Generalized retarded response of nonlinear media and its influence on soliton dynamics

S. Pricking; H. Giessen

doi:10.1364/OE.19.002895

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 CS₂, CCl₄, toluene, and nitrobenzene, apart from their desired huge Kerr index n₂, 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.

Fig. 1 a) Photonic crystal fiber with the central single strand filled with the hypothetic medium. In our simulations the hole diameter is d = 2.5 μm, the hole-to-hole distance Λ = 2.6 μm. The length of the PCF is set to 19 cm. b) Measured retarded responses of commonly used media (CS₂ [15], toluene [18], CCl₄ [19], fused silica [20], and chloroform [21]). The gray lines show the retarded responses of our hypothetic medium for different values of t_R. These values rise quadratically from 2 fs (narrowest) to 200 fs (broadest). Note that all response functions are normalized to their maximum value to provide a better comparability, whereas in the simulations they are normalized such that their integral gives unity.

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The ultrafast pulse propagation in a medium showing a retarded response has been studied theoretically for the cases of a specific medium, especially CS₂ 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 n₂ 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_F. 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.

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

\frac{\partial A (z, T)}{\partial z} = \sum_{k \geq 2} \frac{i^{k + 1}}{k!} β_{k} \frac{\partial^{k}}{\partial T^{k}} A (z, T) + i γ (1 + \frac{i}{ω_{0}} \frac{\partial}{\partial T}) (A (z, T) \int_{- \infty}^{+ \infty} R (t^{'}) {| A (z, T - t^{'}) |}^{2} d t^{'}) .

Here, A is the envelope of the electric field, z the propagation distance, and T the time in the co-moving reference frame.

β_{k} = \frac{\partial^{k} β}{\partial ω^{k}} |_{ω_{0}}

denote the dispersion coefficients with β being the propagation constant. γ = n₂ω₀/(cA_eff) stands for the nonlinear parameter with n₂ being the nonlinear index of refraction, ω₀ 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

R (t) = (1 - f_{R}) δ (t) + f_{R} \frac{h_{R} (t / t_{R})}{t_{R}} .

The first term describes the instantaneous response, and the second covers the retarded response via h_R, which we define with a unitless argument as

h_{R} (x) = \frac{1}{N_{0}} (exp (- x) + \frac{1}{x + 1}) (1 - exp (- x)) exp (- \frac{x}{100}) Θ (x)

with N₀ 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_R) should fit the measured retarded responses of commonly used media (Fig. 1b) for appropriate values of t_R. In a first step we neglect the oscillations imposed on most of the measured data. 2) Apart from the fraction f_R, R(t) should contain only one free parameter t_R to keep the parameter space small. We call t_R the characteristic time and it is used to scale the time axis. 3) R(t) is normalizable such that ∫R(t)dt = 1. This condition leads to N₀ = 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 μm and a hole-to-hole distance of Λ = 2.6 μm. The propagation length z₀ is set to 19 cm. For the calculation of the dispersion coefficients [20] we use the refractive indices of CCl₄ [25], a central wavelength of λ₀ = 1030 nm and include the fiber geometry. This results in β₂ = −4.449 × 10⁻²⁶ s²/m, β₃ = 1.394 × 10⁻⁴⁰ s³/m, β₄ = −8.858 × 10⁻⁵⁶ s⁴/m, and β₅ = 4.351 ×10⁻⁷¹ s⁵/m. Since β₂ is negative, we operate in the anomalous dispersion regime, and soliton formation becomes possible. With the Kerr index of CCl₄ being n₂ = 15 × 10⁻²⁰ m²/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

A (0, T) = \sqrt{P_{0}} sech (\frac{T}{T_{0}}),

where P₀ denotes the peak power and T₀ the pulse width. As central wavelength we choose λ₀ = 1030 nm for the following simulations.

