1. Introduction

The generation of arbitrary optical waveforms with both scalable bandwidth (>1 THz) and long record lengths (> 1 ns) is an ongoing technical challenge with many new and exciting applications in telecommunications, optical signal processing, and microwave photonics [1,2]. Fourier domain pulse shaping techniques such as static optical arbitrary waveform generation (OAWG), also known as line-by-line pulse shaping, adjust the phase and intensity of each line in an optical frequency comb (OFC) using a Fourier domain waveform shaper (see Fig. 1(a) for an example) to produce bandwidth scalable waveforms [1,3,4]. To output a stable waveform, the waveform shapers must maintain coherence between the comb lines. Operating with static modulations, the waveform shaper essentially acts as a highly reconfigurable filter which operates on the OFC. Therefore, the output is a repetitive waveform (i.e., shaped OFC) with a maximum waveform period equal to the input OFC’s period. Temporal multiplexing schemes which combine waveforms from the outputs of multiple waveform shapers [5,6], or high-resolution static waveform shapers [7] have been used to demonstrate incremental increases in the record length of static-OAWG waveforms. However, because of their complexity and size, these techniques are impractical to scale to long record lengths and a different approach is needed.

 

Fig. 1 (a) Dynamic waveform shaper and dynamic-OAWG concept. (b) The spectral slice algorithm used to generate spectral slices. (c) Pre-emphasis of the spectral slices for the multiplexer and generation of temporal modulations. (d) Quadrature and (e) polar modulator.

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The dynamic waveform-shaper architecture shown in Fig. 1(a) applies rapidly updating, or ‘dynamic’, modulations to each comb line to break the periodicity of the OFC and generate continuous, broadband waveforms. Yet, previous studies [8,9] have identified the spectral multiplexer filtering as a severe limiting factor for the waveform update rate. Here, we demonstrate a dynamic-OAWG technique that overcomes those limitations by using a spectral domain algorithm [10,11] to calculate the comb line modulations and a waveform shaper design that incorporates quadrature modulation and a specially designed spectral multiplexer. Our experimental demonstrations using fiber-pigtailed components involving two-line and three-line modulations verify the dynamic-OAWG concept and the spectral domain algorithm. Future demonstrations using a monolithically integrated InP waveform shaper with 100 channels will enable THz bandwidth waveform generation [12].

2. The dynamic-OAWG concept

Figure 1 illustrates the dynamic-OAWG waveform shaper and the spectral-slice algorithm. A high-fidelity and practical dynamic waveform shaper must have a bandwidth-efficient modulator structure sufficiently fast to generate any spectral slice and a multiplexer with gapless spectral transmission. The waveform shaper modulates each line of an OFC to create spectral slices that, when coherently combined, will fill the gaps in the OFC’s discrete spectrum. Time-frequency duality ensures that if the target waveform’s spectrum is generated exactly, then the generated temporal waveform, d(t), is a perfect match to the target, a(t). Figure 1(a) shows the dynamic waveform shaper, which contains the spectral multiplexers and modulators that generate and coherently combine the different spectral slices. A demultiplexer with high adjacent channel isolation directs each comb line to a different channel. Then an array of modulators applies parallel temporal modulations to broaden each comb line into shaped spectral slices, and finally a multiplexer with gapless-transmission combines each slice. For this demonstration, we use in-phase (I) and quadrature-phase (Q) modulators and an arrayed-waveguide grating (AWG) with broadened passbands.

