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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 4 — Feb. 14, 2011
  • pp: 2895–2903
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Generalized retarded response of nonlinear media and its influence on soliton dynamics

S. Pricking and H. Giessen  »View Author Affiliations


Optics Express, Vol. 19, Issue 4, pp. 2895-2903 (2011)
http://dx.doi.org/10.1364/OE.19.002895


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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 Optical Society of America

1. Introduction

Since their first appearance, photonic crystal fibers (PCFs) [1

1. P. St. J. Russell, “Photonic crystal fibers,” Science 17, 358–362 (2003). [CrossRef]

, 2

2. J. C. Knight, “Photonic crystal fibers,” Nature 424, 847–851 (2003). [CrossRef] [PubMed]

] have been successfully used in ultrafast optics as a nonlinear medium for the investigation of soliton dynamics and supercontinuum generation [3

3. W. H. Reeves, D. V. Skryabin, F. Biancalana, J. C. Knight, P. St. J. Russell, F. G. Omenetto, A. Efimov, and A. J. Taylor, “Transformation and control of ultra-short pulses in dispersion-engineered photonic crystal fibres,” Nature 424, 511–515 (2003). [CrossRef] [PubMed]

6

6. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006). [CrossRef]

]. These achievements are based on the freedom of tailoring the temporal and spatial dispersion of a PCF [7

7. P. J. Roberts, B. J. Mangan, H. Sabert, F. Couny, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Control of dispersion in photonic crystal fibers,” J. Opt. Fiber. Commun. Rep. 2, 435–461 (2005). [CrossRef]

, 8

8. C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allan, and K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657–659 (2003). [CrossRef] [PubMed]

] 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

9. F. Benabid, J. C. Knight, G. Antonopoulos, and P. St. Russell, “Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,” Science 298, 399–402 (2002). [CrossRef] [PubMed]

14

14. M. Vieweg, T. Gissibl, S. Pricking, B. J. Eggleton, D. C. Wu, B. T. Kuhlmey, and H. Giessen, “Ultrafast nonlinear optofluidics in selectively liquid-filled photonic crystal fibers,” Opt. Express 18, 25232–25240 (2010). [CrossRef] [PubMed]

]. 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 tR 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 tR. 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.

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

15. R. Zhang, J. Teipel, and H. Giessen, “Theoretical design of a liquid-core photonic crystal fiber for supercontinuum generation,” Opt. Express 14, 6800–6812 (2006). [CrossRef] [PubMed]

, 16

16. H. Zhang, S. Chang, J. Yuan, and D. Huang, “Supercontinuum generation in chloroform-filled photonic crystal fibers,” Optik 121, 783–789 (2010). [CrossRef]

, 17

17. R. V. J. Raja, A. Husakou, J. Hermann, and K. Porsezian, “Supercontinuum generation in liquid-filled photonic crystal fiber with slow nonlinear response,” J. Opt. Soc. Am. B 27, 1763–1768 (2010). [CrossRef]

]. 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

6. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006). [CrossRef]

] 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 TC (C stands for collapse). The spectral broadening is determined mainly by the soliton self-frequency shift (SSFS) [22

22. F. M. Mitschke and L. F. Mollenauer, “Discovery of the soliton self-frequency shift,” Opt. Lett. 11, 659–661 (1986). [CrossRef] [PubMed]

, 23

23. S. M. Kobtsev, S. V. Kukarin, N. V. Fateev, and S. V. Smirnov, “Coherent, polarization and temporal properties of self-frequency shifted solitons generated in polarization-maintaining microstructured fibre,” Appl. Phys. B 81, 265–269 (2005). [CrossRef]

] of the fundamental solitons generated by the break-up [24

24. A. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 87, 203901 (2001). [CrossRef] [PubMed]

] of the launched higher-order soliton after a certain propagation distance, the fission length zF. 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

6. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006). [CrossRef]

] has been proven to accurately describe the propagation of ultrashort laser pulses in PCFs. The NLSE we implemented reads as
A(z,T)z=k2ik+1k!βkkTkA(z,T)+iγ(1+iω0T)(A(z,T)+R(t)|A(z,Tt)|2dt).
(1)
Here, A is the envelope of the electric field, z the propagation distance, and T the time in the co-moving reference frame. βk=kβωk|ω0 denote the dispersion coefficients with β being the propagation constant. γ = n2ω0/(cAeff) 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 Aeff 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)=(1fR)δ(t)+fRhR(t/tR)tR.
(2)
The first term describes the instantaneous response, and the second covers the retarded response via hR, which we define with a unitless argument as
hR(x)=1N0(exp(x)+1x+1)(1exp(x))exp(x100)Θ(x)
(3)
with N0 being a normalization constant and Θ(x) the Heaviside function that ensures causality. The fraction of the retarded response is given by fR. R(t) has been designed such that it fulfills the following requirements: 1) The shape of hR (t/tR) should fit the measured retarded responses of commonly used media (Fig. 1b) for appropriate values of tR. In a first step we neglect the oscillations imposed on most of the measured data. 2) Apart from the fraction fR, R(t) should contain only one free parameter tR to keep the parameter space small. We call tR 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

20. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 1995).

