## Abstract

An ultra-compact surface plasmon polaritons (SPPs) narrow band-pass filter based on a slot cavity is proposed and numerically investigated. Attributed to the coupled resonances in the cavity, the filter demonstrates pass-band selection capability. Also, by varying the positions of output waveguides, the filter shows the spectrally splitting function. Moreover, the combination of the adjustments to the length/width of the slot cavity and to the coupling distance provides more flexibility in design for the locations and widths of the pass-bands of the proposed filter.

© 2011 OSA

## 1. Introduction

Surface Plasmons, which are propagating along a metal-dielectric interface with an exponentially decaying field in both sides, have been considered as energy and information carriers to overcome the diffraction limit of light in conventional optics [1–3]. Among various SPPs based waveguides, metal-insulator-metal (MIM) structure [4,5] has attracted tremendous interests of researchers in recent years, because of its potential applications to manipulate and control light in nanoscale. Numerous MIM waveguide based structures have been numerically and/or experimentally demonstrated, such as bends [6], splitters [7], Mach-Zehnder interferometers [8], Y-shaped combiners [9], etc. In order to realize wavelength selection, several Bragg reflectors of MIM structure [10–12] have been theoretically proposed. But most of these structures have large sizes with a relatively high transmission loss. Subsequently, some simple plasmonic waveguide filters have been proposed and demonstrated, such as tooth-shaped waveguide filters [13,14], coupler-type filters [15], channel drop filters with disk resonators [16], rectangular geometry resonators [17], and ring resonators [16]. Most of them can overcome the complexity of fabrication of Bragg reflectors and operate as good band-stop filters. More recently, F-P cavity [18], microring [19] and nanodisk [20] resonators through a different coupling method have been proposed as band-pass filters. However, all above mentioned can only modify their resonance wavelengths by adjusting the internal parameters of the resonators.

In this paper, a narrow band-pass plasmonic filter based on a slot cavity is proposed and analyzed. Compared with previous researches on plasmonic filters, new adjusting mechanisms are introduced to the structure to expend and enhance the filtering characteristics. In the proposed new filters, the resonance characteristics of the slot cavity and the out-coupling strength can be effectively modified by selecting proper input and output waveguide positions. These properties can be used to achieve the band selection/splitting (selecting the pass-band locations) capabilities. Furthermore, the transmission spectra (including the resonance wavelengths and bandwidths) of the filter can also be easily controlled by modulating the geometrical parameters of the slot cavity and the coupling distance between the waveguides and the slot cavity. The finite difference time domain (FDTD) method with a perfectly matched layer (PML) absorbing boundary condition is employed to simulate and study the property of the filter. Due to its subwavelength scale and very simple configuration, this device can be easily fabricated and highly integrated with other micro/nano devices.

## 2. Device structure and theoretical model

As shown in Fig. 1
, the plasmonic slot filter is composed of two MIM waveguides and a short slot cavity. The materials in the blue and white areas are chosen to be silver and air (${\epsilon}_{d}$ = 1). The widths of input/output waveguides and slot cavity are *w* and ${w}_{t}$, respectively. The length of slot cavity is *L*, and the distance of the input and output waveguides apart from the central line *O* of the slot cavity are $\Delta L$ and *h*, respectively. *d* is the coupling distance between two waveguides and slot cavity. Since the widths of the waveguides are much smaller than the incident wavelength, only a single propagation mode ${\text{TM}}_{0}$ (only ${H}_{y}$,${E}_{x}\mathrm{,\hspace{0.17em}}{E}_{z}$,≠0) can exist in the structure [21], whose complex propagation constant β can be obtained by solving following dispersion equation [10,21]:

*k*are defined as ${k}_{d}={\left({\beta}^{2}-{\epsilon}_{d}{k}_{0}{}^{2}\right)}^{1/2}$ and ${k}_{m}={\left({\beta}^{2}-{\epsilon}_{m}{k}_{0}{}^{2}\right)}^{1/2}$. ${\epsilon}_{d}$ and ${\epsilon}_{m}$ are, respectively, dielectric constants of the insulator and the metal. ${k}_{0}=2\pi /\lambda $ is the free-space wave vector. The effective refractive index of the MIM waveguide can be represented as ${n}_{eff}=\beta /{k}_{0}$. The frequency-dependent complex relative permittivity of metal ${\epsilon}_{m}(\omega )$ can be characterized by Drude mode ${\epsilon}_{m}(\omega )={\epsilon}_{\infty}-{\omega}_{p}/\omega (\omega +i\gamma )$, where ${\epsilon}_{\infty}$ stands for the dielectric constant at the infinite frequency,

_{m}*γ*and ${\omega}_{p}$ are the electron collision frequency and bulk plasma frequency, respectively.

