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# Recursion relations for generalized Fresnel coefficients: Casimir force in a planar cavity 

Rudjer Boˇ skovi´ c Institute, P. O. B. 180, 10002 Zagreb, Croatia 

Marin-Slobodan Tomaˇs ∗ 

(Dated: September 15, 2018) 

We emphasize and demonstrate that, besides using the usual recursion relations involving succes- sive layers, generalized Fresnel coefficients of a multilayer can equivalently be calculated using the recursion relations involving stacks of layers, as introduced some time ago [M. S. Tomaˇs, Phys. Rev. A 51 , 2545 (1995)]. Moreover, since the definition of the generalized Fresnel coefficients employed does not imply properties of the stacks, these nonstandard recursion relations can be used to cal- culate Fresnel coefficients not only for a local but also for a general multilayer consisting of various types (local, nonlocal, inhomogeneous, etc.) of layers. Their utility is illustrated by deriving a few simple algorithms for calculating the reflectivity of a Bragg mirror and extending the formula for the Casimir force in a planar cavity to arbitrary media. 

Generalized Fresnel coefficients are basic ingredients in the theory of electromagnetic processes and effects in layered systems such as light propagation in stratified media [1, 2], molecular fluorescence and energy-transfer near interfaces [3], (dipole) radiation from multilayers [4– 6], spontaneous emission and light scattering at surfaces [7] and in planar cavities[8, 9], the Casimir effect between multilayered stacks [10–12] etc. Correspondingly, the problem of calculating Fresnel coefficients arises in many area of physics: optics, surface physics, spectroscopy of multilayers, cavity QED, theory of the Casimir effect etc. For systems consisting of few layers, these coefficients are standardly calculated using (ordinary) recursion re- lations involving successive layers. With increasing num- ber of layers, however, this method soon leads to cumber- some formulas and thus becomes impractical. Therefore, despite the possibility of a polynomial representation of the generalized Fresnel coefficients [13], for more complex systems these coefficients are conveniently calculated us- ing the transfer matrix method [1, 2, 5, 12]. 

Based on the definition of the generalized Fresnel co- efficients given in Ref. [6], in our consideration of the Green function for a (local) multilayered system [9] we have used the recursion relations for the reflection ( r ) and transmission ( t ) coefficients involving stacks of lay- ers, which enabled us to write the Green function in a simple compact form. For a stack of layers between, say, layers j and m (denoted shortly as j/m ≡ j . . . m ) these recursion relations read [9] (unless necessary, we omit the polarization index q = p, s ) 

where k is an intermediate layer, as depicted in Fig. 1, and where by using the notation r$_{j/k/m}$ we simply stress 

PACS numbers: 42.25.Bs,42.25.Gy,12.20.Ds,42.50.Lc 

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to which intermediate layer we address. As seen, the above recurrence relations look the same as the standard ones [1, 2] (to which they reduce in case of a system jk/m ≡ jk . . . m ), however, this time they generally in- volve Fresnel coefficients for stacks of layers. 

Although Eqs. (1) and (2) have also been known for some time in optics of multilayers [14], it seems that the possibility of grouping the layers in stacks when calcu- lating Fresnel coefficients is not widely recognized. Since this method is particularly convenient in some cases, e.g., when calculating Fresnel coefficients of periodic media, in this work we explicitly derive the above recurrence rela- tions starting from the definition of the generalized Fres- 

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