## Near-perfect absorption in epsilon-near-zero structures with hyperbolic dispersion |

Optics Express, Vol. 22, Issue 6, pp. 7337-7348 (2014)

http://dx.doi.org/10.1364/OE.22.007337

Acrobat PDF (4665 KB)

### Abstract

We investigate the interaction of polarized electromagnetic waves with hyperbolic metamaterial structures, whereby the in-plane permittivity component *ε _{x}* is opposite in sign to the normal component

*ε*. We find that when the thickness of the metamaterial is smaller than the wavelength of the incident wave, hyperbolic metamaterials can absorb significantly higher amounts of electromagnetic energy compared to their conventional counterparts. We also demonstrate that for wavelengths leading to ℜ(

_{z}*ε*) ≈ 0, near-perfect absorption arises and persists over a range of frequencies and subwavelength structure thicknesses.

_{z}© 2014 Optical Society of America

## 1. Introduction

1. D. Sievenpiper, L. Zhang, R. Broas, N. G. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech. **47**(11), 2059–2074 (1999). [CrossRef]

2. M. G. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials”, Phys. Rev. Lett. **97**(15), 157403 (2006). [CrossRef] [PubMed]

6. S. Molesky, C. J. Dewalt, and Z. Jacob, “High temperature epsilon-near-zero and epsilon-near-pole metamaterial emitters for thermophotovoltaics,” Opt. Express **21**(S1), A96–A110 (2013). [CrossRef] [PubMed]

**or permeability**

*ε***. In close connection with these developments, there has also been a substantial amount of research lately involving anisotropic metamaterials, where now**

*μ***and**

*ε***are tensors that have in general differing components along the three coordinate axes. An important type of anisotropic metamaterial is one whose corresponding orthogonal tensor components are of opposite sign, sometimes referred to as indefinite media [7**

*μ*7. D. Schurig and D. R. Smith, “Spatial filtering using media with indefinite permittivity and permeability tensors,” Appl. Phys. Lett. **82**, 2215–2217 (2003). [CrossRef]

8. C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. **14**(6), 063001 (2012). [CrossRef]

7. D. Schurig and D. R. Smith, “Spatial filtering using media with indefinite permittivity and permeability tensors,” Appl. Phys. Lett. **82**, 2215–2217 (2003). [CrossRef]

*λ*in the visible spectrum, an effective HMM was modeled using arrays of metallic nanowires [9

9. Y. Liu, G. Bartal, and X. Zhang, “All-angle negative refraction and imaging in a bulk medium made of metallic nanowires in the visible region,” Opt. Express **16**(20), 15439–15448 (2008). [CrossRef] [PubMed]

*λ*, thus avoiding the usual problems associated with resonances. Periodic arrays of carbon nanotubes [10

10. I. Nefedov and S. Tretyakov, “Ultrabroadband electromagnetically indenite medium formed by aligned carbon nanotubes,” Phys. Rev. B **84**(11), 113410 (2011). [CrossRef]

11. X. Ni, S. Ishii, M. D. Thoreson, V. M. Shalaev, S. Han, S. Lee, and A. V. Kildishev, “Loss-compensated and active hyperbolic metamaterials,” Opt. Express **19**(25), 25242–25254 (2011). [CrossRef]

12. S. Savoia, G. Castaldi, and V. Galdi, “Optical nonlocality in multilayered hyperbolic metamaterials based on Thue-Morse superlattices,” Phys. Rev. B **87**, 235116 (2013). [CrossRef]

13. O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, “Effective-medium approach to planar multilayer hyperbolic metamaterials: Strengths and limitations,” Phys. Rev. A **85**(5), 053842 (2012). [CrossRef]

14. W. Yan, M. Wubs, and N. A. Mortensen, “Hyperbolic metamaterials: nonlocal response regularizes broadband supersingularity,” Phys. Rev. B **86**, 205429 (2012). [CrossRef]

15. T. Tumkur, L. Gu, J. Kitur, E. Narimanov, and M. Noginov, “Control of absorption with hyperbolic metamaterials,” Appl. Phys. Lett. **100**, 161103 (2012). [CrossRef]

16. F. V. Iorsh, I. S. Mukhin, I. V. Shadrivov, P. A. Belov, and Y. S. Kivshar, “Hyperbolic metamaterials based on multilayer graphene structures,” Phys. Rev. B **87**(7), 075416 (2013). [CrossRef]

