In this paper, a novel structure of dynamic optical arbitrary waveform generation (O-AWG) with amplitude controlled by interference of two fiber Bragg grating (FBG) arrays is proposed. The FBG array consists of several FBGs and fiber stretchers (FSs). The amplitude is controlled by FSs through interference of two FBG arrays. The phase is controlled by FSs simultaneously. As a result, optical pulse trains with various waveforms as well as pulse trains with nonuniform pulse intensity, pulse spacing and pulse width in each period are obtained via FSs adjustment to change the phase shift of signal in each array.
©2012 Optical Society of America
Optical arbitrary waveform generation (O-AWG) has aroused people’s attention in recent years due to its flexibility to synthesize arbitrary waveform at optical frequency. O-AWG is realized by manipulating tunable variables such as amplitude and phase to control the waveform of optical pulse . The most prevailing and widely used method of O-AWG is line-by-line pulse shaping method, in which a pulse shaper is used to manipulate amplitude and phase of each spectral line individually. According to Fourier synthesis theory, the waveform of optical pulse is correspondingly manipulated by controlling amplitude and phase of each spectral line in the pulse shaper [1,2]. Due to the development of optical passive devices, such as diffraction grating, arrayed waveguide grating (AWG) and fiber grating, different kinds of line-by-line pulse shapers are proposed. Diffraction gratings combined with a liquid crystal spatial light modulator (SLM) have been used to manipulate more than 100 spectral lines [3,4]. Abundant experiments are performed for high-fidelity O-AWG in pulse shaper based on diffraction gratings [5–9]. However, it has high insertion loss, bulky structure and needs complicated alignment control. AWG combined with integrated phase and amplitude modulator arrays has been used to control tens of spectral lines individually in experiment [10–13]. However, it is difficult to improve device resolution to below 10 GHz and to control the uniformity across the device as frequency resolution increases.
Fiber Bragg gratings (FBGs) have been adopted to achieve various waveforms due to their advantages, such as compactness, free-alignment, low insertion loss and compatibility with fiber. A simple sinusoidal FBG is used to transform bright solitons into dark solitions . A number of cascaded uniform FBGs with equal distance are designed to convert a continuous-wave (CW) with sinusoidal phase modulation into Gaussian pulse train experimentally . A complex superstructed fiber Bragg grating (SSFBG) is fabricated to transform a soliton pulse train into a rectangular pulse train . FBGs are not only used to shape the waveform but also used to multiply the repetition rate of optical pulse train. A sinc-sampled FBG is used to transform 10 GHz pulse train into 40 GHz pulse train experimentally . An eight-passband phase-sampled FBG is employed to generate 160 GHz pulse train from a 40 GHz pulse train . However, O-AWG realized by FBGs is normally static because the structure of FBGs is fixed once they are fabricated. In order to achieve dynamic O-AWG, a FBG array structure constructed with cascaded uniform FBGs, in-line polarization controllers and in-line fiber stretchers (FSs) is proposed to dynamically control the amplitude and phase of each spectral line . Several distinct waveforms are achieved experimentally by manipulating up to 5 spectral lines. In this paper, we propose a novel structure of dynamic O-AWG with amplitude controlled by interference of two FBG arrays with only FSs and FBGs in each array.
The schematic diagram structure of the proposed dynamic O-AWG is shown in Fig. 1 . The pulse shaper consists of an optical coupler and two FBG arrays, which are constructed with FBGs and FSs. Constructions of the two FBG arrays are the same as each other except for FSs adjustment. FBGs in both arrays have high reflectivity. The central wavelengths of FBGs are equally spaced for both arrays, which match the wavelengths of spectral lines in optical frequency comb (OFC). OFC is incident into pulse shaper through port A of the optical coupler. FBGs are used for separating different spectral lines spectrally from incident OFC. FSs are used for changing the phase of each spectral line individually, which can be realized by twisting fiber around a piezoelectric transducer (PZT) and controlled by applying desired voltage to PZT. The optical signals reflected by the two FBG arrays will interfere at the coupler. The spectral amplitude and phase of output signal will be determined by the phase difference and phase shift of the interference signals, respectively, which are modulated by FSs in both arrays. The spectral amplitude and phase control of signal will result in a controllable pulse waveform in time domain, which exports through port D of the coupler.