*ω*is the angular frequency of incident light. The parameters for sliver can be set as ${\epsilon}_{\infty}$ = 3.7, ω

*= 9.1 eV,*

_{p}*γ*= 0.018 eV, which fit the experimental optical constant of silver [22] quiet well in the visible and near-infrared spectral range. The stable standing waves can be exited within the slot cavity only when the following resonance condition is satisfied: $\Delta \varphi ={\beta}_{m}\xb72L+{\varphi}_{r}=2m\pi $, where ϕ

*≡ ϕ*

_{r}_{1}+ ϕ

_{2}, ${\varphi}_{1}$ and ${\varphi}_{2}$ are, respectively, phase shifts of a beam reflected on the upper and lower facets of the slot cavity. Positive integer

*m*is the number of antinodes of the standing waves in this slot cavity. ${\beta}_{m}$ is the propagation constant of SPPs corresponding to the resonance mode of the ${m}^{st}$order of the cavity. Thus, the resonance wavelengths can be obtained as follows:Given the arbitrary input position $\Delta L$ in the structure, the input filed ${H}^{in}$ inside the slot cavity is divided into two nearly identical portions ${H}_{}^{left}$ and ${H}_{}^{right}$ propagating in opposite directions as depicted in Fig. 1. The relation between them is ${\stackrel{\rightharpoonup}{H}}_{}^{left}$ = ${\stackrel{\rightharpoonup}{H}}_{}^{right}$ = ${\stackrel{\rightharpoonup}{H}}^{in}/2$ = ${\stackrel{\rightharpoonup}{H}}_{0}$. We assume the loss coefficient of the slot cavity is

*σ*, which represents the dissipation of the light propagating per round-trip in the cavity, including the absorbing loss by the metal and the loss caused by the power coupled out of the cavity. Since the slot cavity is symmetric with respect to the central line

*x*= 0, we just need to consider the condition that the position of the input waveguide changes above the central line (

*x*>0). Based on the superposition principle of optics [23] and cavity model, we can describe

*H*field inside the cavity with an arbitrary input position $\Delta L$ as follows:

*H*fields inside the cavity are in the form of the standing waves along

*x*direction at the resonance wavelengths. In this paper, we only consider the first and second resonance mode of the slot cavity. Therefore, for the resonance of the first order (

*m*= 1), we can obtain the

*H*field inside the cavity as follows:

*m*= 2), the Eq. (3) can be written as follows:

*H*field inside the cavity is ${\stackrel{\rightharpoonup}{H}}_{1}(x,t)\equiv 0$, which means that the first resonance mode can’t exist inside the slot cavity, only the second resonance mode can be coupled into the cavity. Whereas, when $\Delta L=L/4$, one can obtain the field ${\stackrel{\rightharpoonup}{H}}_{2}(x,t)\equiv 0$, that means the second resonance mode have been highly suppressed in this case. This phenomenon of selectively suppressing the intrinsic resonance mode of the slot cavity will be verified numerically and explained visually latter on.

