Abstract

Sub-wavelength layered metal/dielectric structures whose effective permittivities for different polarizations have different signs are known as hyperbolic metamaterials (HMMs). It is believed they are unique in their ability to support electromagnetic waves with very large wavevectors and, therefore, large densities of states, leading to a strong Purcell enhancement (PE) of spontaneous radiation. To verify this conventional wisdom, we use an analytic Kronig–Penney (KP) model and discover that hyperbolic isofrequency surfaces exist for all combinations of layer permittivities and thicknesses, and the strongest PE of radiation is achieved away from the nominally hyperbolic region. Furthermore, large wavevectors and PE are always combined with high loss, short propagation distances, and large impedances; hence, PE in HMMs is essentially a direct coupling of the energy into the free electron motion in the metal, or quenching of the radiative lifetime. PE in HMMs is not related to the hyperbolicity, per se, and is simply the consequence of strong dispersion of the permittivity in metals or polar dielectrics, as our conclusions are relevant also for the infrared HMMs that occur in nature. Moreover, detailed comparison of field distributions, dispersion curves, and Purcell factors between the HMM and surface plasmon polariton (SPP) guided modes in metal/dielectric waveguides demonstrates that HMMs are nothing but weakly coupled gap or slab SPP modes. Broadband PE is not specific to the HMMs and can be just as easily attained in single or double thin metallic layers. When it comes to enhancement of radiating processes and field concentrations, HMMs are in no way superior to far simpler plasmonic structures.

© 2016 Optical Society of America

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References

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    [Crossref]
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2015 (4)

K. Korzeb, M. Gajc, and D. A. Pawlak, “Compendium of natural hyperbolic materials,” Opt. Express 23, 25406–25424 (2015).
[Crossref]

M. S. Eggleston, K. Messer, L. Zhang, E. Yablonovitch, and M. C. Wu, “Optical antenna enhanced spontaneous emission,” Proc. Natl. Acad. Sci. USA 112, 1704–1709 (2015).

P. Gómez García and J.-P. Fernández-Álvarez, “Floquet-Bloch theory and its application to the dispersion curves of nonperiodic layered systems,” Math. Probl. Eng. 2015, 475364 (2015).

J. B. Khurgin, “How to deal with the loss in plasmonics and metamaterials,” Nat. Nanotechnol. 10, 2–6 (2015).
[Crossref]

2014 (5)

J. D. Caldwell, A. Kretinin, Y. Chen, V. Giannini, M. M. Fogler, Y. Francescato, C. T. Ellis, J. G. Tischler, C. R. Woods, A. J. Giles, M. Hong, K. Watanabe, T. Taniguchi, S. A. Maier, and K. S. Novoselov, “Sub-diffractional volume-confined polaritons in the natural hyperbolic material hexagonal boron nitride,” Nat. Commun. 5, 5221 (2014).

G. M. Akselrod, C. Argyropoulos, T. B. Hoang, C. Cirac, C. Fang, J. Huang, D. R. Smith, and M. H. Mikkelsen, “Probing the mechanisms of large purcell enhancement in plasmonic nanoantennas,” Nat. Photonics 8, 835–840 (2014).
[Crossref]

K. V. Sreekanth, K. H. Krishna, A. De Luca, and G. Strangi, “Large spontaneous emission rate enhancement in grating coupled hyperbolic metamaterials,” Sci. Rep. 4, 06340 (2014).

O. D. Miller, S. G. Johnson, and A. W. Rodriguez, “Effectiveness of thin films in lieu of hyperbolic metamaterials in the near field,” Phys. Rev. Lett. 112, 157402 (2014).
[Crossref]

D. Lu, J. J. Kan, E. E. Fullerton, and Z. Liu, “Enhancing spontaneous emission rates of molecules using nanopatterned multilayer hyperbolic metamaterials,” Nat. Nanotechnol. 9, 48-53 (2014).
[Crossref]

2013 (7)

M. Y. Shalaginov, S. Ishii, J. Liu, J. Irudayaraj, A. Lagutchev, A. V. Kildishev, and V. M. Shalaev, “Broadband enhancement of spontaneous emission from nitrogen-vacancy centers in nanodiamonds by hyperbolic metamaterials,” Appl. Phys. Lett. 102, 173114 (2013).
[Crossref]

Y. Guo and Z. Jacob, “Thermal hyperbolic metamaterials,” Opt. Express 21, 15014–15019 (2013).
[Crossref]

C. Argyropoulos, N. M. Estakhri, F. Monticone, and A. Alù, “Negative refraction, gain and nonlinear effects in hyperbolic metamaterials,” Opt. Express 21, 15037–15047 (2013).
[Crossref]

T. A. Morgado, S. I. Maslovski, and M. G. Silveirinha, “Ultrahigh Casimir interaction torque in nanowire systems,” Opt. Express 21, 14943–14955 (2013).
[Crossref]

B. Liu and S. Shen, “Broadband near-field radiative thermal emitter/absorber based on hyperbolic metamaterials: direct numerical simulation by the Wiener chaos expansion method,” Phys. Rev. B 87, 115403 (2013).
[Crossref]

N. Engheta, “Pursuing near-zero response,” Science 340, 286–287 (2013).
[Crossref]

R. Maas, J. Parsons, N. Engheta, and A. Polman, “Experimental realization of an epsilon-near-zero metamaterial at visible wavelengths,” Nat. Photonics 7, 907–912 (2013).
[Crossref]

2012 (8)

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. USA 109, 8834–8838 (2012).

