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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 12 — Jun. 17, 2013
  • pp: 15048–15064
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Hyperbolic metamaterials: new physics behind a classical problem

Vladimir P. Drachev, Viktor A. Podolskiy, and Alexander V. Kildishev  »View Author Affiliations


Optics Express, Vol. 21, Issue 12, pp. 15048-15064 (2013)
http://dx.doi.org/10.1364/OE.21.015048


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Abstract

Hyperbolic materials enable numerous surprising applications that include far-field subwavelength imaging, nanolithography, and emission engineering. The wavevector of a plane wave in these media follows the surface of a hyperboloid in contrast to an ellipsoid for conventional anisotropic dielectric. The consequences of hyperbolic dispersion were first studied in the 50’s pertaining to the problems of electromagnetic wave propagation in the Earth’s ionosphere and in the stratified artificial materials of transmission lines. Recent years have brought explosive growth in optics and photonics of hyperbolic media based on metamaterials across the optical spectrum. Here we summarize earlier theories in the Clemmow’s prescription for transformation of the electromagnetic field in hyperbolic media and provide a review of recent developments in this active research area.

© 2013 OSA

1. Hyperbolic media in retrospect

Recent advances in nanofabrication, characterization, and high-performance computing, along with developments in the theory of light-matter interaction, have brought to life a new class of multi-scale composite systems, known as metamaterials. In metamaterials, it is material geometry that determines the interaction of these complex systems with electromagnetic fields. Metamaterials offer new avenues for manipulation of light, opening the door for such unusual applications as high-resolution imaging, lithography, and lifetime engineering. This review is focused on a subclass of metamaterials called hyperbolic that provide a flexible platform for manipulation of optical landscape.

Hyperbolic metamaterials (HMM), uniaxial structures that due to their extreme anisotropy combine the properties of transparent dielectrics and reflective metals, first attracted the attention of researchers in the middle of last century. These efforts were stimulated by the problem of propagation of radio waves in the Earth’s ionosphere [1

1. K. G. Budden, Radio Waves in the Ionosphere(Cambridge University, 1961).

4

4. R. H. Ritchie, “Plasma losses by fast electrons in thin films,” Phys. Rev. 106(5), 874–881 (1957). [CrossRef]

], and, more generally, by the behavior of the electromagnetic waves in a plasma of electrons and ions upon the applied permanent magnetic field. Along with the anisotropic plasmas, stratified man-made hyperbolic materials were studied for radiofrequency applications of transmission lines [5

5. S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 10 (1956).

]. Today, hyperbolic metamaterials provide one of the most practical metamaterial platforms. Hyperbolic composites (also known as media with indefinite permittivity and permeability tensors [6

6. 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]

]) and some homogeneous materials with hyperbolic dispersion were experimentally realized across the optical spectrum, from UV to visible, and from near-IR to mid-IR frequencies. Sub-wavelength imaging, focusing, lifetime engineering, and new approaches to enhance nonlinear response of optical structures, have all been demonstrated in hyperbolic structures. The hyperbolic media continue to be of great interest to the research community with possible applications emerging in heat transport and acoustics. The purpose of this review is to provide an outlook of this rapidly developing research area.

The review begins with a historical digest of the hyperbolic material’s properties. Then, we systematically present the recent theoretical and experimental studies of optical hyperbolic metamaterials. While our main goal is to critically revise the existing experimental evidences of the potential applications of hyperbolic media for sub-diffraction imaging and spontaneous emission engineering, we also highlight the vital discrepancies with existing theoretical fundamentals.

2. Fundamentals of hyperbolic media

The vectors of electric displacement D and field E are not always parallel in electrically anisotropic media. They are connected by the constitutive equation D=ε0[ε]E, where [ε] is a symmetric dielectric tensor andε0is the permittivity of vacuum. The unit coordinate vectorsx^, y^, and z^represent the three Cartesian directions - the principal axes – for which [ε]=diag(εx,εy,εz), with εx, εy, and εzbeing the principal dielectric constants. In general, the three principal dielectric constants are distinct εxεyεzreferred to as optically biaxial, and may be wavelength dependent, i.e. exhibit dispersion of the optical axes. In optically uniaxial media, where εx=εy=ε, and εεz, the principal directionz^ must coincide with the optic axis, which typically defines the three-fold, four-fold or six-fold axial symmetry of a given crystal, so that [ε]=diag(ε,ε,εz).

While in crystal optics a uniaxial medium corresponds to a wide class of crystalline structures with mainly elliptic dispersion, for magneto-ionic media it specifies the medium with an infinitely strong magnetostatic field bias [2

2. P. Clemmow, “The theory of electromagnetic waves in a simple anisotropic medium,” Proc. IEEE 110(1), 101–106 (1963).

], which, similar to hyperbolic metamaterials, can also have different signs of the principal dielectric constants, e.g.ε>0, εz<0.Here, we restrict ourselves to uniaxial hyperbolic media, as the main known theoretical analyses have been done for the hyperbolic uniaxial plasmas in [3

3. L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Wiley-IEEE, 1973).

, 7

7. V. Bunkin, “On radiation in anisotropic media,” Sov. Phys. JETP 5(2), 277–283 (1957).

, 8

8. H. Kogelnik, “On electromagnetic radiation in magneto-ionic media,” J. Res. Nat. Bur. Stand. D. 64D, 515 (1960).

].

