Analysis of the effects of jitter, relative intensity noise, and nonlinearity on a photonic digital-to-analog converter based on optical Nyquist pulse synthesis

Christian Kress; Meysam Bahmanian; Tobias Schwabe; J. Christoph Scheytt

doi:10.1364/OE.427424

1. Introduction

In the worldwide digital communication networks the demand for data transmission capacity is continuously increasing. Quite often it is the bandwidth and the resolution of the data converters, i.e. the digital-to-analog converters (DAC) and the analog-to-digital converters which limit the achievable link capacity. Broadband electronic DACs can be implemented using binary and unary weighted current switches [1], time-interleaving techniques [2] [3], with state-of-the-art performance of 4.1 effective number of bits (ENOB) at an analog bandwidth of 58.6 GHz [4], and digital-bandwidth-interleaving, with state-of-the-art performance of less than 3 ENOB at an analog bandwidth of 100 GHz [5]. Photonic DACs have the potential to outperform electronic DACs, especially in terms of bandwidth [6]. Furthermore, photonic integration using silicon photonics technology enables the integration of photonic and electronic circuits on a single chip which allows to combine the advantages of electronic and photonic signal processing and greatly reduces system complexity, size, and cost.

This has motivated research into different electronic-photonic DAC concepts. Several groups have published DACs with segmented Mach-Zehnder modulators (MZM) with binary electronic drivers [7] [8] [9]. However these concepts are limited by the electro-optical bandwidth of integrated MZMs and the electronic drivers and, hence, don‘t provide significant speed improvements compared to electronic DACs. Furthermore, segmented MZMs provide quite limited resolution of the photonic DAC. Others have presented photonic DACs using incoherent addition of multiple, digitally modulated MZMs from either one laser diode, using suitable splitting and attenuation in each MZM branch [10] or using multiple laser diodes with specific powers and a wavelength division multiplexer [11]. This method has advantages in jitter and resistance in electro-magnetic interferences, but is similarly limited by the utilized MZM’s electro-optical bandwidths. The best reported ENOB for this method was only 3.55 at a bitrate of 12.5 Gb/s [12]. Another method is optical pulse generation with time-interleaved MZMs, being driven by electronic DACs allowing to effectively add the individual DAC bandwidths. One approach uses Mode-locked lasers (MLLs) to generate sharp, low-jitter amplitude-modulated pulses for signal synthesis [13][14]. This DAC architecture increases complexity, because precise optical filtering of the ultra-broadband MLL pulses is required to generate the DAC output signal. Furthermore, to this date, low-jitter MLLs have not been integrated in silicon photonics technology, which increases system complexity and cost significantly.

Another promising approach was recently introduced as Nyquist pulse synthesizer [15][16], using optical time-interleaving and Nyquist pulse synthesis from continuous-wave laser diodes (CW-LD). This method allows to achieve precise Nyquist pulse properties without filtering as the frequency comb is generated from an electrical signal generator (SG) with a fixed frequency applied to a MZM under certain bias conditions [17]. Another advantage is that the optical output signal bandwidth is up to three times higher than the applied electrical frequency [18]. Furthermore, the MZM can be operated well beyond its electro-optical 3 dB-bandwidth for the purpose of pulse generation as long as the optical output signal exhibits sufficient power [19]. Using ultra-broadband MZMs, e.g. in LNOI technology [20], will potentially allow for sampling rates beyond 300 GSa/s and continuous optical bandwidths above 300 GHz without frequency interleaving.

In this contribution, we analyze the proposed Nyquist pulse synthesizer DAC with regards to relative intensity noise (RIN) of the utilized laser diode, phase noise of the electrical signal generator, and the influence of unwanted side bands resulting from the MZM’s non-linear transfer function. A formula predicting the effective number of bits (ENOB) for the proposed photonic Nyquist pulse synthesizer DAC is derived for the first time.

