “Mechanisms of spectral shift in ultrashort-pulse laser oscillators,” J. Opt. Soc. Am. B 18(11), 1732–1741 (2001).
[Crossref]

M. Kadleikova, J. Breza, and M. Vesely, “Raman spectra of synthetic sapphire,” Microelectron. J. 32(12), 955–958 (2001).
[Crossref]

2000 (1)

T. Brabec and F. Krausz, “Intense few-cycle laser fields: frontiers of nonlinear optics,” Rev. Mod. Phys. 72(2), 545–591 (2000).
[Crossref]

1999 (1)

1998 (2)

H. A. Haus, I. Sorokina, and E. Sorokin, “Raman-induced redshift of ultrashort mode-locked laser pulses,” J. Opt. Soc. Am. B 15(1), 223–231 (1998).
[Crossref]

I. T. Sorokina, E. Sorokin, E. Wintner, A. Cassanho, and H. P. Jenssen, “Raman induced pulse self-frequency shift in the sub-20 fs Kerr-lens mode-locked Cr:LiSGaF and Cr:LiSAF lasers,” OSA TOPS 19, 359–361 (1998).

1997 (1)

P. Christov, M. M. Murnane, and H. C. Kapteyn, ““High-Harmonic Generation of Attosecond Pulses in the “Single-Cycle” Regime,” Phys. Rev. Lett. 78(7), 1251–1254 (1997).
[Crossref]

1996 (1)

1967 (1)

S. P. S. Porto and R. S. Krishnan, “Raman effect of Corundum,” J. Chem. Phys. 47(3), 1009–1012 (1967).
[Crossref]

Ablowitz, M. J.

Angelow, G.

T. R. Schibli, O. Kuzucu, J. Kim, E. P. Ippen, J. G. Fujimoto, F. X. Kaertner, V. Scheuer, and G. Angelow, “Toward single-cycle laser systems,” IEEE J. Sel. Top. Quantum Electron. 9(4), 990–1001 (2003).
[Crossref]

Apolonski, A.

A. Poppe, R. Holzwarth, A. Apolonski, G. Tempea, Ch. Spielmann, T. W. Hänsch, and F. Krausz, “Few-cycle optical waveform synthesis,” Appl. Phys. B 72, 373–376 (2001).

Baltuška, A.

A. Baltuška, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. 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(6923), 611–615 (2003).
[Crossref] [PubMed]

Benedick, A.

Birge, J.

Birge, J. R.

Brabec, T.

T. Brabec and F. Krausz, “Intense few-cycle laser fields: frontiers of nonlinear optics,” Rev. Mod. Phys. 72(2), 545–591 (2000).
[Crossref]

L. Xu, Ch. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T. W. Hänsch, “Route to phase control of ultrashort light pulses,” Opt. Lett. 21(24), 2008–2010 (1996).
[Crossref] [PubMed]

Breza, J.

M. Kadleikova, J. Breza, and M. Vesely, “Raman spectra of synthetic sapphire,” Microelectron. J. 32(12), 955–958 (2001).
[Crossref]

Cassanho, A.

I. T. Sorokina, E. Sorokin, E. Wintner, A. Cassanho, and H. P. Jenssen, “Raman induced pulse self-frequency shift in the sub-20 fs Kerr-lens mode-locked Cr:LiSGaF and Cr:LiSAF lasers,” OSA TOPS 19, 359–361 (1998).

Chen, J.

Chen, Y.

Cho, S. H.

Christov, P.

P. Christov, M. M. Murnane, and H. C. Kapteyn, ““High-Harmonic Generation of Attosecond Pulses in the “Single-Cycle” Regime,” Phys. Rev. Lett. 78(7), 1251–1254 (1997).
[Crossref]

Crespo, H. M.

Cundiff, S. T.

M. J. Ablowitz, B. Ilan, and S. T. Cundiff, “Carrier-envelope phase slip of ultrashort dispersion-managed solitons,” Opt. Lett. 29(15), 1808–1810 (2004).
[Crossref] [PubMed]

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(4), 1018–1024 (2003).
[Crossref]

T. M. Fortier, J. Ye, S. T. Cundiff, and R. S. Windeler, “Nonlinear phase noise generated in air-silica microstructure fiber and its effect on carrier-envelope phase,” Opt. Lett. 27(6), 445–447 (2002).
[Crossref]

S. T. Cundiff, “Phase stabilization of ultrashort optical pulses,” J. Phys. D Appl. Phys. 35(8), 201 (2002).
[Crossref]

Eikema, K. S. E.

