## Abstract

We propose a coupled metal gap waveguide structure for realizing plasmonic wavelength sorters. Theoretical analysis from the coupled-wave theory reveals that wavelength dependent coupling length of guided surface plasmon polaritons contributes to the routing of different wavelengths to different output ports with reasonable high extinction ratio. The analytical results are confirmed by the finite-difference time-domain numerical simulations. Our result may provide an alternative way to construct nanoscale frequency multiplexers, routers, and sorters for nanophotonic integration and optical communication.

©2008 Optical Society of America

## 1. Introduction

The future integrated optical circuits require the size of optical devices to be reduced to overcome the optical diffraction limit. To satisfy this requirement, surface plasmon polaritons (SPPs) are widely regarded as a feasible candidate to guide the light in subwavelength structures [1-3]. SPPs are the electromagnetic surface waves that travel along the interface between metals and dielectrics [4] with the electromagnetic energy well confined at the surface and decaying exponentially with the distance from the interface. Up to now, a class of SPPs-based passive and active integration photonic elements and devices such as nanoguides [5-8], beam splitters [5,8-10], Bragg reflectors [11], directional couplers [12-14], switches [13,14], and Mach-Zehnder interferometers [8,12,15] have been demonstrated for controlling over and manipulating the propagation of light beams. On the other hand, plasmonic frequency multiplexers [16], dispersion elements [17], wavelength selective add-drop multiplexers and Bragg gating filters [18], and spectral sorters [19] have also been demonstrated recently based upon plasmonic crystals, channel plasmon polaritons, or hole arrays in the metal films.

In this article, we propose a metal gap waveguide (MGW) as an alternative plasmonic sorter for routing the optical information to different output ports according to the wavelengths of SPPs. The sorting effect is revealed by the coupled-wave theory and further established by the finite-difference time-domain (FDTD) numerical simulations. We attribute the above spectral sorting function to wavelength dependent coupling length of guided SPPs in the MGWs, and will demonstrate two plasmonic wavelength sorters for realizing the functions of routing the SPPs respectively excited by two and three wavelengths to different output ports.

## 2. Theoretical analysis

Considering two adjacent two-dimensional MGWs with the gap width *h* and the thickness of the metal films *d* [Fig. 1(a)], the dispersion equation of SPPs in the guides can be read as: [20, 21]

with *b*=⌊*ε*
_{2}
*k*+*ε*
_{1}
*p*-(*ε*
_{2}
*k*-*ε*
_{1}
*p*)*e*
^{-2kh}⌋*ε*
_{2}
*k*/⌊*ε*
_{2}
*k*+*ε*
_{1}
*p*+(*ε*
_{2}
*k*-*ε*
_{1}
*p*)*e*
^{-2kh}⌋*ε*
_{1}
*p*, where
$k=\sqrt{{\beta}^{2}-{k}_{0}^{2}{\epsilon}_{1}}$
and
$p=\sqrt{{\beta}^{2}-{k}_{0}^{2}{\epsilon}_{2}}$
are the transverse propagation constants of SPPs in dielectric layer and metal films, respectively, *k*
_{0}=2*π*/*λ* is the wave number of the incident light with wavelength *λ* in vacuum, *β* is the propagation constant of SPPs along the guiding direction, *ε*
_{1} and *ε*
_{2} represent the relative permittivities of dielectric in guide regions and metal films, respectively. Positive and negative signs in Eq. (1) correspond to symmetry (*β _{s}*) and antisymmetry (

*β*) SPP modes in the waveguides [20]. In the cases where a real metal (Ag) is used, only the above two modes are needed to consider in our analysis [21]. Therefore, the propagation constant

_{a}*β*can be given as: [20, 22]

The coupling length of SPPs in two adjacent waveguides, which determines the distance over which the SPP energy is completely coupled from one waveguide into the adjacent one, is defined as: [23]

in which *C*=(*β _{s}*−

*β*)/2 is the coupling coefficient of SPPs [20, 22]. From Eqs.(1)–(3), we see that

_{a}*L*is dependent on the

_{c}*λ*,

*ε*

_{1},

*ε*

_{2},

*d*, and

*h*simultaneously. This means that, in the case where the dielectric (

*ε*

_{1}) and metal (

*ε*

_{2}) materials as well as the geometric parameters (

*d*and

*h*) of the MGWs are fixed, the coupling length

*L*can be real time modulated by the wavelength (

_{c}*λ*) of the incident light. As the coupling length of SPPs excited by some lights matches the MGW length

*L*[

*L*=(2

*n*+1)

*L*,

_{c}*n*is an integer], the SPPs will completely transport its energy from the exciting waveguide to the adjacent one and finally export from the adjacent guide, while that excited by other lights with the coupling length satisfying with

*L*=2

*nL*will be transported from the incident guide.

