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To reconstruct the optical field, a carrier is necessary to obtain the desired carrier-signal beating term. We denote the carrier and signal field as C and S, respectively. Assuming that the responsivity of the photodiode equals 1 for simplicity, after square-law detection, the received photocurrent I can be expressed as
$$ I = \left| {C + S} \right|^2 \, =\, \left| C \right|^2 \, + \, \left| S \right|^2 \, + \, 2{\mathrm{Re}} [C^ \ast \cdot S] $$ (1) where * stands for conjugation, and Re[·] stands for the real part. For the right-hand side of the above equation, only the last term 2Re[C*·S] is the desired term. Since this term represents the real value, SSB signals with real and imaginary parts satisfying the Hilbert transform can be recovered, while complex-valued DSB signals with no such property cannot be recovered merely via the term 2Re[C*·S].
Figure 1a depicts the structure for the proposed CADD receiver to recover complex-valued DSB signals. The input of the CADD receiver consists of the carrier and the signals, denoted by C + S(t). An optical coupler is utilized to split the input into two paths, with an optical delay of time τ on one path, corresponding to C + S(t - τ). Without loss of generality, we have assumed that the carrier C is a real-valued constant. This is justified, as the optical delay τ is on the order of the baud period, and the carrier phase change would be insignificant with a delay of τ. The delayed path is further split into two branches, one of which is fed into a single-ended photodiode. The photocurrent of the single-ended photodiode R1 is expressed as
$$ \begin{array}{l}R_1 = \left| {C + S(t - \tau )} \right|^2 \, =\, \left| C \right|^2 \, +\, \left| {S(t - \tau )} \right|^2\\ + \, C[S(t - \tau ) + S^ \ast (t - \tau )]\end{array} $$ (2) Fig. 1
a Receiver scheme for CADD; b DSP for OFDM modulated signals using the CADD receiver. Inset (i) is the spectrum of signals fed to the CADD receiver, where S1 and S2 are lower and upper sideband signals, respectively. PD photodiode, BPD balanced photodiode, FFT fast Fourier transform, IFFT inverse fast Fourier transformThe two optical signals at the output of the coupler, C + S(t) and C + S(t - τ), are input into an optical hybrid and then fed into two balanced photodiodes (BPDs). The photocurrents of the two BPDs, I1 and I2, are thus given by
$$ \begin{array}{l}I_1 = 4{\mathrm{Re}} \{ [C + S(t - \tau )]^ \ast \cdot [C + S(t)]\}\\ I_2 = 4{\mathrm{Im}} \{ [C + S(t - \tau )]^ \ast \cdot [C + S(t)]\} \end{array} $$ (3) where Re{·} and Im{·} represent the real and imaginary parts, respectively. It is worth noting that a 3 × 3 coupler can serve the same function as the 90° optical hybrid26 with a lower cost. We reconstruct a complex-valued signal from I1 and I2 as
$$ \begin{array}{l}R_2 = (I_1 + jI_2)/4 = \left| C \right|^2 \, +\, C[S(t) + S^ \ast (t - \tau )]\\ + S(t) \cdot S^ \ast (t - \tau )\end{array} $$ (4) Strictly, C and S should be expressed as $Ce^{j2\pi f_0t}$ and $Se^{j2\pi f_0t}$, where f0 is the carrier frequency. As such, there exists an additional common phase term in R2, which can be easily estimated and compensated with receiver digital signal processing (DSP). We subtract R1 from R2 and obtain
$$ R = R_2 - R_1 = C[S(t) - S(t - \tau )] + S_2 $$ (5) where $S_2 = S(t)S^ \ast (t - \tau ) - \left| {S(t - \tau )} \right|^2$, which is the second-order SSBI. It follows from Eq. (5) that the desired linear term S(t) - S(t - τ) can be expressed as
$$ S(t) - S(t - \tau ) = \left( {R - S_2} \right){\mathrm{/}}C $$ (6) Taking the Fourier transform of Eq. (6), we obtain
$$ \begin{array}{ccccc}\\ S(f) = \left( {1 - e^{j2\pi f\tau }} \right)^{ - 1}{\cal{F}}\{ \left( {R - S_2} \right)/C\} \cr \\ = H(f)^{ - 1}{\cal{F}}\{ \left( {R - S_2} \right)/C\} \\ \end{array} $$ (7) where S(f) is the Fourier transform of S(t), namely, $S(f) = {\mathcal{F}}\{ S(t)\}$. We define the transfer function of the CADD receiver as H(f), which equals 1 - ej2πfτ, in essence, the transfer function of an interferometer with a delay of τ. As shown in Eq. (5), despite SSBI distortions S2, the desired term of $S(t) - S(t - \tau )$ is amplified by the carrier C. Thus, a strong carrier mitigates the effects of SSBI distortions, which in turn enables relatively accurate preliminary symbol decisions using R. Equation (7) is the main formula used to reconstruct S(f). The SSBI term S2 can be reconstructed via the preliminary symbol decision and then subtracted from R iteratively. The preliminary symbol decision is made by setting S2 to zero. Further discussion on the transfer function H(f) can be found in the "Materials and methods" section.
