# Research Progress on Carrier-Free Phase-Retrieval Receivers
## 2 Principle of CF-PR Receivers  
Compared with the self-coherent phase reconstruction scheme with the aid of a strong CW carrier, the CF-PR receiver based on the modified Gerchberg-Saxton (MGS) [48; 49; 50; 51] algorithm consists of only two single-ended PDs and one dispersive element, which could recover the full-field information of the optical QAM signal without relying on any optical carrier. Theoretically, the CF-PR receiver can retrieve the phase of arbitrary QAM signals, and transmission impairments such as CD can be mitigated after phase retrieval [47]. As shown in Figure 1, the received QAM signal is divided into two parts, one part is directly detected using a PD, and the other part of the QAM signal is firstly passed through a dispersive element and then detected by another PD. The CF-PR algorithm is generally utilized to recover the full optical field from two intensity measurements, \(\left|s(t)\right|^{2}\) and \(\left|d(t)\right|^{2}\), where \(s(t)\) and \(d(t)\) are the undispersed signal and the dispersed signal at the Rx, respectively. We set \(a(t)=\left|s(t)\right|\) and \(b(t)=\left|d(t)\right|\) for the ease of discussion. Without considering the nonlinearity of optical fiber, the undispersed signal after standard single-mode fiber (SSMF) transmission, and the dispersed signal after both SSMF and the dispersive element transmission can be expressed as  
\[s(t)=x(t)\otimes h_{CD}=\left(\sum_{k}x_{k}\cdot g(t-kT)\right)\otimes h_{CD}(t) \tag{1}\]  
\[d(t)=s(t)\otimes h_{CD}(t)\otimes h_{D}=\left(\sum_{k}x_{k}\cdot g(t-kT)\right) \otimes h_{CD}(t)\otimes h_{D}(t) \tag{2}\]  
where \(x(t)\) is the output signal of the transmitter side (Tx), \(\otimes\) is the convolution operation, \(h_{CD}\) and \(h_{D}\) represent the complex transfer functions resulting from the use of SSMF links and the dispersive element, respectively. \(x_{k}\) represents the transmitted complex-valued data symbols, \(T\) is the symbol duration, and \(g(t)\) represents the basic pulse waveform of the transmitter. The frequency response of \(h_{CD}(t)\) and \(h_{D}(t)\) can be expressed as  
\[H_{CD}(f)=F_{T}(h_{CD}(t))=\exp\Bigl{(}-j\pi D_{CD}\lambda^{2}f^{2}/c\Bigr{)} \tag{3}\]  
\[H_{D}(f)=F_{T}(h_{D}(t))=\exp\Bigl{(}-j\pi D_{CD}\lambda^{2}f^{2}/c\Bigr{)} \tag{4}\]  
where \(F_{T}\) stands for the Fourier transform function, \(\lambda\) is the laser center wavelength, \(c\) is the speed of light propagation in a vacuum, and \(D_{CD}\) and \(D_{D}\) are the cumulative \(CD\) value originating from the SSMF link and the cumulative \(CD\) value of the dispersive element, respectively. The solution for optical field recovery is to run an iterative CF-PR algorithm and iterate out the phase of the signal \(\angle(t)\) so that it satisfies the following condition:  
\[\underset{\angle(t)}{argmin}\sum_{m=1}^{P}\left(\left|\sum_{k}(a(k)exp\bigl{(} j\angle(t)\bigr{)}\cdot h_{D}(mT-kT)\right|-b(mT)\right)^{2} \tag{5}\]After the projection, by adding additional \(CD\), the signal intensities are decorrelated and thus the signal phase can be retrieved when the reconstructed signal intensities on all projection planes are consistent with the measurements. However, CF-PR receivers often need a dispersive element with a large CD to ensure the phase recovery accuracy and reduce the number of iteration times of the CF-PR algorithm [46]. We should note that the requirement of the larger CD can be significantly relaxed if higher symbol rate signals are applied. Additionally, because the CF-PR algorithm is implemented in a block-wise manner with a block size of \(N\), and handles the signal propagation and shaping in the frequency domain using the classical radix-2 algorithm for the fast Fourier transform [45; 46; 47], the CF-PR receiver also suffers from high computational complexity for each iteration when the accumulated CD is large. Moreover, CF-PR algorithms based on MGS are prone to fall into the local minima, which results in a slow convergence with nearly a thousand iterations and reduced PR accuracy.