Theory of thermal emission from periodic structures Please share
advertisement
Theory of thermal emission from periodic structures The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Han, S. E. “Theory of thermal emission from periodic structures.” Physical Review B 80.15 (2009): 155108. © 2009 The American Physical Society As Published http://dx.doi.org/10.1103/PhysRevB.80.155108 Publisher American Physical Society Version Final published version Accessed Thu May 26 08:42:01 EDT 2016 Citable Link http://hdl.handle.net/1721.1/52485 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Detailed Terms PHYSICAL REVIEW B 80, 155108 共2009兲 Theory of thermal emission from periodic structures S. E. Han* Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455, USA 共Received 22 May 2009; revised manuscript received 3 August 2009; published 5 October 2009兲 We provide a momentum space formulation for thermal emission from photonic crystals and corroborate Kirchhoff’s law for periodic structures. This formalism allows us to calculate the optical coherence in the far and near fields of photonic crystals. Calculations in the far field show that the exchange of momentum between a linear grating and the surface polaritons can significantly decrease the coherence length for the grating compared to that of a flat surface. Considerations in the near field of photonic crystals show that, unlike observations of flat surfaces, the electric energy density does not necessarily decrease as 1 / z3 for a distance z from the surface. DOI: 10.1103/PhysRevB.80.155108 PACS number共s兲: 42.70.Qs, 44.40.⫹a, 73.20.Mf II. CALCULATING GREEN’S DYADIC FROM THE TRANSFER MATRIX I. INTRODUCTION Control of thermal emission is essential for various applications including lighting,1 imaging,2 and heat-to-electrical conversion systems such as in thermophotovoltaics.3,4 Thermal emission can be dramatically modified when the heated objects are periodically structured on an optical length scale.5–7 In these structures, known as photonic crystals,8 optical diffraction leads to a modification of the photonic density of states. This affects the thermal emission. A recent experimental study reported that thermal emission from a photonic crystal could exceed Planck’s blackbody limit.3 However, it was subsequently argued that this would violate the second law of thermodynamics.9 This argument assumed thermal equilibrium between the photonic crystal and the radiation field. Consequently, a potential explanation for the experiments was that they represented the effect of nonequilibrium processes. Out of equilibrium, the emission could potentially be higher than the black body.10–12 A theoretical study13 dealt with a nonequilibrium thermal emission where a photonic crystal is emitting into vacuum. Specifically, the photonic crystal is not in equilibrium with the electromagnetic field but equilibrium is assumed to be achieved inside the material. In this case, calculations predicted not only that the black-body limit should be maintained but also that Kirchhoff’s law should be obeyed. Subsequent experiments supported this conclusion.14,15 In fact, before photonic crystals were introduced, Kirchhoff’s law had already been proved for the more general case of any arbitrary scattering object.16,17 This suggests that any periodic structures should obey Kirchhoff’s law. In this study, we provide a momentum space formulation within the framework of fluctuational electrodynamics for the proof of Kirchhoff’s law for periodic structures. The formalism used can also be extended to optical coherence. Using the formalism, we investigate the optical coherence in the far and near fields of a periodic structure. We also show that the behavior of the electric energy density in the near field of a photonic crystal is different from a flat surface. For example, at a distance z from the surface, the electric energy density does not necessarily scale as 1 / z3 in the near field, as is the case for a flat surface.17,18 1098-0121/2009/80共15兲/155108共10兲 In the formulation of fluctuational electrodynamics, we will use the Green’s dyadic which relates a point source to the resulting fields. Because many methods for calculating electromagnetic fields for periodic structures generate transmission and reflection coefficients, it is useful to obtain the Green’s dyadic in terms of these optical coefficients. A matrix constructed from the optical coefficients for the plane waves of different wave vectors and polarizations is called the transfer matrix. In this section, we will find a relation between the Green’s dyadic and the transfer matrix. Consider light thermally emitted into vacuum from a periodically structured film. For simplicity, we assume that the film is composed of nonmagnetic materials, lies in the xy plane, and extends in z from −L to 0. Due to the periodicity in the xy plane, Bloch waves will be developed. Thus, the fields can be represented by the sum of the Bloch waves as V共r储,z, 兲 = 兺 Vk储共r储,z, 兲, 共1兲 k储 where V is a field, r储 and k储 are the position and the wave vectors parallel to the surface 共xy plane兲, Vk储 is the Bloch wave with the wave vector component k储, and is the angular frequency. Note that k储 lies in the first Brillouin zone and takes discrete values because the size of the structured film is finite in the xy plane. The Bloch wave Vk储 should satisfy the Bloch theorem Vk储共r储 + a储,z, 兲 = Vk储共r储,z, 兲eik储·a储 , 共2兲 where a储 is the lattice vector in the xy plane. If we expand the Bloch wave using the reciprocal-lattice vector g储 as Vk储共r储,z, 兲 = 兺 Ṽ共k储 + g储,z, 兲ei共k储+g储兲·r储 , 共3兲 g储 the Bloch theorem 关Eq. 