## Effective medium description of plasmonic metamaterials

Optics Express, Vol. 15, Issue 11, pp. 6994-6999 (2007)

http://dx.doi.org/10.1364/OE.15.006994

Acrobat PDF (382 KB)

### 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

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]

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]

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]

*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. 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]

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]

*k*= 2π/λ, μ = √

*k*

^{2}- (π/2)

^{2}

*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*= 0.5

*L,h*= 0.25

*L*, 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

*T´*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],

*T*and

*T´*become identical when the periodic structure is of subwavelength scale (λ >

*L*) with the following identifications:

*W*. For λ >

*L*, it can be shown that reflections

*R*and

*R´*for each cases satisfy

*R*

^{2}+

*T*

^{2}= 1(

*R´*

^{2}+

*T´*

^{2}= 1) and reflections

*R*and

*R´*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*=

*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}≈ 4

*L*, which is in agreement with the curve of

*d*

_{1}in Figure 2. With layer thickness fixed to

*d*

_{1}= 4

*L*and

*d*

_{2}= 10

*L*, we find the transmission spectrum, ∣

*T*´∣, for the three layer film that is expressed by crosses in Figure 1. This approximated

*T´*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. 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]

*z*= ±

*h*/2, specifying the locations for the two sides of PM. Thus, we require that

*E*(

^{int}_{y}*z*=

*h*/2) =

*e*

^{iα}*E*(

^{int}_{y}*z*= -

*h*/2), where

*E*is the electric field inside a hole and α is a phase factor to be determined. An explicit calculation of

^{int}_{y}*E*shows that intensities at both sides are equal if

^{int}_{y}*W*= 8

*a*

^{2}/π

^{2}

*d*

^{2}+

*i*β and

*μ*=

*iμ¯*. This perfect transmission condition can be solved for β=β

_{±}and α=α

_{±}such that

*k*=

*k*

_{±}for

*k*

_{±}satisfying Eq.(6) with ± sign respectively. For

*a*/

*d*≪ 1,

*e*≈ ± 1 and thus the internal modes

^{iα±}*E*(

^{int}_{y}*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

*h*, and for

_{c}*h*>

*h*the perfect transmission ceases to exist. This behavior can be readily understood from the expression of β

_{c}_{±}where the critical thickness

*h*makes the term in square root vanish, i.e.,

_{c}*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.

*h*, perforated with an array of slits with lattice constant

*L*and slit width

*a*. The transmission

*T*of PM,

*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.

## References and links

1. | see for example, N. Engheta and R. W. Ziolkowski (eds.), |

2. | J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking Surface Plasmons with
Structured Surfaces,” Science |

3. | A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Experimental Verification of
Designer Surface Plasmons,”
Science , |

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 |

5. | L. L. Chang, L. Esaki, and R. Tsu, “Resonant tunneling in semiconductor
double barriers,” Appl. Phys. Lett. |

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 |

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. |

8. | H. Lochbihler, “Surface polaritons on gold-wire
gratings, ” Phys. Rev. B |

9. | K.G. Lee and Q-Han Park, “Coupling of Surface Plasmon
Polaritons and Light in Metallic Nanoslits,”
Phys. Rev. Lett. |

10. | M. Born and E. Wolf, |

11. | R. Dragila, B. Luther-Davies, and S. Vukovic, “High Transparency of Classically
Opaque Metallic Films, ” Phys. Rev. Lett. |

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. |

13. | I. R. Hooper, T. W. Preist, and J. R. Sambles, “Making Tunnel Barriers (Including
Metals) Transparent,” Phys. Rev. Lett. |

**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

- See for example, N. Engheta and R. W. Ziolkowski, eds., Metamaterials, Physics and Engineering Applications, (IEEE Wiley Interscience, 2006).
- J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847-848 (2004). [CrossRef] [PubMed]
- A. P. Hibbins, B. R. Evans, and J. R. Sambles, "Experimental verification of designer surface plasmons," Science 308, 670-672 (2005). [CrossRef] [PubMed]
- 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]
- L. L. Chang, L. Esaki, and R. Tsu, "Resonant tunneling in semiconductor double barriers," Appl. Phys. Lett. 24, 593-595 (1974). [CrossRef]
- 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]
- 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]
- H. Lochbihler, "Surface polaritons on gold-wire gratings," Phys. Rev. B 50, 4795-4801 (1994). [CrossRef]
- 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]
- M. Born and E. Wolf, Principles of Optics, (Cambridge U. P., 1999).
- R. Dragila, B. Luther-Davies, and S. Vukovic, "High transparency of classically opaque metallic films, " Phys. Rev. Lett. 55, 1117-1120 (1985). [CrossRef] [PubMed]
- 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]
- 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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