] we use the refractive indices of CCl4 [25

25. H. H. Marvin, “The selective transmission and the dispersion of the liquid chlorides,” Phys. Rev. 34, 161–186 (1912).

], 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

26. P. P. Ho and R. R. Alfano, “Optical Kerr effect in liquids,” Phys. Rev. A 20, 2170–2187 (1979). [CrossRef]

], we obtain for the nonlinear parameter γ = 0.38 (Wm)−1.

To solve Eq. 1 we implement the symmetrized split-step Fourier method [20

20. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 1995).

], 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)=P0sech(TT0),
(4)
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 fR and the characteristic time of the retarded response tR. 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 Ep = 2T0P0 still rises. The input pulse duration at which the spectrum collapses in width is called TC. We find that TC decreases for increasing tR and fR. Furthermore, we observe that the maximum red-shift of the solitons λSol, which is visible for small tR, also decreases with increasing tR, but increases for growing fR followed by slightly smaller values for fR = 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 fR, the characteristic time of the retarded response tR, and the input pulse duration T0. The input peak power is fixed to 500 W.

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=T0γP0|β2|.
(5)
During the propagation, this higher-order soliton self-compresses temporarily while simultaneously increasing its peak power [27

27. C.-M. Chen and P. L. Kelley, “Nonlinear pulse compression in optical fibers: scaling laws and numerical analysis,” J. Opt. Soc. Am. B 19, 1961–1967 (2002). [CrossRef]

]. After a certain distance, namely the fission length zF, higher-order dispersion and the retarded response trigger the break-up of the soliton into N fundamental solitons of the order 1 [24

24. A. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 87, 203901 (2001). [CrossRef] [PubMed]

]. Depending on the appropriate phasematching condition [28

28. N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995). [CrossRef] [PubMed]

, 29

29. A. V. Yulin, D. V. Skryabin, and P. St. J. Russell, “Four-wave mixing of linear waves and solitons in fibers with higher-order dispersion,” Opt. Lett. 29, 2411–2413 (2004). [CrossRef] [PubMed]

], 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 fR at around 600 nm. The shortest of the fundamental solitons has a pulse duration of TS with [6

6. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006). [CrossRef]

]
TS=T02N1=T0|β2|2T0γP0|β2|.
(6)
TS 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 zF 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 zF = z0 is found to be equal to TC.

To quantify this evolution we first focus on the fission length zF. The ansatz presented by Chen and Kelley [27

27. C.-M. Chen and P. L. Kelley, “Nonlinear pulse compression in optical fibers: scaling laws and numerical analysis,” J. Opt. Soc. Am. B 19, 1961–1967 (2002). [CrossRef]

] leads to a fission length z˜F=LD/2N with LD being the dispersion length defined by LD=T02/|β2|. This result yields TC=2γP0|β2|z0. This is independent of fR and tR, which is in contrast to the observation of Fig. 2. Therefore we expand their ansatz given by
zF(τ)=τT02β2ΦNL(zF,τ)τ,
(7)
with τ = T/T0 ≥ 0 and ΦNL denoting the nonlinear phase. Including the retarded response we find
ΦNL(z,τ)=γzR(t)|A(0,τT0t)|2dt.
(8)
Using Eqns. 24, we obtain
zF(τ)=z˜F[(1fR)sinhττcosh3τ+fR0(xex+ex+1)(1ex)ex100sinh(ττRx)N0(1+x)τcosh3(ττRx)dx]12
(9)
with τR = tR/T0. We define the minimum of the function zF(τ) as the fission length zF. Taking this result we see that the fission length found by Chen and Kelley z̃F has to be corrected by a dimensionless factor fF if the retarded response is taken into account. fF is only dependent on fR and τR and becomes 1 for fR = 0 or τR = 0, reproducing F as expected. In Fig. 3a we have plotted fF(fR,τR), which demonstrates that for a value tR ≈ 1.5T0 the correction factor may become larger than 20. In that case it cannot be neglected anymore.