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(1), 5–12 (2004).
[Crossref]

Ell, R.

Falcão-Filho, E. L.

Fortier, T. M.

Fujimoto, J. G.

T. R. Schibli, O. Kuzucu, J. Kim, E. P. Ippen, J. G. Fujimoto, F. X. Kaertner, V. Scheuer, and G. Angelow, “Toward single-cycle laser systems,” IEEE J. Sel. Top. Quantum Electron. 9(4), 990–1001 (2003).
[Crossref]

Y. Chen, F. X. Kärtner, U. Morgner, S. H. Cho, H. A. Haus, E. P. Ippen, and J. G. Fujimoto, “Dispersion-managed mode locking,” J. Opt. Soc. Am. B 16(11), 1999–2004 (1999).
[Crossref]

Gohle, Ch.

A. Baltuška, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. 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(6923), 611–615 (2003).
[Crossref] [PubMed]

Goulielmakis, E.

A. Baltuška, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. 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(6923), 611–615 (2003).
[Crossref] [PubMed]

Hänsch, T. W.

A. Baltuška, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. 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(6923), 611–615 (2003).
[Crossref] [PubMed]

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
[Crossref] [PubMed]

A. Poppe, R. Holzwarth, A. Apolonski, G. Tempea, Ch. Spielmann, T. W. Hänsch, and F. Krausz, “Few-cycle optical waveform synthesis,” Appl. Phys. B 72, 373–376 (2001).

L. Xu, Ch. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T. W. Hänsch, “Route to phase control of ultrashort light pulses,” Opt. Lett. 21(24), 2008–2010 (1996).
[Crossref] [PubMed]

Haus, H. A.

Helbing, F. W.

Hentschel, M.

A. Baltuška, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. 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(6923), 611–615 (2003).
[Crossref] [PubMed]

Hogervorst, W.

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(1), 5–12 (2004).
[Crossref]

Holman, K. W.

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(4), 1018–1024 (2003).
[Crossref]

Holzwarth, R.

A. Baltuška, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. 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(6923), 611–615 (2003).
[Crossref] [PubMed]

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
[Crossref] [PubMed]

A. Poppe, R. Holzwarth, A. Apolonski, G. Tempea, Ch. Spielmann, T. W. Hänsch, and F. Krausz, “Few-cycle optical waveform synthesis,” Appl. Phys. B 72, 373–376 (2001).

Ilan, B.

Ippen, E. P.

T. R. Schibli, O. Kuzucu, J. Kim, E. P. Ippen, J. G. Fujimoto, F. X. Kaertner, V. Scheuer, and G. Angelow, “Toward single-cycle laser systems,” IEEE J. Sel. Top. Quantum Electron. 9(4), 990–1001 (2003).
[Crossref]

H. A. Haus and E. P. Ippen, “Group velocity of solitons,” Opt. Lett. 26(21), 1654–1656 (2001).
[Crossref]

Y. Chen, F. X. Kärtner, U. Morgner, S. H. Cho, H. A. Haus, E. P. Ippen, and J. G. Fujimoto, “Dispersion-managed mode locking,” J. Opt. Soc. Am. B 16(11), 1999–2004 (1999).
[Crossref]

Jenssen, H. P.

I. T. Sorokina, E. Sorokin, E. Wintner, A. Cassanho, and H. P. Jenssen, “Raman induced pulse self-frequency shift in the sub-20 fs Kerr-lens mode-locked Cr:LiSGaF and Cr:LiSAF lasers,” OSA TOPS 19, 359–361 (1998).

Jones, R. J.

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(4), 1018–1024 (2003).
[Crossref]

Kadleikova, M.

M. Kadleikova, J. Breza, and M. Vesely, “Raman spectra of synthetic sapphire,” Microelectron. J. 32(12), 955–958 (2001).
[Crossref]

Kaertner, F. X.