2.1 Clemmow’s prescriptions

Anisotropy of material constants results in a transformation of the electromagnetic field distribution in space under diffraction or point source radiation. Because of the foreseeable complexity of the general analysis presented in [3

3. L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Wiley-IEEE, 1973).

, 7

7. V. Bunkin, “On radiation in anisotropic media,” Sov. Phys. JETP 5(2), 277–283 (1957).

, 8

8. H. Kogelnik, “On electromagnetic radiation in magneto-ionic media,” J. Res. Nat. Bur. Stand. D. 64D, 515 (1960).

] we review the Clemmow approach [2

2. P. Clemmow, “The theory of electromagnetic waves in a simple anisotropic medium,” Proc. IEEE 110(1), 101–106 (1963).

] and isolate the simple cases of TM and TE waves in hyperbolic plasma, which are then applied to the refraction and diffraction of light in novel artificial hyperbolic media. The analysis in [2

2. P. Clemmow, “The theory of electromagnetic waves in a simple anisotropic medium,” Proc. IEEE 110(1), 101–106 (1963).

] is restricted to a media of unit permeability, and starts with known E- and H-fields in free space (denoted respectfully as E0(r0) and H0(r0)), with r0=x^x0+y^y0+z^z0 defining an observation point. Then, the scaling procedureE(r)=[e]E0([n]r) and H(r)=[h]H0([n]r) is derived to find the corresponding fields, E(r)and H(r) in a given uniaxial media, where [e],[n], and [h]are linear transforms. The above procedure is impossible unless one of the z-components of either H0 or E0is dropped, leading to solutions valid solely for either transverse magnetic (TM, z^H0=0) or transverse electric (TE, z^E0=0) waves.

From the symmetry considerations for TM-waves it follows that the metric and the E-field scaling transforms repeat the structure of the dielectric tensor, [ε]=diag(ε,ε,εz), so that [e]=diag(e,e,ez), and, while the transverse components of H-field are scaled uniformly, i.e. [h]=h[i], where [i] is the identity matrix. From the Maxwell curl equations for the TM waves, it then follows that nnz[n]1[s]1=h1[ε] andnnz[n]1[s]=h[ε], or eez[n][e]1=h[ε]. Multiplying the above equations we first arrive at n2=e2=εz,nz2=ez2=ε, and h2=εεz, and then, at the remarkably simple Clemmow’s TM-prescription
E(r)=[n]E0([n]r),H(r)=εεzH0([n]r),
(1)
where [n]=diag(εz,εz,ε).

In contrast, for TE-waves it can be expected that since z^E0=0, then εz should not appear in the Maxwell equations for the uniaxial medium, as if that medium were isotropic with dielectric constant εz. Indeed, the final Clemmow’s TE-prescription is, (n=ε)

E(r)=E0(nr),H(r)=nH0(nr).
(2)

2.2 Dispersion relations in a uniaxial hyperbolic medium

Consider the behavior of monochromatic plane waves in uniaxial hyperbolic media before addressing the complex fields generated there by elementary sources. Following the general Clemmow approach, the space-time dependence f(r;t)=exp[ι(ωckrωt)] is taken here for a plane wave defined for a given angular frequencyω and the free-space speed of lightc using the material wavevector k=[n]k0 with the freespace wave-vector k0(|k0|=1). Then, the phase velocity of the plane wave is vp=c/|k|. A plane wave [9

9. M. Born and E. Wolf, Principles of Optics(Cambridge University, 1999).

] in a uniaxial media splits into two linearly polarized characteristic waves: ordinary waves for which the E-vector is normal to the principal plane, which contains both the wave vector and the optic axis,z, and extraordinary waves with E-vector parallel to the principal plane. For ordinary waves, E is aligned with D, and the phase velocity is independent of propagation direction. As any free-space electromagnetic field can be represented as a superposition of coplanar TM- and TE-waves, there is always a matching superposition of coplanar TM- and TE-waves in the uniaxial medium obtained from the scaling transformations (1) and (2). These transformations are also reversible, since a representation of any uniaxial-medium field as a superposition of coplanar TM- and TE-waves is always possible. For example, the independent characteristic extraordinary (TM) and ordinary (TE) plane waves in the anisotropic medium are obtained from the corresponding free-space plane waves in [2

2. P. Clemmow, “The theory of electromagnetic waves in a simple anisotropic medium,” Proc. IEEE 110(1), 101–106 (1963).

].

