We study the influence of dispersive propagation on picosecond flat-top pulses, which are generated using long period fiber grating (LPG)-based optical differentiators. We suggest an extremely simple scheme to compensate for the dispersion-induced flat-top pulse distortion; this scheme is based on proper tuning the LPG coupling strength. As this coupling strength may be changed via LPG axial straining, the demonstrated device can be tuned to compensate for different levels of the dispersion in a very easy and straightforward fashion. This allows for very fine flat-top pulse shape adjustment, even after propagation through a relatively long section of dispersive optical fiber. In the experimental demonstration reported here, the dispersion tolerance of 1.8-ps flat-top pulses propagating through a standard telecom fiber (SMF-28) was increased from ≈2 m to ≈18 m, giving a 9-fold improvement.
©2007 Optical Society of America
The generation of ultrashort pulses with flat-top temporal intensity profiles is highly desired for a range of non-linear optical switching and frequency conversion applications that are of particular interest in ultrahigh-bit-rate fiber optics communication systems [1–4]. The most usual way to obtain short flat-top pulses is based on temporal re-shaping of a short optical pulse generated by a mode-locked laser (e.g., of Gaussian or Sech2 shape) using a linear filtering stage [5, 6]. One of the main challenges in handling short flat-top pulses in a practical experiment is that their optical spectral bandwidth is considerably wide (the bandwidth may be up to several times larger compared to, e.g., a Gaussian pulse of the same duration) and thus these pulses are very sensitive to dispersion. This becomes critical for flat-top pulses suitable for 160–640 Gbit/s systems with the corresponding timeslot durations of 6–1.5 ps. The bandwidth of such flat-top pulses can reach several THz - these extremely-high-bandwidth pulses are severely distorted even when propagated through sub-meter lengths of a standard telecom fiber. In practice, however, it is necessary to deliver the flat-top pulses from their generation stage to the specific location where they are going to be used, preferably through the standard telecom fiber. This implies the need for additional, preferably tunable, dispersion control that would allow for fine tuning to obtain the optimum flat-top waveform shape at the fiber output. However, existing compensation techniques either lack fine tunability (e.g., dispersion compensating fibers) or are designed for signals of much smaller bandwidth (e.g., Fiber Bragg gratings, Gires-Turnois etalons) and/or considerably increase complexity of the system (reflection-operated chirped Bragg gratings, prism-based dispersers, etc.). Another drawback is that they may give rise to additional, detrimental non-linear effects (e.g., dispersion compensating fibers).
Recently, we reported a novel method for flat-top pulse generation that is based on optical differentiation of an input Gaussian-like optical pulse . Our optical differentiator was implemented using a long period fiber grating (LPG) . We showed that this scheme allowed for highly energetically efficient generation of flat-top waveforms with the flat-top pulse duration tuning ability . In this report we investigate how flat-top pulses generated using this new filtering scheme are affected by chromatic dispersion. Further, we show that the dispersion-induced distortion can be pre/post compensated by a proper adjustment of the LPG coupling strength. As the coupling strength may be practically changed via simple LPG straining, the demonstrated device can be tuned to compensate for different levels of dispersion in a very easy and straightforward fashion, allowing for very fine flat-top pulse shape adjustment, even after propagation through a relatively long section of dispersive optical fiber. We emphasize that this is a unique capability of this recently reported LPG-based flat-top pulse generation technique, which is not present in any of the previously proposed flat-top pulse re-shaping methods , . Finally, we provide a detailed physical explanation of this unique feature; in particular, we show that the proposed dispersion compensation mechanism is predominantly based on variations of the LPG phase characteristics, more specifically on variations of the value of the phase jump across the LPG resonance, which are induced by the LPG coupling strength tuning. The same concept could be easily generalized to obtain similar functionality with other differentiator implementations (e.g., using two arm interferometers ).
2. Theoretical analysis
In this section, we briefly review the principle of the flat-top pulse generation based on optical differentiation . Subsequently, we study how this flat-top pulse is distorted by the first-order chromatic dispersion and how this degradation can be compensated by a proper modification of the coupling strength characteristics of the fiber LPG.
