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Energy streamlines in near-field radiative heat transfer between hyperbolic metamaterials

T. J. Bright, X. L. Liu, and Z. M. Zhang

Open Access Open Access

Abstract

Metallodielectric photonic crystals having hyperbolic dispersions are called indefinite materials because of their ability to guide modes with extremely large lateral wavevectors. While this is useful for enhancing near-field radiative heat transfer, it could also give rise to large lateral displacements of the energy pathways. The energy streamlines can be used to depict the flow of electromagnetic energy through a structure when wave propagation does not follow ray optics. We obtain the energy streamlines through two semi-infinite uniaxial anisotropic effective medium structures, separated by a small vacuum gap, using the Green functions and fluctuation-dissipation theorem. The lateral shifts are determined from the streamlines within two penetration depths. For hyperbolic modes, the predicted lateral shift can be several thousand times of the vacuum gap width.

© 2014 Optical Society of America

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Figures (7)
Fig. 1
Fig. 1 Illustration of near-field radiative transfer between two hyperbolic metamaterials whose optical axis is normal to the surface. A vacuum gap d separate the two semi-infinite media. Cylindrical coordinates are used in which ρ specifies the component parallel to the plane. The wavevector k is decomposed into a parallel component β and a perpendicular component γ. The source (medium 1) is at 300 K and the receiver (medium 2) is at 0 K.

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Fig. 2
Fig. 2 Real part of the dielectric function predicted by EMT: (a) doped Si and Ge with f = 0.5; (b) SiC and Ge with f = 0.3. The multilayered structure is illustrated by the inset for each case. These filling ratios are used in the property calculations throughout this paper.

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Fig. 3
Fig. 3 Power transmission factor and spectral heat flux with a gap spacing d = 10 nm. Contour plots of the transmission factor for p-polarization between two semi-infinite hyperbolic metamaterials made of (a) D-Si/Ge and (b) SiC/Ge multilayered structures; (c) Spectral heat flux between D-Si/Ge multilayered structures plotted together with that between blackbodies in the far field; (d) Spectral heat flux between SiC/Ge multilayered structures. The hyperbolic bands for the multilayered structures are highlighted. Note that the unit of Sω is heat flux [W/m2] per frequency [rad/s] interval.

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Fig. 4
Fig. 4 Poynting vector and cumulative heat flux: (a) z-component of the Poynting vector versus lateral wavevector at select frequencies for (a) D-Si/Ge and (b) SiC/Ge multilayer metamaterials; Integration of the total heat flux over the wavevector space for (c) D-Si/Ge and (d) SiC/Ge. The horizontal line in (c) and (d) shows where 50% of the heat flux falls above and below the median wavevector for each frequency. Note that line styles in (c) and (d) correspond to the line styles and frequencies in (a) and (b), respectively. The unit of Sz is the heat flux [W/m2] per unit frequency [rad/s], wavevector [rad/m], and azimuthal angle [rad].

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Fig. 5
Fig. 5 Penetration depth δ ( ω , β ) and spectral penetration depth δ ω ( ω ) for d = 10 nm: (a,b) δ / d for selected frequencies as a function of β/k0 for D-Si/Ge and SiC/Ge, respectively; (c,d) δ ω / d as a function of frequency for D-Si/Ge and SiC/Ge structures, respectively. Note that the hyperbolic bands are highlighted in (c,d).

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Fig. 6
Fig. 6 Energy streamlines at select frequencies based on the median wavevector for (a,b) D-Si/Ge and (c,d) SiC/Ge multilayer structures at d = 10 nm. Here, β * = β median / k 0 that corresponds to 50% of the cumulative heat flux for each frequency as shown in Figs. 4(c) and 4(d).

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Fig. 7
Fig. 7 Average lateral displacements in vacuum and emitter for (a) D-Si/Ge structure and (b) SiC/Ge structure, with a gap spacing d = 10 nm. The highlighted regions correspond to the hyperbolic band (type I or type II).

