We propose and demonstrate a novel linear-optics method for high-fidelity parabolic pulse generation with durations ranging from the picosecond to the sub-nanosecond range. This method is based on dispersion-induced frequency-to-time mapping combined with spectral shaping in order to overcome constraints of previous linear shaping approaches. Temporal waveform distortions associated with the need to satisfy a far-field condition are eliminated by use of a virtual time-lens process, which is directly implemented in the linear spectral shaping stage. Using this approach, the generated parabolic pulses are able to maintain most energy spectrum available from the input pulse frequency bandwidth, regardless of the target pulse duration, which is not anymore limited by the finest spectral resolution of the optical pulse spectrum shaper. High-quality parabolic pulses, with durations from 25ps to 400ps and output powers exceeding 4dBm before amplification, have been experimentally synthesized from a picosecond mode-locked optical source using a commercial optical pulse shaper with a frequency resolution >10GHz. In particular, we report the synthesis of full-duty cycle parabolic pulses that match up almost exactly with an ideal fitting over the entire pulse period.
© 2015 Optical Society of America
An optical pulse with a parabolic intensity profile and a linear frequency chirp is well known for its capability to resist the deleterious effect of optical wave-breaking in a normal dispersion regime [1,2]. As such, parabolic pulses have attracted a great deal of attention over the years for a wide range of applications, including high power pulse amplification and ultrashort pulse generation [3–10], highly coherent continuum sources development [10–13], customized pulse synthesis , and spectral compression [15,16]. Parabolic pulses were successfully generated in normal dispersive amplifiers using rare-earth doped fiber amplification [2–9,11] or Raman amplification [17,18]. Alternative methods that do not target signal amplification are particularly interesting for applications in the context of optical telecommunications, such as time-lens-based optical regeneration methods [19–21] and spectral self-imaging operations . In this context, generation of parabolic pulses have been proposed using dispersion varying nonlinear fibers [23–25]; however, the needed fibers are not easily accessible, and waveform distortions are induced by the impact of third-order dispersion and attenuation in the fibers .
Alternatively, more widely accessible linear pulse shaping techniques have been demonstrated based on the use of a super structure fiber Bragg grating (SSFBG) , an acousto-optic device , and arrayed waveguide gratings (AWGs) . These methods are based on a direct Fourier-domain synthesis technique, where the spectrum of a short optical pulse is re-shaped according to the Fourier transform of the target temporal waveform. However, direct Fourier-domain synthesis of parabolic pulses has two critical limitations. First, the maximum duration of the generated pulses is limited by the finest spectral resolution of the pulse shaper. Typical pulse shaping methods offer frequency resolutions above 10-GHz [12,26] and this has restricted experimental linear generation of parabolic pulses with durations shorter than 25ps. Longer pulse durations are however needed for a range of important applications, for instance for time lensing of optical pulse sequences [20–23]. Moreover, in direct Fourier-domain synthesis methods, the spectral pulse width of the output parabolic pulse is dictated by the target pulse duration and as such, a large portion of the input pulse spectrum, i.e., energy, may need to be filtered out. Again, the energy efficiency is reduced for longer pulse durations, an issue that can be of critical importance for nonlinear optics applications. The possibility of generating parabolic pulses with longer durations was also demonstrated using dispersion combined with spectral amplitude shaping  or temporal intensity modulation , but without any further discussions on the capabilities and limitations of these methodologies, and potential optimization strategies.
In this work, we propose and demonstrate a novel linear-optics method for parabolic pulse generation capable of overcoming the mentioned critical limitations of previous approaches. The proposed method provides a greatly increased flexibility to achieve a desired parabolic pulse duration, independently of the frequency resolution of the linear spectral shaping stage. This method is based on dispersion-induced frequency-to-time mapping, including the possibility of incorporating a virtual time lens so that to relax further the constraints imposed by the far-field condition . Previously, similar strategies were proposed in order to increase the time-bandwidth product in ultrabroadband arbitrary RF waveform generation . Here, we show how this scheme is ideally suited for parabolic pulse generation, enabling to synthesize high-quality parabolic pulses over a wide range of durations, from a few picoseconds to the sub-nanosecond range in the designs shown here, with high energy efficiency, by maintaining most of the energy spectrum available from the input pulse source.
