## Abstract

We have numerically studied a hollow-core photonic crystal fiber, with its core filled with highly nonlinear liquids such as carbon disulfide and nitrobenzene. Calculations show that the fiber has an extremely high nonlinear parameter γ on the order of 2.4/W/m at 1.55 μm. The group velocity dispersion of this fiber exhibits an anomalous region in the near-infrared, and its zero-dispersion wavelength is around 1.55 μm. This leads to potentially significant improvements and a large bandwidth in supercontinuum generation. The spectral properties of the supercontinuum generation in liquid-core photonic crystal fibers are simulated by solving the generalized nonlinear Schrödinger equation. The results demonstrate that the liquid-core PCF is capable to generate dramatically broadened supercontinua in a range from 700 nm to more than 2500 nm when pumping at 1.55 μm with subpicosecond pulses.

©2006 Optical Society of America

## 1. Introduction

Photonic crystal fibers (PCF), where supercontinuum (SC) generation takes place in the telecommunication window, have a large number of applications in pulse compression, parametric amplifiers, supercontinuum-based WDM telecom sources, etc. [1–6]. The mechanisms responsible for the SC generation are known to be soliton fission [7, 8], self-phase modulation, four-wave-mixing, stimulated Raman scattering [9], and cross-phase modulation [10, 11]. Supercontinuum radiation is easily generated when the fiber is pumped in the anomalous dispersion region. A large spectral broadening requires a large nonlinear parameter γ. Therefore, highly nonlinear photonic crystal fibers (with large γ) have long been sought after. At present, methods such as reducing the diameter of the fiber core, enlarging the air-filling fractions, and extruding the fiber preforms in soft lead-glass [12–14] are used to enhance the nonlinear effect of the fiber. Recently, highly broadband supercontinuum radiation from 350 nm to beyond 3000 nm was generated by using a short piece of high-nonlinearity soft-glass PCF, which was made of Schott SF6 glass [14].

In this paper, we demonstrate a new way to realize a highly nonlinear fiber by filling the core of a hollow-core PCF with highly nonlinear liquids, such as carbon disulfide or nitrobenzene, as shown in Fig.1. We term this kind of fiber a “liquid-core photonic crystal fiber” (LCPCF). Our studies can be extended to other liquids as well, as they demonstrate that it is important to also consider the delayed nonlinear optical response of these liquids.

Nearly two decades ago, highly nonlinear liquids such as carbon disulfide have been used to generate spectral broadening by filling the liquid in a hollow fiber [15, 16]. However, the resulting spectrum was not a supercontinuum. The non-continuous generation, which consists of only some distinct peaks, results from the fact that a crucial requirement to generate a supercontinuum was not fulfilled, namely that the GVD should be anomalous and close to zero at the pump wavelength. The liquid-core PCF designed in this paper on the other hand shows a zero GVD wavelength around 1.55 μm and an anomalous dispersion in the near infrared spectral region.

The technique to fill the hole of a hollow-core PCF with liquids has already been taken into practice [17–21]. Yiou *et al*. presented the first experiment to obtain stimulated Raman scattering by using a hollow microstructured fiber with the core filled with low refractive index nonlinear liquid [20]. Recently a liquid-filled hollow core microstructured polymer fiber was realized experimentally [21].

## 2. Physical properties of the high-nonlinearity liquids

Nitrobenzene and carbon disulfide, due to their high nonlinearity, are chosen as candidates to fill into the core of the hollow PCF. We first study their physical properties, such as nonlinear optical coefficients, refractive index dispersion, and transmission properties.

#### 2.1 Nonlinear coefficient and refractive index dispersion

Nitrobenzene and carbon disulfide have very high nonlinear coefficients n_{2} of 250∙10^{-13}
*esu* and 120∙10^{-13}
*esu* [22] (measured with nanosecond pulses), which are more than 200 times and 100 times larger than that of silica, respectively.

The refractive index dispersion of carbon disulfide is given by [23]

*n*
_{cs2}(*λ*) = 1.580826+1.52389∙10^{-2}/*λ*
^{2}+4.8578∙10^{-4}/*λ*
^{4}-8.2863∙10^{-5}/*λ*
^{6}+1.4619∙10^{-5}/*λ*
^{8},

where λ, is the wavelength of light in μm.

