## Abstract

We demonstrate by means of numerical simulations of the generalized Nonlinear Schrödinger Equation that the retarded response of a nonlinear medium embedded in a single hole of a photonic crystal fiber crucially affects the spectrum generated by ultrashort laser pulses. By introducing a hypothetic medium with fixed dispersion and nonlinearity and with a variable retarded response, we are able to separate the influence of the retarded response from other effects. We show that the fission length of a launched higher-order soliton dramatically increases if the characteristic time of the retarded response is close to the input pulse duration. Furthermore, we investigate the effect of the retarded response on the soliton self-frequency shift and find that the optimum input pulse duration for maximizing the spectral width has to be shortened for a larger characteristic retarded response time. Our work has important implications on future studies of spatiotemporal solitons in selectively liquid-filled photonic crystal fibers.

© 2011 OSA

## 1. Introduction

Since their first appearance, photonic crystal fibers (PCFs) [1, 2] have been successfully used in ultrafast optics as a nonlinear medium for the investigation of soliton dynamics and supercontinuum generation [3–6]. These achievements are based on the freedom of tailoring the temporal and spatial dispersion of a PCF [7, 8] as well as on the possibility to provide very long interaction lengths for nonlinear effects, which are mainly given by the Kerr effect. To increase the efficiency of these nonlinear effects, several publications have reported about the filling of a highly nonlinear medium into single or multiple strands of a PCF (Fig. 1a), which then acts as the carrier medium for the launched pulses [9–14]. However, in case of highly nonlinear liquids such as CS_{2}, CCl_{4}, toluene, and nitrobenzene, apart from their desired huge Kerr index *n*_{2}, the molecular response is characteristically delayed when compared with the case of fused silica. The characteristic delay time of these responses *t _{R}* depends on the molecular structure and may vary from a few femtoseconds up to several picoseconds. The almost instantaneous responses can be attributed to electronic contributions, whereas the slower ones are determined by, e.g., molecular reorientation.

The ultrafast pulse propagation in a medium showing a retarded response has been studied theoretically for the cases of a specific medium, especially CS_{2} and chloroform [15, 16, 17]. It could be shown that the retarded response helps to enhance the coherence of the output light. However, most of these investigations have targeted the high Kerr index *n*_{2} of these media and treated the associated retarded response as a collateral effect.

In this paper, we want to set our focus especially on the retarded responses and their influence on the ultrafast pulse propagation in a waveguide formed by a single strand. This is filled with a certain medium with a retarded response and embedded in the photonic structure of a PCF. In order to be able to separate the effects caused by the retarded response from other sources we introduce a hypothetic medium with a variable retarded response but fixed dispersion and nonlinearity. Using this ansatz our investigation is not restricted to the given retarded response of a specific medium, but allows us to study its effects in general.

Performing simulations of the generalized Nonlinear Schrödinger Equation (NLSE) [6] of the structure presented in Fig. 1a we find that with increasing input pulse duration and fixed input peak power the spectral width first increases to a maximum, then decreases again until a full collapse of the spectral width at a certain input pulse duration *T _{C}* (

*C*stands for collapse). The spectral broadening is determined mainly by the soliton self-frequency shift (SSFS) [22, 23] of the fundamental solitons generated by the break-up [24] of the launched higher-order soliton after a certain propagation distance, the fission length

*z*. We find that the fission length as well as the SSFS strongly depend on the actual retarded response function. With the expansion of the already known theories of the SSFS and fission length, we are able to reproduce the behavior described above with good accuracy.

_{F}## 2. Simulation method

The generalized Nonlinear Schrödinger Equation (NLSE) [6] has been proven to accurately describe the propagation of ultrashort laser pulses in PCFs. The NLSE we implemented reads as

*A*is the envelope of the electric field,

*z*the propagation distance, and

*T*the time in the co-moving reference frame. ${\beta}_{k}=\frac{{\partial}^{k}\beta}{\partial {\omega}^{k}}{|}_{{\omega}_{0}}$ denote the dispersion coefficients with

*β*being the propagation constant.

