## 2 Microwave Photonic Filters  
### Background  
Figure 1 shows generic implementations for an incoherent finite impulse response MWP filter using taps created from (a) a single wavelength optical source or (b) a multiple wavelength optical source (we assume operation in a regime where coherent interference effects can be ignored). In the former, the output from the optical source is modulated using an electro-optic modulator (EOM), then split to create multiple taps. The amplitude and delays of each tap are adjusted before being recombined and detected. Successive taps are delayed by multiples of the delay unit \(\Delta T\). Assuming amplitude modulation, the magnitude of the MWP filter frequency response \(|H(f_{RF})|\) is proportional to [14]  
\[|H(f_{RF})|\propto\left|\sum_{m=1}^{N}P_{m}exp[-j(m-1)f_{RF}\Delta T]\right| \tag{1}\]  
where \(N\) is the number of taps and \(P_{m}\) denotes the amplitude of the \(m^{th}\) tap. For the case of a multiple wavelength optical source, each wavelength defines a tap. The taps are modulated and then propagated through a dispersive medium characterized by a dispersion \(D\) [ps/nm]. For taps that are separated equally in wavelength by an amount \(\Delta\lambda\), the corresponding delay is \(\Delta T=D\Delta\lambda\). The MWP filter response \(|H(f_{RF})|\) can also be calculated using Equation (1).  
Figure 1: \(Cont\).  
The MWP filter response described by Equation (1) is a periodic function of \(f_{RF}\) with a free spectral range (FSR) given by  
\[FSR=\frac{1}{\Delta T}=\frac{1}{D\Delta\lambda} \tag{2}\]  
If the spectral distribution of the taps from a multiple wavelength optical source follows a continuous and sinusoidal function of wavelength, then the MWP filter response features a single passband and is not periodic [15]. This passband is located at a central frequency \(f_{0}=1/(D\Delta\lambda)\).  
The multiple wavelength optical source can be obtained using FBGs to spectrally slice a broadband source (BBS). The FBGs themselves can be sensors that are subjected to varying environmental conditions, such as strain, temperature, etc. If we assume a fixed dispersion \(D\), then tuning the wavelength separation \(\Delta\lambda\) of the taps by applying strain or temperature to the FBG sensors will cause \(|H(f_{RF})|\) to shift. Figure 2 illustrates the principle of detecting changes in the MWP filter response \(|H(f_{RF})|\) by monitoring the change in RF power at a specific RF frequency. These changes can then be correlated to changes applied to the FBG sensor (e.g., strain or temperature). If \(|H(f_{RF})|\) shifts to higher frequencies, then choosing a frequency along the rising (\(f_{RF1}\)) or falling edge (\(f_{RF2}\)) of \(|H(f_{RF})|\) will result in a decrease or increase in detected RF power, respectively. The relationship between the change in RF power as a function of strain or temperature depends on several factors, including (1) the shape of the MWP filter response, (2) the RF frequency used for monitoring, and (3) the amount that \(|H(f_{RF})|\) will shift. The relationship can be linear, but it does not need to be so.  
Figure 1: Generic implementations of an incoherent finite impulse response microwave photonic (MWP) filter, based on (**a**) a single wavelength optical source and (**b**) a multiple wavelength optical source.  
Figure 2: Principle of monitoring the radio frequency (RF) power at selected frequencies as the MWP filter response \(|H(f_{RF})|\) shifts.