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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 11 — May. 28, 2007
  • pp: 6994–6999
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Effective medium description of plasmonic metamaterials

Q-Han Park, J. H. Kang, J. W. Lee, and D. S. Kim  »View Author Affiliations


Optics Express, Vol. 15, Issue 11, pp. 6994-6999 (2007)
http://dx.doi.org/10.1364/OE.15.006994


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Abstract

We demonstrate that the electromagnetic properties of a plasmonic metamaterial, composed of a perfectly conducting metal film perforated with an array of holes, can be effectively described by a structureless, three layer film. The enhanced transmission, first observed by Ebbessen, is identified with resonant tunneling in the equivalent three layer system and perfect transmission is shown to be possible below the critical thickness of a metamaterial. The nature of modes mediating perfect transmission is clarified.

© 2007 Optical Society of America

Despite the success in describing novel optical properties, effective permittivity and effective permeability fail to capture certain important features of PM. The well-known phenomenon of resonantly enhanced optical transmission through PM [4

4. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998). [CrossRef]

], in particular, can not be explained in terms of effective quantities. In fact, the metal-like refractive index of PM suppresses optical transmission that occurs through tunneling. The resonant behavior in transmission is absent in a structureless metal film and the most important feature of PM is missing in the effective medium description.

In this paper, we show that this problem can be nicely resolved by considering a hybrid-type effective medium, a structureless film made of three layers of dispersive materials. We demonstrate that the transmission property of PM is the same as that of a three layer film with specifically chosen effective refractive indices and thicknesses. The extraordinary transmission through PM, first observed by Ebbessen [4

4. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998). [CrossRef]

], is identified with the process of resonant tunneling [5

5. L. L. Chang, L. Esaki, and R. Tsu, “Resonant tunneling in semiconductor double barriers,” Appl. Phys. Lett. 24, 593–595 (1974). [CrossRef]

] occurring in the equivalent three layer system. In fact, nearly 100 percent, perfect transmission is known to happen for PM [6

6. J. W. Lee, M. A. Seo, J. Y. So, Y. H. Ahn, D. S. Kim, S. C. Jeoung, C. Lienau, and Q. H. Park, “Invisible plasmonic meta-materials through impedance matching to vacuum,” Opt. Express 13, 10681–10687 (2005); M. Tanaka, F. Miyamaru, M. Hangyo, T. Tanaka, M. Akazawa, and E. Sano, “Effect of a thin dielectric layer on terahertz transmission characteristics for metal hole arrays” Opt. Lett. 30, 1210–1212 (2005) [CrossRef] [PubMed]

] and this perfect transmission occurs, as we show, if the thickness of PM is below the critical value, determined by the opening ratio of a square hole and the wavelength of an incoming wave. We find that perfect transmission occurs at two distinct resonance frequencies [7

7. similar double peak feature has been also observed in A. P. Hibbins, M. J. Lockyear, I. R. Hooper, and J. R. Sambles, “Waveguide arrays as plasmonic metamaterials: transmission below cut-off” Phys. Rev. Lett. 96, 073904 (2006) [CrossRef] [PubMed]

], mediated by a symmetric and an antisymmetric modes of PM, or the symmetric modes of a three layer film. This feature shows a good agreement with an experimental result.

Fig. 1. Transmission spectrum of PM(thick line) with a = 0.5L,h = 0.25L. Insets illustrate structures of (A) PM and (B) the three layer film. The cross symbols represent the transmission of three layer film with thickness d 1 = 4L,d 2 = 10L.