共2兲兴 is satisfied because eig储·a储 = 1. 共4兲 Ṽ can be obtained by the two-dimensional inverse Fourier transform of Eq. 共3兲 as 155108-1 ©2009 The American Physical Society PHYSICAL REVIEW B 80, 155108 共2009兲 S. E. HAN Ṽ共k储 + g储,z, 兲 = 1 Suc 冕 uc dr储Vk储共r储,z, 兲e−i共k储+g储兲·r储 , 共5兲 where uc stands for the two-dimensional unit cell in the xy plane and Suc is the surface area of the unit cell. Substituting Eq. 共3兲 into Eq. 共1兲, we have V共r储,z, 兲 = 兺 兺 Ṽ共k储 + g储,z, 兲ei共k储+g储兲·r储 . k储 Ṽ共k储 + g储,z, 兲 = 1 S El共r储,z, 兲 = i0 Ẽl共k储 + g储,z, 兲 = 兺 g⬘储 冕 dr储V共r储,z, 兲e−i共k储+g储兲·r储 , 冕 ⬘冕 dr储 共7兲 dz⬘ ⫻ Glm共k储 + g储,k⬘储 共10兲 where Glm共k储 + g储,k⬘储 + g⬘储 ;z,z⬘ ; 兲 dr储 dr⬘储 ⫻ Glm共r储,r⬘储 ;z,z⬘ ; 兲e 共14兲 J 共r储 , r⬘ ; z , z⬘ ; 兲 can be expressed by taking the Fourier G 储 transform of Eq. 共11兲 and using Eq. 共14兲 as = 1 兺 i 0S k储 兺 Gml共− k − g⬘,− k − g ;z⬘,z; 兲 储 储 储 储 g储,g⬘储 ⫻ei共k储+g储兲·r储−i共k储+g⬘储 兲·r⬘储 . 共15兲 To consider the polarization property of thermal emission, we define s- and p-polarization unit vectors as21 ŝq储,⫾ ⬅ q̂储 ⫻ ẑ, 共9兲 + g⬘储 ;z,z⬘ ; 兲j̃m共k⬘储 + g⬘储 ,z⬘, 兲, 冕 冕 J 共− k储 − g⬘储 ,− k储 − g储 ;z⬘,z; 兲. J T共k储 + g储,k储 + g⬘储 ;z,z⬘ ; 兲 = G G 共8兲 Equation 共9兲 is the consequence of the electrodynamic reciprocity theorem.17,20 This theorem roughly states that a fluctuating current and the electric field that it creates at a distant location can be interchanged without affecting their relationship. If we consider a point current in Eq. 共8兲 and compare this to the relationship when the current and the field are interchanged, we can obtain Eq. 共9兲 from the reciprocity theorem. Substituting Eq. 共6兲 for V = j into Eq. 共8兲 and the resulting equation into Eq. 共7兲 for V = E, we have k⬘储 ,g⬘储 共13兲 Glm共r储,r⬘储 ;z,z⬘ ; 兲 J T共r储,r⬘储 ;z,z⬘ ; 兲 = G J 共r⬘储 ,r储 ;z⬘,z; 兲. G 冕 dz⬘ ⫻ Glm共k储 + g储,k储 J 共k储 + g储 , k储 + g⬘ ; z , z⬘ ; 兲 relates the elecThe Green’s dyadic G 储 tric current component of parallel wave vector k储 + g⬘储 at the z = z⬘ plane to the electric field component of parallel wave vector k储 + g储 at the z = z plane at frequency . Taking the transpose of Eq. 共11兲 and using Eqs. 共9兲 and 共12兲, we have dz⬘ where the subscripts l , m = x , y , z and 0 is the permeability J 共r储 , r⬘ ; z , z⬘ ; 兲 relates of free space. The Green’s dyadic G 储 the electric current at position 共r⬘储 , z⬘兲 to the electric field at position 共r储 , z兲 at frequency . The Green’s dyadic is known to satisfy19 兺 冕 + g⬘储 ;z,z⬘ ; 兲j̃m共k储 + g⬘储 ,z⬘, 兲. ⫻ Glm共r储,r⬘储 ;z,z⬘ ; 兲jm共r⬘储 ,z⬘, 兲, i0 S and Eq. 共10兲 becomes 共6兲 where S is the surface area of the film. Consider the electric field in vacuum that results from the thermal fluctuations in the electric current j inside the structure. They are related to each other through the Green’s dyadic G as = 共12兲 g储 Therefore Ṽ can also be obtained by the inverse Fourier transform of Eq. 共6兲 as Ẽl共k储 + g储,z, 兲 = Glm共k储 + g储,k⬘储 + g⬘储 ;z,z⬘ ; 兲 = Glm共k储 + g储,k储 + g⬘储 ;z,z⬘ ; 兲␦k储,k⬘储 p̂q储,⫾ ⬅ c 关⫿kz,v共q储, 兲q̂储 + 兩q储兩ẑ兴, 共16兲 where q储 = k储 + g储, q̂储 = q储 / 兩q储兩, and + and − indicate whether the light is propagating or evanescent in the +z and −z directions, respectively. In Eq. 共16兲, kz,v is the z component of the wave vector in vacuum and is obtained by kz,v共k储 + g储, 兲 = 冑 2 − 兩k储 + g储兩2 , c2 共17兲 and kz,v is taken such that its imaginary part is positive but, if the imaginary part vanishes, the real part is non-negative. This sign convention is chosen to ensure that the wave is propagating or evanescent in the ⫾z direction for the ⫾ sign in Eq. 共16兲. To relate the Green’s dyadic to the transfer matrix, consider an electric field in vacuum at the z ⬎ 0 plane generated by a current at the same plane. This electric field incident on the structure is21 dẼinc,l共− k储 − g储,z, 兲 i关共k⬘储 +g⬘储 兲·r⬘储 −共k储+g储兲·r储兴 . 共11兲 In Eq. 共10兲, Ẽ and j̃ should have the same k储 because of the conservation of parallel momentum. Thus the Green’s dyadic satisfies 155108-2 =− 0 2kz,v共k储 + g储, 兲 ⫻ 兺 ␣ˆ −k储−g储,l␣ˆ −k储−g储,m j̃m共− k储 − g储,z, 兲dz, ␣ 共18兲 PHYSICAL REVIEW B 80, 155108 共2009兲 THEORY OF THERMAL EMISSION FROM PERIODIC… ˆ = ŝ− or where the subscript inc denotes incident light and ␣ p̂−. The electric field at a different plane z⬘ ⬍ 0 with parallel wave vector −k储 − g⬘储 resulting from the incident wave is exJ by pressed in terms of the transfer matrix M dẼl共− k储 − g⬘储 ,z⬘, 兲 = 兺 M lm共− k储 − g⬘储 ,− k储 − g储 ;z⬘,z; 兲 g储 ⫻ dẼinc,m共− k储 − g储,z, 兲. 共19兲 Plugging Eq. 共18兲 into Eq. 