For a plane wave defined by f(r;t), the Maxwell equations give,

kD=0,μ0kH=0,k×E=μ0cH,k×E=μ0cH,andk×H=cD
(3)

While in general the Poynting vector S=12Re(E×H*) is not parallel tok, the vectors within the triads(k,H and D) and (S,E and H) are mutually orthogonal and vectorH is normal to coplanar vectorsk,D,Eand S, as shown in Fig. 1(a)
Fig. 1 Typical vector diagram and dispersion relations in uniaxial media. (a) Plane-wave vectors. (b), Isofrequency cross-sections for a negative elliptic media, ε<εz. c, d Isofrequency curves for different types of ideal, lossless hyperbolic media: dielectric, εz<0,ε>0 (c), metallic, εz>0,ε<0, (d). (e), (f) Isofrequency curves for different types of non-ideal, absorbing hyperbolic media: dielectric type, with εx,y=0.57+ι0.13, εz=4.22+ι2.03 (e); metallic type, with εx,y=2.78+ι0.13, εz=6.31+i0.09(f).
.

The eigenvalue problem k×(k×E)+[ε]E=0, obtained from the last two equations of (3), gives two distinct characteristic equations respectively for ordinary and extraordinary waves:

kx2+ky2+kz2ε=1,andkx2+ky2εz+kz2ε=1.
(4)

Note that any “nonideality” of the hyperbolic medium results in a closed form of the iso-frequency curves instead of hyperboloid as shown in Figs. 1(e) and 1(f) for absorbing hyperbolic media.

2.3 Volume plasmon polaritons in hyperbolic media

Since the permittivity tensor has metallic properties for one of the principal components and dielectric for another, there should be a peculiarity in the angular dependence of wave propagation, namely the permittivity for extraordinary wave satisfies the condition Re ε(φc)= ε(φc)=0at the critical angle. This condition determines an angular boundary between “metal” and “dielectric” types of propagation. Thus a coupling between plasmon and polariton can occur at this virtual boundary similar to the surface plasmon polariton at a metal-dielectric interface. In this case the hyperbolic media support plasmon-polariton waves propagating across the interfaces of real metal and dielectric structures, which are called here volume plasmon polaritons,. Consider a slab of hyperbolic uniaxial medium with optical axis in thezdirection. Let displacement vectorD lie in the principal plane containing both optic axis and wave vectork.

Component of electric field directed along Dis given by ED=ED/D=D/ε(φ), where
1ε(φ)= sin2(φ)εe+cos2(φ)εo,
(5)
εo=εx=εy,εe=εz, and  φ is the angle between the wave vector and optical axis. The wave vector refraction at the crystal-isotropic medium interface formally obeys the Snell’s law. Once we know the wave vector direction, then the angleθbetween the ray and optical axis is defined as follows

tanθ=εoεetanφ.
(6)

In a hyperbolic media the angular dependence of the permittivity has a resonance behavior as shown in Fig. 2(a)
Fig. 2 Volume plasmon-polariton. a) Angular dependence of the permittivity for extraordinary wave, imaginary (brown) and real (blue) parts. Critical angle between wave vector and optic axis φc=57°; b) Angle between Poynting vector and optic axis θ(φ),θc=33°; c) magnetic field angular dependence localized at the critical angle (calculated at 650 nm from the source) [24]. Figures reproduced with permission from ©2013 Wiley-VCH
, where we have chosen exemplary values for multilayer structures εo=2.78+ι0.22, and εe=6.31+ι0.15. It can be shown that for the critical angle, the angles betweenD and E as well as between k and are both about 90°, which is clear also from Figs. 2(a) and 2(b). Interestingly the resonance behavior results in a field confinement in the critical direction due to high values ofIm(ε)as shown in Fig. 2(c).

2.4 Radiation patterns from elementary sources

Note that all the discussed types of hyperbolic metamaterials can be realized with the same structure by varying the wavelength of the incident light. Figures 3(e) and 3(f) show permittivity spectra for two components of multilayer structure with optic axis denoted by z. For the multilayer dispersion calculations, the Rytov effective medium theory with nonlocal corrections is employed [5

5. S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 10 (1956).

]. According to [25

25. J. Kim, V. P. Drachev, Z. Jacob, G. V. Naik, A. Boltasseva, E. E. Narimanov, and V. M. Shalaev, “Improving the radiative decay rate for dye molecules with hyperbolic metamaterials,” Opt. Express 20(7), 8100–8116 (2012). [CrossRef] [PubMed]

], all structures containing one or more metal-dielectric interfaces can be qualitatively considered as a hyperbolic layer.

Depending on composition and component dimensions, multilayer systems provide hyperbolic dispersion at the UV (Ag/Al2O3) [13

13. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006). [CrossRef] [PubMed]

], near-IR(Al/ZnO) [26

26. 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), doi:. [CrossRef] [PubMed]

], and mid-IR (InGaAs/AlInAs) [27

27. A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007). [CrossRef] [PubMed]

] frequency ranges. Hyperbolic dispersion has been also demonstrated in nanowire systems [28

28. M. A. Noginov, H. Li, Y. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett. 35(11), 1863–1865 (2010). [CrossRef] [PubMed]

30

30. J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008). [CrossRef] [PubMed]

], and homogeneous materials [31

31. J. Sun, J. Zhou, B. Li, and F. Kang, “Indefinite permittivity and negative refraction in natural material: graphite,” Appl. Phys. Lett. 98(10), 101901 (2011). [CrossRef]

33

33. E. Gerlach, P. Grosse, M. Rautenberg, and W. Senske, “Dynamical conductivity and plasmon excitation in Bi,” Phys. Status Solidi B 75(2), 553–558 (1976). [CrossRef]

].

