## Abstract

A new scheme for supercontinuum generation covering more than one octave and exhibiting extraordinary high coherence properties has recently been proposed [Phys. Rev. Lett. **110**, 233901 (2013)]. The scheme is based on two-pulse collision at a group velocity horizon between a dispersive wave and a soliton. Here we demonstrate that the same scheme can be exploited for the generation of supercontinua encompassing the entire transparency region of fused silica, ranging from 300 to 2300nm. At this bandwidth extension, the Raman effect becomes detrimental, yet may be compensated by using a cascaded collision process. Consequently, the high degree of coherence does not degrade even in this extreme scenario.

© 2014 Optical Society of America

## 1. Introduction

White-light supercontinuum (SC) generation [1] is one of the most intriguing phenomena in nonlinear fiber optics and has revolutionized the field of frequency metrology [2]. Moreover, this process has led to a number of technological advances in various fields, including but not limited to spectroscopy, optical communications, and medical imaging [3–5]. Depending on the fiber and input pulse parameters, spectral broadening during SC generation may be caused by distinctively different mechanisms. To this end, the most efficient way is given by injection of the seed pulse into the anomalous dispersion regime. Here the interplay of soliton fission (SF) [6] and the generation of phase-matched radiation in the normal dispersion regime enables extremely broad spectra at relatively low input powers. This generic mechanism has already been studied extensively, and recently new insight into this phenomenon has been brought up with unexpected analogies to other fields in physics, such as the dynamics of ultracold atoms and ocean waves [7, 8]. Furthermore, it turned out that the complex transformation processes between dispersive radiation and solitons have remarkable impact on the spectrum (for a review, see [9]). Nevertheless, spectral broadening via SF comes with a severe drawback. This mechanisms makes the SC highly susceptible to noise in the input pulse. This input noise is amplified through the modulation instability (MI), which is also always inherently provided in the anomalous dispersion regime [10–12], even if it may play a subdominant role for SC generation. For example, the resulting poor spectral coherence was identified as the reason for the observed temporal incompressibility of these extremely broad spectra [1]. Injection of the seed pulse in the normal dispersion regime leads to spectral broadening chiefly by self-phase modulation and the accompanying Raman effect. In this dispersion regimen and for fs pulses, the sensitivity to input noise is strongly reduced, and one can achieve stable spectra that are virtually free of the spectral modulations and fluctuations seen for anomalous dispersion. Using ps pulses in the normally dispersive regime, one often observes significant noise seeding by spontaneous Raman scattering [13]. Using shorter seed pulses, however, a normally dispersive seed helps to avoid the complicated quantum noise seeded temporal pulse splitting, yet the spectral broadening also appears strongly reduced in comparison to soliton fission. This seemingly unavoidable tradeoff between broadening efficiency and loss of coherence describes the fundamental dilemma of fiber-based supercontinuum generation.

In Ref. [14], a two-color scheme for SC generation has been presented that first offered a perspective towards overcoming the above mentioned dilemma. This method relies on a collision process between a fundamental soliton and a dispersive wave. The spectral broadening is provided by a combination of soliton compression in the anomalous dispersion regime and nonlinear frequency conversion in the normal dispersion regime. In contrast to other SC generation mechanisms, the output spectra exhibit high coherence and homogeneous spectral power density as the underlying generation mechanism is independent of SF and MI. These favorable properties make the spectra highly interesting for applications, e.g., the compression into a coherent single-cycle pulse, i.e., a topic that remained rather discouraging to date. It is important to note that the exploited specific resonant interaction is given by XPM between solitons and dispersive waves, which, in turn, is inherently contained in the standard SC scenario accompanying SF [15, 16]. Moreover, this enhanced XPM mechanism does not only support the SC process, but provides a possibility of its own for SC generation encompassing more than an octave from the anomalous to the normal dispersion regime between the utilized two seed wavelengths. This principle is related to an optical analog of the event horizon [17–19]. This analog relies on the ubiquitous wave blocking phenomenon known from fluid dynamics [20, 21]. In optics, exploitation of this principle has first been suggested for a fiber Bragg grating and is known as the “optical push broom effect” [22]. Recently, this scenario has been revisited from a more general perspective, with two-beam collision processes [26] or collisions of an optical pulse with a moving inhomogeneity [24, 25].