共19兲 and comparing the resulting equation with dẼl共− k储 − g⬘储 ,z⬘, 兲 = 兺 Glm共− k储 − g⬘储 ,− k储 − g储 ;z⬘,z; 兲 g储 ⫻ j̃m共− k储 − g储,z, 兲dz, tures whose structural components are comparable to or less than the wavelength in size but whose overall dimension is much larger than the wavelength. The second assumption of the far field is still imposed. In this case, Kirchhoff’s law is obeyed as will be shown below. A Fourier component of the electric field in vacuum above the structure 共z ⬎ 0兲 can be expressed in terms of the field at the surface 共z = 0兲 by Ẽ共k储 + g储,z, 兲 = Ẽ共k储 + g储,0, 兲eikz,v共k储+g储,兲z . Substituting Eq. 共24兲 into Eq. 共6兲 with V = E and using the method of stationary phase,22 we can obtain the far field form of the electric field as 共20兲 we obtain 共24兲 E共r储,z, 兲 → − eir/c S Ẽ共,0, 兲, i cos r 2 c 共25兲 where r = 兩r储 + zẑ兩, = c r is the wave vector parallel to the film surface, and is the angle between the propagation direction and the z axis. Note that is for the propagating wave and its magnitude is always less than or equal to / c. To take the polarization into account, we define the J ␣⫾ 共␣ = s or p兲 as polarization-dependent Green’s dyadic G r储 Glm共− k储 − g⬘储 ,− k储 − g储 ;z⬘,z; 兲 =− 0 2kz,v共k储 + g储, 兲 ␣− 共− k储 − g⬘储 ,− k储 − g储 ;z⬘,z; 兲, ⫻ 兺 M lm ␣ 共21兲 J 共,k储 + g⬘储 ;0,z⬘ ; 兲 J ␣+共,k储 + g⬘储 ;0,z⬘ ; 兲 ⬅ ␣ ˆ␣ ˆ ·G G ␣− where M lm is defined by ␣− M lm 共− k储 − g⬘储 ,− k储 − g储 ;z⬘,z; 兲 and ⬅ 兺 M ln共− k储 − g⬘储 ,− k储 − g储 ;z⬘,z; 兲␣ˆ −k储−g储,n␣ˆ −k储−g储,m . J 共− k储 − g⬘储 ,− ;z⬘,0; 兲 · ␣ J ␣−共− k储 − g⬘储 ,− ;z⬘,0; 兲 ⬅ G ˆ␣ ˆ, G n 共22兲 J ␣−共−k储 − g⬘ , −k储 − g储 ; z⬘ , z ; 兲 can be viewed as the electric M 储 field of the parallel wave vector −k储 − g⬘储 at z = z⬘ resulting from the ␣-polarized electric current of unit amplitude and the parallel wave vector −k储 − g储 at z = z共⬎z⬘兲 and frequency . Substituting Eq. 共21兲 into Eq. 共15兲, we obtain Glm共r储,r⬘储 ;z,z⬘ ; 兲 = 兺 兺 ␣ 兺 k储 g ,g⬘ 储 储 共26兲 ˆ␣ ˆ =␣ ˆ −,−␣ ˆ −,− = ␣ ˆ ,+␣ ˆ ,+ with ␣ ˆ = ŝ or p̂ and using where ␣ the same notation as in Eq. 共16兲. The Fourier component of the electric field of the ␣-polarized wave going in the ⫾z direction at z = 0 is obtained as ˆ␣ ˆ · Ẽ共,0, 兲. Ẽ␣⫾共,0, 兲 = ␣ From Eqs. 共7兲, 共13兲, 共26兲, and 共27兲 letting k储 + g储 = , we obtain Ẽ␣+ using the inverse Fourier transform of j̃ as i 2kz,v共k储 + g储, 兲兰dr储 ␣− ⫻M ml 共− k储 − g⬘储 ,− k储 − g储 ;z⬘,z; 兲 ⫻ ei共k储+g储兲·r储−i共k储+g⬘储 兲·r⬘储 . 共27兲 Ẽl␣+共,0, 兲 = 共23兲 Hence, we can find the real-space Green’s dyadic from the transfer matrix using Eq. 共23兲. The Green’s dyadic will now be used to derive Kirchhoff’s law 共Sec. III兲 and to calculate the field correlations 共Sec. IV兲. ⬘储 冕 冕 0 dz⬘e−i共k储+g⬘储 兲·r⬘储 dr⬘储 −L ␣+ ⫻Glm 共,k储 + g⬘储 ;0,z⬘ ; 兲jm共r⬘储 ,z⬘, 兲. 共28兲 The ensemble average of Ẽ␣+ⴱ共 , 0 , 兲 · Ẽ␣+共 , 0 , ⬘兲 is obtained from Eq. 共28兲 as III. KIRCHHOFF’S LAW FOR PHOTONIC CRYSTALS Kirchhoff’s law is derived with the assumption that both 共1兲 the length of the object and 共2兲 the distance from the object is much larger than the wavelength. The first condition is imposed in order to use geometric optics and the second to exclude near fields where evanescent waves exist. As the two assumptions break down, interesting phenomena that do not necessarily obey Kirchhoff’s law can be expected. Here, we modify the first assumption by considering periodic struc- 1 兺 Sg 具Ẽ␣+ⴱ共,0, 兲 · Ẽ␣+共,0, ⬘兲典 = 0␦ 共 − ⬘兲 兺 兺 S l,m g ,g ⬘储 ⬙储 冕 0 dz⬘ f̃共g⬙储 − g⬘储 ,z⬘, 兲 −L ␣+ⴱ ␣+ ⫻Glm 共,k储 + g⬘储 ;0,z⬘ ; 兲Glm 共,k储 + g⬙储 ;0,z⬘ ; 兲, 共29兲 where 0 is the permittivity of free space and 155108-3 PHYSICAL REVIEW B 80, 155108 共2009兲 S. E. HAN f̃共g⬙储 − g⬘储 ,z⬘, 兲 = 1 S 冕 ⫻ dr⬘储 ei共g⬘储 −g⬙储 兲·r⬘储 I共r⬘储 ,z⬘, 兲 ប eប/kT共r⬘储 ,z⬘兲 − 1 . P ␣+共 , 兲 = ⬘储 ⬙储 共31兲 where I is the imaginary part of the dielectric function and ⌰共 , T兲 = eបប/kBT−1 is the mean energy of the harmonic oscillator minus the zero-point energy. Equation 共31兲 says that the thermal currents are delta correlated in position. This is the consequence of the assumption that the medium is without spatial dispersion. The magnitude of the ensemble-averaged Poynting vector 具S␣+共r储 , z兲典 of light of ␣ polarization and frequency propagating in the +z direction is given by23 具S␣+共r储,z兲典␦共 − ⬘兲 = 2 具E␣+ⴱ共r储,z, 兲 · E␣+共r储,z, ⬘兲典. 0c 共32兲 r2 具S␣+共r储,z兲典r/c→⬁ . S cos P ␣+共 , 兲 = P ␣+共 , 兲 = 3 cos 0 兺兺 23c3 0 l,m g ,g ⬘储 ⬙储 冕 0 dz⬘ f̃共g⬙储 − g⬘储 ,z⬘, 兲 −L ␣+ⴱ ␣+ 共,k储 + g⬘储 ;0,z⬘ ; 兲Glm 共,k储 + g⬙储 ;0,z⬘ ; 兲. ⫻Glm 共34兲 Now, as in Kirchhoff’s law, we wish to relate the emission rate to the parameters in the absorption process. For this, we use the electrodynamic reciprocity theorem Eq. 共14兲 for ˆ␣ ˆ with Eq. 共14兲 and use k储 + g储 = and z = 0. We premultiply ␣ Eq. 共26兲 to find the relation for polarization ␣ as Substituting Eq. 共35兲 into Eq. 共34兲, we have 3 兺兺 8 c cos l,m g ,g 3 3 ⬘储 ⬙储 冕 0 dz⬘ f̃共g⬙储 − g⬘储 ,z⬘, 兲 −L ␣− ⫻M lm 共− k储 − g⬙储 ,− ;z⬘,0; 兲. 共38兲 Hence the emission rate can be calculated using the transfer matrices. By expressing transfer matrices in terms of the electric fields, the physics becomes clearer. Similarly to Eq. 共19兲, the electric field at a plane z = z⬘ ⬍ 0 with parallel wave vector −k储 − g⬘储 resulting from the incident wave of polarization ␣ and parallel wave vector − is 共␣−兲 dẼ共− ,0兲共− k 储 − g⬘储 ,z⬘, 兲 J ␣−共− k储 − g⬘储 ,− ;z⬘,0; 兲 · dẼ␣− 共− ,0, 兲. =M inc 共39兲 In Eq. 