We report the experimental realization of an ultrafast all-optical temporal differentiator. Differentiation is obtained via all-fiber filtering based on a simple diffraction grating-assisted mode coupler (uniform long-period fiber grating) that performs full energy conversion at the optical carrier frequency. Due to its high bandwidth, this device allows processing of arbitrary optical signals with sub-picosecond temporal features (down to 180-fs with the specific realizations reported here). Functionality of this device was tested by differentiating a 700-fs Gaussian optical pulse generated from a fiber laser (@ 1535nm). The derivative of this pulse is an odd-symmetry Hermite-Gaussian waveform, consisting of two linked 500-fs-long, π-phase-shifted temporal lobes. This waveform is noteworthy for its application in advanced ultrahigh-speed optical communication systems.
©2006 Optical Society of America
The implementation of all-optical circuits for computing and networking could overcome the severe speed limitations currently imposed by electronics-based system , . In photonics, however, there are still no equivalents of fundamental devices that form basic building blocks in electronics, where most of functionalities like logic operations, differentiation, and integration are realized using a combination of operational amplifiers, resistors, and capacitors. Here, we report the experimental realization of one of these fundamental devices, namely a universal all-optical temporal differentiator that operates over arbitrary optical waveforms at terahertz speeds .
Several schemes for performing real-time derivation in the optical domain have been previously proposed at the theoretical level , . The experimental device demonstrated here has been realized according to a novel all-fiber design we recently proposed , which is based on the use of a single uniform long-period fiber grating (LPG) and is suitable for operation over the entire bandwidth of arbitrary signals with sub-picosecond temporal features (corresponding bandwidths of a few terahertz). The main advantages of the realized device are inherent to its all-fiber geometry, namely simplicity, relatively low cost, low losses, and full compatibility with fiber optics systems. Additionally, the same concept could be easily transferred to integrated optics platforms, where affordable and robust all-optical differentiators could be readily matched to current semiconductor lasers and amplifiers.
Besides the intrinsic interest of an all-optical temporal differentiator for all-optical computing and information processing, a direct impact in diverse fields such as ultrahigh-speed optical communications , , femtosecond pulse shaping ,  and ultrafast sensing and control , can be anticipated. Specifically, to showcase its power, we have applied the newly-developed all-optical differentiator to re-shape an input Gaussian-like optical pulse into a sub-picosecond odd-symmetry Hermite-Gaussian (OS-HG) waveform, which is of particular interest for next-generation optical communications , . To our knowledge, the experiments reported here are also the first demonstration of direct generation of this complex temporal waveform in the sub-picosecond regime using an all-fiber setup.
2. Operation principle
The underlying concept of the studied component is based on the fact that the spectrum associated with the derivative of the temporal envelope of a given signal centered at frequency ω0 (carrier frequency) E(ω-ω0) (represented in the Fourier domain) is given by i(ω-ω0)E(ω-ω0) , where ω is the optical frequency, and ω-ω0 is the base-band frequency. Thus, a first-order temporal differentiator is essentially a linear filtering device providing a spectral transfer function of the form H(ω-ω0) = i(ω-ω0). Consequently, the two key features of the filter’s transmission are (i) it depends linearly on the base-band frequency, and (ii) it is zero at the signal central frequency ω0. It is worth noting that these two key features imply an exact π phase shift across the central frequency ω0. Figure 1 shows a schematic of the basic Fourier relationships leading to the mathematical expression of a differentiator’s spectral transfer function. The ideal complex transfer function of an optical differentiator is shown in Fig. 2 (dash-dotted green line).
The required energy depletion at the signal central frequency can be produced by resonance-induced complete energy transfer elsewhere. Specifically, in waveguide optics, which is a platform particularly suitable for compact, robust, and low-loss (and thus practical) devices, this can be achieved by resonant transfer of light between two spatially close waveguides, or between two modes of the same waveguide (e.g. cladding and core modes of an optical fiber). Resonant light coupling is induced when the light propagates through both waveguides (modes) with identical speeds, which is practically attainable e.g. by an increase or decrease of the light speed in one of the waveguides (modes) using a suitable diffraction grating . The specific diffraction grating used in our experiments, which is realized as a periodic change of the refractive index along the direction of light propagation within a single waveguide (optical fiber), induces resonant coupling between two co-propagating modes and is commonly called long-period fiber grating (LPG) , . The term ‘long’ refers to its period, which typically varies from tens to hundreds of micrometers, as opposed to short-period gratings (Bragg gratings), where the light is backscattered, resulting in coupling between modes traveling in opposite directions (in a Bragg geometry, the corrugation period is approximately half the optical wavelength, micrometers or less ).