Although the CADD architecture shown in Fig. 1a comprises three photodetectors, the required bandwidth of each one is reduced by half compared to SSB signal detection. For photonics integrated circuits (PICs), such as silicon photonics, the cost of an integrated circuit is mainly dominated by the electrical bandwidth of the circuit; namely, adding two more photodetectors into the PIC would not significantly increase the cost but double the receiver bandwidth would. Additionally, the system performance analysis is based on the optical signal-to-noise ratio (OSNR) sensitivity, and as such, the impact of receiver passive component loss and noise are immaterial and are not considered in this paper.
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Orthogonal frequency division multiplexing (OFDM) modulation is adopted to demonstrate the CADD receiver scheme. To enhance the SE, no cyclic prefix is inserted. As shown on the right-hand side of Eq. (7), transfer function H(f) has a null point at f = 0, and SSBI can be dramatically amplified around zero frequency. Hence, a small frequency gap (e.g., 10% of the signal bandwidth) is inserted in the vicinity of zero frequency. It is noted that such a frequency gap is not implemented to fully accommodate SSBI. In practice, the frequency of the gap for CADD can be merely approximately 10% of the signal bandwidth. For example, the frequency gap Δf can be as narrow as 2.5 GHz for 25-Gbaud signals. The spectrum of signals along with the carrier is shown in insert (i) of Fig. 1b.
As shown in Fig. 1b, DSB signals along with a carrier are fed into the CADD receiver, with the same structure as depicted in Fig. 1a, which outputs the OFDM signal S(f) using Eq. (7) in the frequency domain. To eliminate SSBI S2 in an iterative manner, preliminary symbol decisions are made in the frequency domain for OFDM signals. IFFT is utilized to transform symbol decisions into the time domain signal S(t), and then, SSBI is reconstructed by using the relation $S_2 = S(t)S^ \ast (t - \tau ) - \left| {S(t - \tau )} \right|^2$. Since the output of CADD S(f) is in the frequency domain, FFT is needed to transform SSBI to the frequency domain and then subtract it from S(f). After several iterations (e.g., four iterations), the system performance converges, indicating that the SSBI has been effectively mitigated.
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The gist of the CADD receiver is to eliminate SSBI. As shown in Eqs. (5)-(7), we estimate $S(t) - S(t - \tau )$ via R by assuming that SSBI S2 equals zero for the first iteration. Although SSBI can be reconstructed and then eliminated iteratively, it is highly preferable to suppress SSBI before iterative cancellation. For the CADD receiver, its unique transfer function (which will be discussed further in "Materials and methods") suggests a tactful approach to suppress SSBI by inserting a small frequency gap (e.g., 10% of the signal bandwidth). The benefit of the frequency gap in CADD is twofold: (ⅰ) SSBI, which is generally more severe in the vicinity of the zero frequency region, does not totally overlap with the signal spectrum due to the gap and hence produces less distortion for information-bearing signals; (ⅱ) when the magnitude of the transfer function H(f) is greater than 1, SSBI is suppressed via the transfer function of CADD. Since the transfer function of the CADD receiver H(f) is also a function of optical delay τ, a desired frequency region with suppressed SSBI can be obtained by adjusting the optical delay τ.