T. R. Schibli, O. Kuzucu, J. Kim, E. P. Ippen, J. G. Fujimoto, F. X. Kaertner, V. Scheuer, and G. Angelow, “Toward single-cycle laser systems,” IEEE J. Sel. Top. Quantum Electron. 9(4), 990–1001 (2003).
[Crossref]

Kalashnikov, V. L.

Kapteyn, H. C.

P. Christov, M. M. Murnane, and H. C. Kapteyn, ““High-Harmonic Generation of Attosecond Pulses in the “Single-Cycle” Regime,” Phys. Rev. Lett. 78(7), 1251–1254 (1997).
[Crossref]

Kärtner, F. X.

Keller, U.

Kim, J.

T. R. Schibli, O. Kuzucu, J. Kim, E. P. Ippen, J. G. Fujimoto, F. X. Kaertner, V. Scheuer, and G. Angelow, “Toward single-cycle laser systems,” IEEE J. Sel. Top. Quantum Electron. 9(4), 990–1001 (2003).
[Crossref]

Kim, J. W.

Krausz, F.

A. Baltuška, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. 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(6923), 611–615 (2003).
[Crossref] [PubMed]

A. Poppe, R. Holzwarth, A. Apolonski, G. Tempea, Ch. Spielmann, T. W. Hänsch, and F. Krausz, “Few-cycle optical waveform synthesis,” Appl. Phys. B 72, 373–376 (2001).

T. Brabec and F. Krausz, “Intense few-cycle laser fields: frontiers of nonlinear optics,” Rev. Mod. Phys. 72(2), 545–591 (2000).
[Crossref]

L. Xu, Ch. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T. W. Hänsch, “Route to phase control of ultrashort light pulses,” Opt. Lett. 21(24), 2008–2010 (1996).
[Crossref] [PubMed]

Krishnan, R. S.

S. P. S. Porto and R. S. Krishnan, “Raman effect of Corundum,” J. Chem. Phys. 47(3), 1009–1012 (1967).
[Crossref]

Kuzucu, O.

T. R. Schibli, O. Kuzucu, J. Kim, E. P. Ippen, J. G. Fujimoto, F. X. Kaertner, V. Scheuer, and G. Angelow, “Toward single-cycle laser systems,” IEEE J. Sel. Top. Quantum Electron. 9(4), 990–1001 (2003).
[Crossref]

Marian, A.

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(4), 1018–1024 (2003).
[Crossref]

Matos, L.

Morgner, U.

Mücke, O. D.

Murnane, M. M.

P. Christov, M. M. Murnane, and H. C. Kapteyn, ““High-Harmonic Generation of Attosecond Pulses in the “Single-Cycle” Regime,” Phys. Rev. Lett. 78(7), 1251–1254 (1997).
[Crossref]

Poppe, A.

A. Poppe, R. Holzwarth, A. Apolonski, G. Tempea, Ch. Spielmann, T. W. Hänsch, and F. Krausz, “Few-cycle optical waveform synthesis,” Appl. Phys. B 72, 373–376 (2001).

L. Xu, Ch. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T. W. Hänsch, “Route to phase control of ultrashort light pulses,” Opt. Lett. 21(24), 2008–2010 (1996).
[Crossref] [PubMed]

Porto, S. P. S.

S. P. S. Porto and R. S. Krishnan, “Raman effect of Corundum,” J. Chem. Phys. 47(3), 1009–1012 (1967).
[Crossref]

Sander, M. Y.

Scheuer, V.

T. R. Schibli, O. Kuzucu, J. Kim, E. P. Ippen, J. G. Fujimoto, F. X. Kaertner, V. Scheuer, and G. Angelow, “Toward single-cycle laser systems,” IEEE J. Sel. Top. Quantum Electron. 9(4), 990–1001 (2003).
[Crossref]

Schibli, T. R.

T. R. Schibli, O. Kuzucu, J. Kim, E. P. Ippen, J. G. Fujimoto, F. X. Kaertner, V. Scheuer, and G. Angelow, “Toward single-cycle laser systems,” IEEE J. Sel. Top. Quantum Electron. 9(4), 990–1001 (2003).
[Crossref]

Scrinzi, A.

A. Baltuška, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. 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(6923), 611–615 (2003).
[Crossref] [PubMed]

Sorokin, E.

