True thermal antenna with hyperbolic metamaterials - OSA Publishing

Vol. 25, No. 19 | 18 Sep 2017 | OPTICS EXPRESS 23356

True thermal antenna with hyperbolic metamaterials G RÉGORY B ARBILLON , 1 E MILIE S AKAT, 1 J EAN -PAUL H UGONIN , 1 S VEND -AGE B IEHS , 2 AND P HILIPPE B EN -A BDALLAH 1,3,* 1 Laboratoire

Charles Fabry, UMR 8501, Institut d’Optique, CNRS, Université Paris Sud, 2 Avenue Augustin Fresnel, 91127 Palaiseau Cedex, France 2 Institut für Physik, Carl von Ossietzky Universität, D-26111 Oldenburg, Germany 3 Université de Sherbrooke, Department of Mechanical Engineering, Sherbrooke, PQ J1K 2R1, Canada * [email protected]

Abstract: A thermal antenna is an electromagnetic source that emits in its surrounding a spatially coherent field in the infrared frequency range. Usually, its emission pattern changes with the wavelength so that the heat flux it radiates is weakly directive. Here, we show that a class of hyperbolic materials of type II possess a Brewster angle, which is weakly dependent on the wavelength, so that they can radiate like a true thermal antenna with a highly directional and p-polarized heat flux. The realization of these sources could open a new avenue in the field of thermal management in far-field regime. c 2017 Optical Society of America

OCIS codes: (160.3918) Metamaterials; (290.6815) Thermal emission.

References and links 1. M. Planck, “Law of energy distribution in normal spectra,” Ann. Phys. 4(3), 553–563 (1901). 2. G. Kirchhoff, “Monatsberichte der Akademie der Wissenschaften zu Berlin,” sessions of Dec., 783 (1859). 3. P. J. Hesketh, J. N. Zemel, and B. Gebhart, “Organ pipe radiant modes of periodic micromachined silicon surfaces,” Nature 324(6097), 549–551 (1986). 4. P. J. Hesketh, J. N. Zemel, and B. Gebhart, “Polarized spectral emittance from periodic micromachined surfaces .2. doped silicon - Angular variation,” Phys. Rev. B 37(18), 10803 (1988). 5. M. Kreiter, J. Oster, R. Sambles, S. Herminghaus, S. Mittler-Neher and W. Knoll, “Thermally induced emission of light from a metallic diffraction grating, mediated by surface plasmons,” Opt. Commun. 168(1-4), 117–122 (1999). 6. Greffet, J. J., R. Carminati, K. Joulain, J. P. Mulet, S. Mainguy, and Y. Chen, “Coherent emission of light by thermal sources,” Nature 416(6876), 61–64 (2002). 7. O. G. Kollyukh, A.I. Liptuga, V. Morozhenko and V. I. Pipa, “Thermal radiation of plane-parallel semitransparent layers,” Opt. Commun. 225(4-6), 349–352 (2003). 8. P. Ben-Abdallah, “Thermal antenna behavior for thin-film structures,” J. Opt. Soc. Am. A 21(7), 1368–1371 (2004). 9. I. Celanovic, D. Perreault and J. Kassakian, “Resonant-cavity enhanced thermal emission,” Phys. Rev. B 72(7), 075127 (2005). 10. B. J. Lee, C. J. Fu and Z. M. Zhang, “Coherent thermal emission from one-dimensional photonic crystals,” Appl. Phys. Lett. 87(7), 071904 (2005). 11. J. Drevillon and P. Ben-Abdallah, “Ab initio design of coherent thermal sources,” J. Appl. Phys. 102(11), 114305 (2007). 12. A. Battula and S. C. Chen, “Monochromatic polarized coherent emitter enhanced by surface plasmons and a cavity resonance,” Phys. Rev. B 74(24), 245407 (2006). 13. K. Joulain and A. Loizeau, “Coherent thermal emission by microstructured waveguides,” J. Quant. Spectro. Rad. Trans 104(2), 208–216 (2007). 14. S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95(13), 137404 (2005). 15. F. M. Wang, H. Liu, T. Li, Z. G. Dong, S. N. Zhu and X. Zhang, “Metamaterial of rod pairs standing on gold plate and its negative refraction property in the far-infrared frequency regime,” Phys. Rev. E 75(1), 016604 (2007). 16. S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics 3 (Springer-Verlag, 1989). 17. P. Ben-Abdallah and K. Joulain, “Fundamental limits for non contact transfers between two bodies,” Phys. Rev. B 82(12), 121419 (2010). 18. S.-A. Biehs, E. Rousseau, and J.-J. Greffet, “Mesoscopic description of radiative heat transfer at the nanoscale,” Phys. Rev. Lett. 105(23), 234301 (2010). 19. S.-A. Biehs and P. Ben-Abdallah, “Revisiting super-Planckian thermal emission in the far-field regime,” Phys. Rev. B 93(16), 165405 (2016).