Fig. 3 a) Correction factor fF for the fission length z˜F=T0/2γP0|β2|, if the retarded response is taken into account. b) TC in dependence of the fraction fR and the characteristic time tR of the retarded response. The symbols stand for data points extracted from our simulations, the lines describe the result of our model.

Applying our model to the calculation of TC we find
TC(fR,τR)=2γP0|β2|z0fF(fR,τR).
(10)
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 TC, 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 zF < z0, the launched higher-order soliton breaks up into fundamental solitons, which are red-shifted due to the SSFS while propagating the remaining distance zR = z0zF. The rate of this frequency shift for a pulse in the form of Eq. 4 is given by [30

30. J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11662–664 (1986). [CrossRef] [PubMed]

, 31

31. A. C. Judge, O. Bang, B. J. Eggleton, B. T. Kuhlmey, E. C. Mägi, R. Pant, and C. Martijn de Sterke, “Optimization of the soliton self-frequency shift in a tapered photonic crystal fiber,” J. Opt. Soc. Am. B 26, 2064–2071 (2009). [CrossRef]

]
ωz=|β2|TSπ20ImR˜(Ω)Ω3sinh2(π2TSΩ)dΩ.
(11)
Here, TS is the pulse duration of the shortest (dominant) fundamental soliton and Im denotes the imaginary part of the Fourier-transformed retarded response R(t). Using Eq. 2, we obtain
ωz=|β2|TStR4π2fR0ImhR˜(x)x3sinh2(π2TStRx)dx
(12)
with x substituting tRΩ. We note that the frequency shift is linear in fR. The calculated frequency shift rate implementing Eq. 3 is demonstrated in Fig. 4a in dependence of tR and T0, which is connected to TS 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 tR 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 tR. 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 fR and tR, respectively.

In a linear approximation we are now able to calculate the maximum frequency shift at a final propagation distance z0:
Δωmax=ωz|ω0zRΘ(zR).
(13)
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

31. A. C. Judge, O. Bang, B. J. Eggleton, B. T. Kuhlmey, E. C. Mägi, R. Pant, and C. Martijn de Sterke, “Optimization of the soliton self-frequency shift in a tapered photonic crystal fiber,” J. Opt. Soc. Am. B 26, 2064–2071 (2009). [CrossRef]

], which in our case is not of importance. The final wavelength of the soliton λSol is determined by λSol = 2πc/(ω0 + Δωmax) with ω0 being the central frequency and c the speed of light in the vacuum. Fig. 4b shows the calculated λSol in dependence of T0 for a selection of fR and tR 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

32. A. A. Voronin and A. M. Zheltikov, “Soliton self-frequency shift decelerated by self-steepening,” Opt. Lett. 33, 1723–1725 (2008). [CrossRef] [PubMed]

] 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 TC and λSol for the special case of fR = 1, tR = 50 fs, and T0 = 500 fs. As expected, λSol and hence the spectral width increase for increasing peak power. We note that TC 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 TC, whereas for λSol we observe the correct tendency but a deviation of the values. The reasons for that deviation are described above.

Fig. 5 a) TC (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, tR = 50 fs, and fR = 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, fR = 1 and tR = 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.

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
ROsci(t)=(1fR)δ(t)+fRNOsci(hR(t/tR)+AetτOscisin(ωOscit)Θ(t)).
(14)
Again, the normalization constant NOsci has to be chosen such that ∫ROsci (t)dt = 1. For the damping constant τOsci we choose 0.7 ps, for the frequency ωOsci 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 tR and fraction fR 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.

References and links

1.

P. St. J. Russell, “Photonic crystal fibers,” Science 17, 358–362 (2003). [CrossRef]

2.

J. C. Knight, “Photonic crystal fibers,” Nature 424, 847–851 (2003). [CrossRef] [PubMed]

3.

W. H. Reeves, D. V. Skryabin, F. Biancalana, J. C. Knight, P. St. J. Russell, F. G. Omenetto, A. Efimov, and A. J. Taylor, “Transformation and control of ultra-short pulses in dispersion-engineered photonic crystal fibres,” Nature 424, 511–515 (2003). [CrossRef] [PubMed]

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7.

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8.

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allan, and K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657–659 (2003). [CrossRef] [PubMed]

9.