17. W. Li, Z. Liu, X. Zhang, and X. Jiang, “Switchable hyperbolic metamaterials with magnetic control,” Appl. Phys. Lett. **100**, 1611084 (2012). [CrossRef]

18. C. Rizza, A. Ciattoni, E. Spinozzi, and L. Columbo, “Terahertz active spatial filtering through optically tunable hyperbolic metamaterials,” Opt. Lett. **37**, 3345–3347 (2012). [CrossRef]

19. C. Rizza, A. Ciattoni, L. Columbo, M. Brambilla, and F. Prati, “Terahertz optically tunable dielectric metamaterials without microfabrication,” Opt. Lett. **38**, 1307 (2013). [CrossRef] [PubMed]

20. W. Yan, L. Shen, L. Ran, and J. A. Kong, “Surface modes at the interfaces between isotropic media and indefinite media,” J. Opt. Soc. Am. A **24**, 530 (2007). [CrossRef]

_{2}HMM structure, creating a “hypergrating” capable of exciting both surface and bulk plasmons [21

21. K.V. Sreekanth, A. De Luca, and G. Strangi, “Experimental demonstration of surface and bulk plasmon polaritons in hypergratings,” Sci. Rep. **3**, 3291 (2013). [CrossRef] [PubMed]

22. E. Narimanov, M. A. Noginov, H. Li, and Y. Barnakov, “Darker than Black: Radiation-absorbing Metamaterial,” in Quantum Electronics and Laser Science Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper QPDA6. [CrossRef]

23. T. U. Tumkur, J. K. Kitur, B. Chu, L. Gu, V. A. Podolskiy, E. E. Narimanov, and M. A. Noginov, “Control of reflectance and transmittance in scattering and curvilinear hyperbolic metamaterials,” Appl. Phys. Lett. **101**, 091105 (2012). [CrossRef]

*ε*

_{x}*>*0,

*ε*

_{z}*<*0 or

*ε*

_{x}*<*0,

*ε*

_{z}*>*0 respectively (see Fig. 1). We show that for those

*λ*leading to the real part of the permittivity component perpendicular to the interfaces (

*ε*) nearly vanishing, an intricate balance between material loss and structure thickness (

_{z}*τ*) yields a broad range of incident angles

*θ*and

*τ*in which nearly the entire EM wave is absorbed. These findings are absent in conventional anisotropic “elliptical” structures.

## 2. Methods

*x*−

*z*plane with polarization (

*E*,

_{x}*E*,

_{z}*B*) (

_{y}*p*-polarized) or (

*E*,

_{y}*B*,

_{x}*B*) (

_{z}*s*-polarized). Once the wave enters the anisotropic medium, its polarization state can then split into linear combinations of both TE and TM polarizations [24

24. J. M. Zhao, Y. Chen, and Y. J. Feng, “Polarization beam splitting through an anisotropic metamaterial slab realized by a layered metal-dielectric structure,” Appl. Phys. Lett. **92**, 071114 (2008). [CrossRef]

*ε*,

_{x}*ε*,

_{y}*ε*) and (

_{z}*μ*,

_{x}*μ*,

_{y}*μ*), where it is always possible to choose principal coordinate axes so that the permittivity and permeability are diagonal. Assuming a harmonic time dependence, exp(−

_{z}*iωt*), for the EM fields, Maxwell’s equations give the corresponding wave equations for the electric field components

*E*and

_{x}*E*: Equations (1) and (2) illustrate that the wave equations are different for

_{y}*E*and

_{x}*E*, resulting in two different wave vectors. In this work, we focus exclusively on

_{y}*p*-polarization from which the nature of the HMM dispersion can be qualitatively understood. From Eq. (1),

*ω*/

*c*). For this discussion we assume real valued material parameters and positive

*μ*. Focusing on

_{y}*ε*>0, we consider two scenarios (a)

_{x}*ε*>0 and (b)

_{z}*ε*<0, yielding the respective dispersion relations

_{z}*k*= 0). Moreover, for the ellipsoidal case, as

_{y}*k̂*increases there will be a frequency cutoff since

_{x}*k̂*increases, there is no cutoff since

_{x}*ε*

_{x}*<*0 and

*ε*

_{z}*>*0, we then have the possibility of a connected hyperbola (see Fig. 2(b)).