*γ*=

*n*

_{2}

*ω*

_{0}/(

*cA*

_{eff}) stands for the nonlinear parameter with

*n*

_{2}being the nonlinear index of refraction,

*ω*

_{0}the central frequency of the pulse,

*c*the vacuum speed of light and

*A*

_{eff}the effective mode area. For simplicity we neglect the influence of noise.

The response of the medium is governed via *R*(*t*), defined by

*h*, which we define with a unitless argument as

_{R}*N*

_{0}being a normalization constant and Θ(

*x*) the Heaviside function that ensures causality. The fraction of the retarded response is given by

*f*.

_{R}*R*(

*t*) has been designed such that it fulfills the following requirements: 1) The shape of

*h*(

_{R}*t*/

*t*) should fit the measured retarded responses of commonly used media (Fig. 1b) for appropriate values of

_{R}*t*. In a first step we neglect the oscillations imposed on most of the measured data. 2) Apart from the fraction

_{R}*f*,

_{R}*R*(

*t*) should contain only one free parameter

*t*to keep the parameter space small. We call

_{R}*t*the characteristic time and it is used to scale the time axis. 3)

_{R}*R*(

*t*) is normalizable such that ∫

*R*(

*t*)

*dt*= 1. This condition leads to

*N*

_{0}= 3.9788. 4)

*R*(

*t*) should approach 0 sufficiently fast towards the edges of the time window used for the simulations (±24 ps) to avoid numerical artifacts.

The PCF in which we embed the hypothetic medium has a hole diameter of 2.5 *μ*m and a hole-to-hole distance of Λ = 2.6 *μ*m. The propagation length *z*_{0} is set to 19 cm. For the calculation of the dispersion coefficients [20] we use the refractive indices of CCl_{4} [25], a central wavelength of *λ*_{0} = 1030 nm and include the fiber geometry. This results in *β*_{2} = −4.449 × 10^{−26} s^{2}/m, *β*_{3} = 1.394 × 10^{−40} s^{3}/m, *β*_{4} = −8.858 × 10^{−56} s^{4}/m, and *β*_{5} = 4.351 ×10^{−71} s^{5}/m. Since *β*_{2} is negative, we operate in the anomalous dispersion regime, and soliton formation becomes possible. With the Kerr index of CCl_{4} being *n*_{2} = 15 × 10^{−20} m^{2}/W [26], we obtain for the nonlinear parameter *γ* = 0.38 (Wm)^{−1}.

To solve Eq. 1 we implement the symmetrized split-step Fourier method [20], which is well suited for the pulse durations and spectral widths we expect for the results. As the initial pulse we choose a sech-pulse given by

where*P*

_{0}denotes the peak power and

*T*

_{0}the pulse width. As central wavelength we choose

*λ*

_{0}= 1030 nm for the following simulations.

## 3. Simulation results

Fig. 2 shows a selection of output spectra after a propagation distance of *z*_{0} = 19 cm. The variable parameters are the input pulse duration *T*_{0}, the fraction of the retarded response *f _{R}* and the characteristic time of the retarded response

*t*. The input peak power has been fixed to

_{R}*P*

_{0}= 500 W. Firstly we note that for an increasing input pulse duration

*T*

_{0}the spectral width decreases after passing a maximum although the pulse energy given by

*E*= 2

_{p}*T*

_{0}

*P*

_{0}still rises. The input pulse duration at which the spectrum collapses in width is called

*T*. We find that

_{C}*T*decreases for increasing

_{C}*t*and

_{R}*f*. Furthermore, we observe that the maximum red-shift of the solitons

_{R}*λ*

_{Sol}, which is visible for small

*t*, also decreases with increasing

_{R}*t*, but increases for growing

_{R}*f*followed by slightly smaller values for

_{R}*f*= 0.8.

