1. Introduction

In recent years, surface plasmon waveguides have become a subject of intensive research because of their outstanding performance in subwavelength optical waveguiding [1, 2]. A variety of metallic structures have been proposed as waveguides in nano scale [3, 4, 5]. Among them, metal-insulator-metal (MIM) waveguide [5] is considered to possess remarkable advantages, such as strong field localization, simplicity and convenience for fabrication and integration into optical circuits. Many efforts have been made to develop nano scale MIM waveguide based plasmonic devices, such as reflectors [6, 7, 8, 9], filters [10] and ring couplers [11]. From recent literatures, most of the research efforts concerning MIM waveguide reflectors are concentrated on Bragg gratings which are realized by periodic change of the cladding layer material, the insulator width and material, respectively [6, 7, 8] or simultaneously [9]. This idea originates from Bragg reflection of X rays from periodic planes in crystalline solids. Actually, many periodic structures have passband and forbidden band characteristics similar to those of Bragg gratings. In this paper, we investigate a novel reflector constructed with periodic stub structure in MIM waveguide.

Stub structure is one of the common elements in microwave networks. In previous studies, it has been introduced to theMIMwaveguide and functions as a wavelength filter with submicron size [12, 13]. The properties of such filters, which consist of a single- or double-stub structure, can be described using the concept of characteristic impedance and transmission line theory [12]. However, they only have good filtering effect for discrete wavelengths. In this paper, we extend the single-stub structure to a periodic one so that it can function like a Bragg reflector whose forbidden band characteristics can not be realized with a single stub. Lin et al [13] provided some numerical results about this multiple-stub structure, but failed to give analytical prediction about the center wavelength and width of the forbidden band. In this paper, we use the transmission line and Bloch wave theory to obtain a general analytical equation for this structure. First an infinite periodic stub structure is studied, in which the heat-loss due to metals is assumed to be negligible. Then, Finite-Difference Time-Domain (FDTD) method [14] is used to validate our theoretic analysis and investigate the transmission characteristics, such as the transmission valleys, the periodicity and symmetry of spectrum.

2. Serial stub structure

We consider a MIM waveguide with a series of stubs perpendicular to it as shown in Fig. 1. The width w of waveguide and stubs are set to be equal. The length of the stubs and the interval between two stubs are denoted by L and d, respectively. The insulator medium of the waveguide is assumed to be air whose permittivity is ε_i=1. The circumambience is silver. The frequency-dependent complex relative permittivity of silver is characterized by the Drude model ε_m(ω)=ε∞-ω_p/(ω ²+iωγ). Here, ω_p=1.38×10¹⁶Hz is the bulk plasma frequency, γ=2.73×10¹³ Hz is the damping frequency of the oscillations, ω is the angular frequency of the incident electromagnetic radiation, and ε _Ȟ stands for the dielectric constant at infinite angular frequency with a value of 3.7. The Finite-Difference Time-Domain (FDTD) method is used for numerical simulation. The calculated area is divided into uniform Yee cells of Δx=Δy=2 nm and surrounded by perfectly matched layer (PML) absorbing boundary.

 

Fig. 1. Schematic of MIM waveguide reflector with serial stub structure.

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2.1. Modeling of lossless periodic stub structure

Assuming that silver is lossless, which means that the damping due to the metal is ignored, the properties of systems such as serial stub structure can be described using the concept of a distributed constant circuit including a loss-free transmission line with a characteristic impedance of $Z_0=\genfrac{}{}{0.1ex}{}{k_{\mathrm{sp}}w}{\omega {\epsilon }_i}$ [15], where k _sp is the propagation constant of the fundamental propagating TM mode in the MIM waveguide. Based on transmission line theory, the stub structure is equivalent to an open-circuited transmission line [12, 16] with characteristic susceptance Y_s=jY ₀ tan(k _sp L), where j=√-1 and Y ₀=1/Z ₀. An equivalent circuit of the serial stub structure is presented in Fig. 2. Each unit cell of the line consists of a length d of transmission line with a shunt susceptance across the midpoint of the line. The voltage V_n at the edge of the cell can be written as: V_n=V ⁺ _nexp(-jk _sp x)+V ^- _nexp(jk _sp x). We can relate the voltages on either side of the nth unit cell using a 2×2 transfer matrix:

where matrix M _n,n+1 can be written as

For a serial stub structure with N cells, the calculation of the transmission spectrum involves multiplication of N transfer matrices: Π^N _i=1 M _i,i+1.