共39兲 and the following, the brackets in the subscript or superscript are used to describe the incident light that leads to a quantity inside the film, in this case dẼ. The subscripts indicate the parallel wave vector of the incident light on the film at z = 0. The superscripts indicate the polarization and the propagation direction of the incident light. For example, 共␣−兲 the electric field Ẽ共− ,0兲 in Eq. 共39兲 results from the incident light that is ␣ polarized, propagating downward, and having parallel wave vector −. Using Eq. 共39兲, we obtain the transfer matrix as J ␣−共− k储 − g⬘储 ,− ;z⬘,0; 兲 M = J ␣−T共− k储 − g⬘储 ,− ;z⬘,0; 兲. J ␣+共,k储 + g⬘储 ;0,z⬘ ; 兲 = G G 共35兲 共37兲 ␣−ⴱ 共− k储 − g⬘储 ,− ;z⬘,0; 兲 ⫻M lm 共33兲 We find the Poynting vector from Eqs. 共25兲, 共29兲, and 共32兲 and insert this into Eq. 共33兲 to obtain the emission rate. Then the resulting expression is c 0 J ␣− M 共− k储 − g⬘储 ,− ;z⬘,0; 兲, 2 cos where we used kz,v共 , 兲 = c cos . Plugging Eq. 共37兲 into Eq. 共36兲, we find that The emission rate of ␣-polarized light per unit solid angle per unit frequency per unit area in the far field is P ␣+共 , 兲 = 共36兲 J ␣−共− k储 − g⬘储 ,− ;z⬘,0; 兲 G =− ⫻␦共 − ⬘兲, dz⬘ f̃共g⬙储 − g⬘储 ,z⬘, 兲 −L In calculating this emission rate, we may use the Green’s ˆ␣ ˆ with Eq. 共21兲 for dyadic in Eq. 共21兲. Postmultiplying ␣ k储 + g储 = and z = 0, and using Eq. 共26兲 and the orthogonality for different polarizations defined in Eq. 共16兲, we obtain the relation 具jⴱl 共r⬘储 ,z⬘, ,T兲jm共r⬙储 ,z⬙, ⬘,T兲典 ⌰共,T兲 0I共r⬘储 ,z⬘, 兲␦lm␦共r⬘储 − r⬙储 兲␦共z⬘ − z⬙兲 冕 0 ␣−ⴱ 共− k储 − g⬘储 ,− ;z⬘,0; 兲 ⫻Glm ␣− ⫻Glm 共− k储 − g⬙储 ,− ;z⬘,0; 兲. 共30兲 f̃ is the Fourier transform of a function f共r⬘储 , z⬘ , 兲 ⬅ I共r⬘储 , z⬘ , 兲⌰关 , T共r⬘储 , z⬘兲兴. Here f is assumed to be periodic in the xy plane. In Eq. 共30兲, the two integrations can be done for a unit cell without changing the results, so that the calculation of f̃ becomes simple. Note that the integration is evaluated only over the material portion because I = 0 in vacuum. In deriving Eq. 共29兲, we used the fluctuationdissipation theorem assuming local equilibrium17 = 3 cos 0 兺兺 23c3 0 l,m g ,g ␣− 共␣−兲 Ẽ共− ,0兲共− k 储 − g⬘储 ,z⬘, 兲Ẽinc共− ,0, 兲 ␣− 关Ẽinc 共− ,0, 兲兴2 . 共40兲 Inserting Eqs. 共30兲 and 共40兲 into Eq. 共38兲 and using 关see Eq. 共3兲兴 155108-4 PHYSICAL REVIEW B 80, 155108 共2009兲 THEORY OF THERMAL EMISSION FROM PERIODIC… 共␣−兲 共␣−兲 −i共k储+g⬘储 兲·r⬘储 E−k 共r⬘储 ,z⬘, 兲 = 兺 Ẽ共− , ,0兲共− k 储 − g⬘储 ,z⬘, 兲e 储 共− ,0兲 IV. OPTICAL COHERENCE FOR PHOTONIC CRYSTALS g⬘储 共41兲 we obtain 冕 ⬘冕 ប P 共, 兲 = 83c3S cos ␣+ ⫻ 共e兲 具Eⴱl 共r1, 兲Em共r2, ⬘兲典 = Wlm 共r1,r2, 兲␦共 − ⬘兲. 0 4 dz The correlation between the electric fields of a given frequency at two different points is represented by the electric 共e兲 , defined by22 cross-spectral density tensor Wlm This can be obtained from Eqs. 共8兲, 共31兲, and 共48兲 as dr⬘储 −L 共␣−兲 2 I共r⬘储 ,z⬘, 兲 兩E−k储共−,0兲共r⬘储 ,z⬘, 兲兩 eប/kT共r⬘储 ,z⬘兲 − 1 兩Ẽ␣−共− ,0, 兲兩2 共e兲 Wlm 共r储1,r储2 ;z1,z2 ; 兲 = , 1 Suc cos 冕 ⬘冕 0 dz −L uc 共␣−兲 dr⬘储 ⫻ Q共− ,0兲 ␣+ ⫻共r⬘储 ,z⬘, 兲PBB 共r⬘储 ,z⬘, 兲, f共r⬘储 ,z⬘, 兲 = 兺 f̃共g⬘储 ,z⬘兲eig⬘储 ·r⬘储 . Substituting Eq. 共50兲 into Eq. 共49兲 and using Eq. 共23兲, we obtain 共e兲 Wlm 共r储1,r储2 ;z1,z2 ; 兲 2 = 共44兲 Finally, is the black-body radiation intensity for ␣-polarized light given by 1 Suc cos 冕 ⬘冕 0 dz −L uc 冕 dz⬘ 兺 兺 兺 兺 兺 n ␣, k储 g ,g⬘ g ,g⬘ 储1 储1 储2 储2 f̃共g⬘储2 − g⬘储1,z⬘兲 e−i共k储+g储1兲·r储1+i共k储+g储2兲·r储2 4kⴱz,v共k储 + g储1, 兲kz,v共k储 + g储2, 兲 ⴱ ⫻e−ikz,v共k储+g储1,兲z1+ikz,v共k储+g储2,兲z2 ␣−ⴱ ⫻M nl 共− k储 − g⬘储1,− k储 − g储1 ;z⬘,0; 兲 − ⫻M nm 共− k储 − g⬘储2,− k储 − g储2 ;z⬘,0; 兲, 共51兲 where  is the polarization defined similarly to ␣ and we used 共46兲 J 共− k储 − g⬘储 ,− k储 − g储 ;z⬘,z; 兲 M where A共␣−兲 is the absorptivity for an ␣-polarized incident wave given by A共␣−兲共− , 兲 = 3200 S ⫻ 共45兲 Equation 共43兲 is the same as the Eq. 共1兲 of Ref. 24 except for slight changes in notation. When the temperature is uniform, ␣+ is independent of the position and Eq. 共43兲 becomes PBB ␣+ P␣+共, 兲 = A共␣−兲共− , 兲PBB 共兲, 共50兲 g⬘储 ␣+ PBB 2 ប . 83c2 eប/kT共r⬘储 ,z⬘兲 − 1 共49兲 In Eq. 共49兲, we assumed local equilibrium for nonuniform temperatures because the fluctuation-dissipation theorem was used. This is in general justified for high-vacuum 共e兲 for an arbitrary temconditions.24 The calculation of Wlm perature distribution is formidable because of the integration in Eq. 共49兲. However, if f共r⬘储 , z⬘ , 兲 ⬅ I共r⬘储 , z⬘ , 兲⌰关 , T共r⬘储 , z⬘兲兴 is periodic in the xy plane, the calculation is simplified using Eq. 共30兲. In this case, f can be expanded in a Fourier series as 兩E−k储共−,0兲共r⬘储 ,z⬘, 兲兩 共␣−兲 Q共− . ,0兲共r⬘储 ,z⬘, 兲 = I共r⬘储 ,z⬘, 兲 c 兩Ẽ␣−共− ,0, 兲兩2 ␣+ 共r⬘储 ,z⬘, 兲 = PBB dz⬘ ⫻Gmn共r储2,r⬘储 ;z2,z⬘ ; 兲. 共␣−兲 Q共− ,0兲 共␣−兲 dr储 n 共43兲 is the local absorption rate per unit volume where when the film is irradiated with ␣-polarized light of unit intensity with parallel wave vector − incident on the z = 0 plane. This absorption rate is given by20 冕 ⬘冕 ⫻ 兺 Gⴱln共r储1,r⬘储 ;z1,z⬘ ; 兲 ␣− where we suppressed the subscript inc on Ẽ . In Eqs. 共41兲 共␣−兲 has a parallel wave vecand 共42兲, the electric field E−k 储 共− ,0兲 tor −k储 in the first Brillouin zone and results from ␣-polarized light with parallel wave vector − incident on the z = 0 plane. Because we assumed that the temperature distribution has the same periodicity as the structure, the integrations in Eq. 