1. Introduction

Techniques for the generation, control, and manipulation of optical pulses attract considerable interest for numerous applications and have become increasingly important in many scientific areas [1]. Of specific interest are techniques for pulse repetition rate multiplication (PRRM), which are used to generate ultra-fast optical pulse trains from a low repetition rate input pulse train [2, 3], as well as for the generation of arbitrary waveforms [4]. Arbitrary or user-defined waveforms at high repetition rates can be used to generate wide-band RF signals [5], as switching windows or gating signals in all-optical pulse reshaping and retiming systems in optical communications [6], and as probing signals for various studies on spectroscopy and the properties of materials or molecules [7].

Traditional pulse shaping methods are based on frequency domain processing (i.e. spectral filtering) in which we specifically manipulate the different spectral components of the input pulse in amplitude and/or phase. However, the relationship between the input pulse spectrum and the target temporal waveform is not straightforward, especially for phase-only filtering processes. For example, if we need to generate an arbitrary waveform from a given input pulse, say Gaussian, it is difficult to determine the phase response and the corresponding filter that generates the waveform. Efficient Fourier synthesis algorithms are necessary to synthesize the required spectral response as well as the filter technologies for their implementation. Several optimization and synthesis algorithms have been used to generate arbitrary waveforms, including genetic, Gerchberg-Saxton [8], and simulated annealing [9, 10] algorithms. In terms of hardware, one of the most popular implementations is based on a bulk optical 4f pulse shaper, which separates spatially the frequency components of an input pulse and uses amplitude and/or phase masks to process the signal [1].

In this paper, we propose an alternate method to synthesize arbitrary waveforms using the direct temporal domain approach. In this approach, traditional frequency domain processing is not used and the waveform is synthesized purely in the temporal domain. While simple and compact optical filters structures are used, the waveform is synthesized by optimizing the filter parameters to obtain a specific temporal impulse response, without concern for the corresponding frequency response. The direct temporal domain approach was initially proposed to generate an ultra-high repetition rate pulse train from a low repetition rate pulse train using spectrally-periodic (SP) filters, details on the theory can be found in [10]. In this paper, we show that this approach can also be applied for general pulse shaping purposes, for example to generate a square waveform from a hyperbolic secant pulse train or a Gaussian pulse train. This approach is very different from the traditional spectral filtering approach which uses spectral filters to shape the input pulse spectrum into a sinc function (with amplitude and phase information preserved) in order to achieve a square waveform in the temporal domain. For implementation, we choose 2D ring resonator arrays (RRAs) as the optical filter to perform the temporal waveform generation since ring resonators are a class of SP filters.

2. Theory and configuration

High repetition rate pulse trains with uniform or even arbitrary envelopes can be generated using an SP filter and the direct temporal domain approach, as long as the input pulse width is sufficiently narrow [10]. When the free spectral range (FSR) of the SP filter satisfies certain conditions with respect to the new (output) repetition rate, pulse repetition rate multiplication (PRRM) is achieved; moreover, the output multiplied pulses (within the original period) can have very different amplitudes and phases due to the simultaneous amplitude and/or phase filtering process. Moreover, these amplitudes and phases can be set independently of the input pulse shape, see Eqs. 4–7 in [10]. We have found that this characteristic can be used to generate and synthesize arbitrary waveforms at the rate of the original input pulse train. The basic principle of the direct temporal domain approach for arbitrary waveform generation is illustrated in Fig. 1. First, we choose an SP filter to perform PRRM with a uniform output envelope. When an input pulse train with a broader pulse width is launched into the same SP filter, PRRM takes place, but there will be an interference among the output pulses. Next, we can manipulate and optimize the amplitude and phase of each individual output pulse (in the multiplied train) to generate the specified waveform (at the input repetition rate). The key in this approach is that we must find an SP filter which can simultaneously perform PRRM and manipulate the amplitude and phase of each individual output pulse with a range of freedom. The amplitude and phase of each individual output pulse is controlled by specifying the value of the temporal impulse response of the filter. This approach is very different from general spectral filtering approach which normally uses spectral filters to shape the input pulse spectrum into a special shape (with amplitude and phase information preserved) in order to obtain the desired waveform in the temporal domain.

 

Fig. 1. Schematic of arbitrary waveform generation using the direct temporal domain approach; (a) PRRM output with a uniform envelope from an input pulse train with narrow pulse widths; (b) PRRM output from an input pulse train with wide pulse widths; (c) waveform generation with an optimized SP filter and an input pulse train with wide pulse widths.

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Multi-stage ring resonators are a powerful class of optical filters which have been widely investigated for WDM add/drop filters, dispersion compensation, and many other applications that are focused on their spectral characteristics [11, 12]. However, their temporal domain characteristics are equally interesting. Since ring resonators are SP filters, they can be used as key components to perform arbitrary waveform generation by implementing the direct temporal domain approach.

Figure 2(a) shows a typical configuration of an M×N 2D RRA, where M represents the number of rings in the vertical direction and N represents the number in the horizontal direction. Figure 2(b) shows the details of the individual ring elements which incorporate a directional coupler with splitting ratio κ and a phase shifter with an additional phase shift φ.

