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and that r$_{j/l}$r$_{l/j}$ is given by a similar expression, we find that actually [15] 

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] 

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which follows from Eq. (9). In this way, we obtain 

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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] 

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 

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 

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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 

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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)] 

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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]. 

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] 

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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 

