1. Introduction

Optical fibers incorporating a 2-dimensional array of fine air holes running down their length are called photonic crystal fibers (PCF’s). These fibers display a range of unusual optical properties, greatly broadening horizons for fiber optics [1]. One of the most surprising effects which has been demonstrated is that light can be guided with low loss down a large central air hole due to the photonic bandgap of the surrounding air/silica matrix [2]. Such bandgap-guided modes are only confined for a limited range of wavelengths, typically 10–15% of the central wavelength in silica-based structures. Hollow-core photonic bandgap fibers (HC-PBGF’s) are remarkable because they are the first demonstrated low-loss waveguides which can guide light single-mode in gas or vacuum cores [3]. The performance of such fibers is substantially freed from the limitations imposed by the solid core material found in all conventional fibers. Many – even most – of the limits of conventional fibers are imposed by the core material: the dispersion, the attenuation, the nonlinearity and the damage threshold of state-of-the-art conventional fibers are all dictated by their solid cores. In HC-PBGF’s these limitations are relieved, sometimes by several orders of magnitude, enabling a number of previously impossible applications.

One of the most dramatic demonstrations of this is the report of Megawatt femtosecond soliton propagation over 3m of photonic bandgap fiber at a wavelength of 1550nm [4]. In conventional fibers, such short, high-power pulses are very rapidly torn apart by self-phase modulation and Raman scattering, combined with the group-velocity dispersion (GVD) of the fiber. As a result they cannot propagate even over a few centimeters without significant distortion. On the other hand, solitons are stable propagating pulses which form due to a balance between the competing effects of anomalous group velocity dispersion and self-phase modulation. Self-phase modulation arises from the intensity-dependent refractive index n₂ of the fiber [5] which depends on the material through which the light travels. The value of n₂ is roughly three orders of magnitude less for air than it is for solid silica. In HC-PBGF’s, the nonlinear response is thus substantially reduced when compared to conventional fibers, because most of the light is trapped in the hollow core. Ouzounov et al [4] have used the reduced n₂ together with the fact that the GVD in HC-PBGF’s is anomalous over much of the transmission band, to deliver pulses over a short piece of HC-PBGF without significant temporal distortion at 1550nm wavelength. This demonstrates that HC-PBGF’s can be used to deliver high-power ultrashort optical pulses over relatively long lengths in a single spatial mode. In this paper we report delivery of ultrashort pulses using a HC-PBGF in the more common 800nm spectral region using a Titanium-Sapphire laser system. Whereas in the previous work [4] the Raman self-frequency shift [6] imposed a limit of less than 3 meters on the useable fiber length in air-core fiber, we have demonstrated transmission over 5 meters of HC-PBGF with only a relatively small Raman shift. Our experiments at 800nm are in a regime where the material dispersion of bulk silica is such as to preclude the transmission of short optical pulses through any significant length of solid-core fiber, even in the absence of nonlinear effects. Scaling the experiments to 800nm significantly reduces the fundamental soliton energy, making it conceivable that HC-PBGF could be used for soliton delivery of unamplified Titanium-Sapphire laser pulses.

2. Properties of the fiber

A range of properties of HC-PBGF’s for the 800nm band have been reported in a previous publication [7]. In this section we focus on those fiber properties which are directly relevant for the experiments being described, specifically the GVD and the Kerr nonlinear effect of self-phase modulation. The GVD in our fiber is shown in Fig. 1, along with the attenuation and the measured group index.

The attenuation was measured with a tungsten-halogen lamp and an optical spectrum analyzer. The minimum attenuation of 270dB/km is at a wavelength of 800nm. Attenuation in HC-PBGF rises as the wavelength decreases because of the strong inverse dependence of scattering processes on wavelength, so that the attenuation here is far higher than the 13dB/km previously reported for a similar type of fiber at 1500nm wavelength [8]. The group index [5] was measured using a low-coherence Michelson interferometer equipped with a supercontinuum source [9]. The minimum of 1.006 is located on the short-wavelength side of the low-loss band, and the index increases slowly towards the center of the transmission band. The GVD curve is derived from a fit to the measured group index data, and is anomalous (with positive slope) over most of the low-loss window, so enabling soliton propagation over much of this band. Our experiments were performed around 800nm, where the GVD was around 140 ps.nm^-1.km^-1 and the dispersion slope was 3ps.nm^-2.km^-1.