Note that the rings are assumed to be identical in size and are coupled in the vertical direction only. While there is no coupling between the rings in the horizontal direction, the signals propagate in the horizontal direction through the two waveguide buses that are placed at the top and the bottom of the RRA. Since the waveguide bus introduces time delays above and beyond those of the rings, the length of the bus must satisfy specific conditions in order to generate a pair of pulse trains at the two output ports simultaneously. In particular, if L denotes the distance between the centers of two adjacent rings, then L must be a positive multiple of πR where R is the radius of the ring. Larger values of L will lead to more loss and an overall increase in the size of the structure; hence, we choose L = πR. The free spectral range (FSR) of the device is given by FSR = c /(n_e 2π∙R), where R is the radius of the ring and n_e is the effective index of the waveguide.

 

Fig. 2. (a) General configuration of an M×N RRA and (b) detailed view of the individual rings, r is the radius of the ring resonator, κ is the coupling coefficient and φ is an additional phase shift.

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The RRA transfer function can be calculated by dividing the M×N array into N horizontally-cascaded M×1 arrays, in which each M×1 array can be calculated separately using transfer matrices and Z-transforms as described in [11, 12]. For the n^th column, the transfer matrix in the vertical direction can be expressed by

where I₁, I₂, O₁ and O₂ represent the inputs and outputs in the two waveguide buses as shown in Fig. 2(a); t_p is the through-amplitude coefficient which is defined as t_p = $\sqrt{1-{\kappa }_p}$ ; γ is the waveguide loss and β is the propagation constant in the device. This matrix can then be converted to another transfer matrix which functions in the horizontal direction using the following transformation:

where θ_ij are the elements in the vertical transfer matrix. The two waveguide buses can also be expressed with a transfer matrix in the horizontal direction: ${\Phi }_{\mathit{bus}}=\left[\begin{array}{cc}{e}^{-\mathit{j\beta }{\mathit{L}}_{\gamma }}& 0\\ 0& e^{-\mathit{j\beta L}·d_{\gamma }}\end{array}\right]·d$ is the signal propagation direction and is equal to -/+ 1 when the signal counter-/co-propagates with the input, i.e. when M is odd/even. The overall RRA transfer function is then given by the system matrix Θ = Φ _N ∙ Φ _bus ∙Φ _N-1 ⋯ Φ _bus∙Φ ₁. By properly designing the parameters t_p and φ_p, the RRA can perform PRRM and simultaneously manipulate amplitudes and phases for the newly generated output pulses, thereby generating a special waveform at output. It is important to stress that the parameters of the RRA are optimized to generate a specific temporal impulse response of the filter in order to control the amplitude and phase of the output pulses.

In order to find a proper set of values of t_p and φ_p to generate a specified waveform, a powerful optimization algorithm is highly desirable. We use a similar optimization process as described in [10], but with some modifications. The major difference is that all sampling points within a period at the output (i.e. the input period) are compared with the target waveform and a weighted root mean square error function is used to evaluate the optimized results. Furthermore, in our optimization process, we use quantized rather than arbitrary values for t_p and φ_p, since values with a high precision may be difficult to realize in practice.

3. Simulation results

As an example, we consider a 5×5 RRA for arbitrary waveform generation. A 10 GHz train of hyperbolic secant input pulses is launched at input 1; we assume no input signal at input 2. The full width half maximum (FWHM) of each hyperbolic secant pulse is 8 ps as shown in Fig. 3(a). The objective is to transform this hyperbolic secant pulse into a square waveform with 5 ps rise and fall times, and a 40 ps flat-top. Square or rectangular shaped pulses are useful for enhancing the operation of nonlinear optical switches for reshaping ultrashort optical pulses [6].

In our simulation, we set the FSR of the filter to 160 GHz (corresponding ring radius is 0.199 mm for an effective index n_e = 1.5) and assume that the total loss in the whole device is 30%. We use the simulated annealing algorithm and a weighted root mean square error function to optimize the filter parameters, namely the through-amplitude t_p and the additional phase shift φ_p for each ring element. We quantize t_p and φ_p in steps of 0.01 and π/16, respectively. The weighted root mean square error function is defined as error = $\sqrt{\left(\sum _{n=1}^N{w}_n{\mid I_{\mathit{output}}-I_{t\arg \mathit{et}}\mid }^2\right)}$ where I is the intensity, N is the total sampling number in one repetition period, and w_n is the weight for each sampling point.

The results are shown in Fig. 3(b) and (c); the corresponding filter parameters are given in Table 1. Figure 3(b) shows the generated square-like waveform. The rising and falling edges are well matched to the target pulse and there are small oscillations in the flat-top portion. Figure 3(c) is a re-plot of (b) but in dB scale. It can be seen that the pulse extinction ratio (ER), defined as the ratio of the smallest intensity in flat-top and the largest intensity in background, is around 25 dB. The peak-to-peak variation in the flat-top is 0.9 dB.

