Reconfigurable symmetric pulses generation using on-chip cascaded optical differentiators

We report a type of programmable pulse shaping method based on cascaded frequency-detuned optical differentiators. By properly adjusting the central wavelength of each differentiator, a large variety of symmetric pulses can be generated from a transformlimited Gaussian-like pulse. We numerically and experimentally demonstrate the generation of flat-top, parabolic and triangular pulses with tunable pulse widths from a 20-ps Gaussianlike pulse, using no more than three cascaded differentiators. It can be found that as more differentiators are used, higher synthesized accuracy and larger tuning range of pulse widths can be obtained in general. 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Introduction
Optical pulse shaping is an important signal processing technique which has been widely studied in optical communications, coherent control, ultra-broadband radio-frequency (RF) waveform generation and some nonlinear optical applications [1][2][3][4].Many efforts have been made during the past decades to exploit the generation of arbitrary pulse shapes precisely and flexibly.The linear spectral shaping technique based on spatial light modulator (SLM) [1,5] is the most effective way to date and has been commercially available, but bulk optical devices are always needed.In the recent years, shaping methods utilizing fiber grating [6] or integrated devices [7][8][9] have got more attention due to their compactness and stability.However, it is always hard to control the shaping system when we need to deal with the frequency comb line by line or obtain very high accuracy limited by the resolution of shaping devices.Apart from these proposals, generation of some special pulses such as flat-top pulse, triangular pulse and parabolic pulse are especially interesting since their wide range of applications in nonlinear optical signal processing [10][11][12][13][14].Many different schemes have been investigated to synthesize these waveforms [15][16][17][18][19][20], which usually show small errors.Whereas, only fixed pulse shape with fixed pulse width may limit their applications to some extent.For these reasons above, we need to develop a kind of shaping method which can simply generate these special shapes with high quality and flexibility.
Optical differentiator (OD) is one of the basic building blocks of optical signal processing, which has been studied extensively and can be easily fabricated utilizing fiber-based devices [21][22][23][24] or integrated devices [25][26][27][28][29][30].It has a large variety of applications [31][32][33], and among which the optical pulse shaping is rather attractive.A shaping technique based on multi-arm optical differentiators has been proposed previously [34] which is able to realize arbitrary waveform generation in theory.But limited by different time delays among parallel arms and relatively low energy efficiency of differentiators, it still haven't been accomplished practically to date.In addition, flat-top pulse generation using single frequency-detuned differentiator has been demonstrated based on long-period fiber grating (LPG) and silicon Mach-Zehnder interferometer (MZI) [15,16].
In this paper, we propose a reconfigurable pulse shaper based on cascaded optical differentiators.We theoretically demonstrate that when we concatenate a series of optical differentiators, many symmetric pulses can be generated from a transform-limited Gaussianlike pulse by simply adjusting the central wavelength of each differentiator.Then we validate the precisely generation of flat-top, parabolic and triangular pulses with tunable pulse widths using up to three differentiators through simulation and experiment.The performances of the shaping system using different numbers of differentiators have been evaluated and compared in detail.It can be found that as more differentiators are used, higher shaping accuracy and larger tuning range of pulse widths can be obtained in general.Furthermore, the optical differentiators used in our experiment are implemented with integrated thermally tunable delay interferometers (DIs).Some previous works have used cascaded DIs for pulse shaping [35][36][37][38] by taking advantage of their time-delay property or generating arbitrary transfer functions.Here, our shaping principle is based on the addition of derivatives of a Gaussian pulse in the time domain and the shapes of output pulses are irrelevant with the time-delay characteristic.Additionally, it should be noted that not only DI can be used as differentiators, but also many other devices such as microring, fiber grating, directional coupler and so on [21][22][23][24][25][26][27][28][29][30].The reason why we choose DI here is that it can be easily designed and fabricated on a SOI platform.Therefore, the pulse shaper has the advantages of compactness, stability and compatible with electronic integrated circuit technology.The working principle of our proposal is illustrated in Fig. 1.An input Gaussian pulse can be transformed into different pulse shapes after the cascaded differentiators.Each differentiator has a specific frequency detuning, which is defined as  ( )