Figures 1(b), 1(c) describe the spectral slice algorithm for computing the complex temporal comb line modulations, m_n(t), that are compatible with the multiplexer [10,11]. First, the target waveform, a(t), is defined across many OFC periods. Then its spectrum, A(ω), is computed with a discrete Fourier transform (DFT). A set of overlapping spectral slice filters, S_n(ω), where n is the comb line index, divides the spectrum into spectral slices, C_n(ω) = A(ω)S_n(ω). There are many possible sets of slice filters; yet, it is required that the summation of all of the spectral slices equals the target spectrum. The next step determines the comb line modulations, M_n(ω), by pre-emphasizing each slice for the multiplexer transmission, H_n(ω), and the n-th comb line’s amplitude and phase, R_n. In the spectral domain, the modulations are M_n(ω) = [C_n(ω)/H_n(ω)]/R_n. The inverse DFT of M_n(ω) provides the required complex temporal modulations, m_n(t). To compute A(ω), the entire temporal waveform, a(t), must be specified a priori. Adding temporal slicing prior to spectral slicing can enable continuous computation of m_n(t) [10,11]. The shape of M_n(ω) follows S_n(ω)/H_n(ω) which implies that adjusting the slice filters creates many different sets of m_n(t) that still produce the target waveform. Detailed studies of the algorithm, and how to optimize its parameters such as the shape of S_n(ω), are described in [10,11]. Note that coherent wavelength-division-multiplexing [13] and optical orthogonal-frequency-division-multiplexing [14,15] systems share a similar transmitter structure to dynamic-OAWG, but often use a star coupler rather than a gapless spectral multiplexer to combine the channels. Compared to a gapless multiplexer, a star coupler requires no pre-emphasis, yet it is impractical to use in large channel dynamic-OAWG systems due to increased insertion loss and decreased signal-to-noise ratio.

Figures 1(d), 1(e) show how to apply m_n(t) to a comb line using the indicated drive signals for the quadrature (i.e., I/Q) modulator and polar (i.e., amplitude and phase) modulator. In an I/Q modulator (Fig. 1(d)), the two electrical drive signals that modulate the I and Q components of an optical carrier are simply equal to the real and imaginary parts of m_n(t). They produce a spectral slice centered on a comb line, and it has twice the optical bandwidth as the electrical modulation bandwidth [16] (i.e., positive and negative with respect to the carrier). In contrast, Fig. 1(e) shows a polar modulator, which is less efficient for producing any arbitrary spectral slice because it maps electrical signals to the magnitude and the phase of an optical carrier. In most cases, the driving signals are more complicated and not bandlimited. For example, frequency shifting a comb line requires endless, or modulo-2π, phase modulation.

3. Two-line dynamic-OAWG experiment

Figure 2(a) shows an experimental demonstration of dynamic-OAWG involving two spectral slices. The essential components of this experiment are the I/Q modulators and a spectral multiplexer with overlapping passbands. The two line, 10 GHz OFC generator (OFCG) is a single-frequency laser (100 kHz linewidth) and a high extinction-ratio Mach-Zehnder modulator biased at its null point, and driven by a 4.74 GHz sine wave (i.e., double-sideband, suppressed-carrier modulation). Figure 2(b) shows the transmission of the delay interferometer which functions as the spectral demultiplexer (adjacent passband crosstalk is less than −25 dB). An electronic arbitrary waveform generator (eAWG) with two independent 12 GS/s outputs produces the temporal I/Q signals, I(t) and Q(t), that are split, amplified by linear RF drivers (30 kHz to 5.5 GHz), and drive the I and Q ports of the two I/Q modulators (4 signals total). This scheme produces slices with 11 GHz of optical bandwidth. The eAWG produces a periodic waveform repeating every 16.67 ns (i.e., 60 MHz repetition rate). To ensure independent modulation of each comb line, modulations on line 2 are delayed by 8 ns from those on line 1. In this time-interleaving scheme, the first 8 ns of I(t) are designed for line 1 and the second 8 ns of I(t) are designed for line 2. The spectral multiplexer is a custom 100 channel × 10 GHz silica AWG with thermal-optic phase shifters on each array arm. Applying a quadratic phase profile across the array arms purposely broadens the AWG passbands. Figure 2(c) shows the measured transmission which has an adjacent channel crossing point of −6 dB, −17 dB of adjacent passband crosstalk, and −30 dB of background crosstalk. The extinction ratio of the waveform shaper, which is primarily related to the extinction ratio of I/Q modulators, is 43 dB (−7 dBm to −50 dBm).

 

Fig. 2 (a) Two line dynamic-OAWG experimental arrangement. Measured transmission of the (b) deinterleaver and (c) broadened AWG multiplexer. (d) Heterodyne measurement technique. AOM: Acousto-optic modulator. IQM: I/Q modulator.