3. Refraction in hyperbolic media

As described above, in hyperbolic media propagation of the energy, given by the Poynting vector, is not collinear to the propagation of the phase fronts, described by the wavevector, which has profound implications for refraction of the waves to and from the hyperbolic structures. Consider first a single plane wave that is incident at an HMM boundary. Electromagnetic field can be described as a linear combination of incident, reflected, and refracted electromagnetic waves. Existence of the continuous planar interface requires conservation of the component of the wavevector parallel to this interface [9

9. M. Born and E. Wolf, Principles of Optics(Cambridge University, 1999).

], which fixes the directions of the reflected and refracted waves. Application of the boundary conditions results in the amplitudes of these waves.

Consider now the situation when the interface is illuminated not by a single plane wave, but by a rather wide, but finite-sized monochromatic beam. The electromagnetic field of the beam can be represented as a linear combination of the plane waves, and refraction/reflection of each component of the beam can be calculated using the single-wave formalism described above. When the beam is substantially wide, its plane wave decomposition will contain a relatively narrow spectrum of the wavevector components. Consequently, the Poynting vectors of all the refracted components of the beam will be aligned with each other. This common direction of the Poynting flux describes the direction of the propagation of the refracted beam.

Light refraction at the interface with anisotropic crystals is often non-trivial. Even with naturally-occurring materials it is possible to achieve negative refraction for a limited range of angles [34

34. Y. Zhang, B. Fluegel, and A. Mascarenhas, “Total negative refraction in real crystals for ballistic electrons and light,” Phys. Rev. Lett. 91(15), 157404 (2003). [CrossRef] [PubMed]

]. When the light is incident from isotropic material onto hyperbolic media, the beam can be refracted in the negative direction [35

35. P. Belov, “Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis,” Microw. Opt. Technol. Lett. 37(4), 259–263 (2003). [CrossRef]

]. Negative refraction of the beam has been experimentally validated in layered metamaterials at mid-IR [27

27. A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007). [CrossRef] [PubMed]

] and near-IR frequencies [26

26. 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), doi:. [CrossRef] [PubMed]

], in nanowire-based metamaterials at visible frequencies [30

30. J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008). [CrossRef] [PubMed]

], and in homogeneous media at UV frequencies [31

31. J. Sun, J. Zhou, B. Li, and F. Kang, “Indefinite permittivity and negative refraction in natural material: graphite,” Appl. Phys. Lett. 98(10), 101901 (2011). [CrossRef]

].

Negative refraction has been historically considered the hallmark of metamaterials, where it enables planar lenses that are not limited by spherical aberrations [36

36. V. Veselago, “The electrodynamics of substances with simultaneously negative values of epsilon and mu,” Sov. Phys. Usp. 10(4), 509–514 (1968). [CrossRef]

]. A classic Veselago lens should have an angle- and polarization-independent index of refraction, and thus requires use of an isotropic magnetic media. Nevertheless, hyperbolic materials can mimic the performance of a Veselago lens in waveguide geometry [37

37. V. Podolskiy and E. Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left-handed system,” Phys. Rev. B 71(20), 201101(R) (2005). [CrossRef]

]. Practical applications of 3D lenses based on hyperbolic materials are affected by the dependence of their refractive index on the angle [38

38. L. V. Alekseyev and E. Narimanov, “Slow light and 3D imaging with non-magnetic negative index systems,” Opt. Express 14(23), 11184–11193 (2006). [CrossRef] [PubMed]

], which re-introduces image distortion similar to spherical aberration. Such distortions, however, can be limited in ENZ and in canalization imaging [13

13. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006). [CrossRef] [PubMed]

,14

14. A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations,” Phys. Rev. B 74(7), 075103 (2006). [CrossRef]

,39

39. Y. Xiong, Z. Liu, and X. Zhang, “Projecting deep-subwavelength patterns from diffraction-limited masks using metal-dielectric multilayers,” Appl. Phys. Lett. 93(11), 111116 (2008). [CrossRef]

].

4. Diffraction in hyperbolic media: sub-wavelength imaging and nanolithography

Any inhomogeneity inside the material or any inhomogeneity along the boundary between two media necessarily leads to diffraction. Qualitatively, the profile of the diffracted beam can be calculated from the Huygens-Fresnel principle. Note that in contrast to conventional isotropic materials, emission of waves in hyperbolic systems is highly directional (Figs. 1 and 2). The unique dispersion allows hyperbolic systems to preserve high-wavevector components of the wave-packets that carry information about the subwavelength features of the source. Therefore, hyperbolic structures are capable of focusing radiation to subwavelength spots, a phenomenon that has been proposed in [19

19. S. Thongrattanasiri and V. A. Podolskiy, “Hypergratings: nanophotonics in planar anisotropic metamaterials,” Opt. Lett. 34(7), 890–892 (2009). [CrossRef] [PubMed]