Sorokina, I.

Sorokina, I. T.

V. L. Kalashnikov, E. Sorokin, and I. T. Sorokina, “Mechanisms of spectral shift in ultrashort-pulse laser oscillators,” J. Opt. Soc. Am. B 18(11), 1732–1741 (2001).
[Crossref]

I. T. Sorokina, E. Sorokin, E. Wintner, A. Cassanho, and H. P. Jenssen, “Raman induced pulse self-frequency shift in the sub-20 fs Kerr-lens mode-locked Cr:LiSGaF and Cr:LiSAF lasers,” OSA TOPS 19, 359–361 (1998).

Spielmann, Ch.

A. Poppe, R. Holzwarth, A. Apolonski, G. Tempea, Ch. Spielmann, T. W. Hänsch, and F. Krausz, “Few-cycle optical waveform synthesis,” Appl. Phys. B 72, 373–376 (2001).

L. Xu, Ch. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T. W. Hänsch, “Route to phase control of ultrashort light pulses,” Opt. Lett. 21(24), 2008–2010 (1996).
[Crossref] [PubMed]

Steinmeyer, G.

Stenger, J.

Telle, H. R.

Tempea, G.

A. Poppe, R. Holzwarth, A. Apolonski, G. Tempea, Ch. Spielmann, T. W. Hänsch, and F. Krausz, “Few-cycle optical waveform synthesis,” Appl. Phys. B 72, 373–376 (2001).

Udem, T.

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
[Crossref] [PubMed]

Udem, Th.

A. Baltuška, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. 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(6923), 611–615 (2003).
[Crossref] [PubMed]

Uiberacker, M.

A. Baltuška, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. 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(6923), 611–615 (2003).
[Crossref] [PubMed]

Vesely, M.

M. Kadleikova, J. Breza, and M. Vesely, “Raman spectra of synthetic sapphire,” Microelectron. J. 32(12), 955–958 (2001).
[Crossref]

Windeler, R. S.

Winter, A.

Wintner, E.

I. T. Sorokina, E. Sorokin, E. Wintner, A. Cassanho, and H. P. Jenssen, “Raman induced pulse self-frequency shift in the sub-20 fs Kerr-lens mode-locked Cr:LiSGaF and Cr:LiSAF lasers,” OSA TOPS 19, 359–361 (1998).

Witte, S.

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(1), 5–12 (2004).
[Crossref]

Xu, L.

Yakovlev, V. S.

A. Baltuška, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. 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(6923), 611–615 (2003).
[Crossref] [PubMed]

Ye, J.

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(4), 1018–1024 (2003).
[Crossref]

T. M. Fortier, J. Ye, S. T. Cundiff, and R. S. Windeler, “Nonlinear phase noise generated in air-silica microstructure fiber and its effect on carrier-envelope phase,” Opt. Lett. 27(6), 445–447 (2002).
[Crossref]

Zinkstok, R. T.

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(1), 5–12 (2004).
[Crossref]

Appl. Phys. B (2)

A. Poppe, R. Holzwarth, A. Apolonski, G. Tempea, Ch. Spielmann, T. W. Hänsch, and F. Krausz, “Few-cycle optical waveform synthesis,” Appl. Phys. B 72, 373–376 (2001).

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(1), 5–12 (2004).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (2)

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(4), 1018–1024 (2003).
[Crossref]

T. R. Schibli, O. Kuzucu, J. Kim, E. P. Ippen, J. G. Fujimoto, F. X. Kaertner, V. Scheuer, and G. Angelow, “Toward single-cycle laser systems,” IEEE J. Sel. Top. Quantum Electron. 9(4), 990–1001 (2003).
[Crossref]

J. Chem. Phys. (1)

S. P. S. Porto and R. S. Krishnan, “Raman effect of Corundum,” J. Chem. Phys. 47(3), 1009–1012 (1967).
[Crossref]

J. Opt. Soc. Am. B (5)

J. Phys. D Appl. Phys. (1)

S. T. Cundiff, “Phase stabilization of ultrashort optical pulses,” J. Phys. D Appl. Phys. 35(8), 201 (2002).
[Crossref]

Microelectron. J. (1)