Unfortunately, no data in the visible and near infrared are available so far for nitrobenzene. We have therefore measured the refractive indices of nitrobenzene with an Abbe refractometer, as shown in Fig. 2. Limited by the instrument, we were not able to perform the measurement at wavelengths larger than 1.1 μm.

From the measured data we extract the Sellmeier fit for the refractive index for nitrobenzene

*n*_{nitrobenzene}
(*λ*) = 1.5205+0.79∙10^{-2}/*λ*
^{2}+1.670∙10^{-3}/*λ*
^{4}-3.1∙10^{-4}/*λ*
^{6}+3.0∙10^{-5}/*λ*
^{8} (*λ* in μm).

Although the Sellmeier equation for nitrobenzene is precise only for wavelengths smaller than 1.1 μm, the dispersion of organic compounds varies only slowly in the infrared, so that we can extrapolate them into the longer wavelength region with reasonable accuracy.

#### 2.2 Transmission properties

The transmission spectra of a 9.8 mm thick cuvette filled with carbon disulfide and nitrobenzene in the visible and near infrared region are shown in Fig. 3 (corrected for the cuvette reflection). Carbon disulfide exhibits no absorption in the visible and near infrared region. Nitrobenzene has a strong absorption at wavelengths larger than 1600 nm, however, it is transparent in the whole visible region and in the range between 1200 nm and 1600 nm.

## 3. Mode properties of the liquid-core photonic crystal fiber

To comprehensively understand the pulse-propagation problem in a liquid-core PCF, we studied the mode properties of the guiding wave.

#### 3.1 Evaluation of the propagation constant β

Many methods for modeling are known to calculate the propagation constant *β*, e.g., the effective-index method (EIM), the multipole method (MPM), and the beam-propagation method (BPM) [24–26]. In this paper, we evaluate the propagation constant using a fully analytical vector approach of the effective-index method, which gives accurate results [27] [28].

The propagation constant β can be characterized theoretically by solving the propagation equation in fibers [29]:

where *κ*
^{2} = ${n}_{\mathit{\text{core}}}^{2}$
${k}_{0}^{2}$ - *β*
^{2}; *γ*
^{2} = *β*
^{2} - ${n}_{\mathit{\text{eff}}}^{2}$
${k}_{0}^{2}$; *J*_{l}*(x)* and *K*_{l}*(x)* are the *l*^{th}
order Bessel function and modified Bessel function, respectively. *l*=1 corresponds to the fundamental mode HE_{11} in the waist region; *n*_{core}
and *n*_{eff}
are the refractive indices of the core and the effective index of the cladding, respectively; *r*_{core}
is the core radius, which is approximately equal to 0.625∙Λ in silica-core PCFs. However, in our case, it is equal to the radius of the air hole *a*.

The cladding effective index *n*_{eff}
is evaluated by the following equations [27]:

where *n*
_{1} and *n*
_{2} are the refractive indices of air and silica, respectively, *a* is the radius of the air hole, *R* is the half of the pitch of the hexagonal lattice, *J*_{l}
and *Y*_{l}
are the *l*
^{th} order Bessel function of the first order and second order, respectively, *I*_{l}
is the *l*
^{th} order modified Bessel function of the first kind, and *l* is equal to 1 for the fundamental mode *EH*_{11}
in the cladding.

#### 3.2 Group velocity dispersion of the fundamental mode

The group velocity dispersion (GVD) is proportional to the second derivative of the propagation constant *β* with respect to ω,
$D=-\frac{2\mathrm{\pi c}}{{\lambda}^{2}}\frac{{d}^{2}\beta}{d{\omega}^{2}}$. Knowing the propagation constant, we can evaluate the group velocity dispersion of the fundamental mode in the liquid-core PCF. We designed the structures of our liquid-core PCFs to be capable to yield zero GVD in the infrared. The GVD curves are shown in Fig. 4 using the following parameters: (a) the core is filled with nitrobenzene, the diameter of the core and air holes *d* is 4 μm, and the lattice pitch (the distance between centers of neighboring holes) *Λ* is 6 μm; (b) the core is filled with carbon disulfide, *d* is 3 μm, and *Λ* is 4.5 μm. The zero-dispersion wavelengths occur around 1.55 μm in the near infrared, and the GVD curves exhibit slow variations and small values.

