一种基于矩形腔的MIM波导滤波器的滤波特性研究
0 Introduction
Surface Plasmon Polaritons (SPP) are electromagnetic waves coupled to electron oscillations and propagating at the interface between an insulator and a conductor[1]. Because the fields of SPP decay exponentially in both sides, SPP can overcome the conventional diffraction limit and manipulate light on a subwavelength scale. Therefore, SPP have promising applications in highly integrated optical circuits[2-3]. Several SPP waveguiding structures have been proposed and studied, such as V-groove waveguides[4], dielectric-loaded waveguides[5], long-range waveguides[6], and Metal-Insulator-Metal (MIM) waveguides[7], to propagate and control SPP effectively. Among these structures, MIM waveguide can focus the light into the insulator core and allow the manipulation and propagation of light at the nanoscale. Hence, MIM waveguide has attracted tremendous interests of researchers in recent years, and various devices based on SPP have been proposed and demonstrated theoretically and experimentally, such as beam splitters[8], couplers[9], Mach-Zehnder interferometers[10], and filters[11-14].
MIM waveguide filters are essential components in SPP integrated circuits. Kinds of filters with filtering bandwidth ranging from tens to hundreds of nanometers based on resonance interference effect have been proposed, including the T-shape structure[13], ring cavity structure[14], Fabry-Perot structure[15], and rectangle cavity structure[16-17]. The effects of cavity structure parameters on the resonance peak and extinction ratio have been studied as well. Since the easy fabrication, the MIM waveguide filters based on microcavity structure have been well-studied. However, filtering bandwidth of those filters is generally wide, that is difficult to meet the requirements of narrow-bandwidth application. Moreover, in fact, the filtering characteristics of microcavity structure filters are based on the coupling between the microcavity and the waveguide, which shows that the coupling will affect the filtering bandwidth of the filter indeed. Unfortunately, there is no relevant research on this aspect at present.
In this study, an MIM waveguide filter based on a rectangular cavity is constructed, and the filtering characteristics of this type of filter are studied. First, a transfer matrix theoretical model for the transmission of electric fields in the filteris established. The effects of the coupling length, rectangular cavity length, and propagation loss on the filtering bandwidth are studied and analyzed. Finally, the narrow 3-dB filtering bandwidth of the filter is obtained by optimizing the coupling length and the rectangular cavity length with loss compensation with the introduction of a gain medium. This work may provide some reference for the research and development of SPP filters.
1 Theoretical model
Here, ε∞=3.7 is the interband transition contribution, ω is the angular frequency, ωp=1.38×1016 Hz is the plasmon oscillation frequency of the free electrons in the metal, and γ=2.7×1013 Hz is the damped oscillation frequency. W and h denote the outside width and height of the rectangular cavity, respectively. The distance (t) between the rectangular cavity and the waveguide is 10 nm. To reduce the propagation loss of the SPP in the rectangular cavity and improve the Q value of the cavity, we used a rectangular cavity with a smooth bend structure on the outer portion[18]. The radius (r) of the arc is 50 nm. When the initial SPP (the electric field is Ein) are incident at Port 1 of the straight waveguide, a portion of the energy is directly outputted from Port 2 (the electric field is
where κ is the coupling coefficient of the coupling region, and τ is the transmission coefficient. κ+τ < 1 because of the existence of loss in the coupling region. According to the Maxwell equations,
Here, β is the propagation constant of the SPP mode in the MIM waveguide. According to Eq (2) and (3), the transmittance of the filter can be expressed as follows
The propagation constant β can be calculated from the dispersion relationship[13]
where
2 Results and discussions
We first calculated the propagation constant of the SPP mode in the MIM waveguide. Without loss of generality, we assume that the wavelength is 1 550 nm and that the waveguide supports only one SPP mode with a propagation constant β=(8.26×106+2.5×104i)m-1, which is calculated using Eq (5). We conducted a numerical simulation and analysis combined with a finite element method (FEM) to investigate the properties of the coupling region[9].
图 2. Characteristics of the filter varying with the h
Fig. 2. Characteristics of the filter varying with the h
The figure shows that with the increase in h from 100 to 1 000 nm, three filtering drop peaks are observed at h=190, 570, and 950 nm, and the corresponding transmissivity values are 0.23, 0.18, and 0.12, respectively. This indicates that 1 550 nm SPP achieve resonance in the cavity at the three heights. To clearly show the resonance characteristics, we give the distribution of the z-direction component of the magnetic field (
The coupling length significantly influences the filtering characteristics, particularly the filtering bandwidth. The sum of h and W is fixed at 870 nm, which corresponds to a total propagation length L1 of 1 540 nm in the rectangular cavity and ensures the length of twice SPP wavelengths (as shown in
图 3. Variation of transmissivity spectrum with different coupling lengths W when L 1 is 1 540 nm
Fig. 3. Variation of transmissivity spectrum with different coupling lengths W when L 1 is 1 540 nm
We studied the filtering characteristics of the filter in cases when the propagation length L1 is equal to one and three times the SPP wavelength, as shown in
图 4. Transmissivity spectra and electric field of the filter
Fig. 4. Transmissivity spectra and electric field of the filter
The previous analysis showed that in addition to the coupling coefficient κ, the propagation loss of the SPP in the rectangular cavity affects the filtering bandwidth of the device. To further realize a filter with a narrower filtering bandwidth, we introduced an optical gain medium to compensate for the propagation loss. Lead sulfide (PbS) quantum dots were used as the gain medium for the 1 550 nm SPP[20]. They were doped into the PMMA material at the intermediate layer of the rectangular cavity. As such, the PbS quantum dots provided gain for the SPP under the external pumping light, thus overcoming the problem of wide filtering bandwidth due to excessive cavity loss.
3 Conclusion
The filtering characteristics of an MIM waveguide filter based on a rectangular cavity are investigated in this study. A transfer matrix theoretical model for the transmission of electric fields in the filter is established, and the effects of the coupling length, rectangular cavity length, and propagation loss on the filtering bandwidth are analyzed. The simulation results indicate that for different resonance cavity lengths, there is an optimal coupling coefficient at which the filtering bandwidth is the narrowest. Moreover, the greater the resonance cavity length and the lower the cavity loss, the narrower the filter bandwidth. For an optimum coupling length of 250 nm and a rectangular cavity length of 2 300 nm, the 3 dB filtering bandwidth of the MIM waveguide filter is found to be 30 nm at a wavelength of 1 550 nm. The filtering bandwidth can be narrowed down to 15 nm by introducing PbS quantum dots as the gain medium into the rectangular cavity to compensate for the propagation loss. This paper may provide some reference for the research and development of SPP filters.
龙虎, 陈红艺, 陆小微, 蔡懿, 曾选科, 李景镇. 一种基于矩形腔的MIM波导滤波器的滤波特性研究[J]. 光子学报, 2020, 49(2): 0223001. Hu LONG, Hong-yi CHEN, Xiao-wei LU, Yi CAI, Xuan-ke ZENG, Jing-zhen LI. Filtering Characteristics of an MIM Waveguide Filter Based on a Rectangular Cavity[J]. ACTA PHOTONICA SINICA, 2020, 49(2): 0223001.