共42兲 can be performed over a surface unit cell. Then Eq. 共42兲 can be written as 3200 ⫻I共r⬘储 ,z⬘, 兲⌰关,T共r⬘储 ,z⬘兲兴 共42兲 P ␣+共 , 兲 = 共48兲 J 共− k储 − g⬘储 ,− k储 − g储 ;z⬘,0; 兲eikz,v共k储+g储,兲z =M 共52兲 and 共␣−兲 dr⬘储 Q共− ,0兲共r⬘储 ,z⬘, 兲. 兺 f̃共g⬘,z⬘兲␦g⬘+共k 储 g⬘储 共47兲 Equation 共46兲 is Kirchhoff’s law for a periodically structured object with an arbitrary length scale. Thus, we proved that the Kirchhoff’s law is obeyed for photonic crystal films if local equilibrium is assumed. 储 储1 +g⬘储1兲−共k储2+g⬘储2兲,0 = f̃共g⬘储2 − g⬘储1,z⬘兲␦k储1,k储2 . 共53兲 Thus the electric cross-spectral density tensor can be calculated from the knowledge of the transfer matrices using Eq. 共51兲. 155108-5 PHYSICAL REVIEW B 80, 155108 共2009兲 S. E. HAN Now we wish to replace the transfer matrices by the electric fields as in the previous section. We replace in Eqs. 共40兲 and 共41兲 by k储 + g储1 and k储 + g储2 and substitute the resulting equations into Eq. 共51兲 to obtain 共e兲 Wlm 共r储1,r储2 ;z1,z2 ; 兲 3200 = S2 冕 ⬘冕 dr储 ⫻兺 兺 兺 n ␣, k储 tance A = 兺␣A共␣−兲 / 2 defined in Eq. 共47兲, Eq. 共56兲 can be written as J 共e兲共R储,z, 兲 = c0⌰共,T兲 兺 A共− k储, 兲 eik储·R储 . 共57兲 TrW 2S cos k储 Equation 共57兲 indicates that, apart from a constant factor, the J 共e兲 is equal to the Fourier transform coherence function TrW of A / cos . Conversely, the absorptance A can be obtained in ៝ 共e兲 by the inverse Fourier terms of the coherence function TrW transform as dz⬘ I共r⬘储 ,z⬘, 兲⌰关,T共r⬘储 ,z⬘兲兴 ⴱ 4kz,v共k储 + g储1, 兲kz,v共k储 + g储2, 兲 储1 储2 兺 g ,g ⴱ ⫻e−i共k储+g储1兲·r储1+i共k储+g储2兲·r储2e−ikz,v共k储+g储1,兲z1+ikz,v共k储+g储2,兲z2 A共− k储, 兲 = 共␣−兲ⴱ 共−兲 ⫻E−k 共r⬘储 ,z⬘, 兲E−k 共r⬘储 ,z⬘, 兲 储 共−k 储 −g 储 1,0兲,n 储 共−k 储 −g 储 2,0兲,n ⫻ − 共− k储 − g储2,0, 兲 Ẽl␣−ⴱ共− k储 − g储1,0, 兲Ẽm Ẽ ␣−ⴱ2 共− k储 − g储1,0, 兲Ẽ −2 共− k储 − g储2,0, 兲 , 0 SSuc 冕 ⬘冕 dr储 共54兲 dz⬘ uc ⫻兺兺兺 n ␣, k储 I共r⬘储 ,z⬘, 兲⌰共,T兲 ik储·R储 e 4 cos2 共␣−兲ⴱ 共−兲 ⫻ E−k 共r⬘储 ,z⬘, 兲E−k 共r⬘储 ,z⬘, 兲 储 共−k 储 ,0兲,n 储 共−k 储 ,0兲,n ⫻ − 共− k储,0, 兲 Ẽl␣−ⴱ共− k储,0, 兲Ẽm Ẽ␣−ⴱ2共− k储,0, 兲Ẽ−2共− k储,0, 兲 , 共55兲 where R储 = r储2 − r储1, 兩k储兩 ⱕ / c, and we used kz,v共−k储 , 兲 = c cos . Note that W共e兲 ij depends only on the difference in J 共e兲 in Eq. 共55兲 bethe position in this case. The trace of W comes J 共e兲共R储,z, 兲 = 0 TrW SSuc 冕 ⬘冕 dr储 k储 dz⬘I共r⬘储 ,z⬘, 兲⌰共,T兲 uc ⫻兺 兺 ␣ 冕 J 共e兲共R储,z, 兲e−ik储·R储 . dR储TrW 共58兲 where we removed f̃ using Eq. 共30兲. Equation 共54兲 is the general equation that describes the correlations of any two points over periodic structures with a periodic temperature distribution. The location of the two points can be in the near field where evanescent fields exist. Often, we are interested in the coherence on a plane, that is, parallel to the surface of the film and in the far field. In this case, the coherence is related to the angular dependence of the thermal emission.22 We set z1 = z2 = z and assume that z is large such that kzI,vz Ⰷ 1 for all g储’s, where kzI,v is the imaginary part of kz,v. We also assume that the temperature is uniform and the wavelength is larger than the periodicity of the structure. For this wavelength, only zero-order reflections occur for an incident light. With these assumptions, Eq. 共54兲 is reduced to 共e兲 Wlm 共R储,z, 兲 = 2c cos ⫻ 20⌰共,T兲 共␣−兲 2 eik储·R储 兩E−k储共−k储,0兲共r⬘储 ,z⬘, 兲兩 , 4 cos2 兩Ẽ␣−共− k储,0, 兲兩2 共56兲 ˆ −k储,− is real. Using the absorpwhere we used the fact that ␣ J 共e兲 is obtained for a black An analytical expression for TrW body. In Eq. 共57兲, setting A = 1 and using cos = c kz,v共k储 , 兲 from Eq. 共17兲, we have J 共e兲共R储,z, 兲 = c0⌰共,T兲 兺 TrW 2S k储 = = c0⌰共,T兲 83 冕 冑 eik储·R储 2 − 兩k储兩2 c2 k储⬍/c c0⌰共,T兲 sin , 42 dk储 eik储·R储 冑 2 − k2储 c2 共59兲 where = c R储. In Eq. 共59兲, the transition from summation to integral is valid when the linear dimension of the black body is much larger than the wavelength = 2c / . Equation 共59兲 agrees with Ref. 25 and shows that the coherence length of a black body is on the order of . Note that in general W共e兲 ij in Eq. 共55兲 cannot be obtained in terms of A共␣−兲 because the electric fields resulting from incident s- and p-polarized light 共E共s−兲 and E共p−兲兲 are not necessarily orthogonal in the photonic crystal. This is possible only when they are orthogonal as J 共e兲 can be exin a homogeneous flat film. However, TrW pressed in terms of A for photonic crystals as seen in Eq. J 共e兲 is in general a complex 共57兲. Equation 共57兲 shows that TrW number. This can be real if A共−k储 , 兲 = A共k储 , 兲 which happens for structures with mirror symmetry. However, even in J 共e兲 in the near field is in general complex. this case, TrW Using Eq. 共57兲 and the transfer matrix method,26 we calJ 共e兲 for the SiC grating reported in Ref. 27. At culated TrW = 11.36 m, surface phonon polaritons can be excited on the SiC film and coupled to light by the grating. Because the surface phonon polaritons can propagate a long distance, it was shown that the coherence length was large. This coherence has already been calculated in Ref. 28 but only the contributions from the surface phonon polaritons were considered. Here we investigate the coherence considering all the propagating fields. The structural parameters were slightly modified such that the groove period, width, and depth are 6.248 m, 2.083 m, and 250 nm, respectively. 