Without introducing any additional assumptions, we have solved the Eqs. (1
) numerically by the symmetrized split-step Fourier method. To provide high spectral resolution, the simulation local time window contains 222
points with the mesh interval 2.5 fs. The simulation parameters for the cubic-quintic version of Eq. (1)
are presented in . The GDD parameter β2
= −1600 fs2
provides a stable single pulse with FWHM ≈100 fs. The single low-power seeding pulse converges to a steady-state solution during z
The pulse propagation within a linear medium (e.g. an absorbing gas outside an oscillator, a passive fiber containing some impurities, or microstructured fiber filled with a gas) is described by Eqs. (1
) with zero α
, and the initial A
) corresponding to output oscillator pulse. The obvious effect of the absorption lines on a pulse spectrum are the dips at ωl
(), that simply follow the Beer’s law. This regime allows using the ultrashort pulse for conventional absorption spectroscopy [1
]. The nonzero real part of an absorber permittivity (i.e.
) does significantly change the pulse in time domain [22
], but does not alter the spectrum. The pulse spectrum reveals only imaginary part of an absorber permittivity (i.e.
FIG. 4 (Color online) Central parts of the dissipative soliton spectra in an oscillator with the single absorption line centered at ω =0. (a) = −0.05; Ω = 4 GHz (solid curve, open circles and crosses) and 1 GHz (dashed curve, open squares (more ...)
Introducing the nonzero SPM coefficient
with zero α
transforms Eq. (1)
to a perturbed nonlinear Schrödinger equation and results in the true perturbed soliton propagation. In this case, as shown in , the situation becomes dramatically different. Besides the dips in the spectrum shown by the gray curve corresponding to a contribution of
only, there is a pronounced contribution from the phase change induced by the dispersion of absorption lines (solid curve in corresponds to the complex profile of
in Eq. (2)
). As a result, the spectral profile has the sharp bends with the maximum on the low-frequency side and the minimum on the high-frequency side of the corresponding absorption line. At the same time, the dips in the spectrum due to absorption are strongly suppressed. In addition to spectral features, the soliton decays, acquires a slight shift towards the higher frequencies, and its spectrum gets narrower due to the energy loss.
FIG. 2 (Color online) Part of the pulse spectrum after: (a) linear propagation for 25 dispersion lengths inside a fiber with two absorption lines and (b) perturbed soliton propagation for 100 dispersion lengths. Gray curve in (b) corresponds to the contribution (more ...)
The soliton spectrum reveals in this case the real part of the absorber permittivity. However, the continuous change of the soliton shape due to energy decay renders the problem as a non-steady state case. The situation becomes different in a laser oscillator, where pumping provides a constant energy flow to compensate the absorption loss.
Let us consider the steady-state intra-cavity narrowband absorption inside a passively modelocked femtosecond oscillator, where the pulse is controlled by the SPM and the SAM, which is described by the cubic-quintic
in Eq. (1)
modeling the Kerr-lens mode-locking mechanism [12
]. Such an oscillator can operate both, in the negative dispersion regime [12
] with an chirped-free soliton-like pulse, and in the positive dispersion regime [18
], where the propagating pulse acquires strong positive chirp. In this study, we consider only the negative dispersion regime, the positive dispersion regime will be a subject of following studies.
The results of the simulation are shown in Figs. and , and they demonstrate the same dispersion-like modulation of the pulse spectrum. demonstrates action by three narrow (Ω =2 GHz) absorption lines centered at −10, 0 and 10 GHz in the neighborhood of ω = 0. One can see (), that the absorption lines do not cause spectral dips at ωl, but produce sharp bends, very much like the case of the true perturbed Schrödinger soliton considered before. One can also clearly see the collective redistribution of spectral power from higher- to lower-frequencies, which enhances local spectral asymmetry (). Such an asymmetry suggests that the dominating contribution to a soliton perturbation results from the real part of an absorber permittivity, which, in particular, causes the time-asymmetry of perturbation in the time domain. This asymmetry is seen in time domain as a ns-long modulated exponential precursor in .
The simulated effect of a single narrow absorption line centered at ω
= 0 is shown in for different values of peak absorption
and width Ω. In ,
= −0.05 and Ω = 4 GHz (solid curve, open circles and crosses) and 1 GHz (dashed curve, open squares and triangles). The solid and dashed curves demonstrate the action of complex profile (2)
, whereas circles (squares) and crosses (triangles) demonstrate the separate action of
, respectively. One can see, that the profile of perturbed spectrum traces that formed by only
(i.e. it traces the real part of an absorber permittivity). One can say, that the pure phase effect (
, circles and squares in ) strongly dominates over the pure absorption (
, crosses in ), like that for the Schrödinger soliton. Such a domination enhances with a lowered
(, crosses), however the |
| growth increases the relative contribution of
and causes the frequency downshift of the bend (). Amplitude of the bend traces the
value, while its width is defined by Ω.
The SAM considered above is modeled by the cubicquintic nonlinear term
. Such a SAM is typically realized by using the self-focusing inside an active medium (Kerr-lens modelocking). For reliable self-starting operation of mode-locked oscillator it is often desirable to use a suitable saturable absorber (SA), e.g. a semiconductor-based SESAM [23
]. Such an absorber can be described in the simplest case by a single-lifetime two-level model, giving the time-dependent loss coefficient Λ (t
is the loss coefficient for a small signal, Aeff
is the effective beam area on the SA, Ts
are the SA relaxation time and the saturation energy fluency, respectively. Eq. (4)
supplements Eq. (1)
and the SAM term
in the latter has to be replaced by (Λ0
)). When the pulse width is longer than the SA relaxation time, one can replace Eq. (4)
by its adiabatic solution so that
where ξ Ts
is the inverse saturation power.
We have simulated Eqs. (1
) in the case of Js
=0.5 ps, that correspond to the measurement in . Two cases have been considered: weak focusing (Aeff
or saturation energy Es
=2 nJ) and hard focusing (Aeff
or saturation energy Es
=0.5 nJ). We also considered Eq. (5)
for the same peak saturation level as weakly focused SA (i.e. ξ−1
=4 kW). In the latter case the SA effectively becomes instantaneous, and the perturbed soliton spectrum is the same as for the Kerr-lens modelocking, i.e. as for the cubic-quintic
. When the saturation energy is sufficiently large, there is no difference between the models expressed by Eqs. (4)
. The effect of a narrow absorption line is similar to that of the soliton of the cubic-quintic Eq. (1)
. Decrease of the saturation energy Es
causes down-shift of the pulse spectrum as a whole, but the narrow bend on the soliton spectrum reproduces the real part of the absorber permittivity. One can thus conclude that the type of the SAM is irrelevant for an effect of the narrowband absorption lines on a dissipative soliton spectrum.
Another important conclusion from the numerical simulations is the demonstrated stability of the dissipative soliton against perturbations induced by narrowband absorption. In the following analytical treatment we shall therefore omit the stability analysis.