], [39

39. Y. Xiong, Z. Liu, and X. Zhang, “Projecting deep-subwavelength patterns from diffraction-limited masks using metal-dielectric multilayers,” Appl. Phys. Lett. 93(11), 111116 (2008). [CrossRef]

] and recently realized in [24

24. S. Ishii, A. V. Kildishev, E. Narimanov, V. Shalaev, and V. P. Drachev, “Sub-wavelength interference pattern from volume plasmon polaritons in a hyperbolic medium,” Laser Photonics Rev. 7(2), 265–271 (2013). [CrossRef]

]. In this experiment, a double-slit interference of resonant cones produced by slits in Cr film results in a sub-diffraction pattern [19

19. S. Thongrattanasiri and V. A. Podolskiy, “Hypergratings: nanophotonics in planar anisotropic metamaterials,” Opt. Lett. 34(7), 890–892 (2009). [CrossRef] [PubMed]

], which is highly beneficial for nanolithography applications [24

24. S. Ishii, A. V. Kildishev, E. Narimanov, V. Shalaev, and V. P. Drachev, “Sub-wavelength interference pattern from volume plasmon polaritons in a hyperbolic medium,” Laser Photonics Rev. 7(2), 265–271 (2013). [CrossRef]

]. The experiment scheme is shown in (Figs. 4(b)
Fig. 4 Imaging (a) [15] and nanolithography (b), (c) [24] with hyperbolic metamaterials. Figures reproduced with permissions: (a) Ref [15] from ©2007 AAAS and (b),(c) Ref [24]. from ©2013 Wiley-VCH.
and 4(c)) where the flat Ag/SiO2 hyperbolic structure forms a line of 90 nm width on the photoresist layer, which was simply brought in contact with the hyperbolic layer and detected then with an Atomic Force Microscope (AFM) [24

24. S. Ishii, A. V. Kildishev, E. Narimanov, V. Shalaev, and V. P. Drachev, “Sub-wavelength interference pattern from volume plasmon polaritons in a hyperbolic medium,” Laser Photonics Rev. 7(2), 265–271 (2013). [CrossRef]

].

Further, the hyperbolic structures can be used to magnify the subwavelength objects and thus enable far-field super-resolution imaging. Super-imaging is a challenging problem that is typically addressed by near-field optical microscopy. A super-lens made of metamaterial with both negative permittivity and permeability [40

40. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]

] provides a way to translate the information about subwavelength objects. This approach was further developed to incorporate alternating layers with optical gain [41

41. J. B. Pendry, “Perfect cylindrical lenses,” Opt. Express 11(7), 755–760 (2003). [CrossRef] [PubMed]

] and to enlarge the objects in acylindrical geometry [42

42. S. A. Ramakrishna and J. B. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B 67(20), 201101 (2003). [CrossRef]

,43

43. N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of partially resonant composites,” Phys. Rev. B Condens. Matter 49(12), 8479–8482 (1994). [CrossRef] [PubMed]

].An approach to magnify subwavelength objects in 2D plasmon-polariton systems was proposed in [44

44. I. I. Smolyaninov, J. Elliott, A. V. Zayats, and C. C. Davis, “Far-field optical microscopy with a nanometer-scale resolution based on the in-plane image magnification by surface plasmon polaritons,” Phys. Rev. Lett. 94(5), 057401 (2005). [CrossRef] [PubMed]

].

Curved hyperbolic metamaterials enable far-field magnification of 3D objects when subwavelength information, encoded in resonant cones [11

11. H. H. Kuehl, “Electromagnetic radiation from an electric dipole in a cold anisotropic plasma,” Plasma Phys. Fluids 5(9), 1095 (1962). [CrossRef]

,45

45. R. K. Fisher and R. W. Gould, “Resonance cones in the field patterns of a short antenna in anisotropic plasma,” Phys. Rev. Lett. 22(21), 1093–1095 (1969). [CrossRef]

] (see Sec. 2) is gradually translated into propagating information similar to the magnification by anisotropic structure [46

46. S. Feng and J. M. Elson, “Diffraction-suppressed high-resolution imaging through metallodielectric nanofilms,” Opt. Express 14(1), 216–221 (2006). [CrossRef] [PubMed]

48

48. D. Schurig and D. R. Smith, “Sub-diffraction imaging with compensating bilayers,” New J. Phys. 7, 162 (2005). [CrossRef]

]. Magnifying lens based on hyperbolic metamaterials was proposed in [13

13. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006). [CrossRef] [PubMed]

,14

14. A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: theory and simulations,” Phys. Rev. B 74(7), 075103 (2006). [CrossRef]

]; it was then realized in [15

15. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007). [CrossRef] [PubMed]

] for UV imaging in Ag/Al2O3 curved multilayer metamaterial (Fig. 4(a)).