#### 3.3 Nonlinear parameter γ of the fundamental mode

The nonlinear parameter *γ* which is important for the understanding of the nonlinear properties, can be evaluated by Eq. (3) [29]:

where *n*
_{2} is the nonlinear refractive index of the fiber material, *ω* and *c* are the given frequency and the speed of light in vacuum, respectively, and *F*(*x*,*y*) is the modal distribution function. In the evaluation of the modal distribution function, we applied the vector approach of effective-index method and the scalar wave equation [30], which give accurate results of the nonlinear parameters [27, 33]

Figure 5 illustrates the nonlinear parameter *γ* of the liquid-core PCFs in dependence of wavelength with (a) the core filled with nitrobenzene and (b) the core filled with carbon disulfide. We used the same fiber parameters as in Fig. 4. The figure shows that the liquid-core PCF have extremely high nonlinear parameters even in the infrared region, e.g., *γ*=2.4/W/m at 1.55 μm. When compared with the nonlinear parameter of 0.11/W/m in a silica PCF, which has a core diameter of 1.4 μm, and a pumping wavelength of 800 nm [31], the carbon disulfide filled PCF with a core diameter of 3 μm shows a nonlinear parameter of 5.7/W/m at 800 nm.

#### 3.4 Guided modes in liquid-core photonic crystal fibers

The high refractive index of the liquid core leads to multi-mode propagation in a liquid-core PCF. The V-parameter $V=\frac{\pi \bullet d}{\lambda}\bullet \sqrt{\left({n}_{\mathit{core}}^{2}-{n}_{\mathit{eff}}^{2}\right)}$ is around 4.4 in the case of the core being filled with carbon disulfide. Despite the capability to generate a supercontinuum provided by a highly nonlinear multi-mode fiber, one difficulty arises, namely the generation of an output beam with good spatial characteristics and little mode dispersion, or more specifically, single-mode transverse characteristics. Single mode propagation and output can be obtained using adiabatic coupling in the following way: A single-mode PCF with the same propagation constant as that of the fundamental mode in the LCPCF can be used to generate a single mode beam, which subsequently is launched into the LCPCF. As the incident mode can couple only to modes with the same azimuthal symmetry and with a similar propagation constant [32], the fundamental mode becomes the dominant guided mode in the LCPCF. In the process of propagation inside the LCPCF, negligible coupling and small power loss from the fundamental mode to the higher order modes occur, because the small core and large core-cladding index difference of the LCPCF leads to a requirement of a large perturbation to couple to the higher order modes. Therefore, the fundamental mode propagates approximately adiabatically in the LCPCF, similar to light propagating in a single mode along a tapered fiber [33].

## 4. Supercontinuum simulation

Although both of the liquid-core PCF filled with carbon disulfide and nitrobenzene exhibited suitable GVD curves and showed large potential to generate supercontinuum, we simulated the SC generation of liquid-core PCF with carbon disulfide only because of the broad transparency of this liquid in the infrared region, as shown in Fig. 3. In the simulation, the generalized scalar propagation equation [7] [34] [31] was solved to study the pulse propagation properties in the liquid-core PCF by using a split-step Fourier algorithm:

where *A* = *A*(*z*,*t*) is envelope of the electric field, *β*_{k}
is the *k*^{th}
order dispersion coefficient at center frequency *ω*
_{0}, *γ* is the nonlinear parameter, and *R(t)* is the response function of carbon disulfide. The timescale *τ*_{shock}
includes the influence of the frequency-dependent effective mode area and can be written as [31]:

where *n*_{eff}
(*ω*) and *A*_{eff}
(*ω*) are the effective index and effective area of the guided mode, respectively.