To demonstrate the effectiveness of SSBI suppression, we investigate the detection of 25-Gbaud 16QAM OFDM signals using the CADD receiver with a sampling rate of 50 Gsample/s. The optical delay is 50 ps, and the frequency gap is 2.5 GHz. The information-bearing signals occupy the bandwidths of [-13.75 GHz, -1.25 GHz] and [1.25 GHz, 13.75 GHz], indicating a frequency gap of 10% of the signal bandwidth. In Fig. 2a, the green dotted line represents the spectrum of S(t) - S(t - τ), with no distortions due to SSBI. After implementing transfer function H(f) as shown in Eq. (7), the spectrum of recovered signal S(t) is shown as the blue solid line in Fig. 2a. Spectra of SSBI are shown in Fig. 2b. Due to the transfer function of the CADD receiver, SSBI is significantly enhanced at frequencies of 0 and ±20 GHz. Since the null frequency of ±20 GHz is not within the information-bearing signal spectrum, these singularity spikes do not affect information-bearing signals. It is noted that in the frequency regions of [-16.7 GHz, -3.3 GHz] and [3.3 GHz, 16.7 GHz], SSBI can be suppressed by up to 6 dB. In addition, it can be concluded that SSBI suppression corresponds to an interplay between the frequency gap and optical delay, indicating that the CADD receiver can be optimized by inserting a frequency gap and tactfully adjusting the optical delay according to the signal bandwidth.
Fig. 2
a Signal spectra before and after implementing transfer function H(f). b SSBI spectra before and after implementing transfer function H(f)To optimize the optical delay, we keep the frequency gap as a fixed value, which is 2.5 GHz for the 25-Gbaud signals. The carrier-to-signal power ratio (CSPR) is set to 8 dB, four iterations are implemented to cancel SSBI, and the BER as a function of the OSNR is shown in Fig. 3a. Since the carrier is transmitted along with signals, both the carrier and information-bearing signal power are considered as the "signal" power when calculating the OSNR. Among the various optical delay values shown in Fig. 3a, the delay of 60 ps is found to be the optimal.
Fig. 3
BER performance a versus OSNR for various delays and b versus the number of iterations @ a CSPR of 8 dB, an optical delay of 60 ps, and a frequency gap of 10%. Insets are the corresponding constellations for each iteration @ OSNR = 28 dBThe crux of optimizing the optical delay is to fit the information-bearing signals in the frequency region with SSBI suppression where the magnitude of the transfer function H(f) is greater than 1. When the delay is small, it is anticipated that the system performance will improve as the delay increases. This is the nature of differential detection, in which the signal difference from the interferometer is enhanced by using a larger delay. This is also manifested by the fact that the SSBI suppression region moves to a lower frequency when increasing the delay. However, when the interferometer delay becomes excessive, the second null point of the transfer function moves into the signal spectrum, degrading the system performance. As such, there exists an optimal delay for the receiver performance. To obtain the above results, we have used four iterations to mitigate SSBI. Figure 3b shows the BER as a function of iteration number. When the iteration number equals zero, we obtain the preliminary symbol decisions, which are used to reconstruct the SSBI, followed by iteration of SSBI mitigation. When the iteration number is greater than 4, more iterations do not bring about a substantial improvement. As such, in the following results, the iteration number is set to four.
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In addition to the frequency gap and optical delay, the CSPR is another key factor to optimize for the CADD receiver. A high CSPR enlarges the desired term $S(t) - S(t - \tau )$ relative to the SSBI. However, this reduces the effective signal power due to the high carrier power and hence degrades the OSNR. Taking the CSPR into consideration, optimization of the CADD receiver is a three-parameter process involving varying the CSPR, frequency gap and optical delay. To maximize the electrical SE, it is preferable to narrow the frequency gap. For a given frequency gap, we sweep the CSPR from 6 to 14 dB to identify the optimal value. For the 25-GBaud DSB 16QAM signals, a 20% frequency gap indicates that the gap occupies 5 GHz [-2.5 GHz, 2.5 GHz], and signals occupy the frequencies of [-15 GHz, -2.5 GHz] and [2.5 GHz, 15 GHz]. Figure 4a, b depicts the BER as a function of CSPR for the 25-Gbaud 16QAM signals with frequency gaps of 5% and 20%, respectively.
