## 9 Stokes Space Polarization and Field Recovery Principle
### Stokes Space Field Recovery
#### 9.2.2 2D Direct-Detection-Based Field Recovery  
The 2D direct-detection-based field recovery can be based on an SCOH Stokes space modulation (SCOH-SSM) or PDM-SSB modulation. The associated concept of these 2D direct-detection-based field recoveries is presented in this subsection.  
Self-Coherent Stokes Space Modulation  
As previously mentioned, the Stokes parameters are second-order terms of the field signal \(X\) and \(Y\). Furthermore, the four Stokes parameters can be classified into two groups. The first group contains \(S_{0}\) and \(S_{1}\), which are the self-beating of \(X\) and \(Y\). Similarly, the second group comprises \(S_{2}\) and \(S_{3}\), which are the cross-beating between \(X\) and \(Y\)[13]. Nevertheless, in order to linearize the direct detection optical channel, it is highly desirable to modulate the \(X\) polarization only with a complex DSB signal, while the \(Y\) polarization is launched as a constant carrier. The process helps ensure that the cross-beating terms are directly linearized. Consequently, after polarization rotation, the linear replica of \(X\) (a phase diverse signal) can be promptly recovered from \(S_{2}+iS_{3}=S\cdot C^{*}\) by the Stokes vector receiver. This is comparable to the output that can be realized by COH detection [11; 13].  
Polarization Division Multiplexing Single-Sideband Modulations  
The SSB-FR exploits the corresponding double receiver bandwidth to convert a complex signal into an intensity signal after the square law detection. In this process, \(|U|^{2}\) is transformed into \(U\). For the PDM-SSB scheme, the dual-polarized optical field can be recovered by the SSB modulation of both polarizations. Consequently, after Stokes space polarization rotation, the field can then be recovered from \(S_{0}\) and \(S_{1}\) (i.e., \(|X|^{2}\) and \(|Y|^{2}\)) [13].  
Moreover, the PDM-SSB can be adapted by omitting the carrier at the \(Y\)-POL. With this approach, about a 3 dB carrier power at the transmitter can be conserved. Based on this, the system OSNR sensitivity can be improved. The implementation is known as PDM-SSB no-carrier \(Y\)-POL (PDM-SSB-NCY). Conceptually, rather than being retrieved from \(|Y|^{2}\), the field signal, \(Y\), is retrieved from \([S_{2}\,S_{3}]\) based on procedures that are similar to those of the 3D Stokes space field recovery as discussed in the following subsection [13].  
In relation to the PDM-COH systems, the SCOH-SSM can attain 50% optical spectrum efficiency and electrical spectrum efficiency owing to the unmodulated \(Y\)-POL. However, with the PDM-SSB, the assessment is not straightforward. This is due to the fact that it is the direct detection system that is capable of recovering the dual-polarized optical field, and based on this, the optical spectrum efficiency is maximized to 100%. Nevertheless, being a 2D format in the Stokes space, only 50% electrical spectrum efficiency can be realized as the required receiver bandwidth is doubled by the SSB detection [13].  
Furthermore, it is remarkable that, unlike the long-haul transmission in which high optical spectrum efficiency is usually taken into consideration in order to maximize the fiber capacity, in the short-reach applications, the primary consideration is electrical spectrum efficiency instead of optical spectrum efficiency. This is owed to the fact that the AIR per wavelength can be maximized by high electrical spectrum efficiency for a band-limited transceiver. This helps lessen the cost per bit and consequently results in transceiver cost reduction. In the short-reach networks that are deployed for commercial purposes, the optical spectrum resource is not a scarce commodity. Consequently, it is highly essential to focus on the electrical spectrum efficiency maximization in the 3D Stokes space field recovery system design [13; 20].