## Abstract

A structure based on plasmonic nano-capillary resonators for optical wavelengths demultiplexing is proposed and numerically investigated. The structure consists of main/bus waveguide connected with series of nano-capillary resonators, each of which tuned at different wavelength transmission band. A model based on resonator theory is given to design the working wavelength of the structure. Both analytical and simulation results reveal that the demultiplexing wavelength of each channel has linear and nonlinear relationships with length and width of the nano-capillary structure.

© 2010 OSA

## 1. Introduction

Surface plasmons polaritons (SPPs) have been considered as energy and information carriers to significantly overcome the classical diffraction limit for their ability of confining and propagating the electromagnetic energy in a subwavelength limit [1,2]. The prospect of integration has motivated significantly recent activities in exploring plasmonic waveguide structures. A number of plasmonic waveguide structures have been proposed such as metallic strips and nanowires [3] as well as V grooves in metal substrates [4], plasmon slots [5], and metal wedges [6,7]. Among those structures, waveguides consisted of an insulator sandwiched between two metals serve as metal-dielectric-metal (MDM) waveguides or metal-insulator-metal (MIM) support propagating surface plasmon modes that are strongly confined in the insulator region with an acceptable propagation length [8]. Therefore, MDM waveguides are promising for the design of nanoscale all-optical devices with a relatively easy fabrication according to the current state of the art [9,10]. Some metal-dielectric-metal waveguides based on SPPs, such as ring resonators [11], Y-shaped combiners [12], couplers [13], Mach-Zehnder interferometers [14] have been designed theoretically and demonstrated experimentally.

Wavelength selecting is one of key technologies in fields of optical communication and computing. Plasmonic Bragg reflectors [15–17], side coupled nanocavity filter [18] and gap plasmon filter waveguide with stub structure or tooth-shaped structure [19–21] have been investigated recently. These filters allow majority of wavelengths to pass through the structure while one or several wavelengths are stopped and reflected. The reflected or selected waves in these structures are in the same channel of entrance waves and not easily to separate. Recently, we proposed an asymmetrical structure to realize the function of a narrow-passband filter with single selective wavelength [22]. In many cases such as WDM systems, it is required to select several specific wavelengths. Noual et al. [23] designed a plasmonic two-wavelength-demultiplexer based on a Y-bent integrated with two rejective or selective waveguide filters around telecommunication wavelengths to drop two different wavelengths received in two output branches. The device requires at least 1.5 μm separation-distance between Y-junction and the cavities to avoid the coupling between them. Photonic crystal-based multi-channel drop filters based on resonant tunneling/coupling mechanism and a plasmonic demultiplexer structure using metallic grating in 3D free space have been reported [24,25]. Considering that the mode confinement in photonic crystals relies on the formation of Bloch wave states by interference of waves diffracted from an array of periodic elements and metallic grating in 3D free space, total sizes of the two structures would be over wavelength scale.

In this paper, a sub-micrometer multi-wavelength demultiplexing structure based on MDM nano-capillary resonators is proposed and demonstrated numerically for the first time. The characteristics of a MDM nano-capillary waveguide are firstly studied. The dependences of demultiplexed wavelength of each channel on geometrical parameters of the structure are discussed. The finite-difference time-domain (FDTD) method with perfectly matched layer (PML) absorbing boundary conditions is used in simulation.

## 2. Theory model

The inset of Fig. 1
shows the nano-capillary resonators composed of two parallel metal plates with a dielectric core. Obviously, the structure can be treated as two MDM waveguides with different widths. Because the width of the lower MDM waveguide is much smaller than that of the upper part, here we call the lower (narrower) part as nano-capillary. When gap width *w* of the MDM waveguide is reduced below the diffraction limit, only a single propagation mode TM_{0} can exist. The complex propagation constant $\beta ={\beta}_{R}+j{\beta}_{I}$ of surface plasmon polaritons in every MDM waveguide can be obtained by solving the dispersion equation [26,27]:

*k*

_{z}_{1}and

*k*

_{z}_{2}defined as: ${k}_{z1}^{2}={\epsilon}_{d}{k}_{0}^{2}-{\beta}^{2}$and ${k}_{z\text{2}}^{2}={\epsilon}_{m}{k}_{0}^{2}-{\beta}^{2},$ where ${\epsilon}_{d}$and ${\epsilon}_{m}$are respectively dielectric constants of the dielectric medium and the metal. ${k}_{0}=2\pi /\lambda $ is the wave number of light with wavelength

