3 and that r$_{j/l}$r$_{l/j}$ is given by a similar expression, we find that actually [15] single-interface transmission coefficients t$_{jk}$ and t$_{km}$ . As- suming in Eq. (2) the same symmetry properties of the generalized coefficients t$_{j/k}$ and t$_{k/m}$ , we immediately find that Eq. (18) is obeyed. Accordingly, using the in- duction argument, we may conclude that this equation is for local systems generally valid. Now we argue that Eq. (18) is also valid for piecewise nonlocal systems as it actu- ally ensures the equality of the stack’s transmittances for waves incident on it from either side. Indeed, the Poynt- ing vector of the upward/downward-propagating wave in a layer l is according to Eqs. (3) and (15) given by Accordingly, the right-hand side of Eq. (10) is, upon using Eq. (9), indeed identified as r$_{j/l/m}$ . To prove the equality of the transmission coefficients in Eq. (8) we first apply Eq. (2) to the coefficient t$_{k/m}$ = t$_{k/l/m}$ and then use the identity ( r$_{k/m}$ = r$_{k/l/m}$ ) [11] which follows from Eq. (9). In this way, we obtain which is according to Eq. (2) equal to t$_{j/l/m}$ . In most textbook approaches to the wave propagation in layered media the basic ingredients in the calculation of the generalized Fresnel coefficients are coefficients r$_{jk}$ and t$_{jk}$ for the interface between two neighbouring local media j and k . With unit polarization vectors in Eq. (3) [9] with η p l = η s l ∗ = ε ∗ l /µ ∗ $_{l}$. Accordingly, assuming the out- most layers j and m transparent, transmittances of the j/m stack for waves incident upward and downward on it [given by the ratios of the respective transmitted- to incident-energy fluxes T j/m = P + $_{mz}$( k , ω, 0) /P + $_{jz}$( k , ω, 0) and T m/j = P − $_{jz}$( k , ω, 0) /P − $_{mz}$( k , ω, 0)] can only be equal if Eq. (18) is fulfilled. Thus, being a consequence of the reciprocity property of the electro- magnetic field, this equation is valid for a quite general class of (nongyrotropic) media [16]. and applying the usual boundary conditions for the field, it is straightforward to show that the above definition of Fresnel coefficients leads to the standard single-interface coefficients As follows from the above results, recurrence relations for Fresnels coefficients of a multilayered system can be written in a number of ways depending on number of in- termediate local layers. We illustrate this by calculating Fresnel coefficients of few simple systems. We first con- sider a 12 / 3 system (that is, we allow for an unspecified stack of layers between layers 2 and 3 of a three-layer local system 123). Since from Eq. (17) a$_{12}$ = 1, Eq. (9) leads to standard forms of the recurrence relations for Fresnel coefficients [2] where γ p jk = ε$_{j}$/ε$_{k}$ and γ s jk = µ$_{j}$/µ$_{k}$ . As can be easily verified, these coefficients obey the Stokes relation We note that the symmetry property of the single- interface transmission coefficient, as expressed by Eq. (16b), implies for local systems the same symmetry prop- erty of the generalized transmission coefficient t$_{j/m}$ , that is [2, 9] Indeed, through Eq. (2), it certainly holds for a three- layer system jkm owing to the symmetry property of the which, for a three-layer system 123, reduce to the well- known results usually quoted in textbooks [1, 2]. Next, we consider Fresnel coefficients of a 123 / 4 system. Ac- cording to Eq. (9), they can be calculated from two equivalent sets of recurrence relations differing in the choice of the intermediate layer. The first set is the stan- dard one [2] and is given by Eq. (21), with the replace- ment of indices 2 / 3 → 23 / 4. The second set of recurrence t$_{j/m}$ = µ$_{m}$β$_{j}$ µ$_{j}$β$_{m}$ t$_{m/j}$. (18) a$_{jk}$ ≡ t$_{jk}$t$_{kj}$ − r$_{jk}$r$_{kj}$ = 1 . (17) t$_{jk}$ = √ γ$_{jk}$ γ s jk (1 + r$_{jk}$ ) = µ$_{k}$β$_{j}$ µ$_{j}$β$_{k}$ t$_{kj}$, (16b) r$_{jk}$ = β$_{j}$ − γ$_{jk}$β$_{k}$ β$_{j}$ + γ$_{jk}$β$_{k}$ = − r$_{kj}$, (16a) ˆ e ± pl = ∓ β$_{l}$ ˆ k + k ˆ z k$_{l}$ , ˆ e ± sl = ˆ k × ˆ z , (15) t$_{j/k/m}$ = t$_{j/k}$t$_{k/l}$t$_{l/m}$e i ( β$_{k}$d$_{k}$ + β$_{l}$$^{d$_{l}$}$) (1 − r$_{k/j}$r$_{k/l}$e 2 iβ$_{k}$$^{d$_{k}$}$)(1 − r$_{l/j}$r$_{l/m}$e 2 iβ$_{l}$$^{d$_{l}$}$) , (14) (1 − r$_{l/k}$r$_{l/m}$e 2 iβ$_{l}$$^{d$_{l}$}$)(1 − r$_{k/j}$ r$_{k/m}$e 2 iβ$_{k}$$^{d$_{k}$}$) = (1 − r$_{k/j}$r$_{k/l}$e 2 iβ$_{k}$$^{d$_{k}$}$)(1 − r$_{l/j}$r$_{l/m}$e 2 iβ$_{l}$$^{d$_{l}$}$) , (13) a$_{j/l}$ = t$_{j/l}$t$_{l/j}$ − r$_{j/l}$r$_{l/j}$ = ˜ a$_{j/l}$. (12) P ± l ( k , ω, z ) = c 8 π Re √ η$_{l}$ k ± β$_{l}$ ˆ z k$_{l}$ | E ± l e ± iβ$_{l}$$^{z}$| $^{2}$, (19) T j/m = µ$_{j}$β$_{m}$ µ$_{m}$β$_{j}$ | t$_{j/m}$ | $^{2}$, T m/j = µ$_{m}$β$_{j}$ µ$_{j}$β$_{m}$ | t$_{m/j}$ | $^{2}$, (20) r$_{12}$$_{/}$$_{3}$ = r$_{12}$ + r$_{2}$$_{/}$$_{3}$e 2 iβ$_{2}$d$_{2}$ 1 − r$_{21}$r$_{2}$$_{/}$$_{3}$e 2 iβ$_{2}$d$_{2}$ , t$_{12}$$_{/}$$_{3}$ = t$_{12}$t$_{2}$$_{/}$$_{3}$e iβ$_{2}$d$_{2}$ 1 − r$_{21}$r$_{2}$$_{/}$$_{3}$e 2 iβ$_{2}$d$_{2}$ , (21)