## 9 Stokes Space Polarization and Field Recovery Principle
### Stokes Space Field Recovery
#### 9.2.3 3D Direct-Detection-Based Field Recovery  
As the Stokes space supports three distinct degrees of freedom for modulation, the 3D clock has been extensively investigated in an effort to maximize the Stokes vector receiver electrical spectrum efficiency. Basically, a conventional 3D clock comprises one interpolarization differential phase modulator and two IMs for dual-polarized optical intensities. Nevertheless, the 3D clock subsystems are not capable of recovering the whole set of fields for the three possible dimensions. Based on this, the attainable reach of the 3D clock subsystems is limited at the \(C\)-band when chromatic dispersion precompensation is not implemented. A notable way of addressing the challenge so as to ensure field recovery at the receiver is to perform linear complex modulations in the Jones space. It should be noted that due to the limitation being presented by the 3D modulation in the Stokes space, the dimension of one polarization in the Jones space has to be reduced to 1D and then the 1D field recovery exploited. This can be achieved by modulating the \(X\)-POL with a 1D signal, which can either be an SSB signal or a DSB signal with blockwise phase switching. On the other hand, the \(Y\)-POL can be a 2D DSB complex signal. Subsequently, the entire 3D field can be retrieved by the Stokes space field recovery after polarization rotation based on the following steps [13]:  
* The intensity \(|X|^{2}=(S_{0}+S_{1})/2\) can be recovered, and subsequently, the field \(X\) can be retrieved from the intensity information using 1D field recovery;
* Furthermore, the field \(Y=(S_{2}+iS_{3})^{*}/X^{*}\) can then be retrieved. Although all the 3D Stokes space field recovery schemes employ the aforementioned procedures for retrieving all 3D fields, the main difference between them can be perceived from the adopted 1D field recovery method. For instance, there is a stringent impact on the SNR with SSB signal polarization employment because of the noise folding effect in the sideband of interest. This results in about a 3 dB OSNR penalty for the SSB-FR when the noise from the unoccupied sideband is folded into the utilized one after the square law detection. It should also be noted that the effect will also impact the other polarization. On the other hand, with the BPS-FR scheme implementation, DSB signals are modulated on both polarizations. This helps address the aforementioned penalty and related challenges of the SSB-FR [13]. Moreover, when the 1D systems such as the SSB-FR and BPS-FR that require SSBN cancellation at the \(X\)-POL are employed, then the field recovery depends on digital demodulation. However, the associated bit-error can impact the \(Y\)-POL field recovery. On the other hand, the related error propagation can be avoided by the SSB-FR. This can be achieved through direct signal phase reconstruction from the photocurrent in the polar coordinates [13]. In addition, the optical spectrum efficiency is halved by the BPS-FR in the 1D field recovery system compared with the SSB-FR. Nevertheless, when a 3D Stokes space field recovery scheme is implemented with either the BPS-FR or SSB-FR at the \(X\)-POL, the same optical spectrum efficiency will be achieved. This is a result of the unused sideband at the \(X\)-POL that is really challenging to share with another WDM channel since DSB modulation is implemented at the \(Y\)-POL. Moreover, with respect to the 4D PDM-COH systems, 75% optical spectrum efficiency can be achieved by the 3D Stokes space field recovery, and the electrical spectrum efficiency can be maximized to 75% as well. This value is the highest realizable electrical spectrum efficiency currently offered by the direct-detection-based receivers [13]. The related features of different dimensional detection field recovery schemes are presented in Table 10.  
In general, the ability of a 1D direct detection scheme to recover a single-polarization-based optical field in the Jones space can be enhanced by duplicating the 1D field recovery scheme to realize a DP-FR system.