#301982 Journal © 2017

https://doi.org/10.1364/OE.25.023356 Received 7 Jul 2017; revised 11 Aug 2017; accepted 11 Aug 2017; published 14 Sep 2017


20. L. Hu and S. T. Chui, “Characteristics of electromagnetic wave propagation in uniaxially anisotropic left-handed materials,” Phys. Rev. B 66(8), 085108 (2002). 21. L. Sun and K. W. Yu, “Strategy for designing broadband epsilon-near-zero metamaterials,” J. Opt. Soc. Am. B 29, 5 (2012). 22. L. Sun, X. Yang and J. Gao, “Loss-compensated broadband epsilon-near-zero metamaterials with gain media,” Appl. Phys. Lett. 103, 201109 (2013). 23. P. Yeh, Optical Waves in Layered Media (John Wiley & Sons, New Jersey, 2005). 24. M. Tschikin, S. A. Biehs, R. Messina and P. Ben-Abdallah, “On the limits of the effective description of hyperbolic materials in the presence of surface waves,” J. Opt. 15(10), 105101 (2013). 25. J. Frigerio, A. Ballabio, G. Isella, E. Sakat, P. Biagioni, M. Bollani, E. Napolitani, C. Manganelli, M. Virgilio, A. Grupp, M. P. Fischer, D. Brida, K. Gallacher, D. J. Paul, L. Baldassarre, P. Calvani, V. Giliberti, A. Nucara and M. Ortolani, “Tunability and Losses of Mid-infrared Plasmonics in Heavily Doped Germanium Thin Films,” Phys. Rev. B 94, 085202 (2016).

1.

Introduction

The thermal radiation [1, 2] emitted by a hot body into its surrounding results from a well-known incoherent emission process. The local charges in the medium (electrons, ions or partial atomic charges) oscillate thanks to thermal fluctuations and as the corresponding oscillators are usually delta-correlated, they radiate incoherently in their surrounding. A direct consequence of this mechanism is the absence of favored directions of emission. However, in 1986 and 1988, Hesketh et al. [3, 4] showed that a textured surface of a doped silicon sample could behave as a thermal antenna that is a spatially coherent source thanks to the presence of a surface plasmon polariton, corresponding to a surface wave whose electromagnetic field is spatially correlated. Since this pioneer work, numerous spatially coherent sources [5–15] have been proposed. However, sources with extremely directional emission patterns have been mainly achieved so far at a given single frequency. Generally, the emission angle of these sources significantly changes with respect to the wavelength throughout the Planck window. It follows that the heat flux they radiate, resulting from the spectral integration of the directional monochromatic emissivity weighted by the Planck distribution function, is not notably directional. Today, the development of broadband angular selective sources in the infrared range remains a challenging problem. In this paper, we show that a class of hyperbolic materials (HM) can be used to achieve a ’true thermal antenna’, which radiates a highly directional and p-polarized heat flux in its surrounding. 2.