 

Fig. 2. Equivalent circuit of the serial stub structure.

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For an infinite periodic stub structure, the waves at the edges of the unit cells is similar to Bloch waves that propagate through periodic crystal lattices. The Bloch wave dispersion relation for this periodic stub structure can be written as

where K=α+jβ is the Bloch wave number. Since the right-hand side of Eq. (3) is purely real, we must have either β=0 or αd=απ (m is a non-negative integer). The case of β=0, α≠0 corresponds to a nonattenuating propagating wave. The magnitude of the right-hand side of Eq. (3) is less than or equal to unity. For the case of αd=mπ, β≠0, the wave does not propagate, but is reflected completely. Eq. (3) is reduced to

This corresponds to the forbidden band of the structure. Within the forbidden band, βd varies from zero at the band edges to its maximum value at the center of the band.

Using the multiplication operation of N transfer matrices Eq.(2), we can get the optical transmittance through N stubs in the periodic stub structure

where

From Eqs. (5) and (6), it can be seen that there exist some low transmission points when F →±∞ which requires

or

The solutions of Eq. (7) are simply

where m is a non-negative integer. In the forbidden band, Kd is a complex number Kd=mπ +jβd, the left-hand side of Eq. (8) becomes sinh²(βd)/sinh²(Nβd) which approaches zero exponentially as e ^-2(N-1)βd.

Strictly speaking, there is no rigorous Bloch wave and forbidden band for a finite serial stub structure which is constructed with lossy metal. To investigate the reflection characteristics of these “real” stub structures, FDTD method is used in the following section.

2.2. Simulation and results

From Eqs. (5) and (6), it is clear that the transmission spectra are closely related to L and d. For simplicity, we investigate the case of d/L=1 first. Eq. (3) is reduced to cos(Kd)=[3cos(k _sp L)-sec(k _sp L)]/2. The period of the transmission spectra is π and the symmetric center of the first period is located at k _sp L=π/2. To demonstrate the validity of our analysis, we construct a 4 stub structure with “real” lossy metal. Other geometric parameters are set as: w=50 nm, d=L=400 nm. FDTD method is used to calculate the transmission spectrum which covers the first two periods. Transfer matrix method (TMM) deduced from Eq.(1) and Eq. (2) is also used for lossless case as a comparison. The calculation results, which are depicted in Fig.3(a), agree very well with each other. The main deviation only exists at the high transmission regions. The transmission valleys are located around k _sp L=0.5π and 1.5π, and the spectra possess good periodicity and symmetry as expected. For comparison, the dispersion relation Eq. (3) is solved numerically and shown in Fig. 3(c). Although there is no strict forbidden band for finite serial stub structure, the low transmission regions in Fig. 3(a) correspondent to the forbidden bands in Fig. 3(c). The serial stub structure with L=d can be used as a wavelength selective reflector whose transmission spectrum is analogous to that of the conventional Bragg gratings.

 

Fig. 3. (a) The first two periods of the transmission spectra for serial stub structure with L=d=400 nm and N=4 by FDTD (lossy case, blue) and TMM (lossless case, red). (b) The first period of the transmission spectra for serial stub structure with L=400 nm, d=600 nm and N=4. (c) and (d) are the corresponding dispersion curves for lossless infinite periodic stub structure with the same geometric parameters as (a) and (b), respectively.