Working principle
where k a is an arbitrary constant which has no influence on the pulse shape, i b is the coefficient of the corresponding derivative and can be expressed as follows: Assuming that the complex envelope of a transform-limited Gaussian pulse is ( ) in E t .Thus, the calculated temporal intensity of the output signal can be written as: It has been anticipated that any desired pulse shape could be synthesized as a linear superposition of a Gaussian pulse and its successive time derivatives with specific relative weights [34].From Eq. ( 2) and Eq. ( 3) we know that based on our proposal, the relative weights of derivatives can be simply changed by varying the frequency detuning of each differentiator, thus different pulse shapes can be generated.In addition, it should be noted that i b is a complex number and there is a phase difference of 2 π between the odd term and even term.Therefore, Eq. ( 3) can be rewritten into: As we know, the even-order derivatives of a Gaussian pulse are even symmetric, while the odd orders are odd symmetric.It can be found from Eq. ( 4) that the even terms and odd terms are superposed in intensity because of their phase differences, which means that all the output pulses of our shaping system are even symmetric.Theoretically, our shaping system is able to generate output pulses whose time features are not limited to the input pulse.Or in other words, the temporal variations of the output waveforms can be much faster or slower than the input waveform.It can be qualitatively explained from the frequency domain that when we change the frequency detunings of the optical differentiators, the low-frequency components of the input pulse may be suppressed while the high-frequency components are extracted, which leads to the faster change of pulse and vice versa.However, for a practical consideration, the amounts and operating bandwidths of optical differentiators won't be infinite, which may lead to an increase of shaping error.Fortunately, not all the pulses need so many differentiators.Waveforms such as flat-top pulse, parabolic pulse and triangular pulse can be generated precisely with a highest differential order of about two or three [34].After choosing different numbers of optical differentiators and calculate the corresponding frequency detunings, we show that three differentiators are enough to realize all of these interesting pulses flexibly.In addition, aiming at the input Gaussian pulse, we have optimized the working bandwidth of the differentiators in consideration of shaping error and energy efficiency (EE).