Theory

To start, let us consider an arbitrary semi-infinite planar anisotropic medium at temperature T surrounded by a bosonic field at zero temperature. According to the theory of fluctuational electrodynamics [16], thepradiative heat flux lost in its surrounding by this medium in the direction u = ωc (κ, γ0 ) with γ0 = ω2 /c2 − κ 2 can be written in a Landauer-like form as [17–19] Z ∞ dω d2 κ dΦ(u) = 2 Θ(ω, T )T (ω, κ) , (1) (2π) 2 0 2π where κ := (k x , k y ) t is the parallel component of wavevector (with the constraint |κ| < ω/c for the propaging waves), Θ(ω, T ) := e ~ω~ωβ −1 is the mean energy of a harmonic oscillator in thermal equilibrium at temperature T and T denotes the transmission coefficient associated to each mode (ω, κ) which reads 1 Tr (1 − R† R) . 2 Here, we have introduced the reflection operator for both polarization states (s, p) ! r ss r s p R := . r ps r p p T (ω, κ, d) :=

(2)

(3)


When further introducing Iω0 (T ) := Θ(ω, T ) 4πω2 c 2 , the spectral intensity of a blackbody at the frequency ω and using the generalized emissivity 2

(ω, u) := T (ω, κ) 1 = 2 − |r ss | 2 − |r p p | 2 2 − |r s p | 2 − |r ps | 2 , the flux radiated by the source in the direction u can be written as Z ∞ c2 d2 κ dΦ(u) = dω 2 (ω, u)Iω0 (T ) . π ω 0

(4)

(5)

Notice that expression (4) includes the emissivity of the two different polarization states s and p, and the emissivity of the cross-polarized states sp and ps as well allowing us to deal with arbitrary anisotropic sources. When the source displays an azimuthal symmetry, the directional heat flux simplifies to Z ∞ dΦ(θ) = 2 cos θ sin θdθ dω (ω, θ)Iω0 (T ) . (6) 0

Both expressions (5) and (6) are the classical expressions predicted by Kirchoff’s theory [2]. An inspection of these expressions clearly shows that an angular drift in the emission pattern for the source with respect to the frequency leads to a heat flux, which is less directive. Therefore, in order to get a highly directional heat flux, we need to have a source with a spatial degree of coherence, which does not change significantly all over the Planck window. In other words, this control requires to design the dispersion curve ω = ω(κ) of the source so that the structure supports only modes of parallel component κ = ωc sin e θ for a given emission angle e θ over the whole Planck window. Below, we show that a class of HM precisely behaves like that. To this end, we consider a uniaxial anisotropic medium with a dielectric permittivity tensor = | | (x ⊗ x + y ⊗ y) + ⊥ z ⊗ z , | | being the permittivity parallel to the surface and ⊥ the permittivity along the normal to its surface. For this medium, the components of the reflection operator are γ0 − γs , γ0 + γs | | γ0 − γ p = , | | γ0 + γ p

r ss = r pp

r ps = r s p = 0

(7) (8) (9)

q p with γs = | | ω2 /c2 − κ 2 and γ p = | | ω2 /c2 − ⊥| | κ 2 . From these expressions, it can be directly seen that by imposing the condition r p p = 0 and using the identity κ = ω/c sin θ B , the Brewster angle θ B is given by s ⊥ ( | | − 1) θ B = arcsin . (10) ||⊥ − 1 In Fig. 1, this angle is plotted in the ( | | , ⊥ ) plane for the case of arbitrary uniaxial media when losses are negligible. Each quadrant corresponds to a specific class of anisotropic medium. When both parameters | | and ⊥ are positive, the medium is a standard uniaxial crystal with an ellipsoidal iso-frequency surface. On the contrary, if both parameters are negative, the medium is


kz

kz

kx

ky

ky

kx

Hyperbolic type I

𝜺∥ Hyperbolic type II kz ky kx

𝜺⊥ Fig. 1. Brewster angle (in degree) in the ( ⊥ , | | ) plane for transparent uniaxial media. The black zones correspond to media which have no Brewster angle. The insets show the iso-frequency surfaces of different uniaxial crystals.

a metallic-like anisotropic medium. In this case, the iso-frequency surface is purely imaginary and the medium does not support propagating modes. In the two other quadrants, | | and ⊥ have κ 2 +κ 2