Experimentally this specific XPM action has been demonstrated in a photonic crystal fiber [17], a fiber Bragg grating [23] as well as in a nonlinear optical mesh lattice [27]. Experimental proof of principle has been demonstrated directly for the interaction between a fundamental soliton and a dispersive wave in a microstructured fiber, too [28].

For our purposes it is important that this kind of interaction especially between a fundamental soliton and a dispersive wave is able to induce a frequency shift not only for a weak dispersive wave but also for a strong fundamental soliton [29]. Especially this possibility to induce a center frequency for both pulses can be exploited for a vastly stronger manipulation of optical pulses [30, 31]. To this end, however, appropriate material parameters have to be chosen. The induced center frequency shift then results in adiabatic soliton reshaping, i.e., effectively enables control of peak power and pulse width of the soliton. Qualitatively similar reshaping behavior results from the Raman-induced soliton self-frequency shift [32] or from ionization [33], with the former causing a red-shift and the latter a blue-shift. In our case, however, the frequency shift is induced by the dispersive wave and may be used to shift the center wavelength to either direction. The resulting supercontinua exhibit a high degree of coherence.

## 2. Propagation model in terms of the analytical signal

In this section we outline derivation of the basic propagation equation [34,35] using the Hamiltonian framework [36]. We start with the pulse propagation in a single-mode fiber which is described by an effectively one-dimensional propagation equation for the real-valued optical field *E*(*z*, *t*)

*c*is the speed of light,

*χ*

^{(3)}is the third-order nonlinear susceptibility. Both bulk and geometric contributions to the fiber dispersion are encoded in the linear operator

*ε̃*, which is defined via convolution with some effective dispersion function

*ε*

_{eff}(

*ω*). For the time being the fiber is dissipation-free, the Raman effect will be accounted for later on. For the Fourier components

*E*(

_{ω}*z*) we obtain

*β*

^{2}(

*ω*)

*c*

^{2}=

*ω*

^{2}

*ε*

_{eff}(

*ω*), and (

*E*

^{3})

*denotes Fourier transform of the nonlinear terms. The key idea is to consider Eq. (2) as a system of weakly coupled nonlinear oscillators with the “frequency” |*

_{ω}*β*(

*ω*)| and to transform the left-hand-side of Eq. (2) to the normal form,

*i*∂

*+ |*

_{z}𝒜_{ω}*β*(

*ω*)|

*𝒜*, for a suitable normal variable

_{ω}*𝒜*(

_{ω}*z*). Equation (2) is then treated as a Hamiltonian

*H*

_{nonlin}generates the nonlinear terms in Eq. (2). Equation (3) provides a

*z*-propagation version of the standard time-evolution Hamiltonian equation of the form

*𝒜*(

*z*,

*t*) is introduced by the relations

*E*(

*z*,

*t*) = Re[Ψ(

*z*,

*t*)

*e*

^{−iω0t}], where it is impossible to express Ψ(

*z*,

*t*) in terms of

*E*(

*z*,

*t*) without further assumptions such as the slowly-varying envelope approximation (SVEA). Equation (1) is transformed to the form

*β*=

*β*(

*ω*) does not support the third-harmonics generation process. Neglecting then all non-resonant terms, one obtains a much more simple expression

The above Hamilton equations provide an easy way to derive the constants of motion. For a sequence of pulses propagating with the time period *T* the Hamiltonian (8) can be transformed to the form

We are in a good position to stress that equations (6) and (10) look like the generalized nonlinear Schrödinger equations (GNSE) and can be solved in the same way, e.g., with the split-step method. However, Eq. (6) is an exact reformulation of Eq. (1), the transformation (5) is just a change of variables. Also a non-resonant dispersion law that is required to derive Eq. (10) is a weaker assumption than the SVEA. In particular, both (6) and (10) are *bidirectional* equations. The standard GNSE is unidirectional, it appears if one additionally applies an unidirectional approximation. Indeed, let us consider how the differential operator in Eq. (5) acts on a forward (backward) wave

*𝒜*(

*z*,

*t*). If the forward waves dominate, the negative-frequency part of

*𝒜*(

*z*,

*t*) can be neglected. The complex field

*𝒜*(

*z*,

*t*) is then directly related to the analytic signal for the electric field

*ℰ*(

*z*,

*t*)

*ℰ*(

*z*,

*t*) is just a sum of the positive-frequency components of

*E*(

*z*,

*t*).