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Equations (28)

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ε ¯ ¯ = ε t I ¯ ¯ − ( ε t − ε z ) z ^ z ^ = [ ε t 0 0 0 ε t 0 0 0 ε z ]
γ o 2 + β 2 = ε t k 0 2   ,  for ordinary waves
γ e 2 ε t + β 2 ε z = k 0 2 , for extraordinary waves
E( r,ω )=iω μ 0 ∫ V ′ G ¯ ¯ ( r, r ′ ,ω )· J r ( r ′ ,ω ) d 3 r ′
H( r,ω )= ∫ V ′ Γ ¯ ¯ ( r, r ′ ,ω )· J r ( r ′ ,ω ) d 3 r ′
〈 J r, i ( r,ω ) J r,k * ( r ′ ,ω ) 〉= Θ( ω,T ) ε 0 ε ″ ik ( ω )ω π δ( r− r ′ )
G ¯ ¯ ( r, r ′ ,ω )= 1 2π ∫ 0 ∞ g ¯ ¯ ( β,z, z ′ ,ω ) e iβ(ρ− ρ ′ ) βdβ
g ¯ ¯ 1 ± ( β,z, z ′ ,ω )= i 2 γ o,1 ( e ±i γ o,1 ( z− z ′ ) + R s e i γ o,1 ( z ′ −z ) ) o ^ o ^ + i F(β,ω)( e ^ 1 ± e ^ 1 ± e ±i γ e,1 ( z− z ′ ) + R p e i γ e,1 ( z ′ −z ) e ^ 1 − e ^ 1 + )
g ¯ ¯ 0 ( β,z, z ′ ,ω )= i 2 γ o,1 T s e i γ o,1 z ′ t 02,s ( e i γ 0 (z−d) + r 02,s e −i γ 0 z ) o ^ o ^ + i F(β,ω) Z 10 (β,ω) T p e i γ e,1 z ′ t 02,p ( e i γ 0 (z−d) e ^ 0 + + r 02,p e −i γ 0 z e ^ 0 − ) e ^ 1 +
g ¯ ¯ 2 ( β,z, z ′ ,ω )= i 2 γ o,1 T s e i γ o,1 z ′ e i γ o,2 ( z−d ) o ^ o ^ + i F(β,ω) Z 12 ( β,ω ) T p e i γ e,1 z ′ e i γ e,2 ( z−d ) e ^ 2 + e ^ 1 +
F(β,ω)= k 0 2 ( ε t,1 + ε z,1 )− k e,1 2 2 γ e,1 k 0 2 ε z,1
Z 10 (β,ω)= k 0 / γ e,1 2 ε t,1 2 + β 2 ε z,1 2 ; Z 12 (β,ω)= γ e,2 2 ε t,2 2 + β 2 ε z,2 2 / γ e,1 2 ε t,1 2 + β 2 ε z,1 2
e ^ l ± = ∓ γ e,l ε z,l ρ ^ +β ε t,l z ^ ( γ e,l ε z,l ) 2 + ( β ε t,l ) 2
e ^ 0 ± = ∓ γ 0 ρ ^ +β z ^ k 0
〈 S z ( z,ω,β ) 〉= k 0 2 Θ( ω,T )β 2 π 3 Re[ i ∫ −∞ 0 ( ε ″ t,1 g 11 h 21 * + ε ″ z,1 g 13 h 23 * − ε ″ t,1 g 22 h 12 * )d z ′ ]
〈 S ρ ( z,ω,β ) 〉= k 0 2 Θ( ω,T )β 2 π 3 Re[ i ∫ −∞ 0 ( ε ″ t,1 g 22 h 32 * − ε ″ t,1 g 31 h 21 * − ε ″ z,1 g 33 h 23 * )d z ′ ]
m= 〈 S ρ 〉 〈 S z 〉
q ″ net = 1 4 π 2 ∫ 0 ∞ [ Θ( ω, T 1 )−Θ( ω, T 2 ) ] dω ∫ 0 ∞ β ∑ j=s,p ξ j ( ω,β ) dβ
ξ j ( ω,β )={ ( 1− | r 10,j | 2 ) 2 / | 1− r 10,j 2 e 2i γ 0 d | 2 , β< k 0 4 [ Im( r 10,j ) ] 2 e 2i γ 0 d / | 1− r 10,j 2 e 2i γ 0 d | 2 , β> k 0 }
ε t =f ε m +( 1−f ) ε d
ε z = ε d ε m f ε d +( 1−f ) ε m
δ(ω,β)= 1 ±Re( ε t / ε z ) 1 2β
δ ω (ω)= ∫ 0 ∞ β ξ p δ(ω,β)dβ ∫ 0 ∞ β ξ p dβ
ζ ≡ { z / δ for z < 0 z / d for 0 < z < d 1 + ( z − d ) / δ for z > d
Δ(ω,β)=ρ( z 2 )−ρ( z 1 )
Δ ω (ω)= ∫ 0 ∞ β ξ p Δ(ω,β))dβ ∫ 0 ∞ β ξ p dβ
e c = 1+ | ε ′ z | ε ′ t for type I hyperbolic dispersion
e c = 1+ | ε ′ t | ε ′ z for type II hyperbolic dispersion
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