2.1 Effect of dispersive propagation
In  it was shown that an optical differentiator with the spectral transfer function of -j(ω - ω 0), where ω 0 is the central frequency of the optical differentiator (frequency of zero transmission), can re-shape a transform-limited input optical pulse of real temporal envelope (spectrally centered at ωcar) into an optical pulse of temporal envelope v(t) (centered at the same carrier frequency ωcar) as follows:
In this notation, the average time delay induced by the LPG is not considered. From (1) and (2), it is obvious that for ωcar = ω 0 the device operates as an optical differentiator. However, for ωcar ≠ ω 0, |v(t)|2 consists of the sum of the differentiated waveform and the original waveform, with a relative weight given by the detuning factor (ωcar-ω 0)2. When |u(t)|2 is a temporally symmetric pulse (like a Sech2 or a Gaussian), the differentiated pulse is a symmetric double-pulse . By properly adjusting the detuning factor (ωcar-ω 0)2, the valley in between the two peaks of the double-pulse (differentiated waveform) can be completely filled with the contribution coming from the second term of the right side of (2) (which has the shape of the original pulse u(t)) leading to the formation of a single flat-top pulse .
In the presence of the first-order chromatic dispersion D, the original, transform-limited waveform u(t) changes to:
where w(t) is a real (transform-limited) function. For example, for u(t) Gaussian, w(t) is also of Gaussian shape of longer duration. As a result, Eq. (1) can be re-written as:
and in terms of intensity,
By comparing (5) with (2) we see that besides u(t) being replaced by w(t), the so-called detuning factor has been modified to ((ω car-ω 0)-2Dt)2, which depends on the time variable due to the presence of the introduced term Dt. This term is odd-symmetric in t (i.e. it is negative for t<0 and positive for t>0 with the same absolute value for ±|t|), thus inducing an asymmetric distortion in the otherwise temporally symmetric output pulse waveform. It is worth noting that the term Dt does not cause the mentioned asymmetric distortion in (5) when the device is operated as an optical differentiator, which corresponds to (ωcar-ω 0)=0.
The dispersion-induced flat-top waveform distortion predicted by (5) can be better illustrated by numerical simulations. Through all our numerical analyses, we consider an input pulse generated by a mode-locked laser (with Sech2 intensity temporal shape) with a full-width at half of maximum (FWHM) time duration of 1.8 ps and a central (carrier) optical wavelength of 1550 nm. We assume an ideal differentiator with a spectral transfer function ∝-j(ω-ω 0). Further, we assume a value of the differentiator-pulse detuning (ωcar-ω 0)2 that fills 90% of the gap between the two peaks of the differentiated waveform (double-pulse), i.e. the generated flat-top consists of two peaks with a 10% valley in between them. Although the pulse can be ideally flat-top (valley of 0%) , the chosen value allows for a better visualization of the principle of operation. Fig. 1 shows the temporal profile of the pulse generated at the differentiator output after propagation through different lengths of a standard telecom fiber (SMF-28, dispersion of 0.023ps2/m, Corning Inc.) when the central wavelength of the pulse is shorter than the central wavelength of the differentiator (ωcar -ω 0>0). As predicted by (5), the amplitude of the leading-edge peak of the flat-top pulse (occurring at t<0) decreases as the chromatic dispersion increases while the opposite tendency is observed for the trailing-edge peak of the flat-top pulse (occurring at t>0). When considering the opposite sign of the dispersion, -D, (5) predicts that the situation is inverted in time (the leading-edge peak increases in intensity while the trailing-edge peak decreases in intensity as the pulse propagates through the dispersive fiber). Obviously, as evidenced by (5), when the pulse carrier wavelength is longer than the central wavelength of the differentiator (ωcar -ω 0 <0), the induced asymmetric waveform distortion follows the opposite trend as that described above.
2.2 Principle of LPG tuning for counterbalancing the effect of the dispersion
Here, we provide a brief heuristic analysis that predicts that the dispersion-induced distortion of a flat-top pulse generated with an LPG-based optical differentiator  could be compensated by properly adjusting the coupling strength of the LPG. The physics behind this interesting feature is further investigated is Section 2.3.