*r*. The corresponding reflectance

*R*is then given by

*R*=

*|r|*

^{2}. For a

*p*-polarized plane wave incident at an angle

*θ*relative to the normal of a planar layer of thickness

*τ*, we find,

*j*labels the regions 1,2 or 3 (see Fig. 1). In all cases below, the incident beam is in vacuum (region 3) so that

*k̂*= sin

_{x}*θ*, which is conserved across the interface. The frequency dispersion in the HMM takes the Drude-like form:

*ε*

_{z}_{2}=

*a*+

*ib*, where

*a*= 1 −

*α*

^{2}/[1+(

*αf*)

^{2}], and

*b*=

*α*

^{3}

*f*/[1 + (

*αf*)

^{2}]. Here,

*α*≡

*λ*/

*λ*,

_{z}*f*= 0.02, and the characteristic wavelength,

*λ*= 1.6

_{z}*μ*m. When discussing the two types of HMM, the permittivity parallel to the interface is described using

*ε*

_{x}_{2}= ±4 + 0.1

*i*for type-1 (+) and type-2 (−). The wavelength range considered here, where the system exhibits HMM behavior is consistent with experimental work involving HMM semiconductor hybrids [25

25. G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Natl. Acad. Sci. U.S.A. **109**, (23)8834–8838 (2012). [CrossRef] [PubMed]

*R*= 0): where

*n*is an integer. The

*k̂*in Eq. (7) corresponds to the classic Brewster angle condition for isotropic media:

_{x}*k̂*in Eq. (8) corresponds to a standing wave condition in the

_{x}*z*-direction. In either case, when Eqs. (7) or (8) is satisfied, a minimum in

*R*arises. Under the Brewster angle condition in Eq. (7), a simple rearrangement shows that

*ε*

_{x}_{2}=

*μ*

_{y}_{2}and

*ε*

_{x}_{2}

*ε*

_{z}_{2}= 1, then we should have

*R*= 0 for any value of

*k̂*= sin

_{x}*θ*. This choice of anisotropic material parameters is similar to the perfectly matched layer (PML) approach to eliminating unwanted reflection from absorbing computational domain boundaries, especially in time-domain [26

26. J. P. Berenger, “A perfect matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. **114**, 185–200 (1994). [CrossRef]

27. Z. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Ant. Prop. **43**, 1460–1463, (1995). [CrossRef]

28. J. Merle Elson, “Propagation in planar waveguides and the effects of wall roughness,” Opt. Express **9**, 461–475 (2001). [CrossRef] [PubMed]

**= 1). The reflection coefficient in Eq. (3) then becomes, where**

*μ**ϕ*≡ (

*ω*/

*c*)

*k̂*, and

_{z}τ*k̂*= cos

_{z}*θ*. It is readily verified that for lossless media, Eq. (9) yields perfect reflection (

*|r|*

^{2}=1) as expected. In the absence of transmission, the absorption,

*A*, is simply written as

*A*= 1 −

*R*.

## 3. Results

*θ*for both types of HMM: type-1, ℜ{

*ε*

_{2}

*}*

_{x}*>*0, ℜ{

*ε*

_{2}

*}*

_{z}*<*0 (panels a and c), and type-2, ℜ{

*ε*

_{2}

*}*

_{x}*<*0, ℜ{

*ε*

_{2}

*}*

_{z}*>*0 (panels b and d). Since

*λ*≈

*λ*, we have also the condition, ℜ(

_{z}*ε*

_{2}

*) ≈ 0. There cannot be any substrate transmission and thus*

_{z}*R <*1 is due to intrinsic HMM losses. In terms of practical designs, it is important to determine the range of sub-wavelength HMM layer thicknesses that can admit perfect absorption. Thus Figs. 2(a) and (b) explore differing

*τ*ranging from 0.001 to 0.156

*μ*m. Although the relative sign of

*ε*

_{x}_{2}and

*ε*

_{z}_{2}usually plays a pivotal role, for extremely thin HMM widths this is not the case. Indeed in the regime of small

*ϕ*

_{2}, Eq. (9) simplifies to, which is independent of

*ε*

_{x}_{2}. Here we have introduced the dimensionless thickness:

*τ̂*≡

*ωτ*/

*c*. Setting the numerator of

*r*to zero gives the angle,

*θ*, where the reflectance vanishes: In Fig. 2(a), for the incident wavelength of 1.601

_{c}*μ*m, the approximate absorption angles are found from taking the real part of Eq. (11), giving,