Fig. 4
BER versus CSPR for 25-Gbaud signals a with a 5% frequency gap and b with a 20% frequency gapFor the 5% gap case, at the OSNR of 30 dB, the lowest BER occurs at a CSPR of 9 dB. As the OSNR decreases to 22 dB, the CSPR of 8 dB tends to have a similar performance as the 9-dB CSPR. For the 20% gap case, the optimal CSPR is 7 dB, as shown in Fig. 4b. This phenomenon indicates that the implementation of a frequency gap can relieve the requirement of high carrier power. This is because SSBI does not fully overlap with information-bearing signals, and a high CSPR is not required to obtain a strong replica of signals (e.g., $C[S(t)- S(t- \tau )]$ for CADD receivers).
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To demonstrate the OSNR sensitivity of the CADD receiver with various frequency gaps, we optimize the optical delay and CSPR as discussed above. The frequency gap ranges from 5 to 25% for 25-Gbaud signals. The optimized parameters are listed in Table 1. The step sizes of the optimized optical delay and CSPR are 10 ps and 1 dB, respectively.
Frequency gap (%) Optical delay (ps) CSPR (dB) 5 60 9 10 60 8 15 60 8 20 50 7 25 50 7 Table 1. Optimal delay and CSPR for various frequency gaps
The optimal CSPRs shown in Table 1 generally decrease as the frequency gap becomes wider, which agrees with the analysis previously discussed. Meanwhile, the optical delay decreases from 60 to 50 ps, which is due to the interplay between the frequency gap and SSBI suppression. As we insert a wider gap in the low frequency region, information-bearing signals are pushed into the higher frequency region, and hence, the optical delay should be correspondingly decreased to guarantee that signals are in the SSBI suppressed region. We simulate the transmission of 25-Gbaud OFDM 16QAM DSB signals using our proposed CADD receiver scheme. As a representative SSB case for reference, the performance of the KK receiver is also presented. Considering the requirement of a high oversampling rate for KK receivers27, the sampling rate for the KK receiver is set to 100 Gsample/s, while for the CADD receiver, the sampling rate is 50 Gsample/s. With the same data rate of 100 Gbit/s, the OSNR sensitivities of the SSB case (e.g., KK receiver) and DSB case (e.g., CADD receiver) are presented in Fig. 5 in terms of the BER and mutual information (MI). For the CADD receiver, the OFDM modulation format is adopted. In contrast, a single carrier is adopted for the KK receiver due to the low peak-to-average power ratio (PAPR) of the single-carrier modulation format, which is beneficial for the KK receiver28, and no PAPR reduction technique is employed29. The CSPR of the KK receiver is 6 dB, which is the optimal value, and the corresponding optimal parameters for CADD are listed in Table 1. Aiming to avoid sophisticated wavelength stabilization and control, no optical filters are implemented for the KK and CADD receivers. Figure 5a shows that both the KK and CADD receivers are effective in mitigating SSBI. For the CADD receiver, even with a narrow frequency gap (e.g., merely 5%), the receiver algorithm still works properly. As the frequency gap increases, the CADD receiver can achieve better OSNR sensitivity with the adjustment of the optical delay and CSPR. At the BER threshold of 1 × 10-3, the OSNR sensitivity of the CADD receiver with a 10% gap is 28 dB. By inserting a wider frequency gap, the OSNR sensitivity of CADD is further improved. For example, without sacrificing much of the SE, the OSNR sensitivity of the CADD receiver with a 25% gap is approximately 26 dB. It can be concluded that for CADD receivers, there exists a trade-off between the SE and OSNR sensitivity. In addition to the BER, the mutual information (MI) of the CADD and KK receivers is depicted in Fig. 5b. Since the MI greatly converges when the OSNR is high, the inset of Fig. 5b displays the zoom-in detailed MI at high OSNRs.