3. Simulation results

Fig. 2 shows a selection of output spectra after a propagation distance of z₀ = 19 cm. The variable parameters are the input pulse duration T₀, the fraction of the retarded response f_R and the characteristic time of the retarded response t_R. The input peak power has been fixed to P₀ = 500 W. Firstly we note that for an increasing input pulse duration T₀ the spectral width decreases after passing a maximum although the pulse energy given by E_p = 2T₀P₀ still rises. The input pulse duration at which the spectrum collapses in width is called T_C. We find that T_C decreases for increasing t_R and f_R. Furthermore, we observe that the maximum red-shift of the solitons λ_Sol, which is visible for small t_R, also decreases with increasing t_R, but increases for growing f_R followed by slightly smaller values for f_R = 0.8.

Fig. 2 Output spectra for an input pulse centered at 1030 nm after a propagation distance of z₀ = 19 cm in the single strand filled with our hypothetic medium in dependence of the retarded response fraction f_R, the characteristic time of the retarded response t_R, and the input pulse duration T₀. The input peak power is fixed to 500 W.

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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

N = T_{0} \sqrt{\frac{γ P_{0}}{| β_{2} |}} .

During the propagation, this higher-order soliton self-compresses temporarily while simultaneously increasing its peak power [27]. After a certain distance, namely the fission length z_F, higher-order dispersion and the retarded response trigger the break-up of the soliton into 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_R at around 600 nm. The shortest of the fundamental solitons has a pulse duration of T_S with [6]

T_{S} = \frac{T_{0}}{2 N - 1} = \frac{T_{0} \sqrt{| β_{2} |}}{2 T_{0} \sqrt{γ P_{0}} - \sqrt{| β_{2} |}} .

T_S 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 z_F is larger than the fiber length z₀ 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 = z₀ 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_D being the dispersion length defined by $L_{D} = T_{0}^{2} / | β_{2} |$ . This result yields $T_{C} = \sqrt{2 γ P_{0} | β_{2} |} z_{0}$ . This is independent of f_R and t_R, which is in contrast to the observation of Fig. 2. Therefore we expand their ansatz given by

z_{F} (τ) = \frac{τ T_{0}^{2}}{β_{2} \frac{\partial Φ_{NL} (z_{F}, τ)}{\partial τ}},

with τ = T/T₀ ≥ 0 and Φ_NL denoting the nonlinear phase. Including the retarded response we find

Φ_{NL} (z, τ) = γ z \int_{- \infty}^{\infty} R (t^{'}) {| A (0, τ T_{0} - t^{'}) |}^{2} d t^{'} .

Using Eqns. 2–4, we obtain

z_{F} (τ) = {\tilde{z}}_{F} {[(1 - f_{R}) \frac{sinh τ}{τ {cosh}^{3} τ} + f_{R} \int_{0}^{\infty} \frac{(x e^{- x} + e^{- x} + 1) (1 - e^{- x}) e^{- \frac{x}{100}} sinh (τ - τ_{R} x)}{N_{0} (1 + x) τ {cosh}^{3} (τ - τ_{R} x)} d x]}^{- \frac{1}{2}}

with τ_R = t_R/T₀. 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_F is only dependent on f_R and τ_R and becomes 1 for f_R = 0 or τ_R = 0, reproducing z̃_F as expected. In Fig. 3a we have plotted f_F(f_R,τ_R), which demonstrates that for a value t_R ≈ 1.5T₀ the correction factor may become larger than 20. In that case it cannot be neglected anymore.

Fig. 3 a) Correction factor f_F for the fission length ${\tilde{z}}_{F} = T_{0} / \sqrt{2 γ P_{0} | β_{2} |}$ , if the retarded response is taken into account. b) T_C in dependence of the fraction f_R and the characteristic time t_R of the retarded response. The symbols stand for data points extracted from our simulations, the lines describe the result of our model.

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Applying our model to the calculation of T_C we find

T_{C} (f_{R}, τ_{R}) = \sqrt{2 γ P_{0} | β_{2} |} \frac{z_{0}}{f_{F} (f_{R}, τ_{R})} .

