Dynamic line-by-line pulse shaping with GHz update rate.

We introduce a novel scheme for dynamic line-by-line pulse shaping with GHz update rates. Four lines of an optical frequency comb source are used to injection-lock four individual VCSEL, which are subsequently electrically modulated at 0.4 to 1 GHz through current modulation. This concept could be considered a completely new way of pulse shaping as the light is not simply modified, but rather regenerated with the desired properties. We also discuss an important drawback of line-by-line pulse shapers that ultimately limits the modulation speed capability.


Introduction
Ability to generate large-bandwidth, arbitrarily-shaped waveforms in optical and microwave domains serve many applications, among which are bandwidth-efficient telecommunications, parallel signal processing and pattern recognition, remote sensing, and physical sciences. To date, the most mature and versatile way to generate large-bandwidth arbitrary waveforms is through line-by-line ultrafast optical pulse shaping [1]. In this technique, an arbitrary, complex optical filter function H(ω) is designed and applied to individual lines or groups of lines of a optical frequency comb source in order to yield the desired waveform in timedomain. A variety of periodic arbitrary waveforms have been generated with line-by-line pulse shapers based on liquid-crystal based spatial light modulators [1] and resistive-heating of waveguides [2]. Nevertheless these techniques are limited to ~kHz update rates which restricts their use. Recently, fast dynamic pulse shaping schemes based on hybrid, on-chip electro-optic waveguides were shown [3][4][5]. However, these schemes suffer from practical limitations in scalability due to increasing complexity of fabrication and electrical/optical cross-connects as the number of comb lines, hence the waveform bandwidth, increase.
In this paper, we introduce a novel scheme for line-by-line pulse shaping with GHz update rates, and straight forward scalability to large number of lines. The concept is based on injection-locking of semiconductor lasers. The individual lines of a frequency comb source are used to injection-lock individual Vertical Cavity Surface Emitting Lasers (VCSELs). The VCSELs subsequently are current-injection-modulated in order to regenerate the lines of the frequency comb with the desired amplitude and phase functions. In fact, this concept could be considered a completely new type of pulse shaping, where the source light is not simply modified, but rather regenerated with the desired properties. Owing to the large currentmodulation bandwidth of VCSELs, potentially tens of GHz update rates could be achieved.
We experimentally show line-by-line pulse shaping through injection-locking of 4 individual VCSELs with a frequency comb source with 12.5 GHz repetition rate. Each regenerated line was sine-wave modulated at 0.4 to 1 GHz, generating an arbitrary optical waveform with a 5 ns period and 37.5 GHz bandwidth. The modulation speed was mainly limited by the passband of the dispersive optical element in the line-by-line pulse shaper. We believe that this limitation is one of the biggest hurdles in increasing the modulation speed of any continuous spectral-dispersion based arbitrary waveform generation architecture.

