Surface plasmon polaritons (SPPs) are essentially oscillation modes of free electrons coupled with the electromagnetic field at the interface between metallic and dielectric materials.^[1] Because they can confine the optical field to the micro-nanometer scale and breakthrough the diffraction limit, SPPs have immense potential to constitute a new generation of all-optical integrated circuits.^[2–6] In order to effectively propagate and control SPPs, various SPP waveguides have been studied and designed, including dielectric-loaded waveguides, V-groove waveguides, nanowire waveguides, and metal–insulator–metal (MIM) waveguide.^[7,8] Among them, the MIM waveguide can limit the light field at the nanometer scale, which has drawn the attention of researchers. Therefore, devices based on MIM waveguide structures have been proposed, such as couplers, beam splitters, wavelength multiplexers, filters, and reflectors.^[9–12] Wherein the reflectors are crucial components in applications such as redirecting SPPs and SPP laser resonators. Therefore, how to obtain reflectors with high-performance and simple structure has become the focus of research.

Currently, different types of SPP reflectors have been developed, including those with a Bragg structure,^[13–15] T-shape structure,^[16–18] and microcavity structure.^[19] For Bragg structure, the reflection can be realized by periodic change of the cladding layer material, the insulator width or material. For example, Ming et al. proposed a metal-embedded MIM structure plasmonic Bragg reflector with the transmittance of 86% and filter range from to .^[13] For T-shape structure, the filters consist of waveguide with a single stub or serial stubs structure. For instance, Liu et al. studied the plasmonic reflectors based on serial stub structure with the developed transmission line model.^[16] For the microcavity structure, Wang et al. studied the flat-top reflection characteristics based on a metal–dielectric–metal (MDM) plasmonic waveguide side coupled with cascaded double cavities.^[19] All types of structural filters mentioned above are fabricated the microstructure in or near MIM waveguides. Although those filters have achieved a variety of excellent filtering characteristics, how to fabricate and apply them is still a challenge.

In this study, a new type and easy-to-fabricate MIM waveguide SPP reflector based on Sagnac loop is designed and investigated. The transfer matrix model, used for describing the relationship between the input and output optical fields of each port in the SPP reflector, is established for the first time to our knowledge. The reflection and transmission properties of the reflector are studied and analyzed by combining the transfer matrix model with finite element method (FEM) simulation. Finally, based on the theoretical and simulation results, an optimal SPP reflector is designed to achieve high reflectivity, high isolation, and a large bandwidth simultaneously. This can provide some reference for the research and development of SPP devices.

Figure 1 shows the structure of the MIM waveguide SPP reflector. The metal used is silver and the intermediate insulator layer is poly (methyl methacrylate) (PMMA). The relative permittivities of the silver and PMMA are and , respectively. The width of waveguide is W. The SPPs are coupled to the reflector from port 1 as . Then SPPs enter the coupling region through position 3 after propagating for a distance of 0.8 m. In the coupling region, t is the separation between the two waveguides and L is the length of the coupling region. The SPPs are split into two beams after passing through the coupling region and they are output from positions 5 and 6. Then, the two beams propagate along a loop with length l and return to the coupling region from positions 6 and 5. Finally, the two beams interfere with each other in the coupling region and are output from port 1 as and port 2 as . According to the principle of the transfer matrix theory,^[20] the theoretical model for the transmission of electric fields in the MIM waveguide reflector can be expressed as follows:

Fig. 1.

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	Fig. 1. Structure of the MIM waveguide SPP reflector.

It can be seen from the above theory that the reflectivity of our device is related to the coupling splitting ratio K and the loss of the coupling region. However, the analytical solution for K and is more complicated. Therefore, numerical simulation and analysis combined with finite element method (FEM) are applied to investigate the properties of the coupling region.^[21]

When the SPPs are incident from port 1, the variation in the transmissivity of positions 5 and 6 with the coupling length L for different wavelengths of , , and is shown in Fig. 2(a), assuming t = 20 nm and W = 50 nm. As shown in the figure, transmission characteristics exhibit the same trend for different wavelengths. The transmissivity of position 5 tends to decrease initially and then increase periodically as L increases. However, the opposite trend is observed for position 6. Taking the wavelength of as an example, initially L is 100 nm, which is relatively short, hence most of the power from the SPP input from port 1 is directly output via position 5, and the transmissivity is as high as 88%. Only a small amount of power is coupled to the other branch and output from port 4, and the transmissivity in this case is 4%. The distribution of the absolute value of the electric field ( ) in the waveguide coupler is shown in Fig. 2(b). As is shown, the SPPs are mostly output from position 5. When L = 750 nm, the transmissivities of positions 5 and 6 are equal, and the corresponding electric field distribution is shown in Fig. 2(c), which indicates that a 50:50 splitting ratio is realized at this coupling length. For , the electric field distribution is shown in Fig. 2(d), and the transmissivity of position 6 reaches a maximum of 84%, while that of position 5 is 0.5%. However, the minimum transmissivity of position 5 does not occur at , but at , which is 0.1%. The different L values corresponding to the extreme values of transmissivity for each port are attributed to the additional phase introduced by the imaginary part of the dielectric constant of the metal.^[21] Owing to the transmission loss of SPPs, the transmissivity of the two ports is gradually reduced as the length of the coupling region increases. In order to reduce the loss, the required splitting ratio K is generally selected in the first period. The coupling splitting ratio can be calculated based on the results shown in Fig. 2. In addition, can be deduced from the relationship between amplitude reduction and length increase. Although the transmission characteristics exhibit the same trend, the coupling strengths for different wavelengths are different, resulting in transmissivities of position 5 and position 6 at the same coupling length. For instance, the 50:50 splitting ratio is realized at the short coupling length of 400 nm for the wavelengths of , however, for the wavelengths of , the coupling length increases to to achieve the same splitting ratio. It can be seen that the SPPs with short wavelength can achieve strong coupling at a short coupling length. Moreover, the properties of the coupling region are affected by the waveguide separation distance and the smaller the value of t, the stronger the coupling effect.^[21,22] The theoretical analysis of the MIM waveguide reflector can be realized by applying K and to the transfer matrix model, as described in formula (1).