2. System analysis

2.1 System and ideal periodic Nyquist pulse description

Nyquist pulses are widely used in communication and signal processing applications due to their inherent properties in time and frequency domain. In the frequency domain, a sinc-shaped Nyquist pulse is a rectangular function. If this rectangular function is multiplied with a frequency comb, the sinc-shaped pulses are convoluted with a Dirac Delta sequence. Since sinc-shaped pulses possess the property of zero intersymbol-interference (ISI), the different time shifted copies of the sinc-shaped pulses do not interfere with each other and the result is a perfect sinc-pulse sequence [21], [22]. These sinc-pulse sequences are called periodic Nyquist pulses in the following. The electric field amplitude of an ideal optical Nyquist comb with N numbers of lines can be mathematically written as

(1)$$E_{Nyquist}(t,N)= E_0\sum_{n={-}\frac{N-1}{2}}^{\frac{N-1}{2}} \frac{e^{i 2\pi \left(f_c+n\Delta f\right)t+i\phi}}{N} $$

(2)$$ = E_0 \frac{\sin(\pi N\Delta f t) }{N \sin(\pi \Delta f t)}e^{i 2\pi f_c t+i\phi} $$

where $f_c$ and $\Delta f$ are the optical carrier and the frequency offset between the lines. The peak amplitude is normalized to $E_0$ [21]. The bandwidth of the centered Nyquist pulse for a fixed frequency offset scales with N (s. Figure 1(b)).

Fig. 1. (a) Optical Nyquist pulse synthesizer system: A continuous-wave laser diode (CW-LD) launches light into a MZM. A signal generator (SG) applies a high-frequency sinusoidal to the MZM, which generates a precise sinc-sequence under certain bias and RF power conditions. The sinc-sequence is splitted into N branches and modulated with electro-optical systems with N times less bandwidth demand. Suitable delays and Nyquist properties allow for perfect, time-aligned combination of the branches. (b) Normalized periodic Nyquist sinc-sequence in respect of $T=\frac {1}{\Delta f}$. The bandwidth of the (isolated) Nyquist sinc-sequence scales with the number of comb lines N. (c) Exemplary visualization of a Nyquist pulse synthesizer system with $N=5$ arms. The different color codes represent the respective signal in each arm. The black curve depicts the superposition of those signals.

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Periodic Nyquist pulses allow for a time-interleaving system as proposed in [15]. A continuous-wave laser diode emits light to a MZM. A signal generator modulates the laser line and, under certain bias and drive conditions, generates periodic Nyquist pulses (s. sec. 2.2). The periodic Nyquist pulse is then splitted into N arms. The pulses of each arm needs to be delayed by $kT/N$ with $k\in \{ 0,1,\dots ,N-1\}$ progressively from top to bottom regarding Fig. 1(a), where $T=1/\Delta f$ is the repetition period coming from the electrical signal from the SG. A weighting in each arm is performed by variable attenuators or electro-optic modulators. For this purpose electronic DACs need to be used. Subsequently, the suitably delayed signals of each branch are superposed using combiners. In Fig. 1(a) optical delays are used while in [15] electrical phase shifters are proposed for the Nyquist pulse generation. Although the system principles are identical, the advantage of the system in 1(a) is that Nyquist pulse generation has to be performed only once, thus simplifying the overall system. Furthermore, from a standpoint of chip integration controlling the biasing of multiple MZMs and electronic phase shifters adds complexity and is more prone to cross-talk. The proposed Nyquist pulse synthesizer allows to add the effective bandwidths of the electronic DACs, whose individual bandwidths only need to be $\Delta f$. Then, the total bandwidth of the system amounts to $N\Delta f$.