Consider PM consisting of a perfectly conducting metal film perforated with an array of square holes of size a × a, thickness h, and lattice constant L as illustrated in the inset (A) of Figure 1. An analytic expression for the transmission T through PM can be found by solving the Maxwell equation in terms of diffraction orders [8

8. H. Lochbihler, “Surface polaritons on gold-wire gratings, ” Phys. Rev. B 50, 4795–4801 (1994). [CrossRef]

, 9

9. K.G. Lee and Q-Han Park, “Coupling of Surface Plasmon Polaritons and Light in Metallic Nanoslits,” Phys. Rev. Lett. 95, 103902, (2005). [CrossRef] [PubMed]

]. By assuming the single mode approximation inside a hole [9

9. K.G. Lee and Q-Han Park, “Coupling of Surface Plasmon Polaritons and Light in Metallic Nanoslits,” Phys. Rev. Lett. 95, 103902, (2005). [CrossRef] [PubMed]

], we obtain

T=32ka2μπ2L2[(1+μ1kW)2eiμh(1μ1kW)2eiμh]1
(1)

where k = 2π/λ, μ = √k 2 - (π/2)2

W=m,n=1(2kL)21(m2+n2)(2πkL)2a22L2[sinc(π2mπaL)+(π2+mπaL)]2[sinc(nπaL)]2,
(2)

and the sinc function is defined by sinc(x) ≡ sin(x)/x for x ≠ 0 and sinc(x) ≡ 1 for x = 0. The transmission spectrum, ∣T∣, of an exemplary structure of PM, with a = 0.5L,h = 0.25L, is given in Figure 1 (thick line). The feature of perfection transmission is demonstrated through two peaks, one sharp and one broad, which we explain later. These peaks may arise from the coherent buildup of evanescently diffracted, surface bound wave that is in resonance with the periodic structure. However, instead of seeking any further microscopic origins of these resonances, we follow the spirit of a metamaterial. We demonstrate that the resonance feature, including the full transmission behavior itself, can be simply recovered from an equivalent system made of a structureless three layer film as illustrated in the inset (B) of Figure 1. The transmission of a three layer film, with film thicknesses d 1,d 2,d 1 and refractive indices n 1,n 2,n 1 respectively, can be obtained by using the transfer matrix method [10

10. M. Born and E. Wolf, Principles of Optics, (Cambridge U.P., 1999).

],

T´=16n2n12D+2ein2kd2D2ein2kd2,D±(n11)(n1±1)eikn1d1(n1+1)(n1n2)eikn1d1.
(3)

A remarkable fact is that transmissions T and become identical when the periodic structure is of subwavelength scale (λ > L) with the following identifications:

n2=μhkd2,n12=8ha2π2L2d2+h2π2L2β28ha2d2π2L2d22,cot(kn1d1)=hπ2L2βn1(π1L2d28ha2).
(4)

Here, β denotes the imaginary part of W. For λ > L, it can be shown that reflections R and for each cases satisfy R 2 + T 2 = 1( 2 + 2 = 1) and reflections R and become also identical. The physical meaning of these identifications is clear. Since the internal momentum μ is pure imaginary, n 2 is also a pure imaginary refractive index indicating that the middle layer of thickness d 2 is metallic and presents a tunnel barrier. The thickness d 2 can be determined by demanding a particular phase change after the transmission. In this paper, we will focus only on the equivalence of transmission magnitude and leave d 2 arbitrary. The refractive index n 1 of layers surrounding the tunnel barrier is dispersive due to the β -dependent term, and dispersion becomes most pronounced near λ = L. The condition for d 1 in (4) appears to be unphysical, since it requires a “dispersive” (wavelength length dependent) thickness d 1. The dispersive behaviors of refractive indices n 1 and n 2 and the thickness d 1 are shown in Figure 2 for a typical PM structure. However, if we do not adhere to the exact identity, T = , this seemingly unphysical condition for d1 can be approximated to a physical one, cot(kn¯ 1 d 1) ≈ 0, where k is a constant wave number at the center of the region of interest and n 1 is the nondispersive part of n 1 from (4) and β is assumed to be negligible. For example, in the spectral region as shown in Figure 2, this consideration gives the value, d 1≈ 4L, which is in agreement with the curve of d 1 in Figure 2. With layer thickness fixed to d 1 = 4L and d 2 = 10L, we find the transmission spectrum, ∣T´∣, for the three layer film that is expressed by crosses in Figure 1. This approximated shows a good qualitative agreement with the transmission T of PM. In particular, the perfect transmission feature of PM with double resonance peaks is nicely reproduced in the three layer system. Thus, we find that in the spectral region of subwavelength scale structures with λ > L (after all, this is the region where metamaterials make sense), PM can be effectively described by a structureless three layer film. Perfect transmission in a nondispersive three layer film has been found previously and physically attributed to the process of resonant tunneling [11

