# Research Progress on Carrier-Free Phase-Retrieval Receivers
## 3 Modified CF-PR Receiver Scheme
### Improve the Performance of the CF-PR with a Reduced Number of Iterations and Better Steady Accuracy
#### 3.1.2 Enhanced CF-PR Algorithms
In order to accomplish the polarization demultiplexing in dual-polarization CF-PR receiver, the scheme of using training sequence to estimate the fiber transfer matrix is proposed in [47], as shown in Figure 12. This scheme uses 300 consecutive training symbols to obtain the fiber transfer matrix \(h=h_{CD}*h_{PM}\)[75; 76; 77; 78], by multiplying the received training symbols with the pseudo inverse of time-aligned transmitted training symbols. By running the phase retrieval algorithm and channel estimation iteratively, the fiber transfer matrix \(h\), containing both the polarization rotation and CD, can be properly recovered. Note that polarization dependent loss and polarization mode dispersion is ignored for simplicity considering the short-reach applications. After removing the CD contribution, \(h_{CD}\), the transfer matrix from polarization rotation can be obtained [43]. After obtaining the transfer matrix, the PR-PR algorithm is used to reconstruct the polarization multiplexed signal in the entire field. Experimental results have shown that the polarization-diversity CF-PR receiver can reconstruct the full field of 30 Gbaud dual-polarization QPSK signals

Figure 6:0 Schematic of the polarization-diversity CF-PR receiver. Figure 6:1 Complexity comparison, in terms of (**a**) the number of adders; and (**b**) number of multiplications. after transmitting over a 520 km SSMF, with BERs of \(0.8\times 10^{-2}\) and \(2.2\times 10^{-3}\) with 10% and 20% pilot symbols, respectively. #### 3.3.2 CF-PR Receiver with Both Polarization and Spatial Mode Diversity

In the absence of any optical carrier, full-field reconstruction of six spatial and polarization modes was achieved using intensity measurements only after 30 km multimode fiber transmission [44]. The three spatial modes are \(LP_{01}\), \(LP_{11a}\), and \(LP_{11b}\), each with two polarizations. Figure 13 shows the schematic diagram of the receiver architecture for the reception of those spatial and polarization modes. After mode demultiplexing, each spatial and polarization mode is detected by two single ended PDs. The difference between the two intensity measurements lies in the addition of dispersion elements in one path. Firstly, the 6 \(\times\) 6 transmission matrix, h, for polarization multiplexing is estimated through a training sequence, followed by the use of PR-PR to recover optical field information, and finally, DSP including 6 \(\times\) 6 MIMO is performed. Thus, achieving full-field reconstruction of mode and polarization multiplexed signals with six spatial and polarization modes is accomplished. Figure 6:2 Illustration of the time domain transmission matrix, \(h\), estimation (\(TS\): training sequence, \(*\): convolution, \(U\): unitary matrix, \(h_{ei}\): estimation of \(h\) at iteration \(i\)). Figure 6:3 Schematic diagram of the receiver architecture for the full field of the spatial and polarization modes. ## 4 Conclusions

In order to meet the requirements of low cost, low power consumption, and high sensitivity for short-range optical communication receivers, reconstructing the optical signal field from the intensity measurements has attracted widespread attention. The CF-PR receiver is a potential candidate and can reconstruct the full field of QAM signals from two intensity measurements by running an iterative PR algorithm, without the assistance of a strong CW optical carrier. However, the PR algorithm is prone to falling into local minima, slowing low convergence speed, low PR accuracy, and high computational complexity. This article reviews the research progress on how to improve the PR performance, in terms of reducing the iteration number along with the complexity of each iteration, and enhancing the steady BER performance. In addition, we also review the work on polarization and spatial mode diversity CF-PR receivers which can reconstruct the full field of multi-dimensional signals. We should note that there are still some challenges we need to overcome in future studies.