# Comparison of Basic Notch Filters for Semiconductor Optical Amplifier Pattern Effect Mitigation
## 2 Basic Optical Notch Filters' Configurations  
The considered basic optical notch filters differ in their construction and operation and accordingly in the way that their spectral response is obtained and tailored for suppressing the SOA pattern effect. Figure 3: Data amplification in SOA with pattern-dependent output distortion and compensation by optical notch filtering. The ODI's principle of operation relies on splitting, delaying and recombining the amplified signal, which is a process realized by means of intensity discrimination. Other waveguide-based technologies, which have been used for performing these functions in the wavelength domain, are reported in [34; 35; 36; 37; 38]. As shown in Figure 4a, it is constructed from two 3-dB couplers by interconnecting the output ports of the first one to the respective input ports of the second one. The upper arm of the formed interferometric configuration has a relative delay, \(\Delta\tau\), against the second arm, into which a phase bias, \(\Phi_{b,ODI}\), is introduced. When the amplified signal is inserted in the ODI from the first coupler, it is halved into two identical copies, which follow distinct paths along the ODI and acquire a wavelength-dependent phase difference analogous to their relative delay. Thus, when these components recombine at the second coupler, they interfere either constructively or destructively and produce at the ODI crossed output port the spectral response shown in Figure 5 with red color. The wavelength separation between adjacent maxima (peaks), or the free spectral range (FSR), is inversely proportional to the ODI relative delay, i.e., \(FSR_{ODI}=\lambda_{amp}^{2}/(c\Delta\tau)\)[32], where \(\lambda_{amp}\) is the reference optical wavelength of the amplified signal, which lies in the vicinity of 1550 nm, and \(c\) is the speed of light in a vacuum. The minima (notches) are located halfway between maxima, and their exact wavelength position with respect to the shorter (blue) or longer (red) sideband, indicated by \(\lambda_{notch,ODI}\) in Figure 5, depends on the phase bias, \(\Phi_{b,ODI}\). This determines the wavelength detuning of the data carrier from the nearest transmission peak, \(\Delta\lambda_{max,ODI}\), according to \(\Phi_{b,ODI}=-2\pi\times(\Delta\lambda_{max,ODI}/FSR_{ODI})\)[39]. Figure 4: (**a**) Optical delay interferometer (ODI) and (**b**) microring resonator (MRR) structure. Figure 5: ODI (red color) and MRR (green color) spectral response together with spectrum at SOA output (black color). The red-colored oblique line and the yellow-colored zones show the red-shifted spectral components, which are filtered by the ODI and MRR, respectively. The MRR is a waveguide shaped into a ring structure that is coupled to a bus waveguide, as shown in Figure 4b. When a signal enters this configuration, part of it is fed back to the MRR input while the rest is transferred at the MRR output. The exact signal magnitude and wavelength that penetrate into the ring depend on the strength of coupling between the straight and circular waveguides and whether the condition of resonance is satisfied. The amplified signal components that enter the ring subject to these conditions travel around it and after a time delay proportional to the ring circumference interfere with those components that pass directly to the exit. Then, provided that the MRR operates in the so-called critical coupling regime, where the field transmission coefficient, \(r\), and the amplitude attenuation factor, \(l\), are equal, the circulating intensity is maximized, and the transmitted intensity becomes minimum [33]. As a result, the spectral response shown with green color in Figure 5 is obtained. The notches, which are located at \(\lambda_{notch,MRR}\), occur at resonance according to the condition \(2\pi Rn_{eff}=m\lambda_{notch,MRR}\), where \(R\) is the MRR radius, \(n_{eff}\) is the waveguide effective refractive index and \(m\) is a non-zero integer.