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For another case of d/L=3/2, Eq. (3) can be rewritten as cos(Kd)=cos(3k _sp L/2)-(1/2) tan(k _sp L) sin(3k _sp L/2). The period of k _sp L for transmission spectrum is 2π, and the symmetric center of the first period is k _sp L=π. The transmission spectra with L=400 nm and d=600 nm are shown in Fig. 3(b). The corresponding dispersion relation is depicted in Fig. 3(d). According to the forbidden band condition Eq.(4), centers of the first four forbidden bands are located at k _sp L=0.5π 0.732π, 1.268π and 1.5π. This agrees well with the simulation results. In Fig.3(b), it is also worth noting that the transmittances at the first and fourth valleys are obviously lower than those of the second and third ones. To have a clearer comparison, more cases with different stub number N are studied. The transmission spectra by FDTD with N=1, 2, 4 and 8 are presented together in Fig. 4. As N decreases from 8 to 1, the transmission coefficient at the valley points increases gradually. This is analogous to the conventional Bragg reflector. But at k _sp L=0.5π and 1.5π, the transmission coefficients are fixed at zero no matter how much the stub number changes. According to Eq. (5), there are two categories of low transmission points which are described by Eqs. (7) and (8), respectively. The transmission coefficient T reaches zero, if Eq. (7) is satisfied, while T only approaches zero with an increasing N at the situation of Eq. (8). We present the optical field at the first two valley points (λ=1503 nm and λ=2191 nm) in Figs. 4(b) and (c), respectively. At λ=1503 nm (Fig. 4(b)), the optical field vanishes gradually after about 4 stubs, while at λ=2191 nm (Fig. 4(c)), the transmission becomes negligible just after the first stub, which is simply the single-stub case presented in Ref.[12]. The existence of these zero transmission points make the transmission characteristics of the serial stub structure quite different from those of conventional Bragg reflectors.

 

Fig. 4. (a) The transmission spectra by FDTD for serial stub structure with different stub number N=1, 2, 4 and 8, respectively. Other paramters remain unchanged as in Fig.3(b). (b) and (c) The corresponding optical field distributions (|H_z|²) for the first two transmission valleys in (a): λ=1503 nm (b) and λ=2191 nm (c).

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In more general cases d/L=q/p, where q and p are positive integers, it is easy to see that the transmittance T in Eq.(5) is a periodic function of k _sp L with period of pπ. The numerator and denominator of Eq.(6) have different period (π and pπ, respectively), which means the two categories of transmission valleys have different intervals. If d/L can not be expressed as a fraction of two integers, the transmission spectrumlosses its periodicity, or the period is infinity.

It can also be seen from the numerical results that the loss-free approximation is reasonable for micro-dimensional serial stub structure. In Fig. 3, this model can predict accurately the positions of both the forbidden and the pass bands for L and d with submicron size. In practical photonic circuits, to shift the first forbidden band center to the desired telecommunication wavelengths (1550 nm or 1310 nm), the structure size (involving d and L) will be reduced further and the heat-loss will decrease consequently. Furthermore, at the incident wavelength 1550 nm, light intensity of the primary propagation TMmode drops at about 0.1dB/micron in straight Silver-Air-Silver waveguide (50 nm wide). Therefore, the model presented here can be used to analyze the transmission characteristics of the serial stub structure in MIM waveguide and facilitate the designs of stub based reflectors with micro-dimensional size in telecommunication circuits.

3. Conclusion

In this paper, the plasmonic reflectors based on serial stub structure are studied both theoretically and numerically. A generalmodel of periodic stub structure using transmission line model is developed. The transmission characteristics, e.g., periodicity and symmetry of the spectra, are closely related to the ratio of structure period to stub length. Investigation reveals that the transmission valleys of the spectra could be divided into two categories, which is quite different from conventional Bragg reflectors. It is expected that the model and numerical results may provide useful information in the design of serial stub based reflectors in photonic circuits.