M. Kadleikova, J. Breza, and M. Vesely, “Raman spectra of synthetic sapphire,” Microelectron. J. 32(12), 955–958 (2001).
[Crossref]

Nature (2)

A. Baltuška, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. 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(6923), 611–615 (2003).
[Crossref] [PubMed]

T. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature 416(6877), 233–237 (2002).
[Crossref] [PubMed]

Opt. Express (2)

Opt. Lett. (5)

OSA TOPS (1)

I. T. Sorokina, E. Sorokin, E. Wintner, A. Cassanho, and H. P. Jenssen, “Raman induced pulse self-frequency shift in the sub-20 fs Kerr-lens mode-locked Cr:LiSGaF and Cr:LiSAF lasers,” OSA TOPS 19, 359–361 (1998).

Phys. Rev. Lett. (1)

P. Christov, M. M. Murnane, and H. C. Kapteyn, ““High-Harmonic Generation of Attosecond Pulses in the “Single-Cycle” Regime,” Phys. Rev. Lett. 78(7), 1251–1254 (1997).
[Crossref]

Rev. Mod. Phys. (1)

T. Brabec and F. Krausz, “Intense few-cycle laser fields: frontiers of nonlinear optics,” Rev. Mod. Phys. 72(2), 545–591 (2000).
[Crossref]

Other (3)

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

M. Y. Sander, and F. X. Kärtner, “Carrier-envelope phase dynamics of octave-spanning Titanium:sapphire lasers,” Conference on Lasers and Electro Optics (CLEO), Baltimore, MD, June 1–5, 2009, CMN4.

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

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Figures (6)

Fig. 1
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.
Fig. 2
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.
Fig. 3
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).
Fig. 4
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.
Fig. 5
Fig. 5 Frequency shift Δωfilter due to a rectangular filter with bandwidth Ωf = 1.4 · 1015 rad/s for a spectrum shifted by Δω from the carrier frequency for three values of soliton pulse width τsech .
Fig. 6
Fig. 6 Frequency shifts Δωfilter induced by a rectangular filter of bandwidth of Ωf = 1.4 ·1015 rad/s on a pulse with τsech = 3.5 fs. The linear approximation according to Eq. (15) is plotted as a dashed line.

Tables (2)

Tables Icon

Table 1 Time constants for relaxation behavior from filtering by mirrors and output coupler τFilter and from gain filtering τGain for a given pulse duration τsech .

Tables Icon

Table 2 Raman shift ΔωRaman , value of convolution function G(ωRτsech), resulting frequency shift ΔωTotal and CEPS contribution for each Raman line ωR for different pulse parameters τsech .

Equations (17)