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M. Vieweg, T. Gissibl, S. Pricking, B. J. Eggleton, D. C. Wu, B. T. Kuhlmey, and H. Giessen, “Ultrafast nonlinear optofluidics in selectively liquid-filled photonic crystal fibers,” Opt. Express 18, 25232–25240 (2010). [CrossRef] [PubMed]

15.

R. Zhang, J. Teipel, and H. Giessen, “Theoretical design of a liquid-core photonic crystal fiber for supercontinuum generation,” Opt. Express 14, 6800–6812 (2006). [CrossRef] [PubMed]

16.

H. Zhang, S. Chang, J. Yuan, and D. Huang, “Supercontinuum generation in chloroform-filled photonic crystal fibers,” Optik 121, 783–789 (2010). [CrossRef]

17.

R. V. J. Raja, A. Husakou, J. Hermann, and K. Porsezian, “Supercontinuum generation in liquid-filled photonic crystal fiber with slow nonlinear response,” J. Opt. Soc. Am. B 27, 1763–1768 (2010). [CrossRef]

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21.

T. F. Laurent, H. Hennig, N. P. Ernsting, and S. A. Kovalenko, “The ultrafast optical Kerr effect in liquid fluoro-form: an estimate of the collision-induced contribution,” Phys. Chem. 2, 2691–2697 (2000). [CrossRef]

22.

F. M. Mitschke and L. F. Mollenauer, “Discovery of the soliton self-frequency shift,” Opt. Lett. 11, 659–661 (1986). [CrossRef] [PubMed]

23.

S. M. Kobtsev, S. V. Kukarin, N. V. Fateev, and S. V. Smirnov, “Coherent, polarization and temporal properties of self-frequency shifted solitons generated in polarization-maintaining microstructured fibre,” Appl. Phys. B 81, 265–269 (2005). [CrossRef]

24.

A. Husakou and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 87, 203901 (2001). [CrossRef] [PubMed]

25.

H. H. Marvin, “The selective transmission and the dispersion of the liquid chlorides,” Phys. Rev. 34, 161–186 (1912).

26.

P. P. Ho and R. R. Alfano, “Optical Kerr effect in liquids,” Phys. Rev. A 20, 2170–2187 (1979). [CrossRef]

27.

C.-M. Chen and P. L. Kelley, “Nonlinear pulse compression in optical fibers: scaling laws and numerical analysis,” J. Opt. Soc. Am. B 19, 1961–1967 (2002). [CrossRef]

28.

N. Akhmediev and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995). [CrossRef] [PubMed]

29.

A. V. Yulin, D. V. Skryabin, and P. St. J. Russell, “Four-wave mixing of linear waves and solitons in fibers with higher-order dispersion,” Opt. Lett. 29, 2411–2413 (2004). [CrossRef] [PubMed]

30.

J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11662–664 (1986). [CrossRef] [PubMed]

31.

A. C. Judge, O. Bang, B. J. Eggleton, B. T. Kuhlmey, E. C. Mägi, R. Pant, and C. Martijn de Sterke, “Optimization of the soliton self-frequency shift in a tapered photonic crystal fiber,” J. Opt. Soc. Am. B 26, 2064–2071 (2009). [CrossRef]

32.

A. A. Voronin and A. M. Zheltikov, “Soliton self-frequency shift decelerated by self-steepening,” Opt. Lett. 33, 1723–1725 (2008). [CrossRef] [PubMed]

OCIS Codes
(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers
(060.5530) Fiber optics and optical communications : Pulse propagation and temporal solitons
(060.4005) Fiber optics and optical communications : Microstructured fibers

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: December 21, 2010
Revised Manuscript: January 28, 2011
Manuscript Accepted: January 28, 2011
Published: January 31, 2011

Citation
S. Pricking and H. Giessen, "Generalized retarded response of nonlinear media and its influence on soliton dynamics," Opt. Express 19, 2895-2903 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-4-2895