11. R. Dragila, B. Luther-Davies, and S. Vukovic, “High Transparency of Classically Opaque Metallic Films, ” Phys. Rev. Lett. 55, 1117–1120 (1985). [CrossRef] [PubMed]

, 12

12. L. Zhou, W. Wen, C. T. Chan, and P. Sheng, “Electromagnetic-Wave Tunneling Through Negative-Permittivity Media with High Magnetic Fields,” Phys. Rev. Lett. 94, 243905 (2005). [CrossRef]

, 13

13. I. R. Hooper, T. W. Preist, and J. R. Sambles, “Making Tunnel Barriers (Including Metals) Transparent,” Phys. Rev. Lett. 97, 053902 (2006). [CrossRef] [PubMed]

]. We emphasize that the dispersive nature of surrounding layers is essential in proving the equivalence. Nevertheless, perfect transmission can be still attributed to resonant tunneling.

Fig. 2. Dispersion of refractive indices n 1,n 2 and thickness d 1.

In order to understand the nature of perfect transmission of PM in double peaks, we first observe that symmetry is present in the system due to the absence of reflection. The electric field intensity for a reflectionless, perfect transmission is symmetric under the spatial inversion with respect to PM. This, in particular, requires the same field intensity at z = ±h/2, specifying the locations for the two sides of PM. Thus, we require that Einty(z = h/2) = e Einty (z = -h/2), where Einty is the electric field inside a hole and α is a phase factor to be determined. An explicit calculation of Einty shows that intensities at both sides are equal if

tanh(μ̅h)=2μ̅k2W2+μ̅2,
(5)

where W = 8a 22 d 2 +iβ and μ= iμ¯. This perfect transmission condition can be solved for β=β± and α=α± such that

β±μkcoth(μ̅h)±[μ2k2sinh2(μ̅h)64a4π2d4]12,
tan(α±)±8a2π2d2[μ̅2k2sinh2(μ̅h)64a2π2d212.
(6)

A couple of notes are in order. Firstly, the plus and minus sign indicates that we have two cases of perfect transmission, mediated by resonant tunneling, with the wave number k = k ± for k ± satisfying Eq.(6) with ± sign respectively. For a/d ≪ 1, e± ≈ ± 1 and thus the internal modes Einty(k = k ±) are symmetric (upper sign) and anti-symmetric (lower sign) in z. Secondly, otherwise spectrally separated peaks of symmetric and anti-symmetric modes get closer and merge into one as thickness h becomes larger and reaches the critical value hc, and for h > hc the perfect transmission ceases to exist. This behavior can be readily understood from the expression of β± where the critical thickness hc makes the term in square root vanish, i.e.,

μ̅2k2sinh2(μ̅hc)64a2π2d2=0.
(7)

The thickness h-dependence of transmission T is illustrated in Figure 3. Clearly, it justifies the observation made for peak positions and the existence of critical thickness for perfect transmission.

Fig. 3. Transmission vs. thickness for PM. Two perfect transmission peaks merge into a single peak and then reduce as thickness h passes the critical value hc= 0.75 L.