## 3. Simulation results and analysis

In the following FDTD (commercial package) simulations, the grid size in the *x* and *z* directions are set to be 4 nm × 4 nm for good convergence of the numerical calculations. The fundamental TM mode of the MIM waveguide is excited by a pulse dipole source from the left waveguide. Two power monitors *P* and *Q* are set to detect the reflected and transmitted powers of ${P}_{ref}$ and ${P}_{tr}$ at the locations, the transmittance and reflectance are defined as $T={P}_{tr}/{P}_{in}$ and $R={P}_{ref}/{P}_{in}$, respectively. The absorption parameter is simply given by $A=1-R-T$, which represents the dissipation of the power in the device. The parameters of the structure are set to be *w* = ${w}_{t}$ = 50 nm, *d* = 15 nm, and *L* = 500 nm. Firstly, the positions of input/output waveguides are fixed as $\Delta L$ = *h* = 225 nm, which means they are kept on the top end of the slot cavity. Figure 2(a)
shows the spectra of the transmission, reflection and the absorption of the proposed filter. It can be seen that two resonance peaks at the wavelengths *λ* = 0.74 μm and 1.47 μm are located in the wavelength range 0.6-1.7 μm of interest, and the corresponding maximum transmittances are 70% (−1.5 dB) and 46% (−3.37 dB), respectively. The quality factor (defined as $Q=\frac{\lambda}{\Delta \lambda}$, where *λ* is the resonance wavelength of the cavity and $\Delta \lambda $ is the full width at half maximum of transmission spectra [10]) at 0.74 μm and 1.47μm are, respectively, 35 and 37 in this case (*d* = 15 nm). It is also shown that two peaks appear around the resonance wavelengths in the absorption curve, because the SPPs coupled into the slot cavity would propagate backwards and forwards inside the cavity and thus undergo great absorption caused by the metal. The counter profiles of fields $\left|{H}_{y}\right|$ at the different wavelengths are depicted in Fig. 2(b)–2(d). According to the Eq. (1), the effective index ${n}_{eff}$ of the MIM waveguide at 0.74 μm and 1.47 μm are calculated to be 1.41 and 1.376, respectively. Given the total phase shift ${\varphi}_{r}$, one can estimate the resonance wavelengths from Eq. (2). Submitting ${\lambda}_{1}$ = 1.47 μm and ${n}_{eff}$ = 1.376 into Eq. (2) gives ${\varphi}_{r}$ = 0.22 for *m* = 1 and *L* = 500 nm. Therefore, the wavelength for *m* = 2 can be approximately calculated as 0.73 μm for ${n}_{eff}$ = 1.41 and ${\varphi}_{r}$ = 0.22 with the formula, which agrees reasonable well with the simulation result for ${\lambda}_{2}$ = 0.74 μm. The deviation between FDTD simulation and the result from Eq. (2) could be attributed to the neglecting of wavelength dependence of ϕ* _{r}*.

Based on the assumptions and analysis in above section, a few novel characteristics of the proposed filter will be demonstrated as follows.

Firstly, in order to verify the phenomenon that the intrinsic resonance modes are suppressed alternatively inside the slot cavity, the input waveguide is moved to the central position *O* (*x* = 0) and the output waveguide is kept still on the top end of the slot cavity. Figure 3(a)
shows the spectra of the transmission and the reflection of the structure in this case. It can be seen that there is only one narrow dip in the curve of reflection at the second resonance mode and that the first resonance mode is completely suppressed inside this cavity, which is highly in conformity with our theoretical analysis above. Similarly, when input waveguide position is chosen to be $\Delta L$ = 132 nm, one can see clearly that the second intrinsic resonance mode of the slot cavity is suppressed as depicted in Fig. 3(b), and that only the first resonance mode at 1.47 μm can be coupled into the slot cavity. The simulation results ($\Delta L$ = 132 nm) are consistent with our theoretical analysis ($\Delta L$ = 125 nm) reasonably well. For clarity, we can visually explain this phenomenon in detail from propagation behavior of SPPs inside the slot cavity as shown in Fig. 1. Taking the first resonance mode at 1.47 μm for example, when the input waveguide is located in the central position *O*, the input field ${H}^{in}$ is divided into two equal parts ${H}_{}^{left}$ and ${H}_{}^{right}$ with the identical initial phase ${\varphi}_{0}$. One portion of field ${H}_{}^{left}$ propagates to the upper facet of the slot cavity and returns back to the input position *O* with a phase ${\varphi}_{0}+\pi $, thus, it will couple and interfere with another part of filed ${H}_{}^{right}$ destructively due to phase difference *π* between them. The similar condition would also happen for the second resonance mode at 0.74 μm when input waveguide moves to the position Δ*L* = 132 nm. These two resonances in opposite directions coupled with each other are the physical reasons that the intrinsic resonance modes of the slot cavity can be alternatively suppressed by choosing proper input waveguide positions, which have never been explicitly studied in the previous researches on plasmonic filters [18–20]. By introducing these coupled resonances to modify the resonance characteristics of the slot cavity, the pass-band selection can be achieved without changing the parameters of the cavity. Moreover, a single channel pass-band transmission in a broad wavelength range can be obtained as depicted in Fig. 3(a) and 3(b), which may have promising applications in photonics and nanoscale optics.