For simplicity, we postulate the incident OFC consists of discrete comb lines with ideal linewidth, which can be expressed as 
After propagating in the FBG arrays, the optical signals recombine and interfere at the optical coupler. Optical signals at port A and port D of the coupler are determined by the following matrix [1,20]Eq. (3), which are expressed asEquation (4) implies that the amplitude of the nth spectral line of output signal is influenced by |Sn|, α, |ρ1n|, |ρ2n| and the phase difference Δφn. The changing range of the amplitude is maximum when |ρ1n| = |ρ2n|. Equation (5) implies that the phase of the nth spectral line of output signal is influenced by phase shift φn and phase difference Δφn. |Sn| and α do not influence the output waveform, only influence the amplitude of output pulse. For a given incident OFC and a given desired output waveform, the phase difference Δφn between the nth spectral lines in the two arrays and the phase shift φn of the nth spectral line caused by the first array can be derived from Eq. (4) and Eq. (5), which are expressed asEq. (4) and Eq. (5), the adjustment of phase difference Δφn and phase shift φn will result in variation of amplitude |Fn| and phase pn of the nth spectral line of output signal. Thus, the pulse shaper will produce different optical waveforms by adjusting FSs in the two arrays.
The pulse shaper is not only capable of controlling the waveform of pulse, but also capable of controlling other features such as pulse intensity, pulse spacing, pulse width and repetition rate. The spectrum of controllable pulse can be divided into k groups, which are corresponding to k pulses within each period. The delay and waveform of each pulse are user-defined according to the following filter function 
3. Simulation and discussion
Aiming for demonstrating the pulse shaper specifically, different waveforms such as Gaussian pulse, rectangular pulse are converted from incident OFC by only controlling the phase shift caused by the two arrays via FSs adjustment. Moreover, pulse intensity, pulse spacing and pulse width for individual pulses within each period can also be altered by only controlling the phase shift caused by the two arrays via FSs adjustment.
In the simulation, optical pulse shapers under ideal and non-ideal assumptions are investigated for comparison. For ideal assumptions, all FBGs have uniform reflective coefficient as |ρ1n| = |ρ2n| = 1. The power-splitting ratio of the coupler is set as α = 0.5. The incident OFC has 25 spectral lines with uniform amplitude as |Sn| = 1 and constant phase as ψn = 0 centered at 1550 nm with wavelength spaced by 1 nm, which is shown in Fig. 2(a) . For non-ideal assumptions, the incident OFC is shown in Fig. 2(b) and the power-splitting ratio of the coupler is set as α = 0.52. Realistic parameters of FBGs in the lab are used in simulation. The reflectivities of all FBGs are above 80%. The reflectivity difference of the corresponding FBGs with same Bragg wavelength in the two arrays varies from 0.8% to 10%.
3.1 Optical pulse generation with various waveforms
In order to get a Gaussian pulse with 0.35 ps pulse width under ideal and non-ideal assumptions, the required values of phase difference Δφn and phase shift φn are calculated according to Eq. (6) and Eq. (7). The phase difference Δφn caused by the two FBG arrays and phase shift φn caused by the first FBG array are realized by adjusting the FSs. After being reflected by the arrays and interfering at the coupler, the amplitudes of spectral lines at port D of the coupler are diverse for both assumptions, which are shown in Fig. 3(a) , due to different values of Δφn. The phases of spectral lines pn are constant for both assumptions, which are shown in Fig. 3(b). According to Fourier transform, output signal is Gaussian waveform pulse for both assumptions, which are illustrated in Fig. 3(c). As a result, the Gaussian waveform is achieved by only controlling FSs in both FBG arrays to result in different spectral amplitudes and constant spectral phases. It can be seen from Fig. 3, the Gaussian waveforms under ideal and non-ideal assumptions have little difference, which means that the incident OFC, power-splitting ratio and FBG reflectivity have no obvious effect on the output pulse waveform. From Fig. 3(a), the spectral amplitude under non-ideal assumptions is obviously smaller than that under ideal assumptions, which is mainly caused by the smaller amplitude of incident OFC.
Likewise, a rectangular waveform with 2 ps pulse width and 0.5 duty cycle can also be achieved under both assumptions by setting the phase difference Δφn and phase shift φn according to Eq. (6) and Eq. (7). The amplitude, phase of spectral lines and waveform of the output signal under both assumptions are shown in Figs. 4(a) , 4(b) and 4(c), respectively. As a result, the rectangular shape pulse is also achieved by only adjusting FSs in both FBG arrays. The ripples on the top of rectangular pulse are caused by the finite number of FBGs, which can be reduced by increasing the number of FBGs in the arrays to manipulate more spectral lines. It can be seen from Fig. 4(c) that the tops of rectangular pulses for both assumptions are slightly different from each other, which results from amplitude difference shown in Fig. 4(a) caused by the reflectivity difference as indicated in Eq. (4). In order to optimize the amplitude, the reflectivities of corresponding FBGs in two arrays should be equal to each other.
Figures 3 and 4 demonstrate that this pulse shaper has converted a given OFC into two distinct waveforms by only adjusting FSs, which manifests its potential towards generating optical pulse with arbitrary waveform. They also verify that power-splitting ratio and input OFC have negligible effect on waveform. The nonuniform reflectivities of FBGs will result in waveform distortion due to non-zero extinction for spectral lines, which can be optimized by introducing extra loss to achieve equal amplitude interfering signals.