Injection-Locking concept and experimental results
Injection-locking is commonly used to generate coherent laser sources that share a phase relationship. When a free-running slave laser cavity is injected with the light from a master laser, the slave cavity locks its frequency to the master laser. Moreover, the relative phase of the slave laser obeys Adler's equation (for small injection): where φ is the optical phase difference of the slave and master lasers, ω s and ω m are freerunning slave and master laser frequencies, and B is the half locking bandwidth. It is evident from Eq. (1) that, once the injection-locking occurs, any perturbation of the slave cavity will result in a relative phase change. Especially for semiconductor lasers, this perturbation could be through current-injection, which in turn yields a high-speed, intrinsically nonlinear phase modulator with applications in optical sampling and telecommunications [6]. Normally, bias current is utilized to tune the frequency of semiconductor lasers at high speeds. However, as the injection-locking process fixes the frequency (defined by the master laser), the bias current ends up tuning the optical phase. The phase of an injection-locked semiconductor laser can be tuned for a full π through the bias current. The bias current range for this tuning is called as -referred by the free-running slave laser frequency change-the injection-locking bandwidth. Injection-locking bandwidth depends on the master/slave optical injection power ratio. Additionally, the response speed of the phase tuning process goes from DC to a few tens of GHz based on the semiconductor device structure and electrical connections. We have previously demonstrated such operation of a VCSEL up to a few GHz as shown in [6]. Since the VCSEL cavities are very sensitive to bias current, the full π phase swing can be achieved with a very small bias current range, without any significant additional amplitude modulation. Obviously, this requires a very small master/slave optical injection power ratio (usually below −25 dB). If necessary, stronger bias current modulation can be achieved (without the slave laser falling out of injection-lock) by increasing the master laser optical injection power. However, this results in additional direct intensity modulation of the slave laser output along with the phase modulation from the injection-locking process. In this paper, we utilize these high-speed phase and intensity modulation schemes for dynamic lineby-line pulse shaping.
The pulse-shaper setup is shown in Fig. 1. A 1538nm single-frequency laser was modulated with a 12.5 GHz sine-wave by cascaded electro-optic intensity modulators in order to generate a frequency comb source with ~5 comb lines of equal power (Fig. 1). Alternatively, a frequency-stabilized 12.5 GHz repetition-rate modelocked laser could be used as the source [7]. The signal was then amplified and fed into a fiber-pigtailed Virtually-Imaged Phase Array (VIPA) based spectral demultiplexer with 6.25 GHz channel separation and 100 GHz free spectral range. Four odd channel outputs of the demultiplexer (frequency separation of 12.5 GHz each) were connected to 4 individual commercial fiber-pigtailed single-mode VCSELs at ~1538nm center wavelength. No effort was spent on actively stabilizing VCSEL temperatures, fiber lengths and polarization fluctuations. The VCSELs were tuned with DC-current to fall within the locking bandwidth of the corresponding injecting comb line, and directly-modulated with various RF power levels at 0.4GHz, 0.6GHz, 0.8GHz, and 1GHz with sine-wave generators that share a common reference clock. The peak-to-peak optical intensity modulation depth of the VCSELs ranged from 2% to 20% with various RF power levels, and the optical injection ratio varied from −23dB to −17dB. The VCSEL optical outputs were isolated from the input demultiplexer with circulators in order to prevent the reflections from flat fiber tips that could destabilize the injection-locking process. An output spectral multiplexer that is identical to the input demultiplexer was used in the VCSEL output signal path to observe the resultant shaped arbitrary waveforms. The waveforms were characterized with a high-resolution Optical Spectrum Analyzer (OSA), Radio Frequency (RF) spectrum analyzer, a sampling oscilloscope and two different speed real-time oscilloscopes. Although simple sine-wave modulation was implemented here as a proof-of-principle, work is ongoing with complex modulation waveforms involving highspeed bit pattern generators. The measured high resolution optical spectra and RF spectra are shown in Fig. 2 with and without injection-locking for −25dBm, −30dBm, and −35dBm RF modulation powers. It can be easily seen from the RF traces that there was no phase relationship between free-running VCSELs before injection-locking, however injection-locked traces show clear tones with high Signal to Noise Ratio (SNR). Furthermore, the high resolution optical spectrum trace confirms that the linewidths of the VCSELs were narrowed by injection-locking as expected. Initially, we tried to observe the time-domain waveform with a high-speed sampling oscilloscope. Although certain periodicity of the generated waveform could be seen on the sampling scope traces, clear waveforms could not be observed due to the slow triggering speed of the scope relative to the environmental fluctuations of the VCSEL pigtail fibers. The time-domain data from a two different speed real-time oscilloscopes are shown in Fig. 3 for −25dBm and −30dBm RF modulation power levels. A 1GHz, 5Gs/s real-time oscilloscope was used to obtain the overall time envelope data, which was subsequently Fouriertransformed to yield clean RF frequencies with >30dB SNR (Fig. 3(d)) confirming the RF spectrum analyzer measurement of Fig. 2. Additionally, a 16GHz, 40Gs/s real-time oscilloscope (LeCroy WaveMaster 816zi) was utilized to observe a periodic arbitrary waveform with 5ns period, which corresponds to the lowest common multiple of the individual VCSEL modulation periods (Fig. 3(a)). Again, the noise in the time-domain trace is attributed to the environmental fluctuations of the VCSEL pigtail fibers causing fast phase and polarization fluctuations. Nevertheless, subsequent Fourier-transform of the data yields clean RF frequencies at 0.4,0.6,0.8,1 GHz, and 12.5 ± 0.4,0.6,0.8,1 GHz (Fig. 3(b)). Moreover, intermixing products and higher order RF beats are also observed up to 20 GHz, limited by the 10 GHz photodetector and 16 GHz oscilloscope bandwidth. A sample simulation of the system is also shown in Fig. 3(c). that is qualitatively similar the observed waveform. We believe that the stronger portion of the modulation was due to the direct intensity modulation of the lines with moderate modulation depth, while some portion came from the phase modulation. In fact, injection-locking bandwidth, intensity modulation bandwidth, and phase modulation depth are all functions of the power injection ratio, and we are working on identifying these parameters in order to calibrate and model our system more accurately.