155108-6 PHYSICAL REVIEW B 80, 155108 共2009兲 THEORY OF THERMAL EMISSION FROM PERIODIC… FIG. 1. Angular dependence of the absorptance of a SiC linear grating for s and p polarizations at = 11.36 m. The groove period, width, and depth are 6.248 m, 2.083 m, and 250 nm, respectively. is the angle from the surface normal in the plane perpendicular to the grating direction. Figure 1 shows the calculated absorptance for s and p polarizations as a function of the inclination angle from the surface normal. We set the Cartesian coordinate system such that the y axis is the direction in which the grating is ruled and the z axis is normal to the surface. The inclination angle is in the xz plane. The peak in the p polarization is due to the surface phonon polaritons. The peak width is about 2.5° and the approximate formula for the coherence length l l⬃ ⌬ cos 共60兲 yields l ⬃ 32. However, the coherence calculated from Eq. 共57兲 shown in Fig. 2 shows that the coherence length is much J 共e兲 is normalized to its value at R = 0. shorter. In Fig. 2, TrW x J 共e兲 decays to 1 / e at ⬃. The envelope of the normalized TrW Thus the coherence length is ⬃ even though the thermal emission is highly directional. This is partly due to the contributions of the modes that are not related to the surface phonon polaritons. These contributions are also manifested J 共e兲 as a function of FIG. 2. Normalized electric coherence TrW the separation of the two points at = 11.36 m in the far field of the grating in Fig. 1. The distance between the two points 共Rx兲 is normalized to the wavelength . in the oscillating features with a periodicity of ⬃ in Fig. 2. J 共e兲 of a black body as These features are present in the TrW shown as a dashed line in Fig. 2. However, this periodicity of ⬃ is also contributed by the surface phonon polaritons. The peak in Fig. 1 at = 46.7° corresponds to kxax / 2 = 0.4, where ax is the grating period. At this wave vector, the exJ 共e兲 is R / = 1.375 from Eq. 共57兲 if only pected period in TrW x the peak in Fig. 1 is considered. We also observe in Fig. 2 the beats with a period Rx / ⬃ 9. This cannot be explained by the surface phonon polariton modes in Fig. 1 or other modes different from surface phonon polaritons. The surface phonon polaritons couple to the light not only in the xz plane but also in all the other planes perpendicular to the surface. These surface phonon polaritons have different wave vectors than k储ax / 2 = 0.4 and contribute to the beats in Fig. 2. The exact locations of k储 can be found in Ref. 29. Because these k储’s cover a broad range, this also contributes to the short coherence length in Fig. 2. Therefore, the large difference in the coherence length between Eq. 共60兲 and Fig. 2 comes partly from the fact that Eq. 共60兲 considers only the surface phonon polaritons in the xz plane. In Ref. 28, the coherence length of a grating shorter than that of a flat film was attributed to the radiative losses. Here, we find that the surface phonon polaritons with a range of wave vectors couple to the light for the grating and this is the main reason for the decrease in the coherence length. In this case, many different wave vectors are summed up in Eq. 共57兲 resulting in dephasing which implies a decrease in the coherence length. The range of wave vectors is the consequence of the directional momentum transfer from the grating to the surface phonon polaritons. If we consider only the surface phonon polaritons of the wave vectors in the x direction, Eq. 共57兲 yields l ⬃ 32 which agrees with the prediction of Eq. 共60兲 and is not very different from l ⬃ 36 predicted for a flat film. When we considered all the surface phonon polariton modes excluding the background emission, the calculated coherence was l ⬃ 5 共not shown兲. Therefore, we conclude that the main reason for the decrease in the coherence length for the grating is not the radiation damping but the momentum transfer in the x direction from the grating. If the grating has a cylindrical symmetry as in a bull’s eye pattern, the momentum transfer occurs in the radial direction. Further, if the resulting momentum of the light is such that k储 ⯝ 0, the coherence length due to the surface polaritons does not decrease by the momentum transfer because a very narrow range of wave vectors is involved. In this case, the light due to the surface polaritons will be beaming in the direction of surface normal.30 The calculation of coherence in the near field can be demanding even for simple structures. This is because large parallel wave vectors contribute to the coherence significantly in the near field. To see this more clearly, we note that Eq. 共54兲 contains a factor exp关−kzI,vz兴. From Eq. 共17兲, kzI,v共k储 + g储 , 兲 increases as 兩k储 + g储兩 increases. When z is not small, the contributions of large 兩k储 + g储兩’s become negligible because exp关−kzI,vz兴 becomes small. Thus, in this case, we do not need large 兩g储兩’s in numerical calculations. However, when z is small, we need to include many terms in the summation over g储’s in Eq. 