A scalable and accurate technique for measuring the group index and dispersion of optical fibers is used to provide the first accurate measurements of dispersion slope in hollow-core photonic band-gap fibers. We present data showing group index, group-velocity dispersion and dispersion slope in hollow-core fibers guiding at both 800 nm and 1064 nm wavelength.
©2009 Optical Society of America
The advent of microstructured fibers has given additional freedom to fiber design and allowed the tailoring of dispersion to a degree which was not possible in conventional fibers. In hollow-core photonic crystal fibers (HC-PCFs) the dispersion properties come almost solely from structural dispersion, due to the waveguiding core and the surrounding photonic band-gap material with little contribution from bulk material dispersion. This not only gives rise to anomalous dispersion across much of the restricted range of wavelengths over which light is guided but allows the dispersion curve to be shifted in wavelength by scaling the cladding structure . The anomalous dispersion of HC-PCFs is used in pulse recompression [2,3], modelocked fiber lasers  and soliton delivery . For these applications accurate knowledge of the dispersion is necessary.
To measure the dispersion of HC-PCF’s we previously used a method in which a low-coherence interferometer with a test fiber in one arm was used to measure the wavelength dependence of group index. By differentiating this data we can derive the group-velocity dispersion, and in principle, a second numerical derivative provides the dispersion slope. However in practice this is not possible. When a second derivative of any of our previously published data was taken even apparently smooth dispersion data is in fact shown to be rough. Therefore it is not possible to deduce the dispersion slope from these seemingly accurate plots, without making additional assumptions. This problem is exacerbated with the current generation of HC-PCFs which have reduced dispersion compared to those previously studied , and so the change in their group index with wavelength is smaller and the signal-to-noise ratio of the measurement decreases. Using longer lengths of test fiber would increase the absolute value of group delay, thus increasing the accuracy of the measurement. However this would require a long free-space delay arm which becomes impractical for long lengths of test fiber.
In our past measurements, the wavelength-tunable low-coherence source for the interferometer consisted of a filtered supercontinuum from a low repetition rate (kHz), sub-nanosecond, microchip laser at 1064 nm pumping a nonlinear fiber. If a mode-locked laser is used instead as a pump source, it is possible to interfere a pulse which has been delayed by travelling though a long length of test fiber in one arm with the next pulse from the oscillator in the shorter arm; allowing the use of an almost arbitrarily long test fiber without the use of a similarly long delay arm. This technique is scalable; the length of test fiber can be increased by integer multiples of the pump laser’s cavity length.
The supercontinuum to be used in this measurement has to have a high level of shot-to-shot coherence. Supercontinua generated in photonic crystal fibers (PCF’s) have widely varying noise characteristics depending on the characteristics of the nonlinear fiber and the pump source. Commonly, broadband supercontinua rely on soliton propagation, over some part of the bandwidth, for spectral broadening. If modulation instability is also present, it amplifies low-level noise present in the system, causing the solitons to have varying amplitudes and durations and rendering the source useless for the type of measurement described here. To overcome, this shorter pulses are often used to pump the supercontinuum, so that self phase modulation plays a greater role in the spectral broadening and the effects of modulation instability are reduced .
Before a dispersion measurement was attempted, the coherence of the supercontinuum was checked using the interferometer setup in Fig. 1a . In this setup the arms of the interferometer were unbalanced by a length corresponding to that of the laser cavity, to interfere adjacent continuum pulses in a manner similar to Lu et al . The Ti-Sapphire oscillator emits 200 fs pulses at 75.3 MHz, with a total power of 550 mW. The output was passed though a band-pass filter and isolator to minimize feedback.
With an equal intensity of light in both arms, different fibers were tested as nonlinear elements to establish over what wavelength range it was possible to obtain fringes. The supercontinua were attenuated before entering the endlessly single mode (ESM) fiber which fed the interferometer so that the all subsequent propagation was linear, with wavelength selection being provided by a 3 nm band-pass monochrometer.
After testing various fibers and lengths a long wavelength limit of ~1080 nm was found; no fiber tested produced a coherent continuum beyond this wavelength. The best nonlinear fiber had a core diameter of 1.5 μm. Larger core fibers have a zero dispersion wavelength far from the pump and modulation instability has less effect in the supercontinuum generation process , making any continuum generated more coherent. However the increased dispersion combined with the reduced nonlinear response (γ) of the larger core mean that the generated continuum is not as broad [8,9]. A similar effect is seen for different lengths of supercontinuum fiber, whereby longer lengths generate a broader spectrum but are not as coherent (Fig. 2 ).
