Ann. Phys. (Berlin) 523, No. 12, 1029 – 1044 (2011) / DOI 10.1002/andp.201100141 Equilibrium ensembles of electromagnetic thermal radiation in layered media Bair V. Budaev∗ and David B. Bogy∗∗ University of California, Etcheverry Hall, Berkeley, CA 94720, USA Received 19 June 2011, revised 12 October 2011, accepted 8 November 2011 by F. W. Hehl Published online 5 December 2011 Key words Thermal radiation, thermal equilibrium, Planck’s law. This paper describes equilibrium ensembles of thermally excited electromagnetic fields in layered media. The obtained results complement Planck’s law of thermal radiation that determines the spectrum of the radiation but supplies little information about the ensemble of eigenfields (normal modes) excited in the medium. The developments regarding these ensembles presented in this paper make it possible to apply perturbation techniques for the analysis of the ensembles of radiated fields in layered media with a steady heat flux. c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction For several decades layered structures have been used as efficient means for thermal management, and until recently the theoretical predictions of thermal resistance of such structures were reasonably close to some experimental observations, although the predictions of other observations were obviously incorrect. For example, the conventional theory implies that the thermal resistance of a vacuum gap between two halfspaces of identical materials does not depend on the gaps’s width, which means that even a vanishingly narrow gap has a finite non-vanishing thermal resistance, which obviously contradicts common sense. This deficiency of the theory implies that there is a fundamental flaw in the conventional approach to radiative heat transport that shows up in cases involving micro and nano scale structures. In order to address this flaw it is necessary to understand which assumptions of the conventional theory do not hold in nano and micro scales and then revise the theory retaining as few restrictive assumptions as possible. It has long been established that thermal radiation is a consequence of electrodynamics, thermodynamics and the atomistic theory of matter. According to atomistic theory matter is composed of particles with a somewhat complicated pattern of electric charges performing perpetual thermal motion. On the other hand the theory of electrodynamics implies that accelerating electric charges radiate electromagnetic waves that affect the motion of other charged particles, so that the particles exchange energy even if they never contact each other. Then, according to thermodynamics, any isolated domain of space that does not exchange energy with its exterior eventually comes to thermodynamical equilibrium, which, in particular, means that the average amount of electromagnetic energy emitted per unit time from any sub-domain equals the average amount of electromagnetic energy absorbed by this sub-domain, and that the spectra of such emitted and absorbed radiations can be described by the celebrated Planck’s theory of thermal radiation. The most known formula of Planck’s theory Et (ω, T ) = ∗ ∗∗ ω 3 π 2 c30 (eω/κT − 1) , (1.1) Corresponding author E-mail: bair@berkeley.edu E-mail: dbogy@berkeley.edu c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1030 B. V. Budaev and D. B. Bogy: Radiation in layered media where c0 is the speed of light in vacuum and k is the Boltzmann constant, represents the spectrum of electromagnetic thermal radiation in a sufficiently large equilibrium system at temperature T . To understand the meaning of this formula it is convenient to consider a material body that contains a vacuum cavity. Due to thermal radiation the cavity contains electromagnetic waves, and the average amount of electromagnetic energy contained in the cavity of volume V can be computed by the integral of a product ∞ Pt2 (ω, T )dNV (ω), (1.2) EV = 2 0 where Pt2 (ω, T ) is the average energy of a single wave at frequency ω, the factor “2” accounts for two possible polarizations, and NV (ω) is the number of waves with frequencies below ω, so that dNV (ω) = NV (ω + dω) − NV (ω) represents the number of waves with frequencies in the band (ω, ω + dω). Then, quantum mechanics [1, 2] provides that the average energy of a single wave has the value Pt2 (ω, T ) = ω eω/κT −1 . (1.3) Weyl’s theorem [3–5] implies that if the volume V of the cavity is sufficiently large compared to the cubed dominant wavelength of thermal radiation then V ω3 V 2πc0 , (1.4) + o NV (ω) = , λ= 6π 2 c30 λ3 ω and from (1.2)–(1.4) it follows that the average thermal energy stored in a cavity of sufficiently large volume V can be approximated by the formula ∞ ∞ V ω 3 dω Et (ω, T )dω, (1.5) ≡ V EV ≈ 2 3 π c0 0 eω/κT − 1 0 which shows that Et (ω, T ) in (1.1) may be treated as the spectrum of thermal radiation (energy per unit volume per unit frequency band) in equilibrium at temperature T . Planck’s law is very general in the sense that it does not depend on such details as the material properties or the shape of the cavity. This generality accounts for the power and beauty of Planck’s law, but at the same time, due to its generality, Planck’s law does not provide much information about the structure of thermally excited electromagnetic fields, and this makes it difficult to study radiative heat transport in the considered system by use of perturbation methods on an equilibrium system, since this equilibrium system is not well-described. To illustrate the insufficiency of the information provided by Planck’s law we consider an ensemble of plane electromagnetic waves described by the formula Aei(ex x+ey y+ez z)ω/c0 −iωt , (1.6) where e = (ex , ey , ez ) is a random vector of unit length, c0 is the speed of light in a vacuum, and A is a 2 random complex-valued amplitude. Each of these random fields has the energy density E = 0 |A| , where iβ 0 is the dielectric permittivity of the vacuum. Therefore, if A = e Et (ω, T )/0, where β is a random phase shift, then the average energy density of the ensemble of waves (1.6) has Planck’s spectrum (1.1) of thermal radiation in a sufficiently large equilibrium system. This observation suggests that ensembles of waves (1.6) with random e and β can be used