An optical fiber-based LPG induces gradual coupling at a rate of κ per unit length between the core guided mode and cladding mode(s) . To obtain efficient coupling between these modes, the period of the LPG must be properly adjusted to cause light diffraction from the core mode into the chosen cladding mode. Due to the different dispersion slopes of these two modes, the resonant coupling occurs only at a specific frequency ω0, referred to as the LPG resonance frequency. It is known that if the device is designed to exactly satisfy the condition κL = π/2, where L is the grating length, then the grating induces a total (100%) energy coupling from the input guided core mode into the cladding mode (at ω0) . We have recently shown that an optical fiber-based LPG specifically designed to provide 100% coupling between the fiber core mode and one of its cladding modes at the resonance frequency provides both the required π phase shift and the transmission linear dependence that is necessary for first-order time differentiation (assuming that the input optical signals are centered at the LPG resonance frequency) . Such an LPG, coincidentally, has the required spectral linear response over a bandwidth as broad as several terahertz.
The fact that a single uniform LPG provides just the spectral features required for optical differentiation around its resonance frequency is an extraordinary and fortunate occurrence which cannot be generalized to other types of basic resonant structures, such as uniform Bragg gratings.
While a single LPG can be used to realize first-order time differentiation of an incoming optical signal, the technique can be also easily extended to implement higher-order differentiation. Specifically, an N-order optical differentiator, able to calculate the N-th temporal derivative of the input signal, can be implemented by connecting in series N first-order optical differentiators ; this can be achieved by simply concatenating N uniform LPGs, each one operating in full-coupling conditions, along the same optical fiber. In practice, some additional technique should be used in between the gratings in order to remove (or significantly attenuate) the energy coupled into the cladding mode by the LPGs (e.g., by deposition of a high-refractive index layer).
3. Fabrication of LPG-based optical differentiators
To implement an LPG-based optical differentiator, stringent control of the fiber LPG coupling strength (which must be fixed exactly at κL = π/2) is required. The LPGs used in our experiments were made in a standard fiber sample (SMF-28, Corning Inc.), using the established point-by-point technique with a CO2 laser . The two LPG examples discussed here have physical lengths of 2.6 cm (sample S1) and 8.9 cm (sample S3), and grating period of 415 μm. Based on numerical analysis , this grating period corresponds to coupling into the 5th odd cladding mode at a resonance wavelength of 1540 nm, and the corresponding refractive index change amplitude (half of the peak-to-peak value) is 7.4×10-5 and 2.6×10-4 for S3 and S1, respectively. To perform a very fine tuning of the grating strength (coupling coefficient), we used a technique which takes advantage of the coupling coefficient dependence on the fiber longitudinal strain .
The amplitude and phase characteristics of the fiber LPG filters produced were measured by an Optical Vector Analyzer (Luna Technologies), and are shown in Fig. 2. The linear and quadratic terms in the phase curve – caused by the delay and linear chromatic dispersion, respectively – have been subtracted. We measured an exact π phase shift at the filter resonance wavelength (1534 nm for S1, 1535 nm for S3), which is an essential feature to obtain the desired filtering operation. The LPGs exhibited an extremely deep attenuation, breaking the 60 dB limit (incidentally, this is also the deepest LPG resonance ever reported ), confirming operation at almost exact full-coupling condition, as required by our application. The usable (“operational”) bandwidths of the fabricated LPGs were approximately 19 nm (S1) and 5.5 nm (S3). Notice that the “operational” bandwidth is the LPG resonance bandwidth over which the fiber filter provides the desired filtering function (i.e. a linear function of frequency). This corresponds approximately to the bandwidth over which the LPG transmission (in intensity) is lower than 10 %. Assuming input Gaussian pulses, these LPG samples could be used to process pulses as short as 700 fs (S3) and 180 fs (S1), where we considered the LPG operational bandwidth matched to the pulse bandwidth given at 10% of its peak power.
