Enhanced nonlinear spectral broadening and sub-picosecond pulse generation by adaptive spectral phase optimization of electro-optic frequency combs

: We utilize adaptive optimization to enhance the spectral broadening of an ampliﬁed electro-optic frequency comb with a 25 GHz repetition rate in a highly nonlinear ﬁber and subsequently generate sub-picosecond pulses. The spectral phase of the comb is adaptively optimized by a Fourier pulse shaper in a closed control loop with the HNLF output spectrum as the process variable to be optimized. Enhanced spectral broadening also increases the stimulated Brillouin scattering threshold allowing increased power scaling and thereby boosting the bandwidth by a factor of more than 13 times over the initial comb. System versatility to varying conditions is demonstrated by achieving consistent bandwidth enhancement (nearly or more than 100 lines) in varying operating conditions that distort the temporal proﬁle of the comb. In all cases, the optimization yields a near transform limited pulse that enters the nonlinear ﬁber. Sub-picosecond pulse generation is achieved with a short length of single mode ﬁber post the nonlinear ﬁber.


Introduction
Optical frequency combs (OFC) form vital infrastructure in high bandwidth communications where they meet the increased data demands through wave length division multiplexing (WDM) [1], and orthogonal frequency division multiplexed (OFDM) super channels [2] in a compact and efficient manner. In these systems, OFC avoid the need for multiple laser sources and complicated locking techniques [3] . Further, OFC find applications in spectroscopy, microwave arbitrary waveform generation, bridging the microwave and optical time standard, calibration of astronomical spectrographs and coherent LIDAR [4]. Traditionally, mode locked lasers based OFC [5] were used, but they lack independent tuning of central wavelength and repetition rate [6], an essential requirement for flex-grid WDM. Moreover, they also suffer from frequency instability and increased complexities at high repetition rates. External modulation through electro-optic (EO) modulators [7][8][9][10][11][12][13][14][15][16] achieves independent tuning of repetition rate and central wavelength with increased system stability, but has low bandwidth, limited by the RF power handling of the modulator. Cascading modulators [11][12][13] will only increase the bandwidth (BW) linearly with number of modulators and is unfeasible for large BW scaling, due to increased number of RF amplifiers and passive components needed. Wide-band combs can be generated by four-wave mixing of lasers in a highly non-linear fiber (HNLF) [6,17] but requires the HNLF to be strained [17] to overcome stimulated Brillouin scattering (SBS). Further, the repetition rate of the comb is directly determined by the spacing between lasers which is coupled to the extent of spectral broadening. Use of single mode fiber (SMF) compression stages in between HNLF stages to increase the peak power results in poor spectral flatness. Also, this multi-stage design requires precise dispersion engineering of various stages to achieve spectral broadening [17] and cannot adapt to varying conditions. Electro-optic combs can be spectrally broadened in nonlinear media [18] enabling few cycle pulse generation [19] and optical frequency synthesis [20] among other applications. However, BW scaling of amplified EO combs in HNLF is severely constrained by non-optimal temporal profile of the EO combs resulting in poor spectral broadening at a given power. Also, the resulting spectral power is mostly concentrated among few lines. This results in low SBS threshold of the system that limits power scaling and thus the maximal BW scaling. BW at a given power can be enhanced by optimizing the temporal profile of the pulse entering HNLF for enhanced self-phase modulation (SPM). This would also allow further power scaling of the system due to better distribution of the power across the comb increasing the maximal achievable BW. Temporal profile can be modified through a Fourier pulse shaper [21] that controls the spectral phases of EO comb lines. However, the determination of optimum phase profile for a given system is not a convex optimization problem and the search space is vast. Adaptive optimization has been used to solve such problems in various applications like phase distortion correction of wave front [22], beam combining [23], phase correction for distorted orbital angular momentum beams [24] and also for BW scaling [25][26][27].