Finally, we point out that our effective description of PM using a three layer film can be extended to PM composed of a perfectly conducting metal film of thickness h, perforated with an array of slits with lattice constant L and slit width a. The transmission T of PM,

T=4aL[eikh(1+kW)2eikh(1kW)1]1,W=akL+n=1L(1cos(2πnaL))π2n2ak2(2πnL)2
(8)

is identical with the transmission of a three layer film provided that

n2=hd2,n12=[ad2Lh+β2k2d22Lad2hLh2]1,cot(kn1d1)=n1d2Lad2Lh,
(9)
Fig. 4. Measured transmission spectrum in magnitude ∣T∣(filled circles) and in normalized argument Arg(T)/π (open circles). Sample parameters are d = 400μm,a = 200μm, thickness h = 17μm. Insets show a theoretical result from Eq. (1) with a close-up image.

where β is the imaginary part of W in (8). Note that n 2 is real in this case indicating that slit supports a propagating TEM mode. Here, the identity holds only for d 2 > Lh/a, since otherwise n 1 becomes imaginary.

Fig. 5. FDTD calculation of transmission spectra for PM made of perfector, silver, gold with L = 800nm,a = 400nm,h = 200nm.

In conclusion, we have demonstrated that electromagnetic properties of PM can be effectively described by a structureless, dispersive three layer film. The transmission resonance of PM is identified with resonant tunneling occurring in the three layer film system. We point out that a finite-difference time domain(FDTD) calculation of transmission spectrum for PM composed of a real metal, when compared with the perfect metal case, does not make a significant change in the resonance behavior except for the reduced magnitude and shifts of transmission peaks as illustrated in Figure 5. Thus at the qualitative level, a real metal PM may be also identified with a three layer film though an explicit quantitative identification remains as an open problem. We conclude that a metamaterial is characterized by effective permeability and effective permittivity in the spectral range far off from resonance, but near resonance additional surrounding effective materials causing resonant tunneling is a natural choice.

We thank C. Lienau for discussion. This work is supported by q-Psi, KOSEF, MOST, MOCIE and the Seoul R&BD Program.

References and links

1.

see for example, N. Engheta and R. W. Ziolkowski (eds.), Metamaterials, Physics and Engineering Applications, (IEEE Wiley Interscience, 2006).

2.

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking Surface Plasmons with Structured Surfaces,” Science 305, 847–848 (2004). [CrossRef] [PubMed]

3.

A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Experimental Verification of Designer Surface Plasmons,” Science , 308, 670–672 (2005). [CrossRef] [PubMed]

4.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998). [CrossRef]

5.

L. L. Chang, L. Esaki, and R. Tsu, “Resonant tunneling in semiconductor double barriers,” Appl. Phys. Lett. 24, 593–595 (1974). [CrossRef]

6.

J. W. Lee, M. A. Seo, J. Y. So, Y. H. Ahn, D. S. Kim, S. C. Jeoung, C. Lienau, and Q. H. Park, “Invisible plasmonic meta-materials through impedance matching to vacuum,” Opt. Express 13, 10681–10687 (2005); M. Tanaka, F. Miyamaru, M. Hangyo, T. Tanaka, M. Akazawa, and E. Sano, “Effect of a thin dielectric layer on terahertz transmission characteristics for metal hole arrays” Opt. Lett. 30, 1210–1212 (2005) [CrossRef] [PubMed]

7.

similar double peak feature has been also observed in A. P. Hibbins, M. J. Lockyear, I. R. Hooper, and J. R. Sambles, “Waveguide arrays as plasmonic metamaterials: transmission below cut-off” Phys. Rev. Lett. 96, 073904 (2006) [CrossRef] [PubMed]

8.

H. Lochbihler, “Surface polaritons on gold-wire gratings, ” Phys. Rev. B 50, 4795–4801 (1994). [CrossRef]

9.

K.G. Lee and Q-Han Park, “Coupling of Surface Plasmon Polaritons and Light in Metallic Nanoslits,” Phys. Rev. Lett. 95, 103902, (2005). [CrossRef] [PubMed]

10.

M. Born and E. Wolf, Principles of Optics, (Cambridge U.P., 1999).

11.

