4 T ( j ) zz = ¯ h 2 π 2 ∫ ∞ 0 dξ ∫ ∞ 0 dkkκ ∑ p,s r$_{j}$$_{−}$r$_{j}$$_{+}$e − 2 κd$_{j}$ 1 − r$_{j}$$_{−}$r$_{j}$$_{+}$e − 2 κd$_{j}$ (23) r$_{123}$$_{/}$$_{4}$ = r$_{123}$ + a$_{123}$r$_{3}$$_{/}$$_{4}$e 2 iβ$_{3}$d$_{3}$ 1 − r$_{321}$r$_{3}$$_{/}$$_{4}$e 2 iβ$_{3}$d$_{3}$ , a$_{123}$ = t$_{123}$t$_{321}$ − r$_{123}$r$_{321}$ = e 2 iβ$_{2}$d$_{2}$ − r$_{12}$r$_{32}$ 1 − r$_{21}$r$_{23}$e 2 iβ$_{2}$$_{d$_{2}$}$, t$_{123}$$_{/}$$_{4}$ = t$_{123}$t$_{3}$$_{/}$$_{4}$e iβ$_{3}$d$_{3}$ 1 − r$_{321}$r$_{3}$$_{/}$$_{4}$e 2 iβ$_{3}$$_{d$_{3}$}$, (22) r$_{1+(2}$$_{−}$$_{)}$ = r + t $^{2}$R$_{2(1)}$ e − 2 κd$_{2(1)}$ 1 − rR$_{2(1)}$e − 2 $_{κd$_{2(1)}$}$, (24) F = ¯ h 2 π 2 ∫ ∞ 0 dξ ∫ ∞ 0 dkkκ ∑ p,s r R$_{2}$e − 2 κd$_{2}$ − R$_{1}$e − 2 κd$_{1}$ N , N = 1 − r ( R$_{1}$e − 2 κd$_{1}$ + R$_{2}$e − 2 $^{κd$_{2}$}$) + ( t 2 r $^{2}$) R$_{1}$R$_{2}$e − 2 κ ( d$_{1}$ + d$_{2}$ $^{)}$. (25) F = T (2) zz − T (1) zz , relations for r$_{123}$$_{/}$$_{4}$ and t$_{123}$$_{/}$$_{4}$ reads with T ( j ) zz being the relevant component of the vacuum- field (Minkowski) stress tensor in the cavity region j . Here κ = √ n 2 $_{c}$( iξ ) ξ $^{2}$/c2 + k 2 is the perpendicular wave vector at the imaginary frequency ( ω = iξ ) in the cavity and r$_{j}$$_{±}$ ( iξ, k ) are the reflection coefficients of the right and left stack of layers bounding the region j . We ob- serve that according to the identity Eq. (13) the tensor components T (1) zz and T (2) zz are related to each other [11]. Noting that r$_{1}$$_{−}$$_{(2+)}$ = R$_{1(2)}$ and using the recurrence relation [cf. Eq. (1)] where r$_{123}$ , r$_{321}$ , t$_{123}$ and t$_{321}$ are given by Eq. (21). In our consideration of the Casimir effect in a planar cavity [11] we have used this result to calculate the reflection coefficient of a slab in front of a cavity mirror. As mentioned, grouping the layers is particularly useful when calculating Fresnel coefficients of periodic media. Consider, for example, the reflection coefficient of the central segment 121 . . . 121 of a Bragg mirror formed by two alternating quarter-wavelength ( d$_{i}$ = λ/ 4 n$_{i}$ ) dielec- tric layers 1 and 2. Noting that at normal ( k = 0) inci- dence e 2 iβ$_{i}$d$_{i}$ = − 1 and a$_{121}$ = − 1, we find from Eq. (22) (with 3 and 4 → 1) that the normal reflection coefficient R$_{N}$ of the segment with N type 2 layers can be calculated using the algorithm: R$_{N}$ = ( R$_{1}$ + R$_{N}$$_{−}$$_{1}$ ) / (1 + R$_{1}$R$_{N}$$_{−}$$_{1}$ ), with R$_{0}$ = 0 and R$_{1}$ ≡ r$_{121}$ = 2 r$_{12}$/ (1 + r 2 $_{12}$) as in- put. Of course, grouping the layers in a different way leads to a different algorithm. Thus, one may show that starting with R$_{2}$ the following algorithm holds: R$_{4}$$_{N}$ = 2 R$_{2}$$_{N}$/ (1 + R 2 $_{N}$), where in each step the num- ber of type 2 layers is doubled [17]. F can be expressed as [11] Since properties of the slab and mirrors are not specified this result is valid for arbitrary media. For local uniform media described by the corresponding refraction indexes, Fresnel coefficients r = r$_{1}$$_{s}$$_{2}$ and t = t$_{1}$$_{s}$$_{2}$ are given by Eq. (21) and those of the mirrors by Eq. (16). In that case, of course, the above result agrees with the one obtained through a conventional way [18]. When removing a mir- ror (letting, say, d$_{2}$ → ∞ ), the above result gives the Casimir force between two arbitrary planar objects, as also obtained recently through a different approach [19]. Evidently, the above nonstandard recurrence relations are particularly convenient when the Fresnel coefficients of a stack are already known (being either calculated sep- arately or measured) or are to be calculated by some method at a later stage. To illustrate this, we (re)derive and generalize to media with arbitrary properties the for- mula for the Casimir force on a slab in a planar cavity [11]. Referring to Fig. 2 for the description of the sys- tem, the Casimir force (per unit area) acting on the slab To summarize, we have emphasized and demonstrated the possibility of grouping the layers into stacks when calculating Fresnel coefficients of a multilayered system. This enables one to consider wave propagation in local as well as in piecewise nonlocal stratified media on an equal footing. As an example, we have shown that the formula for the Casimir on a slab in a planar cavity derived con- sidering local media [11] is also valid when the objects involved have arbitrary properties. The author is indebted to I. Brevik for useful interac- tions and encouragement. This work was supported by the Ministry of Science, Education and Sport of the Re- public of Croatia under Contract No. 098-1191458-2870. z d$_{1}$ d$_{2}$ s d 1 2 M M FIG. 2: A slab (s) in a planar cavity ( n$_{1}$ = n$_{2}$ = n$_{c}$ ) schemat- ically. Cavity mirrors are described by their reflection co- efficients R$_{1}$ and R$_{2}$ and the slab by its Fresnel coefficients r ≡ r$_{1}$$_{/}$$_{2}$ = r$_{2}$$_{/}$$_{1}$ and t ≡ t$_{1}$$_{/}$$_{2}$ = t$_{2}$$_{/}$$_{1}$ . is given by [11] ∗ Electronic address: tomas@thphys.irb.hr [1] M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1980).