Recursion relations for generalized Fresnel coefficients: Casimir force in a planar cavity
aa r X i v : . [ qu a n t - ph ] A p r Recursion relations for generalized Fresnel coefficients: Casimir force in a planarcavity
Marin-Slobodan Tomaˇs ∗ Rudjer Boˇskovi´c Institute, P. O. B. 180, 10002 Zagreb, Croatia (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 therecursion relations involving stacks of layers, as introduced some time ago [M. S. Tomaˇs, Phys. Rev.A , 2545 (1995)]. Moreover, since the definition of the generalized Fresnel coefficients employeddoes 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 varioustypes (local, nonlocal, inhomogeneous, etc.) of layers. Their utility is illustrated by deriving a fewsimple algorithms for calculating the reflectivity of a Bragg mirror and extending the formula forthe Casimir force in a planar cavity to arbitrary media. PACS numbers: 42.25.Bs,42.25.Gy,12.20.Ds,42.50.Lc
Generalized Fresnel coefficients are basic ingredientsin the theory of electromagnetic processes and effects inlayered systems such as light propagation in stratifiedmedia [1, 2], molecular fluorescence and energy-transfernear 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 betweenmultilayered stacks [10–12] etc. Correspondingly, theproblem of calculating Fresnel coefficients arises in manyarea of physics: optics, surface physics, spectroscopy ofmultilayers, cavity QED, theory of the Casimir effect etc.For systems consisting of few layers, these coefficientsare 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 ofthe generalized Fresnel coefficients [13], for more complexsystems 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 theGreen function for a (local) multilayered system [9] wehave 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 asimple compact form. For a stack of layers between, say,layers j and m (denoted shortly as j/m ≡ j . . . m ) theserecursion relations read [9] (unless necessary, we omit thepolarization index q = p, s ) r j/m ≡ r j/k/m = r j/k + t j/k t k/j r k/m e iβ k d k − r k/j r k/m e iβ k d k , (1) t j/m ≡ t j/k/m = t j/k t k/m e iβ k d k − r k/j r k/m e iβ k d k . (2)where k is an intermediate layer, as depicted in Fig. 1,and where by using the notation r j/k/m we simply stress to which intermediate layer we address. As seen, theabove recurrence relations look the same as the standardones [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. n m n j n k z=d k z=0z=0z=0 FIG. 1: Stack considered when deriving recursion relations forgeneralized Fresnel coefficients shown schematically. Layers j , k and m are local and described by (complex) refractionindexes n a ( ω ) = p ε a ( ω ) µ a ( ω ), a = j, k, m , whereas stacksbetween them are unspecified. Arrows indicate propagationof a wave incident on the stack. A shifted- z coordinate systemis adopted, as explained in the text. Although Eqs. (1) and (2) have also been known forsome time in optics of multilayers [14], it seems that thepossibility of grouping the layers in stacks when calcu-lating Fresnel coefficients is not widely recognized. Sincethis method is particularly convenient in some cases, e.g.,when calculating Fresnel coefficients of periodic media, inthis work we explicitly derive the above recurrence rela-tions starting from the definition of the generalized Fres-nel coefficients [6] (this derivation was omitted in Ref.[9]) and demonstrate their equivalence with the standardrecursion relations involving successive layers. We usethis opportunity to emphasize that Eqs. (1) and (2)are also valid for piecewise nonlocal stratified media andadapt their derivation accordingly. We illustrate theirusefulness on a few simple examples including the theoryof the Casimir effect in (nonlocal) multilayers.Consider a linear system consisting of isotropic layers0 . . . n ≡ /n . The electric field of a wave incident onthe system with the parallel wave vector k can in a locallayer l be written as E l ( k , ω, r ) = E l ( k , ω, z ) exp( i k · r k ) , E l ( k , ω, z ) = ˆ e + l ( k ) e iβ l z E + l + ˆ e − l ( k ) e − iβ l z E − l , (3)where β l = p k l − k , with k l = n l ω/c = √ ε l µ l ω/c . Fora local system, the coefficients E ± l are determined bymatching the field at boundaries of each layer and canbe entirely expressed in terms of the incident-wave am-plitude and the generalized reflection and transmissioncoefficients 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 incomingwave electric-field amplitude (factors multiplying ˆ e ’s) atthe corresponding stack’s boundary considering only lay-ers of the stack, i.e. as if there are no other layers in thesystem. Similarly, a transmission coefficient t of a stackis the ratio of the transmitted to incident wave amplitudecalculated at the corresponding stack’s boundaries as ifthere are no other layers present in the system. Notethat these definitions do not imply any property of thestack. Therefore, we can use these coefficients to describewave