2.2 Nyquist Comb Generation in Mach-Zehnder Modulators

A single Mach-Zehnder modulator can be used to generate a three-line Nyquist comb by setting a suitable input voltage amplitude and bias condition of the modulator [22]. The output characteristic for the electrical field vector in a push-pull driven MZM can be written as

(3)$$\underline{E}_{\textrm{out}}=\underline{E}_0 \cos\Big(\frac{\pi}{V_\pi}v_i +\Phi_{\textrm{MZM}}\Big)$$

where $\underline {E}_0 =E_0 e^{i(\omega _c t + \phi _c)}$ is the normalized carrier, $\omega _c$ the optical carrier frequency, $\phi _c$ the (random) optical phase, $V_\pi$ the required voltage to achieve a $\pi$ shift in the MZM, and $\Phi _{\textrm {MZM}}$ the accumulation of all influences and adjustments regarding the MZM’s phase. When applying a sinusoidal input to the MZM $v_i=V_{S}\cdot \sin (\omega _e t +\phi _e)$, the real part of the MZM’s output can be written as:

(4)$$\begin{aligned} E_{\textrm{out}}&=E_0 \Bigg[ \cos(\Phi_{\textrm{MZM}}) J_0(\alpha)\\ & + 2\sum_{k=1}^\infty \cos(\Phi_{\textrm{MZM}}) J_{2k}(\alpha) \cos{\bigg(}2k (\omega_e t + \phi_e ){\bigg)}\\ &\left. - 2\sum_{k=1}^\infty \sin(\Phi_{\textrm{MZM}}) J_{2k-1}(\alpha) \sin{\bigg(}(2k-1) (\omega_e t + \phi_e ){\bigg)} \right] \end{aligned}$$

with the normalized amplitude $\alpha =\frac {\pi }{V_\pi }V_{S}$ and $J_n$ the Bessel function of n-th order. In order to generate a three-line comb one has to find suitable $\Phi$ and $\alpha$ so that the amplitudes at the carrier $\omega _c$ and first order harmonics $\omega _c \pm \omega _e$ are equal while the higher order harmonics need to be respectively small, ideally zero. In order to make the amplitudes of carrier and first harmonics equal, the bias phase $\Phi$ needs to be set to

(5)$$\Phi_{\textrm{MZM}}=\tan^{{-}1} \left( \frac{J_0(\alpha)}{J_1(\alpha)} \right).$$

This optical phase adjustment can be achieved in a LiNbO3 MZM by means of applying a differential voltage to the electrodes. For silicon MZMs it relies on phase shifters based on the plasma dispersion or temperature effect. There is a trade-off in finding the a suitable amplitude $V_S$: Too small $V_S$ results in a too small overall power of the periodic Nyquist pulse. But the higher $V_S$, the more power is transferred to the higher order harmonics, which deteriorate the Nyquist pulse waveform, loosing Nyquist properties, which are needed for the time-interleaved superposition of signals in the branches of the optical Nyquist pulse synthesizer (s. Figure 1(a) and (c)). As metric for the quality of the MZM-generated 3 line-comb the signal-to-distortion-ratio (SDR) can be used:

(6)$$ SDR_{\textrm{Nyquist}}= \dfrac{(\cos(\Phi)J_0(\alpha))^2+2 (\sin(\Phi)J_1(\alpha))^2}{\mathop{\sum}\limits_{\substack{k={-}\infty \\ k\neq 0,1}}^\infty (\sin(\Phi)J_{2k-1}(\alpha))^2 + \mathop{\sum}\limits_{\substack{k={-}\infty \\ k\neq 0}}^\infty (\cos(\Phi)J_{2k}(\alpha))^2} $$

where the weights of the carrier and 1st order harmonics contribute to the signal power and the amplitudes of higher order harmonics are taken into account for the distortions. Although the periodic Nyquist pulses itself are not the signal of interest in a DAC configuration as described in Fig. 1(a), the harmonics deteriorate the signal to be synthesized ($s_1,\dots ,s_N$) through convolution in the frequency domain, and thereby will affect the achievable ENOB. The ENOB depending on SDR can be calculated as $\textrm {ENOB(SDR)}=\frac {10\log (\textrm {SDR})-1.77}{6.02}$. Figure 2(a) shows that the ENOB limitation by harmonics is severe, even when neglecting noise and other undesirable effects. While additional reasons will be discussed in following paragraphs, the influence of the harmonics can be concluded as a major degradation factor.