#### 4.1. Response function R(t) of carbon disulfide

Unlike fused silica, the main mechanism responsible for the huge nonlinear refractive index of carbon disulfide is a non-instantaneous response due to molecular reorientation, which is induced by the tendency of molecules to align in the electric field of an applied optical wave. The nonlinear refractive index therefore depends highly on the pulse duration. In this paper, we derived the response function of carbon disulfide, which, as summarized by McMorrow *et al*. [35], includes instantaneous response due to electronic hyperpolarizability, subpicosecond response due to molecular librational motion, subpicosecond response due to collision-induced molecular polarizability, and long-time response due to molecular reorientation. The contribution of the intramolecular Raman effect in the carbon disulfide is not taken into account in our simulations, because pulses with extremely large bandwidth and short pulse duration (less than 20 fs) are required to excite those Raman-active modes in the molecule. In our case, where pulse durations are longer than 100 fs, the intramolecular Raman effect is considered to be negligible.

Using the Born-Oppenheimer approximation [36], the response function for optical pulses far from electronic resonance can be written in the following way:

where *f*_{e}
∙*δ*(*t*) and (1 - *f*_{e}
)∙*h*_{m}
(*t*) are electronic hyperpolarizability and molecular effect (including molecular librational motion, collision-induced molecular polarizability, and molecular reorientation), respectively, with *δ*(*t*) being the Dirac function and
$\underset{0}{\overset{\infty}{\int}}{h}_{m}\left(t\right)\mathit{dt}=1.{f}_{e}$ representing the fractional contribution of the electronic hyperpolarizability.

According to the measurements on the third-order hyperpolarizability, the electrical fractional contribution *f*_{e}
is determined to be *11%* [37].

The molecular contribution to the nonlinear optical response including molecular librational motion, collision-induced molecular polarizability, and molecular reorientation, can be determined by time-resolved optical Kerr-effect experiments. Based on the Born-Oppenheimer approximation, the molecular time-scale response of carbon disulfide was determined, as shown in Fig. 6, and a model functional form to express the molecular response was obtained [38], although this form was lacking rigorous physical justification. After normalization, the functional form of *h*_{m}*(t)* can be written as follows:

$$\phantom{\rule{.2em}{0ex}}+1.633\bullet \mathrm{exp}(-\frac{{\alpha}^{2}{t}^{2}}{2})\mathrm{sin}\left({\omega}_{0}t\right),\phantom{\rule{18.2em}{0ex}}$$

where *τ*_{diff}
= 1.68 *ps*, *τ*_{rise}
= 0.14 *ps*, *τ*_{int}
= 0.4 *ps*, *α*= 5.4 /*ps*, and *ω*_{0}
= 6.72 /*ps*. According to the estimate of *f*_{e}
, the fractional contribution of the molecular dynamics is determined to be 89%.

#### 4.2 Supercontinuum simulation results

Knowing the time-dependent response function of carbon disulfide, we are able to solve the generalized nonlinear Schrödinger equation and simulate the supercontinuum generation. From the molecular contribution in the response function, it is obvious that a pulse with longer pulse duration can lead to stronger nonlinear effects. We therefore simulated the spectra in the CS_{2}-core PCF with different input pulse durations.

### 4.2.1 Spectral properties with different input pulse durations and the same initial peak intensity

We first study the spectral properties in the liquid-core PCF with different pulse durations and the same initial peak intensity. The fiber used in our simulations has a core diameter of 3 μm, a lattice pitch of 4.5 μm, and a length of 5 cm. Its center hole was filled with carbon disulfide. The dispersion effects are considered up to the 14^{th} order, and the dispersion coefficients are shown in the appendix. The fiber is pumped at 1.55 μm. Actually, the inexact evaluation of dispersion coefficients and the neglected Raman effect can lead to imprecise results of the supercontinuum generation, however, it is reasonable to consider the simulated bandwidths of the spectra as indicative. Figures 7 and 8 show the SC spectra generated by the CS_{2}-core PCF with pulse durations of 100 fs and 500 fs, respectively. The figures demonstrate that a liquid-core PCF is capable to generate supercontinua with dramatically broadened spectra, which cover a range from 700 nm to more than 2500 nm. The output spectra depend on the input pulse duration, with the longer pulse duration causing a broader spectrum. Increasing the fiber length can further broaden the spectra.

### 4.2.2 Spectral properties with different input pulse durations and the same pulse energy

We also studied theoretically the spectral properties when the fiber is pumped with different input pulse durations, however, keeping the same pulse energy. In the simulations, we used a CS_{2}-core PCF with a core diameter of 3 μm, a lattice pitch of 4.5 μm, and a fiber length of 10 cm. The high nonlinearity of carbon disulfide enables the fiber to generate supercontinua with low input power, therefore, we applied a very low pulse energy of only 0.4 nJ in the simulations. For a hyperbolic-secant pulse, the peak power can be related to the pulse duration and pulse energy using the following formula:

where *P*
_{0} is the peak power, *W* represents the pulse energy, and *T*
_{0} is the pulse duration.

