## Abstract

The fundamental limits to the compressibility of broadband supercontinuum spectra generated in photonic crystal fiber are examined using numerical simulations based on a stochastic extended nonlinear Schrödinger equation. An ensemble average over multiple simulations performed with random quantum noise on the input pulse and spontaneous Raman noise during propagation allows a quantitative study of the effects of pulse to pulse fluctuations on the ability to obtain few-cycle pulses after compensation of the supercontinuum spectral phase. We study the dependence of the supercontinuum compressibility on the input pulse duration, the photonic crystal fiber length, and the spectral resolution of the pulse compressor employed.

© 2004 Optical Society of America

## 1. Introduction

Although supercontinuum (SC) generation in photonic crystal fibers (PCF) has now been intensively studied by many groups [1], it is only recently that its application to pulse compression has been considered in detail. SC can be routinely generated with spectral widths around 500 THz, suggesting that compression to the sub-5 fs regime should be possible. However, experimental reports of compression in PCF have been limited to SC generation regimes associated with significantly smaller spectral widths around 30 THz, and the compressed pulse durations reported have been in the range of only 20–30 fs [2, 3].

These limited experimental results can be understood by noting that recent numerical studies have shown that ultrabroadband SC with 100’s of THz bandwidth can exhibit significant spectral phase instabilities, whereas efficient SC compression is only possible when the SC spectral phase remains stable from shot-to-shot [4, 5]. Reference [5] in particular studied the effect of spectral phase instabilities on SC compressibility by considering two distinct SC generated by independent input pulses with peak powers differing by 0.2%. It was shown that even this small variation in input power was sufficient to induce differences in the spectral phase characteristics such that compressing the SC obtained in one case using an ideal compressor designed for the other resulted in significant jitter in the compressed pulse that was obtained.

The objective of this paper is to consider this problem in more detail, studying in particular the expected compressed pulse characteristics in the presence of noise. Our calculations take into account quantum noise on the input pulse and spontaneous Raman scattering during propagation. Note that we do not consider in this work the effect of technical noise sources that are also likely to play a limiting role under many experimental conditions, but which can be reduced by appropriate technological improvements. In this way, we are able to determine the fundamental limits to SC compressibility, and to discuss some general features of the SC compression problem that may prove useful in further studies. In particular, we investigate the compressibility as a function of input pulse duration and fiber propagation distance, and we also study the effect on the compression efficiency of the finite resolution of the spectral phase compensation employed.

## 2. Numerical model and single-shot compressibility

Our analysis is based on numerical simulations of a stochastic nonlinear Schrödinger equation that rigorously includes input pulse quantum noise and spontaneous Raman noise during propagation [6]. The inclusion of higher order dispersion and nonlinearity permits accurate modeling in the few-cycle regime, and the simulations have previously been demonstrated to yield results in quantitative agreement with experiments [7]. We consider typical experimental conditions for SC generation in a highly nonlinear PCF with zero dispersion wavelength at 780 nm, and we consider pumping in the anomalous dispersion regime at 850 nm. Using the parameters of an equivalent 2.5 µm diameter fused silica strand, we have at this pump wavelength, a nonlinearity coefficient of *γ*=77 W^{-1} km^{-1}, and dispersion coefficients of: *β*
_{2}=-12.759 ps^{2} km^{-1}, *β*_{3}
=8.1186×10^{-2} ps^{3} km^{-1}, *β*_{4}
=-1.3215×10^{-4} ps^{4} km^{-1}, *β*_{5}
=3.0322×10^{-7} ps^{5} km^{-1}, *β*_{6}
=-4.1959×10^{-10} ps^{6} km^{-1}, *β*_{7}
=2.5696×10^{-13} ps^{7} km^{-1}. The input pulses have hyperbolic secant profiles with a range of pulse durations from 25–150 fs as described below. The pulse energy used was 1 nJ in all cases corresponding to peak powers in the range 35.3-5.9 kW.

We first consider the general spectral features of the SC, and the corresponding compressed pulse characteristics that might be expected under ideal conditions. Figure 1(a) shows the SC output spectrum obtained from one simulation for an injected 150 fs input pulse propagating in 10 cm of PCF. The spectrum at the fiber output (left axis) exhibits the typical features of SC generation in the case of anomalous dispersion regime pumping, showing complex wavelength-dependent structure, and a series of distinct Raman soliton peaks at wavelengths greater than 900 nm [8]. The SC spectral phase is conveniently examined in terms of the wavelength-dependent group delay (right axis), and the dashed line also shows the expected group delay based only on dispersive propagation in the PCF. It is apparent that, although the general structure of the group delay follows the intrinsic PCF dispersion over a large wavelength range, the combined effects of nonlinearity and dispersion result in significant fine structure, as well as distinct flat regions of wavelength-independent group delay across the generated Raman solitons.