Simulation results and discussions
As we have mentioned before, our target is to realize the flat-top, parabolic and triangular pulses precisely and flexibly.The intensity profiles of these pulses [35] can be expressed as: where f T is the full width at a quarter of maximum of flat-top pulse, and α ( 0 ) is a roll-off factor which determines the width of the flat region.p T and t T represent the full width at half maximum (FWHM) of parabolic pulse and triangular pulse respectively.In our simulation, a Gaussian pulse with FWHM of about 20 ps is chosen as an input pulse, considering the 10-GHz repetition rate of our pulse source and the requirement to the working bandwidth of differentiator.Actually, optical pulses with other pulse widths can also be processed, as optical differentiators with bandwidth from tens of GHz to tens of THz has been realized [24,29,30].After specifying the features of the input and the target pulses, we then need to optimize the frequency detunings to find the most accurate output that coincides with the target.A least-square optimizing algorithm [35] has been used to find the right values.
That is, we sweep the frequency detunings and compare the errors between calculated results in Eq. ( 3) and targets in Eq. ( 5)- (7), when the error reaches the minimum, the corresponding frequency detunings are considered to be the best.The error of the calculated result is defined as: where m is the sampling points in the time domain and ( ) w t is an weight function.For different pulse shapes, the weight functions have different forms, which are used to ensure smaller errors in the specific regions.For example, the flat regions of flat-top pulses, the central regions of parabolic pulses and the linear-variation regions of triangular pulses.The simulation results using different numbers of differentiators are illustrated in Fig. 2, which show the maximum pulse widths that can be realized by the pulse shaper.As we know, when using one differentiator, a flat-top pulse can be generated [15].Here, a flat-top pulse with a flat duration of 12 ps has been realized.In addition, a 30-ps parabolic pulse can also be realized when we change the frequency detuning.But the error is rather big when synthesizing the 25-ps triangular pulse.When we use two or three differentiators, all the three kinds of pulses can be obtained with small deviations, as Figs.2(b) and 2(c) illustrate.In order to further compare the performances when using different numbers of differentiators, we show the shaping errors in Fig. 3.If we define 5% as the acceptable error, it is easy to find out that larger tuning range of pulse widths can be obtained when we use more differentiators.When three differentiators are used, triangular pulses from 30 ps to 35 ps, parabolic pulses from 25 ps to 45 ps and flat-top pulses with flat durations from 4 ps to 32 ps can be synthesized accurately.Besides, aiming at a certain pulse, the shaping accuracy is generally higher based on the high order shaping system.Small parts of pulses such as the 25ps triangular pulse and the 20-ps flat-top pulse don't follow this rule, which are limited by the tuning range of i b .It can be anticipated that if we are able to use more differentiators, we can get better shaping performances.However, the energy efficiency will become even lower and we may need amplifiers among the differentiators, which will increase complexity of the shaping system.Apart from the amounts of differentiators, the working bandwidth is another important factor which will affect the shaping performance.Here, aiming at the 3rd-order shaping system, we calculate the errors and energy efficiencies under different working bandwidths (80 GHz, 120 GHz and 160 The results are shown in Fig. 4, as the working bandwidth increase, the shaping error and energy efficiency will decrease in general.The shaping error when using 160-GHz differentiators is close to the ideal case in Fig. 3, but the energy efficiency is rather low.When the bandwidth reduce to 80 GHz, the energy efficiency has a promotion of about 15 dB, but the accuracy may become unacceptable.In order to realize a high accuracy ( 5% error < ) with acceptable energy efficiency ( 35 EE dB > − ), we finally choose the 120-GHz differentiators in our following experiment.From the simulations above, we know that the shaping system based on three cascaded differentiators can implement the reconfigurable generation of those special waveforms.However, limited by the number of differentiators, the tuning ranges of their pulse widths are relatively small.In order to obtain an output pulse several times wider (or narrower), an alternative way is to change the FWHM of the input pulse [15] in equal proportion, while change the frequency detunings and working bandwidths of differentiators in inverse proportion at the same time.This is an interesting property which can help us redesign the shaping system without extra calculations.For example, a 30-ps triangular pulse can be generated from a 20-ps Gaussian pulse with frequency detunings of −41.30GHz, 9.45 GHz and 22.15 GHz.Then it can be easily worked out that a 15-ps triangular pulse can be generated from a 10-ps Gaussian pulse with frequency detunings of −82.60 GHz, 18.90 GHz and 44.30GHz, and a 60-ps triangular coming from a 40-ps Gaussian pulse with frequency detunings of −20.65 GHz, 4.73 GHz and 11.08 GHz, as shown in Fig. 5.The working bandwidths of differentiators are respectively 240 GHz, 120 GHz and 60 GHz for the 15-ps, 30-ps, and 60-ps triangular pulses, and shaping errors are respectively 3.59%, 3.62% and 3.58%.The small deviations among shaping errors verify the correctness of this property.