*E*(

*z*,

*t*) = Re[

*ℰ*(

*z*,

*t*)]. The propagation equation for the analytic signal reads

*ω*> 0 and

*ℰ*

_{ω}_{<0}vanishes by construction. In what follows, we ignore generation of the third harmonics and simplify Eq. (17) to the form

*β*(

*ω*) in the linear term by

*iβ̂*, to approximate

*β*(

*ω*) in the nonlinear term by

*ωn*(

*ω*

_{0})/

*c*for some reference index of refraction

*n*(

*ω*

_{0}), and to use a standard notation for the nonlinear refraction,

*n*

_{2}= (3/8)

*χ*

^{(3)}/

*n*(

*ω*

_{0}). Equation (18) yields

_{+}denotes the positive-frequency part. The range of applicability of Eq. (19) is similar to that of the unidirectional forward Maxwell equation [6], but with the benefit of a clear separation of third-harmonic generation terms. If the Raman effect is important, Eq. (19) is replaced by

*f*and

_{K}*f*= 1 −

_{R}*f*describe relative contributions of the Kerr and Raman effects and

_{K}*ĥ*denotes a standard convolution with the Raman response function

*f*= 0.82,

_{R}*τ*

_{1}= 12.2 fs and

*τ*

_{2}= 32 fs for silica fibers [1].

Figure 1 depicts an exemplary group delay profile *β*_{1} = *β′*(*ω*) and related group-velocity dispersion profile *β*_{2} = *β″*(*ω*) for a 5 micron strand of silica in air [37]. The presence of absorptive resonances in the ultraviolet as well as vibrational resonances in the infrared generally causes a characteristic concave group-velocity dispersion profile, which is modified by the waveguide dispersion. For the exemplary case shown here, this results in a zero-dispersion wavelength (ZDW) located at 997nmm wavelength. For numerical solution of Eq. (20) we use either a de-aliased pseudospectral method, with the implementation of the Runge-Kutta integration scheme in the frequency domain and adaptive step-size control, or a more direct split-step Fourier approach. Accuracy of the numerical scheme is monitored by tracing the constants of motion. Considering ultrashort optical pulses with carrier frequencies of several hundreds THz, we use a maximum temporal discretization step of Δ*t* = 0.6fs. Sufficiency of the discretization is ensured by exemplary runs that involve a considerably larger number of harmonics.

For the calculation of the coherence properties we generated a total of 512 supercontinua instances, using different quantum noise seeds. Namely, the noise seeds are numerically implemented by adding one photon with random phase per mode to each discrete frequency bin of the input signal [1].

## 3. Optical event horizon inducing collisions between dispersive waves and solitons

The concave group-velocity profile of solid dielectric materials always enables co-propagation of a dispersive wave (DW) and a soliton at exactly identical group velocity, see Fig. 1 for the linear optical contributions. For an exact match, nonlinear contributions to the group velocity have to be considered. In particular, the group velocity is modified in the immediate vicinity of the soliton. Launching a DW and a soliton at nearly identical group velocities may then lead to a condition that has been termed “optical group-velocity horizon”. Once the dispersive wave reaches the nonlinear index modification induced by the soliton, both pulses remain temporally locked to each other. Consequently, the effective XPM interaction length between the two pulses is significantly increased, and any walk-off effect is prevented or at least substantially delayed.

The according group velocity dispersion *β*_{2} profile and the relative group delay *β*_{1} = 1/*v _{g}* profile are shown in Fig. 1, together with a selection of three exemplary frequency combinations. The fiber group index is taken from [37]. Accordingly, for the given situation, we launch two synchronized pulses with similar pulse parameters

*ℰ*,

_{s,p}*t*,

_{s,p}*ω*correspond to the initial field, the temporal width, and the center frequency, respectively. The pulses are injected at an initial time delay of Δ

_{s,p}*T*= 450 fs into the fiber on either side of the zero dispersion wavelength. For demonstration of the main mechanism, we first neglect the Raman effect. In principle, such pure Raman-free behavior can be realized in hollow-core fibers filled with noble gases [33].