Discussion
In this work, we used direct current-modulation of the VCSELs to achieve intensity and phase modulation. Additionally, one can carefully modulate the VCSELs with a very small depth as to achieve phase-only modulation as in [6]. Furthermore, combination of a portion of the original master signal with the injection-locked phase-only modulated slave signal can yield precisely controlled intensity and phase modulation.
Another, more preferred implementation of our pulse shaping scheme can be realized with a 1-Dimensional free space VCSEL array. This would eliminate the non-common fiber pigtail paths and improve the long term stability. However, in this case, alignment of the pulse shaper becomes more challenging due to the increased number of dimensions. Namely, every source frequency line should overlap with the individual VCSEL frequency injection-locking band, and individual VCSEL spatial location. Besides the alignment challenges, this implementation allows straightforward scaling of the number of VCSELs, especially via 2-Dimensional VCSEL arrays and 2-Dimensional pulse shaping [8]. Fabrication and electrical cross-connect wiring of 2-Dimensional VCSEL arrays are well-established techniques in the industry [9]. One advantage of the two dimensions is that scaling from a single channel to N channels requires a maximum of N 1/2 increase in the physical dimensions of the system, therefore enabling scaling to hundreds of channels with minimized engineering challenges. It is evident in Fig. 2 and Fig. 3 that, as the modulation frequency was increased, the sideband power ratio diminished in the dynamic pulse shaper. This evidence points to a upper limitation of the modulation frequency of the optical comb lines. The commercial VCSELs used in the experiment had a free-running rise time of ~100ps, and the injection-locking process is expected to reduce the rise time even more to reach tens of GHz modulation bandwidth [10]. Thus, the main limitation of the upper modulation frequency of the optical comb lines is not the VCSELs, but rather the spectral multiplexer. In order to explain this, let's assume a standard, reflective, "zero-dispersion 4-f" diffraction grating based line-by-line pulse shaper setup (Fig. 4). The pulse shaper includes two lenses for imaging the input light to the plane of the optical modulator array. In Fig. 4, for simplicity, only a single modulator pixel is shown corresponding to a single optical frequency comb line. The grating equation for a single comb line at the optical frequency of ν 0 can be written as (Eq. (2)): where c is the speed of light, G is the groove density, θ in is the input angle of the beam, and θ d is the diffraction angle of the comb line of interest. The individual pixels of the spatial light modulator in a regular "static" pulse shaper are passive, and do not change the optical frequencies -they just introduce a static optical phase. However, in the case of active VCSELs along with modulation, new frequencies are generated. For a VCSEL injection-locking to the input optical frequency of ν 0 , a simple small-signal sine-wave modulation at f m generates the frequency components "ν 0f m " and "ν 0 + f m ". The intensities of these components will depend on the strengths of phase modulation and intensity modulation of the VCSEL. The problem arises from the fact that, these new frequencies are emitted by the same VCSEL aperture that regenerates the original ν 0 signal. Therefore, on the return path, the diffraction grating directs "ν 0f m " and "ν 0 + f m " at slightly different angles (θand θ + ) compared to ν 0 . Consequently, the newly generated frequencies do not couple back into the pulse shaper input aperture with the same efficiency as ν 0 . The angles θand θ + are given by Eq.
If we assume a thin-lens with focal length FL 1 at distance d, a single mode fiber with core diameter 2a as the aperture, and perfect coupling of the ν 0 component back into the single mode fiber, the power coupling ratio of the new frequencies can be simply estimated in dBscale, shown below as Pand P + . Here, we show simplified analytical derivations of the phenomena, while a more generalized and detailed study can be found in [11]. We assume a flat beam profile for simplicity of the analytical derivation, however this can easily be extended to a Gaussian beam profile by the use of "Error Function" in the overlap integral calculations. In fact, the flat beam profile assumption gives a rather optimistic estimate of the coupling efficiencies compared to a Gaussian beam profile. Additionally, the numerical aperture (NA) of the single mode fiber is not considered due to the difference between (θand θ + ) and θ in being much smaller than the NA for practical implementations. Recently, new AWG designs yielding raised cosine filter functions were proposed for lineby-line pulse shaping [12]. These AWGs are optimized for fixed discrete channels, and the filter function of each channel has a large overlap with adjacent channels. For standard high speed line-by-line modulation, these overlaps would cause crosstalk between channels, and limit modulation speed in a similar fashion as above. In order to avoid the crosstalk, more complicated modulation schemes such as I/Q modulation and pre-emphasis are required. Although this combination alleviates the problem, it increases the system complexity and cost, therefore we believe that it is not the best solution.

Conclusion
In conclusion, we have introduced a novel line-by-line pulse shaping scheme that enables potentially tens of GHz update rates with straight forward scalability for channel count. We have experimentally shown 4 individual lines sine-wave modulated at 0.4 to 1 GHz, generating an arbitrary waveform with a 5 ns period and 37.5 GHz bandwidth. We also discuss a simple fundamental mechanism that ultimately limits the modulation speed capability of any line-by-line pulse shaper that is based on continuous spectral dispersion.