*θ*≈ 79°,66°,52°,28°, and 11°, in order of increasing

_{c}*τ*. Deviations in the angle predicted from Eq. (11) arise for larger

*τ*as higher order corrections are needed. As the thickness

*τ*decreases, near-perfect absorption shifts towards grazing incidences, in agreement with Eq. (11) where as

*τ*→ 0,

*θ*→

_{c}*π*/2. For the type-2 HMM, similar trends are seen in Fig. 2(b), where

*λ*= 1.59

*μ*m and the near-perfect absorption angles were found to agree well with Eq. (11). It is apparent that for a type-2 HMM, the angular range of near-perfect absorption exhibits a greater sensitivity to

*τ*than the type-1 case shown. In both cases (a) and (b), near-perfect absorption can be controlled over nearly the whole angular range by properly choosing the effective material thicknesses. When calculating the regions of high absorption, ℑ(

*ε*

_{z}_{2}) plays a significant role when ℜ(

*ε*

_{z}_{2}) ≈ 0. This is consistent with anisotropic leaky-wave structures [4

4. K. Halterman, S. Feng, and V. C. Nguyen, “Controlled leaky wave radiation from anisotropic epsilon near zero metamaterials,” Phys. Rev. B **84**, 075162 (2011). [CrossRef]

5. S. Feng and K. Halterman, “Coherent perfect absorption in epsilon-near-zero metamaterials,” Phys. Rev. B **86**, 165103 (2012). [CrossRef]

30. Y. Jin, S. Xiao, N.A. Mortensen, and S. He, “Arbitrarily thin metamaterial structure for perfect absorption and giant magnification,” Opt. Express **19**, 11114 (2011). [CrossRef] [PubMed]

11. X. Ni, S. Ishii, M. D. Thoreson, V. M. Shalaev, S. Han, S. Lee, and A. V. Kildishev, “Loss-compensated and active hyperbolic metamaterials,” Opt. Express **19**(25), 25242–25254 (2011). [CrossRef]

*τ*is set to 0.16

*μ*m for both the type-1 and type-2 HMM cases respectively. For the type-1 HMM (panel c), as

*λ*increases beyond

*λ*, the wavelength-dependent

_{z}*ε*

_{z}_{2}shifts so that its real part becomes more negative. The corresponding absorption peaks then migrate towards

*θ*= 90°. The opposite trend occurs for the type-2 case, where increasing

*λ*from

*λ*= 1.5

*μ*m causes ℜ(

*ε*

_{z}_{2}) (which is positive at this wavelength) to approach zero. Consequently, the observed double-peaked absorption shifts towards normal incidence, consistent with the trends above and Eq. (11), where as

*λ*→

*λ*(and hence ℜ(

_{z}*ε*

_{2}

*) → 0), the angle of near-perfect absorption tends to zero. It is worth noting that if the HMM is replaced by an isotropic metallic layer like silver, the condition where the permittivity is near zero is consistent with the generation of bulk longitudinal collective oscillations of the free electrons. This type of excitation can produce moderate (but less than 100%) absorption when there is minimal intrinsic material loss.*

_{z}*ϕ*

_{2}= 0, then

*R*= 0. If on the other hand, both substrate and superstrate are perfectly conducting, then setting the denominator of (3) equal to zero also yields sin

*ϕ*

_{2}= 0, which coincides with the dispersion relation for guided waves in an HMM layer. Equation (8) shows that when

*n*= 0,

*x*-direction. Thus if

*ϕ*

_{2}=

*k̂*

_{z}_{2}

*τ̂*=

*n*π, this assertion is valid if

*ϕ*

_{2}<<

*n*π. If however

*ε*

_{z}_{2}/

*ε*

_{x}_{2}< 0, Eq. (8) reveals that there is no guided mode cutoff for

*λ*and

*θ*. The HMM thickness is fixed at

*τ*=0.16

*μ*m, as in Figs. 2(c) and (d). In Fig. 3(a)

*ε*

_{x}_{2}= (4,0.1), so that the HMM region where ℜ(

*ε*

_{z}_{2}) < 0 corresponds to

*λ*>

*λ*(recall that

_{z}*λ*=1.6

_{z}*μ*m). Similarly for (b),

*ε*

_{x}_{2}= (−4, 0.1), and thus the HMM region there corresponds to

*λ*< 1.6

*μ*m. Figs. 4(a) and (b) are slices from Figs. 3(a) and (b). In Fig. 4(a) near-perfect absorption occurs at