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Figure 2(d) shows the heterodyne measurement technique used to simultaneously measure the full field (i.e., amplitude and phase) of the three separate outputs, A, B, and C [17]. The average amplitude and phase of a repetitive signal, s(t), is measured against a reference line, r(t) = exp(j2πf_Rt), with approximately 32 GHz of optical measurement bandwidth. Here, s(t) is periodic with a 60 MHz repetition rate. The reference line is generated by frequency shifting the single-frequency laser by 35 MHz using an acousto-optic modulator. The balanced coherent receiver downconverts the signal comb to baseband and the detected photocurrent is digitized at 50 GS/s with 16 GHz of electrical bandwidth for 4 µs. The electrical spectrum (Fig. 2(d), bottom right) contains two sets of spectral comb lines with 60 MHz spacing: one set has frequencies greater than f_R with a 25 MHz offset from dc, and the other set has frequencies less than f_R with a 35 MHz offset from dc. The amplitude and phase of each comb line of S(ω) is selected from the electrical spectrum to recover the optical spectrum. Picking the peak value of each comb line effectively averages 240 of the 16.67-ns waveforms together (i.e., 4 µs averaging time). This heterodyne technique provides 32-GHz bandwidth, full-field measurements at a 15-Hz update rate (limited by computer processing time). Other techniques, such as optical arbitrary waveform measurement (OAWM) [18], are necessary for characterization of terahertz-bandwidth waveforms in real-time. Note that the fiber-pigtailed components between the deinterleaver and the AWG are susceptible to environmental fluctuations that cause a slowly varying phase drift between the two spectral slices which are not present in integrated devices. Therefore, we only show measurements with the correct relative phase which constitute approximately 8% of the measurements in the two-line experiment, and 1% of the measurements in the three-line experiment.

Figure 3 shows the measured results of a 20 GHz target waveform with a 6 ns record length consisting of a transform-limited pulse, a pulse with cubic spectral phase, and a pulse with quadratic spectral phase each with a 20th-order super-Gaussian spectral amplitude. The waveform has no resemblance to the input OFC; the waveform’s spectrum is continuous, without dips, and the temporal domain waveform periodicity is unrelated to the OFC period. Figure 3(a) shows the full time record (16.67 ns) which contains the fully specified arbitrary waveform in the first 8 ns. Due to the time multiplexing scheme used to provide the four signals to the two I/Q modulators, the second 8 ns contains another waveform that cannot be independently defined. Figures 3(b), 3(c) show the transform-limited pulse generated with and without applying pre-emphasis for the spectral multiplexer. With pre-emphasis for H_n(ω), the spectrum is flat and the time-domain ripple is reduced. Figure 3(d) shows the measured temporal modulations, m_n(t) at points A and B that produce the two spectral slices shown in Fig. 3(e). The 1 GHz overlap of the modulated lines results from using 11-GHz wide spectral slice filters. Figure 3(f) shows that the measured waveform is a high-fidelity reproduction of the target waveform. Two figures of merit help quantify the fidelity of the generated waveforms, a normalized average energy error (NEE) [11] which represents a mean-squared error or total signal-to-noise ratio (SNR) and the normalized maximum energy error (MEE) that expresses the single largest deviation from the target waveform. The NEE is defined as the ratio between the error waveform’s [i.e., e(t) = d(t) − a(t)] energy and the energy in the target waveform, a(t), and the MEE is equal to the ratio between the peak energy of the error waveform and the peak energy of the target waveform. The waveform in Fig. 3(f) has an NEE less than 2% and a MEE of 4%. This level of accuracy is achieved using a thorough calibration of the waveform shaper components and it does not rely on any iterative algorithms or feedback. Waveform fidelity is increased by using feedback from this measurement to calculate a new set of modulations and Figs. 3(g), 3(h) shows the resulting waveform with NEE less than 0.6% and a MEE of 0.4%.

 

Fig. 3 Measurement results for the two line dynamic-OAWG experiments. (a) The entire 16 ns time record. Boxed area indicates the specified arbitrary waveform. (b,c) Isolation of the transform limited pulse with pre-emphasis (solid lines) and without pre-emphasis (dashed lines) for multiplexer filtering, H_n(ω). (d) Measured modulations, m_n(t) at points (A) and (B) in Fig. 2(a). Comparison of the generated arbitrary waveform to the target waveform (grey) with (f) no iterations, and then (g,h) one iteration.

