1. Introduction

Self-similar scaling and intermediate asymptotic evolution are fundamental physical properties of a large number of very diverse phenomena ranging from groundwater flow and flooding processes to blastwave expansion during the explosion of nuclear weapons [1]. In nonlinear optics, research in this field has led to the prediction and observation of a novel type of ultrashort optical pulse with parabolic intensity profile generated in optical fiber amplifiers with normal group velocity dispersion [2–7]. Such parabolic similariton pulses are of fundamental interest because they represent a particular class of solution to the nonlinear Schrödinger equation (NLSE) with gain that is associated with self-similar propagation such that the pulse retains its parabolic intensity profile and resists the deleterious effects of optical wavebreaking. Moreover, they are of much practical significance since they are generated asymptotically in the amplifier independent of the input pulse shape, and they possess a strictly linear chirp which leads to efficient pulse compression. A number of experimental studies have now reported high power similariton generation in fiber amplifiers based on Yb³⁺ and Er³⁺ doped fiber [4, 8–12], as well as amplifiers exploiting the Raman effect [13–15]. Although these results have firmly established the technological importance of similariton amplifiers, research into the fundamental nature of the self-similar evolution in this regime has remained largely unexplored. In some cases for example, the similariton pulse characterization has been performed using only spectral measurements [8–12] and, even when complete characterization using frequency resolved optical gating (FROG) has been used, measurements have been restricted only to the amplifier output pulses [4, 13–15].

The major objective of this paper is to report a series of experiments completely characterizing the pulse evolution dynamics in an Er³⁺ doped fiber based similariton amplifier using high dynamic range FROG. In particular, a cutback procedure on the Er³⁺ fiber has allowed us to perform detailed measurements of the self-similar evolution phase within the amplifier gain medium itself. The experimental results are shown to agree well with numerical simulations, and we have also been able to confirm previous theoretical and numerical studies that have predicted that the self-similar evolution in the amplifier is associated with the development of intermediate asymptotic wings on the parabolic pulse core. Our measurements have then been used to optimize additional self-similar propagation in highly-nonlinear fiber (HNLF) and subsequent compression in a hollow-core bandgap guiding photonic crystal fiber (PCF) [16].

2. Simulations

The Er³⁺ similariton amplifier in this study was designed and optimized via a generalized NLSE model used in a number of previous studies [4,5,9,11]:

$\frac{\partial E(z,T)}{\partial z}=-i\frac{{\beta }_2}{2}\frac{{\partial }^{2}E(z,T)}{\partial T^2}+\text{i}\gamma E(z,T){\mid E(z,T)\mid }^2+\frac{g\left(z\right)}{2}E(z,T).$.

Here, E(z,T) is the pulse envelope in a co-moving frame, and parameters γ=6×10^-3 W^-1 m^-1 and β₂=40×10^-3 ps² m^-1 were used to model the particular Er³⁺ doped fiber (OFS R37003) used in our experiments. The amplifier was pumped at 1480 nm with the pump co-propagating with near transform-limited picosecond input pulses from a Pritel FFL Series passively modelocked fiber laser. The input pulses had close to a hyperbolic secant intensity profile with FWHM and energy 1.4 ps and 100 pJ respectively, but the simulations used the complex pulse envelope obtained from our FROG measurements described below. With a gain of 13.6 dB over a 9 m length of fiber (corresponding experimentally to 140 mW of 1480 nm pump), the gain distribution was well-modeled by: g(z)=g₀ exp(-z/z_g) with g₀=0.68 m^-1 and z_g=7m. For these parameters, Fig. 1(a) shows the simulated evolution of the input pulses in the amplifier, and the amplification and reshaping of the input pulse to a pulse with a parabolic profile after approximately 4 m of propagation is very apparent.

 

Fig. 1. (a) Numerical simulations of parabolic pulse evolution in a normal dispersion amplifier as described in the text. (b) Corresponding experimental results obtained from FROG measurements. (c) Experimental results using a logarithmic intensity scale to illustrate the temporal broadening and the growth of intermediate asymptotic wings. (d) Experimental results using a logarithmic intensity scale to illustrate the associated spectral evolution.

Download Full Size | PDF

3. Experimental setup and results

It is clear that the dynamical evolution shown in Fig 1(a) should be observable in suitably designed experiments. To this end, we constructed a 9 m amplifier as described above and performed experiments where the amplifier length was cut back in 50 cm increments. At each length, the intensity and chirp were measured using second harmonic generation FROG with the Er³⁺ fiber coupled directly into the FROG to avoid unwanted distortion from extra fiber segments. The use of a high dynamic range spectrum analyzer and data averaging yielded a dynamic range of over 40 dB at the second harmonic wavelength, and a comparable dynamic range in the retrieved intensity profile. Extensive tests using synthetic data were used to verify the fidelity of the retrieval of the physically significant features in both the central region and the wings of the pulses studied here [17]. Figure 1(b) shows the experimental results with the peak power calculated from pulse energy measurements. The experimental intensity evolution is in remarkable agreement with the simulations and clearly shows the expected dynamical evolution to the asymptotic parabolic pulse. The pulse duration (FWHM) and peak power of the output pulses in this case are 5.2 ps and 560 W respectively.