In [25], spectral phase of transform limited gaussian pulses from a mode locked laser-based frequency comb was optimized with evolutionary strategy algorithms to increase BW by defining the phase perturbations with a weighted Chebyshev polynomial sum. The optimized pulse shape in this system has considerable chirp and deviates significantly from a transform limited pulse which was the initial pulse shape. In [26], EO comb line amplitude control is used to achieve gaussian spectrum followed by spectral phase control to achieve transform-limited gaussian pulses with low background noise. Spectral phase control is implemented through genetic algorithm with feedback from an autocorrelator. These synthesized pulses were later spectrally broadened in HNLF. This method targets gaussian pulses with low background noise, which may not always be the required pulse shape for maximal broadening. The optimal pulse shape in each system is a complicated function of system parameters such as power, initial spectral phase of comb, modulator driving conditions, length and dispersion profile (sign, magnitude of various orders of dispersion) of various fibers following the Fourier pulse shaper. As spectral broadening in HNLF is done in an open loop and not in a closed control loop, the pulse shape is not always optimized as variations in system parameters change the optimal pulse shape.
In the present work, we adaptively optimize EO comb spectral phase in a completely closed loop to directly target the HNLF output magnitude spectrum as the process variable to be optimized. The EO comb output is significantly different from a mode locked laser-based transform limited pulse. The EO comb requires major modifications to the spectral phase based on the operating conditions of the modulators (for example: DC bias of intensity modulator and time dependent drifts). Therefore, restricting the search space to spectral phases having the functional form of a smooth polynomial is not enough and true line by line spectral phase shaping with precise control is necessary. In order to do this, the perturbations are applied directly to the spectral phases of individual comb lines without any assumptions about the nature of the spectral phase function. A modified variation of evolutionary strategy algorithm is utilized where the vector of phase perturbations and its explement are separately applied to the phase vector and the optimal of three phase vectors is selected for next iteration. The system is optimized at moderate power (225 mW) to avoid catastrophic damage to the system from SBS due to occurrence of undesirable spectral phase profiles during optimization that compress the comb. The optimized system is power scaled to an output power of ∼700 mW to achieve a bandwidth scaling factor of ∼8 over the unoptimized case and >13 times over the initial EO comb. The optimal pulse shape strongly depends on the magnitude and sign of various orders of dispersion of fibers following the spectral phase control unit (Fourier pulse shaper). Hence, limiting the pulse profile to a specific shape such as a gaussian [26] is not always optimal for spectral broadening. Further, it has been shown that the optimal pulse shape can be far from a transform limited pulse [25]. Thus, closed loop systems that evolve the pulse profile according to the prevailing system parameters are necessary to achieve consistent spectral broadening under various operating conditions. The system versatility is demonstrated by varying the EO comb pulse shape through DC bias control of intensity modulator and achieving nearly 100 lines in almost all cases. The completely closed loop system is capable of reliably adapting to various system conditions and configurations. Zero delay implementation of spectral shearing interferometry [28] is used to determine the spectral phase profile and thus the temporal profile of the optimized pulse and output comb. The optimized pulse profile in our case is a near-transform limited pulse, whereas the optimized solution for mode locked lasers in reference 25 significantly deviates from the transform limited pulse (which was the initial pulse shape). Sub-picosecond pulses (730 fs) with 25 GHz repetition rate were measured after a short length of single mode fiber(∼8 m) following HNLF. The intrinsically phase locked nature of the C-band comb along with repetition rate and center frequency tunability allowed by the electro-optic comb makes it an excellent source for orthogonal frequency division multiplexed super channels, optical sampling, RF arbitrary wave form generation, coherent LIDAR and in calibration of astronomical spectrographs. The demonstrated technique to generate high repetition rate pulses can be implemented at a wide variety of wavelengths and repetition rates with equivalent components to potentially achieve high ablation rate of tissues and material targets with minimal thermal damage by operating in the so called ablation cooled regime [29]. Figure 1 shows the architecture of adaptive spectral phase optimization of EO frequency combs for enhanced spectral broadening. A 1550 nm laser source (∼100 kHz linewidth) is externally modulated by a cascade of lithium niobate intensity and phase modulators driven by a 25 GHz sinusoid. The intensity modulator has a DC V Π of 6.6V. The comb line spacing of 25 GHz is tunable for flexgrid implementation and is chosen here for compatibility with the existing International Telecommunication Union (ITU) DWDM grid. The spectral phase control unit (Fourier pulse shaper) is a Finisar waveshaper with a frequency resolution of 10 GHz that controls individual spectral phases with a resolution of 0.01 rad. An inhouse-built Erbium doped fiber-amplifier (EDFA) prior to waveshaper compensates the system losses. Power scaling is achieved with a cascade of a pre-amplifier EDFA (output power ∼40 mW) and an in-house built erbium-ytterbium co-doped fiber amplifier (EYFA) that increase the output power to ∼700 mW. The HNLF of ∼290 m length with nonlinear coefficient of 11.3/W-km and 0.7 dB/km loss has the zero-dispersion wavelength of 1550 nm with 0.017 ps/nm 2 -km dispersion slope.