According to a previous study , the transfer function of a LPG that is set to operate close to the full coupling condition (zero transmission at the resonance frequency ω 0) can be approximated in the vicinity of the resonance frequency as:
where κ is the LPG coupling coefficient, α is a constant, L denotes the grating length, κL is the coupling strength, and β is the core mode propagation constant. In order to implement an optical differentiator, the LPG must be set to operate in the full-coupling condition (cos(κL)=0, giving κL = π/2 within multiples of 2π) . Now, let us consider a slight detuning Δκ in the LPG coupling coefficient with respect to the value for which κL = π/2. Considering the first two terms of the corresponding Taylor series, we can use the following approximations
and rewrite (6) as
For the sake of simplicity, we will first consider zero detuning between the input pulse carrier frequency ωcar and the filter central frequency ω 0; in this case, the filtering operation in the temporal domain is given by
We recall that u(t) and v(t) are the complex temporal envelopes of the input and output optical pulses (both spectrally centered at ωcar). The key aspect of (9) is that there is no phase difference between the two terms on the right hand side, so the output intensity includes an interference term:
In deriving (10), a transform-limited input optical pulse (with real temporal envelope) has been assumed. Further assuming that this input pulse is also temporally symmetric (e.g. Gaussian or Sech2), the last term in (10) exhibits temporal odd symmetry (i.e. it is negative for t<0 and positive for t>0 with the same absolute value for ±|t|), thus inducing a temporally asymmetric waveform distortion similar to that caused by the dispersion (see discussions above). This odd symmetry comes from the differentiated waveform (∂u(t)/∂t) that consists of two pulses that have identical temporal field profiles (inverted in time), but are phase shifted by π. This π-phase shift implies that the field intensities of the two pulses are of opposite signs . As a result, (10) indicates that depending on the sign of Δκ, the leading-edge peak of the generated optical pulse will exhibit a higher intensity (Δκ>0) or a lower intensity (Δκ<0) than that of the trailing-edge peak.
When considering (i) a finite detuning of the pulse carrier frequency ωcar from the filter central frequency ω 0 and (ii) the effect of dispersive propagation, we obtain an equation that includes two asymmetry terms – one that is associated with the pulse dispersive propagation (present in (5)) and another one that is associated with the effect of LPG coupling strength tuning (present in (10)):
By a proper choice of the sign and magnitude of Δκ, the two asymmetry terms in (11) can, to a certain degree, cancel each other out. In this way, the dispersion-induced waveform asymmetry could be compensated by a proper tuning of the LPG coupling strength (to operate outside the full-coupling condition). As will be shown, our numerical simulations and experimental results show that the dispersion-induced waveform distortion can be counterbalanced almost entirely using this simple mechanism even in the presence of a significant amount of the dispersion.
The cos(κL) term of the LPG spectral transfer function in (6) influences predominantly the phase filtering characteristics close to the LPG resonance wavelength, as shown in Fig. 2. Thus, any variation of the LPG coupling strength around its full-coupling value will essentially affect the value of the phase jump induced by the LPG filter across its resonance frequency. We will show in what follows that this phase jump is actually the key factor in providing the above anticipated dispersion compensation feature of the LPG filter.
To confirm this claim, we considered a notch filter that has a spectral transfer function with the amplitude identical to that of an ideal optical differentiator (i.e. linear spectral amplitude variation with zero transmission at the filter’s central frequency) but with a phase variation having an arbitrary jump across the filter central frequency. The time waveform obtained at the output of this notch filter is shown in Fig. 3, where we considered ωcar = ω 0 (same conditions as for the analytical study presented above, Eq. (9) and (10)). The results present in Fig. 3 can be understood by considering the spectral characteristics of the filtered pulse. The output pulse spectrum consists of two adjacent spectral lobes that are formed by filtering the input symmetric (single-lobe) pulse spectrum with the considered notch filter. The corresponding temporal waveform is the result of the temporal beating between these two spectral lobes. The temporal beating period is inversely proportional to the spectral distance of the two spectral lobes; the duration of the entire waveform is then given by the pulse bandwidth (more precisely by the inverse Fourier transform of any of the two pulse spectral lobes). Typically, only two or three temporal periods are visible, Fig. 3. The relative phase between the two pulse spectral lobes, which is set by the phase jump at the filter central frequency, determines the temporal features of the generated beating waveform. For instance, in the case of a phase jump value of π, the destructive interference between the two spectral lobes occurs just at the center of the time envelope and, as a result, a temporally symmetric waveform that consists of two time lobes with the same amplitude (the exact differentiation) is obtained , Fig. 3(c). However, for a zero phase jump, the constructive interference occurs at the center of the envelope and we obtain a symmetric temporal waveform consisting of the main peak and two smaller sidelobes, Fig. 3(c). It is also straightforward to understand the situation when the phase jump is decreased (Fig. 3(a)) or increased (Fig. 3(b)) from the value of π: simply, the position of destructive interference is moved to longer (Fig. 3(a)) or shorter (Fig. 3(b)) times, which, following the time envelope of the entire beating waveform (shown in Fig. 3), results in the leading- edge peak amplitude increase (Fig. 3(a)) or decrease (Fig. 3(b)), while the trailing-edge peak follows just the opposite trend. This behavior is in excellent agreement with that predicted by (10) when the LPG coupling strength is changed around its full-coupling value; we reiterate that this is expected considering that the LPG coupling strength tuning affects predominantly the value of the phase jump induced by the LPG filter across its resonance (central) frequency, as shown in Fig. 2.