*θ*= 65° for both HMM types. For

*λ*= 1.7

*μ*m, ℜ(

*ε*

_{x}_{2}) = 4 and ℜ(

*ε*

_{z}_{2}) = −0.128 corresponding to a Type-1 HMM. For

*λ*= 1.5

*μ*m, ℜ(

*ε*

_{x}_{2}) = −4 and ℜ(

*ε*

_{z}_{2}) = 0.121, corresponding to a Type-2 HMM. In Fig. 4(b), the Type-1 absorption peak occurs at

*λ*= 1.66

*μ*m, where ℜ(

*ε*

_{z}_{2}) = −0.076, and the Type-2 case peaks at

*λ*= 1.55

*μ*m, where ℜ(

*ε*

_{z}_{2}) = 0.062.

**term is absent. In all of the near-perfect absorption examples investigated here, evaluation of Eq. (12) confirmed that the net energy flow into the HMM volume,**

*J**V*, is converted into heat.

**in the HMM as a function of**

*P**θ*for the two cases in Fig. 4(a). Thus, panel (a) is for

*λ*= 1.7

*μ*m (type-1 HMM), and panel (b) corresponds to

*λ*= 1.5

*μ*m (type-2 HMM). The average power along the

*x*and

*z*directions,

*P*

_{x}_{2}and

*P*

_{z}_{2}, is found from averaging the corresponding components of the Poynting vector over the HMM region (see Eqs. (29)). It is evident that the direction of energy flow depends on the sign of

*ε*

_{z}_{2}(or equivalently whether

*λ*is above or below

*λ*). The component of

_{z}**normal to the interfaces (**

*P**P*

_{z}_{2}) must always have the same sign on both sides of the interface [32

32. D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. **90**(7), 077405 (2003). [CrossRef] [PubMed]

*P*

_{x}_{2}) however can be negative if the HMM is of type-1, as seen in Fig. 5(a), and is clearly opposite in direction to

*k*, which is always positive. This manifestation of “negative refraction” was discussed in the context of uniaxially anisotropic media [33

_{x}33. I. V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, “BW media media with negative parameters, capable of supporting backward waves,” Microw. Opt. Technol. Lett **31**, 129 (2001). [CrossRef]

9. Y. Liu, G. Bartal, and X. Zhang, “All-angle negative refraction and imaging in a bulk medium made of metallic nanowires in the visible region,” Opt. Express **16**(20), 15439–15448 (2008). [CrossRef] [PubMed]

25. G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Natl. Acad. Sci. U.S.A. **109**, (23)8834–8838 (2012). [CrossRef] [PubMed]

*|P*

_{z}_{2}

*|*is maximal.

## 4. Conclusion

*ε*

_{x}_{2}) and ℜ(

*ε*

_{z}_{2}) are both of the same sign. Our results show that the incident beam can couple to the HMM structure without recourse for a second compensating layer. We also revealed that the condition ℜ(

*ε*) ≈ 0 leads to near-perfect absorption over a range of frequencies, angles of incidence, and subwavelength structure thicknesses, making the proposed structures experimentally achievable. Alternate methods exist to achieve perfect absorption, including periodic layers of silver and conventional dielectrics that depending on the direction of incident wave propagation and loss, can exhibit anisotropic behavior that cancels the reflected and transmitted waves simultaneously [34

_{z}34. J. Yang, X. Hu, X. Li, Z. Liu, X. Jiang, and J. Zi, “Cancellation of reflection and transmission at metamaterial surfaces,” Opt. Lett. **35**, 16 (2010). [CrossRef] [PubMed]

35. H. Hu, D. Ji, X. Zeng, K. Liu, and Q. Gan, “Rainbow Trapping in Hyperbolic Metamaterial Waveguide,” Sci. Rep. **3**, 1249 (2013). [CrossRef] [PubMed]

36. W. T. Lu and S. Sridhar, “Slow light, open-cavity formation, and large longitudinal electric eld on a slab waveguide made of indenite permittivity metamaterials,” Phys. Rev. A **82**, 013811 (2010). [CrossRef]

37. C. Rizza, A. Ciattoni, and E. Palange, “Two-peaked and flat-top perfect bright solitons in nonlinear metamaterials with epsilon near zero,” Phys. Rev. A **83**, 053805 (2011). [CrossRef]