to model thermally excited electromagnetic fields, and the possibility of representing any field in any domain by a Fourier superposition of the waves (1.6) provides further support for this suggestion. Nevertheless, the fields (1.6) can not be used for the analysis of thermal radiation even in such a simple domain as a vacuum gap 0 < x < H between two half-spaces because c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 523, No. 12 (2011) 1031 no one of such fields, if taken alone, can be continued to a field satisfying the equations of electrodynamics in the entire system consisting of the material half-spaces and the vacuum gap. As follows from the considered example, thermal radiation in layered structures can not be modeled in terms of ensembles of plane waves because no single such wave alone can be excited in such structures. However, the conventional approach to radiative heat transport considers ensembles of plane electromagnetic waves, which implies that this approach is based on an unjustified assumption, and, therefore, it can not lead to reliable results. A similar situation is observed in the acoustic mismatch theory of interface thermal resistance, widely known as the theory of Kapitsa resistance, which studies heat transport carried by acoustic waves across the interface between different materials. This theory is essentially parallel to the theory of radiative heat transport, and in some cases it predicts relatively good approximations, but in other cases it predicts up to two orders of magnitude errors. With the goal of developing a reliable model of Kapitsa interface thermal resistance we recently revisited the conventional theory and found [6–8] that three things needed to be addressed to get a logically consistent model: first we needed to remove the often-used Sommerfeld radiation condition, which is not relevant in problems of heat transport; second modify Plank’s distribution to make it valid for non-equilibrium systems with constant heat flux; and third understand the structure of the equilibrium ensembles of thermally excited wave fields, which can then be used as the foundation for perturbation methods. The results presented in [6–8] show correct orders of magnitude Kapitsa resistance for the first time without any added elements such as surface roughness. Inspired by this success we applied our methodology to another notorious problem, that of nano-scale thermal radiation across a narrow vacuum gap [9], and we obtained agreement with recent experimental observations that the radiative conductance between two identical materials at different temperatures becomes unbounded as O(1/H 2 ) as the gap’s width H vanishes. This was accomplished without employing extraneous point sources, roughness, or non-linearities, which are commonly used without foundation to improve the predictions based on the classical approach.. Numerous discussions of our work with reviewers and others made it clear that since it consists of three quite different elements, which independently modify conceptually different parts of the conventional theory, each of these elements should be published in all detail separately, without being attached to any particular problem, whether it is Kapitsa resistance or radiative transport across a narrow gap. Therefore, in addition to the present paper devoted to equilibrium ensembles of wave fields in layered structures, we have published a paper devoted to the extension of Planck’s distribution to non-equilibrium systems with a constant heat flux [10]. In the next paper of the sequel (under preparation) we will describe non-equilibrium ensembles of wave fields in arbitrary layered structures. This paper is devoted to the analysis of equilibrium ensembles of electromagnetic eigenfields in basic layered structures. In Sect. 2 we outline the main results and discuss their importance for understanding radiative heat transfer in such structures. Section 3 describes the ensemble of eigenfields (2.2) in the entire space. The results of this section are neither novel nor especially interesting if considered alone, but they play an important role in the subsequent developments of the next three sections dealing with two halfspaces separated by a stack of layers of different materials. Although most of the material of this section is well-known, it is spread in the literature devoted to very different fields, including engineering, acoustics, electromagnetism, general physics and mathematics. Correspondingly, the results and the concepts are formulated in such different forms that in order for one to be able to use them together it is necessary to introduce a unified notation and summarize the assumptions, because otherwise, approximations that can be taken for granted in some fields are likely to be used in the fields where they are not valid. In Sect. 4 we describe a complete system of eigenfields in a sandwich-like structure, and in Sect. 5 this system is used to obtain the main result of the paper, which is the description of equilibrium ensembles of thermally radiated fields in a sandwich-like system with a stack of layers between two half-spaces of possibly different materials. The paper is concluded by Sections 6 and 7 which discuss the importance of the obtained results for overcoming deficiencies of the conventional approach to radiative heat transfer across a nanoscale gap. www.ann-phys.org c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1032 B. V. Budaev and D. B. Bogy: Radiation in layered media 2 Preview of the main results and their importance The above example with the ensemble of plane waves (1.6) serves as a reminder of a well known but often overlooked fact that a statistical model of a thermally excited electromagnetic field must consist of eigenfields (also called normal modes) of the considered domain, which satisfy all of the field equations including appropriate boundary and interface conditions. Since no single one of the fields (1.6) satisfies the interface conditions, these fields cannot be used for modeling thermal radiation in any layer of finite width. Therefore, it becomes clear that a study of thermal radiation is impossible without a thorough analysis of the ensembles of eigenfields of the domains of interest. It