2. Principle of dispersion-induced frequency-to-time mapping
Linear frequency-to-time mapping (FTM), which is the time-domain analogous of spatial-domain Fraunhofer diffraction in the far-field, essentially involves a linear distribution of the input signal’s frequency components along the time domain, which is induced by the linear group velocity variation as a function of optical frequency that is characteristic of a predominantly second-order dispersive medium. After certain amount of propagation in this dispersive medium, the output temporal intensity profile is expected to be closely related to the shape of the input pulse spectrum, as illustrated in Fig. 1. The relationship between the input temporal waveform and output waveform (t is for the time variable) of an optical pulse linearly propagating through a second-order dispersive medium can be mathematically approximated as follows (obviating average group delay) Eq. (1) can be approximated asEquation (2) clearly shows that under the prescribed condition, the output temporal intensity profile is a scaled replica of the input spectrum. The condition for FTM can be deduced from the above-defined phase approximation :Eq. (3) is the temporal far-field condition, and the linear FTM obtained in this case is usually referred to as far-field FTM (FF-FTM). The approximation in Eq. (3) has been also expressed as a function of the frequency bandwidth of the input pulse through its TBP.
3. Design guidelines
3.1 Far-field frequency-to-time mapping (FF-FTM)
By using this phenomenon, parabolic pulses can be simply generated by shaping the input pulse spectrum to be parabolic followed by propagation through a dispersive medium satisfying Eq. (3). The duration of the generated parabolic pulses is determined by12,26], where the temporal width of the generated parabolic pulse is inversely proportional to the shaped spectrum width only. This translates into a lower energy efficiency for synthesizing longer parabolic pulses, as this requires filtering out a larger portion of the input pulse energy spectrum, as shown in Fig. 2(a). In sharp contrast, the proposed FTM method enables utilizing most of the input pulse energy spectrum, regardless of the target duration or the input pulse source bandwidth, through optimization of the dispersion amount only.
However, the far-field condition in Eq. (3) also restricts the minimum duration of pulses that can be generated with a sufficiently high quality using the proposed scheme. In particular, introducing the far-field condition, Eq. (3), into Eq. (4), we estimate thatEq. (5).
In order to confirm this anticipated constraint, a numerical calculation was carried out for generation of a 100-ps full-width duration parabolic pulse. A Gaussian pulse with intensity FWHM of ~2 ps was assumed as the input source. After spectral shaping, the input spectrum was modified to exhibit the desired parabolic shape, as shown in Fig. 2(b); the target was to use most of the input spectral bandwidth and as a result, the frequency bandwidth of the reshaped spectrum (zero-to-zero full width) was GHz. The corresponding second-order dispersion coefficient to achieve an output parabolic pulse duration of 100 ps, as per Eq. (4), was 67 ps2. Notice that it has been previously shown that the far-field condition in Eq. (3) is too restrictive , i.e., in most cases it is sufficient that the amount of dispersion is only a few times (~2-3) higher than the right-hand side terms in the inequality. Nonetheless, our numerical simulations show that the output temporal intensity waveform, Fig. 2(e), still exhibits visible distortions, particularly in regards to the presence of undesirable temporal oscillations. These distortions can be directly attributed to the non-negligible quadratic phase term in the integrand of Eq. (1). As predicted, this is due to the fact that the input signal (parabolic spectrum) exhibits a relatively high TBP (~9, in our numerical example), so that the inequality in Eq. (3), even in its less strict form, require the use of a notably larger dispersion; correspondingly, using the given input pulse specifications, the inequality in Eq. (5) indicates that one should target generation of a pulse with a duration significantly longer than 100 ps. Notice that also in line with the trend expressed by Eq. (5), this restriction can be relaxed by using a wider input spectral width. To give a reference, the same level of distortions as those observed in Fig. 2(e) would appear in a generated parabolic pulse with half the duration (50 ps) if the input parabolic spectral bandwidth was twice larger (GHz). However, the fact that the required input pulse transform-limited time-width must be significantly longer (about 25-50 times longer) than the target output pulse duration represents a key limitation for practical uses of the FF-FTM methodology in the problem of parabolic pulse synthesis.
