On the explicit parametric description of waves in periodic media

Parametric representation of wave propagation in non‐uniform media (both in transmission and stop bands)

“…In such a way, we construct a closed‐form parametric solution that reveals some quantitative relations inherent in wave propagation in non‐uniform media, not obvious in classical solutions of the corresponding differential equations . In particular, for any periodic refraction index n ( x ) = n ( x + χ ) defined implicitly by a Fourier series, that is, where a 0 , a 2 m , b 2 m are some constants, we obtain Floquet solution with the minus sign in the exponent and the characteristic exponent μ = ν ∕ χ with the explicit formulae for the period χ and the attenuation per period ν given as follows : These analytical relations, giving the very simple description of the wave field attenuation in a periodic structure, are useful for the optimal design of multilayer mirrors and Bragg fibre claddings . However, from the theoretical point of view, this solution remains incomplete until a similar parametric representation is found for the propagating waves in the corresponding transmission bands of a periodic medium.…”

“…Let us recall the results obtained in the previous papers concerning the parametric representation of the fundamental system of solutions to wave equation . If we introduce an admittance function that is, the solution of Equation can be represented as follows: where w 0 is an integration constant, then it is easy to observe that primary wave equation can be equivalently rewritten in the following form: that is, …”

“…Let us recall the results obtained in the previous papers [3][4][5] concerning the parametric representation of the fundamental system of solutions to wave equation (1). If we introduce an admittance function…”

“…where a 0 , a 2m , b 2m are some constants, we obtain Floquet solution (2) with the minus sign in the exponent and the characteristic exponent D = with the explicit formulae for the period and the attenuation per period given as follows [3][4][5][6]:…”

“…In many applications, the inverse problem (a kind of optimal control) arises when one needs to find such a dependence n ( x ) that provides the maximal value of the characteristic exponent μ . In our previous papers , a simple analytical theory of parametric (anti)resonance was developed, and explicit formulae for field attenuation in stop bands were derived. However, for the sake of completeness, we need also to know the behaviour of the Bloch wave in a transmission band where the characteristic exponent is purely imaginary.…”