 

Fig. 3. (a) Input and target waveform in one repetition period; (b) The generated square-like waveform; (c) The generated waveform in dB scale.

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Fig. 4. Spectrum of the 10 GHz pulse train at the RRA input and of the square-like waveform at he output. Amplitude (a) and (c); phase (b) and (d).

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Figure 4 shows the spectra (amplitudes and phases) of the 10 GHz hyperbolic secant pulse train at the input and the square-like waveform at the output. It can be seen that while the amplitude response of the output waveform exhibits the same 10 GHz spectral spacing as the input, the magnitudes for each individual mode have been altered which shows the amplitude-filtering nature of the RRA. In addition, the input and output phases are completely difference which shows the phase-filtering nature of the RRA. Since our approach only process the waveform in the temporal domain, the spectral shape of the output (and consequently of the filter) is not important in our approach and optimization algorithm; both magnitude and phase can be manipulated by RRA with a large scale of freedom (note in particular that the output spectrum does not exhibit any sinc-like features). This demonstrates the flexibility of the temporal domain approach over amplitude and phase filtering approaches which needs to alter the spectrum to a sinc shape (with appropriate phase jumps), or the phase-only filtering which is constrained to maintaining a constant magnitude or a uniform loss for all wavelengths.

 

Fig. 5. (a) The input hyperbolic secant pulse train at 10 GHz and the target waveform; (b) the target waveform and the generated triangular waveform form. the RRA.

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Table 1. Parameters of the RRA configuration for generation of a square waveform and a triangular waveform from a 10 GHz hyperbolic secant pulse train.

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As a second example to demonstrate the capabilities of the approach, we use the RRA to generate a triangular waveform which has 30 ps rising and falling times from the same input hyperbolic secant pulse train. The FSR of the RRA is still 160 GHz. The optimized RRA parameters t_p and φ_p are also given in Table 1. Figure 5 shows the generated triangular waveform, which is well matched to the target. We obtain similar results when generating triangular waveforms of different widths, which demonstrate that this approach is extremely powerful for generating arbitrary waveforms

We have also investigated the impact of the input pulse width (FWHM) on the quality of the generated waveform. If the FWHM varies in a range of ± 0.8 ps from its ideal value (8 ps), a good square-like waveform can be maintained with the same set of filter parameters. Otherwise, we need to re-optimize the ring resonator parameters. In particular, when the FWHM of the input pulse is between 6 ps to 9 ps, a good square-like waveform can be synthesized using the optimization algorithm. However, if the FWHM < 6 ps, we cannot limit the peak-to-peak variations to < 1 dB in the flat-top, even after re-optimization. Moreover, if the FWHM > 9 ps, the rising and falling times (5 ps) cannot be achieved. We also investigated the use of Gaussian input pulses to generate the square-like waveform. The result is similar to that obtained using hyperbolic secant pulses, which demonstrate that 2D RRAs have a big potential to synthesize square-like waveforms and even arbitrarily defined waveforms using the direct temporal domain approach.

 

Fig. 6. Contour plots for fabrication errors in the RRA for the square waveform generation; (a) the average extinction ratio; (b) peak-to-peak intensity variation in the flat-top portion.

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Next, we investigate the impact of fabrication errors in t_p and φ_p of each ring element. For illustrative purposes, we consider the same 5×5 RRA that generates the square-like waveform. We assume that the fabrication errors are uniformly distributed in a range of 0 to 2.5% for t_p and 0 to 1% for φ_p of all rings in the RRA and obtain both ER and the peak-to-peak intensity variations in the flat-top portion based on these imperfect parameters. We repeat the simulations 10,000 times and determine the average ER for the device. Figure 6(a) shows a contour plot for the average extinction ratio as the fabrication errors in t_p and φ_p vary from 0 to 2.5% and 0 to 1%, respectively; Fig. 6(b) shows the contour plot for the peak-to-peak variations in the flat-top portion. It can be seen that in order to maintain both an ER greater than 15 dB and peak-to-peak variations less than 2.5 dB, the fabrication error of t_p and φ_p should be within 2% and 0.8%, respectively. We note that programmable ring-resonator-based integrated photonic circuit with an accurate vertical coupling structure for t_p and independent controls for phase shift φ_p have been demonstrated [15], which indicate the practicality and feasibility of our temporal domain approach for arbitrary waveform generation.

4. Summary

We have demonstrated the use of 2D RRAs to synthesize square and triangular waveforms with the direct temporal domain approach. A square-like waveform with a 5 ps rising time, a 40 ps flat-top period, and a 5 ps falling time is generated by this filter from a hyperbolic secant input pulse with a FWHM of 8 ps. A high extinction ratio (25 dB) is observed in the generated square-like waveform and the peak-to-peak variation in the flat-top portion is 0.9 dB. In addition, we have shown the generation of a triangular waveform with a 30 ps rising time and a 30 ps falling time. These results show that the direct temporal domain approach can be applied to perform optical signal processing and pulse shaping and arbitrary waveforms using 2D RRAs.