An exemplary collision process of a DW at the leading edge of a fundamental soliton is demonstrated in Fig. 2(a,b). The DW is injected earlier into the fiber at a frequency with a slightly slower group velocity as the FS, leading to a collision process that induces a blueshift of the soliton center frequency and a redshift of the DW [Fig. 2(b)]. Both pulses are accelerated by the collision [30]. This process has recently also been referred to as diametrically driven self-acceleration [38]. For a comparison of this enhanced nonlinear interaction and the standard XPM, we illustrate the resulting collision process for the latter case, i.e., without the repulsion of the DW in Fig. 2(c,d). The initial wavelength of the DW is shifted away from the “resonant condition” for nearly equal group-velocities and is changed from 627 nm to 607 nm. The pulses quickly pass each other, and only a marginal change of the pulse parameters is observed. However, there exists a range of DW frequencies that establish a collision process. Given the chosen frequency of the DW, different collision scenarios can be generated while keeping all other parameters of both pulses fixed. Moving farther away from the exact resonant condition, only a part of the DW is reflected. This behavior suggests use of the DW frequency as an effective control parameter for all-optical manipulation. Other control parameters are provided by the amplitude and the width of the DW as well as by the initial time delay between both pulses. Among all these parameters, the DW amplitude represents the simplest adjustable and the most efficient control parameter.

The main idea behind the manipulation by the event horizon relies on exploitation of the induced center frequency shift of the soliton in a suitably chosen dispersion profile. Namely, the soliton has to be injected into a region with a large *β*_{3} coefficient. The elementary collision process is completely elastic, causing a mutual shift of optical frequencies, yet without transferring photons from the normal dispersion regime into the soliton regime or vice versa. With the photon number of the soliton being conserved, the soliton blueshift is automatically accompanied by a mild energy increase. However, the blue-shifted soliton is located closer to the zero-dispersion wavelength and therefore also experiences a reduced second-order dispersion after the collision. Depending on the initial frequency of the soliton, an induced frequency shift of only a few nm may result in a considerable change of the dispersion parameter, depending on the strength of the *β*_{3} coefficient. Considering that the energy *E* of a fundamental soliton relates to *P*_{0} and *β*_{2} via

*β*

_{2}cannot be compensated by a decrease of

*E*because

*E*is nearly constant and may even slightly grow due to the increase of photon energy. As

*γ*also does not vary appreciably with wavelength, Eq. (23) therefore affects a massive growth of

*P*

_{0}as a result of the blue shift. Moreover, as the pulse energy is nearly conserved, this peak power growth is automatically accompanied by a decrease in pulse width. In summary, one can understand the reshaping process of the soliton as an adiabatic compression mechanism that is induced by the DW via mutual four-wave mixing processes. A reverse scenario where the DW is reflected at the trailing edge of the soliton also results in the opposite effect. In the latter case both pulses are decelerated due to their interaction. Consequently, the soliton is not compressed, which renders this situation unsuitable for SC generation.

## 4. SC generation by soliton collision with dispersive waves

The importance of DWs, in particular their initial generation and subsequent transformation by the solitons, has already been demonstrated in the soliton-fission dominated SC generation, where it leads to additional spectral broadening [9, 15, 39].

However, the underlying process may also be employed for direct generation of the super-continuum without the fission process. To this end, one has to provide a continuous repulsion of dispersive waves at the temporal location of the soliton. Background radiation with a suitable frequency for a repulsion process may be created by the dispersion of an intense and ultrashort pulse in the normal dispersion regime. The time delay between the initial DW and the soliton can then be chosen in such a way to ensure a reflection of radiation at the soliton edge over longer propagation distances, but one has to take care that the radiation amplitude is too low for inducing a soliton frequency shift at any point of the reflection process.

For the three frequency combinations marked in Fig. 1, time domain representations of the SC propagation dynamics are shown in Fig. 3 together with the corresponding evolution of the spectra. Detailed pulse parameters are given in Table 1. In all three cases, the width of the solitons are chosen to display an equal number of cycles at the given wavelength. Scattering of the background radiation at the edge of the fundamental soliton can be realised over long propagation distances in all three cases. In parallel, we observe that the constant acceleration of the soliton leads to a parabolic trajectory in the (*t*, *z*)-plane, which is typical for the interaction behavior at an optical event horizon. However, we have a natural limitation of the acceleration process as the group-velocity must not exceed its maximum value given by the minimum of *β*_{1} at the ZDW. At this point the soliton can no longer be accelerated, which terminates soliton compression. Subsequently, we finally observe a transfer of energy from the soliton into the normal dispersion regime. This final process also contributes to the SC generation.

