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We demonstrate with realistic numerical simulations that fiber optical parametric chirped pulse amplification is able to amplify ultra-short optical pulses. Such amplifiers driven by two-pump waves can amplify pulse bandwidth twice as large as the one of a single pump configuration. We show that pulses as short as 50 fs can be directly amplified. In addition, we take benefit from the saturation regime to achieve spectral broadening which makes possible to reduce pulse duration down to 15 fs.
©2013 Optical Society of America
1. Introduction
Fiber optical parametric amplifiers (FOPAs) are a promising tool for all optical signal processing, signal amplification and wavelength conversion [1–3]. The most striking features of this kind of amplifiers are related to their wide gain bandwidth [4] and to their quasi-instantaneous response. This makes them very attractive for high bit rate telecommunication systems, in which bit rates as well as data streams increase from year to year. The basic principle of operation of such amplifiers involves a single strong pump or two and a small signal which is to be amplified. The phenomenon of amplification relies on the phase-matched four-wave mixing process [5]. From a quantum mechanical point of view, two pump photons or one photon for each pump are transferred to the signal and to another wave, called the idler wave. This last one is symmetric to the signal with regard to the pump in single pump configuration, or with regard to the average pump frequency in a two-pump scheme [6]. The gain band of such amplifiers is optimistically defined for a monochromatic signal which frequency is tuned around the pump(s) [1,2,4], since it encompasses the idler plus the signal bands. The useful gain band is in reality twice narrower, one half for the signal and the other half for the idler. This property is of great interest to perform frequency conversion for telecommunication applications, but it represents an important drawback when one wants to exploit the widest achievable bandwidth in order to amplify ultra-short optical pulses in a chirped pulse amplification scheme. In this last case, the experimental setup is known as a fiber optical parametric chirped pulse amplifier (FOPCPA) [7–11]. It constitutes a promising alternative of usual rare-earth doped fiber amplifiers. Indeed, although ytterbium fiber technology has proven its ability to deliver very high energy pulses [10] or few-cycle optical pulses [11] around 1 µm, spectral gain narrowing makes high gain amplification of ultra-short pulses very challenging and clearly restricts their use to relatively large pulse widths [12]. However, up to now, in all FOPCPA setups, only half of the bandwidth has been used, which severely limits the duration of amplified pulses to a few hundred of fs at minimum.
In this paper, we numerically demonstrate that the whole gain band can be exploited in a two-pump FOPCPA configuration by launching a chirped signal in the middle of the pumps. Moreover, we show that spectral broadening originating from the saturation regime makes it possible to compress amplified pulses down to 15 fs in a realistic all-fiber device.
2. Model
Our numerical simulations are performed by integrating the extended nonlinear Schrödinger equation [13]:
with β2,3,4 the second, third and fourth order dispersion terms, γ the nonlinear coefficient and hR(t) the Raman response deduced from experimental measurements. It has been numerically integrated by using an adaptative split step method [14], with a precision of 10−8, 1021 points over 60 THz which leads to a temporal resolution of 4.15 fs. One photon per mode has been added on the input signal in order to account for quantum fluctuations. We can note that this is the same equation used in Ref [9], where the good agreement with experiments confirms that it accurately models these kinds of setups. The signal is a Gaussian pulse centered at λC = 1065 nm with a duration of 50 fs at full width at half maximum (FWHM) and an energy of 1 pJ. Such ultra-short pulses are now available with ytterbium fiber oscillator technology [11]. It is then stretched to a duration of 4.5 ns at −20 dB (1.74 ns at - 3 dB) with a purely quadratic stretcher (for instance, a very striking option for a compact stretcher is to use long chirped fiber Bragg grating [15]). It is coupled into an optical fiber serving as the parametric gain medium together with two pumps. These pumps are square pulses of 6 ns duration at −20 dB with a peak power of 200 W and are frequency shifted of +/−11 THz from the central frequency, i.e. at 1025 and 1108 nm respectively. Note that these wavelengths are located within the ytterbium gain band and can then easily be generated by using standard ytterbium-doped fiber amplifiers [16]. The input overall spectrum is represented in red line in Fig. 1(a).