*λ*in vacuum. The effective refractive index is represented as${n}_{eff}={\beta}_{R}/{k}_{0}.$ We carry out calculation treating the surrounding metal as silver with Drude-Lorentzian model for its dielectric constant [28]:

*ε*

_{∞}= 2.3646,

*ω*= 8.7377 eV,

_{D}*γ*= 0.07489 eV,

_{D}*Δε*= 1.1831,

*g*= 0.26663,

_{L1}*ω*= 4.3802 eV,

_{L1}*γ*= 0.28 eV,

_{L1}*g*= 0.7337,

_{L2}*ω*= 5.183 eV, and

_{L2}*γ*= 0.5482 eV provide a good description of empirical dielectric constant data for silver [29], over the wavelength range of

_{L2}*λ*= 350 nm~1800 nm of interest. Equation (2) thus adequately describes the silver absorption loss, related to Im[

*ε*], and frequency (or wavelength) dispersion. The dielectric in the core of the structure is assumed to be air with a permittivity

_{m}*ε*= 1.

_{d}To fully understand how the width of the nano-capillary structure influences the SPPs propagation, the dependences of the effective index of SPPs on the width *w* at various wavelengths of the incident light are calculated and shown in Fig. 1. From the Fig. 1, one can see that the effective index of the waveguide decreases with increasing of *w* at same wavelength. The effective index at short wavelength is larger than that at long wavelength, for a given width *w*. The effective index *n _{eff}*

_{2}of the nano-capillary can be larger than

*n*

_{eff}_{3}of upper MDM part and

*n*

_{1}of air. As shown in inset of Fig. 1, the waves will flow into the nano-capillary due to its higher effective index, when SPP waves propagate along the interface between metal and air. The wave transmitted into the capillary will be partly reflected at two ends of nano-capillary, because of index differences between

*n*and

_{eff2}*n*

_{eff}_{3}as well as

*n*

_{1.}One can expect the nano-capillary operates as a resonator. Resonance waves can be formed only in some appropriate conditions within nano-capillary segment. Defining $\mathrm{\Delta}\varphi $ to be the phase delay per round-trip in the nano-capillary, one has$\mathrm{\Delta}\varphi =4\pi {n}_{eff}d/\lambda +{\varphi}_{r},$where ${\varphi}_{r}\equiv {\varphi}_{1}+{\varphi}_{2},$ ${\varphi}_{1}$and ${\varphi}_{2}$ are respectively the phase shifts of a beam reflected on the entrance of the capillary and the junction connecting the nano-capillary and the upper MDM waveguide, and

*d*is the length of the capillary. The waves propagating through the structure will be trapped within the nano-capillary when the following resonant condition is satisfied: $\mathrm{\Delta}\varphi =m\cdot 2\pi .$ Here, positive integer

*m*is the number of antinodes of the standing SPPs wave. The resonant wavelengths can be obtained as follows:

It can be seen that the wavelength *λ _{m}* is linear to the length and the effective index of the nano-capillary, respectively.

Obviously, only the waves with the wavelength *λ _{m}* can stably exist in the nano-capillary, and thus partly transmit or drop into the output end of the nano-capillary. When wideband SPP waves incident into the structure, only the resonance waves with the wavelength

*λ*can be selected and dropped by the nano-capillary. In other words, a transmission peak with the wavelength

_{m}*λ*will be formed in the output section.

_{m}## 3. Discussion of Multiple-nano-capillary resonators structure for wavelength demultiplexing

Figure 2(a)
shows a typical schematic of a 1 × 3 wavelength demultiplexing structure based on MDM nano-capillary resonators. The wavelength demultiplexing structure consists of three nano-capillary resonators perpendicularly connected to a bus waveguide. *w*
_{1} and *d*
_{1} stand for the width and the length of the first nano-capillary, respectively. Since the width of the bus waveguide is much smaller than the operating wavelength in the structure, only the excitation of the fundamental waveguide mode is considered. The FDTD method with perfectly matched layer (PML) absorbing boundary conditions is employed to calculate the transmission spectra. The incident light used to excite SPP is a TM-polarized (the magnetic field is parallel to y axis) fundamental mode. In the following FDTD simulation, the grid sizes in the x and the z directions are chosen to be Δ*x =* 5 nm, Δ*z =* 1.5 nm. Power monitors are respectively set at the positions of *P* and *Q* to detect the incident power of *P _{in}* and the transmitted power of