3.2 Optical pulse train generation with nonuniform intensity, pulse spacing and pulse width
In addition to controlling waveform for identical pulse train, the pulse shaper can also control pulse intensity, pulse spacing and pulse width for individual pulses within each period.
Under ideal assumptions and perfect delay adjustment, the required phase difference Δφn and phase shift φn for generating 4 separated Gaussian pulses with a decreasing pulse intensity pattern [1 0.49 0.25 0.10] within 8 ps period are calculated according to Eq. (6) and Eq. (7). The spectral amplitude and phase are illustrated by blue solid stems in Fig. 5 (a-1) and blue solid stems in Fig. 5(a-2), respectively. The corresponding output Gaussian pulse train is illustrated by blue solid line in Fig. 5(a-3). In addition to nonuniform pulse intensity, nonuniform pulse spacing in each period can also be realized by phase control via FSs. Corresponding control is implemented to achieve 4 uniform Gaussian pulses spaced by different spacings within each period as 1.6 ps, 3.0 ps, 1.4 ps, and 2.0 ps. The spectral amplitude (blue solid stems), phase (blue solid stems) and output pulses (blue solid line) are shown in Figs. 5(b-1), 5(b-2) and 5(b-3), respectively. The intensity and pulse width of 4 equally separated Gaussian pulses can be controlled simultaneously and individually by changing the phase difference Δφn and phase shift φn. The spectral amplitude (blue solid stems) and phase (blue solid stems) of a complex waveform with a decreasing pulse width pattern [0.54 0.38 0.31 0.23] ps combined with a decreasing pulse intensity pattern [1 0.64 0.47 0.24] are shown in Fig. 5(c-1) and Fig. 5(c-2), respectively. The waveform is shown in Fig. 5(c-3) (blue solid line), which verifies intensity and pulse width for individual pulses within each period can be controlled simultaneously and individually. The slight pulse distortion results from overlapping of adjacent pulses.
Due to that amplitude controlled by interference effect is very sensitive to phase variation, a sub-wavelength accuracy on the delays is normally required. In practice, imperfect FSs adjustment, stress and thermal drifts will result in imperfect delay. In this section, the sensitivity against imperfect phase shift resulting from the central FS0 in the first array is discussed due to its significant influence on the output pulse.
If the imperfect adjustment of the central FS0 in the first array results in a phase offset Δθ = 0.01π, the amplitude, phase of three kinds of pulse trains are shown on the left side of Fig. 5 as illustrated by red circles. They show that the amplitude and phase well coincide with those of the ideal waveform with zero phase offset (blue solid stems). The corresponding temporal pulse trains are illustrated by red dotted lines in Figs. 5(a-3), 5(b-3) and 5(c-3), which show that pulse trains for Δθ = 0.01π are almost consistent with pulse trains for Δθ = 0. If phase offset Δθ increases up to 0.1π, the amplitude and phase (black ‘x’ symbols) evidently deviate from those of the ideal waveform, as shown on the left side of Fig. 5. Correspondingly, temporal pulse trains represented by black dotted line are visibly distorted, shown in Figs. 5(a-3), 5(b-3) and 5(c-3). Total integrated error E defined as average deviation from the ideal waveform in each period is introduced to demonstrate waveform distortion against phase offset Δθ, which is illustrated by insets in Figs. 5(a-3), 5(b-3) and 5(c-3). The insets imply that pulse distortion gradually increases versus phase offset Δθ and the pulse train in Fig. 5(a-3) is the most insensitive to phase offset Δθ compared with the two other kinds of pulse trains. They also indicate the tolerance phase offset for the proposed scheme is larger than 0.01π.
Figure 5 demonstrates that pulses in each period can be manipulated independently via only FSs adjustment in the proposed pulse shaper. The generated pulse train with diverse controllable variables such as amplitude, time delay and pulse width can be utilized to generate optical vector code in next generation communication system.
In this paper, a structure of O-AWG with amplitude controlled by interference of two FBG arrays is proposed. For a given incident OFC, optical pulse with various waveforms can be dynamically generated by only adjusting FSs. Moreover, pulse trains with nonuniform pulse intensity, pulse spacing and pulse width in each period are generated by only adjusting FSs as well. It can be concluded from results that power-splitting ratio and nonuniform reflectivities of FBGs have no obvious effect on the waveform of output pulse and the tolerance phase offset of the proposed structure is larger than 0.01π. In this pulse shaper, only simple uniform FBGs and FSs are required to realize arbitrary amplitude and phase control. Besides, it enables dynamic O-AWG in all-fiber configuration. The interference effect is very sensitive to environmental perturbation. In order to minimize the environment effect, a water-based sodium polyacrylate gel can be used to effectively insulate fiber from both temperature fluctuation and vibration disturbance . In addition, due to complex phase control, computerized feedback control system  can be adopted to ensure efficient FSs adjustment.
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