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References

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  8. C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allan, and K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657–659 (2003). [CrossRef] [PubMed]
  9. F. Benabid, J. C. Knight, G. Antonopoulos, and P. St. Russell, “Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,” Science 298, 399–402 (2002). [CrossRef] [PubMed]
  10. C. R. Rosberg, F. H. Bennet, D. N. Neshev, P. D. Rasmussen, O. Bang, W. Krolikowski, A. Bjarklev, and Y. Kivshar, “Tunable diffraction and self-defocusing in liquid-filled photonic crystal fibers,” Opt. Express 15, 12145–12150 (2007). [CrossRef] [PubMed]
  11. B. T. Kuhlmey, B. J. Eggleton, and D. K. C. Wu, “Fluid-filled solid-core photonic bandgap fibers,” J. Lightwave Technol. 27, 1617–1630 (2009). [CrossRef]
  12. A. Bozolan, C. J. S. de Matos, C. M. B. Cordeiro, E. M. dos Santos, and J. Travers, “Supercontinuum in a water-core photonic crystal fiber,” Opt. Express 16, 9671–9676 (2008). [CrossRef] [PubMed]
  13. J. Bethge, A. Husakou, F. Mitschke, F. Noack, U. Griebner, G. Steinmeyer, and J. Herrmann, “Two-octave supercontinuum generation in a water-filled photonic crystal fiber,” Opt. Express 18, 6230–6240 (2010). [CrossRef] [PubMed]
  14. M. Vieweg, T. Gissibl, S. Pricking, B. J. Eggleton, D. C. Wu, B. T. Kuhlmey, and H. Giessen, “Ultrafast nonlinear optofluidics in selectively liquid-filled photonic crystal fibers,” Opt. Express 18, 25232–25240 (2010). [CrossRef] [PubMed]
  15. R. Zhang, J. Teipel, and H. Giessen, “Theoretical design of a liquid-core photonic crystal fiber for supercontinuum generation,” Opt. Express 14, 6800–6812 (2006). [CrossRef] [PubMed]
  16. H. Zhang, S. Chang, J. Yuan, and D. Huang, “Supercontinuum generation in chloroform-filled photonic crystal fibers,” Optik (Stuttg.) 121, 783–789 (2010). [CrossRef]
  17. R. V. J. Raja, A. Husakou, J. Hermann, and K. Porsezian, “Supercontinuum generation in liquid-filled photonic crystal fiber with slow nonlinear response,” J. Opt. Soc. Am. B 27, 1763–1768 (2010). [CrossRef]
  18. P. Wiewior, and C. Radzewicz, “Dynamics of molecular liquids studied by femtosecond optical Kerr effect,” Opt. Appl. 30, 103–120 (2000).
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  20. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 1995).
  21. T. F. Laurent, H. Hennig, N. P. Ernsting, and S. A. Kovalenko, “The ultrafast optical Kerr effect in liquid fluoroform: an estimate of the collision-induced contribution,” Phys. Chem. 2, 2691–2697 (2000). [CrossRef]
  22. F. M. Mitschke, and L. F. Mollenauer, “Discovery of the soliton self-frequency shift,” Opt. Lett. 11, 659–661 (1986). [CrossRef] [PubMed]
  23. S. M. Kobtsev, S. V. Kukarin, N. V. Fateev, and S. V. Smirnov, “Coherent, polarization and temporal properties of self-frequency shifted solitons generated in polarization-maintaining microstructured fibre,” Appl. Phys. B 81, 265–269 (2005). [CrossRef]
  24. A. Husakou, and J. Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” Phys. Rev. Lett. 87, 203901 (2001). [CrossRef] [PubMed]
  25. H. H. Marvin, “The selective transmission and the dispersion of the liquid chlorides,” Phys. Rev. 34, 161–186 (1912).
  26. P. P. Ho, and R. R. Alfano, “Optical Kerr effect in liquids,” Phys. Rev. A 20, 2170–2187 (1979). [CrossRef]
  27. C.-M. Chen, and P. L. Kelley, “Nonlinear pulse compression in optical fibers: scaling laws and numerical analysis,” J. Opt. Soc. Am. B 19, 1961–1967 (2002). [CrossRef]
  28. N. Akhmediev, and M. Karlsson, “Cherenkov radiation emitted by solitons in optical fibers,” Phys. Rev. A 51, 2602–2607 (1995). [CrossRef] [PubMed]
  29. A. V. Yulin, D. V. Skryabin, and P. St. J. Russell, “Four-wave mixing of linear waves and solitons in fibers with higher-order dispersion,” Opt. Lett. 29, 2411–2413 (2004). [CrossRef] [PubMed]
  30. J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986). [CrossRef] [PubMed]
  31. A. C. Judge, O. Bang, B. J. Eggleton, B. T. Kuhlmey, E. C. Mägi, R. Pant, and C. Martijn de Sterke, “Optimization of the soliton self-frequency shift in a tapered photonic crystal fiber,” J. Opt. Soc. Am. B 26, 2064–2071 (2009). [CrossRef]
  32. A. A. Voronin, and A. M. Zheltikov, “Soliton self-frequency shift decelerated by self-steepening,” Opt. Lett. 33, 1723–1725 (2008). [CrossRef] [PubMed]

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