## Abstract

As is well-known, the sub-wavelength transmissions and field enhancement are related to the surface plasmon excitation. Here we demonstrate that in two-layer metallic gratings the deep sub-wavelength transmissions is supported with the TE polarization where the surface plasmon mode is forbidden. The new mechanism to the sub-wavelength transmission is discovered to be completely different from the findings in the literatures. we propose a simple resonance condition to classify the resonance types which are responsible for those sub-wavelength transmissions and confirm with numerical simulations. To give a complete explanation of underlying physics, we inspect the near-field phenomenon within the grating slits.

© 2009 Optical Society of America

## 1. Introduction

There has been great interest in high transmission of light through metallic thin film in the past decade [1–5]. It has been discovered that the extraordinary transmission through two-dimensional (2-D) sub-wavelength hole arrays is led by surface plasmon polaritons [6]. The maximum transmission occurs at resonance when the momentum of surface plasmon matches the reciprocal lattice vector (lattice momentum) [7].

In contrast to 2-D hole arrays, the optical gratings (periodically-arranged metallic rods) allow waves to pass when electric fields are polarized perpendicular to the gratings (the TM polarization). On the other hand, those with electric fields parallel to the grating lines (the TE polarizations) are mostly reflected [8]. Such concept has been widely used for polarizers [9]. Recent theoretical works found that down to the deep sub-wavelength (rod-spacing in comparison to wavelength) regime, high transmission only appears at discrete frequencies with TM polarizations. Those frequencies are known as the consequence of guided-wave resonance within the slits [10–12]. Furthermore, these guided-waves are revealed to be coupled surface plasmon modes [11,12].

Regarding the TE polarization, 1-D gratings do not excite any surface plasmon modes as they can only be supported with major electric field polarized perpendicular to the metal/dielectric surface. Therefore, the low transmission is expected in the sub-wavelength regime. However, with the two-layer gratings, a great opportunity is open for the deep sub-wavelength transmission. It is known that the two-layer structure creates a vertical cavity, leading to distinct cavity modes [13,14]. In this paper, we found that the horizontal resonance with the grating periods may be supported if additional momentum matching condition is satisfied. We suggest a simple formula to estimate the resonance frequency and identify two resonance types: the Fano (horizontal) and Fabry-Perot (vertical) resonances. All the above studies are confirmed by the finite-difference time-domain (FDTD) simulations.

## 2. Results and discussion

'Figure 1 depicts the basic configuration of two-layer gratings. The electromagnetic waves are normally impinged on the top of the gratings with electric fields parallel to the grating lines. The Drude model is adopted to simulate gold rods with plasmon frequency *f _{p}* =

*ω*/(2

_{p}*π*) = 2172 THz and relaxation time

*τ*= 27.4 fs [15]. The surrounding index is chosen to be

*n*

_{0}= 1.4. The detailed structure parameters are listed below: the period

*a*= 500 nm,

*w*= 150 nm,

*h*= 80 nm, and

*L*= 800 nm. The wavelengths we are interested in range from 0.5 μm to 2.5 μm, which are above the cutoff condition of the grating slits (

*λ*

_{cutoff}≈2

*w*·

*n*

_{0}= 420 nm). This is a strictly 2-D problem where only

*E*,

_{y}*H*, and

_{x}*H*components need to be considered [16]. Since the only electric field component

_{z}*E*is obviously parallel to any geometry drawn in the

_{y}*xz*-plane, there is no surface plasmon mode included.

The FDTD-computed spectrum is shown in Fig. 2(a), where many resonant peaks are observed. Figure 2(b) is the zoom-in diagram for cases of the asymmetric lineshapes. In addition to the resonance formed in the vertical direction [13], we observed that the periodicity of the gratings provides an alternative resonance medium where the similar idea have been demonstrated in the 2-D hole arrays [1] and photonic crystal slabs [17,18]. Since the two-layer structure opens the possibility for a variety of resonances, we propose a momentum matching condition to estimate the resonance frequencies:

where (*i*, *j*) is the mode index, and *k _{ν}* and

*k*are defined as

_{a}Equation (2) reveals that the vertical momentum matching condition requires the integer number of *k _{ν}* while the horizontal one obeys the multiple of

*k*. Combining both the vertical and horizontal conditions result in Eq. (1), which predicts the resonance frequencies. The resonance wavelengths are thus given by

_{a}The predicted wavelengths as well as FDTD simulated wavelengths are tabulated in Table 1. The under-estimated values from the theoretical prediction are the consequence of neglecting the tunneling effect in the slits, which contributes some phases. This issue has been discussed in Ref [13]. To overcome the tunneling phases, one can decrease the slit width ( *w* ) or the ratio of slit width to grating period ( *w* / *a* ). In observation of the resonance lineshapes, one can find that modes of (1,0) , (2,0) , and (3,0) are symmetric while (1,1) , (2,1) , and (3,1) are asymmetric.