Experimental results and discussions
In order to demonstrate the shaping method practically, we design an on-chip pulse shaper with three cascaded optical differentiators.The schematic diagram of the pulse shaper is shown in Fig. 6(a), the differentiators are realized by thermally tunable DIs.All the input ports and output ports of DIs were connected with the TE-mode grating couplers, thus the number of differentiators can be changed flexibly by moving the coupling fibers.The thicknesses of the top silicon layer and the buried oxide layer are respectively 220 nm and 2 μm.The width of the silicon waveguides is 500 nm.A silica layer of nearly 2 μm was deposited on the silicon layer for the fabrication of micro-heaters.As shown in Fig. 6(b), the TiN heaters are 3-μm wide and located right above the upper arms of DIs.Then another 500nm-thick silica layer was further deposited on the heaters, followed by the fabrication of aluminum wires and pads.In order to simplify the tests, the metal electrodes were connected to larger pads on a film circuit through wire bonding technology.The measured transmission spectrum of a single DI is shown in Fig. 7(a) with blue solid line, the free spectrum range (FSR) is about 200 GHz and the insertion loss is about 1 dB.Comparing with the transmission spectrum of an ideal differentiator shown by the red dash line, it can be estimated that the effective differentiating bandwidth of the DI is about 120 GHz. Figure 7(b) illustrates the variation of the transmission spectrum of DI when applying different voltages.The tuning efficiency is calculated to be 45 pm/mW and the frequency shift can reach a FSR with a voltage less than 7 V.The experimental setup is shown in Fig. 8.A wavelength tunable mode locked laser with a repetition rate of 10 GHz is used as a pulse source.Then a 0.18-nm Gaussian filter programmed by the waveshaper (Finisar Waveshaper 1000S) is used to produce the 20-ps Gaussian pulse.Before inputting into the pulse shaper, the Gaussian pulse is amplified by an EDFA to pre-compensate the insertion loss of the device.The optical power after the EDFA is about 15 dBm, the coupling loss of each grating coupler is about 3.5 dB and the insertion loss of each DI is about 1dB.After the pulse shaper, a low noise EDFA is employed to amplify the output signal and an 1.6-nm BPF is further used to extract the signal as well as suppress the out-of-band noise.The measured outputs using one differentiator are shown in Fig. 10.A flat-top pulse with 13.4-ps flat duration and a 30-ps parabolic pulse have be generated.In Figs.10(a-1) and 10(b-1), the green dot lines represent the target pulses, the blue solid lines are experimental results and the red dash curves show simulation results.The simulation results are calculated using the frequency detunings shown in Fig. 10(a-2) and 10(b-2), where the blue lines are the spectra of input signals, the green lines are the transmission spectra and red lines show the output spectra.In order to better fit with the experimental results, the frequency detunings which are predefined through the optimizing algorithm have been modified according to the measured transmission spectra and waveforms.The small depression at the top of flat-top pulse is caused by the deviation of frequency detuning.Besides, it should be noted that the output pulses are not transform-limited, which can be deduced from the optical spectra.Fortunately, their phase variations always have little influence on the nonlinear applications we have mentioned above [10][11][12][13][14], where only specific intensity profiles are required.
When we increase the number of differentiators from one to two, more kinds of pulses with large tuning range can be generated.Figure 11 illustrates the corresponding waveforms and spectra.Flat-top pulses with flat durations from 11.4 ps to 20 ps, parabolic pulses from 30 ps to 35 ps and a 30-ps triangular pulses can be synthesized accurately.At last, all the three cascaded differentiators have been utilized and larger tuning ranges of pulse widths have been realized, as shown in Fig. 12. Flat-top pulses with flat durations from 6.8 ps to 25 ps, parabolic pulses from 28 ps to 40 ps and triangular pulses from 30 ps to 35 ps have been generated.In addition, the shaping error should be decreased in general according to our simulation.For instance, it can be easily observed that the rising and falling edges of the 30-ps triangular pulse have a better linearity compared with that using two differentiators.However, compared with the simulation results, the practical tuning ranges are still smaller.Wider pulses always have larger deviations, such as the 25-ps flat-top pulse and 40-ps parabolic pulse which show obvious asymmetries.This is caused by the finite resonance depths of differentiators, which will bring asymmetries to the derivatives in Eq. ( 4).Narrower pulses usually show smaller errors because they are mainly determined by the original input pulse and the weights of derivatives are relatively small.In our experiment, the measured resonance depths of DIs are around 35 dB, which should be higher actually, considering the limited resolution of the optical spectrum analyzer (OSA).Fortunately, the resonance depth can be further improved by using variable power splitters [39,40], which will help to realize higher shaping accuracy.