Because of this group delay complexity, efficient compression of the SC would not be possible using simple dispersive delay-line type elements, and more advanced techniques for spectral phase manipulation would be required [9]. Optimal compression would correspond to exact compensation of the spectral phase, and this can be readily performed numerically in order to determine the compressed pulse characteristics expected on a single-shot basis under ideal conditions. These results are shown in Fig. 1(b). Despite residual substructure arising from the complex spectral intensity, the compressed pulse nonetheless shows a well defined central lobe of sub-3 fs duration, indicating that obtaining few-cycle pulses after spectral phase compensation is, in principle, possible.

## 3. Effect of noise and multi-shot compressibility

Although the single-shot result above suggests the potential compressibility of PCF-generated SC, it nonetheless represents a lower limit on the compressed pulse duration. Achieving such compression in practice, however, would require a high degree of spectral stability, because existing phase compensation techniques (based on spatial light modulators for example) would be unable to adaptively follow noise-induced shot-to-shot fluctuations in the spectral phase occurring at the MHz repetition rates of typical pump laser sources.

Spectral phase instabilities in SC generation arise from the sensitivity of the spectral broadening mechanisms to input pulse noise, and depend strongly on parameters such as the input pulse duration and propagation distance [10, 11]. To examine the consequences for SC compressibility, we carried out multiple simulations to obtain an ensemble of typically 100 SC generated from different initial noise seeds. As shown in Ref. [10], the SC spectral phase stability can be rigorously quantified through the modulus of the complex degree of mutual coherence: |${{\mathrm{g}}^{\left(1\right)}}_{12}$(λ,t_{1}-t_{2})|=|<E_{1}*(λ, t_{1})E_{2}(λ,t_{2})>/[<|E_{1}(λ,t_{1})|^{2}> <|E_{2}(λ,t_{2})|^{2}>]^{1/2}|. Here, the angle brackets indicate an ensemble average over independently generated SC pairs [E_{1}(λ,t_{1}), E_{2}(λ,t_{2})] and we consider zero relative delay between the fields so that t_{1}-t_{2}=0. To examine the correlation between the spectral decoherence and compressibility for a particular set of input parameters, the results from the ensemble were used to not only calculate the mutual coherence function as described above, but also to simulate the pulse compression process that would occur in a multi-shot experiment. To this end, we calculated the median spectral phase over the ensemble, and used the corresponding phase conjugate as the transfer function of an ideal compressor that numerically compensates the spectral phases from each individual SC in the ensemble. Taking the inverse Fourier transform of each resulting compensated spectrum then yields a shot-to-shot distribution of the temporal characteristics of the resulting compressed pulses. By taking the mean over this distribution, we can predict the average compressed pulse that would be expected in a realistic experiment.

Figure 2 shows the results obtained using this procedure for input pulses of (a) 150 fs, (b) 50 fs and (c) 25 fs duration. The top graphs show the mean spectrum and coherence, while the bottom graphs show the corresponding mean compressed pulses obtained using the procedure described above. Due to the variation in the SC spectral width with pulse duration, the single-shot lower limits on the achievable compressed pulse durations are slightly different at: (a) 2.8 fs, (b) 2.5 fs and (c) 2.2 fs. However, although this suggests that sub-5 fs pulses are potentially obtainable for all three cases, the strong dependence of the coherence on input pulse duration translates directly into differences in the corresponding mean compressed pulses. This is clearly shown in the bottom graphs. Quantitatively, we have found it convenient to correlate the width of the mean compressed pulse for each case with the median value of the coherence |g12| calculated over the -20 dB points of the spectrum which yields: (a) 20 fs and med(|g_{12}|)≈0.2, (b) 2.6 fs and med(|g_{12}|)≈0.75, (c) 2.2 fs and med(|g_{12}|)≈0.99.

Although these results might suggest that obtaining few-cycle pulses after SC compression necessarily requires initial injected pulses in the sub-50 fs regime, it should be noted that the spectral coherence degradation also depends strongly on propagation distance, and that the use of shorter fiber lengths has previously been demonstrated to lead to improved spectral stability and coherence during SC generation [10]. To illustrate this, we consider the case of an initial 150 fs pulse where the highest coherence degradation after 10 cm was observed, but we consider propagation in shorter fiber segments of (a) 2.5 cm and (b) 5 cm. Results similar to those above are shown in Fig. 3. Here, we obtain for the duration of the mean compressed pulse and the median coherence (a) 3.5 fs and med(|g_{12}|)≈0.7 and (b) 10 fs and med(|g_{12}|)≈0.50. It is interesting to note that a median coherence as low as 0.5 can still yield 10 fs pulses after compression, but it is clear that the overall pulse quality in this case is low, with the central 10 fs pulse lobe sitting atop a strong pedestal component.