_{c}Figure 1(b) shows the dependence of the propagation constant *β* (real part) of SPPs on the thickness of metal film *d* with fixed guide width (*h*=30 nm) and guide length (*L*=500 nm) by Newton iterative method from Eq. (1) as the incident light is with wavelength *λ*=520 nm and 580 nm, respectively. From the figure, we can see that, on the one hand, for fixed *d* and *λ* Eq. (1) produces two solutions, which correspond to the symmetric (*β _{s}*) and antisymmetric modes (

*β*) of SPPs. On the other hand, for a fixed

_{a}*d*(for instance,

*d*=10 nm, the horizontal red dashed line in the figure), different incident lights will produce SPPs different propagation constants

*β*and

_{s}*β*(blue solid line:

_{a}*λ*=520 nm, green dashed line:

*λ*=580 nm). Therefore, the coupling length

*L*of SPPs in a given MGW is determined by the exciting wavelength according to Eq. (3), which indicates that different incident light excited SPPs will be routed to different output waveguides. When a polychromic light beam is injected into one leading guide of the MGWs [upper port in Fig. 1(a)], the SPPs excited by different light wavelengths would be sorted into the output ports

_{c}*a*or

*b*.

## 3. Two wavelength plasmonic sorters

To verify the above conclusion we numerically simulate the propagation behavior of SPPs in the MGWs as shown in Fig. 1(a) by the FDTD method, where the thicknesses of air layer (*ε*
_{1}=1) and Ag film between the two adjacent waveguides are *h*=30 nm and *d*=10 nm (thinner than the skin depth 25 nm of Ag in the visible range [4]), respectively. The length of the MGW is *L*=500 nm. The distance between two adjacent input ports and output ports are separated by a 70 nm thick Ag block to prevent the interaction of electromagnetic fields of the ports [8]. The relative permittivity of Ag is from the measured data [24]. In our simulations, the magnetic field intensity (|*H _{y}*|

^{2}) is used to represent the field intensity distributions in all cases. The unit length of FDTD cell Δ

*x*=Δ

*z*=2

*nm*and time step Δ

*t*=Δ

*x*/2

*c*(

*c*is the velocity of light in air) are used, respectively. The incident light is TM-polarized waves whose magnetic field is parallel to the

*y*direction. Figure 2 presents the dependence of the output field intensity at ports

*a*(red—*—) and

*b*(blue—∇—), respectively, on the exciting wavelengths. One can see that the SPPs excited by

*λ*=580 nm and 520 nm incident lights will be sorted into the output ports

*a*and

*b*, respectively.

Figures 3(a)-3(b) show the gray distributions of SPPs passing through the MGW as the exciting light is with *λ*=520 nm [part (a)] and *λ*=580 nm [part (b)], respectively. The results show that SPPs are well confined in the MGWs. The difference of the coupling lengths of SPPs excited by two incident wavelengths can be clearly observed: that excited by *λ*=520 nm are sorted into the output port *b* after three times coupling process between the two adjacent waveguides while that by *λ*=580 nm is exported from the port *a* after twice coupling process.

Calculated results from Eq. (3) reveal that the coupling length of SPPs in the MGWs is *L _{c}*=194 nm and 236 nm as the incident light is with

*λ*=520 nm and 580 nm, respectively. The values of

*L*seem not to match the MGW length

_{c}*L*. This is due to the influence of the corners between the input and output ports of the MGWs on the exact evaluation of the guide length, which can be understood in detail from the Figs. 3(a) and 3(b). Figure 3(c) shows the corresponding intensity profiles of SPP field across the output end of the MGWs as the incident light is with

*λ*=520 nm and 580 nm, respectively. We can see that the extinction ratio [defining as 10lg|

*H*|

_{y}^{2}

_{0}/|

*H*|

_{y}^{2}

_{1})(

*dB*), where |

*H*|

_{y}^{2}

_{0}and |

*H*|

_{y}^{2}

_{1}denote the field intensities of SPPs output from port

*0*and the cross-talk from port

*1*to port

*0*, respectively] is 15.9 dB at port

*a*for

*λ*=580 nm and 15.4 dB at port

*b*for

*λ*=520 nm, respectively.