3.2 Near-field frequency-to-time mapping (NF-FTM)
An efficient way to overcome the described constraint of the FF-FTM method, without replacing the input pulse source, involves using a time lens process, namely the time-domain analogous of producing the desired Fourier transform in the focal plane of a spatial lens. This approach is known as near-field frequency-to-time mapping (NF-FTM).
We assume that the time lens provides a phase modulation with the same magnitude but opposite sign to the quadratic phase term in the integrand of Eq. (1) asEq. (1), the quadratic phase term is canceled out and the output waveform is then a scaled replica of the input spectrum , regardless of the magnitude of the dispersion coefficient, which means that the generated pulse is no longer restricted by the far-field condition in Eq. (3). The time-lens modulation process can be ‘emulated’ in the linear spectral shaping stage . The main idea here is to reshape linearly the original input pulse spectrum with an amplitude and phase mask that includes the parabolic spectrum reshaping process and spectral changes from the virtual time-lensing process, as shown Fig. 3. In particular, the desired complex spectrum after the linear spectral shaping stage should be the Fourier transform of the time-lensed signal . It is important to mention that the time-lensing process generally induces spectral distortion and broadening of the original parabolic spectrum and this should be kept into account in designing the entire linear pulse shaping stage so that the resulting frequency bandwidth after the virtual time lens does not exceed the bandwidth available from the input pulse source. To be more concrete, the spectral broadening induced by the time-lensing process is inversely proportional to the subsequent dispersion coefficient, which is directly related to the target pulse duration. Fortunately, for a fixed duration pulse, the selection of a narrower spectral width at the linear shaping stage would require a relatively higher dispersion, which in turns results in a narrower spectral broadening, and viceversa. The frequency bandwidth solely induced by the time lens should be narrower than the lens frequency chirp excursion over the entire input pulse period, T, in turn fixing the maximum duration of the output parabolic pulses:Eq. (7) was within the input Gaussian spectral bandwidth. We reiterate that Eq. (7) provides an estimate for the maximum possible bandwidth to be induced by the time-lens over the entire input pulse period; in our designs, most of the energy of the signal (sinc-like temporal intensity profile) is strongly concentrated near the center () so that the estimate provided by Eq. (7) is usually well above the actual frequency broadening induced by the time-lensing process.
Figure 2 shows the results of numerical simulations on the generation of 100-ps duration parabolic pulses from an ~2-ps input Gaussian pulse by direct spectral shaping (a, d), spectral shaping with FF-FTM (b, e), and NF-FTM (c, f). The direct spectral shaping requires the use of a pulse shaper with a very high spectral resolution, higher than 10 GHz; additionally, a large portion of the Gaussian input energy spectrum has to be filtered out in this case. This input pulse-energy loss is significantly reduced by use of the FF-FTM; however, as discussed in detail above, in this case, the output temporal waveform is significantly distorted due to the fact that the far-field condition is not fully satisfied. By using linear spectral shaping combined with NF-FTM, one can generate a parabolic intensity profile with a nearly ideal temporal shape, as shown in Fig. 2(f) while still using most of the input energy spectrum. Recall that the virtual time-lensing process is lossless. The spectral shape in Fig. 2(c) corresponds to Fourier transforming , which is a sinc-like temporal waveform (Fourier transform of a parabolic function) modulated by the quadratic phase term from the virtual time-lensing process.
