## 7 Towards a Simplified Optical Transceiver
### Full Coherent Optical System
#### 7.1.2 Optical Coherent Detection Principles  
The principle of optical signal detection in the COH system can be explained by considering the configuration shown in Figure 15. The receiver measures the beat between the LO and signal. In general, when the product of the electric fields of the CW LO and modulated signal is estimated, the complex electric fields of the transmitted optical signal, \(E_{s}(t)\), and that of the LO, \(E_{\text{LO}}\), can be defined respectively as [10]:  
\[E_{s}(t)=A_{s}(t)\exp(j\omega_{\text{s}}t), \tag{25a}\] \[E_{\text{LO}}(t)=A_{\text{LO}}\exp(j\omega_{\text{LO}}t), \tag{25b}\]  
where \(A_{s}(t)\) denotes the complex amplitude of the optical signal, \(A_{\text{LO}}(t)\) represents the constant complex amplitude of the LO, and \(\omega_{\text{s}}\) and \(\omega_{\text{LO}}\) are the angular frequency of the optical signal and LO, respectively. Figure 14: Digital coherent receiver configurations: (**a**) polarization- and phase-diverse intra-dyne/homodyne detection, (**b**) polarization- and phase-diverse heterodyne detection, and (**c**) phase-diversity heterodyne detection. ADC: analog-to-digital converter, BPD: balanced photodetector, TIA: transimpedance amplifier, PDM: polarization division multiplexed signal (adapted from [7; 10; 22]). In addition, apart from its ability to suppress the DC component, balanced detection also helps maximize the beat. The beating can be accomplished by a 3 dB optical coupler. The coupler ensures a 180\({}^{\circ}\) phase shift between the two output ports. Hence, the co-polarized signal and LO, the incident electric fields on the upper and lower photodiodes, can be expressed respectively as [134]:  
\[E_{1} = \frac{1}{\sqrt{2}}(E_{s}+E_{\text{LO}}), \tag{26a}\] \[E_{2} = \frac{1}{\sqrt{2}}(E_{s}-E_{\text{LO}}), \tag{26b}\]  
It is noteworthy that the photodiode is usually modeled as a square law detector. Its responds to the square of the electric field as:  
\[i(t)=R\Big{\langle}E_{s}(t)^{2}\Big{\rangle} \tag{27}\]  
Furthermore, using Equations (25a), (26a), and (27), as well as (25b), (26b), and (27), the output photocurrents produced by a pair of photodiode when illuminated by the fields \(E_{1}\) and \(E_{2}\) can be expressed as [135; 10]:  
\[I_{1}(t) = \frac{R}{2}\big{[}P_{s}(t)+P_{\text{LO}}+2\sqrt{P_{s}(t)P_{\text{ LO}}}\cos\{\omega_{lF}t+\theta_{lF}(t)\}\big{]}, \tag{28a}\] \[I_{2}(t) = \frac{R}{2}\big{[}\underbrace{P_{s}(t)+P_{\text{LO}}}_{\text{DD term}}-\underbrace{2\sqrt{P_{s}(t)P_{\text{LO}}}\cos\{\omega_{lF}t+\theta_{lF}(t)\}}_{ \text{Coherent detection term}}\big{]}, \tag{28b}\]  
where \(\omega_{lF}\) denotes the intermediate angular frequency (IF) and \(\theta_{lF}(t)\) represents the phase information and given respectively as [135]:  
\[\omega_{IF}=|\omega_{s}-\omega_{\text{LO}}|, \tag{29a}\] \[E_{s}(t)=A_{s}(t)\exp(j\omega_{\text{s}}t), \tag{25a}\]0  
where \(\theta_{sig}(t)\) denotes the phase of the transmitted signal and \(\theta_{\text{LO}}\) is the phase of the LO. Furthermore, to obtain the aggregate photocurrent, the output photocurrents of the photodiode pair are subtracted. Hence, using Equation (28), the output of the balanced detector can be expressed as:  
\[E_{s}(t)=A_{s}(t)\exp(j\omega_{\text{s}}t), \tag{25a}\]1 \[E_{s}(t)=A_{s}(t)\exp(j\omega_{\text{s}}t), \tag{25a}\]2  
It can be deduced from Equation (30) that balanced detection measures just the carrier-signal beating, while direct detection terms and excess intensity noise produced  
\[E_{s}(t)=A_{s}(t)\exp(j\omega_{\text{s}}t), \tag{25a}\]3 Coherent detection configuration. Note: The derivations are based on the assumption that the LO and the signal have the same polarization. by the LO are eliminated. It is noteworthy that the photocurrent is proportional to \(\sqrt{P_{\text{LO}}}\). Consequently, an increase in the LO power results in a relative gain in the photocurrent. Based on this, the receiver's thermal noise becomes negligible [10; 135].