is well known that eigenfields are determined by the equation, by the domain and by its boundary conditions. Thus, the eigenfields in the entire space are represented by (1.6). The eigenfields of a vacuum half space x > 0 are delivered by a more complex formula ex > 0, (2.1) A ei(ex x+ey y+ez z)ω/c0 ± ei(−ex x+ey y+ez z)ω/c0 e−iωt , where the choice of the sign “±” is determined by the type of boundary conditions on the face x = 0, which depends on the polarization of the considered wave. As for the eigenfields of a layered medium, they have a more complex structure that cannot be described by a single analytic expression. Consider, for example, the case illustrated in Fig. 1, where the space is subdivided into N domains by the planes x = aν , where a1 < a2 < · · · < aN −1 . Assuming that cν is the wave speed in the ν-th layer aν−1 < x < aν , it is straightforward to derive that the eigenfields of this structure can be represented by the expressions ν ν ν ν ν ν aν−1 < x < aν , (2.2) Aν ei(ex x+ey y+ez z)ω/cν + Bν ei(−ex x+ey y+ez z)ω/cν e−iωt , where the unit vectors eν = (eνx , eνy , eνz ) and the pairs of amplitudes (Aν , Bν ) associated with each of the N domains are uniquely defined in terms of the unit vector em and the pair (Am , Bm ), associated with any pre-assigned m-th layer, which can be selected for convenience. (A1 , B1 ) (A2 , B2 ) ... (Am , Bm ) . . . (AN , BN ) eN em e3 e2 e1 a1 eN−1 em−1 ... a2 am−1 aN−1 Fig. 1 (online colour at: www.annphys.org) Eigenfields of layered structures. Direction of propagation em in the m-th layer determines the directions of propagation eν in all other layers. Once m is selected, any vector eν and the pair (Aν , Bν ) are determined by em and (Am , Bm ). To represent the field in the ν-th layer aν−1 < x < aν in terms of the coefficients Am , Bm associated with the layer am−1 < x < am , it suffices to recall [11] that in adjacent domains the directions of propagation eν−1 and eν are related by Snell’s law, and the amplitudes are related by Aj−1 Aj = Tj−1,j , (2.3) Bj−1 Bj where Tj−1,j is the transmission matrix of the (j − 1)-th interface defined by simple formulas involving parameters of both of the involved layers. Then, applying this relationship recursively, we find that the fields in the ν-th and the m-th layers are related by Aν Am = Tνm , (2.4) Bν Bm c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 523, No. 12 (2011) where Tνm = ⎧ ⎨T ν,ν+1 Tν+1,ν+2 ⎩T−1 , 1033 . . . Tm−1,m , if ν ≤ m, if m > ν, mν (2.5) may be viewed as the transmission matrix between the ν-th and m-th layers. It should be noticed that this reasoning is a particular case of a more general theory surveyed in [12]. This example shows that in order to describe a complete set of eigenfields of a layered medium it suffices to describe these fields in any one particular layer. Although this observation is quite apparent, it leads to a critically important conclusion that the analysis of an equilibrium ensemble of electromagnetic fields in a multilayered medium can be reduced to the analysis of the fields in one basic layer, which may be selected for convenience. This implies that the eigenfields of a layered medium admit parametrization in terms of the unit vector em and the pair (Am , Bm ) of the complex amplitudes of the field (2.2) in the m-th layer. Therefore, in order to describe a statistical ensemble of eigenfields of the layered medium we assume that em and (Am , Bm ) are random, and then we find their distributions, which are compatible with the relevant laws of wave propagation. This paper analyzes such distributions, and its main result can be summarized by the statement that the pair of amplitudes (Am , Bm ) describing an equilibrium ensemble of eigenfields of a multilayered media at temperature T admits the representation Am = Λm cos ηm eiβm +iαm , Bm = Λm sin ηm eiβm −iαm , (2.6) where Λm = Λm (ω) is a real-valued amplitude determined by the energy density of the selected m-th layer, αm and βm are random phase shifts, and ηm = ηm (ω) is a random number uniformly distributed over a domain Rm on the (ω, ηm )-plane, which is symmetric, as shown in Fig. 2, with respect to the axis η = π4 , and that the width of this domain is determined by the material parameters. Thus, in the limiting case when the half-spaces have identical parameters forming a homogeneous space this domain expands to the maximal possible strip 0 ≤ η ≤ π/2. In the opposite limiting case when there is no transmission between the half-spaces the domain Rm collapses to the line η = π/4. ηm ηm π 2 π 2 π 4 0 π 4 Rm ω Rm 0 ω Fig. 2 (online colour at: www.ann-phys.org) Distribution of ηm (ω) in equilibrium ensembles. In an equilibrium ensemble ηm (ω) is distributed over a domain Rm symmetric with respect to the axis η = π4 . The width of Rm is determined by the contrast between materials. The higher contrast corresponds to narrower Rm . 3 An equilibrium ensemble of fields in a homogeneous space To study the spectra of thermal radiation it is convenient to adopt the point of the view that electric charges interact with each other indirectly, through the electromagnetic field. Thus, the radiation of energy may be considered as the process whereby a particle passes some of its energy to the electromagnetic field, while the absorbtion of energy is the process whereby a particle takes energy from the field. From this point of view the energy concentrated in a domain G consists of the energy of particles located in G and www.ann-phys.org c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1034 B. V. Budaev and D. B. Bogy: Radiation in layered media the energy of the field in G. If the system is in thermodynamic equilibrium then each form of energy is itself thermodynamically balanced [1, 2, 13], which implies that in equilibrium thermal radiation may be considered as an equilibrium ensemble of electromagnetic oscillators that can be studied independently of any other form of heat. It is well known [1,2,13] that electromagnetic energy is evenly divided between two fields with different polarizations and that each of these fields can be described by the equation 1 ∂2Φ = ∇2 Φ, c2 ∂t2 1 c= √ , μ (3.1) where c is the speed of light in the material with permittivity and permeability μ. If the