Experiment
Adaptive optimization decision unit implements the modified evolution strategy algorithm (Fig. 2) in a MATLAB environment, processing data from optical spectrum analyzer and controlling the waveshaper. The algorithm initializes waveshaper with zero additional spectral phase and obtains the magnitude spectrum. The target spectral envelope is defined as a peak normalised gaussian that is sampled at the comb wavelengths and is compared with the peaks of the measured spectrum (also peak normalized). Defining the target spectral envelope as a flat line [27] leads to reduced spectral broadening as the algorithm may prefer flatter spectrum with lesser bandwidth over a wider spectrum with poor flatness. The RMS error function is defined as The error is evaluated in logarithmic scale rather than linear scale to give proper weightage to the contribution of the comb lines at the edges of the spectrum in the error. In each iteration, a stochastic phase perturbation vector (∆Φ) with gaussian distribution is generated. The waveshaper adapts the current spectral phase vector Φ with the perturbation and its explement to generate new vectors Φ+∆Φ and Φ-∆Φ respectively. The vector with least error among the three is retained and the process is repeated. The search space i.e. the extent of perturbation is controlled by the standard deviation of the perturbation vector and is updated according to one-fifth rule [25] [30] with a 10% change. The number of iterations before search space updation depends on the number of variables to be optimized and is made equal to the number of spectral phases to be optimized. The algorithm makes no assumptions about the nature of phase function allowing unbiased evolution of the system to evolves towards the target. As the BW scaling capability of the system is initially unknown, the target gaussian is initially defined with a narrow width (standard deviation) that is progressively increased each time the measured spectrum reaches it. When the target is redefined, the error and standard deviation of perturbed phase vector are re-initialized. The algorithm is stopped upon no improvement over more than 100 iterations. To test the system versatility to varying system conditions, the pulse shape of EO comb is changed through DC bias control of intensity modulator and a new optimization run is done for each case. Though the initial optimization took 1300 iterations, the system capabilities were understood and the widest target spectrum and initial search space definitions to which the system responds rapidly were obtained from observation. Using these parameters, the optimization runs for DC bias variations were done in 300-400 iterations to achieve similar BW scaling.