共54兲, which leads to a large compu- 155108-7 PHYSICAL REVIEW B 80, 155108 共2009兲 S. E. HAN length in Fig. 3 is mainly due to the surface phonon polaritons which are p polarized. J 共e兲 for The electric energy density is proportional to TrW Rx = 0. In Fig. 3, we can see that the electric energy density increases as z decreases. It is well known that, for a flat surface, the electric energy density decreases with increasing z as 1 / z3 in the near field.17,18 However, this is not in general true for photonic crystals. To see this more clearly, we set r储1 = r储2 and z = z1 = z2 in Eq. 共54兲 and define the electric energy density averaged over a unit cell ū共e兲 as ū共e兲共z, 兲 = 1 Suc 冕 uc dr储 兺 W共e兲 ll 共r 储,r 储 ;z,z; 兲. 共61兲 l Substitution of Eq. 共54兲 for a uniform temperature into Eq. 共61兲 yields ū共e兲共z, 兲 = 0⌰ e−2kzI,v共k储+g储,兲z 兺 兺 兺 4S ␣, k储 g储 兩kz,v共k储 + g储, 兲兩 ⫻A共␣−−兲共− k储 − g储, 兲, 共62兲 where the generalized absorptance A共␣−−兲 is defined similarly to Eq. 共47兲 as A共␣−−兲共− k储 − g储, 兲 = c兩kz,v共k储 + g储, 兲兩Suc ⫻ FIG. 3. Real 共solid line兲 and imaginary 共dashed line兲 part of the J 共e兲 as a function of the separation of the two electric coherence TrW points at = 11.36 m in the near field at 共a兲 z = 0.1, 共b兲 z = , and J 共e兲’s are normalized to their 共c兲 z = 10 of the grating in Fig. 1. TrW value at Rx = 0 and z = 10. The distance between the two points 共Rx兲 is normalized to the wavelength . tation time. However, these calculations are suitable for a parallel computing approach because the calculations for each g储 can be done in parallel. Here we present calculations of optical coherence in the near field over a periodic structure. The grating used in the calculations is the same as the one in Fig. 1. In such structures with a mirror symmetry, the summation in k储 can be done only for kx ⱖ 0. Because of a large computation time in the near-field calculations, we investigate the coherence on the planes at z ⱖ 0.1. J 共e兲 for the grating at z = 0.1, , and Figure 3 shows TrW J 共e兲’s are normalized to their values for R = 0 at 10. All TrW x z = 10. At z = 10, the evanescent field is almost absent and J 共e兲 is almost real the coherence is very similar to Fig. 2. TrW at this far field. However, at z = 0.1 and , we observe that J 共e兲, which is real for a flat surface, is now complex. TrW Moreover, the imaginary part does not necessarily decrease as z increases even though it disappears for a sufficiently J 共e兲 decreases as z large z. The period of the oscillation in TrW decreases, which indicates that large wave vectors become important as the surface is approached. W共e兲 yy , which is associated with the fields polarized in the grating direction, behaves similar to the black body in the range of z’s investigated 共not shown兲. This indicates that the large coherence 冕 ⬘冕 dz uc 共␣−−兲 dr⬘储 Q共−k 共r⬘储 ,z⬘, 兲 储 −g 储 ,0兲 共63兲 and the generalized absorption rate per unit volume is 共␣−−兲 Q共−k 储 −g 储 ,0兲 共␣−−兲 Q共−k 共r⬘储 ,z⬘, 兲 储 −g 储 ,0兲 =兺 l,n ⫻ I共r⬘储 ,z⬘, 兲 c Ẽl␣−ⴱ共− k储 − g储,0, 兲Ẽl−共− k储 − g储,0, 兲 Ẽ␣−ⴱ2共− k储 − g储,0, 兲Ẽ−2共− k储 − g储,0, 兲 共␣−兲ⴱ 共−兲 共r⬘储 ,z⬘, 兲E−k 共r⬘储 ,z⬘, 兲. ⫻E−k 储 共−k 储 −g 储 ,0兲,n 储 共−k 储 −g 储 ,0兲,n 共64兲 共␣−−兲 Q共−k 储 −g 储 ,0兲 is the local absorption rate per unit volume for the two sources that have polarizations ␣ and  and the same parallel wave vector −k储 − g储 at z = 0. A共␣−−兲 is the corresponding absorptance. When 1 / z Ⰷ / c, it follows from Eq. 共62兲 that ū共e兲共z, 兲 ⬃ 冕 兺冕 0⌰ 163 ⫻ 冤 ␣, ⬁ dq储q储e−2q储z 0 2 dA共␣−−兲共− q储, 兲 0 兩kz,v共q储, 兲兩 冥 , 共65兲 where q储 = k储 + g储 and is the azimuthal angle on the xy plane. For a flat surface, the factor in the bracket in Eq. 共65兲 155108-8 PHYSICAL REVIEW B 80, 155108 共2009兲 THEORY OF THERMAL EMISSION FROM PERIODIC… u共e兲共x = ax / 2兲 at z / = 1 but becomes larger at z / = 0.4. Overall, the energy density undergoes a significant change when z ⬃ which marks the transition region between the near and the far fields. In the far field, the energy density of the grating is roughly twice that of the flat surface. However, the energy density averaged over the unit cell can be smaller or larger than the flat surface in the near field depending on the distance. These results imply that the distance dependence of heat transfer can be modified with periodic structures. V. CONCLUSION FIG. 4. The ratio of the electric energy density u共e兲 for the grating in Fig. 1 to u共e兲 flat for the flat film as a function of the distance z from the top of the grating at = 11.36 m. The distance z is normalized to the wavelength . u共e兲 was calculated at different positions in the x direction. The coordinate system and the geometry of the unit cell are shown in the inset. x is measured from the center of the groove. Square and triangular symbols are for x = 0 and x = ax / 2 = 3.124 m, respectively, and circular symbols show the average ū共e兲 / u共e兲 flat over the unit cell 0 ⱕ x ⱕ ax. is proportional to q储 and ū共e兲 ⬀ 1 / z3 in the near field.18 However, this does not happen in general for periodic structures. Therefore, the near-field heat transport can be modified when a uniform film is periodically structured. The condition for ū共e兲 ⬀ 1 / z3 is typically satisfied for a flat surface when z / ⬍ 0.01.18 Due to the previously discussed long-time computation for the near field of photonic crystals in this range, we instead investigate the electric energy density behavior for 0.1ⱕ z / ⱕ 10. Figure 4 shows how the ratio of the electric energy density u共e兲 for the grating in Fig. 1 to u共e兲 flat for the flat surface varies with the distance z from the top of the grating. While the energy density of the flat surface is uniform on a plane parallel to the surface, at small distances from a corrugated surface it is inhomogeneous in the x direction 共see Fig. 4 inset for the coordinate system兲. As can be seen from Eq. 