γ2

opposite signs. In both cases, the iso-frequency relation x ⊥ y + |z| = ωc 2 defines hyberboloidal surfaces. When | | > 0 and ⊥ < 0, the HM is called type I HM while in the case where | | < 0 and ⊥ > 0 it is of type II. Two different iso-frequency surfaces are associated to these two types of HM: a two-sheeted hyperboloid for type I HM and an one-sheeted hyperboloid for type II HM. As shown in Fig. 1, both types of HM possess a Brewster angle in some regions of ( | | , ⊥ ) plane. For the type I HMs, this angle only exists when | | ≥ 1 and we observe its drift towards grazing angles as the value of | | increases. On the contrary for type II HMs, a Brewster angle only exists when ⊥ ≤ 1. But what is worthwhile to note is that the Brewster angle changes very little with the value of | | making these media potentially good candidates for designing a coherent thermal antenna with a weak angular variation of emission angle with respect to the frequency. To confirm this prediction, let us investigate the reflection coefficients of HMs. These reflections coefficients are plotted in Figs. 2 and 3. In s-polarization, while the reflection of type I HMs is relatively weak for any angle of incidence, we see that the reflection of type II HMs is very close to 1 showing so that their thermal emission is almost entirely p-polarized. Moreover, the reflection coefficient in p-polarization of type I and II HMs, plotted in Fig. 3 for two different values of ⊥ , confirm the weak variation of Brewster angle of HMs of type II when ⊥ is smaller than one. They also demonstrate according to relation (4), that the thermal emission of these media, which is proportional for semi-infinite media to 1 − |r p p | 2 , is restricted to an angular 2


Hyperbolic type I

𝜀∥ Hyperbolic type II

(°) Fig. 2. Reflection in s-polarization of HM for the type I ( | | > 0) and type II ( | | < 0). The light zone corresponds to the region where the reflection is close to 1.

sector beyond a critical angle. (a)

(b)

𝜀∥

𝜀∥

(°)

(°)

Fig. 3. Reflection in p-polarization of HM for type I ( | | > 0) and type II ( | | < 0) when (a) (| ⊥ |= 0.5) and (b) (| ⊥ |= 5) .

This critical angle increases (not shown in Fig. 3) with the value of ⊥ . It can be derived by √ √ the condition that γ p = 0, which is equivalent to κ = ⊥ ω/c or θ = arcsin( ⊥ ) [20] showing again that the condition 0 ≤ ⊥ ≤ 1 must be necessarily fulfilled. For ⊥ = 0.5 (see Fig. 3(a)), this angle is located around 45◦ . 3.

Results and discussions

Finding a natural HM of type II that displays the required properties over a broad spectral range in the Planck frequency window is a tricky task. However, a metamaterial could be designed

ε2 ε2 = ε1-f-f ε1 1

ε2 = 1-f f

ε1

y

x

ε1 ε2 ε1 ε2

ε2 = f ε1 f-1

Fig. 4. Dielectric properties of materials given by an alternating layered medium as sketched in the inset and satisfying the inequalities (13) and (14). The set of solutions of this system is shown in grey.

for that purpose [21, 22]. To this end, we consider here an artificial composite structure formed by alternating layers of materials of permittivity 1 and 2 . In the long-wavelength limit, the structure behaves like an homogeneous uniaxial crystal [23, 24] with ||

=

⊥

=

f 1 + (1 − f ) 2 , 1 2 , f 2 + (1 − f ) 1

(11) (12)

where f denotes the filling factor with respect to medium 1. Considering, for clarity reasons, the ideal situation of non-lossy media, it is easy to show that to obtain a HM of type II with ⊥ < 1, the two dielectric constants must satisfy the following inequalities f 1 + (1 − f ) 2 < 0, 1 ( 1 − f ) < , (1 − f ) 1 2

(13) (14)

A graphical solution of this system (see Fig. 4) shows that the possible values of 1 and 2 are restricted to a special region, which itself depends on the filling factor. On the other hand, when fixing 1 and 2 this system of conditions leads to limits of the filling factor for which we can expect to have a broadband Brewster angle. Guided by these conditions, which require to use a near ENZ material for medium 1 and a metallic-like material for medium 2, we consider below a numerical example of a hyperbolic thermal antenna made with a metamaterial composed by alternating layers of one non-dispersive dielectric material of permittivity 1 and one metallic layer made with heavily doped germanium (n-Ge) whose the imaginary part is weak enough in order to allow an analytical continuation of the conditions derived previously into the complex plane. The permittivity is described by the simple Drude-like model [25]   ω2p   , Ge = ∞ 1 − (15) ω(ω + iγ) 


