## 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

Metamaterial is one of the most intriguing object in optics because of its unprecedented optical behaviors not found in naturally-formed substances [1]. Electromagnetic properties of metamaterials, composed of metallic structures at the subwavelength scale, are characterized by effective permittivity and effective permeability that can be designed artificially. A perfectly conducting metal film perforated with an array of holes is an important case of metamaterial, known as plasmonic metamaterial(PM), for which the effective refractive index has been shown to mimic that of an ordinary metal [2]. This effective refractive index allows PM to possess a finite skip depth and a surface bound wave, so called the spoof surface plasmon [2], which has been verified experimentally [3].

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], 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], is identified with the process of resonant tunneling [5] occurring in the equivalent three layer system. In fact, nearly 100 percent, perfect transmission is known to happen for PM [6] 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], 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.

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

where *k* = 2π/λ, μ =
√*k*
^{2} - (π/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* = 0.5*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],

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

Here, β denotes the imaginary part of *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, 12, 13]. 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.

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
*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}where *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

$$\mathrm{tan}\left({\alpha}_{\pm}\right)\equiv \pm \frac{8{a}^{2}}{{\pi}^{2}{d}^{2}}{[\frac{{\stackrel{\u0305}{\mu}}^{2}}{{k}^{2}{\mathrm{sinh}}^{2}\left(\stackrel{\u0305}{\mu}h\right)}-\frac{64{a}^{2}}{{\pi}^{2}{d}^{2}}}^{-\frac{1}{2}}.$$

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 ^{iα±}*
≈ ± 1 and thus the internal modes

*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}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.

The claim that the perfect transmission is mediated by symmetric and anti-symmetric
modes is verified experimentally. We have measured transmission by using coherent
THz waves in the range of 0.1 to 2.5THz, generated by a semi-insulating GaAs emitter
biased with a 50 kHz and 300 V square voltage and a standard THz time-domain
spectroscopy system with a ZnTe crystal as a detector. The PM sample is made of a
metal with structural constants, *L* = 400
*μm*,*a* = 200
*μm*,*h* = 17
*μm*. Figure 4 shows an experimentally measured transmission
spectrum of PM where the magnitude and the normalized argument of *T*
are drawn in filled circles and open circles respectively. The transmission peak
reaching to 1 (100 percent transmission) and possessing zero argument clearly
indicates the perfect transmission mediated by a symmetric mode. The peak
corresponding to the anti-symmetric mode is too sharp to be visible within the
spectral resolution of the experimental measurement. Nevertheless, the argument is
close to -π around the region of the anti-symmetric mode peak indicating
the presence of an anti-symmetric mode. All these features agree well with
theoretical predictions shown in the inset(bottom) of Figure 4. Even the theoretical result shows a diminished
anti-symmetric mode peak and argument due to the resolution of a graphical plot. The
close-up image in the inset(top) reveals the sharpness of peak corresponding to the
anti-symmetric mode. It is remarkable that the theoretical result based on the
single mode approximation shows a good qualitative agreement with the experimental
result, confirming the claim on perfect transmission, since the single mode
approximation is relatively poor for a thin sample such as ours compared to the hole
opening, *h*/*a* < 1. In the case of the
three layer film, the anti-symmetric mode guided perfect transmission is indirectly
related to the material parameter, the dispersive refractive index
*n*
_{1} of surrounding films and the corresponding mode
in the three layer film is not necessarily anti-symmetric and should be
distinguished from the anti-symmetric mode in the non dispersive three layer system [13]. On the other hand, the symmetric mode guided perfect
transmission comes from resonant tunneling in the same manner as described for the
non dispersive three layer system.

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,

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

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.

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

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