Fig. 1 (a) Input spectrum (red line) and small signal gain curve (dotted black line) calculated from a numerical integration of the nonlinear Schrödinger equation (Eq. (1)). (b) Input temporal characteristics normalized to unity, stretched signal in red and pump in blue.
The input temporal profiles of the pump and stretched signal are displayed in blue and red lines respectively in Fig. 1(b). The fiber is an air/silica microstructured optical fiber with the following realistic parameters: β2(λC) = −1.05 × 10−29 s2/m, β3(λC) = 0.6 × 10−40 s3/m, β4(λC) = −1 × 10−55 s4/m, γ = 9 /W/km, fiber length L = 2 m, linear attenuation α = 13.5 dB/km and the zero dispersion wavelength is 1064.9 nm. This fiber is similar to the one used in the experimental work presented in Ref [9]. The small signal gain curve calculated by numerically integrating the nonlinear Schrodinger equation [1] is represented in black dots in Fig. 1(a). It shows that more than 80 nm bandwidth (central region) can be achieved with a 40 dB maximum amplification gain.
3. Results
4.1 Utilization of the whole bandwidth of the amplifier
Figure 2(a) represents the spectrogram of the stretched signal at the fiber input. The whole spectrum, represented vertically on the right side of the spectrogram, has been filtered with a super-Gaussian spectral filter of order 18 centered at λC with a width of 68 nm at – 3dB (76 nm at −20 dB) in order to extract the signal and remove pump waves. We obtain a slanting trace that corresponds to the linearly chirped signal originating from a quadratic stretcher. The spectrogram of the output signal is represented in Fig. 2(b) after 2 m of propagation within the fiber which corresponds to the upper limit of the linear regime of the amplifier. Characteristics of the amplifier in the saturated regime will be discussed in the next section. It rapidly exhibits an “X” shape which formation can be decomposed in the following steps (see the green points in Fig. 2(b)).
Fig. 2 Spectrograms of the signal at the fiber input (a) and at the fiber output (b). The gate duration is 100 ps.
Firstly, let us remind that, since the input signal is chirped, each spectral component is related to a different temporal part of the pulse. For example, we focus our attention to a spectro-temporal point labeled “1” in Fig. 2(b) (−1 ns, 1040 nm). During the amplification process, the corresponding idler wave is generated around 1080 nm at the same time but with a conjugate phase (spectro-temporal point labeled point “2” in Fig. 2(b)). This idler wave exhibits a linear chirp with an opposed sign with regard to the signal. These two opposed linear chirps (occurring for each time) lead to the formation of an “X” trace in the spectro-temporal plane. Note that this property is not specific to two-pump FOPCPAs but it is also the case in single pump FOPCPA configurations [1,2] in which the idler indeed exhibits an opposite chirp to the one of the signal. It is very important to note that this setup is a phase insensitive parametric amplifier despite the fact that spectral components are present both in the signal and idler gain bands at the input of the amplifier. We remind that phase-sensitive and phase-insensitive configurations are commonly differentiated by looking at the input of the amplifier: in the first case, both the idler and the signal are launched with a specific phase relationship, while in the second case, only the signal is launched inside the amplifier [17]. In our case as the signal is chirped, each of its spectro-temporal points is clearly separated from the ones of the idler. It is then possible to consider each spectro-temporal point of the signal individually and as a consequence to consider that the amplifier operates in a phase-insensitive regime.
At the output of the amplifier, the signal was extracted by means of a spectral filter to remove the pumps (whose characteristics are similar to the ones used for the spectrogram) and then, in a second step it was perfectly recompressed. This includes a perfect compensation of the whole set-up dispersion up to fourth order. This is not unrealistic since, experimentally, besides compressor (using either diffraction gratings or Bragg gratings), one has to properly manage dispersion via a careful design of a dispersion compensating fiber [18] and/or via active spectral phase shaping with either a spatial light modulator [19] or an acousto-optic programmable filter [20] or an electro-optic phase modulator [21] to achieve such perfect compensation.