Figure 3 visualizes the resonance mode profiles with *j* =1 . The red and blue colors denote the positive and negative field values, respectively. Clearly, the mode indices *i* and *j* represent the resonance orders for vertical and horizontal directions, respectively. For example, the (1,1) mode suggests one peak along *z* -axis (*i* = 1) and one propagation wavelength along *x* -axis within a grating period (*a*). At those resonances, the incident waves are generated from the top of the gratings with evanescent tunneling through the slits. A portion of electromagnetic fields tunnel into the two-layer cavity and start to accumulate the field intensity. The gratings diffract some of the incoming light to the horizontal direction, offering the opportunity to “resonate” with *a*. Reaching their steady-states, the resonance patterns are grown. Cases of *j* =1 are all asymmetric lineshapes, which possibly result from the Fano resonance. To ensure this, two ingredients of the Fano resonance must be identified: the direct pathway and the resonance-assisted indirect pathway [17]. The above resonance nature is evidently the indirect pathway. The direct pathway comes from the light that tunnels straight through the two-layer gratings and goes out without diffraction (Note that some of the incoming light is diffracted into the *x*-direction and the rest of it does not). Interference of the resonance mode (the diffracted fields that propagate along *x*-axis) and the direct tunneling fields (the undiffracted fields propagate along *z*-axis) lead to the Fano lineshape as illustrated in Fig. 2(b). We ever simulated the one-layer gratings with TE polarization in the sub-wavelength regime. Of course, there is no resonance mode, but the tunneled fields contribute to some transmissions, say, few percents. This is a good support to our explanation on the direct pathway part.

Figure 4 is the mode profiles of the other categorized resonance type: the Fabry-Perot resonance. This occurs with *j* = 0, where the periodicity does not play a role. The electromagnetic field tunnels through the top layer, accumulates the phases vertically, and directly goes out from the bottom layer. The whole dynamics is much like the conventional Fabry-Perot resonance, which produces the Lorentzian lineshape. One can see that the mode profiles have constant amplitudes along *z* -axis, indicative of the absence of the horizontal resonance.

By far, two resonance types associated with the two-layer gratings are distinguished. Without the resonance related to the periodicity ( *j* = 0), the two-layer gratings behave as a vertical cavity. The Fabry-Perot resonance is then realized. If the periodicity (*j* ≠ 0) is involved, the standing waves are formed with equal forward and backward propagations in the *x* -direction. This interaction picture is an analogue to 2-D photonic crystal slab, of which the detailed description was provided by Fan *et al*. [18]. Instead of the two-layer structure, the resonance is mediated by waveguide modes within the slab, which is known as the guided resonance. In the spectrum of photonic crystal slab, light is excited from the top of slab and a number of asymmetric resonance peaks are observed. The lattice momenta of the crystal are closely related to our **k**
_{a}. In addition, the guided wave within the slab has to satisfy the standing wave boundary condition in the vertical direction, which is conceptually equivalent to our **k**
_{ν} in some senses. When the lattice momenta and standing-wave boundary condition are both met, the resonances are obtained just like our case.

The tunneling strength can be controlled by the slits as they directly leak the electromagnetic energy into the two-layer structure. By decreasing the slit width (*w*), slit to period ratio (*w* / *a*), or grating thickness (*h*), the tunneling strength is reduced. This leads to the narrower full-width-at-half-maximum (FWHM), longer time to reach the steady-state, and the resonances closer to results of the proposed formula. Figure 5(a)
(b) are the collective plots of FDTD-computed resonances with respect to a variety of slit widths for (2,0) and (1,1) modes, respectively. As is seen, the resonance wavelengths decrease with narrower slit widths and converge to the formula-computed wavelengths (the dashed line). The tunneling strength is also weakened if the realistic metal is replaced by the perfect electric conductor (PEC). To investigate this aspect, we introduce the scaling factor δ and replace *ω _{p}* and

*τ*by

*ω*′

_{p}=

*ω*·

_{p}*δ*and

*τ*′ =

*τ*/

*δ*, respectively. The PEC is realized when δ is sufficiently large, e.g.

*ε*(

_{r}*ω*) = 1-

*ω*′

^{2}

_{p}/ (

*ω*

^{2}+

*j*(

*ω*/

*τ*′)) ≈ 1 +

*j*

*ω*

^{2}

_{p}

*τ*′ /

*ω*≈

*j*∞. Unlike realistic metals wherein the electromagnetic energy is present with skin depth, the PEC reflects light with no penetration. If this is the case, the tunneling strength is largely reduced, leading to the blue-shift of the resonances. Figure 5(c) (d) plot the collective resonance data versus the scaling factor δ. With larger δ, the PEC is approached and the resonances move to that of the proposed formula.

## 3. Conclusion

In summary, the deep sub-wavelength transmissions are realized with the two-layer metallic gratings where surface plasmons are excluded. According to our study, two kinds of resonances are identified: the Fano and Fabry-Perot resonances. We discussed the detailed dynamics of those resonances and proposed a simple formula to estimate the resonance frequencies. With the assistance of formula, the resonance types, the orders, and the mode profiles are clearly understood.

## Acknowledgments

This work was supported in part by the National Science Council of the Republic of China under grant NSC97-2221-E002-043-MY2, in part by the Excellent Research Projects of National Taiwan University under grant 97R0062-07, and in part by the Ministry of Education of the Republic of China under “The Aim of Top University Plan” grant.

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