# A DFB-SOA Based Optical Vector Network Analyzer for Characterization of Bandpass Optical Devices
## 2 Operating Principle
Its elements are given by  
\[F_{11}^{i}=\left(F_{22}^{i}\right)^{*}=\cosh(\gamma L_{i})-j(\sigma/\gamma) \sinh(\gamma L_{i}) \tag{2}\]  
\[F_{12}^{i}=\left(F_{21}^{i}\right)^{*}=-j(\kappa/\gamma)\sinh(\gamma L_{i}) \tag{3}\]  
where \(*\) denotes complex conjugation, \(L_{i}\) is the length of \(i\)th grating section, \(\gamma=\sqrt{k^{2}-\sigma^{2}}\), \(\kappa\) represents the \"ac\" coupling coefficient and \(\sigma\) represents the general \"dc\" self-coupling coefficient which is defined by  
\[\sigma=n_{eff}\frac{\omega}{c}-\frac{\pi}{\Lambda}-\frac{g}{2}\alpha-j\frac{g -\alpha_{\text{int}}}{2} \tag{4}\]  
where \(n_{eff}\) is the effective mode refractive index, \(c\) is the speed of light in a vacuum, \(\Lambda\) is the grating period, \(\alpha\) symbolizes the linewidth enhancement factor of active layer. The coefficient of power gain under unsaturated conditions is denoted as \(g=\Gamma aN_{0}(I/I_{0}-1)\). \(\Gamma\) denotes the confinement factor, \(a\) represents the differential gain, \(N_{0}\) is the carrier density at transmission, \(I_{0}\) signifies the current required to reach transparency, and \(\alpha_{\text{int}}\) corresponds to the internal loss coefficient. The gain spectrum of the DFB-SOA can be expressed as  
\[G(\omega)=1/\left\{j\frac{k^{2}}{\gamma^{2}}\text{sinh}^{2}(\gamma I)-j\left[ \cosh(\gamma I)-j\frac{\sigma}{\gamma}\text{sinh}(\gamma I)\right]^{2}\right\} \tag{5}\]  
Certainly, Equations (4) and (5) shows that the gain spectrum of the DFB-SOA is determined by the injected current. The manipulation of the resonant wavelength and power gain can be achieved through the precise regulation of the injected current. Mathematically, the optical field of the optical signal output from the PM can be described as  
\[E_{PM}(t)=A\exp[i\omega_{c}t+j\beta\sin(\omega_{m}t)] \tag{6}\]  
The amplitude of the optical carrier from the TLS is denoted as \(A\). Let angular frequency \(\omega_{c}\) represents the optical carrier, and assign \(\omega_{m}\) for the microwave signal's angular frequency. The first kind Bessel functions of the \(n\)th order are represented by \(J_{n}(\cdot)\) while the phase modulation index of the PM is denoted by \(\beta\). The Fourier transform is utilized to manipulate Equation (6). The optical signal output from the phase modulator in the frequency domain can be expressed as  
\[E_{PM}(\omega)=2\pi A\sum_{n=-\infty}^{\infty}J_{n}(\beta)\delta[\omega-( \omega_{c}+n\omega_{m})] \tag{7}\]  
Subsequently, the phase-modulated signal is transmitted to the ODUT. It is postulated that the ODUT transmission response is characterized by \(H(\omega)\) with a unitary passband centered at \(\omega_{c}-\omega_{m}\), effectively attenuating the optical carrier and any unwanted sidebands. The phase-modulated signal is transmitted to the ODUT. Consequently, the lower sideband at \(\omega_{c}-\omega_{m}\) is able to transmit through the ODUT with minimal obstruction. Disregardingthe negligible undesired sidebands of minimal intensity, the optical carrier-suppressed single-sideband (CS-SSB) signal emanating from the ODUT can be represented as  
\[E_{ODUT}(\omega)=2\pi AIJ_{0}(\beta)H(\omega_{c})\delta(\omega-\omega_{c})+2\pi AIJ _{-1}(\beta)H(\omega_{c}-\omega_{m})\delta[\omega-(\omega_{c}-\omega_{m})] \tag{8}\]  
The CS-SSB signal is subsequently injected into the DFB-SOA. Given that the gain peak is located at \(\omega_{c}\), The emitted optical signal from the DFB-SOA solely comprises the regenerated optical carrier and the scanning lower sideband. Thus, the equation presented in Equation (8) can be rewritten as  
\[E_{out}(\omega)=2\pi AIJ_{0}(\beta)G(\omega_{c})H(\omega_{c})\delta(\omega- \omega_{c})+2\pi AIJ_{-1}(\beta)H(\omega_{c}-\omega_{m})\delta[\omega-(\omega_ {c}-\omega_{m})] \tag{9}\]  
where \(G(\omega)\) is the gain spectrum of the DFB-SOA which can be tuned by the injected current.