Fig. 2. (a) Signal-to-distortion ratio of 3-line Nyquist pulse generated in a MZM by a sinusoidal input without filtering. (b) Visualization of the achievable synthesis signal power: The power of a 3-line periodic Nyquist pulse (black, dotted) is evenly splitted into 3 arms, whereas the pulses are suitably delayed to increase the sampling rate (red,blue). The signal power for the synthesis is therby reduced by factor 3.

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2.3 Synthesizer signal power

The optical source for periodic Nyquist pulses is a CW-LD having a laser power of $P_{CW}$. As indicated in Fig. 2(b), the Nyquist pulse intensity is scaled to $P_{CW}$ as peak value. However, in the synthesizer system, where the periodic Nyquist pulse in split into N arms, the power will be evenly distributed. This will reduce the maximum signal power in each branch to $P_{CW}/N$. Figure 2(b) visualizes that argument for a system with $N=3$ branches. In a real system, additional losses have to considered like MZM insertion loss, which typically ranges from 3-8 dB, optical coupling losses, if integrated MZMs are used, and excess losses from splitters and combiners. Although the optical signals can be amplified, the signal-to-noise-and-distortion (SINAD) ratio will not improve. It was shown in Fig. 2(a) that SDR increases for small $\alpha$, but it is clear that maximum achievable signal amplitude scales with the nominator of Eq. (6). Hence, when other signal degradation effects are taken into account, which are discussed in the following, decreasing $\alpha$ doesn‘t necessarily improve the ENOB. Although the signal power of the synthesizer system is distributed into N arms, the SINAD does not degrade because the powers of all deterioration effects will be as well evenly split. Therefore, the SINAD is the same before splitting, in each arm, and as well after combining as the power is contained.

2.4 Phase noise transfer in MZMs

Equation (4) indicates that the electrical phase $\phi _e$ of the input signal is simply passed on to the Bessel-weighted output of the optical signal. Hence, the phase noise of the signal generator is likewise passed on to the optical signal $E_{out}$. This is true for the optical intensity and therefore for the photo detector current as well:

(7)$$\begin{aligned} I &= R P_{out} = R \left\| E_{out}\right\|^2\\ &= \frac{E_0^2}{2}\Bigg[ 1 + \cos(2\Phi_{\textrm{MZM}})J_0(2\alpha))\\ & + 2\sum_{k=1}^\infty \cos(2\Phi_{\textrm{MZM}}) J_{2k}(2\alpha) \cos{\bigg(}2k (\omega_e t + \phi_e ){\bigg)}\\ & - 2\sum_{k=1}^\infty \sin(2\Phi_{\textrm{MZM}}) J_{2k-1}(2\alpha) \sin{\bigg(}(2k-1) (\omega_e t + \phi_e ){\bigg)} \Bigg] \end{aligned}$$

We have also proven this experimentally by comparing the phase noise of our signal generator (Anritsu MG3694B) and the phase noise when applying the signal generator to the MZM, followed by a detection process by a 70 GHz photodiode and electrical amplification (SHF S807 C), which was necessary for the phase noise analyzer‘s sensitivity (Anapico APPH20G) at 10 GHz input frequency (see Fig. 3). We have used a LiNbO3 and an integrated silicon MZM with linear, segmented drivers on-chip to evaluate the additional deterioration effects of the silicon modulator. For comparison, we have performed this experiment with a second signal source (Keysight AWG M8194A).

Fig. 3. Phase noise measurement of a signal generator (SG) / arbitrary signal generator (AWG): simple electrical (blue) and with transition to optical using continuous-wave laser, Mach-Zehnder modulator (MZM) and photodiode (PD). The electrical signal was amplified (AMP) and it’s phase noise measured via phase noise analyzer (PNA).