4. Experimental results
The derived design guidelines were experimentally validated. Figure 4 shows the experimental setup for parabolic pulse generation with durations ranging from 25ps to 400ps. Pulses below 100ps were generated using the NF-FTM approach whereas an example of generation of parabolic pulses, with a duration of 400ps, using the FF-FTM approach is also reported. The initial optical pulses were generated from an actively mode-locked fiber laser producing ~2 ps (intensity FWHM) Gaussian-like pulses at a repetition rate of ~10GHz. The input pulse power was ~10 dBm and the central wavelength was 1550.2 nm. The input source was spectrally shaped into the desired complex spectra by using a commercial free-space optical pulse shaper (Finisar WaveShaper 4000S, resolution >10 GHz), with user-defined amplitude and phase masks for each generation case. The dispersive medium was created through a simple combination of readily available single mode optical fibers (Corning SMF-28: dispersion, dispersion slope at 1550 nm, and SMF-LEAF: dispersion, dispersion slope at 1550 nm). Given the spectral bandwidth of the input pulses and dominant second-order dispersion values of the fibers, contributions from the third-and higher-order dispersion terms are negligible. In the same context, polarization mode dispersion (maximum PMD) can be also neglected in our analysis, considering the relatively long parabolic pulse durations (from 25ps to 400ps) and short lengths of SMF fibers used in our designs (from 0.7 km to 20 km). The input and output parabolic pulses were characterized in the spectral and temporal domains by using an Optical Spectrum Analyzer (OSA) with a wavelength resolution of 0.01 nm, and an Optical Sampling Oscilloscope (PSO-100, EXFO) with a bandwidth of 500 GHz, respectively.
4.1 Far-field frequency-to-time mapping (FF-FTM)
Parabolic pulses with 400ps duration were generated using the FF-FTM approach. The original pulse repetition rate (~10 GHz) was decreased down to 2.5GHz by using a Mach-Zehnder intensity modulator (MZM) to prevent interference between consecutive output pulses. An erbium-doped fiber amplifier (EDFA) was additionally used after the dispersive medium to compensate for the losses introduced by the intensity modulation combined with propagation losses along the dispersive medium. The zero-to-zero full width of the parabolic shape spectrum was adjusted to 2π × 147 GHz, whereas a ~20-km long section of SMF-28 was used as the dispersive medium, introducing a second-order dispersion coefficient of −433 ps2. The parabolic spectrum after the pulse shaper is shown in Fig. 5(b). The intensities of 10 GHz longitudinal modes were slightly higher than the 2.5-GHz spaced spectral lines due to imperfect pulse suppression from the intensity modulation. The measured output intensity profiles are plotted in Fig. 5(c) (log scale) and Fig. 5(d) (linear scale). The experimentally generated full-duty cycle parabolic pulses matched up well with an ideal fitting; the observed minor distortions in the synthesized pulses can be attributed to slight deviations from the strict far-field condition.
4.2 Near-field frequency-to-time mapping (NF-FTM)
Parabolic pulses with durations ranging from 25 ps to 100 ps were generated using the NF-FTM technique. No amplification was used in any of the experiments reported here below. The time lens, providing the required quadratic phase variation for each dispersion coefficient, was virtually implemented into the linear spectral shaping stage. The insertion loss of the used waveshaper was ~4 dB.
For generation of 100-ps duration parabolic pulses, the dispersive medium was implemented using an ~20 km long section of SMF-LEAF fiber, for a total dispersion coefficient of ~106 ps2, which was slightly larger than the optimal amount inferred from the numerical calculation. This required the use of a narrower spectral bandwidth than that available at the input, reducing the energy efficiency. The total frequency bandwidth (full-width of 1% at the maximum intensity) of the output spectrum, including the time-lensing process, was ~184GHz, fitting well in the input source bandwidth (~617GHz). The input and output spectra from the pulse shaper are plotted in Figs. 6(a)-6(b), respectively. The experimentally generated full-duty cycle parabolic pulses perfectly matched up with an ideal fitting over the entire pulse period, without any sign of detrimental effects, such as pulse-to-pulse interaction in the wings, as shown in Figs. 6(c)-6(d). These specific pulse waveforms are highly desired for a range of applications [15,27]. The output average power was ~0 dBm, which is relatively low due to the propagation loss along the dispersive medium (~4 dB) and the selection of the narrow bandwidth in the spectral shaping stage; the obtained output power could be easily increased by replacing the dispersive medium by a ~3 km long SMF-28 fiber section.