## 4. Three wavelength plasmonic sorters

To sort three plasmonic wavelengths we propose a three symmetric MGWs as shown in Fig. 4(a), in which the thicknesses of air layer and Ag films and the guide length are the same as that of Fig. 1(a) (*h*=30 nm, *d*=10 nm, and *L*=500 nm). Figures 4(b)-4(d) present the gray distributions of SPP passing through the system as the exciting light with *λ*=460 nm, 510 nm, and 650 nm, respectively, is incident from the upper input port of the MGWs. We see that SPP field is coupled from the input guide to the middle one and then to the lower one after a certain coupling length. This coupling process can in turn take place from the lower waveguide to the middle and then to the upper (input) one. The SPPs excited by different wavelengths exhibit different coupling lengths and consequently, will export from different output ports as the MGWs are with a fixed length (*L*=500 nm). From the figures, we can see clearly that SPPs excited by *λ*=460 nm is firstly coupled into the middle guide, and then into the lower one, and in turn re-coupled to the middle and the upper one [Fig. 4(b)]. While for the SPPs excited by *λ*=510 nm and 650 nm, the coupling lengths are longer than that by *λ*=460 nm. So the SPP fields are just coupled into ports *b* and *c*, respectively. Figure 4 (e) shows the corresponding intensity profiles of SPP fields across the output end of the MGWs as the incident light is with *λ*=460 nm, 510 nm, and 650 nm, respectively. We can see that the SPP energy excited by *λ*=460 nm, 510 nm, and 650 nm can be exported from ports *a*, *b*, and *c*, respectively, with distinguishable extinction ratios of 10.6 dB and 12.9 dB (port *a* for *λ*=460 nm), 15.3 dB and 16.1 dB (port *b* for *λ*=510 nm), and 9.1 dB and 18.6 dB (port *c* for *λ*=650 nm).

## 5. Discussion

As we know that, at the present, it is difficult for the current lithography to realize nanometer size air gaps between Ag films. Fortunately, deposition of nanometer metal and dielectric films is greatly feasible for conventional techniques to construct MGW. Here, we fill the gap of the MGWs instead with SiO_{2} (*ε*
_{1}=2.25) layers. As an example, we just explore the two wavelengths plasmonic sorting behavior of SiO_{2} filled MGWs by the FDTD simulation. Results by using these MGW structures to sort multiple wavelength SPPs can be achieved similarly. The simulated structure is chosen as what schematically depicted in Fig. 1(a) (*h*=30 nm, *d*=10 nm, and *L*=500 nm). Figure 5(a) presents the dependence of the output field intensity of SPPs at ports *a* (red dashed line) and *b* (blue solid line) on the exciting wavelengths. We can see that the SPPs excited by *λ*=850 nm will be sorted into the output port *a* while that by *λ*=700 nm into port *b*. Figures 5(b)-5(c) show the gray distributions of SPPs passing through the MGW as the exciting light is with wavelength *λ*=700 nm [part (b)] and *λ*=850 nm [part (c)], respectively. Similar process of SPPs propagation in the MGWs to that shown in Figs. 3(a)-3(b) can be observed. Figure 5(d) shown is the corresponding intensity profiles of SPP field across the output end of the MGWs as the incident light is with *λ*=700 nm and 850 nm, respectively. From the figure we can get the extinction ratio is 8.3 dB at port *a* for *λ*=850 nm and 15.7 dB at port *b* for *λ*=700 nm.

## 6. Conclusion

To sum up, we have proposed coupled MGW structures for realizing plasmonic wavelength sorters. Theoretical analysis from the rigorous coupled-wave theory reveals that wavelength dependent coupling length of guided SPPs contributes to routing different wavelength SPPs to different output ports. The analytical results are confirmed by the FDTD numerical simulations. Our result provides an alternative way to construct nanoscale multiplexers, routers, and sorters for nanophotonic integration and optical communication.

## Acknowledgments

This work is financially supported by the National Basic Research Program (Grant No. 2007CB935300), NSFC (Grant Nos. 10774116, 60736041, and 10574101) and the program of NCET (Grant No. 04-0678).

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