38. M. A. Vincenti, D. de Ceglia, A. Ciattoni, and M. Scalora, “Singularity-driven second- and third-harmonic generation at ε-near-zero crossing points,” Phys. Rev. A **84**, 063826 (2011). [CrossRef]

40. A. Ciattoni, C. Rizza, and E. Palange, “Transmissivity directional hysteresis of a nonlinear metamaterial slab with very small linear permittivity,” Opt. Lett. **35**, 2130 (2010). [CrossRef] [PubMed]

41. R. J. Pollard, A. Murphy, W. R. Hendren, P. R. Evans, R. Atkinson, G. A. Wurtz, A.V. Zayats, and V.A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett. **102**, 127405 (2009). [CrossRef] [PubMed]

42. A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Y. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B **84**, 045424 (2011). [CrossRef]

43. D. J. Bergman, “The dielectric constant of a composite material - a problem in classical physics,” Phys. Rep. **43**, 377–407 (1978). [CrossRef]

## Appendix: Poynting’s Theorem

*ε*

_{x},*ε*

_{z},*μ*). We also assume that EM wave propagation and polarization is in the

_{y}*x*-

*z*plane. The wave equation for

*E*is thus, Taking into account that ▿·

_{x}**= ▿·**

*D***= 0, this yields the electric field solutions in their respective media as, Similarly, the components of the magnetic field are written, The**

*B**I*terms represent the incident field. The quantities

*k̂*and

_{zj}*ϕ*are defined in Eqs. (5) and (6). Utilizing matching boundary conditions for the tangential components of the electric and magnetic fields permits calculation of the coefficients, were

_{j}*β*is given in Eq. (4). We also define, where

*ε*describe the media for regions

_{xj}*j*= 1, 2, 3, and

*k*is defined in Eq. (6). The caret symbol signifies that wavenumber components

_{zj}*k*and

_{x}*k*have been normalized to

_{zj}*ω*/

*c*. In general,

*k̂*can be any value, but for the case of an incident plane wave in vacuum,

_{x}*k̂*= sin

_{x}*θ*.

**=**

*D***·**

*ε***and**

*E***=**

*B***·**

*μ***. Inserting Eqs. (15) and (18) into Eq. (24) yields the following energy conservation relationships,**

*H**x*and

*y*integrations over

*V*are omitted.

## References and links

1. | D. Sievenpiper, L. Zhang, R. Broas, N. G. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microwave Theory Tech. |

2. | M. G. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials”, Phys. Rev. Lett. |

3. | K. Halterman and S. Feng, “Resonant transmission of electromagnetic fields through subwavelength zero-ε slits,” Phys. Rev. A |

4. | K. Halterman, S. Feng, and V. C. Nguyen, “Controlled leaky wave radiation from anisotropic epsilon near zero metamaterials,” Phys. Rev. B |

5. | S. Feng and K. Halterman, “Coherent perfect absorption in epsilon-near-zero metamaterials,” Phys. Rev. B |

6. | S. Molesky, C. J. Dewalt, and Z. Jacob, “High temperature epsilon-near-zero and epsilon-near-pole metamaterial emitters for thermophotovoltaics,” Opt. Express |

7. | D. Schurig and D. R. Smith, “Spatial filtering using media with indefinite permittivity and permeability tensors,” Appl. Phys. Lett. |

8. | C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. |

9. | Y. Liu, G. Bartal, and X. Zhang, “All-angle negative refraction and imaging in a bulk medium made of metallic nanowires in the visible region,” Opt. Express |

10. | I. Nefedov and S. Tretyakov, “Ultrabroadband electromagnetically indenite medium formed by aligned carbon nanotubes,” Phys. Rev. B |

11. | X. Ni, S. Ishii, M. D. Thoreson, V. M. Shalaev, S. Han, S. Lee, and A. V. Kildishev, “Loss-compensated and active hyperbolic metamaterials,” Opt. Express |

12. | S. Savoia, G. Castaldi, and V. Galdi, “Optical nonlocality in multilayered hyperbolic metamaterials based on Thue-Morse superlattices,” Phys. Rev. B |

13. | O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, “Effective-medium approach to planar multilayer hyperbolic metamaterials: Strengths and limitations,” Phys. Rev. A |