Fig. 5
Back-to-back performance of the a OSNR sensitivity and b mutual information of CADD for various frequency gaps. The KK receiver is also included for referenceIt is also worth noting that the transfer function of the CADD receiver is not uniform, leading to the signal-to-noise ratio (SNR) over the signal bandwidth not being uniform. Given that the OFDM modulation format is adopted for CADD receivers, we illustrate the SNR as a function of frequency for each iteration in Fig. 6. Since the frequency gap is 10%, the SNR is not displayed in the region of [-1.25 GHz, 1.25 GHz]. For the preliminary decision (e.g., no iteration is conducted), the SNR in the low frequency region is low, indicating that the SSBI in this region is severe. However, this is a colored-SNR channel; in the SSBI suppressed region, the SNR can be improved by more than 10 dB. After performing several iterations, the SNR gradually improves, and the characteristics of the colored SNR are mitigated. This is because for OFDM signals, SSBI is reconstructed in the time domain, while symbol decisions are made in the frequency domain, revealing the low correlation between the reconstructed SSBI and symbol decisions, and as such, SSBI can be sufficiently eliminated. This phenomenon fundamentally empowers effective SSBI mitigation for CADD receivers. After the fourth iteration, the average SNR is 19.7 dB, with the lowest SNR in the low frequency region of approximately 15 dB. We also observe that when inserting a wider frequency gap (e.g., a 20% gap), the SNR curve over the signal bandwidth is more uniform than that in the 10% gap case shown in Fig. 6.
Architecture of the CADD receiver
Digital signal processing for CADD
System impact of the transfer function
Carrier-to-signal power ratio of CADD
OSNR sensitivity
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After thoroughly studying the system performance of CADD, we are in a good position to reproduce the comparison between different coherent and direct detection schemes presented in ref. 30 using some relevant cost metrics, as shown in Table 2. For a fair comparison, we also assume that all the detection schemes aim to achieve 200 Gb/s per polarization per wavelength at an OSNR of 30 dB. In our view, the optoelectronic bandwidth and whether a coherent laser is required are the two most important contributing factors to the implementation cost of photonic integrated circuits. As shown in Table 2, coherent homodyne detection outperforms all the other modulation formats in the bandwidth requirement. Although it requires twice as many components as coherent heterodyne detection, the reduced electrical bandwidth for homodyne detection is such a predominant advantage that all the field-deployed coherent systems use the homodyne architecture. Similarly, in the direct detection domain, although CADD requires more components, due to the unique capability of detecting DSB signals, the required electrical bandwidth for CADD is reduced by almost half, and therefore, it is greatly positioned to be implemented in photonic integrated circuits. The advantage of CADD over the KK receiver is analogous to that of homodyne over heterodyne receivers in coherent detection. As such, we believe that our proposed receiver architecture opens a new class of direct detection schemes that are suitable for photonic integration analogous to homodyne receivers in coherent detection.
Modulation format BW per ADC (GHz) Requirement of stable lasers Number of ADCs Coherent (homodyne) 9.7 Yes 2 Coherent (heterodyne) 19.4 Yes 1 CADDa 16.0 No 3 KKb 31.6 No 1 Stokesc 25.1 No 3 Gapped SSBd 50.2 No 1 Interleaved SSBe 50.2 No 1 This table is reproduced from ref. 30 and the OSNR is set to 30 dB
aA 10% frequency gap is employed for the CADD receiver, with a CSPR of 8 dB
bThe CSPR is 6 dB for the KK receiver19
cFor the Stokes receiver13, modulated signals are in the X polarization, and the Y polarization is occupied by the carrier, with a CSPR of 0 dB. Since this comparison table is based on single polarization, while both polarizations are loaded with either signals or the carrier for the Stokes receiver, we include a multiplication factor of 2 for the bandwidth. In other words, all the bandwidths should be reduced by half when two polarizations are used to obtain the same net interface rate
dThe frequency gap is as wide as the signal bandwidth, and the CSPR is 0 dB for the gapped SSB scheme15
eOdd-numbered subcarriers are loaded with signals, and even-numbered subcarriers are null. The CSPR is 0 dB for such an interleaved SSB scheme31Table 2. Cost metrics of the 200-Gb/s net interface rate per wavelength per polarization detection system with field recovery
In summary, we have proposed a novel receiver scheme called CADD to recover the field of IQ modulated DSB signals via direct detection. CADD enables digital compensation of chromatic dispersion and almost doubles the electrical SE of KK or IC receivers. The transfer function of the CADD receiver is theoretically analyzed. It is shown that SSBI can be judiciously suppressed by taking advantage of the transfer function and further mitigated via iterative cancellation. Several key parameters, including the optical delay, frequency gap, and CSPR, are discussed and optimized. Additionally, the receiver sensitivity of CADD is presented, showing that the CADD receiver is robust to CD. This is the first realization of the field recovery of complex-valued DSB signals via direct detection with a low receiver bandwidth at almost half of the baud rate.