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Figure 4(a) indicates that the phase noise through the optical system ideally follows the signal generator‘s phase noise up to an offset frequency of 1 MHz, which is the dominant part when calculating the jitter in the time domain. Above 1 MHz the optical systems (LiNbO3 and silicon MZMs) reach a plateau which was identified as shot noise limit of the photodiode. We see a little bit worse performance for the silicon MZM, as we had higher optical coupling losses in the integrated chip. There is no plateau for the AWG measurement, as we have boosted the AWG signal power by an electrical amplifier. It is worth noting that the laser linewidth, although it contributes to the laser phase noise, can be neglected as it vanishes in a detection process. In contrast to laser phase noise, the phase noise of the signal generator transforms into amplitude noise by means of the MZM.

Fig. 4. (a) Phase noise measurement comparing a signal generator (SG) / arbitrary signal generator (AWG) and transfer in the optical system at 10 GHz input frequency. (b) ENOB limitation depending on Nyquist pulse signal generator jitter.

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2.5 Sampling error due to jitter

The power of a random jitter process can be modeled as a mean-free Gaussian-distribution with variance $\sigma _{t_j}$. In a sampling process the error caused by jitter of the clock signal can be estimated using a first-order Taylor series approximation, leading to a analytical expression of the signal-to-noise ratio (SNR) of

(8)$$\textrm{SNR}_{\textrm{X,Jit}} = 10\log \left(\frac{\rho_{XX}(0)}{-\rho_{XX}^{\prime \prime }(0)\cdot \rho_{t_jt_j}(0)}\right)$$

where $\rho$ denotes the auto-correlation function, $X$ and $t_j$ referring to the data signal and jitter distribution function respectively [23]. Although this expression was derived for analog-digital converters, the linear approximation holds true for digital-analog conversion as well. When we assume $X$ to be a sinusoidal signal with frequency $f_X$, Eq. (8) can be derived as:

(9)$$\begin{aligned}\textrm{SNR}_{\textrm{sin,Jit}}&=10\log\left( \frac{1}{(2\pi f_X)^2 \cdot \sigma_{t_j}^2} \right)\\ &= 20\log\left( \frac{1}{2\pi f_{X} \cdot \sigma_{t_j}} \right). \end{aligned}$$

The absolute jitter of a signal is related to the phase noise by integrating it’s spectrum between certain frequency limits:

(10)$$\sigma_{tj} = \frac{\sqrt{2{f_{1}}^{f_{2}}S_{PS}(f) df }}{2\pi f}$$

Since many systems can reconstruct a carrier up to kHz-range, we have integrated the measured phase noise (s. Figure 4(a)) from 1 kHz to 100 MHz. The calculated jitter rms values for the signal generator have been 32 fs, 41 fs and 60 fs for just the SG, the SG and a LiNbO3 MZM, and the SG and a silicon MZM respectively. For the AWG we have measured 156 fs, 160 fs, and 162 fs for the respective cases. We can conclude that there is a slight increase of the jitter in respect to the optical systems. When there is no detection limit (as for the measurement with the AWG), the main contribution of the jitter is the signal source, which is the main conclusion here. Using Eq. (8) and feeding it with the calculated jitter values leads to an estimation of the achievable ENOB using periodic Nyquist pulses with $N=3$ lines, which means 3 times the bandwidth of the regular signal generator output sinusoid (s. Figure 4(b)).

2.6 Relative intensity noise

Another limiting factor in the optical signal quality is relative-intensity-noise (RIN) of the optical source (s. Figure 1(a)). RIN comprises random fluctuations in the intensity of a laser. RIN is usually defined as a mean square power fluctuation:

(11)$$\overline{ \delta P_o^2}(t)= \int_0^\infty S_{\textrm{RIN}}(f)df$$

with $S_{\textrm {RIN}}$ being it’s one-sided spectral density [24]. When applying an optical bandpass filter centered at the carrier $f_c$, we get approximate integration limits $B=f_{max}-f_{min}$, with $f_{max,min}=f_c \pm B/2$, and calculate the RIN as $\overline { \delta P_o^2}(t)= B S_{\textrm {RIN}}$. The bandwidth of the filter should obviously not be smaller that the bandwidth of the generated comb for the Nyquist pulses: $N\Delta f_{SG}$, when the filter is placed at the output of the optical DAC.