As anticipated above, the potential dispersion in the system could be ‘pre-compensated’ by introducing a suitable phase jump (different from π) across the central frequency of the optical differentiator filter, e.g. by properly tuning the LPG coupling strength, during the flattop pulse re-shaping process. Specifically, an introduction of a phase jump smaller than π (Fig. 3(a)) across the differentiator’s central frequency will generate an asymmetrically distorted flat-top pulse with the leading-edge peak of higher amplitude than that of the trailing-edge peak. During the dispersive propagation through a conventional telecom fiber, the amplitude of the leading-edge peak is expected to decrease while the amplitude of the trailing-edge peak is expected to increase (assuming that ωcar>ω0, see results in Fig. 1); in this way, at a certain fiber propagation distance, the original asymmetric distortion (at the pulse re-shaper output) will be almost fully compensated resulting in the optimum flat-top pulse shape.
Our predictions have been first confirmed by numerical simulations. Fig. 4 shows the results of our numerical analysis considering (a) the ideal optical filter of Fig. 3 and (b) the simulated LPG filter with characteristics shown in Fig. 2. First, we observe that the performance of both filters is almost identical, fully confirming our central hypothesis that the value of the phase jump at the filter’s center frequency is the key parameter providing the dispersion-compensation feature. Fig. 4 shows that the dispersive propagation through up to ≈7.5 meters of SMF-28 fiber can be almost fully compensated (the flat-top waveform can be made almost identical to that obtained at the pulse re-shaper output) by properly tuning the phase jump across the filter’s center frequency. Specifically, in order to compensate the dispersion introduced by 7.5 meters of SMF-28 fiber, the filter’s phase jump needs to be reduced to 0.75π (the corresponding LPG filter has a resonant transmission of -20 dB). When a longer section of SMF-28 is used (e.g. 15 m in the example shown), the flat-top shape may be still obtained (for phase jump of 0.5π or LPG resonant transmission of -16 dB), however, a certain level of distortion is already apparent.
3. Experiments and discussion
In our previous work  we demonstrated that the coupling strength of a LPG made by CO2 irradiation  can be varied by means of the axial strain applied to the fiber where the LPG is inscribed. Here, we have used this extremely simple and practical technique to implement the described tunable dispersion-compensation mechanism for the optimal flat-top pulse generation using a LPG-based optical differentiator.
The LPG used in our proof-of-concept experiments was made in a standard telecom fiber (SMF-28, Corning Inc.) using the point-by-point technique with a CO2 laser . Specifically, the used fiber LPG was 70 mm long, with a period of 530 μm, which corresponds to coupling into the 3rd odd cladding mode at the resonance wavelength of 1550 nm. The measured and predicted field magnitude responses are shown in Fig. 5. We see that the predicted resonance dip is slightly narrower than that experimentally achieved; however, the shape of the spectral response of the realized LPG filter is very close to the theoretical predictions, especially the linear spectral dependency around the resonance wavelength.