Adaptive optimization for bandwidth scaling
The 9-line (20 dB BW) EO comb (DC bias of intensity modulator 3.5V) shown in Fig. 3(a) without optimization gives negligible bandwidth scaling to 16 lines [ Fig. 3(b)]. The output power is ∼350 mW and is limited by SBS. Spectral evolution of the frequency comb across several runs during adaptive optimization is shown in Figs. 4(a)-4(d). The evolution of error with adaptive optimization in the third is shown in Fig. 5(a). After each run, the target spectrum is redefined by increasing the standard deviation of the gaussian envelope in steps of 0.2. The error is reset to a value close to the initial error and standard deviation (std) of the perturbation phase vector is reset to 0.5. This value is obtained from observed adaptive search space optimization in the system. A smaller value will also approach the target by gradually increasing search space because of adaptive search space correction but with longer convergence time. Values greater than 0.5 are observed to only contract in our setup with no improvements and thus are not initialized with. The initial optimization took 9 runs with a total of 1300 iterations and 11 spectral phase variables (1549 nm-1551 nm) are optimized. The other lines in the EO comb are weaker by more than 36 dB relative to the peak and are not considered because of their negligible influence on the pulse shape. Whenever the measured spectrum is close to the target spectrum, the current run is terminated, and a wider target is defined. Continuing with the older run would lead the system to reject perturbations that would increase the output bandwidth beyond the defined target. This is because of the definition of error [Eq. (1)] which is made in relation to the target spectrum. The unwrapped optimized spectral phase function (linear component corresponding to fixed delay is removed) applied by the spectral phase control unit is shown in Fig. 5(b). Though the quadratic fit is close, the deviations are significant and cannot be ignored as these systems have very high phase sensitivity of the order of π/100 [26]. This indicates that the common practice of pulse compression with SMF or similar fiber, which generates primarily a quadratic spectral phase, of EO combs prior to spectral broadening in HNLF will not provide the optimal phase compensation to achieve maximal spectral broadening. Spectral phase of EO combs in  general is not a pure quadratic [13] and has significant contributions of higher order dispersion terms (especially for the usual RF sinusoid driven modulators). The spectral phase of the EO combs strongly depends on the DC bias of the intensity modulator and the RF drive used for the modulators. These factors along with variations in system parameters make quadratic phase compensation inadequate and necessitate precise control of line by line spectral phase. Further, transform limited pulses may not even be the optimal solution in all systems. This has been the case in a previous work in a system based on mode locked lasers [25] where a broadened pulse with considerable chirp is obtained as the optimal solution. Thus, the best approach to enhance the spectral broadening is in a closed loop system without any limitations on the possible spectral phases. The optimized phase spectrum takes into consideration all the system parameters and adaptively tailors the synthesized pulse shape accordingly for a given system for maximal spectral broadening.
The output spectrum optimization is done at a lower output power of ∼210 mW to avoid SBS initiated damage and has 47 lines in 20 dB bandwidth [ Fig. 6(a)]. With the optimization lowering the SBS threshold by reducing peak spectral power, the system was power scaled up to ∼700 mW output power (SBS limited) and the obtained spectrum is shown in Fig. 6(b) with 120 lines in 20 dB bandwidth thus having a bandwidth scaling factor of ∼8 over the unoptimized case and has over 13 times the number of lines in the initial EO comb. The absolute bandwidth achieved strongly depends on the properties of experimental components used. Therefore, this by itself is not a measure of the performance of the adaptive optimization system. To compare the performance across a wide variety of systems, we define a figure of merit that considers the nonlinear parameter of the HNLF, its length and the power pumped into it apart from the bandwidth scaling factor.
Here, the 10-dB (from the peak) bandwidths before the nonlinear fiber (input bandwidth) and after the nonlinear fiber (output bandwidth) are compared. γ (/W-km) is the nonlinear parameter of fiber, Pin (W) is the power entering the nonlinear fiber and L (km) is the length of the nonlinear fiber. The normalization to the factor (γ*Pin*L) is essential for comparing spectral broadening between different systems having different powers and nonlinear fiber parameters. Table 1 compares the spectral broadening efficiency among systems implementing closed loop adaptive control. The FOM of the present system is significantly higher than the earlier systems demonstrating the superiority of the technique. The spectral phase function obtained is repeatable and gives nearly same spectrum when the undisturbed system is switched on after few days. Though the optimization is done at lower powers, the optimal spectral phase profile does not change with power and is confirmed by tweaking the spectral phase at higher powers and observing deterioration of the spectral broadening. The reason for this is that the optimal solution is always found to be a near transform limited pulse entering the HNLF which is independent of operating power. This is further discussed in the next section. To make the algorithm convergence robust, we initialize the system with zero additional spectral phase instead of beginning with an initial phase with a definite functional form (such as a polynomial) which can bias the system evolution. It was observed that at the end of a successful run we always get the same optimal spectral phase independent of the initially applied additional spectral phase.