5. Photonic density of states and radiative rate engineering

Q=Γrτ,  τ=(Γr+κnr)1=Γ1.
(7)

Here,Γr is the radiative decay rate, κnris the non-radiative decay rate, τ is the excited state lifetime, andΓis the total decay rate. For Ag nanoantennae [62

62. R. M. Bakker, V. P. Drachev, Z. Liu, H. K. Yuan, R. H. Pedersen, A. Boltasseva, J. Chen, J. Irudayaraj, A. V. Kildishev, and V. M. Shalaev, “Nanoantenna array-induced fluorescence enhancement and reduced lifetimes,” New J. Phys. 10(12), 125022 (2008). [CrossRef]

],direct measurements of both the lifetime and the quantum yield changes are necessary for conclusive results of the radiative decay rate and Purcell factor. The quantum yieldis simply the ratio of the emitted to the absorbed photons and can be determined by measuring the absorption, emission, reflection, and lifetime of the dye molecules relative to a reference dye film. Then the results are compared with the radiative decay rate of control samples that are similar to those used in classic experiments, namely thin and thick gold films. By using the reference method, the quantum yield can be experimentally obtained through absorption and fluorescence measurements for the samples under study relative to the corresponding dye/epoxy reference sample [25

25. J. Kim, V. P. Drachev, Z. Jacob, G. V. Naik, A. Boltasseva, E. E. Narimanov, and V. M. Shalaev, “Improving the radiative decay rate for dye molecules with hyperbolic metamaterials,” Opt. Express 20(7), 8100–8116 (2012). [CrossRef] [PubMed]

]. Purcell factors are shown in Table 1

Table 1. Radiative, nonradiative decay rates, apparent quantum yield, and fluorescence and absorption enhancements in layered HMM are shown in the table for four samples, each at 89 and 21 nm dielectric spacer [25]. Table reproduced with permission from ©2012 OSA.

table-icon
View This Table
. Our maximum changes in the radiative decay rates relative to Rh800 in methanol are about 1.2 for dye molecules on glass, 1.35 for thin gold films, 1.73 for thick gold, and 2.7 for multilayer HMM samples.

The photonic density of states (PDS) in emitters can be modified due to the interference of emitted and reflected waves near metal films [63

63. K. Drexhage, “Influence of a dielectric interface on fluorescence decay time,” J. Lumin. 1, 693–701 (1970). [CrossRef]

,64

64. W. Barnes, “Fluorescence near interfaces: the role of photonic mode density,” J. Mod. Opt. 45(4), 661–699 (1998). [CrossRef]

], which have led to the development of PDS engineering that uses metal-dielectric interfaces [64

64. W. Barnes, “Fluorescence near interfaces: the role of photonic mode density,” J. Mod. Opt. 45(4), 661–699 (1998). [CrossRef]

,65

65. G. Ford and W. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. 113(4), 195–287 (1984). [CrossRef]

], metal-film interfaces [66

66. J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B Condens. Matter 33(8), 5186–5201 (1986). [CrossRef] [PubMed]

68

68. G. Winter and W. L. Barnes, “Emission of light through thin silver films via near-field coupling to surface plasmon polaritons,” Appl. Phys. Lett. 88(5), 051109 (2006). [CrossRef]

].Non-radiative decay can be modified due to dipole-image interaction and excitation of the surface-plasmon polaritons or wave-guiding modes. Indeed, similar to SPP, the limiting factor of the radiative decay is in the out-coupling of the HMM modes to the low PDS of free space [69

69. J. B. Khurgin, G. Sun, and R. A. Soref, “Enhancement of luminescence efficiency using surface plasmon polaritons: figures of merit,” J. Opt. Soc. Am. B 24(8), 1968–1980 (2007). [CrossRef]

]. Thus the ratio between leaky and bound modes [66

66. J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B Condens. Matter 33(8), 5186–5201 (1986). [CrossRef] [PubMed]

] should be a critical parameter in theory, which would allow to determine effect of HMM on both, the radiative and nonradiative rates. This is the main limitation of the existing theories.

6. Beyond the effective medium theory: nonlocality corrections and additional waves

Hyperbolic metamaterials provide unprecedented opportunities for controlling the flow of optical information. However, the majority of exciting applications of hyperbolic systems can be traced to extreme (either vanishingly small, or infinitely big) values of the components of the effective permittivity tensor or to extreme values of effective refractive indexneff=kzc/ω.To name a few, traditional implementations of hyperlens and canalization imaging systems rely on εeεo. Modulation of photonic density of states, as well as numerous designs leading to subwavelength focusing [19

19. S. Thongrattanasiri and V. A. Podolskiy, “Hypergratings: nanophotonics in planar anisotropic metamaterials,” Opt. Lett. 34(7), 890–892 (2009). [CrossRef] [PubMed]

,70

70. A. Govyadinov and V. Podolskiy, “Metamaterial photonic funnels for subdiffraction light compression and propagation,” Phys. Rev. B 73(15), 155108 (2006). [CrossRef]

] rely on the existence of propagating modes atneff1. In fact, “extreme” photonics is deeply interweaved with the novel applications of metamaterials, and this relationship extends far beyond the area of hyperbolic systems to cloaking and light transmission through subwavelength channels [71

71. M. 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]

73

73. W. Cai, U. Chettiar, A. Kildishev, and V. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1(4), 224–227 (2007). [CrossRef]

] and other applications.