_{R}To explain these results we take a look at the pulse evolution when propagating through the fiber: In the presence of anomalous dispersion the launched pulse forms a higher-order soliton of the order *N*, defined by

*z*, higher-order dispersion and the retarded response trigger the break-up of the soliton into

_{F}*N*fundamental solitons of the order 1 [24]. Depending on the appropriate phasematching condition [28, 29], this break-up leads to the generation of non-solitonic radiation in the short-wavelength region, which is visible in Fig. 2 for smaller values of

*f*at around 600 nm. The shortest of the fundamental solitons has a pulse duration of

_{R}*T*with [6]

_{S}*T*is on the order of a few femtoseconds and gives rise to a strong SSFS, which red-shifts the soliton up to several hundreds of nanometers. If for a given input pulse duration the fission length

_{S}*z*is larger than the fiber length

_{F}*z*

_{0}used in the simulation, the break-up and hence the red-shift of the solitons does not occur and the spectrum remains narrow. The input pulse duration which yields

*z*=

_{F}*z*

_{0}is found to be equal to

*T*.

_{C}To quantify this evolution we first focus on the fission length *z _{F}*. The ansatz presented by Chen and Kelley [27] leads to a fission length
${\tilde{z}}_{F}={L}_{D}/\sqrt{2}N$ with

*L*being the dispersion length defined by ${L}_{D}={T}_{0}^{2}/\left|{\beta}_{2}\right|$. This result yields ${T}_{C}=\sqrt{2\gamma {P}_{0}\left|{\beta}_{2}\right|}{z}_{0}$. This is independent of

_{D}*f*and

_{R}*t*, which is in contrast to the observation of Fig. 2. Therefore we expand their ansatz given by

_{R}*τ*=

*T*/

*T*

_{0}≥ 0 and Φ

_{NL}denoting the nonlinear phase. Including the retarded response we find

*τ*=

_{R}*t*/

_{R}*T*

_{0}. We define the minimum of the function

*z*(

_{F}*τ*) as the fission length

*z*.

_{F}*Taking this result we see that the fission length found by Chen and Kelley z̃*

_{F}*has to be corrected by a dimensionless factor f*

_{F}*if the retarded response is taken into account. f*is only dependent on

_{F}*f*and

_{R}*τ*and becomes 1 for

_{R}*f*= 0 or

_{R}*τ*= 0, reproducing

_{R}*z̃*as expected. In Fig. 3a we have plotted

_{F}*f*(

_{F}*f*,

_{R}*τ*), which demonstrates that for a value

_{R}*t*≈ 1.5

_{R}*T*

_{0}the correction factor may become larger than 20. In that case it cannot be neglected anymore.

Applying our model to the calculation of *T _{C}* we find

*T*, whose value is the result of a fitting procedure. The results of this calculation are demonstrated in Fig. 3b as lines, which show an excellent agreement with the values extracted from the simulations (symbols).

_{C}If *z _{F}* <

*z*

_{0}, the launched higher-order soliton breaks up into fundamental solitons, which are red-shifted due to the SSFS while propagating the remaining distance

*z*=

_{R}*z*

_{0}−

*z*. The rate of this frequency shift for a pulse in the form of Eq. 4 is given by [30, 31]

_{F}*T*is the pulse duration of the shortest (dominant) fundamental soliton and Im

_{S}*R̃*denotes the imaginary part of the Fourier-transformed retarded response

*R*(

*t*). Using Eq. 2, we obtain

*x*substituting

*t*Ω. We note that the frequency shift is linear in

_{R}*f*. The calculated frequency shift rate implementing Eq. 3 is demonstrated in Fig. 4a in dependence of

_{R}*t*and

_{R}*T*

_{0}, which is connected to

*T*by Eq. 6. For larger input pulse durations

_{S}*∂ω*/

*∂z*becomes almost independent of

*T*

_{0}, whereas for very short pulses the SSFS rate approaches 0. Furthermore, we see that smaller

*t*lead to larger shift rates.