System adaptability to varying conditions
The system adaptability to varying operating and environment conditions is tested by varying the EO comb pulse profile by changing the DC bias of intensity modulator. Zero delay spectral shearing interferometry [28] is used to obtain the temporal pulse profiles of the EO comb [ Fig. 7(a)]. The adaptive optimization algorithm is run separately for each of these cases with the starting value of the standard deviation of phase perturbation as 0.5 and the target spectrum as the widest target that was used in the previous run. The system optimization took only 350-450 iterations in each case and is nearly 3 times faster than the initial run due to the optimal starting parameters for the system obtained. The optimized pulse profiles after the waveshaper are shown in Fig. 7(b). In each case, the system consistently shaped the temporal profile into a sharp pulse with pulse widths (FWHM) of 4.66 ps (0V), 5.03 ps (2V) and 5.59 ps (3V). The adaptively optimized spectra at ∼220 mW output power for these cases have 45 lines [0V case, Fig. 8(a) Fig. 7(a)], demonstrating its versatility. Our comprehensive work, which does not assume a specific pulse shape as optimal for spectral broadening reiterates the generally held notion that a near transform limited pulse entering the HNLF yields optimal spectral broadening. The method described in this work thus can also be used as a pulse measurement tool together with the spectrum of the pulse measured through the OSA. When the system is optimized, the pulse entering the nonlinear fiber is a near transform limited pulse and the applied spectral phase for optimal spectral broadening is the negative of the spectral phase of the input pulse coming out of the EO comb generator. The optimal pulse shape would change only if a different HNLF or a fiber with different dispersion properties were to be used. As the section of system post the spectral phase control unit is not perturbed, the optimal pulse shape for this system does not change. This can be seen in Fig. 7(b) where the optimized pulse shapes bear strong similarity irrespective of changes to the pulse shape introduced by the EO comb generator section.

Sub-picosecond pulse generation
The adaptively optimized spectrum at 220 mW for 0V IM DC bias case [ Fig. 8(a)] is passed through short length of single mode fiber (SMF)(∼8 m) and pulse profile is measured. The pulse after waveshaper had an FWHM of 4.66 ps [ Fig. 9(a)], almost close to a transform-limited pulse (obtained with zero spectral phase for the same magnitude spectrum). After HNLF and SMF, the pulse is compressed by more than 6 times to generate 730 fs pulses [ Fig. 9(b)] corresponding to a peak power of ∼35 W with 25 GHz repetition rate. The output pulses slightly deviate from transform-limited pulse and the asymmetric pulse profile is due to the residual third and higher order dispersion which has not been compensated by SMF.

Conclusions
Adaptive spectral phase optimization was used to enhance the bandwidth by over 13 times the initial BW to 2.975 THz (20 dB BW) and over 8 times the unoptimized case (0.375 THz). System adaptability to varying conditions is demonstrated through consistent BW scaling factors and optimized pulse profiles. We have rigorously demonstrated that a near transform limited pulse is the optimal temporal profile for spectral broadening. The adaptively optimized, broadened spectrum is compressed to generate sub-pico second pulses with high repetition rate. The source is an optimal solution for high bandwidth optical communications through super channels, optical sampling and LIDAR. The search space of feasible pulse shapes is partially limited as only the spectral phase of the comb lines is being varied. Varying the spectral amplitude would allow realization of more pulse shapes. However, we have not used amplitude control in our system as this allows the perturbations where the signal power entering the fiber amplifier is too low resulting in ASE based damage to the fiber amplifier.

Funding
Ministry of Electronics and Information Technology (Visvesvaraya PhD Scheme); Department of Science and Technology, Ministry of Science and Technology, India (NNetra); Office of the Principal Scientific Advisor, GOI (Prn.SA/ADV/Photonics/2015-16).