Fig. 2.

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	Fig. 2. (a) Variation of the transmissivity of positions 5 and 6 over the coupling length L when t = 20 nm for the wavelengths of , , and ; Distribution of the absolute value of the electric field ( ) at each port under the coupling length of (b) 100 nm, (c) 750 nm, and (d) for .

Without loss of generality, the wavelength of and is chosen and the reflection and transmission characteristics of the device are investigated. The variation curves of reflectivity R of port 1 and transmissivity T of port 2 in the MIM reflector over the coupling length L calculated using both the transfer matrix model and FEM are shown in Fig. 3(a).

Fig. 3.

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	Fig. 3. (a) Variation of reflectivity of port 1 and transmissivity of port 2 over the coupling length L when the wavelength is and (points for the transfer matrix model results; lines for the FEM results ); (b) Distribution of the absolute value of the electric field in the reflector under L of 300 nm and 750 nm when ; (c) Variation of reflectivity of port 1 and transmissivity of port 2 for and .

As shown in the figure, both R and T tend to vary periodically and alternately as L increases. When L = 300 nm, the splitting ratio of ports 3 and 4 is 77:13 (shown in Fig. 2(a)), and the reflectivity equals transmissivity, indicating that the reflection with 50% is achieved at this coupling length. The left image in Fig. 3(b) shows the distribution of a normalized electric field of the reflector. The light and dark fringes of the electric field of SPPs are observed in the device, indicating that the electric fields in both directions interfere. And for L = 750 nm, R reaches a maximum of 78% when T is approximately 0.4% as shown in Fig. 3(a), corresponding to an extinction ratio of 23 dB. At this length, the coupling region realizes a 50:50 splitting ratio. The right image in Fig. 3(b) shows the distribution of a normalized electric field. As is shown in the figure, there is a very obvious interference phenomenon in the device, and the light input to port 1 is completely output from port 1 after propagating through the device, which demonstrates its function of total reflection. It can be seen that the reflection and transmission characteristics of the device strongly depend on the coupling splitting ratio, while the splitting ratio in turn depends on the coupling length. Reflectivity can be varied by changing the coupling length. When the splitting ratio is 50:50, the strongest interference condition is observed, and the total reflection can be achieved. In addition, figure 3(a) shows that the results obtained from the transfer matrix model and FEM simulation agree well in spite of small discrepancies caused by the reflection of the boundary condition in the FEM simulation. This confirms the accuracy of our proposed theoretical model. In addition, the length of the loop also affects the reflection and transmission characteristics of the device. Figure 3(c) shows the variation of reflectivity in the port 1 and transmissivity of port 2 for and . It can be seen from the figure that the change in length l does not affect the overall trends of R and T, but affects the amplitudes of R and T due to different losses. The short length helps reduce the loss, and the transmittance and reflectivity are improved. However, the length cannot be reduced directly because the bending loss increases when the length reduces to a certain level, therefore the length of the loop must be optimized.

The simulation results of the reflectivity of the device for different wavelengths of the incident light when L is 400 nm, 750 nm, and are shown in Fig. 4. It can be seen that the reflectivity initially increases and then decreases as the wavelength increases for all coupling lengths. However, the peak position and bandwidth of the reflectivity are different for different coupling lengths. When L = 400 nm, because a 50:50 splitting ratio is realized when the wavelength is (shown in Fig. 2(a)), the reflectivity reaches its maximum at , and the 3-dB reflection bandwidth is . And for L = 750 nm and , the peak positions shifts to and , and the reflection bandwidth has been extended to and , respectively. Therefore, it is necessary to design the length of the coupling region according to the interesting wavelength, and the shorter wavelength has a narrower bandwidth.

Fig. 4.

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	Fig. 4. Simulation results of reflectivity under different wavelengths of incident light when L = 400 nm, 750 nm, and .

A new type and easy-to-fabricate metal–insulator–metal waveguide reflector based on Sagnac loop was designed and investigated. The transfer matrix theoretical model for the transmission of electric fields in the reflector was established for the first time and the properties of the reflector were studied and analyzed. The simulation results indicate that the reflectivity of the reflector strongly depends on the coupling splitting ratio determined by the coupling length. For an optimum coupling length of 750 nm when the wavelength is , the reflectivity and isolation were 78% and 23 dB, respectively. Moreover, the peak position and bandwidth of the reflection spectrum also varied with the coupling length. For a larger coupling length becomes longer, the peak position became to longer wavelengths and the bandwidth became wider.