The spectrogram illustrating this recompression process is represented in Fig. 3(a). The “X” trace evolves into a strait vertical line with a slanting component. As only half of the spectro-temporal points are concerned by the phase law of the compressor, the spectrogram trace can be divided into two parts. The vertical trace corresponds to spectro-temporal components of the signal for which the chirp has been quasi-perfectly cancelled by the linear recompression process. Indeed, we do not obtain a perfect recompression due to gain narrowing process because the bandwidth of the amplifier, defined in the small signal regime, is shorter than the FWHM of the signal. This is clearly illustrated in Fig. 1(a) where input signal spectrum and gain curve are superimposed. The oblique part of the spectrogram corresponds to the spectro-temporal components of the idler wave exhibiting a chirp which is now twice larger than the one of the input signal. Indeed, we add a chirp which is of the same sign than the one of the input trace. Note that it is quite possible to alternatively compress either the spectro-temporal components of the signal or of the idler by modifying the sign of the dispersion of the compressor. A close-up of the recompressed signal is represented in Fig. 3(b) in black dash-dot line and for comparison the input signal has been superimposed in red dashed lines. The duration of the amplified signal after recompression is 90 fs which is reasonably close to the input duration of 50 fs. Another important issue concerns the contrast of the amplified signal which is of primary importance for laser matter experiments [22]. In fact, it is one of the main drawbacks of ytterbium fiber amplification where good temporal quality is obtained only at the expense of temporal cleaning systems with high losses, instability, increased complexity and loss of compactness [23]. Insets in Fig. 3(b) represent the output signal in logarithmic scales over large temporal windows of 2 ps and 100 ps respectively. We can see that a quite good temporal contrast of 26 dB can be achieved in the ps range and more than 42 dB in the hundred ps range. The slight degradation of the quality of amplification is due to all spectro-temporal components located in the slanting branch of the “X” trace after recompression which are located at the feet of the pulse, without significant gain narrowing effect.
Fig. 3 (a) Spectrogram of the recompressed signal after parametric amplification. (b) Temporal traces of the recompressed signal at L = 2 m (black dash-dot line) and at L = 4 m (blue solid line)) compared to the initial signal before stretching (red dashed line), normalized to unity. Insets: temporal profile of the recompressed signal in log scale (left inset) and over a larger temporal span (right inset). Data correspond to L = 2 m.
4.2 Saturation to reach the gain broadening effect
Results presented above are related to the linear regime of the amplifier. Here, we investigate the behavior of the amplifier in the saturated regime. In standard FOPA theory, it is well known that saturation regime leads to a flattening of gain curves. In order to identify the limit between linear and nonlinear regimes of amplification, Fig. 4(a) represents the gain in energy versus fiber length (red points, right axis). An exponential amplification gain is achieved until 2 m length, and then the gain starts to saturate at about 55 dB at 4 m.
Fig. 4 (a) Evolution of the FWHM pulse width of recompressed pulses (blue circles, left axis) and of the gain in energy (red dots, right axis) versus fiber length. The horizontal dashed line represents the initial pulse duration. (b)-(d) Output spectra for different fiber lengths (blue lines). For the sake of clarity, input spectrum (red line) and different filter bandwidths (68 nm at −3 dB in black solid line, 137 nm at −3 dB in solid green line and 274 nm at −3 dB in solid pink line) are superimposed in the middle of the picture.
On the same figure, the FWHM duration of the recompressed pulse is superimposed in blue circles. In the linear regime, due to the gain narrowing process, the signal duration increases to reach a maximum value of 90 fs at L = 2m (see the signal shape in Fig. 3(b) in black dashed-dotted line). Figures 4(b) and 4(c) clearly illustrate that the spectrum of the signal in between the pumps at 1.2 m and 2 m (blue curves) is thinner than the input signal spectrum which is superimposed in red lines for the sake of clarity. When the amplifier starts to saturate, this leads to a lowering of the gain per unit length, i.e. the gain in energy is no longer exponential and in addition, the saturation leads to a distortion of the spectral gain curve. This can be seen in Fig. 4(d) where the power of spectral components located in the wings of the signal grows more rapidly that the central part of the signal. This process leads to a gain broadening mechanism and to the shortening of the signal duration. As a consequence, the signal FWHM evolves from 90 fs at L = 2 m to 52 fs at L = 4 m. This value is very close to the initial signal duration of 50 fs, showing that these kinds of setups are capable of amplifying such ultra-short pulses.
