# Flexible Thermo-Optic Variable Attenuator based on Long-Range Surface Plasmon-Polariton Waveguides
## 2 Device Design and Simulation  
The schematic diagram of the VOA based on LRSPP waveguide is shown in Figure 1. The bottom left inset is the cross-section of the device. It consists of a gold (Au) layer, lower polymer cladding,thin metal strip core, upper polymer cladding and a heater aligned on top of the upper cladding. The thicknesses of the upper and lower polymer cladding layers are both 15 \(\upmu\)m. The silver (Ag) strip core of the LRSPP waveguide is carefully designed with an optimized parameter and the width and thickness is 4 \(\upmu\)m and 12 nm, respectively. The heater is made of aluminum (Al). Its cross-section dimension is 10 \(\upmu\)m wide and 150 nm thick and the length is 10 mm. The refractive indices of the polymer claddings are both 1.45 and that of the Ag strip core is 0.15 + 11.38i at 1550 nm. The calculated optical field distribution by the finite element method (FEM) is shown in the top right inset of Figure 1. It indicates that the optical field is symmetrically distributed in the interface between the claddings and the strip core, and most of the energy is distributed in the claddings next to the interface, which ensures a long-range propagation mode.  
The principle of the VOA is based on changing the radiation loss of the LRSPP waveguide resulting from the refractive index difference between the upper and lower claddings. When a voltage is applied to the heater, thermal energy being produced by resistive heating is transferred to the upper cladding, Ag strip, and lower cladding in turn. Then, a temperature gradient is established by the thermal energy and it increases with the applied voltage, as shown in Figure 2. The temperature gradient subsequently results in the refractive index difference between the upper and lower cladding due to the thermo-optic effect. Finally, the asymmetry of refractive indices results in the radiation loss of the LRSPP mode. Therefore, the optical output power of the device can be controlled by changing the temperature gradient by applying different voltages to the heater. The thermo-optic coefficient of the polymer is \(-1.758\times 10^{-4}/^{\circ}\)C. The negative thermo-optic coefficient results in the refractive index decrease with the increase in temperature.  
Figure 1: The schematic diagram of the variable optical attenuator (VOA). The bottom left inset is the cross-section and the top right inset is the optical field distribution of the long-range surface plasmon-polariton (LRSPP) waveguide.  
Figure 2: The calculated thermal distribution with applied voltage of (**a**) 1 V, (**b**) 2 V, (**c**) 3 V, (**d**) 4 V to the heater. (**e**) Temperature gradient in the depth of the device at different applied voltage.  
According to the dimension of the heater, we can calculate resistance of the heater. By using FEM, we calculate the thermal distribution in the device while applying different voltages to the heater, as shown in Figure 2a-d. The corresponding temperature distribution is shown in Figure 2e. It shows a clear asymmetry distribution of the temperature in the upper and lower claddings when the voltage is applied. Temperature gradient also increases rapidly with the increasing applied voltage (see Figure 2e), which will result in a larger refractive index gradient and eventually enlarge the radiation loss of the LRSPP mode. Therefore, the propagation loss of the device can be adjusted by changing the applied voltage to the heater.