*P*. The transmittance is defined to be $T={P}_{out}/{P}_{in}.$ The width

_{out}*w*´ of the bus waveguide is set to be 250 nm while the length of

*L*

_{1}and

*L*

_{2}are fixed to be 50 nm and 500 nm. As an example, three nano-capillaries have been designed to split the first, the second and third optical transmission windows, although more nano-capillaries can be added. The parameters of the structure are set to be

*w*= 15 nm,

*w*

_{1}= 250 nm,

*d*

_{1}= 202 nm,

*d*

_{2}= 290 nm,

*d*

_{3}= 347 nm in calculation. Figure 2(b) shows transmission spectra at the outputs of the three channels, and inset of Fig. 2(b) shows transmittance and reflectance of the bus waveguide. From it, one can see channels l-3 can select 980 nm, 1310 nm, 1550 nm bands, respectively, and the maximum transmittance in three bands can exceed 30% (−5.2 dB). And there is also another high transmission in channel 3 around 820 nm wavelength for

*m*= 2. Given the total phase shift

*ϕ*, one can estimate the resonance wavelength from Eq. (3). Submitting

_{r}*λ*1310 nm into Eq. (3) gives

_{m}=*ϕ*= 0.35 for

_{r}*d*= 290 nm and

*n*2.01. Other resonance wavelengths can be approximately calculated with the formula. For the lengths of the nano-capillaries of 347 nm and 202 nm, resonance wavelengths are simply estimated to be 1559 nm and 926 nm. The deviation between FDTD simulation and the result from Eq. (3) could be partly attributed to the neglecting of wavelength dependence of

_{eff}=*ϕ*. And it is partly due to the fact that Eq. (3) is derived based on the effective index approximation that SPP waves with the phase factor of $\text{exp(i2}\pi {n}_{eff}x/\lambda )$ travel back and forth within a capillary, similar to a 3-dimentional plane wave with $\mathrm{exp}(i2\pi nx/\lambda )$traveling in a bulk medium with refractive index

_{r}*n*.

The FWHM (full width at half maximum) of channel 1-3 are 75 nm, 130 nm, 160 nm, respectively. Obviously, the FWHM of the channel 2 and channel 3 are larger than that of channel 1. The reason is that, from the calculation in Fig. 1, the effective index at short wavelength with a fixed width of nano-capillary is higher compared with the one at long wavelength, thus the waves at short wavelength have a higher reflectivity at two ends of nano-capillary and its Q factor is higher. Cross-talk is defined as the ratio between the power of the undesired and desired bands at the outputs. The cross-talk between channel 1 and channel 2 is around −19.7 dB for the 980 nm branch, and the cross-talk between them is −13.1 dB for the 1310 nm branch. The cross-talk between channel 1 and the whole channel 3 is around −19.2 dB for the 980 nm branch, and is −16.6 dB for the 1550 nm branch, although there is also another high transmission in channel 3 around 820nm wavelength for *m* = 2. Therefore, this structure is suitable for wideband wavelengths demultiplexing.

Equation (3) indicates that the transmission behavior of each nano-capillary (channel of the demultiplexing structure) depends mainly on two parameters: the length of the nano-capillary, and the effective index of SPPs in the nano-capillary, which is determined by its width. Figure 3
shows the central wavelength of the nano-capillary resonator as a function of nano-capillary length *d*. One can see that the central wavelength of nano-capillary shifts toward longer wavelengths with the increasing of nano-capillary length *d*, as expected from Eq. (3). Therefore, one can realize the demultiplexing function at arbitrarily wavelengths through the nano-capillary resonator by means of properly choosing the parameters of the structure, such as nano-capillary length and width.

Finally, Fig. 4
shows the propagation of field *H*
_{y} for two monochromatic waves with different wavelengths of 980 nm and 1550 nm launched into nano-capillary resonator demultiplexing structure. The demultipexing effect is clearly observed. From the figure, one can see the wave with wavelength of 980 nm passing through the first nano-capillary and the wavelength of 1550 nm wave transmitting from the third nano-capillary. This is in good agreement with the transmission spectra shown in Fig. 2(b).

## 4. Conclusion

A compact wavelength demultiplexing structure based on MDM nano-capillary resonators is introduced and its performance is analyzed. The structure is suitable for demultiplexing of wideband signals. The dependences of demultiplexed wavelength of each channel on geometrical parameters of the structure are discussed. The wavelength demultiplexing structure might become a choice for the design of all-optical integrated architectures for optical computing and communication, especially in WDM systems in the nanoscale.

## Acknowledgments

The authors acknowledge the financial support from the Natural Science Foundation of Guangdong Province, China (Grant No. 07117866).

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