A Modified Transmission Line Model for Extraordinary Optical

Transmission Through Sub-wavelength Slits

Qilong Wang1 & Yusheng Zhai1 & Shengqi Wu1 &

Zhiyang Qi1 & Lihui Wang2 & Xiaohua Li1

Received: 9 February 2015 /Accepted: 20 April 2015 # Springer Science+Business Media New York 2015

Abstract Amodified transmission line model with the equivalent capacitors is proposed to describe extraordinary optical transmission (EOT) of the sub-wavelength slit with different inner walls. The equivalent capacitors can effectively explain the discontinuity of the interface at which the surface wave propagates. The transmittance calculated with this model is in accordance with finite difference time domain (FDTD) calculations. This modified analytical model provides a methodology to analyze the transmission properties of sub-wavelength slits with different inner walls. The results show that if welldesigned and manufactured, sub-wavelength curved slits have the ability to enhance the optical transmission in the visible region or the infrared region.

Keywords Transmission linemodel . Extraordinary optical transmission . Sub-wavelength . FDTD

Introduction

Since the first observation of extraordinary optical transmission (EOT) through periodic sub-wavelength holes or slit arrays on the metallic substrate [1, 2], it has attracted extensive investigations for its broad applications in sensors [3, 4], color filters [5, 6], photolithography [7, 8], polarizers [9], and others. In order to realize the broadband transmission, unique nano-slits with the linear or nonlinear tapered walls have been designed [10–13]. With the rapid development of variable applications, the investigations on the physics underlying

EOT are also prospering. Some researchers thought that

EOT through the metallic screens perforated by the periodic hole arrays is probably related to surface plasmons (SPs) [1, 14]. The other researchers reported that metallic screens with the positive real part of the permittivity behave as the perfect conductors rather than the solid plasma in the microwave or

THz band [15, 16]. However, some researchers still attempted to explain EOT with considering the surface waves as the spoof plasmons even in the perfect conductor [17]. Recently,

Francisco Medina and the colleagues proposed the comprehensive equivalent circuit model to describe EOT [18, 19].

The transmittance of the sub-wavelength periodic structures can be analytically calculated via their model, and the results have been found to be in accordance with that from the EOT experiments and the numerical simulations. Usually, the equivalent circuit model is only applied to the periodic hole or slit array with the inner straight wall. Some researchers have attempted to modify the model in order to analyze the tapered grating [11]. They improved the equivalent circuit model into the cascaded transmission lines (TLs) and found it reasonable compared to the numerical analysis. However, they ignored the influence of the discontinuity of the interface along the propagation of the wave.

In this paper, the transmission line model with the equivalent capacitors is proposed to simulate the discontinuity of the interface. Besides, the transmittance of the slit with the inner curved wall has been calculated based on our modified model.

The analytical results are more accordant with those got with the numerical finite difference time domain (FDTD) simulations, especially in the infrared region. For the analytical calculation and the numerical simulation, the subtle discrepancy of the transmission spectra is carefully discussed. * Qilong Wang northrockwql@seu.edu.cn 1 School of Electronic Science and Engineering, Southeast University,

Nanjing, Jiangsu 210096, China 2 School of Instrument Science and Engineering, Southeast University,

Nanjing, Jiangsu 210096, China

Plasmonics

DOI 10.1007/s11468-015-9952-z

Modeling

Figure 1a, b shows the schematic diagrams of a unit of 1-day metallic sub-wavelength grating with curved slits on the silica (SiO2) substrate. The metallic slits are characterized by the shape of cross section, the thickness t, the period P, and the widths of slit entrance win and exit wout. In our analysis, TMpolarized plane wave normally incidences on the surface of the sub-wavelength structure. The metallic slits are made of silver, and the surrounding dielectric is air. The waist width w(z) of the cross section is described as the following exponential function with the index α, w zð Þ ¼ win þ wout−winexp αð Þ−1 exp α z t −1 h i ð1Þ where z is the coordinates of slit in the z direction. The profile of the slit walls varies with the index α of the equation, for example, α=−5 for convex, α=5 for concave, and α→0 for linearly tapered slits. In our research, the calculations of the transmittance are both carried out with our proposed modified transmission line model and the FDTD numerical method.

The modified transmission line model is shown in Fig. 1c.

When λ>P>>w, the higher-order transmission modes could be ignored and only the zero order diffraction will be considered in the analysis [20]. So the slit can be equally divided into several TL segments with the characteristic length ti and the width wi. In order to simulate the interface discontinuity around the entrance and the exit, equivalent capacitors CTM1 and CTM2 are added. In short, the sub-wavelength periodic structure proposed is considered as one cascaded transmission lines included CTM1 and CTM2.

The surrounding and substrate are modeled as semi-infinite

TLs characterized by the wave number βu and characteristic impedance per unit length Zu (Here, uwill be in that means the surrounding or s that means substrate.). The wave number is described as β u=k0nucosθ, and the characteristic impedance is Zu ¼ ffiffiffiffiffiffiffiffiffiffiffi μ0=ε0 p

Pcosθ=nu, where θ is the incidence angle.

Therefore, in the slits, we have the propagation constant and the characteristic impendence per unit length shown in (2), (3), tanh ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi β2−k20n22 q w=2 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi β2−k20n21 q n22ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi β2−k20n22 q n21 ð2Þ