propagation not only in a local system but also ina general system consisting of various (local, nonlocal,inhomogeneous, unspecified etc.) layers. In calculatingthese coefficients it is convenient to adopt a (shifted- z )representation for the field [9] in which 0 ≤ z ≤ d l in anyfinite layer l , whereas −∞ < z ≤ ≤ z < ∞ inexternal layers l = 0 and l = n , respectively.On the basis of the above definition we can straightfor-wardly calculate the recurrence relations for generalizedFresnel coefficients r j/m and t j/m of the stack of layersbetween local layers j and m . Indeed, according to itsdefinition, to calculate the reflection coefficient r j/m weconsider the wave incident from the layer j upon the sys-tem j/m (see Fig. 1). Then E − m = 0 and by definition E − j = r j/m E + j . (4)However, r j/m can also be calculated by consideringtransmission of the wave to an intermediate (local) layer k and its subsequent partial transmission back to thelayer j . Considering the field at the relevant boundariesand using the above definitions of the generalized Fresnel coefficients, the amplitudes of the wave are then relatedby the following set of equations E − j = r j/k E + j + t k/j E − k , (5a) E + k = t j/k E + j + r k/j E − k , (5b) e − iβ k d k E − k = r k/m e iβ k d k E + k . (5c)Eliminating E ± k and comparing the ratio E − j /E + j withthat from Eq. (4), we arrive at Eq. (1). Similarly, thetransmission coefficient t j/m of the stack is by definitiongiven by E + m = t j/m E + j . (6)On the other hand, by considering transmission of thewave 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 E + m = t k/m e iβ k d k E + k . (7)Proceeding as before, we obtain Eq. (2).Clearly, Fresnel coefficients must not depend on thechoice of the intermediate layer in Eqs. (1) and (2).Therefore, to prove the consistency of these recurrencerelations, we must show that r j/k/m = r j/l/m , t j/k/m = t j/l/m , (8)where l denotes some other intermediate local layer. Toprove this for r j/m , we rewrite Eq. (1) in the form r j/m = r j/k/m = r j/k + a j/k r k/m e iβ k d k − r k/j r k/m e iβ k d k ,a j/k = t j/k t k/j − r j/k r k/j = a k/j , (9)consider the layer l between layers k and m and applyEq. (9) to the reflection coefficient r k/m = r k/l/m in thisvery same equation. Rearranging the terms and usingagain Eq. (9) to recognize reflection coefficients r j/l and r l/j , we find r j/k/m = r j/l + ˜ a j/l r l/m e iβ l d l − r l/j r l/m e iβ l d l , ˜ a j/l = a j/k a l/k e iβ k d k − r j/k r l/k − r k/j r k/l e iβ k d k (10)Now, noting from Eqs. (2) and (9) that t j/l t l/j = ( a j/k + r j/k r k/j )( a l/k + r l/k r k/l ) e iβ k d k (1 − r k/j r k/l e iβ k d k ) (11)and that r j/l r l/j is given by a similar expression, we findthat actually [15] a j/l = t j/l t l/j − r j/l r l/j = ˜ a j/l . (12)Accordingly, the right-hand side of Eq. (10) is, uponusing Eq. (9), indeed identified as r j/l/m . To prove theequality of the transmission coefficients in Eq. (8) wefirst apply Eq. (2) to the coefficient t k/m = t k/l/m andthen use the identity ( r k/m = r k/l/m ) [11](1 − r l/k r l/m e iβ l d l )(1 − r k/j r k/m e iβ k d k ) =(1 − r k/j r k/l e iβ k d k )(1 − r l/j r l/m e iβ l d l ) , (13)which follows from Eq. (9). In this way, we obtain 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 iβ k d k )(1 − r l/j r l/m e iβ l d l ) , (14)which is according to Eq. (2) equal to t j/l/m .In most textbook approaches to the wave propagationin layered media the basic ingredients in the calculationof the generalized Fresnel coefficients are coefficients r jk and t jk for the interface between two neighbouring localmedia j and k . With unit polarization vectors in Eq. (3)[9] ˆ e ± pl = ∓ β l ˆ k + k ˆ z k l , ˆ e ± sl = ˆ k × ˆ z , (15)and applying the usual boundary conditions for the field,it is straightforward to show that the above definition ofFresnel coefficients leads to the standard single-interfacecoefficients r jk = β j − γ jk β k β j + γ jk β k = − r kj , (16a) t jk = s γ jk γ sjk (1 + r jk ) = µ k β j µ j β k t kj , (16b)where γ pjk = ε j /ε k and γ sjk = µ j /µ k . As can be easilyverified, these coefficients obey the Stokes relation a jk ≡ t jk t kj − r jk r kj = 1 . (17)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 , thatis [2, 9] t j/m = µ m β j µ j β m t m/j . (18)Indeed, through Eq. (2), it certainly holds for a three-layer system jkm owing to the symmetry property of the single-interface transmission coefficients t jk and t km . As-suming in Eq. (2) the same symmetry properties of thegeneralized coefficients t j/k and t k/m , we immediatelyfind that Eq. (18) is obeyed. Accordingly, using the in-duction argument, we may conclude that this equation isfor 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 forwaves incident on it from either side. Indeed, the Poynt-ing vector of the upward/downward-propagating wave ina layer l is according to Eqs. (3) and (15) given by P ± l ( k , ω, z ) = c π Re √ η l k ± β