To consider the relationship between achievable pulse duration and pulse quality in more detail, we define the fraction of the total pulse energy contained in the central pulse lobe as a useful figure of merit for the pulse quality. Using this criterion, Fig. 3(c) plots as a function of propagation distance (over the first 5 cm) the duration of the mean compressed pulse, the associated median coherence, and the fractional energy within the central lobe. For propagation distances less than 1 cm, the pulse coherence and quality are both very high but, because significant spectral broadening has not yet occurred, the compressed pulse durations are relatively long (> 30 fs). With increasing propagation distance, the duration of the compressed pulse decreases as the SC spectral broadening increases; however, because the spectral coherence itself also decreases, significant temporal jitter appears in the compressed pulse and the energy within the pulse pedestal increases. As a result, after around 3.5 cm, the compressed pulse duration itself begins to increase. From the figure, we can identify a propagation distance of around 2.5 cm with med(|g_{12}|)≈0.7 as an optimal fiber length yielding a sub-5 fs central pulse lobe containing over 70 % of the total pulse energy.

## 4. Effect of compressor resolution

The numerical phase compensation carried out above is, of course, based on an ideal compressor with spectral resolution equal to the simulation precision (typically <0.1 nm around the pump wavelength.) In this case, phase compensation is always possible at the level of the finest wavelength-dependent structure that is present on the spectral phase. In practice, however, the compressed pulse quality would be strongly dependent on the particular compressor resolution available in a particular experiment, even in the case where there were negligible shot-to-shot phase fluctuations. To illustrate this, we consider the case shown above in Fig. 2(b) where a 50 fs pulse injected in 10 cm of PCF yielded a 2.6 fs compressed pulse assuming simulation-limited compressor resolution. Here, however, we calculate the compressed pulse characteristics for the case of imperfect spectral phase compensation with (a) 1600, (b) 800 and (c) 400 compressor elements over an 800 nm bandwidth covering more than the -20dB extremes of the SC. This corresponds to spectral resolutions of 0.5 nm, 1 nm and 2 nm respectively. The mean pulse characteristics obtained for these three cases are shown in Fig. 4. These results clearly show the degradation in the quality of the compressed pulse as the compressor resolution varies from 0.5 nm/element to 2 nm/element. The quantitative increase in the FWHM of the central lobe is given in the caption.

It is particularly significant to note that the resolutions used in these numerical simulations greatly exceed those that are commonly attainable with spatial light modulators where typically only 50–100 pixels are used. This highlights the potential difficulties in compressing complex broadband SC using available technology. In this context, we note that attempting the numerical compression of the generated SC in Fig 2(b) using only 50 elements over the 800 nm bandwidth yielded a “compressed” pulse whose duration of 500 fs was significantly broader than the original 50 fs pulse used to generate the SC.

## 5. Conclusions

There are several important conclusions to be drawn from this work. Firstly, even in the presence of input pulse and propagation noise at the fundamental quantum level, fluctuations in the spectral phase of SC can be sufficient to seriously hinder the compression of the SC to the few cycle regime. Our simulations have confirmed that the compressed pulse quality is closely related to the spectral coherence of the SC, and a study of compression as a function of propagation distance has identified a median spectral coherence of med(|g_{12}|)≈0.7 as yielding multi-shot compressed pulse characteristics close to the theoretical limit expected from single-shot compression. Extensive additional simulations suggest that a median coherence of around 0.7 can be used as a convenient benchmark for the potential compressibility of a SC generated over a wide range of experimental conditions. However, our simulations also stress the need to use compressor technology capable of adequately compensating the fine structure present in the SC group delay. Nonetheless, with the use of compressors of sufficiently high resolution, octave spanning SC generated under quantum-limited conditions should indeed yield few cycle pulses after compression. We anticipate that the numerical procedure for simulating SC compression and the benchmarks for assessing potential compressibility will find application in further studies considering a wider range of experimental parameters and including additional technical sources of noise.

## Acknowledgments

The work of J. M. Dudley has been supported in part by a *Fonds National pour la Science* contract ACI-Photonique PH43.

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