14. | W. Yan, M. Wubs, and N. A. Mortensen, “Hyperbolic metamaterials: nonlocal response regularizes broadband supersingularity,” Phys. Rev. B |

15. | T. Tumkur, L. Gu, J. Kitur, E. Narimanov, and M. Noginov, “Control of absorption with hyperbolic metamaterials,” Appl. Phys. Lett. |

16. | F. V. Iorsh, I. S. Mukhin, I. V. Shadrivov, P. A. Belov, and Y. S. Kivshar, “Hyperbolic metamaterials based on multilayer graphene structures,” Phys. Rev. B |

17. | W. Li, Z. Liu, X. Zhang, and X. Jiang, “Switchable hyperbolic metamaterials with magnetic control,” Appl. Phys. Lett. |

18. | C. Rizza, A. Ciattoni, E. Spinozzi, and L. Columbo, “Terahertz active spatial filtering through optically tunable hyperbolic metamaterials,” Opt. Lett. |

19. | C. Rizza, A. Ciattoni, L. Columbo, M. Brambilla, and F. Prati, “Terahertz optically tunable dielectric metamaterials without microfabrication,” Opt. Lett. |

20. | W. Yan, L. Shen, L. Ran, and J. A. Kong, “Surface modes at the interfaces between isotropic media and indefinite media,” J. Opt. Soc. Am. A |

21. | K.V. Sreekanth, A. De Luca, and G. Strangi, “Experimental demonstration of surface and bulk plasmon polaritons in hypergratings,” Sci. Rep. |

22. | E. Narimanov, M. A. Noginov, H. Li, and Y. Barnakov, “Darker than Black: Radiation-absorbing Metamaterial,” in Quantum Electronics and Laser Science Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper QPDA6. [CrossRef] |

23. | T. U. Tumkur, J. K. Kitur, B. Chu, L. Gu, V. A. Podolskiy, E. E. Narimanov, and M. A. Noginov, “Control of reflectance and transmittance in scattering and curvilinear hyperbolic metamaterials,” Appl. Phys. Lett. |

24. | J. M. Zhao, Y. Chen, and Y. J. Feng, “Polarization beam splitting through an anisotropic metamaterial slab realized by a layered metal-dielectric structure,” Appl. Phys. Lett. |

25. | G. V. Naik, J. Liu, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Demonstration of Al:ZnO as a plasmonic component for near-infrared metamaterials,” Proc. Natl. Acad. Sci. U.S.A. |

26. | J. P. Berenger, “A perfect matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. |

27. | Z. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Ant. Prop. |

28. | J. Merle Elson, “Propagation in planar waveguides and the effects of wall roughness,” Opt. Express |

29. | J. M. Elson, Proc. SPIE 4780, Surface Scattering and Diffraction for Advanced Metrology II, 32, (October1, 2002). |

30. | Y. Jin, S. Xiao, N.A. Mortensen, and S. He, “Arbitrarily thin metamaterial structure for perfect absorption and giant magnification,” Opt. Express |

31. | See, for example, J. D. Jackson, |

32. | D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. |

33. | I. V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, “BW media media with negative parameters, capable of supporting backward waves,” Microw. Opt. Technol. Lett |

34. | J. Yang, X. Hu, X. Li, Z. Liu, X. Jiang, and J. Zi, “Cancellation of reflection and transmission at metamaterial surfaces,” Opt. Lett. |

35. | H. Hu, D. Ji, X. Zeng, K. Liu, and Q. Gan, “Rainbow Trapping in Hyperbolic Metamaterial Waveguide,” Sci. Rep. |

36. | W. T. Lu and S. Sridhar, “Slow light, open-cavity formation, and large longitudinal electric eld on a slab waveguide made of indenite permittivity metamaterials,” Phys. Rev. A |

37. | C. Rizza, A. Ciattoni, and E. Palange, “Two-peaked and flat-top perfect bright solitons in nonlinear metamaterials with epsilon near zero,” Phys. Rev. A |

38. | M. A. Vincenti, D. de Ceglia, A. Ciattoni, and M. Scalora, “Singularity-driven second- and third-harmonic generation at ε-near-zero crossing points,” Phys. Rev. A |

39. | A. Ciattoni, C. Rizza, and E. Palange, “Transverse power flow reversing of guided waves in extreme nonlinear metamaterials,” Opt. Lett. |