Figures 9 and 10 show the simulated spectra in the liquid-core PCF with pulse durations of 50 fs and 200 fs, respectively. The fiber is pumped at 1.55 μm. The figure demonstrates that pulses with the same input pulse energy generate a similar output spectrum. The supercontinuum ranges in both cases from about 1000 nm to more than 2000 nm, with the 200 fs pulse generating a slightly narrower spectrum.

### 4.2.3 Discussion

The supercontinuum spectrum generated in a liquid-core PCF is dramatically broadened. The generation mechanisms leading to such broadened spectra are similar as that in a silica-core PCF. Namely, the broadened spectra are generated by soliton splitting due to anomalous group velocity dispersion in combination with a very strong self-phase modulation (SPM) component, which, however, is mainly induced by the delayed molecular effects. The spectra also present a quite smooth spectrum, which is owed to the fact that the liquid-core PCF is pumped very close to the zero dispersion wavelength (ZDW). Pumping the fiber at a wavelength farther away from the ZDW leads to a slightly broader spectrum, however, at expense of a widened spectral dip in the near infrared region. Figure 11 shows a simulated output spectrum of a liquid-core PCF when the fiber is pumped at 1.7 μm. Except for the pump wavelength, the parameters used in the simulation are the same as in Fig. 10. It is clear that when compared with Fig. 10 the generated spectrum using the longer pump wavelength covers a wider region from 1000 nm to more than 2100 nm, however, lacking of a smooth output spectrum.

## 5. Conclusion

Theoretical calculations reveal that a liquid-core photonic crystal fiber is well suited as a highly nonlinear fiber for supercontinuum generation. The liquid-core PCF with carbon disulfide and nitrobenzene filled into the core exhibits an extremely high nonlinear parameter γ on the order of 2.4/W/m at 1.55 μm. The GVD curves of the liquid-core PCF display slow variation and small absolute values in the anomalous dispersion region, and the zero dispersion wavelength lies around 1.55 μm in the near infrared. The simulations of supercontinuum generation demonstrate that the liquid-core PCF is capable to generate dramatically broadened spectra in a range from 700 nm to more than 2500 nm.

## Appendix

The dispersion coefficients of the liquid-core PCF used in the simulations are evaluated as follows: *β*
_{2} = -1.077×10^{-2}
*ps*
^{2}/*m*, *β*
_{3} = 3.569×10^{-4}
*ps*
^{3}/*m*, *β*
_{4} = -7.658×10^{-7}
*ps*
^{4}/*m*, *β*
_{5} = 2.787×10^{-9}
*ps*
^{5}/*m*, *β*
_{6} = -1.096×10^{-11}
*ps*
^{6}/*m*, *β*
_{7} = 4.158×10^{-14}
*ps*
^{7}/*m*, *β*
_{8} = -7.924×10^{-17}
*ps*
^{8}/*m*, *β*
_{9} = -1.127×10^{-18}
*ps*
^{9}/*m*, *β*
_{10} = 2.488×10^{-20}
*ps*
^{10}/*m*, *β*
_{11} = -3.600×10^{-22}
*ps*
^{11}/*m*, *β*
_{12} = 4.472×10^{-24}
*ps*
^{12}/*m*, *β*
_{13} = -4.788×10^{-26}
*ps*
^{13}/*m*, and *β*
_{14} = 3.794×10^{-28}
*ps*
^{14}/*m*.

## Acknowledgements

The authors would like to thank J. Dudley for many important suggestions, A. Leitenstorfer for a crucial hint, A. Laubereau for valuable advice, C. Milne, R. J. Dwayne Miller, and L. Kaufman for the discussion of the response function of carbon disulfide, K. Buse’s group for use of their CARY spectrometer, H. C. Guo for the help of transmission measurements, and D. Meiser for critical reading. The work has been supported by DFG (FOR557, SPP1113) and BMBF (13N8340).

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