However, the very composites that bring to life the extreme behaviors usually provide tight limitations on what can be realized in realistic systems. The optics of nanolayered composites can be understood analytically, with the help of transfer matrix formalism [5

5. S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 10 (1956).

,74

74. P. Yeh, A. Yariv, and C. Hong, “Electromagnetic propagation in periodic stratified media.I. General theory,” J. Opt. Soc. Am. 67(4), 423 (1977). [CrossRef]

].

The effective parameters for a multilayer, periodic structure with a period d=b+a and containing both a metal (with permittivityεm and permeabilityμm) and a dielectric(εd,μd)are given by [5

5. S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 10 (1956).

]:

εo=ε¯o(1ιkab4dμdεmμmεdε¯oμ¯o),ε¯o=(bεm+aεda+b),
(8)
μo=μ¯o(1+ιkab4dμdεmμmεdε¯oμ¯o),μ¯o=(bμm+aμda+b),
(9)
εe=ε˜e,ε˜e1=a/εd+b/εma+b.
(10)

It was shown in [54

54. O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, “Dipole radiation near hyperbolic metamaterials: applicability of effective-medium approximation,” Opt. Lett. 36(13), 2530–2532 (2011). [CrossRef] [PubMed]

,56

56. B. Wood, J. Pendry, and D. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74(11), 115116 (2006). [CrossRef]

] that increasing the number of layers at the same thickness brings the results for the multilayer structure closer and closer to those of a homogeneous sample. Note that Eqs. (8)-(10) were obtained in [5

5. S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 10 (1956).

] for infinite periodic system with elementary cell made of two layers. For bounded layered systems the key moment of the problems is the boundary conditions. Modification of the boundary conditions by introduction of additional surface currents suggested in [75

75. A. P. Vinogradov, A. I. Ignatov, A. M. Merzlikin, S. A. Tretyakov, and C. R. Simovski, “Additional effective medium parameters for composite materials (excess surface currents),” Opt. Express 19(7), 6699–6704 (2011). [CrossRef] [PubMed]

] returns the conventional permittivity and permeability of metamaterials their usual physical properties. The modified retrieval procedure based on reflection/transmission yields bulk values of effective impedance and refractive index, which are independent of system size and boundary realization, whereas the conductivities of the excess surface currents depend on the property of the interface [75

75. A. P. Vinogradov, A. I. Ignatov, A. M. Merzlikin, S. A. Tretyakov, and C. R. Simovski, “Additional effective medium parameters for composite materials (excess surface currents),” Opt. Express 19(7), 6699–6704 (2011). [CrossRef] [PubMed]

].

Optics of nanowire metamaterials offer unique opportunity to realize regimes where nonlocality not corrects, but rather dominates the response of the system. Nanowire materials are known to provide a flexible platform that can realize elliptic, hyperbolic, and -near-zero (ENZ) regimes in the same material. In the ENZ regime, the optical properties of the [meta]material can be approximated as εe(ω,k)=ιε+δ(k), with ε representing material absorption at ENZ frequency andδ(k) describing nonlocal correction to the EMT response [77

77. S. I. Pekar, “The theory of electromagnetic waves in crystal in which excitons are produced,” Sov. Phys. JETP 6, 785–796 (1958).

].Therefore, when the losses in material are substantially small, the nonlocal “correction” dominates the polarization inside the system.

Direct solutions of Maxwell equations demonstrate that in this case, [meta-]material supports not two, but at least three different waves, at least two of which have identical (TM) polarization. The existence of additional waves fundamentally changes optical response of the system.

Any phenomenon that relies on optical response to change of material parameters can potentially be greatly enhanced. In particular, the interference-based transmission has already provided a new way to enhance nonlinear optical response employing nonlocal metamaterials. A clear manifestation of such an enhancement was reported in [78

78. G. A. Wurtz, R. Pollard, W. Hendren, G. P. Wiederrecht, D. J. Gosztola, V. A. Podolskiy, and A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol. 6(2), 107–111 (2011). [CrossRef] [PubMed]

], where optical nonlocality provides a four-fold enhancement to optical nonlinearity in TM-polarized response in comparison to local TE-polarized response of the same metamaterial.

Strong optical nonlocality and existence of additional waves in ENZ regime in layered structures was reported in [79

79. P. A. Belov, R. Marqués, S. I. Maslovski, I. S. Nefedov, M. Silverinha, M. Simovski, and S. A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B 67(11), 113103 (2003). [CrossRef]

,80

80. A. Chebykin, A. Orlov, A. Vozianova, S. Maslovski, Y. Kivshar, and P. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B 84(11), 115438 (2011). [CrossRef]

], and many other designs of ENZ metamaterials are likely to follow this trend. Optical nonlocality remains an active research area with multiple groups working on better understanding of the collective excitations that underline ENZ response of macroscopic materials

7. Natural hyperbolic materials

A similar example has been demonstrated [81

81. M. J. Thompson, M. J. DeVries, T. E. Tiwald, and J. A. Woollam, “Determination of optical anisotropy in calcite from ultraviolet to mid-infrared by generalized ellipsometry,” Thin Solid Films 313–314, 341–346 (1998). [CrossRef]

] with one of the most common mineral, calcite (CaCO3), which, on top of its text-book famous birefringence, exhibits two very distinct non-overlapping ordinary and extraordinary absorption bands in the mid-infrared spectral range as a result of the internal vibration modes of its planar carbonate ions. Two sets of optical dispersion parameters, ordinary and extraordinary, using just a few terms of the Lorentz oscillator model, have been fit to the experimental data using generalized ellipsometry [82

82. L. J. Prokopeva, D. P. Brown, X. Ni, V. P. Drachev, A. Urbas, and A. Kildishev, “Pulse shaping using optical metamaterials with naturally anisotropic structural elements,” in CLEO: QELS-Fundamental Science, OSA Technical Digest (Optical Society of America), 2012.