A. N. Poddubny, P. A. Belov, P. Ginzburg, A. V. Zayats, and Y. S. Kivshar, “Microscopic model of Purcell enhancement in hyperbolic metamaterials,” Phys. Rev. B 86, 035148 (2012).
[Crossref]

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, 8100–8116 (2012).
[Crossref]

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

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

Z. Jacob, I. I. Smolyaninov, and E. E. Narimanov, “Broadband Purcell effect: radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100, 181105 (2012).
[Crossref]

S.-A. Biehs, M. Tschikin, and P. Ben-Abdallah, “Hyperbolic metamaterials as an analog of a blackbody in the near field,” Phys. Rev. Lett. 109, 104301 (2012).
[Crossref]

H. N. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological transitions in metamaterials,” Science 336, 205–209 (2012).
[Crossref]

2011 (2)

T. Tumkur, G. Zhu, P. Black, Y. A. Barnakov, C. Bonner, and M. Noginov, “Control of spontaneous emission in a volume of functionalized hyperbolic metamaterial,” Appl. Phys. Lett. 99, 151115 (2011).
[Crossref]

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]

2010 (1)

Z. Jacob, J.-Y. Kim, G. V. Naik, A. Boltasseva, E. E. Narimanov, and V. M. Shalaev, “Engineering photonic density of states using metamaterials,” Appl. Phys. B 100, 215–218 (2010).
[Crossref]

2009 (1)

2008 (4)

L. Douillard, F. Charra, Z. Korczak, R. Bachelot, S. Kostcheev, G. Lerondel, P.-M. Adam, and P. Royer, “Short range plasmon resonators probed by photoemission electron microscopy,” Nano Lett. 8, 935–940 (2008).
[Crossref]

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100, 033903 (2008).
[Crossref]

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, 930 (2008).
[Crossref]

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, 125022 (2008).
[Crossref]

2007 (4)

L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. 98, 266802 (2007).
[Crossref]

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. 90, 191109 (2007).
[Crossref]

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

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410 (2007).
[Crossref]

2006 (2)

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

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

2004 (3)

D. R. Smith, J. B. Pendry, and M. C. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[Crossref]

R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,” Phys. Rev. E 70, 046608 (2004).
[Crossref]

K. Okamoto, I. Niki, A. Shvartser, Y. Narukawa, T. Mukai, and A. Scherer, “Surface-plasmon-enhanced light emitters based on InGaN quantum wells,” Nat. Mater. 3, 601–605 (2004).
[Crossref]

2003 (2)

S. Mishra and S. Satpathy, “One-dimensional photonic crystal: the Kronig-Penney model,” Phys. Rev. B 68, 045121 (2003).
[Crossref]

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

2001 (1)

R. W. Ziolkowski and E. Heyman, “Wave propagation in media having negative permittivity and permeability,” Phys. Rev. E 64, 056625 (2001).
[Crossref]

1998 (1)

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

1996 (1)

U. Siegner, R. Fluck, G. Zhang, and U. Keller, “Ultrafast high-intensity nonlinear absorption dynamics in low-temperature grown gallium arsenide,” Appl. Phys. Lett. 69, 2566–2568 (1996).
[Crossref]

1987 (1)

H.-S. Cho and P. R. Prucnal, “New formalism of the Kronig-Penney model with application to superlattices,” Phys. Rev. B 36, 3237–3242 (1987).
[Crossref]

1981 (2)

L. Eyges and P. Wintersteiner, “Modes of an array of dielectric waveguides,” J. Opt. Soc. Am. A 71, 1351–1360 (1981).
[Crossref]

D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47, 1927–1930 (1981).
[Crossref]

1980 (1)

C. Colvard, R. Merlin, M. Klein, and A. Gossard, “Observation of folded acoustic phonons in a semiconductor superlattice,” Phys. Rev. Lett. 45, 298–301 (1980).
[Crossref]

1972 (1)

P. B. Johnson and R.-W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

1968 (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ϵ and μ?” Sov. Phys. Usp. 10, 509–514 (1968).
[Crossref]

1946 (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 37–38 (1946).
[Crossref]

1931 (1)

R. D. L. Kronig and W. Penney, “Quantum mechanics of electrons in crystal lattices,” Proc. R Soc. London A 130, 499–513 (1931).
[Crossref]

Adam, P.-M.