The input pulses in our experiments were almost transform-limited Sech2 pulses generated from a passively mode-locked (10 GHz) wavelength-tunable semiconductor laser (U2t Inc., Germany) with Full-Width-Half Maximum (FWHM) pulse width of 1.8 ps – its spectrum is shown in Fig. 5. The pulses from the laser were first launched into a polarization controller, as the fiber LPG was slightly birefringent, and were subsequently propagated through the LPG-based pulse shaper. We used a fiber-based Fourier-transform spectral interferometry (FTSI) setup to obtain the complex temporal waveform of the output pulse (in this setup, the input pulse itself was used as the reference pulse) . Previously, we found this technique capable of magnitude and phase characterization of more complicated complex pulse waveforms, generated at various repetition rates, processed by LPG filters [8,9,13]. Similarly to our previous work, the FTSI technique allowed us to monitor the obtained temporal waveform and optimize the experiment conditions, namely the wavelength detuning (in our experiment, this was achieved via the laser wavelength tuning) and the adjustment of the LPG coupling strength (via LPG straining ), in order to achieve the desired flat-top temporal waveform.
First, we measured the flat-top waveform generation without any dispersive propagation (this is similar to our previous work ). Results are shown in Fig. 6(a). We confirmed that the LPG was operating close to the full coupling condition by measuring the spectrum of the output pulse. Note that a certain axial strain was applied on the LPG to tune its coupling strength to obtain the full-coupling condition. Subsequently, we added 18 meters of SMF-28 fiber – this was predicted as the maximum value of the dispersion that could be still compensated by our LPG: Beyond this length, the required change of the LPG coupling strength was too high to be realized via straining of the LPG designed to operate at the full-coupling condition. The result (theory and experiment) is shown in Fig. 6(b). As predicted above, the trailing-edge peak amplitude increased considerably whereas the leading-edge peak amplitude decreased, which degraded completely the flat-top shape. Finally, we almost-fully released the strain applied to the LPG, which lowered its coupling strength. This changed also the resonance wavelength, which was compensated for in our experiment by a change in the input laser wavelength (however, in practice, techniques that allow for the resonance transmission tuning without change in the resonant position exist ). For LPG resonance transmission of -19 dB, the flat-top shape was obtained again, Fig. 6(c). Although the experimental shape is slightly distorted (this effect was not theoretically predicted and may be attributed to some non-linear effects, change of the laser spectrum when slightly wavelength tuned, or to imperfect shape of the experimentally realized LPG), it clearly shows that the dispersion-induced asymmetry in the flat-top waveform was properly compensated for.
To quantify the ability of our tunable dispersion-compensation technique, we define ‘critical dispersion’ as the amount of dispersion that causes the ‘maximum acceptable’ deviation from the optimum (dispersion-less) flat-top waveform. Obviously, the ‘maximum acceptable’ deviation depends on the specific application. Here, we consider that the ‘maximum acceptable’ deviation is of 10% in the power intensity (5% in the field intensity), which means that the maximum intensity difference across the flat-top part of the generated pulse cannot exceed 10%. In our particular demonstration, this corresponds to 2 m of SMF-28 fiber. Considering that we were able to counter-balance the dispersion induced by 18 meters of SMF-28, we demonstrated compensation ability of 9 critical dispersions.
We have investigated how (sub-)picosecond flat-top optical pulses generated using a recently reported linear pulse re-shaping scheme based on ultrafast differentiation  are affected by first-order chromatic dispersion. The predominant effect is a temporally asymmetric waveform distortion, which becomes more noticeable for higher dispersion amounts. Further, we have demonstrated that this dispersion-induced distortion can be compensated for by properly adjusting the LPG coupling strength. We have shown that this interesting property is mostly due to the change in the phase shift across the LPG spectral resonance which is induced by the LPG coupling strength variation. As the LPG coupling strength may be practically changed via simple axial straining of the fiber LPG, the demonstrated device can be tuned to compensate for different levels of dispersion in an extremely simple and practical fashion, thus allowing for a very fine flat-top pulse shape adjustment, even after propagation through a relatively long section of a dispersive optical fiber. Specifically, in the experiment reported here, we increased the acceptable level of residual dispersion in the pulse shaping system by nearly one order of magnitude. We believe that besides its intrinsic physical interest, the newly reported, unique feature of a LPG-based flat-top pulse shaper could be very useful for practical optical switching applications based on the use of flat-top waveforms, particularly in ultrahigh-bit-rate optical communication systems.
This work was supported by Natural Sciences and Engineering Research Council of Canada (NSERC) through its Strategic Grants Program and by the Grant Agency of AS, Czech Republic, contract KJB200670601 and Czech Science Foundation (102/07/0999).
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