4.3 Impact of the filter bandwidth
The aim of the 68 nm width spectral filter used in the previous configuration (black solid line in Fig. 4(d)) is to remove both pump waves. Unfortunately, it also filters out higher frequency components of the signal which are located outside the pumps. The shortest pulse duration is then given by the inverse of the spectral bandwidth of the filter, then, by the spectral shift between the pumps. When the amplifier started to saturate (Fig. 4(d)), we see that both pump waves are completely depleted and a quite flat and broad spectrum is achieved at this length. The use of a spectral filter to separate the signal from the pump is then no longer necessary. One can wonder if these spectral components follow the appropriate phase law that would allow a shortening of the pulse duration by using broader spectral filters. We therefore used broader filters (68 nm at −3 dB, 137 nm at −3 dB and 274 nm at −3 dB, respectively represented in black, green line and pink lines in Fig. 4(d)). Corresponding recompressed temporal traces are plotted in Fig. 5(a).
Fig. 5 (a) Temporal traces of the recompressed signal normalized to unity at L = 4 m for different filter bandwidths represented in Fig. 4(d). We used similar color for the filters and for the extracted pulses. The durations of the pulses at FWHM are 53 fs (black solid line), 24 fs (solid green line) and 15 fs (solid pink line). The initial signal before stretching is superimposed in red dashed line and has 50 fs duration at FWHM.
As it can be seen, the pulse is reshaped and by increasing the filter bandwidth, pulses as short as 15 fs are generated. It means that most of the spectral components of the output spectrum of such saturated two pump amplifier follow an appropriate phase law that allows to achieve pulse shortening after the recompression stage. The limit of this shortening effect is fixed by the spectral acceptance of all the optical elements (amplifier, stretcher…) and by the relative level of the signal wings compared to the noise floor. Then, in order to get a complete insight of the impact of the filter bandwidth, we focused our investigations on the contrast of these signals. To this aim, Fig. 5(b) represents temporal signals with a scale in dBm. It shows that the use of filters broader than the pump spacing leads to a contrast degradation of a few dB for the shorter pulse (42 dB to 37 dB). However, this value remains still by far better than ytterbium-doped fiber amplifiers.
4. Conclusion
We numerically demonstrate with realistic numerical simulations that ultra-short optical pulses can be amplified with FOPCPA systems. We show that the available bandwidth of FOPCPAs can be twice as large as in standard setups by using a two-pump configuration and a short chirped signal launched exactly in between them. The amplifier operates in the phase insensitive regime since each generated idlers are temporally separated from signals. This is due to a combination of the frequency chirp of the signal and to the phase conjugation properties of idlers waves in parametric amplifiers. In addition, the gain narrowing process is counterbalanced by the amplifier saturation leading to spectral broadening. As a result, ultra-short pulses can be amplified in such devices. As an example, we show that a launched signal of only 50 fs can be amplified by more than 57 dB with a pretty good signal contrast of 42 dB in the hundreds of ps range. Moreover, at the expense of slight contrast degradation due to the residual pump, we show that we can take advantage of the spectral broadening associated to the use of spectral filters broader than the pump frequency shift to shorten input pulses. In this example, by launching 50 fs pulses we obtain pulses as short as 15 fs. Such a result paves the way toward all-fiber generation and amplification of few cycle optical pulse [24]. This numerical study highlights the benefit of a two-pump configuration for the amplification of ultra-short pulses compared to ytterbium fiber systems where gain narrowing strongly increases the signal duration.
Acknowledgments
This work was partly supported by the Agence Nationale de la Recherche through the ANR FOPAFE project, by the French Ministry of Higher Education and Research, the Nord-Pas de Calais Regional Council and Fonds Européen de Développement Régional through the “Contrat de Projets Etat Région (CPER) 2007-2013” and the “Campus Intelligence Ambiante (CIA)”, the equipex FLUX and the labex CEMPI.
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