electromagnetic field is localized in some domain G then the theory of partial differential equations implies [4] that any solution of Eq. (3.1), accompanied by appropriate boundary conditions, can be represented as a superposition Φ(r, t) = umj (r)e−iωm t (3.2) m j where r = (x, y, z) is the position-vector of the observer, ωm is the spectral frequency, which takes one of the values determined by the shape of the domain G and by the boundary conditions, and umj (r), where j = 1, 2, . . . , are independent solutions of the Helmholtz equation ∇2 u + ω 2 c (3.3) u = 0, with the spectral frequency ω = ωm . The set of all spectral frequencies is referred to as the spectrum, and the fields umj (r), where j = 1, 2, . . . , are referred to as eigenfields, or normal modes, corresponding to the spectral frequency ωm . The theory of partial differential equations implies that to compute the energy of the field in a domain it suffices to determine the spectrum of this domain and to compute the energy of each of the eigenfields. It may be extremely difficult to describe the spectrum and eigenfields of an arbitrary domain, but in the idealized case when G occupies the entire space, both of these problems have tractable solutions that admit many different representations. For example, straightforward inspection shows that the wave Eq. (3.1) is satisfied by any field Φ(r, t; k) = u(r; k)e−iωt , r = (x, y, z), (3.4) where the frequency may take any non-negative value ω ≥ 0 and u(r; k) = A(k)e2πik·r + B(k)e2πik ·r (3.5) is a superposition of two plane waves with complex-valued coefficients A(k) and B(k) depending on the wave-vector k = (kx , ky , kz ) with kx ≥ 0, which satisfies the dispersion relation 2πc|k| = ω, (3.6) and uniquely determines the associated wave-vector k = (−kx , ky , kz ). Let k be a random vector related to the frequency ω by the dispersion Eq. (3.6), and let A(k) and B(k) be random complex numbers. Then, formulas (3.4), (3.5) determine a statistical ensemble of eigenfields at frequency ω whose average energy density can be computed from the point of view of electrodynamics as well as from the point of view of statistical physics. c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 523, No. 12 (2011) 1035 From the point of view of electrodynamics the average energy density of the ensemble of waves (3.4) with all possible wave vectors k and with random amplitudes A(k), B(k) has the value Ω 1 (k) |A(k)|2 | + B(k)|2 k 2 dk dS(e), (3.7) E= 2 S 0 where Ω is the maximal frequency, e = k/k is the unit vector, dS(e) is the area element on the unit half-sphere S = {e : |e| = 1, ex ≥ 0}, the brackets · denote the process of averaging over the random numbers A(k) and B(k), and = (k) is the permittivity of the medium, which may depend on the frequency ω = 2πkc. To compute the average energy density of the ensemble of waves (3.4) from the point of view of statistical physics it suffices to treat the field u(r; k)e−iωt as a single harmonic oscillator at frequency ω. Then, assuming that this oscillator belongs to an equilibrium ensemble at temperature T we recall [2, 13] that its average energy density has the value P 2 (ω, T ) = Pt2 (ω, T ) + P02 (ω, T ), (3.8) where the first term introduced by (1.3) represents the thermal energy while the second term P02 (ω, T ) = ω , 2 (3.9) represents a quantum mechanical zero-point energy that is unrelated to thermal processes. Correspondingly, the average energy density of the entire ensemble can be represented as Ω 2 E= (3.10) Pt (ω, T ) + P02 (ω, T ) dW (ω), 0 where W (ω) is the volume of the domain occupied by the wave vectors k, with kx ≥ 0, corresponding to the frequencies below ω. It is obvious from (3.6) that W (ω) is the volume of the half-sphere of radius |k| = ω/2πc. Therefore, k ω 2π ω 3 , (3.11) ≡ ξ 2 dξ dS(e), k= W (ω) = 3 2πc 2πc 0 S and differentiating (3.11) by k we first find that 2 dW (ω) = k dS(e) dk, ω = 2πck, (3.12) and then, combining (3.12) with (3.10), we get the expression Ω 2 E= Pt (ω, T ) + P02 (ω, T ) k 2 dk dS(e). (3.13) S S 0 Since the integrals (3.7) and (3.13) represent the same quantity, their comparison requires that the amplitudes of the fields (3.5) satisfy the equation (k) ω = 2πck, k = |k|. (3.14) |A(k)|2 | + B(k)|2 = Pt2 (ω, T ) + P02 (ω, T ), 2 Then, assuming that the thermal and zero-point oscillations of the electromagnetic field are statistically independent we split the random coefficients A(k) and B(k) as A(k) = A0 (k) + At (k), www.ann-phys.org B(k) = B 0 (k) + B t (k), (3.15) c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1036 B. V. Budaev and D. B. Bogy: Radiation in layered media where the pairs of random numbers A0 , At and B 0 , B t are uncorrelated in the sense 0 t 0 t |A A | = |B B | = 0, (3.16) which makes it possible to split (3.14) into two independent equations j 2Pj2 (ω, T ) |A (k)|2 | + |B j (k)|2 = , (k) (j = 0 and j = t), (3.17) where the index j may take one of the two symbolic values j = 0 or j = t. The last expression can be significantly simplified by the representation of the complex-valued amplitudes Aj (k) and B j (k) in the “polar” form Aj = Λj cos ηj eiβj +iαj , B j = Λj sin ηj eiβj −iαj , j = 0, j = t , (3.18) with all real-valued parameters, including the amplitude Λj , the relative and absolute phase shifts αj and βj , and ηj , which will be referred to hereafter as a composition index. Adopting this notation we conclude that the equilibrium ensemble of electromagnetic fields in a homogeneous space at temperature T consists of two independent ensembles modeling the the zero-point radiation and the thermal radiation, respectively. These ensembles consist of the fields iα+iex xω/c −iα−iex xω/c + sin ηe (3.19) eiβ+i(ey y+ez z)ω/c−iωt , Λj (ω) cos ηe where ω is a random frequency, α, β, η are uniformly distributed random phases, 2Pj2 (ω, T ) Λj (k) = , (j = 0 or j = t), (k) (3.20) and e = (ex , ey , ez ) is as a random unit vector uniformly distributed over the half-sphere ex ≥ 0. 