].

Dispersion spectra of ɛo and ɛe depicted in Fig. 7(b), indicate that the best figure of merit FOMj=Re(εj)/Im(εj),(j = {o,e}) is around 4.7 at 6.75 µm, and around 6.3 at 11.33 µm for the ordinary and extraordinary components respectively εo(λ=6.75μm)=2.59+ι0.55,εe(λ=11.33μm)=1.90+ι0.30. The major absorption peaks for the ordinary and extraordinary rays are correspondingly located atλ=7.13μm and11.48μm. Strong hyperbolic anisotropy is not limited to resonance phonon excitations that occur for example in calcite. For example in Bi, a Group V semimetal with rhombohedral lattice and trigonal symmetry, such anisotropy is induced by a substantial difference in its electron effective masses along different directions in the crystal. Hence, the most interesting feature of the Bi dispersion - transition from Re(εj)>0 to Re(εj)<0,j = {o, e}, _ is determined by the strong anisotropy of its plasma frequency within a band between λ = 53.7 μm and 63.2 μm as shown in Fig. 7(c). The existence of that 10-μm band has been confirmed experimentally in [33

33. E. Gerlach, P. Grosse, M. Rautenberg, and W. Senske, “Dynamical conductivity and plasmon excitation in Bi,” Phys. Status Solidi B 75(2), 553–558 (1976). [CrossRef]

].

Acknowledgments

VPD and AVK acknowledge the support from AFRL Materials and Manufacturing Directorate - Applied Metamaterials Program. VP acknowledges the support from NSF (grant ## ECCS-1102183, DMR-1209761) and ARO (grant # W911NF-12-1-0533)

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W. Cai, U. Chettiar, A. Kildishev, and V. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1(4), 224–227 (2007). [CrossRef]

74.

P. Yeh, A. Yariv, and C. Hong, “Electromagnetic propagation in periodic stratified media.I. General theory,” J. Opt. Soc. Am. 67(4), 423 (1977). [CrossRef]

75.

A. P. Vinogradov, A. I. Ignatov, A. M. Merzlikin, S. A. Tretyakov, and C. R. Simovski, “Additional effective medium parameters for composite materials (excess surface currents),” Opt. Express 19(7), 6699–6704 (2011). [CrossRef] [PubMed]

76.

V. Agranovich and V. Kravtsov, “Notes on crystal optics of superlattices,” Solid State Commun. 55(1), 85–90 (1985). [CrossRef]

77.

S. I. Pekar, “The theory of electromagnetic waves in crystal in which excitons are produced,” Sov. Phys. JETP 6, 785–796 (1958).

78.

G. A. Wurtz, R. Pollard, W. Hendren, G. P. Wiederrecht, D. J. Gosztola, V. A. Podolskiy, and A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol. 6(2), 107–111 (2011). [CrossRef] [PubMed]

79.

P. A. Belov, R. Marqués, S. I. Maslovski, I. S. Nefedov, M. Silverinha, M. Simovski, and S. A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B 67(11), 113103 (2003). [CrossRef]

80.

A. Chebykin, A. Orlov, A. Vozianova, S. Maslovski, Y. Kivshar, and P. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B 84(11), 115438 (2011). [CrossRef]

81.

M. J. Thompson, M. J. DeVries, T. E. Tiwald, and J. A. Woollam, “Determination of optical anisotropy in calcite from ultraviolet to mid-infrared by generalized ellipsometry,” Thin Solid Films 313–314, 341–346 (1998). [CrossRef]

82.

L. J. Prokopeva, D. P. Brown, X. Ni, V. P. Drachev, A. Urbas, and A. Kildishev, “Pulse shaping using optical metamaterials with naturally anisotropic structural elements,” in CLEO: QELS-Fundamental Science, OSA Technical Digest (Optical Society of America), 2012.

OCIS Codes
(160.1190) Materials : Anisotropic optical materials
(260.2110) Physical optics : Electromagnetic optics
(160.3918) Materials : Metamaterials

ToC Category:
Metamaterials

History
Original Manuscript: May 23, 2013
Revised Manuscript: June 10, 2013
Manuscript Accepted: June 10, 2013
Published: June 17, 2013

Virtual Issues
Hyperbolic Metamaterials (2013) Optics Express

Citation
Vladimir P. Drachev, Viktor A. Podolskiy, and Alexander V. Kildishev, "Hyperbolic metamaterials: new physics behind a classical problem," Opt. Express 21, 15048-15064 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-12-15048


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