Conclusion
We have proposed and experimentally demonstrated the reconfigurable pulse shaper based on cascaded optical differentiators.By adjusting the frequency detunings of the differentiators, many symmetric pulses can be generated in theory.In this paper, flat-top pulses, parabolic pulses and triangular pulses with tunable pulse widths have been realized using up to three differentiators.We have analyzed and compared the shaping performances in detail when changing the numbers or bandwidths of differentiators.And we also show how to change the frequency detunings and bandwidths of differentiators when we want to change the tuning ranges of the output pulses by varying the width of input pulse.Moreover, for practical demonstration, we have designed and fabricated three thermally tunable DIs to work as optical differentiators on a SOI platform, which has the advantages of compactness and stability.And it is easy to change the number of DIs by moving the coupling fibers, so we are able to choose the right number of differentiators under different situations.In addition, through the wire bonding technology, we are able to adjust the working status of the pulse shaper simply and flexibly.The final experiment results show good agreements with the simulation results and can be further improved by optimizing the resonance depths of differentiators.

Funding
The work was supported by the National Science Fund for Distinguished Young Scholars (No. 61125501), the NSFC Major International Joint Research Project (No. 61320106016) and Foundation for Innovative Research Groups of the Natural Science Foundation of Hubei Province (Grant No. 2014CFA004).

Fig. 1 .
Fig. 1.Working principle of the pulse shaper based on cascaded frequency-detuned differentiators.The green lines represent the transmission spectra of ideal differentiators and the blue arrows show the carrier wavelength of the input signal.

Nω
δω .As shown in Fig.1, is the central frequency of the Nth differentiator, and 0 ω is the carrier frequency of the input signal.Then the transfer function of our shaping system can be written as:

Fig. 2 .
Fig. 2. Simulation results (blue solid lines) using (a) one differentiator, (b) two differentiators or (c) three differentiators.The red dash lines represent the target pulses and green dot lines represent the weight functions.The frequency detunings are shown with blue texts.

Fig. 3 .
Fig. 3. Shaping errors for generating triangular, parabolic and flat-top pulses using one OD, two ODs or three ODs.The widths of triangular and parabolic pulses are defined with FWHM, while the widths of flat-top pulses refer to the durations of flat regions.

Fig. 4 .
Fig.4.Shaping errors and energy efficiencies for generating triangular, parabolic and flat-top pulses using a 3rd-order shaping system with different working bandwidths.

Fig. 6 .
Fig. 6.(a) Schematic diagram and (b) micrograph of the on-chip pulse shaper based on cascaded differentiators.

Fig. 7 .
Fig. 7. (a) Transmission spectra of the fabricated DI (blue solid line) and an ideal differentiator (red dash line).(b) Measured transmission spectra of DI under different voltages.

Figure 9
Figure 9 illustrates the measured waveform and spectrum of the input Gaussian pulse.The red dash line in Fig. 9(a) is an ideal 20-ps Gaussian pulse and the blue solid line is the measured one which has been normalized, they show a good agreement.The number of averaging used for the measured waveforms is 25.The inset in Fig. 9(a) shows the corresponding waveform displayed on the OSO.The carrier wavelength of the input signal is about 1550.71 nm as we can see from Fig. 9(b).

Fig. 9 .
Fig. 9. Measured (a) waveform (blue solid line) and (b) spectrum of the input Gaussian pulse.The inset shows the measured waveform displayed on the OSO.

Fig. 10 .
Fig. 10.Measured (a-1), (b-1) waveforms (blue solid lines) compared with simulation results (red dash lines) and target pulses (green dot lines).(a-2), (b-2) Optical spectra of input signal (blue) and output signals (red), the green lines show the transmission spectra using one optical differentiator and the blue texts are the practical frequency shifts.