(b)

(a) Φ Φ𝐵𝐵

Φ Φ𝐵𝐵

(c)

Φ Φ𝐵𝐵

(d)

Φ Φ𝐵𝐵

Fig. 5. Angular heat flux emitted by a dielectric-metal layered structure at T = 300 K for different filling factors f in dielectric of permittivity (a) 1 = 0.9+i0.01, (b) 1 = 0.5+i0.01, (c) 1 = 0.1 + i0.01 and (d) 1 = 1.1 + i0.01. The metallic layers are made with heavily doped germanium of permittivity 2 = Ge [25]. The flux is normalized by the flux radiated by a blackbody at temperature T.

where the permittivity at infinite frequency, the plasma frequency and the electron damping are given by ∞ = 16, ω p = 3.768 × 1014 rad/s and γ = 3.768 × 1013 rad/s, respectively. Note that the plasma frequency of this medium is in the infrared so that it behaves in the Planck window at ambient temperature (T = 300K) as a metal with relatively low losses. In Fig. 5, we plot the angular heat flux emitted at this temperature by these dielectric-metal layered structures with different filling factors and different values of 1 when Re( 1 ) < 1 (Figs. 5(a)–5(c)) and Re( 1 ) > 1 (Fig. 5(d)). At ambient temperature, Wien’s wavelength is λ W = 9.66 µm and about 95% of the radiative energy is emitted by the source between 0.5λ W = 4.83 µm and 4.5λ W = 43.47 µm. At these wavelengths Re( 2 ) < 0 and | 2 | 1 so that we have approximately ⊥ ≈ 1 / f and k ≈ (1− f ) 2 for the chosen filling factors and values of 1 . Then, the first condition (13) is automatically fulfilled and the second condition (14) is approximately given by Re( 1 ) < f . Consequently, for filling factors f < Re( 1 ), the structure does not show any distinguished emission feature. On the other hand, as the filling factor of dielectric increases such that f ≥ Re( 1 ), a lobe of emission for the heat flux appears at oblique incidence reaching a maximum value at 88o in Fig. 5(a) when f = 0.9 (which means that only 10% of the structure is metallic) and around 72o (see Fig. 5(b)) when f = 0.9. Note that these filling factors are realistic from a nanofabrication point of view. In the case where f = 0.9, the directivity observed results from the invariance of Brewster angle as shown in Fig. 1. However, since the reflectivity for the s-polarization is almost equal to 1, the thermal flux radiated at the Brewster angle is mainly


p-polarized and therefore cannot exceed one-half of flux radiated by a blackbody at the same temperature. In Figs. 5(b) and 5(c), we can also observe a wider emission peak in the pattern of flux when the real part of the dielectric permittivity becomes √ smaller, because the emitter can radiate into all angles above the critical angle θ = arcsin( Re( ⊥ )). Finally, as anticipated, we verify in Fig. 5(d) the absence of emission lobe in the flux pattern when Re( 1 ) > 1 and therefore also Re( 1 ) > f for any choice of f . Although the thermal antenna designed above radiates most of its radiative power around a specific direction of space, on one hand its angle of thermal emission is very oblique and on the other hand the emission lobe is not extremely narrow. Reducing the emission angle around the normal incidence still remains a challenging problem, which goes well beyond this preliminary work since it requires to design broadband anisotropic ENZ metamaterials. However, solutions have been already proposed in the literature to design such artificial materials [21, 22]. As the narrowness of the emission pattern is concerned, it is closely related to the absorption of the structure around the Brewster angle. To increase the directivity, this absorption also should be optimized. 4.

Conclusion

In summary, we have predicted that a class of HM of type II can emit most of its radiative power in privileged directions of space. Unlike usual thermal antenna, the emission angle of these sources is almost invariant with respect to the wavelength all over the Planck window. This result paves the way for highly directive radiative heat sources. We believe that these "true" thermal antennas should find broad applications in the field of fundamental sciences and for a number of applications such as thermal management, heat-driven control of chemical reaction or thermal regulation in biology. However, further works are needed to achieve true thermal antenna in arbitrary emission angles.