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We choose to quantify fidelity by using metrics which analyze the energy in e(t) (i.e., NEE and MEE) because they include both phase and amplitude errors between a(t) and d(t) in a single number. Alternatively, phase and intensity errors of the generated waveforms could be calculated separately. For example, in Fig. 3(f) the maximum intensity error (MIE) between a(t) and d(t) (i.e., |d(t)|² − |a(t)|²) compared to the peak intensity of a(t) is ~6% which is greater than the MEE of 0.4%. For other cases, when there is a mismatch in phase between a(t) and d(t) but very small intensity errors, the MEE is larger than the MIE. Additionally, phase and intensity metrics are more dependent on the waveform shape than NEE and MEE and are not as effective for comparing the fidelity of different waveforms.

Figure 4 shows additional examples of dynamic-OAWG waveform generation to further emphasize that the shaped waveforms are arbitrary and do not resemble, or have any relation to, the periodicity to the input OFC. Figure 4(a) shows a waveform with quadratic spectral phase equivalent to a linear frequency chirp of −2.5 GHz/ns. The amount of quadratic spectral phase is large enough that each spectral component maps to a different temporal location. The generated waveform is a close match with the target (NEE = 0.7%, MEE = 4.7%) and the transition between the two spectral slices is nearly seamless. Figure 4(b) depicts a 300-ps spaced pulse train generated from an input OFC with a 105.5-ps period (NEE = 1.5%, MEE = 1.7%).

 

Fig. 4 Additional dynamic-OAWG measurements. Target waveforms are grey. (a) Waveform with a −2.5 GHz/ns linear frequency chirp (20 GHz change over 8 ns). (b) Train of nine transform-limited pulses with a 300-ps spacing.

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4. Three-line dynamic-OAWG experiment

Demonstrating true scalability to THz-bandwidth waveforms requires 100 I/Q modulators at a 10 GHz spacing, and a scheme to maintain the coherence between the slices either through integration, active stabilization, or a common-mode setup (i.e., a bulk-optics shaper). To demonstrate the bandwidth scalability of dynamic-OAWG, Fig. 5(a) shows a three-line experiment, which is an extension of the two-line experiment. The OFCG works by applying both strong amplitude and phase modulation to a single-frequency laser to generate a 10 GHz OFC with three lines of equal amplitude and approximately seven lines within −10 dB [19]. The deinterleaver splits the even and odd comb lines to the upper and lower channels, respectively. A time-interleaving scheme similar to the one described previously, applies six independent modulations (corresponding to three sets of I and three sets of Q modulations) to generate three unique spectral slices. The upper I/Q modulator generates the center slice from the center even comb line, which is directed to input 2 of the AWG. The lower I/Q modulator generates slice 1 and slice 3, and a 1 × 2 splitter is used to direct them to input 1 and input 3 of the AWG multiplexer. At input 1 to the AWG, slice 1 is passed to the output and slice 3 is rejected with 30 dB of extinction [see AWG transmission in Fig. 2(c)]. Likewise, at input 3 of the AWG, slice 3 is passed and slice 1 is rejected.

 

Fig. 5 (a) Three line dynamic-OAWG experiment. Dashed rectangles indicate the slices selected by the AWG. Shaped three-line waveform and comparison to target (grey) in the (b) temporal and (c) spectral domain.

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Figures 5(b), 5(c) show high-fidelity generation of the target waveform. The waveform has approximately 30 GHz of optical bandwidth, 4 ns duration, and an NEE of 4% and a MEE of 4.7%. The greater error when compared to the two-line experiment is primarily attributed to the difficulty in aligning the constant spectral phase between the three slices. For example, the phase of slice 3 is slightly offset from the target phase [see the phase on the right side of Fig. 5(c)]. The waveform does not show degradation in fidelity due to the multiplexer filtering.

5. Conclusions

This experimental demonstration validates the dynamic-OAWG concepts and algorithms, which enable generation of continuous, high-fidelity waveforms in a bandwidth scalable fashion. Scaling to four spectral slices and beyond in a fiber-pigtailed configuration is difficult due to the complexities of coherently combining multiple spectral slices with a stable phase relationship. Implementing dynamic-OAWG in an integrated platform overcomes this issue; however, integration has many challenges including the development of an array of high-speed modulators, and individually accessing the modulators with low electrical crosstalk.

Dynamic-OAWG in conjunction with dynamic-OAWM [18], an analogous counterpart to OAWG in which a signal-spectrum is measured in slices against a reference comb, is a bandwidth scalable system for continuous generation and detection of arbitrary waveforms.