_{R}In a linear approximation we are now able to calculate the maximum frequency shift at a final propagation distance *z*_{0}:

*β*

_{3}: A positive

*β*

_{3}leads to smaller shifts, a negative

*β*

_{3}to larger shifts. The SSFS is limited by the frequency of the second zero dispersion point [31], which in our case is not of importance. The final wavelength of the soliton

*λ*

_{Sol}is determined by

*λ*

_{Sol}= 2

*πc*/(

*ω*

_{0}+ Δ

*ω*

_{max}) with

*ω*

_{0}being the central frequency and

*c*the speed of light in the vacuum. Fig. 4b shows the calculated

*λ*

_{Sol}in dependence of

*T*

_{0}for a selection of

*f*and

_{R}*t*values. In comparison to the results of the simulation of the NLSE given in Fig. 2 we find a good qualitative agreement. However, our model predicts too small SSFS and input pulse durations of maximum spectral width. We attribute this deviation again to the fact that our model does not take into account higher-order dispersion and chirp accumulation, which in a full model cannot be neglected. Furthermore, interference effects and cross-phase modulation between the fundamental solitons as well as self-steepening [32] affect the soliton dynamics, which also are not contained in our model to keep it simple and fast to calculate.

_{R}So far our results were obtained using a fixed input peak power of 500 W. As an example, we demonstrate in Fig. 5a the power-dependence of *T _{C}* and

*λ*

_{Sol}for the special case of

*f*= 1,

_{R}*t*= 50 fs, and

_{R}*T*

_{0}= 500 fs. As expected,

*λ*

_{Sol}and hence the spectral width increase for increasing peak power. We note that

*T*increases with rising peak power, too. The results of our model (lines) show excellent agreement with the data extracted from the simulations (dots) in the case of

_{C}*T*, whereas for

_{C}*λ*

_{Sol}we observe the correct tendency but a deviation of the values. The reasons for that deviation are described above.

The measured retarded response functions of most commonly used media show imposed damped oscillations with frequencies in the order of tens of THz. To check for the influence of these oscillations we have performed simulations with a modified retarded response as given by

*N*

_{Osci}has to be chosen such that ∫

*R*

_{Osci}(

*t*)

*dt*= 1. For the damping constant

*τ*

_{Osci}we choose 0.7 ps, for the frequency

*ω*

_{Osci}45 THz, which is in the typical range of the liquids listed above. The oscillations imposed on the retarded responses are plotted in Fig. 5b (top) for three different amplitudes of the oscillations

*A*. In the bottom part of Fig. 5b we show the corresponding output spectra in dependence of

*T*

_{0}. Although the chosen amplitudes are much higher than the measured ones, there are only slight changes observable in the spectra. Therefore we note that the influence of the oscillations imposed on the retarded response function plays a minor role and might be neglected in further considerations.

## 4. Conclusion

In conclusion, we have demonstrated the influence of the characteristic time and fraction of the retarded response on the propagation of ultrashort pulses. The introduction of a hypothetic medium with variable retarded response has allowed us to study this influence independently of a specific medium. Nevertheless, due to the careful design of our retarded response function *R*(*t*) our results are applicable to real media by choosing an appropriate characteristic time *t _{R}* and fraction

*f*such that

_{R}*R*(

*t*) fits the measured one. We have developed a model which enables us to predict the soliton dynamics with a high accuracy. An important finding of that model is the fact that for media with a large retarded response the conventionally defined soliton fission length must be corrected by a factor of up to 20 for input pulse durations slightly above the characteristic time of the retarded response. This result might be utilized by experiments with a need for suppression of the soliton self-frequency shift by tuning the input pulse duration accordingly to the used propagation medium and its retarded response. This might aid further spatiotemporal nonlinear pulse propagation studies in selectively liquid-filled PCFs.

The authors would like to thank the Landesgraduiertenförderung of Baden-Württemberg for support of this work.

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