As already mentioned by Chen and Kelley, the actual fission length is smaller than the calculated one, since in the model the effects of higher-order dispersion and of chirp accumulation are not taken into account. We correct for this by adding a constant time 309 fs to the calculated T_C, 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).

If z_F < z₀, 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 = z₀ − z_F. The rate of this frequency shift for a pulse in the form of Eq. 4 is given by [30, 31]

\frac{\partial ω}{\partial z} = - | β_{2} | T_{S} \frac{π}{2} \int_{0}^{\infty} \frac{Im \tilde{R} (Ω) Ω^{3}}{{sinh}^{2} (\frac{π}{2} T_{S} Ω)} d Ω .

Here, T_S is the pulse duration of the shortest (dominant) fundamental soliton and ImR̃ denotes the imaginary part of the Fourier-transformed retarded response R(t). Using Eq. 2, we obtain

\frac{\partial ω}{\partial z} = - | β_{2} | \frac{T_{S}}{t_{R}^{4}} \frac{π}{2} f_{R} \int_{0}^{\infty} \frac{Im \tilde{h_{R}} (x) x^{3}}{{sinh}^{2} (\frac{π}{2} \frac{T_{S}}{t_{R}} x)} d x

with x substituting t_RΩ. We note that the frequency shift is linear in f_R. The calculated frequency shift rate implementing Eq. 3 is demonstrated in Fig. 4a in dependence of t_R and T₀, which is connected to T_S by Eq. 6. For larger input pulse durations ∂ω/∂z becomes almost independent of T₀, whereas for very short pulses the SSFS rate approaches 0. Furthermore, we see that smaller t_R lead to larger shift rates.

Fig. 4 a) Frequency shift rate in dependence of input pulse duration T₀ and the characteristic time of the retarded response t_R. P₀ is fixed to 500 W. b) Calculated maximum wavelength of the dominant red-shifted fundamental soliton in dependence of input pulse duration T₀, the fraction and the characteristic time of the retarded response f_R and t_R, respectively.

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In a linear approximation we are now able to calculate the maximum frequency shift at a final propagation distance z₀:

Δ ω_{max} = \frac{\partial ω}{\partial z} |_{ω_{0}} z_{R} Θ (z_{R}) .

Here, Θ again denotes the Heaviside function. The deviation from this linear approximation depends on the sign of β₃: A positive β₃ leads to smaller shifts, a negative β₃ 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 = 2πc/(ω₀ + Δω_max) with ω₀ being the central frequency and c the speed of light in the vacuum. Fig. 4b shows the calculated λ_Sol in dependence of T₀ for a selection of f_R and t_R 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.

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_R = 1, t_R = 50 fs, and T₀ = 500 fs. As expected, λ_Sol and hence the spectral width increase for increasing peak power. We note that T_C 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 T_C, whereas for λ_Sol we observe the correct tendency but a deviation of the values. The reasons for that deviation are described above.

Fig. 5 a) T_C (blue) and maximum wavelength of the dominant red-shifted fundamental soliton (red) in dependence of the input peak power. T₀ is fixed to 500 fs, t_R = 50 fs, and f_R = 1. The dots denote the data extracted from the simulations, the lines are the results of our model. b) Influence of oscillations imposed on the retarded response function. Here, f_R = 1 and t_R = 50 fs. The bottom row shows the output spectra in dependence of T₀ with varying amplitudes of the oscillations: Left A = 0.05, middle A = 0.22, right A = 0.60.

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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

R_{Osci} (t) = (1 - f_{R}) δ (t) + \frac{f_{R}}{N_{Osci}} (h_{R} (t / t_{R}) + A e^{- \frac{t}{τ_{Osci}}} sin (ω_{Osci} t) Θ (t)) .

Again, the normalization constant 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 T₀. 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_R such that 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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Generalized retarded response of nonlinear media and its influence on soliton dynamics

Abstract

1. Introduction

2. Simulation method

3. Simulation results

4. Conclusion

References and links

Cited By

Figures (5)

Equations (14)

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