3. Dispersion Measurements
To measure the dispersion (D)  of a test fiber, setup Fig. 1(b) was used, with ESM fiber being used to eliminate the influence of higher order modes. A lock-in amplifier allowed the fringes to be visible without the intensity of the light in both arms being balanced, and a thin film polarizer allowed one polarization of guided mode to be selected at a time.
HC-PCFs are of particular interest at three wavelengths: 800, 1064 and 1550 nm to match available laser sources. However it was only possible to complete a full measurement on a fiber designed for 800 nm using this technique because of the lack of coherence at longer wavelengths. As well as measuring the dispersion of this fiber, it was also possible to deduce the test fiber’s group index using the following method. Interference fringes were first observed interfering adjacent pulses (with a 4 m length of fiber in the test fiber arm). Then this length of fiber was cut very accurately such that the arms of the interferometer became rebalanced. If the length of the removed fiber and the repetition rate of the laser are known accurately then the group index can be deduced.
The measured group index and group-velocity dispersion curves are shown in Fig. 3 . Our highly accurate measurements also allow us to plot the ratio of dispersion to dispersion slope. Accurate knowledge of this ratio is essential as higher order dispersion has a negative effect on many of the techniques that use the anomalous dispersion of HC-PCFs [2–5]. The group index curve is close to unity and of a magnitude slightly less than given by Bouwmans et al. in 2003 . This is not unexpected as the pitch is larger (2.2 μm compared to 1.94 μm), the cladding air-filling fraction is higher, and the glass wall that surrounds the core is thinner in the fiber used in the current measurements . We believe that the main source of inaccuracy in these measurements is the due to uncertainty in interpreting the center of the recorded interference fringe. If this was done with an accuracy of ± 0.005 mm and points are taken at a spacing of 10nm then it follows that the accuracy of the of the dispersion points is ± 1.6 ps/nm/km, for a 4.14 m fiber length.
An accurate measurement of the dispersion of a fiber guiding at 1064 nm was also made using a similar 4 m length of test fiber (Fig. 4 .). Due to the decoherence of the supercontinuum source in this wavelength range it was not possible overlap a pulse with next repetition of the laser. Therefore a long free space delay arm was used. A microchip laser was used as the supercontinuum pump source for simplicity.
The value of dispersion / dispersion slope for the 1064 nm fiber is much larger than that of the 800 nm fiber, with values of approximately 70 nm and 30 nm respectively. Frequently dispersion is expressed in terms of β 2, Eq. (1). Comparing the two fibers the maximum values of β2/ β3 are approximately −0.1 and −0.08 fs−1 for the 1064 nm and 800 nm fibers respectively. A typical value for a grating pair compressor is −0.7 fs−1 (at 800 nm) .
We present results for a modeled HC-PCF (Fig. 4(d)) with a similar structure to the fiber in Fig. 4(a,b,c). The pitch was 3.3 μm, the strut thickness 100 nm and the size of the glass regions where the cladding struts meet was 310 nm. The curve of the ratio of dispersion to dispersion slope is qualitatively similar to that of the measured fiber. The differences between the two fibers are due to the difficulties in measuring and then replicating an actual fiber’s structure when modeling, as slight (nm scale) changes can have large effects of the fiber’s properties. Using a scanning electron microscope (SEM) we measured the parameters of the 800 nm and 1064 nm fibers, to be as follows, the pitches were 2.2 μm and 3.4 μm, the strut thicknesses were 129 nm and 152 nm and the size of the glass regions where the cladding struts meet were 317 nm and 394 nm respectively.
We present an accurate and scalable technique for measuring the group index of hollow-core fibers. With a proof of principle experiment, accurate measurements of both these quantities have been calculated with an accuracy that allows for the first time a plot of the ratio of dispersion to dispersion slope for a HC-PCF.
The authors acknowledge the help of Alan George and Steve Renshaw in fabricating the fibers. This work was funded by the UK EPSRC, by the FP6 project “NextGenPCF” and the FP7 project “CARS Explorer” funded by the European Commission. WJW is a Royal Society University Research Fellow.
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