40. | A. Ciattoni, C. Rizza, and E. Palange, “Transmissivity directional hysteresis of a nonlinear metamaterial slab with very small linear permittivity,” Opt. Lett. |

41. | R. J. Pollard, A. Murphy, W. R. Hendren, P. R. Evans, R. Atkinson, G. A. Wurtz, A.V. Zayats, and V.A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett. |

42. | A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Y. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B |

43. | D. J. Bergman, “The dielectric constant of a composite material - a problem in classical physics,” Phys. Rep. |

**OCIS Codes**

(160.1190) Materials : Anisotropic optical materials

(160.3918) Materials : Metamaterials

**ToC Category:**

Metamaterials

**History**

Original Manuscript: January 10, 2014

Revised Manuscript: February 28, 2014

Manuscript Accepted: March 4, 2014

Published: March 21, 2014

**Citation**

Klaus Halterman and J. Merle Elson, "Near-perfect absorption in epsilon-near-zero structures with hyperbolic dispersion," Opt. Express **22**, 7337-7348 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-6-7337

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### References

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- T. U. Tumkur, J. K. Kitur, B. Chu, L. Gu, V. A. Podolskiy, E. E. Narimanov, M. A. Noginov, “Control of reflectance and transmittance in scattering and curvilinear hyperbolic metamaterials,” Appl. Phys. Lett. 101, 091105 (2012). [CrossRef]
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- Z. Sacks, D. M. Kingsland, R. Lee, J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Ant. Prop. 43, 1460–1463, (1995). [CrossRef]
- J. Merle Elson, “Propagation in planar waveguides and the effects of wall roughness,” Opt. Express 9, 461–475 (2001). [CrossRef] [PubMed]
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- Y. Jin, S. Xiao, N.A. Mortensen, S. He, “Arbitrarily thin metamaterial structure for perfect absorption and giant magnification,” Opt. Express 19, 11114 (2011). [CrossRef] [PubMed]
- See, for example, J. D. Jackson, Classical Electrodynamics, 3 (Wiley and Sons, 1998).
- D. R. Smith, D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90(7), 077405 (2003). [CrossRef] [PubMed]
- I. V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, S. Ilvonen, “BW media media with negative parameters, capable of supporting backward waves,” Microw. Opt. Technol. Lett 31, 129 (2001). [CrossRef]
- J. Yang, X. Hu, X. Li, Z. Liu, X. Jiang, J. Zi, “Cancellation of reflection and transmission at metamaterial surfaces,” Opt. Lett. 35, 16 (2010). [CrossRef] [PubMed]
- H. Hu, D. Ji, X. Zeng, K. Liu, Q. Gan, “Rainbow Trapping in Hyperbolic Metamaterial Waveguide,” Sci. Rep. 3, 1249 (2013). [CrossRef] [PubMed]
- W. T. Lu, S. Sridhar, “Slow light, open-cavity formation, and large longitudinal electric eld on a slab waveguide made of indenite permittivity metamaterials,” Phys. Rev. A 82, 013811 (2010). [CrossRef]
- C. Rizza, A. Ciattoni, E. Palange, “Two-peaked and flat-top perfect bright solitons in nonlinear metamaterials with epsilon near zero,” Phys. Rev. A 83, 053805 (2011). [CrossRef]
- M. A. Vincenti, D. de Ceglia, A. Ciattoni, M. Scalora, “Singularity-driven second- and third-harmonic generation at ε-near-zero crossing points,” Phys. Rev. A 84, 063826 (2011). [CrossRef]
- A. Ciattoni, C. Rizza, E. Palange, “Transverse power flow reversing of guided waves in extreme nonlinear metamaterials,” Opt. Lett. 18, 11911 (2010).
- A. Ciattoni, C. Rizza, E. Palange, “Transmissivity directional hysteresis of a nonlinear metamaterial slab with very small linear permittivity,” Opt. Lett. 35, 2130 (2010). [CrossRef] [PubMed]
- R. J. Pollard, A. Murphy, W. R. Hendren, P. R. Evans, R. Atkinson, G. A. Wurtz, A.V. Zayats, V.A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett. 102, 127405 (2009). [CrossRef] [PubMed]
- A. A. Orlov, P. M. Voroshilov, P. A. Belov, Y. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84, 045424 (2011). [CrossRef]
- D. J. Bergman, “The dielectric constant of a composite material - a problem in classical physics,” Phys. Rep. 43, 377–407 (1978). [CrossRef]

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