3. Results and discussion

The above discussed effects of MZM distortion, signal generator jitter and RIN on the DAC precision can be merged into a formula to calculate the signal-to-noise-and-distortion (SINAD) which is extended compared to the regular case by the jitter error:

(12)$$\textrm{SINAD}(f)=\frac{\textrm{Signal}(f)}{\textrm{Jitter}(f) + \textrm{RIN}+\textrm{Distortion}(f) }$$

For the signal power we have calculated $\textrm {Signal}(f)=P_{CW} H(f) G(V_S)$, where the maximum signal power is normalized to $P_{CW}$ with as described in section 2.3, $H(f)$ the bandpass filter power transfer function of order 20 and $G(V_S)$ a correction function for the optical peak power depending on the modulation amplitude $V_S$ in accordance to the nominator of Eq. (6). The jitter error can be derived in accordance to the denominator of Eq. (8) for the periodic Nyquist pulse as $\textrm {Jitter}(f)=\frac {2}{3} P_{CW} (2\pi f)^2 \sigma _{t_j}^2$ for a three-line spectral comb. The distortions stemming from the MZM non-linearity can be defined as the denominator in Eq. (6) besides that the weights of the harmonics are additionally multiplied with the suitable values from the bandpass filter function $H(f)$ for the respective frequencies. The RIN is defined as $\textrm {RIN}=\int _0^\infty S_{RIN}(f) H(f) df$, also shaped by the optical bandpass filter. Lastly, we use the SINAD to calculate the ENOB for sinusoidal signals:

(13)$$\textrm{ENOB}(f)=\frac{10\log \left(\textrm{SINAD}(f) \right)-1.77}{6.02}.$$

We have compared the outcomes from the analytical formula with system simulations using Lumerical Interconnect. The analytical and simulation results are in good agreement (s. Figure 5). To differentiate the effects, the graphs contain ENOB calculations of each single deterioration effect, reducing denominator Eq. (12) to a respective deterioration. In the example graphs, we have assumed filter bandwidths that should be easily available as standard lab equipment (0.3 & 1 nm). The jitter values of 53 fs and 160 fs were taken from the phase noise measurement of the system with a signal generator and an arbitrary signal generator respectively (s. section 2.5). Several statements can be concluded from the analysis and several cases: For poor lasers, the RIN is a dominant factor limiting the ENOB. Commercial lasers, having a RIN of -145 dBc, without any narrow filters are limiting the ENOB to around 3.5 bits (s. Figure 5(b)). In order to achieve useful ENOB, the laser RIN should be as small as possible by either using very advanced lasers or inserting a very narrow optical filter after the laser (s. Figure 5(a)). For comparison, RIN of -165 dBc [25], even using a broadband 1 nm band pass filter allows for an ENOB of 7. In all cases the modulation amplitude $V_s$ was maximized so that the deterioration of the distortion was slightly less than for the RIN and the total ENOB was maximized. In order to keep the distortion by the harmonics the least contributor, the signal amplitude should be scaled to less than 0.15 $V_\pi$ of the modulator. When $V_s > 0.36 \ V_\pi$, without any additional counter measures, the distortion will limit the ENOB to 4 for any frequency. The rising of the SDR caused by distortions is due to the fact that the harmonics move outside the optical band pass cut-off frequency when increasing the fundamental frequency. Hence, the power in the higher order harmonics decreases and the SDR increases. The jitter error becomes dominant around 10 GHz for the AWG and around 20 GHz for the SG for this setup. Hence, when focusing on high bandwidth signal synthesizers, improving the phase noise of the electrical source is paramount. In Fig. 5(e) a very low phase noise signal generator (LPN-SG) with 4 fs rms jitter has been used [26]. In that scenario, the electrical signal generator is no longer the limiting factor, allowing for $\ge$ 6 ENOB for the displayed 200 GHz. Figure 5(e) also indicates that the ENOB could go up to 9.6 over a 50 GHz span or up to 8.5 over 100 GHz span when filtering the laser with an additional narrow filter between laser and MZM. The analysis shows that a 1 GHz filter would suffice for that purpose.