共57兲, this inhomogeneity disappears in the far field. This is shown in Fig. 4. The distribution of the electric energy density in the x direction depends on the distance z. For example, u共e兲共x = 0兲 is smaller than *Present address: Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA; sehan@mit.edu 1 S. John and R. Wang, Phys. Rev. A 78, 043809 共2008兲. 2 Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P. A. Lemoine, K. Joulain, J. P. Mulet, Y. Chen, and J. J. Greffet, Nature 共London兲 444, 740 共2006兲. 3 S.-Y. Lin, J. Moreno, and J. G. Fleming, Appl. Phys. Lett. 83, 380 共2003兲. 4 P. Nagpal, S. E. Han, A. Stein, and D. J. Norris, Nano Lett. 8, 3238 共2008兲. 5 C. M. Cornelius and J. P. Dowling, Phys. Rev. A 59, 4736 We have shown using fluctuational electrodynamics that thermal emission in the far field of photonic crystals obeys Kirchhoff’s law. This does not assume that the photonic crystals are in thermal equilibrium with the radiation field but the material components should be in thermal equilibrium. Our proof is applicable to any photonic crystal structure. Thus, in the study of thermal emission, we can avoid the direct calculation of thermal emission from photonic crystals which is computationally intensive.13 Our formulation of fluctuational electrodynamics provides a comprehensive study of thermal emission from photonic crystals both in the far field and the near field. The coherence in the far field of a linear grating that supports surface polaritons is not much improved in comparison to the flat surface due to two reasons: 共1兲 the directional momentum transfer from the grating to the surface polaritons and 共2兲 the modes that are not related to the surface polaritons. These two lead to the far-field coherence length of ⬃ which is not much different from the black body. The thermal near-field behavior of photonic crystals is shown to be different from a uniform film. This implies that near-field heat transport can be controlled by using photonic crystals. ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy 共Grant No. DE-FG02-06ER46348兲 and used resources at the University of Minnesota Supercomputing Institute. The author thanks D. J. Norris and G. Chen for helpful comments, and the Samsung Foundation for financial support. 共1999兲. G. Fleming, S.-Y. Lin, I. El-Kady, R. Biswas, and K. M. Ho, Nature 共London兲 417, 52 共2002兲. 7 S. E. Han, A. Stein, and D. J. Norris, Phys. Rev. Lett. 99, 053906 共2007兲. 8 J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, Nature 共London兲 386, 143 共1997兲. 9 T. Trupke, P. Wurfel, and M. A. Green, Appl. Phys. Lett. 84, 1997 共2004兲. 10 S.-Y. Lin, J. Moreno, and J. G. Fleming, Appl. Phys. Lett. 84, 1999 共2004兲. 11 I. El-Kady, W. W. Chow, and J. G. Fleming, Phys. Rev. B 72, 6 J. 155108-9 PHYSICAL REVIEW B 80, 155108 共2009兲 S. E. HAN 195110 共2005兲. W. Chow, Phys. Rev. A 73, 013821 共2006兲. 13 C. Luo, A. Narayanaswamy, G. Chen, and J. D. Joannopoulos, Phys. Rev. Lett. 93, 213905 共2004兲. 14 J. G. Fleming, Appl. Phys. Lett. 86, 249902 共2005兲. 15 C. H. Seager, M. B. Sinclair, and J. G. Fleming, Appl. Phys. Lett. 86, 244105 共2005兲. 16 A. T. de Hoop, Appl. Sci. Res., Sect. B 8, 135 共1960兲. 17 S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics, Vol. 3: Elements of Random Fields 共Springer-Verlag, Berlin, 1989兲. 18 C. Henkel, K. Joulain, R. Carminati, and J.-J. Greffet, Opt. Commun. 186, 57 共2000兲. 19 C. T. Tai, Dyadic Green Functions in Electromagnetic Theory, 2nd ed. 共IEEE, New York, 1994兲, Secs. 4 and 5. 20 L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed. 共Pergamon, Oxford, 1984兲. 12 W. J. E. Sipe, J. Opt. Soc. Am. B 4, 481 共1987兲. Mandel and E. Wolf, Optical Coherence and Quantum Optics 共Cambridge University Press, Cambridge, 1995兲. 23 L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1 共Pergamon, Oxford, 1984兲, Sec. 58. 24 S. E. Han and D. J. Norris 共unpublished兲. 25 C. L. Mehta and E. Wolf, Phys. Rev. 161, 1328 共1967兲. 26 J. B. Pendry and A. MacKinnon, Phys. Rev. Lett. 69, 2772 共1992兲. 27 J.-J. Greffet, R. Carminati, K. Joulain, J. P. Mulet, S. Mainguy, and Y. Chen, Nature 共London兲 416, 61 共2002兲. 28 F. Marquier, K. Joulain, J.-P. Mulet, R. Carminati, J.-J. Greffet, and Y. Chen, Phys. Rev. B 69, 155412 共2004兲. 29 F. Marquier, C. Arnold, M. Laroche, J.-J. Greffet, and Y. Chen, Opt. Express 16, 5305 共2008兲. 30 S. E. Han, P. Nagpal, and D. J. Norris 共unpublished兲. 21 22 L. 155108-10