L. Douillard, F. Charra, Z. Korczak, R. Bachelot, S. Kostcheev, G. Lerondel, P.-M. Adam, and P. Royer, “Short range plasmon resonators probed by photoemission electron microscopy,” Nano Lett. 8, 935–940 (2008).
[Crossref]

Akselrod, G. M.

G. M. Akselrod, C. Argyropoulos, T. B. Hoang, C. Cirac, C. Fang, J. Huang, D. R. Smith, and M. H. Mikkelsen, “Probing the mechanisms of large purcell enhancement in plasmonic nanoantennas,” Nat. Photonics 8, 835–840 (2014).
[Crossref]

Alekseyev, L. V.

Alù, A.

C. Argyropoulos, N. M. Estakhri, F. Monticone, and A. Alù, “Negative refraction, gain and nonlinear effects in hyperbolic metamaterials,” Opt. Express 21, 15037–15047 (2013).
[Crossref]

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100, 033903 (2008).
[Crossref]

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410 (2007).
[Crossref]

Argyropoulos, C.

G. M. Akselrod, C. Argyropoulos, T. B. Hoang, C. Cirac, C. Fang, J. Huang, D. R. Smith, and M. H. Mikkelsen, “Probing the mechanisms of large purcell enhancement in plasmonic nanoantennas,” Nat. Photonics 8, 835–840 (2014).
[Crossref]

C. Argyropoulos, N. M. Estakhri, F. Monticone, and A. Alù, “Negative refraction, gain and nonlinear effects in hyperbolic metamaterials,” Opt. Express 21, 15037–15047 (2013).
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Supplementary Material (1)

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Figures (12)

Fig. 1.
Fig. 1. (a) Two configurations of HMMs. (b) IFS for Type I, Type II HMMs and normal anisotropic dielectric medium.
Fig. 2.
Fig. 2. (a) Dielectric constants for different TR from EMT. (b), (c) IFS for different TR according to the KP model; in case B, the elliptical IFS obtained from EMT and the KP model nearly overlap each other.
Fig. 3.
Fig. 3. (a) IFS at λ=500  nm for TR=1. (b) PF as a function of the position of the emitter in the dielectric for two polarizations. (c) Fields and (d) energies and Poynting vector for the minimum value of the transverse wavevector and (e), (f) the same for the maximum value of the transverse wavevector.
Fig. 4.
Fig. 4. Change of (a) differential PF, and (b) effective parameters with wavevector at TR=1.
Fig. 5.
Fig. 5. (a) Comparison of EMT PF with the results of the KP model. (b) Change of the mean loss and energy velocity with TR. (c) Change of the mean impedance and propagation length with TR.
Fig. 6.
Fig. 6. (a) IFS for different periods. (b) Change of maximum PF for the two components of the emitting dipole in the dielectric with granularity.
Fig. 7.
Fig. 7. Comparison of HMMs with (a) a dielectric gap waveguide (b) and a metal slab waveguide.
Fig. 8.
Fig. 8. Change of PF and effective parameters of a metal slab and a dielectric gap with thickness of the middle layer. (a), (d) PF; (b), (e) propagation length and effective loss; (c), (f) energy velocity and effective impedance.
Fig. 9.
Fig. 9. (a) IFS at λ=500  nm when the thickness of the dielectric and metal are 24 and 6 nm, respectively. (b) Lateral dispersion relation for HMMs in (a). (c) Dispersion relation of a metal slab waveguide when the thickness of the metal is 6 nm. (d) Normal dispersion relation of HMMs in (a).
Fig. 10.
Fig. 10. Change of PF with frequency for HMMs (a=24  nm,b=6  nm) (red solid line), metal slab (dm=6  nm) SPP waveguide (blue dotted line), and dielectric gap waveguide (dd=6  nm) (green dashed line).
Fig. 11.
Fig. 11. (a), (b) PF calculated using the transfer-matrix-method for different periods. (c) Change of PF with the increase of periods. (d) PF obtained with a metal slab with smaller thickness can be the same as HMMs in Fig. 7(b) (top).
Fig. 12.
Fig. 12. Comparison of (a) IFS and (b) PF of the HMMs with real metal and hypothetical dispersionless metal indicates that most of the density of states and PE originate from the metal dispersion.

Equations (6)

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Hy={(AeKz+BeKz)eikxx;0<z<a(CeQz+DeQz)eikxx;b<z<0,
Q2K2=1+|ϵm|,
Ex={iK(AeKzBeKz)eikxx;0<z<aiQ|ϵm|(CeQzDeQz)eikxx;b<z<0.
cos(kz(a+b))=12(QK|ϵm|+K|ϵm|Q)sinh(Ka)sinh(Qb)+cosh(Ka)cosh(Qb),
PFx(z)=12Ex(z)2(z)ϵgE2+H2kx1νgxdkz.
PFx(z)(kz)=12Ex(z)2(z)ϵgE2+H2kx1νgx,

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