Fig. 5. ENOB is displayed for single effects as well as the total ENOB, which is compared to simulated value. For comparison, three cases are shown: (a) Lumerical simulation setup: RIN is generated in CW-laser. Freqeuncy sweep is done with the signal generator (SG). Distortions are scaled with the SG output amplitude. A jitter source is inserted, fed by measured values. A band pass filter (BPF) filters RIN and Distortions. SINAD is measured by a optical spectrum analyzer (OSA). An optional BPF (dashed) could decrease the RIN significantly. (b) $\textrm {RIN} = -145$ dBc, $\textrm {BW}_{optical} = 1$ nm, $\alpha = 1.13$, $\sigma _{rms} = 53$ fs (SG). (c) $\textrm {RIN} = -165$ dBc, $\textrm {BW}_{optical} = 0.3$ nm, $\alpha = 0.44$, $\sigma _{rms} = 160$ fs (SG) . (d) $\textrm {RIN} = -165$ dBc, $\textrm {BW}_{optical} = 0.3$ nm, $\alpha = 0.44$, $\sigma _{rms} = 53$ fs (AWG). (e) $\textrm {RIN} = -165$ dBc, $\textrm {BW}_{optical} = 0.3$ nm, $\alpha = 0.53$, $\sigma _{rms} = 4$ fs (LPN-SG).

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As future work, the ENOB prediction can be more refined by considering non-idealities of the utilized electro-optic modulators or variable attenuators, which are used in the parallel branches of the Nyquist pulse synthesizer DAC system (s. Figure 1(a)). In addition, since perfect reconstruction, as property of the Nyquist pulses, is only possible at the zero-crossings, errors of the delays in the parallel branches will be transformed to amplitude errors. Hence, circuits or systems that fine-tune the delay and suitable calibration techniques should be taken into account as well. If the transition from the proposed photonic DAC into electrical domain is desired, additional effects such as photo current shot-noise, absolute optical power, linearity and thermal noise of the photo detector and an optional, subsequent transimpedance amplifier need to be considered.

4. Conclusion

We have derived an analysis of the impairments of a Nyquist pulse synthesizer system by laser RIN, distortion by the MZM non-linearity and electrical signal source jitter. The analysis allows to predict an upper limit on ENOB, which is an important metric for high-speed arbitrary waveform signal generators and DACs. For high ENOB the CW laser signal should exhibit a low RIN and the laser signal should be filtered with a high-Q optical filter. In addition, the frequency synthesizer phase noise should be as low as possible as it is the main limiting factor in high-frequency performance. The best electronic DACs achieve 4.1 ENOB over a frequency range of up to 58.6 GHz without frequency interleaving [4]. In conclusion, we have shown that a photonic DAC based on Nyquist pulse synthesis outperforms state-of-the-art electronic DACs both in bandwidth and ENOB. Our analysis predicts that using an ultra-low phase noise SG [26], a laser source with -165 dBc RIN [25] and a 1 GHz pre-filter enables more than 8 ENOB over more than 100 GHz signal bandwidth.

Funding

Deutsche Forschungsgemeinschaft (403154102).

Disclosures

The authors declare no conflicts of interest.

Data availability

Simulation and measurement data underlying the results presented in this paper may be obtained from the authors upon reasonable request.

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Analysis of the effects of jitter, relative intensity noise, and nonlinearity on a photonic digital-to-analog converter based on optical Nyquist pulse synthesis

Abstract

1. Introduction

2. System analysis

2.1 System and ideal periodic Nyquist pulse description

2.2 Nyquist Comb Generation in Mach-Zehnder Modulators

2.3 Synthesizer signal power

2.4 Phase noise transfer in MZMs

2.5 Sampling error due to jitter

2.6 Relative intensity noise

3. Results and discussion

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Equations (13)

Optics Express