The attainable processing bandwidth of a LPG-based optical differentiator is ultimately limited by the presence of a slight non-linear dispersion slope of the core and cladding modes, which starts to deform the resonance dip linear shape when a large bandwidth is considered – using numerical simulations, we have estimated that a differentiator made in a standard telecom fiber could operate up to 10-THz speeds (Gaussian pulses down to 60 fs); the corresponding LPG length is 1 cm. For higher speeds, a special fiber with engineered core/cladding mode dispersion or an apodized LPG would be needed.
4. Numerical and experimental results
The fabricated LPG-based differentiators in this work were used for re-shaping sub-picosecond, Gaussian-like optical pulses into OS-HG waveforms. This specific waveform is of particular interest for next-generation optical communications , . An OS-HG waveform consists of two consecutive pulses in anti-phase inverted to their amplitude form in time and in particular this function can be mathematically expressed as v(t) ∝ t ∙ exp(-q t 2), where t is time and q is an appropriate constant ; the reader can easily prove that this temporal variation corresponds to the first time derivative of a Gaussian pulse. Generating such odd-symmetry pulses in a relatively simple and efficient way poses challenges, since a precise local change in phase (exactly π) is required. OS-HG pulses were first generated in the picosecond regime by filtering short pulses with a complex-profile fiber Bragg grating-based filter, operated in reflection . The drawbacks of this technique are associated with the inherently small bandwidth provided by Bragg grating filters, which are narrower than a few hundreds of gigahertz. This limits their operation to waveforms with temporal features longer than a few tens of picoseconds.
For an input pulse source, we used a passively mode-locked wavelength-tunable fiber laser (Pritel Inc., U.S.A.), which generated nearly transform-limited Gaussian-like optical pulses with a FWHM time width of ≈ 700 fs at a repetition rate of 20 MHz. The pulses were centered at the LPG resonance wavelength of 1535 nm. The input pulse full-width taken at 10% of its peak intensity was measured to be 5 nm. Because the LPGs are slightly birefringent, the light from the laser was passed through a fiber polarization controller before differentiation. Notice that in the demonstrations reported here, we used the optical differentiator based on the long LPG sample S3, because the bandwidth of this LPG was better matched to the bandwidth of the optical pulses generated by the available laser, ensuring a higher energetic efficiency in the differentiation process . After propagation through the LPG-based optical differentiator and a dispersion-compensated amplifier, the output temporal waveform was first analyzed with a conventional optical autocorrelator.
Results from numerical simulations are shown in Fig. 3. Specifically, the intensity profile of the ideal 700-fs (FWHM time-width) Gaussian input pulse (red dashed line) together with the form of the theoretically expected temporal field (blue solid line), and the corresponding intensity profile (red dotted line), at the output of the LPG are shown. The output temporal waveform was calculated as the inverse Fourier transform of the LPG-filtered pulse spectrum, which is the product of the simulated input pulse spectrum and the simulated LPG spectral transmission response. The obtained temporal field profile matched that of an OS-HG pulse and, in particular, it exhibited the predicted π phase shift between the two generated pulses. The computed intensity profile reveals that the peaks of the two lobes in the generated waveform are temporally separated by ≈ 900 fs, and that they have slightly different amplitudes (5% in terms of relative intensities). This difference is caused by a slight deviation of the LPG filtering characteristic from an ideal (linear) distribution. It is worth noting that each peak (FWHM ≈ 500 fs) of the generated temporal waveform is slightly narrower than the original Gaussian (FWHM ≈ 700 fs).
Figure 4 displays the autocorrelation traces of the measured input pulse (solid line) and of an ideal 700-fs Gaussian pulse (dashed line), in the bottom two plots (blue lines). The top two plots (red lines) show the measured and theoretically predicted autocorrelation traces of the generated temporal waveform at the LPG differentiator output. There is a remarkable agreement between the experimental results (solid lines) and the theoretical predictions (dashed lines), corresponding to the simulations shown in Fig. 3 (dotted line).
In our first set of experiments, we verified the accuracy of the phase shift of π between the individual pulses in the generated OS-HG waveform by its linear propagation through a 20-m long section of a SMF-28 fiber (dispersion β̈0 ≈ 0.023 ps2/m), which corresponds approximately to two dispersion lengths LD ≈ Δt 2/β̈0 for the generated individual pulses (Δt ≈ 500 fs). The autocorrelation result is shown in Fig. 5 (green dashed line). If the π phase shift was not sufficiently sharp, due to the local difference between phase and group velocities, some interference beating would have occurred in the long dispersive fiber . However, no interference beating is evident in Fig. 5, even when the individual peaks are significantly broadened by the chromatic dispersion.