The entire SC generation process stems from different effects in the anomalous and normal dispersion regimes. In the former the spectrum is broadened by soliton compression. In contrast, for normal dispersion, the spectrum is mainly generated by the reflection of the DW. Further contributions originate from self-phase modulation of the non-reflected part and partially from the above-mentioned merging of the soliton spectrum into the normal dispersion regime. Here a further advantage of our scheme is visible. Soliton compression may also be realized in a dispersion-decreasing fiber [40] or with higher-order soliton compression [41], and in both cases the soliton fission process is avoided. One receives SC with high coherence, but the SC is restricted to the anomalous dispersion regime. In the presented scheme, the SC encompasses a large fraction in the normal dispersion regime, i.e, spanning over the short-wavelength range. This enables achieving higher spectral densities beyond octave coverage as it would not be possible solely exploiting solitons.

Figure 3(a,b) is equivalent to the SC generation demonstrated in [14], yet for the frequency combination [1550, 627]nm. The attainable bandwidth of the SC is chiefly determined by the initial separation of the two frequencies. The generated spectrum exhibits a 20-dB spectral width of 1050nm, encompassing 1.5 octaves from 570nm to 1620nm. In Fig. 3(c,d) we extend the separation of the frequency between the soliton and the DW from 1800nm to 520nm. To observe the same behavior, we have to increase the energy of the DW by an increase of the peak intensity or the pulse width. Figure 3(e,f) finally discusses an expansion of the wavelength separation close to the total available transparency region of the given strand fiber, i.e., 400–2400nm for fused silica. Also for this limiting example, the entire bandwidth between the two initial frequencies is continuously excited. In contrast to the SC by soliton fission, we do observe any limitation of spectral broadening. Consequently, this has to be considered a huge advantage of our scheme as it enables the generation of a SC over the whole bandwidth between the resonant absorption in the UV and the mid-infrared.

The spectral width can be adjusted by the frequency combination and the energy contents of both pulses. In general, there appears a disadvantage of the less effective spectral broadening in the normal dispersion regime [1]. However, in our case the spectral broadening depends on both pulses, and the spectral width and shape can be adjusted in different ways. It is also possible to excite only a part of the spectrum between the two pump wavelengths with high brightness in the normal dispersion regime.

Without the Raman effect, the main limitation for the obtainable spectral bandwidth is just dictated by the transparency region of the material, i.e., more than 2.5 octaves for fused silica. Depending on the material of the fiber, even broader SC may be generated, reaching into the mid-infrared if one chooses,e.g., fluoride glasses or chalcogenides.

Apart from the possibility for multi-octave SC generation, the resulting spectra exhibit extraordinary coherence properties [14]. In the normal dispersion regime, non-deterministic temporal pulse splitting is typically inherently avoided. Such behavior could translate into spectral modulation, and fluctuations are avoided. Markedly in the anomalous dispersion regime, our scheme does neither exhibit a pronounced susceptibility to noise. Usually noise is amplified in the input pulse through the modulation instability. In our case, this degeneration process is suitably avoided as the soliton pulse width becomes extremely short, which, in turn, substantially reduces the impact of the MI [11].

## 5. Supercontinuum generation under consideration of the Raman effect

The discussed scattering processes in the event horizon have previously reported to also appear in the presence of the Raman effect [7,17,42–45] and the same process may therefore exploited for SC generation. For the particular frequency combination [1550,627] nm, we previously demonstrated that a SC with a similarly high degree of coherence is generated [14]. One has to justify that the properties of the DW may readily serve to pre-compensate the Raman induced soliton-self-frequency-shift (SSFS) [46]. Such pre-compensation requires only minor adjustment of input parameters. As control parameters one may use the peak power, the pulse width of the DW, or the time delay between the DW and the soliton. As only the low intensity parts of the DW are reflected, the intensity and the width of the initial DW need to be chosen such that a suitable low level background is built up by DW broadening. It should be noted that higher intensities of the DW effectively cross the trajectory of the soliton, with an accordingly smaller reflected part. Temporal broadening of the DW can easily be controlled by choosing an appropriate time delay between the soliton and the DW. At the same time, however, the intensity of the DW segments have to be high enough at any interaction point to compensate and overcome the counteracting deceleration by the self-frequency shift. This naturally implies that the group velocities of both pulses should not be chosen too close to each other, effectively limiting the range of the resonant condition for the reflection process. For further extension of the bandwidth between the two initial frequencies, compensation of the SSFS becomes increasingly difficult. Figure 4 depicts simulations that include the Raman effect for the wavelength combination [1800,530] nm and [2200,428] nm. For the spectrum that ranges from 520nm to 1800nm [Fig. 4(a,b)], filling the gap between the two input wavelengths with the same mechanism turns out to be possible. However, in order to do so, we need careful adjustments of several parameters. Apart from a suitable combination of the peak intensity, the width of the DW and the time delay between the DW and the soliton, we also find an adequate modulation of the DW center frequency necessary.