 

Fig. 1. Measured group index and group-velocity dispersion GVD, with inset showing the fiber attenuation. The measured group index points are indicated by crosses: the line is a fit to the data points and is used to derive the dispersion curve.

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3. Experiments

Our experiments were performed using a modelocked Titanium-Sapphire laser which was regeneratively amplified using a Q-switched pump source. The amplified pulse repetition rate was 5 kHz and the output pulses had a pulse length of 140fs. The output pulse length from the laser was roughly 1.5 times bandwidth limited. The central wavelength used was 796nm. The pulse power was controlled in our experiments using a polarizing beamsplitter, and a waveplate was used to align the excitation axis with one of the polarization axes of the fiber [7]. The beam from the amplifier was attenuated and coupled into the PBG fiber using a standard microscope objective. The coupling efficiency obtained was around 30% and the maximum output pulse energy before input endface damage occurred was 320nJ, after transmission through 5m of fiber. Experiments described here were performed at output pulse energies of up to 120nJ to avoid endface damage. Pulse characterization was done using a spectrometer and an autocorrelator based on 2-photon absorption in a GaAsP diode. The 5m length of HC-PBGF used in our experiments had an effective length of 4.2m and a dispersion length for 140fs pulses of 0.13m.

Sample autocorrelation traces for the pulses transmitted through 5m of fiber are shown in Fig. 2(a) for low and high pulse energies. The measured autocorrelation pulse lengths as a function of the output pulse energy are shown in Fig. 2(b). The features are very similar to those reported in [4] for 1550nm wavelength and 3m of fiber. Linear pulse propagation is expected to lead to an output pulse length of over 5ps, comparable to that in a silica fiber at this wavelength. As the input power is increased we observe dramatic pulse shortening to a minimum autocorrelation width of around 450fs at an output pulse energy of 60nJ. The output pulse length then remains almost constant towards higher pulse energies, although this short pulse sits on a pedestal of dispersed energy. Assuming a 1.55 deconvolution factor for the actual pulses gives an output pulse length of about 290fs. Observed output spectra for selected energies are shown in Fig. 3. The spectra show both a solitonic and a dispersive component, with the solitonic component having a bandwidth of roughly 3nm, implying that the compressed pulses shown in Fig. 2(b) are virtually bandwidth-limited. At 60nJ pulse energy, we estimate that more than 80% of the input energy is coupled to the soliton. The solitonic component is shifted to lower frequencies due to the soliton self-frequency shift, higher power resulting in a greater wavelength change. However, even over 5m of fiber and 120nJ output power, the soliton remains well within the low-loss window of the HC-PBGF.

 

Fig. 2. (a) Sample autocorrelation traces at low (28nJ – blue curve) and high (62nJ – red curve) pulse energies. (b) Measured autocorrelation widths (FWHM) as a function of output pulse energy.

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4. Discussion and analysis

Fundamental soliton propagation occurs when the linear anomalous GVD (D) is balanced by the nonlinear effect of self-phase modulation. Quantitatively, the peak power P₀ required for fundamental soliton propagation can be found by matching the dispersion length and the nonlinear length [5], giving

where A_eff is the nonlinear effective area [5] and τ is the FWHM pulse length. In Eq. (1), everything is in SI units. By analysis of guided modes in a similar fiber (see below), we have computed the effective area of our mode as 27µm² using an adaptation of the plane-wave method. Taking n₂ =2.9×10^-23 m²/W (the known value for n₂ for air [10]) and an output pulse length of 290fs, we get a peak power of 208kW and a pulse energy of around 68nJ. This preliminary number is in good agreement with the results in Fig. 2, suggesting that the nonlinearity of the air core plays an important role in determining the soliton energy.