More recently, we have developed a fiber-based Spectral Interferometry (SI) setup  to fully characterize the amplitude and phase of the OS-HG pulses generated via optical differentiation. The specific details of this measurement technique have been reported in Ref. . Briefly, an imbalanced Mach-Zehnder interferometer was formed by two fiber couplers; the fiber differentiator was put into one of its arms together with two polarization controllers for polarization control of the light incident to the component and of the interferometer. The spectral pattern resulting from interference between the input (reference) optical pulse and the waveform at the differentiator output was captured at the interferometer output using an optical spectrum analyzer (OSA); this interference pattern was used for the full (amplitude and phase) reconstruction of the waveform generated at the differentiator output. For this purpose, the well-known Fourier-Transform SI algorithm was applied . The result of one of these measurements (realized under the same conditions as in the previous experiments) together with the theoretical prediction (Fig. 3) is presented in Fig. 6; we observe an excellent agreement between the theory and the experiment and in particular, this result clearly confirmed the presence of an almost exact discrete p phase shift between the two temporal lobes of the generated OS-HG waveform.
5. Brief discussions on potential applications
As mentioned above, OS-HG temporal waveforms are of interest in several fields associated with optical telecommunications. This waveform is a good approximation of the second-order dispersion-managed (DM) temporal soliton, also called a ‘soliton molecule’ . A soliton molecule is the second solution of the non-linear Schrödinger equation (the first solution is well approximated by a Gaussian waveform), which describes the pulse propagation in an optical fiber with periodically varying (positive and negative) dispersion , . Since the first- and second-order DM solitons are temporally orthogonal and generally carry a different amount of energy, the OS-HG waveform could be used as a new communication symbol in DM links, potentially leading to a considerable increase of the information capacity limit in fiber optics communications systems. The generation of second-order DM solitons in the sub-picosecond regime was only recently reported . However, the required waveform was obtained indirectly, using a bulky and relatively expensive setup. The LPG-based direct generation method demonstrated here is obviously simpler and more efficient.
Other potential applications for an OS-HG waveform are related to the fact that this waveform is orthogonal to any temporally symmetric waveform (e.g. Gaussian), which means that these two waveforms can interfere destructively and cancel each other out. This property could be readily exploited for advanced coding applications, such as optical code division multiple access (O-CDMA) . As a generalization of this concept, it has been theoretically shown that the complete set of HG orthogonal temporal functions could be easily generated as a linear superposition of a Gaussian pulse and its time derivatives . This in turn should allow for synthesis of an arbitrary (ultrafast) temporal waveform from a single Gaussian pulse using a proper combination of optical differentiators, e.g. the simple devices demonstrated here . We have recently applied this concept for re-shaping Gaussian-like optical pulses into (sub-) picosecond flat-top temporal waveforms using LPG-based optical differentiation . Other interesting applications of all-optical temporal differentiators, e.g., for ultrafast sensing and control using optical signals, have been also discussed in Ref. .
In conclusion, we have reported the experimental demonstration of ultrafast all-optical (all-fiber) differentiators. The technique demonstrated in this report is based on the use of a simple uniform fiber LPG, and is capable of operation at terahertz processing speeds. In particular, the specific differentiators reported here can process arbitrary optical signals with temporal features as short as 180 fs, corresponding to bandwidths of more than 2 THz. In a relevant demonstration, one of the all-fiber devices was used to generate sub-picosecond odd-symmetry Hermite-Gaussian pulses by temporal differentiation of 700-fs optical Gaussian pulses (@ 1535nm). This implementation offers advantages inherent to all-fiber components: robustness, ease of integration, small size, and low cost. The developed device has the potential to become an invaluable tool for a multitude of important applications in all-optical computing and processing, ultrahigh-speed optical communications, femtosecond optical pulse shaping, and ultrafast sensing and control.
The authors thank Federico Rosei and Tudor Johnston for useful discussions. The authors also thank Sophie LaRochelle, Serge Doucet, and Philippe Giaccari for assistance with LPG amplitude/phase characterization.
This research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) through its Strategic Grants Program, and by the Czech Academy of Sciences Foundation under contract No. B200670601.
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