The experimental evolution dynamics are shown in more detail in Fig. 1(c) and (d) where the temporal and spectral profiles are shown (using logarithmic scales) at 2 m increments. Although the figure shows that the pulse evolution is associated with significant broadening in both temporal and spectral domains, of particular interest here is the temporal reshaping dynamics apparent in Fig. 1(c). Specifically, we see that the initial evolution from 0–4 m is associated with a significant temporal reshaping of the input pulse as it evolves towards the parabolic form with its characteristic rapidly falling leading and trailing edges (when plotted on a logarithmic scale). In contrast, the subsequent evolution from 4–8 m is qualitatively different. Here, although the intensity and duration both increase with propagation distance, the characteristic features of the parabolic pulse shape are preserved. This is a very important experimental observation, as it directly reveals for the first time the self-similar nature of the pulse evolution in this regime

An additional and significant feature in Fig. 1(c) is the presence of low amplitude wings that are apparent about the central parabolic core region after around 4 m propagation. In this context, we note that the linear nature of the wings when plotted on a logarithmic scale is associated with exponential decay as a function of co-moving time. These low amplitude wings represent a fundamentally important aspect of the pulse evolution in the amplifier, as they are associated with the intermediate asymptotic regime of the self-similar propagation. Intermediate asymptotic evolution is of crucial importance in nonlinear physics, as it describes the development of a self-similar system at propagation distances such that fine structure due to the boundaries has disappeared yet the system is still far from its ultimate asymptotic state. For the similariton amplifier, this corresponds to the fact that the self-similar nature of the propagation is sufficiently dominant so that we can clearly identify a parabolic pulse core, yet the presence of the wings indicates that the strictly asymptotic solution has not been reached.

These results confirm previous analytic and numerical predictions of the presence of exponentially-decaying intermediate asymptotic wings [5–7]. These wings are also seen in our simulations, and Fig. 2(a) compares explicitly the experimental results (solid lines) with the simulations (dashed line) for 7 m propagation. Here we plot the intensity profiles on logarithmic scales, and we also show the corresponding chirps. Note that the shading is used to distinguish the central parabolic core region from the wings, where the boundary of the core is determined from a numerical fit to an ideal parabolic pulse. We note excellent quantitative agreement over the central parabolic core region, and good qualitative agreement in the region of the wings. We attribute the difference between simulations and experiment in the amplitude of the wings to inherent approximations of the NLSE modelling such as the neglect of resonant dispersion and amplified spontaneous emission.

 

Fig. 2. (a) Experiment (solid line) and simulation (dashed line) after 7 m propagation, showing intensity (left axis) and chirp (right axis). The shading distinguishes the parabolic pulse core and intermediate asymptotic wings. (b) Transition between the core and wing regions. (c) Experimental results (circles) and best straight line fit (solid line) showing the exponential decrease in the relative energy in the intermediate asymptotic wings with propagation.

Download Full Size | PDF

Our results also allow two additional features of the intermediate asymptotic evolution to be characterized. The first is the presence of oscillations in the transition region between the parabolic pulse core and the wings, with the detailed plot in Fig. 2(b) showing the good agreement between simulation and experiment in this regime. The matching of the transition between asymptotic and intermediate asymptotic regimes is of much theoretical interest, and we anticipate that these results may motivate additional analytical work in this area [7]. Secondly, although we see from Fig. 1(c) that the absolute amplitude of the intermediate asymptotic wings increases with propagation distance in the amplifier, the relative contribution of the wings to the total pulse energy would be expected to decrease as the pulse enters the asymptotic regime [4, 5]. Our ability to distinguish the parabolic pulse core from the exponentially decaying wings in our experimental results allows us to confirm this directly, and Fig. 2(c) plots the fraction of the total energy contained in the wings (for propagation distances greater than 3 m. We see clearly that the energy fraction in the wings decreases exponentially as the pulse evolves to the ideal asymptotic parabolic profile.