R. Dragila, B. Luther-Davies, and S. Vukovic, “High Transparency of Classically Opaque Metallic Films, ” Phys. Rev. Lett. 55, 1117–1120 (1985). [CrossRef] [PubMed]

12.

L. Zhou, W. Wen, C. T. Chan, and P. Sheng, “Electromagnetic-Wave Tunneling Through Negative-Permittivity Media with High Magnetic Fields,” Phys. Rev. Lett. 94, 243905 (2005). [CrossRef]

13.

I. R. Hooper, T. W. Preist, and J. R. Sambles, “Making Tunnel Barriers (Including Metals) Transparent,” Phys. Rev. Lett. 97, 053902 (2006). [CrossRef] [PubMed]

OCIS Codes
(050.1960) Diffraction and gratings : Diffraction theory
(310.6860) Thin films : Thin films, optical properties

ToC Category:
Metamaterials

History
Original Manuscript: April 9, 2007
Revised Manuscript: May 15, 2007
Manuscript Accepted: May 15, 2007
Published: May 22, 2007

Citation
Q-Han Park, J. H. Kang, J. W. Lee, and D. S. Kim, "Effective medium description of plasmonic metamaterials," Opt. Express 15, 6994-6999 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-11-6994


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References

  1. See for example, N. Engheta and R. W. Ziolkowski, eds., Metamaterials, Physics and Engineering Applications, (IEEE Wiley Interscience, 2006).
  2. J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847-848 (2004). [CrossRef] [PubMed]
  3. A. P. Hibbins, B. R. Evans, and J. R. Sambles, "Experimental verification of designer surface plasmons," Science 308, 670-672 (2005). [CrossRef] [PubMed]
  4. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998). [CrossRef]
  5. L. L. Chang, L. Esaki, and R. Tsu, "Resonant tunneling in semiconductor double barriers," Appl. Phys. Lett. 24, 593-595 (1974). [CrossRef]
  6. J. W. Lee, M. A. Seo, J. Y. So, Y. H. Ahn, D. S. Kim, S. C. Jeoung, C. Lienau, and Q. H. Park, "Invisible plasmonic meta-materials through impedance matching to vacuum," Opt. Express 13, 10681-10687 (2005);M. Tanaka, F. Miyamaru, M. Hangyo, T. Tanaka, M. Akazawa, and E. Sano, "Effect of a thin dielectric layer on terahertz transmission characteristics for metal hole arrays" Opt. Lett. 30, 1210-1212 (2005). [CrossRef] [PubMed]
  7. Similar double feature has been observed in A. P. Hibbins, M. J. Lockyear, I. R. Hooper, and J. R. Sambles, "Waveguide arrays as plasmonic metamaterials: transmission below cut-off" Phys. Rev. Lett. 96, 073904 (2006). [CrossRef] [PubMed]
  8. H. Lochbihler, "Surface polaritons on gold-wire gratings," Phys. Rev. B 50, 4795-4801 (1994). [CrossRef]
  9. K. G. Lee and Q-Han Park, "Coupling of surface plasmon polaritons and light in metallic nanoslits," Phys. Rev. Lett. 95, 103902, (2005). [CrossRef] [PubMed]
  10. M. Born and E. Wolf, Principles of Optics, (Cambridge U. P., 1999).
  11. R. Dragila, B. Luther-Davies, and S. Vukovic, "High transparency of classically opaque metallic films, " Phys. Rev. Lett. 55, 1117-1120 (1985). [CrossRef] [PubMed]
  12. L. Zhou, W. Wen, C. T. Chan, and P. Sheng, "Electromagnetic-wave tunneling through negative-permittivity media with high magnetic fields," Phys. Rev. Lett. 94, 243905 (2005). [CrossRef]
  13. I. R. Hooper, T. W. Preist, and J. R. Sambles, "Making tunnel barriers (including metals) transparent," Phys. Rev. Lett. 97, 053902 (2006). [CrossRef] [PubMed]

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