In conventional fibers, n₂ is usually taken as being constant across the area of the guided mode, so that we can define the nonlinear change in the effective index of the guided mode as

where P is the peak power in the fiber. In HC-PBGF’s the guided mode covers both silica and air regions, which have widely differing values for n₂ . The nonlinear phase shift in hollow-core fibers has been variously attributed to exclusively the nonlinear refractive index of air [4] (n₂ =2.9×10^-23m²/W) and that of silica [11] (n₂ =2.4×10^-20m²/W). As the light is concentrated mainly in the air, which has a much lower value of n₂ than the silica, it is not obvious a priori which material contributes most to the observed phase shift. By defining new effective areas $A_{\mathit{\text{eff}}}^i$ for the silica and air regions [12], we can write

where the summation is over the i different materials. This enables us independently to compute the contribution to the nonlinear phase shift arising from the glass and from the air.

 

Fig. 3. Observed output spectra for different output pulse energies.

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We have used computed mode field patterns to evaluate the contribution to the nonlinear phase shift from the air and from the glass as in Eq. (3). Our fiber shown in Fig. 4(a) was modeled as the structure shown in Fig. 4(b). The modeled structure has a 92% air fraction in the cladding and a core formed by a seven-unit-cell defect. The pitch in the cladding was taken as 2.33µm and the thickness of the struts in the cladding was 70nm. The silica interstices seen in the actual fibers were reproduced in the model using arcs of circles. The core wall thickness was defined to be 76nm. The core of our experimental fiber became distorted during the fiber draw, which resulted in a rather larger core than seven unit cells, but we neglect this in our numerical modeling. Modeling of the structure in Fig. 4(b) gives a band gap in imperfect agreement with the experimental observations, but within the uncertainties of the structural parameters. The modeled bandgap covered the wavelength range from 780nm to 970nm, and had an avoided crossing of a “surface mode” in the middle [8]. This surface mode was not observed experimentally (Fig. 1, inset), perhaps because of the distortions around the core. The intensity pattern of the fundamental core-guided mode in the structure is shown in Fig. 4 (c), at a wavelength of 801nm. 99% of the energy travels in air, with just 1% of the energy located in the glass. The highest intensity occurring in the glass is 16% of the peak intensity in the centre of the hollow core.

Evaluation of Eq. (3) using the data shown in Fig. 4(c) gives $\frac{\Delta n_{\mathit{eff}}}{P}=1.28\times {10}^{-12}{W}^{-1}$ for the silica regions and $\frac{\Delta n_{\mathit{eff}}}{P}=1.78\times {10}^{-12}{W}^{-1}$ for the air regions. At first glance, the fact that the air and the glass appear to be making roughly equal contributions appears to contradict our experimental observation that the soliton energy can be predicted by neglecting the nonlinear contribution from the glass. However, the reason for this is immediately apparent on comparing Fig. 4(a) and Fig. 4(b). Our calculated soliton energy was derived from a value for the nonlinear effective area of the modeled fibers, which is significantly smaller (by a factor of around 1.5) than that of the actual fiber used in the experiments, which has an enlarged core. The enlarged core not only increases the effective area, but also increases the light-in-air fraction of the mode. Taking this into account, and also the fact that the pulse energies in the fiber are on average somewhat higher than those measured at the output (due to the linear fiber loss over 5 m length) the agreement between our experiments and modeling is actually very good. The fact that the silica and the air both contribute to the nonlinear phase shift suggests that it should be relatively easy to engineer the nonlinear response to reduce the fundamental soliton energy to be attainable with an unamplified modelocked laser system. For example, further numerical studies have shown that reducing the air-filling fraction to 87.5% in the structure above makes very little difference to the modal field pattern, but increases the silica contribution to the nonlinearity by a factor of more than 2, just because of the extra glass around the core.

 

Fig. 4. Actual (a) and modeled (b) fiber cross-sections, showing the region around the core. (c) shows the intensity pattern of the fundamental guided mode used in the modeling of the nonlinear response.

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5. Conclusions

We have demonstrated delivery of pulses from a regeneratively amplified Titanium-Sapphire laser system over 5m of HC-PBGF with an output pulse length of less than 300fs. The nonlinear phase shift required for soliton formation arises roughly equally from the nonlinear refractive index of air and the relatively small overlap of the guided mode with the more nonlinear silica. Based on our experimental and numerical results, we anticipate that it will be possible to deliver femtosecond pulses from an unamplified laser oscillator using a HC-PBGF with a somewhat smaller core and a lower air-filling-fraction cladding.