4. Self-similar evolution in passive fiber and photonic bandgap fiber compression

Studies using longer amplifier lengths and higher pump powers showed that fiber lengths or integrated gains exceeding 15 m or 16 dB respectively led to significant deviations from the parabolic profile due to gain-bandwidth limiting. With the particular Er³⁺ fiber used in our experiments, the spectral extent of pulses possessing strictly linearly chirped similariton characteristics was limited to 1535–1570 nm at the -10 dB level. For potential compression applications, such bandwidths would, however, limit the achievable Fourier transform limited compressed pulse duration to ~250 fs. To generate additional bandwidth whilst retaining the desired linearly-chirped similariton characteristics, we exploit the fact that self-similar propagation can also be observed in passive undoped optical fibers with normal dispersion [2]. To demonstrate this experimentally, we first constructed a near-identical amplifier to that described above using 10 m of Er³⁺ fiber providing a total gain of 14.5 dB with 170 mW of 1480 nm pump. With identical input pulses as above, Fig. 3 (a) shows the output spectrum and intensity and chirp directly from the amplifier illustrating the expected similariton characteristics. The spectral and temporal widths (FWHM) here are 17 nm and 5.4 ps respectively. The Er³⁺ fiber was then fusion spliced (with 80% coupling efficiency) to 10 m of OFS Speciality HNLF with a flat normal dispersion β₂=3.1×10^-3 ps² m^-1 over the range 1510–1680 nm. Fig. 3(b) shows the results obtained. After propagation in the HNLF, we note both spectral and temporal broadening to FWHM of 35 nm and 6.7 ps respectively, but it is also clear that both the parabolic intensity profile and linear chirp are preserved. The additional self-similar propagation in the HNLF allows approximately a factor of two increase in the pulse bandwidth whilst retaining the linearly-chirped parabolic characteristics.

The similariton pulses after the HNLF were then injected into photonic bandgap air-guiding PCF for compression, and the output compressed pulse characteristics were measured using high dynamic range FROG. The HNLF-PCF coupling was through precise physical contact coupling and 60% efficiency was obtained. The particular bandgap fiber used (Crystal Fiber-BlazePhotonics HC-1550-02) had attenuation of less than 0.03 dB/m and reasonably flat anomalous dispersion of β₂=-0.11 ps² m^-1 over 1500–1600 nm. Although the use of bandgap guiding PCF for pulse compression has been previously reported [16], our complete characterization of the similariton pulses allows numerical propagation to be used to select the precise length of the required bandgap fiber in order to optimize the compressed pulse quality. Fig. 3(c) shows the results obtained using an optimal length of 2 m of PCF. The solid line shows the measured intensity profile in this case which yields a compressed pulse FWHM of 136 fs. This represents a compression ratio of 50× relative to the 6.7 ps duration pulses at the input to the bandgap fiber, and a ratio of 10× relative to the 1.4 ps duration initial pulses from the fiber laser. The experimental results (solid line) are compared with those expected based on ideal compression with compensation of all orders of spectral phase (dashed line), and are plotted on both linear (top) and logarithmic (bottom) scales. It is clear that the PCF provides near-perfect compression over two orders of magnitude with only minimal residual contribution in the compressed pulse pedestal. After all coupling losses are considered, the final peak power of the compressed pulse is 7 kW. Even shorter compressed pulses were obtained using longer lengths of HNLF, but at some cost to pulse quality. For example, with 15 m of HNLF and 1 m of bandgap PCF, the compressed pulse FWHM is only 89 fs, but as seen in Fig. 3(d), the pulse quality is significantly degraded, with the central sub-100 fs peak sitting atop an extensive pedestal component that spans several picoseconds.

 

Fig. 3. Pulse Spectra and intensity and chirp characteristics from (a) the 10 m Er3+ amplifier and (b) after 10 m additional self-similar propagation in HNLF. The spectral intensities (top) are shown using a linear scale whilst the temporal intensities (bottom) use a logarithmic scale. (c) and (d) show pulse characteristics after photonic bandgap fiber compression using linear (top) and logarithmic (bottom) scales. The solid line in (c) corresponds to the experimental compression of the pulse shown in (b) whilst the dashed line shows the ideal Fourier transform limit. The experimental results in (d) show the non-optimal case using 15 m of HNLF.

Download Full Size | PDF

5. Conclusions

There are several important conclusions to be drawn from our results. Firstly, we have confirmed a number of significant features of similariton amplifiers that have not been explicitly studied in previous experiments. In particular, cutback measurements and high dynamic range FROG has allowed the direct observation of the reshaping of an incident hyperbolic secant-like input pulse to a parabolic similariton. We have also confirmed predictions of the presence of intermediate asymptotic exponentially-decaying wings about the parabolic pulse core, and have verified experimentally their decreasing contribution to the total pulse energy with propagation distance. The use of highly-nonlinear and photonic bandgap fiber has been shown to lead to a convenient all-fiber source of compressed pulses in the 100 fs regime, with the use of FROG measurements proving indispensable for the system optimization. Finally, we stress that the entire amplifier-compressor setup studied here has been constructed using only commercially-available fibers and devices, and we expect that similar systems will become valuable generic testbeds for the study of self-similar dynamics. We anticipate that the combination of optical similaritons and bandgap compression in hollow core PCF may come to represent a key new technology in ultrafast optics.