4 Electromagnetic fields in two half-spaces separated by a stack of layers Let the half-spaces x < 0 and x > H be occupied by possibly different materials and let the gap 0 < x < H be filled by a stack of layers of different materials, as shown in Fig. 3. As mentioned in the Introduction, to study the spectrum of thermal radiation in any structure it is necessary to compute the number of eigenfields per unit volume of this structure, and the best way to compute this number is to get an explicit description of the eigenfields. Therefore we start from the analysis of electromagnetic fields in the described sandwich-like medium. We seek an electromagnetic field in the considered sandwich-like structure in the form ⎧ −iωt ⎪ , if x < 0, ⎪ ⎨u− (r, ω, e)e (4.1) u(r, ω, e, t) = u+ (r, ω, e)e−iωt , if x > H, ⎪ ⎪ ⎩ u0 (r, ω, e)e−iωt , if 0 < x < H, where u0 is not specified because it does not play any role in our analysis and u− (r, ω, e) = A− (ω, e)ei(dx x+dy y+dz z)ω/c− + B− (ω, e)ei(−dx x+dy y+dz z)ω/c− , (4.2) u+ (r, ω, e) = A+ (ω, e)ei(ex x+ey y+ez z)ω/c+ + B+ (ω, e)ei(−ex x+ey y+ez z)ω/c+ , (4.3) c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 523, No. 12 (2011) 1037 A stack of layers A+ ei(−ex x+ey y+ez z)ω/c+ θ+ B− ei(dx x+dy y+dz z)ω/c− B+ ei(ex x+ey y+ez z)ω/c+ θ− A− ei(−dx x+dy y+dz z)ω/c− Fig. 3 (online colour at: www.ann-phys.org) Coupling of waves propagating on the different sides of the stack of layers. Amplitudes (A− , B− ) are determined by (A+ , B+ ) through the transmission matrix T(e) which is determined by the reflection and transmission coefficients of the entire sandwich. are specific solutions of the Helmholtz equations 2 1 ω 2 ∇ u± + u± = 0, c± = √ , c± ± μ± (4.4) which consist of plane waves propagating along the directions determined by the unit vectors e = (ex , ey , ez ) and d = (dx , dy , dz ) with the components dx = cos θ− , dy = sin θ− cos φ− , dz = sin θ− sin φ− , ex = cos θ+ , ey = sin θ+ cos φ+ , ez = sin θ+ sin φ+ , (4.5) defined in terms of the spherical angles θ± and φ± , the first of which are shown in Fig. 3. Formulas (4.2) and (4.3) describe electromagnetic fields in the half-spaces x < 0 and x > H, but they do not guarantee that these fields admit extension to a continuous field in the entire space. To formulate the conditions that guarantee such extension is possible we first observe that the directions of propagation d = (dx , dy , dz ) and e = (ex , ey , ez ) must be related by the expressions c− dx = 1 − γ 2 (e2y + e2z ), dy = γey , dz = γez , γ= , (4.6) c+ which are equivalent to Snell’s formulas c+ sin θ− = c− sin θ+ and φ− = φ+ . Next, to derive the relationship between the pairs of the coefficients (A− , B− ) and (A+ , B+ ) we observe that a plane wave ui = ei(dx x+dy y+dz z)ω/c− propagating in the half-space x < 0 generates the reflected wave ur = R ei(−dx x+dy y+dz z)ω/c− and the transmitted wave ut = Kei(ex x+ey y+ez z)ω/c+ , which are coupled in the sense that the field u− = ui + ur propagating in x < 0 and the field u+ = ut propagating in x > H can be considered as parts of the continuous field in the entire layered space. Since the waves ei(dx x+dy y+dz z)ω/c− and ei(−dx x+dy y+dz z)ω/c− appear in (4.2) symmetrically, we may also consider the reversed case when the field in the layered structure is excited by the plane wave ũi = ei(−dx x+dy y+dz z)ω/c− propagating in x < 0. Then, using analytical arguments we readily find that ũi is coupled with the reflected and transmitted waves ũr = R ei(dx x+dy y+dz z)ω/c− and ũt = K ei(−ex x+ey y+ez z)ω/c+ . In both cases the coupling is provided by some fields in the domain 0 < x < H, which are not used in the subsequent analysis. As for the reflection and transmission coefficients R and K they can be computed by standard methods presented in many textbooks, but for the purpose of the current paper it is sufficient to know that these coefficients are related by the identity |R|2 + www.ann-phys.org + e2x |K|2 = 1, − d2x (4.7) c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1038 B. V. Budaev and D. B. Bogy: Radiation in layered media which is equivalent to the assumption that all layers of the medium are lossless. The above implies that the following two special fields ⎧ ⎨ ei(dx x+dy y+dz z)ω/c− + R ei(−dx x+dy y+dz z)ω/c− , if x < 0, u1 = ⎩K ei(ex x+ey y+ez z)ω/c+ , if x > H, and ⎧ ⎨ ei(−dx x+dy y+dz z)ω/c− + R ei(dx x+dy y+dz z)ω/c+ , if u2 = ⎩Kei(−ex x+ey y+ez z)ω/c+ , if (4.8) x < 0, (4.9) x > H, admit continuation to the entire space. Moreover, any pair of the fields u± from (4.1)–(4.3) that admits such continuation can be represented as a linear combination pu1 + qu2 with the coefficients p and q related to the amplitudes A± and B± from (4.2) and (4.3) by the equations p + Rq = A− , Kp = A+ , q + Rp = B− , Kq = B+ . (4.10) Therefore, eliminating p and q from (4.10) we find that the pairs of amplitudes (A− , B− ) and (A+ , B+ ) are related by the equations −1 −1 A− K 0 A+ 1 R = , (4.11) B− B+ 0 K R 1 which couple the fields (4.2) and (4.3). Finally, resolving (4.11) with respect to the pair (A− , B− ) we find that the amplitudes of the waves (4.2) and (4.3) are related by the formula A+ A− =T , (4.12) B− B+ where T= 1 R R 1 K 0 0 K −1 ≡ 1/K R/K R/K 1/K (4.13) is the transmission matrix determined by the reflection and transmission coefficients R and K. The last formulas show that in order to define an eigenfield in the layered structure in Fig. 3 it suffices to specify a frequency ω, a unit vector e and a pair of coefficients A+ (e, ω) and B+ (e, ω). Indeed, if these parameters are known then in the half-space x > H the corresponding eigenfield can be computed by (4.3), and in the half-space x < 0 this field can be defined by the formulas (4.2) with the coefficients A− and B− computed by (4.12). As for the eigenfield inside the gap 0 < x < H, it can be computed by a more complex algorithm, which is not discussed here because we don’t need any information about fields in this domain. 