2012杭州薄膜会议第7个ppt：Floquet-Bloch waves and hybrid floquet modes

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The subject of our presentation goes back to Hill's equation \begin{equation} \frac{d^{2}f(z)}{dz^{2}}+a(z)f(z)=0, \qquad a(z)=a(z+d), \end{equation} where a(z) is a periodic, real or complex valued function of the real variable z . First introduced by Hill in 1877, it has appeared in many applications, including the propagation of electrons in crystals and the propagation of optic waves in periodic thin films. There is always one non-trivial particular solution $F_{1}(z)$, called a Floquet-Bloch wave, which satisfies the relation $F_{1}(z+d)=\rho_{1}F_{1}(z)$, where $\rho_{1}$ is a non-zero, generally complex, constant. A second linearly independent particular solution constitutes either a second Floquet-Bloch wave, $F_{2}(z+d)=\rho_{2}F_{2}(z)$ or a hybrid Floquet mode G(z), with the property $G(z+d)=\rho_{1}G(z)+\rho_{1}dF_{1}(z)$. Using a periodic arrangement of two thin films as an example, we show how to construct two independent Floquet-Bloch waves for propagating light in a lossless periodic structure. Particular attention is given to those special cases where only one Bloch wave develops inside the structure and a hybrid Floquet mode appears instead of a second Bloch wave. TE and TM polarized light are both treated.
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