## 2 Principle of Operation
\begin{array}{l}T=\left[\begin{array}{cc}1&0\\ 0&0\end{array}\right]T_{HBF2}(\theta_{B2})T_{QWR2}(\theta_{Q2})T_{HWR2}(\theta _{H2})T_{HBF1}(\theta_{B1})T_{HWR1}(\theta_{H1})T_{QWR1}(\theta_{Q1})\left[ \begin{array}{cc}1&0\\ 0&0\end{array}\right]\\ +\left[\begin{array}{cc}0&0\\ 0&1\end{array}\right]T_{QWR1}(-\theta_{Q1})T_{HWR1}(-\theta_{H1})T_{HBF1}(- \theta_{B1})T_{HWR2}(-\theta_{H2})T_{QWR2}(-\theta_{Q2})T_{HBF2}(-\theta_{B2}) \left[\begin{array}{cc}0&0\\ 0&1\end{array}\right],\end{array}  \tag{1}\]where \(T_{QWR1}\), \(T_{HWR1}\), \(T_{HBF1}\), \(T_{HWR2}\), \(T_{QWR2}\), and \(T_{HBF2}\) are the Jones transfer matrices of QWR 1, HWR 1, HBF 1, HWR 2, QWR 2, and HBF 2, which have the slow-axis AOA's of \(\theta_{Q1}\), \(\theta_{H1}\), \(\theta_{B1}\), \(\theta_{H2}\), \(\theta_{Q2}\), and \(\theta_{B2}\) with respect to the \(x\) axis, respectively. The Jones transfer matrices (\(T_{QWR}\), \(T_{HWR}\), and \(T_{HBF}\)) of a QWR, an HWR, and an HBF, which have the AOA's of \(\theta_{Q}\), \(\theta_{H}\), and \(\theta_{B}\), respectively, are given as follows. \[T_{QWR}\big{(}\theta_{Q}\big{)}=\frac{\sqrt{2}}{2}\left[ \begin{array}{cc}1- j\cos 2\theta_{Q}&-j\sin 2\theta_{Q}\\ -j\sin 2\theta_{Q}&1+j\cos 2\theta_{Q}\end{array} \right],\ T_{HWR}(\theta_{H})=-j \left[ \begin{array}{cc}\cos 2\theta_{H}&\sin 2\theta_{H}\\ \sin 2\theta_{H}&-\cos 2\theta_{H}\end{array} \right],\]  
and  
\[T_{HBF}(\theta_{B})=\left[ \begin{array}{cc}\cos(\Gamma/2)-j\cos 2\theta_{B} \sin(\Gamma/2)&-j\sin 2\theta_{B}\sin(\Gamma/2)\\ -j\sin 2\theta_{B}\sin(\Gamma/2)&\cos(\Gamma/2)+j\cos 2\theta_{B}\sin(\Gamma/2) \end{array} \right]. \tag{2}\]  
On the basis of (1), \(t_{filter}\) is given by  
\[t_{filter}=\frac{1}{4}(P_{0}+P_{1}\cos\Gamma+P_{2}\sin\Gamma)^{2}+\frac{1}{4}( Q_{0}+Q_{1}\cos\Gamma+Q_{2}\sin\Gamma)^{2}. \tag{3}\]  
\[P_{0} =\sin\alpha\sin\beta-\sin\!\left(\!\gamma-\frac{\pi}{4}\!\right)\! \sin(\delta-2\theta_{B1}),\ Q_{0}=\cos\gamma\cos\beta-\!\cos\!\left(\!\alpha+ \frac{\pi}{4}\!\right)\!\cos(\delta-2\theta_{B1}),\] \[P_{1} =\sin\alpha\sin\beta+\sin\!\left(\!\gamma-\frac{\pi}{4}\!\right)\! \sin(\delta-2\theta_{B1}),\ Q_{1}=\cos\gamma\cos\beta+\!\cos\!\left(\!\alpha+ \frac{\pi\pi}{4}\!\right)\!\cos(\delta-2\theta_{B1}), \tag{4}\] \[P_{2} =\cos\!\left(\!\gamma-\frac{\pi}{4}\!\right)\!\cos\beta+\cos \alpha\cos(\delta-2\theta_{B1}),\ \text{and}\ Q_{2}=-\sin\!\left(\!\alpha+\frac{\pi}{4}\!\right)\!\sin \beta-\sin\gamma\sin(\delta-2\theta_{B1}),\]  
where \(\alpha=\theta_{Q1}-\theta_{Q2}\), \(\beta=2\theta_{H1}-2\theta_{H2}-\theta_{Q1}+\theta_{Q2}\), \(\gamma=\theta_{Q1}+\theta_{Q2}\), and \(\delta=2\theta_{H1}+2\theta_{H2}-\theta_{Q1}-\theta_{Q2}\). The narrowband transmittance function \(t_{narrow}\) can be obtained from this general transmittance \(t_{filter}\) and is given by (5), which can also be drawn out in consideration of the transmittance function of a conventional fan Solc filter with two birefringent components (\(M=2\)) [22]. The additional phase difference \(\psi\) in \(t_{narrow}\) sets the wavelength position of the narrowband transmission spectrum, that is, can be utilized to tune the spectral location. \[t_{narrow}=\left[1+2\cos(\Gamma+\psi)+\cos^{2}(\Gamma+\psi)\right]/4 \tag{5}\]  
To establish the theoretical conditions of the wave retarders, required to continuously tune the wavelength location of \(t_{narrow}\), the AOA's of the four wave retarders, or (\(\theta_{Q1}\), \(\theta_{H1}\), \(\theta_{H2}\), \(\theta_{Q2}\)), which could induce \(\psi\)'s ranging from \(0^{\circ}\) to \(360^{\circ}\) in \(t_{narrow}\), were investigated by quantitatively comparing (3) with (5).