关于Floquet理论与Floquet激励的设置

    Floquet模实质上是电磁场波动方程满足周期性边界条件时，利用数学上分离变量法得到的电磁场的解，周期结构的影响反映在Floquet模中。

    Floquet theory is a branch of the theory of ordinary differential equations relating to the class of solutions to periodic linear differential equations of the form

• x ˙ = A ( t ) x , {\displaystyle {\dot {x}}=A(t)x,

with A ( t ) {\displaystyle \displaystyle A(t)} a piecewise continuous periodic function with period T {\displaystyle T} and defines the state of the stability of solutions.

    The main theorem of Floquet theory, Floquet's theorem, due to Gaston Floquet (1883), gives a canonical form for each fundamental matrix solution of this common linear system. It gives a coordinate change y = Q − 1 ( t ) x {\displaystyle \displaystyle y=Q^{-1}(t)x} with Q ( t + 2 T ) = Q ( t ) {\displaystyle \displaystyle Q(t+2T)=Q(t)} that transforms the periodic system to a traditional linear system with constant, real coefficients.

    In solid-state physics, the analogous result is known as Bloch's theorem.

Note that the solutions of the linear differential equation form a vector space. A matrix ϕ ( t ) {\displaystyle \phi \,(t)} is called a fundamental matrix solution if all columns are linearly independent solutions. A matrix Φ ( t ) {\displaystyle \Phi (t)} is called a principal fundamental matrix solution if all columns are linearly independent solutions and there exists t 0 {\displaystyle t_{0}} such that Φ ( t 0 ) {\displaystyle \Phi (t_{0})} is the identity. A principal fundamental matrix can be constructed from a fundamental matrix using Φ ( t ) = ϕ ( t ) ϕ − 1 ( t 0 ) {\displaystyle \Phi (t)=\phi \,(t){\phi \,}^{-1}(t_{0})} . The solution of the linear differential equation with the initial condition x ( 0 ) = x 0 {\displaystyle x(0)=x_{0}} is x ( t ) = ϕ ( t ) ϕ − 1 ( 0 ) x 0 {\displaystyle x(t)=\phi \,(t){\phi \,}^{-1}(0)x_{0}} where ϕ ( t ) {\displaystyle \phi \,(t)} is any fundamental matrix solution.