5 An equilibrium ensemble of fields in two separated half spaces The formulas of the previous section determine wave fields in the sandwich like structure in terms of an indefinite unit vector e and a pair of indefinite complex-valued coefficients A+ and B+ . Assuming that these parameters are random, we get a statistical ensemble of electromagnetic fields and can study its c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 523, No. 12 (2011) 1039 thermodynamical properties by the method employed in Sect. 3. In particular, if the ensembles of thermally excited fields (4.2) and (4.3) are in equilibrium at temperature T then formulas (3.18) and (3.20) imply that the amplitudes A± and B± have the structure A± = Λ± cos η± eiβ± +iα± , B± = Λ± sin η± eiβ± −iα± , (5.1) where Λ± = P (ω, T )) 2 , ± (5.2) and α± , β± and η± are random phases uniformly distributed in their admissible domains determined by the equation eiβ+ 1/K R/K eiβ− cos η− eiα− cos η+ eiα+ = , (5.3) − + sin η− e−iα− sin η+ e−iα+ R/K 1/K which follows from (4.12) and (4.13). To solve (5.3) we eliminate η− and get a single equation + |K|2 = 1 + |R|2 + 2|R| sin(2η+ ) cos(α+ + ψ), − ψ = arg(R/K 2 ), (5.4) which makes it possible to explicitly specify the admissible domains of all random parameters involved in (5.1). Thus, assuming for transparency that the half-spaces have similar wave speeds we take into account (4.7), which, in this case, implies that (+ /− )|K|2 = 1 − |R|2 , and reduce (5.4) to the equation sin(2η+ ) cos(α+ + ψ) = −|R|, which implies that η+ can take any value from the domain 1 π π R ≡ R(R) : η+ − < − arcsin(|R|), 4 4 2 (5.5) (5.6) where sin 2η+ ≥ |R| so that (5.5) can be enforced by the selection of α+ . Then, observing that β+ has no constraints we conclude that it may take arbitrary values. As for η− , α− and β− these parameters are not random because they are determined by the Eqs. (5.3), where all parameters with the index “+” should be considered fixed. Summing up the above development we arrive at the conclusion that the ensemble of thermally excited electromagnetic fields in the layered structure consists of the fields U− and U+ defined in the half-spaces x < 0 and x > H by the formulas Pt (ω, T ) U± (r, t) = cos η± eiα± +iex xω/c + sin η± e−iα± i−iex xω/c eiβ± +i(ey y+ez z)ω/c−iωt , (k)/2 (5.7) where e = (ex , ey , ez ) is an arbitrary direction considered as a random unit vector uniformly distributed over the unit sphere, β+ is a random number uniformly distributed on the interval [0, 2π), ω is a random frequency, and η+ is a random number uniformly distributed in the interval in Eq. (5.6), defined by the absolute value of the reflection coefficient R of the entire layered structure. Then, the relative phase shift α+ is determined from Eq. (5.5), which completes the description of the ensembles of wave fields in the half-space x > H. www.ann-phys.org c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1040 B. V. Budaev and D. B. Bogy: Radiation in layered media As for the ensemble of fieldscorresponding to zero-point radiations, it is described by the formulas (5.7) with Pt (ω, T ) replaced by ω/2, but we don’t go into further detail here. It is easy to see that the direction of the energy flux carried by the field (4.1)–(4.3) with the coefficients from (5.1) is determined by the value of η+ . Thus, if 0 < η+ < π4 then the energy flows towards x → −∞, if π4 < η+ < π2 the energy flows in the opposite direction x → +∞, and the waves with η+ = π4 carry no energy along the x-axis. Therefore, practically all of the considered fields with η+ ∈ R carry some nonvanishing energy flux, but since the domain R is symmetric with respect to the line η+ = π4 , the ensemble of these fields with η+ uniformly distributed over R has a vanishing average energy flux because the waves carrying energy in different directions compensate each other. This means that the width of the domain R may be viewed as a measure of the “content” of an equilibrium ensemble of electromagnetic fields in the layered medium. Indeed, if R contracts to the line η+ = π/4 then the ensemble consists of waves that do not carry energy along the x-axis. For the opposite extreme, when R expands to the strip 0 < η+ < π2 , the ensemble consists of waves that carry energy in all directions. The above discussion shows that in order to understand the structure of an equilibrium ensemble it is instructive to study the dependence of the width of R(R) on the reflection coefficient R of the layered structure filling the gap 0 < x < H between the half-spaces x < 0 and x > H. Consider first a degenerate case when the gap collapses to an imaginary interface in a homogeneous space. In this case R = 0, K = 1 and the Eq. (5.4) reduces to the identity 1 = 1 which sets no restrictions on the random parameters in (5.1). Therefore, in this case the equilibrium ensembles of fields (4.1) in the pseudo-layered structure coincide with the equilibrium ensembles of fields (3.5) in the homogeneous space. In the opposite degenerate case of an impenetrable stack of layers |R| = 1, and the domain R(1) collapses to the line η+ = π4 , which means that the equilibrium ensemble contains only those waves that do not carry energy across the interface. Correspondingly, in the intermediate cases when |R| decreases from |R| = 1 to |R| = 0, the domain R(R) expands from the line η = π4 to the entire strip 0 < η < π2 , as shown in Fig. 4. η If |R| = 0 then R coincides with the strip 0 ≤ η ≤ π 2 π 2 If |R| 1 then R(R) is wide. If |R| ≈ 1 then R(R) narrows. π 4 If |R| = 1 then R(R) collapses to the axis η = 0 Ω π 2 ω Fig. 4 (online colour at: www.ann-phys.org) Dependence of R(R) on the reflection coefficient of the stack of layers. 6 The role of evanescent waves in equilibrium ensembles of thermal radiation in layered structures Our results show that an equilibrium ensemble of thermally excited electromagnetic fields in two half spaces separated by a gap filled with a stack of layers can be described solely in terms of the reflection and transmission coefficients of the entire stack. This conclusion sharply contradicts the popular notion that in order to analyze the radiative heat exchange between two closely located bodies it is necessary to use information about the fields inside the layer and, in particular, to view evanescent waves as the dominant heat carriers across narrow layers. This contradiction is not a minor issue, it is a serious flaw in c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 523, No. 12 (2011) 1041 the conventional approach to the analysis of the radiative heat exchange, which is discussed below in the simplest layered structure where half-spaces x < 0 and x > H of the same material are separated by a layer 0 < x < H of a different material. Let the entire structure be in thermal equilibrium at temperature T . Then, each half-space