Secondly, the parameter *h*, which stands for the distance of output waveguide apart from the central line *O* of the slot cavity, is also an important factor influencing the output characteristics of the proposed filter, because the out-coupling strength through the end-coupling method is strongly depending on the intensity of *H*field in the out-coupling regions. In another word, the SPPs can hardly be coupled out from the cavity in the position with very low intensity of *H*field. In order to verify the above theoretical analysis, let the input waveguide on the top end of the slot cavity with other parameters unchanged to make sure that two resonance modes exist inside the slot cavity. According to Eq. (3), one can easily find out that the *H*field inside the slot cavity is in the form of standing waves and that the antinodes of the standing waves for the first and second resonance are in the positions *h* = 0 and *h* = 125 nm, respectively, which is also seen in Fig. 2(b) and 2(d). Therefore, when two output waveguides are put in the above positions of the antinodes as shown in Fig. 4(a)
, the two resonance modes are separately coupled into two output waveguides as depicted in Fig. 4(b). It can be seen that only the first (second) resonance mode could be coupled out from the slot cavity in the position of the antinodes of the second (first) resonance. And the crosstalks between the port 1 (0.74 μm) and port 2 (1.47 μm) are −16 dB for the port 1 and −25 dB for the port 2, respectively. This characteristic can be utilized to realize a narrow band-pass filter with spectrally splitting function.

Next, the influence of internal parameters of the slot cavity on the resonance wavelengths is studied by FDTD method in detail. The input/output waveguides are fixed to the position Δ*L* = $L/2-w/2$ to make sure that both two resonance modes exist inside the slot cavity. At the beginning, the length of the slot cavity is set as variable while the other parameters are fixed as above. Figure 5(a)
shows the transmission spectra of the structure corresponding to different cavity lengths. The inset of Fig. 5(a) reveals the wavelengths of each resonance modes have nearly linear relationships with the length of the slot cavity, but with different slope factors (approximate to 1/*m*). This result is in accordance with the solution of Eq. (2). Meanwhile, according to Eq. (2), the resonance wavelengths will also shift when altering the width ${w}_{t}$ of the slot cavity, as shown in Fig. 5(b), resulting from the width-dependent effective index of MIM waveguide. Based on the simulations and analysis above, it is seen that the locations of the pass-bands of the filter can be easily designed by changing both the length and width of the slot cavity.

Now, we study the influence of the coupling distance *d* on the transmission characteristics of the proposed filter, which is also an important factor influencing the intensities of transmission spectra near the resonance wavelengths. Figure 6
shows the transmission curves would change with altering the coupling distance. It is obvious that the resonance wavelengths exhibits slightly blue-shift and transmission peaks decrease simultaneously with increasing the coupling distance, which is consistent with the results in Refs. [15,19]. Moreover, the bandwidths of peaks become a bit of narrower with increased *d* because a large coupling distance would result in small coupling strength which will enhance the “cavity” effect due to small amount of energy coupled out of the slot cavity. Therefore, the bandwidths (*Q* factor) of the resonance spectra can be modified by controlling the coupling distance *d*.

Finally, we make a simple comparison of our proposed structure with those considered in Refs. [15–20]. Since the wavelengths of SPPs correspond to the resonance peaks are allowed to transport efficiently in the output waveguides, while others are forbidden. Our structure can operate as plasmonic band-pass filters, which is very different from the band-stop filters in Refs. [15–17] based on the parallel directional coupling method. Compared with all other band-pass filters in the literature [18–20], the proposed slot filter has a very simple structure and flexible input/output positions (the input/output waveguides can be designed in the same side or different side of the slot cavity). Most importantly, the novel phenomena of suppressing resonance mode and spectrally splitting light have been both theoretically demonstrated and numerically verified for the first time in this paper. Besides, a single channel transmission can be realized in a broad wavelength range, while it’s unachievable in Refs. [18–20].

## 4. Conclusion

In conclusion, a subwavelength plasmonic slot filter is proposed and numerically analyzed by using 2D FDTD method. Several adjustable parameters have been investigated to flexibly modify the filtering characteristics of the proposed plasmonic filter. Both the theoretical analysis and simulation results show the variation of the input/output waveguide positions is an effective method to select pass-band and spectrally split light. Moreover, the transmission spectra, including the resonance wavelength and bandwidth can also be adjusted by modulating the internal parameters of the cavity and the coupling distance between the slot cavity and input/output waveguides. The results above imply that it have potential applications in nanoscale integrated photonic circuits on flat metallic surface.

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