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A ( z , t ) z ​ ​ = [ g ( z , t ) q ( z , t )   ​ j δ { | A ( z , t ) | 2 + j ω C ​ ​ ​ t ( | A ( z , t ) | 2 ) } j ϕ ( j t ) ​ ​ ] A ( z , t ) ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​
E ( t ) = n A ( t n T R n τ )   ​ e j ω c ( t n T R n τ ) e j n Φ C E ​ ​
Φ C E = Φ C E , N L   ​ + Φ C E , L i n =   ​ Φ + ω c τ + Φ C E = = Φ + ω c τ + ω c L ( 1 v g 1 v p ) = Φ + ω c τ + ω c i L i ( k i ω | ω C k i ω c )
Φ C E W   ​ = Φ W   ​ + ω c τ W + τ ω c W ω c i L i 2 k i ω 2 | ω c ω c W
Φ W = 0.06   rad/nJ and τ W = 0.1   fs/nJ
Φ C E W   ​ = Φ W   ​ + ω c τ W = 0.06   rad/nJ + 0.23   rad/nJ = 0.17   rad/nJ
Δ ω R a m a n = i π 16 1 τ s e c h g R δ ω R G ( ω R , i τ s e c h ) L A W
T Δ ω = Δ ω F i l t e r + Δ ω G a i n + Δ ω R a m a n = Δ ω τ F i l t e r Δ ω τ G a i n + Δ ω R a m a n with 1 τ G a i n = 4 3 Ω G 2 τ s e c h 2
Δ ω T o t a l = 1 1 τ F i l t e r + 1 τ G a i n Δ ω R a m a n = i 1 1 τ F i l t e r + 1 τ G a i n π 16 1 τ s e c h g R δ ω R , i G ( ω R τ s e c h ) L A W
Δ f C E Δ P = 1 2 π Δ Φ C E Δ W C Δ f C E , S o l i t o n Δ P   with C 2.6 2.6 L n 2 4 λ A e f f τ s e c h = 30.3   MHz/W
f ¯ ω ( t ) = j A 0 τ 2 tanh ( t τ sech )   sech ( t τ sech )
Δ ω f i l t e r = Re f ¯ ω * Δ a ( t ) d t = Re f ¯ ω * g f i l t e r ( t ) A ( t ) d t
f ¯ ω * Δ a ( t ) d t = f ¯ ω * g f i l t e r ( t ) A ( t ) d t = F { f ¯ ω * } F { g f i l t e r ( t ) } F { A ( t ) } d ω
sech ( t ) π   sech ( π 2 ω ) tanh ( t )   sech ( t ) j π ω   sech ( π 2 ω )
Δ ω f i l t e r = Re f ¯ ω * Δ a ( t ) d t = Re j A 0 τ 2 tanh ( t τ sech )   sech ( t τ sech ) g f i l t e r ( t ) A ( t ) d t = = Re j A 0 τ sech 2 tanh ( t τ sech )   sech ( t τ sech ) F 1 {   rect ( ω Δ ω Δ ω f Ω f ) } A 0   sech ( t τ sech ) d t = Re F { f ¯ ω * } F { g f i l t e r ( t ) } F { A ( t ) } d ω = = Re j A 0 τ sech 2 ( j τ sech 2 ) π ω   sech ( π 2 τ sech ω )   rect ( ω Δ ω Δ ω f Ω f ) ... ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ... ( A 0 ) π τ sech   sech ( π 2 τ sech ω ) d ω = ​ ​ ​ ​ ​ ​ ​ = Ω f 2 + Δ ω + Δ ω f Ω f 2 + Δ ω + Δ ω f π 2 τ sech ω sech 2 ( π 2 τ sech ω ) d ω = = π 2 τ sech [ 4 π 2 τ sech 2   log [ cosh ( π 2 τ sech ω ) ] + 2 ω π τ sech   tanh ( π 2 τ sech ω ) ] | Ω f 2 + Δ ω + Δ ω f Ω f 2 + Δ ω + Δ ω f = = 4 τ sech {  log [ cosh ( π 2 τ sech ( Ω f 2 + Δ ω + Δ ω f ) ) ] +   log  [ cosh ( π 2 τ sech ( Ω f 2 + Δ ω + Δ ω f ) ) ] } + ... + 2 π [ ( Ω f 2 + Δ ω + Δ ω f )   tanh ( π 2 τ sech ( Ω f 2 + Δ ω + Δ ω f ) ) ] ... 2 π [ ( Ω f 2 + Δ ω + Δ ω f )   tanh ( π 2 τ sech ( Ω f 2 + Δ ω + Δ ω f ) ) ]
Δ ω f i l t e r Δ ω f ( Ω f 2 + Δ ω f + Δ ω Ω f 2 + Δ ω f + Δ ω π 2 τ sech ω sech 2 ( π 2 τ sech ω ) d ω | Δ ω = 0 ) Δ ω = π 2 τ sech ( Ω f 2 + Δ ω f + Δ ω ) sech 2 ( π 2 τ sech ( Ω f 2 + Δ ω f + Δ ω ) ) | Δ ω = 0 Δ ω + ... ... + π 2 τ sech ( Ω f 2 + Δ ω f + Δ ω ) sech 2 ( π 2 τ sech ( Ω f 2 + Δ ω f + Δ ω ) ) | Δ ω = 0 Δ ω = π 2 τ sech [ ( Ω f 2 + Δ ω f ) sech 2 ( π 2 τ sech ( Ω f 2 + Δ ω f ) ) ( Ω f 2 + Δ ω f ) sech 2 ( π 2 τ sech ( Ω f 2 + Δ ω f ) ) ] Δ ω
Δ ω f i l t e r = f o r Δ ω f = 0 π 2 τ sech Ω f sech 2 ( τ sech π 2 Ω f 2 ) Δ ω = 1 τ f i l t e r Δ ω

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