simultaneously radiates and absorbs the same amount of electromagnetic energy. The total energy absorbed by each of the domains naturally splits into the part radiated by one domain and the part radiated by the other domain. Let Q> (T ) be the flux of energy radiated from the domain x < 0 and absorbed by the domain x > H, and let Q< (T ) be the flux of energy radiated from x > H and absorbed in x < 0. Since the entire system is in thermodynamical equilibrium, it is known a priori that Q< (T ) = Q> (T ), but Planck’s law and the laws of electromagnetic wave propagation make it possible to compute the values of Q≶ (T ) as functions of temperature T and of the parameters of the structure. The exact values of the directional fluxes Q< (T ) and Q> (T ) make it possible to compute the net flux Q(T ) = Q> (T ) − Q< (T ) = 0, but the obtained result is essentially useless because it contains no information beyond the trivial statement that the net energy flux in the considered equilibrium system vanishes. However, the last formula is often used in a different, not always justified, way which in certain cases makes it possible to estimate the heat transport across a gap between half-spaces that are maintained at different temperatures [14–17]. Let the half-spaces x < 0 and x > H be maintained at different temperatures T = T− and T = T+ , respectively. Then, each of these domains simultaneously radiates and absorbs electromagnetic energy, but the flows of energy in the two different directions do not compensate each other. Let Q> (T− , T+ ) denote the flux of energy radiated from x < 0 and absorbed in the half-space x > H, and let Q< (T− , T+ ) be the flux of energy radiated from the domain x > H but absorbed in the half-space x < 0. Then the net flux across the gap can be represented by the formula Q(T− , T+ ) = Q> (T− , T+ ) − Q< (T− , T+ ), (6.1) which remains useless until the values of Q− (T− , T+ ) and Q+ (T− , T+ ) are known. It is easy to see that the latter task is not trivial because the computations of Q≶ (T− , T+ ) can not be based on Planck’s law which is valid only in equilibrium systems and, therefore, it can not be legitimately applied to the cases when T− = T+ . Nevertheless, it is common practice to assume that [14–17] Q< (T− , T+ ) = Q< (T− ), Q> (T− , T+ ) = Q> (T+ ), (6.2) where Q≶ (T± ) are defined by Planck’s law, and then, combining (6.1) and (6.2) one obtains the formula Q(T− , T+ ) = Q> (T− ) − Q< (T+ ) (6.3) which is claimed to represent the heat flux between the two half-spaces maintained at different temperatures. Although the last formula is not rigorously justified, in many cases when the gap between the halfspaces is sufficiently wide it can provide reasonable agreement with experimental data. However, this formula leads to an obviously wrong result in the degenerate case when the gap collapses to a plane in an unbounded homogeneous material. Indeed, if T− = T+ then the net flux defined by (6.3) has a finite value which remains finite even as the gap’s width reduces to zero. This leads to the conclusion that a fictitious interface has a finite thermal resistance which is obviously incorrect. Since the above reasoning leads to a contradiction, it is apparent that it has a flaw which must be corrected. Although it is obvious that the flaw comes from the unjustified assumption (6.2), a popular way to correct the flaw is based on the idea that the expression (6.3) leads to wrong conclusions because it does not take into account the evanescent waves which propagate in the layer “mediating the enhanced heat transfer” not accounted for by the functions Q≶ (T ) involved in (6.3) [14–17]. This approach, however, is incorrect because evanescent waves are irrelevant to radiative heat transport as follows from the simple example considered next. www.ann-phys.org c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1042 B. V. Budaev and D. B. Bogy: Radiation in layered media Consider a sandwich-like structure shown in Fig. 5 where a layer 0 < x < H of a material with the speed of electromagnetic waves c∗ separates two half-spaces of materials with a higher wave speed c > c∗ . It is well known that evanescent waves appear only on the side of the interface where the speed of propagation is higher. Therefore, in the considered structure, the evanescent waves propagate in the exterior half-spaces but do not appear in the layer, which means that they cannot mediate energy exchange between the half-spaces. However, it is easy to see that the discussed contradiction between common sense and the conventional approach to radiative heat transport remains intact in the considered structure. Indeed, common sense suggests that a vanishingly thin layer of any material may not have any thermal resistance, which means that if two half-spaces separated by an imaginary plane have different temperatures then the heat flux between them should be infinite. However, according to the conventional theory the value of the heat flux between two half-spaces of the same material separated by a layer is represented by a formula which gives a finite value even in cases when the width of the layer vanishes. Faster medium, wave speed c > c∗ wave speed c∗ wave speed c > c∗ Plane waves propagate everywhere. Evanescent waves are localized along the interface There are no evanescent waves in the Evanescent waves Slow layer, Evanescent waves Faster medium, Plane waves propagate everywhere. Evanescent waves are localized along the interface layer Fig. 5 (online colour at: www.ann-phys.org) Transmission through a material layer separating half-spaces with a higher wave speed. As a material layer narrows to nothing its thermal conductance should increase to infinity, but this increase can not be credited to evanescent waves because they do not propagate inside the layer. Although the above comments show that evanescent waves can not be responsible for enhanced radiative thermal transport across narrow layers, these waves, nevertheless, are excited in layered structures and, therefore, affect their properties. In particular, evanescent waves certainly affect electromagnetic fields in a system of two material half-spaces separated by a vacuum gap of width H and, correspondingly, these waves affect the structure of the equilibrium ensembles of thermal radiation in this system. However, the method developed in Sect. 5 takes into account these evanescent waves automatically without the explicit tracing of individual waves. Indeed, as mentioned above, our results make it possible to describe an equilibrium ensemble of electromagnetic fields in the considered structure solely in terms of the transmission and reflection coefficients K(θ, H) and R(θ, H) that already include all necessary information about the evanescent waves, which are generated when the incident angle θ exceeds the critical angle. 