Further increase of the gap between the initial frequencies of DW and the soliton, now for the combination [2200,428] nm, makes it even more difficult to fill all frequencies in the gap as a compensation of the Raman effect is no longer possible for the whole necessary range [Fig. 4(c,d)]. Consequently, the interaction length between the DW and the soliton does not suffice anymore for the generation of a SC over the entire spectral range that was previously discussed in the absence of the Raman effect. Including the Raman effect, the soliton trajectory is strongly influenced by the self-frequency shift. This effect pushes the soliton back towards the red, leading to adverse temporal broadening and deceleration of the soliton. Consequently, in comparison to the Raman free-case, the amplitude of the DW has always to be higher at the reflection point as the SSFS has to be compensated. For longer propagation distances, interaction of a DW with a FS may be enforced by an increase of the initial pulse width and peak intensity. However, there is another important limiting issue as the DW induced blueshift of the soliton also affects the resonance condition implied by the dispersion properties. A rapid soliton frequency sweep therefore quickly terminates effective interaction.

One possibility to overcome this obstacle is suggested in Fig. 5(a,b), where our scheme is used in a cascaded way. We inject now two DWs at different frequencies into the fiber. The wavelength of the first injected DW exhibits a centre wavelength at 470nm that is adjusted to be in resonance with soliton at a center wavelength closer to the ZDW. Despite of being injected first, this DW only interacts with the soliton after having been blue-shifted by the other primary DW. Once the effective interaction with this primary DW has come to an end, the secondary DW takes over and thereby expands the duration of effective nonlinear interaction with the soliton. In this way the whole spectrum in the transparency region can be excited as the Raman effect is compensated over longer distances. Cascaded interaction of the soliton and several DWs at different frequencies does not only enable generation of broader SCs, but also opens up a perspective for conveniently realising the necessary pulse parameters.

One of the advantages of the SC generated by two-pulse collision relates to their superior coherence properties. In the cascaded scheme, there are also no detrimental effects that could cause a degradation of the coherence properties, as MI and pulse splitting are still avoided. To demonstrate this, we calculate the measure of the SC phase stability by the modulus of the complex degree of first-order coherence at each wavelength [47]

*E*

_{1}(

*λ*,

*t*),

*E*

_{2}(

*λ*,

*t*)] obtained from 512 simulations.

Figure 6(a) displays single-shot spectra for selected values of *z* of the fiber, using the same parameters as in Fig. 5(b). The corresponding degree of coherence [Fig. 6(b)] reflects the expected insensitivity to noise. We observe similar coherence properties for a SC that covers the whole transparency region as was previously reported for an only 1.5 octave spanning SC in Ref. [14]. The only notable degradation is found in the vicinity of the zero dispersion wavelength. Otherwise, wide spectral segments exhibit favorable coherence properties, both in the normal and anomalous dispersion regime.

## 6. Conclusion

We presented a novel scheme for generating a multi-octave supercontinua. This scheme is based on the strong pulse reshaping at an optical event horizon, and it consists of the combination of different mechanisms for each dispersive regime. Namely, these mechanisms are soliton compression and frequency conversion due to the reflection of dispersive waves at the effective horizon of a soliton in the anomalous and the normal dispersion regime, respectively. Our scheme bears several possibilities for tailoring properties of the SC, which enables adaption to different applications and demands. Apart from the possibility to generate different spectral coverage ranges, favorable coherence properties may be achieved, in particular also in the anomalous dispersion regime.

## Acknowledgments

We acknowledge support by Deutsche Forschungsgemeinschaft and Open Access Publishing Fund of Leibniz Universität Hannover. Sh.A. gratefully acknowledges support by the DFG Research Center MATHEON (project D 14).

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