In the case of 50-ps pulse duration, an ~1.8 km long SMF-28 fiber section was used as the dispersive medium, which provides a dispersion coefficient of ~37.7 ps2. The value was optimized to minimize the input pulse-energy loss from spectral shaping while ensuring that the frequency bandwidth of the spectrum after time lensing is still within the available input spectral bandwidth. The total frequency bandwidth of the output spectrum, including the time-lens process, was estimated to be ~280 GHz. From the output spectrum in Fig. 7(b), it was confirmed that most of the input frequency bandwidth was utilized. Moreover, the generated pulse was a high-quality parabola, as shown in Figs. 7(c)-7(d), with a measured output average power of 3.7 dBm.
Finally, for comparative purposes, parabolic pulses with 25 ps duration were generated by using both the NF-FTM method and the direct spectral shaping, noting that this target duration is equal to the pulse duration reported in previous parabolic pulse generation experiments using the direct shaping approach [12, 26]. Figure 8(a) shows the output spectrum for the direct spectral shaping approach, which clearly shows that a large portion of the input energy spectrum was filtered out from the spectral shaping stage. On the other hand, by using the NF-FTM, the output spectrum of the parabolic pulse was able to maintain most of the available input energy spectrum, as shown in Fig. 8(b). In this latest experiment, the dispersion coefficient was ~14.1 ps2 (~700-m long SMF-28 fiber) and the total frequency bandwidth was ~405 GHz. The measured output power was ~4.2 dBm, which was about 6 dB higher than the output power of the parabolic pulse generated by the direct spectral shaping approach. The measured output temporal waveforms are illustrated in Figs. 8(c)-8(f). The generated pulses were both very close to the ideal fitting. The observed slight deviations (e.g., temporal asymmetry) in the synthesized parabolic shapes for the NF-FTM case are mainly attributed to the use of an asymmetric input pulse spectrum; this frequency-domain asymmetry was purposely not totally compensated for in the linear spectral shaping stage so that to be able to utilize a larger portion of the available input spectrum.
In summary, we have introduced a novel linear-optics method for parabolic pulse generation. This method is based on dispersion-induced FTM, enabling to save most of the energy spectrum available from the input pulse sources, independently of the target pulse duration. Design guidelines were provided for generation of parabolic pulses over a wide range of pulse durations, where the direct spectral shaping approach is restricted by the finest spectral resolution of the pulse shaping stage and input energy spectrum losses. The generation method based on FF-FTM was first shown to be capable of overcoming the pulse duration and energy limitations of the direct spectral shaping approach, and as proof of a concept, a 400-ps parabolic pulse was successfully synthesized from an ~2-ps input pulse source. In order to relax the waveform distortions imposed by the need to satisfy a strict far-field condition, which becomes more critical as one targets a shorter pulse duration, a virtual time lens was incorporated into the linear spectral shaping stage, effectively implementing a near-field FTM approach. Using this optimized method, generation of parabolic pulses with durations ranging from 25 ps to 100 ps was successfully demonstrated. Experimentally generated pulses were high-quality parabolic pulses with output powers as high as ~4 dBm without using any amplification. The total system losses could be further minimized through optimization of the dispersion and/or by replacing the free-space pulse shaper by an alternative, more efficient spectral shaping technique, such as a super-structured fiber Bragg grating (SS-FBG) . To conclude, Fig. 9 summarizes the key capabilities and limitations of the three studied methods for parabolic pulse generation according to the target pulse duration from a given input pulse source.
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