7 Summary and conclusion The above analysis describes equilibrium ensembles of electromagnetic waves for layered media that can be used as a point of departure for analyzing non-equilibrium steady heat flux across layered structures. This is accomplished by employing Planck’s equilibrium ensembles together with methods of statistical physics and of the theory of wave propagation. The obtained information about the structure of equilibrium ensembles of electromagnetic fields in a layered structure may not be practically important itself because it does not affect such observable macroscopic characteristics of the medium as its temperature and the heat flux, which vanishes in equilibrium. c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.ann-phys.org Ann. Phys. (Berlin) 523, No. 12 (2011) 1043 However, this information becomes critically important for the extension of our analysis to non-equilibrium ensembles, introduced and briefly discussed in [7, 8]. Thus, an ensemble of eigenfields in a layered structure with a small heat flux Q can be modeled by a statistical ensemble of wave fields (5.7) with random composition indices η(ω) distributed over the asymmetrical domain R(Q) shown in Fig. 6, which is obtained from R by a specific non-linear transformation η −→ η suggested in [6, 8] and discussed in detail in [10]. ηm ηm π 2 π 2 Rm (Q) π 4 0 π 4 ω 0 Rm (Q) ω Fig. 6 (online colour at: www.ann-phys.org) Distribution of ηm (ω) in non-equilibrium ensembles. In a non-equilibrium ensemble ηm (ω) is distributed over an asymmetric domain Rm (Q). The extent of asymmetry of Rm (Q) is determined by Q, and by the “width” of the domain Rm (Q) describing the corresponding equilibrium ensemble. As illustrated in Fig. 6 and explained in [7, 10] the extent of the asymmetry of the domain R(Q) is determined not only by the heat flux Q but also by the unperturbed domain R and by the temperature T . This shows that there is a connection between the heat flux Q, the temperature T and the domain R, which determines the unperturbed equilibrium ensemble of electromagnetic fields at temperature T . Reading the last statement from another angle we conclude that the connection between the heat flux Q and the temperature T is determined by the domain R characterizing an equilibrium ensemble of eigenfields. Therefore, computing domains R− and R+ for the half-spaces x < 0 and x > H, we can connect the heat flux Q with the temperatures T− and T+ on the different sides of the sandwich and then eventually estimate the thermal resistance of the entire layered structure. This shows that the analysis of the equilibrium ensembles of thermally excited electromagnetic waves plays an important role in understanding the process of radiative heat transport. The outlined approach to thermal conductance of layered structures was introduced and tested in [7, 8] where it was successfully applied to the analysis of the interface thermal conductance widely known as Kapitsa resistance, which is related with acoustic waves in a similar way as the radiative heat conductance is related with electromagnetic waves. As shown in [7, 8] a description of a non-equilibrium ensemble of eigenfields can be obtained by a universal process consisting of the following three distinctive steps: obtain detailed descriptions of equilibrium ensembles of eigenfields, understand the relationship between the equilibrium and steady-state non-equilibrium ensembles of eigenfields of a homogeneous medium, obtain the ensemble with a steady heat flux, and, finally, combine the results of the first two steps and get the information about the non-equilibrium ensembles in a considered layered structure. In [7,8] all of these critical steps have been proposed and tested in the simplest possible layered structure consisting of two joined half-spaces. In order to extend the promising first results obtained in those papers we first extended Planck’s law of thermal radiation to a case of a homogeneous medium with a constant heat flux [10]. Here we describe equilibrium ensembles of eigenfields in general sandwich-like structures as shown in Fig. 3. The final paper of the planned sequence will describe non-equilibrium ensembles of eigenfields and apply the obtained results to estimations of the radiative thermal resistance of the considered layered media. Acknowledgements This work was supported by the William S. Floyd, Jr., Distinguished Chair in Engineering at the University of California, Berkeley, held by D. B. Bogy. www.ann-phys.org c 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1044 B. V. Budaev and D. B. Bogy: Radiation in layered media References [1] L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Statistical Physics, Part 1. Course of Theoretical Physics, Vol. 5, 3rd edition (Elsevier/Butterworth-Heinemann, Amsterdam, Heidelberg, 2006). [2] R. P. Feynman, Statistical Mechanics: A Set of Lectures, 2nd edition (Westview Press, Boulder, CO, 1998). [3] H. Weyl, Math. Ann. 71(4), 441–479, 1912. [4] R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience Publishers, New York, 1953). [5] W. Arendt, R. Nittka, W. Peter, and F. Steiner, Mathematical Analysis of Evolution, Information, and Complexity, chapter Weyl’s Law: Spectral Properties of the Laplacian in Mathematics and Physics (Wiley, Weinheim, 2009) pp. 20–70. [6] B. V. Budaev and D. B. Bogy, J. Phys. A, Math. Theor. 43(42), 425201, 2010. [7] B. V. Budaev and D. B. Bogy, Ann. Phys. (Berlin) 523(3), 208–225, 2011, DOI 10.1002/andp.201000111. [8] B. V. Budaev and D. B. Bogy, SIAM J. Appl. Math. 70(5), 1691–1710, 2010. [9] B. V. Budaev and D. B. Bogy, Appl. Phys. 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