l ˆ z k l | E ± l e ± iβ l z | , (19)with η pl = η sl ∗ = 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 onit [given by the ratios of the respective transmitted- toincident-energy fluxes T j/m = P + mz ( k , ω, /P + jz ( k , ω, T m/j = P − jz ( k , ω, /P − mz ( k , ω, T j/m = µ j β m µ m β j | t j/m | , T m/j = µ m β j µ j β m | t m/j | , (20)can only be equal if Eq. (18) is fulfilled. Thus, beinga consequence of the reciprocity property of the electro-magnetic field, this equation is valid for a quite generalclass of (nongyrotropic) media [16].As follows from the above results, recurrence relationsfor Fresnels coefficients of a multilayered system can bewritten in a number of ways depending on number of in-termediate local layers. We illustrate this by calculatingFresnel coefficients of few simple systems. We first con-sider a 12 / a = 1, Eq. (9)leads to standard forms of the recurrence relations forFresnel coefficients [2] r / = r + r / e iβ d − r r / e iβ d , t / = t t / e iβ d − r r / e iβ d , (21)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. The second set of recurrencerelations for r / and t / reads r / = r + a r / e iβ d − r r / e iβ d ,a = t t − r r = e iβ d − r r − r r e iβ d ,t / = t t / e iβ d − r r / e iβ d , (22)where r , r , t and t are given by Eq. (21). Inour consideration of the Casimir effect in a planar cavity[11] we have used this result to calculate the reflectioncoefficient of a slab in front of a cavity mirror.As mentioned, grouping the layers is particularly usefulwhen calculating Fresnel coefficients of periodic media.Consider, for example, the reflection coefficient of thecentral segment 121 . . .
121 of a Bragg mirror formed bytwo alternating quarter-wavelength ( d i = λ/ n i ) dielec-tric layers 1 and 2. Noting that at normal ( k = 0) inci-dence e iβ i d i = − a = −
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 calculatedusing the algorithm: R N = ( R + R N − ) / (1 + R R N − ),with R = 0 and R ≡ r = 2 r / (1 + r ) as in-put. Of course, grouping the layers in a different wayleads to a different algorithm. Thus, one may showthat starting with R the following algorithm holds: R N = 2 R N / (1 + R N ), where in each step the num-ber of type 2 layers is doubled [17].Evidently, the above nonstandard recurrence relationsare particularly convenient when the Fresnel coefficientsof a stack are already known (being either calculated sep-arately or measured) or are to be calculated by somemethod at a later stage. To illustrate this, we (re)deriveand 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 z d d d M M
FIG. 2: A slab (s) in a planar cavity ( n = n = n c ) schemat-ically. Cavity mirrors are described by their reflection co-efficients R and R and the slab by its Fresnel coefficients r ≡ r / = r / and t ≡ t / = t / . is given by [11] F = T (2) zz − T (1) zz , T ( j ) zz = ¯ h π Z ∞ dξ Z ∞ dkkκ X p,s r j − r j + e − κd j − r j − r j + e − κd j (23)with T ( j ) zz being the relevant component of the vacuum-field (Minkowski) stress tensor in the cavity region j .Here κ = p n c ( iξ ) ξ /c + k is the perpendicular wavevector at the imaginary frequency ( ω = iξ ) in the cavityand r j ± ( iξ, k ) are the reflection coefficients of the rightand left stack of layers bounding the region j . We ob-serve that according to the identity Eq. (13) the tensorcomponents T (1) zz and T (2) zz are related to each other [11].Noting that r − (2+) = R and using the recurrencerelation [cf. Eq. (1)] r − ) = r + t R e − κd − rR e − κd , (24) F can be expressed as [11] F = ¯ h π Z ∞ dξ Z ∞ dkkκ X p,s r R e − κd − R e − κd N ,N = 1 − r ( R e − κd + R e − κd )+ ( t − r ) R R e − κ ( d + d ) . (25)Since properties of the slab and mirrors are not specifiedthis result is valid for arbitrary media. For local uniformmedia described by the corresponding refraction indexes,Fresnel coefficients r = r s and t = t s 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 obtainedthrough a conventional way [18]. When removing a mir-ror (letting, say, d → ∞ ), the above result gives theCasimir force between two arbitrary planar objects, asalso obtained recently through a different approach [19].To summarize, we have emphasized and demonstratedthe possibility of grouping the layers into stacks whencalculating Fresnel coefficients of a multilayered system.This enables one to consider wave propagation in local aswell as in piecewise nonlocal stratified media on an equalfooting. As an example, we have shown that the formulafor the Casimir on a slab in a planar cavity derived con-sidering local media [11] is also valid when the objectsinvolved have arbitrary properties.The author is indebted to I. Brevik for useful interac-tions and encouragement. This work was supported bythe Ministry of Science, Education and Sport of the Re-public of Croatia under Contract No. 098-1191458-2870. ∗ Electronic address: [email protected][1] M. Born and E. Wolf,
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