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nel coefficients [6] (this derivation was omitted in Ref. [9]) and demonstrate their equivalence with the standard recursion relations involving successive layers. We use this opportunity to emphasize that Eqs. (1) and (2) are also valid for piecewise nonlocal stratified media and adapt their derivation accordingly. We illustrate their usefulness on a few simple examples including the theory of the Casimir effect in (nonlocal) multilayers. 

Consider a linear system consisting of isotropic layers 0 . . . n ≡ 0 /n . The electric field of a wave incident on the system with the parallel wave vector k can in a local layer l be written as 

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where β$_{l}$ = √ k 2 l − k $^{2}$, with k$_{l}$ = n$_{l}$ω/c = √ ε$_{l}$µ$_{l}$ω/c . For a local system, the coefficients E ± l are determined by matching the field at boundaries of each layer and can be entirely expressed in terms of the incident-wave am- plitude and the generalized reflection and transmission coefficients of the corresponding stacks of layers [6, 9]. 

We define generalized Fresnel coefficients of a stack be- tween two (local) layers as follows [6]. A reflection coeffi- cient r of a stack is the ratio of the reflected to incoming wave electric-field amplitude (factors multiplying ˆ e ’s) at the corresponding stack’s boundary considering only lay- ers of the stack, i.e. as if there are no other layers in the system. Similarly, a transmission coefficient t of a stack is the ratio of the transmitted to incident wave amplitude calculated at the corresponding stack’s boundaries as if there are no other layers present in the system. Note that these definitions do not imply any property of the stack. Therefore, we can use these coefficients to describe wave propagation not only in a local system but also in a general system consisting of various (local, nonlocal, inhomogeneous, unspecified etc.) layers. In calculating these coefficients it is convenient to adopt a (shifted- z ) representation for the field [9] in which 0 ≤ z ≤ d$_{l}$ in any finite layer l , whereas −∞ < z ≤ 0 and 0 z < ∞ in external layers l = 0 and l = n , respectively. 

On the basis of the above definition we can straightfor- wardly calculate the recurrence relations for generalized Fresnel coefficients r$_{j/m}$ and t$_{j/m}$ of the stack of layers between local layers j and m . Indeed, according to its definition, to calculate the reflection coefficient r$_{j/m}$ we consider the wave incident from the layer j upon the sys- tem j/m (see Fig. 1). Then E − m = 0 and by definition 

However, r$_{j/m}$ can also be calculated by considering transmission of the wave to an intermediate (local) layer k and its subsequent partial transmission back to the layer j . Considering the field at the relevant boundaries and using the above definitions of the generalized Fresnel 

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coefficients, the amplitudes of the wave are then related by the following set of equations 

Eliminating E ± k and comparing the ratio E − j /E + j with that from Eq. (4), we arrive at Eq. (1). Similarly, the transmission coefficient t$_{j/m}$ of the stack is by definition given by 

On the other hand, by considering transmission of the wave to the intermediate layer k and its subsequent par- tial transmission to the layer m , we find that the ampli- tudes E + m and E + j are related by Eq. (5), with Eq. (5a) replaced by 

Proceeding as before, we obtain Eq. (2). 

Clearly, Fresnel coefficients must not depend on the choice of the intermediate layer in Eqs. (1) and (2). Therefore, to prove the consistency of these recurrence relations, we must show that 

where l denotes some other intermediate local layer. To prove this for r$_{j/m}$ , we rewrite Eq. (1) in the form 

consider the layer l between layers k and m and apply Eq. (9) to the reflection coefficient r$_{k/m}$ = r$_{k/l/m}$ in this very same equation. Rearranging the terms and using again Eq. (9) to recognize reflection coefficients r$_{j/l}$ and r$_{l/j}$ , we find 

Now, noting from Eqs. (2) and (9) that 

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