In this example we calculate the loss and resonant frequency of a waveguide coupled surface plasmon mode as described in Lavers and Wilkinson, "A waveguide-coupled surface-plasmon sensor for an aqueous environment", Sensors and Actuators, B 22, 75-81, (1994).
The file sp_waveguide.lms is setup as described in Lavers and Wilkinson.
The Silver material properties are defined with the silver material "Ag (silver) - Johnson and Christy" included with MODE, rather than the values specified in Lavers and Wilkinson. This could account for some of the differences between these simulations and the results given in the paper.
Select the Analysis - Modal Analysis tab. Set the wavelength to 500nm. Click the Calculate Modes button to search for propagating modes in this structure. Three modes will be found. The first, with an effective index of approximately 1.57, is the surface plasmon mode. We can tell that this is the surface plasmon mode because it has a high loss and most of the field intensity is near the silver layer. The second and third modes, with effective indices close to 1.52, are the waveguide modes. These modes have much lower loss and are primarily confined to the waveguide layer. The following figure show the mode profiles.
The figure on the left shows |E|^2 as a function of y while the right figure shows |Ey|^2 as a function of y. Notice that the fields in Mode 1 and Mode 3 are TM polarized (Ey and Ez only) while Mode 2 is TE (Ex only). This is important because it means the surface plasmon mode (Mode 1) can only couple to one of the waveguide modes (Mode 3).
Effective index vs wavelength
Next, we can calculate the effective index of each mode vs wavelength. Select the Analysis - Frequency analysis tab. Set the Stop wavelength to 600nm. Uncheck the "track selected mode" option, then perform the frequency sweep. The following figure on the left will be created. The figure on the right shows a close up of the crossing region. Note, before doing the frequency sweep, make sure that the material fitting is continuous and reasonably accurate.
The line with the steep slope is the surface plasmon mode. At approximately 547nm, the effective index of the surface plasmon mode becomes equal to the effective index of the waveguide modes. At this point, the TM modes will have the highest coupling efficiency. We expect the TM waveguide mode to suffer from high loss at this wavelength because energy from the waveguide mode will couple into the high loss surface plasmon mode.
It is also interesting to notice the drastic change in slope of the surface plasmon mode at 555nm. At this point, the effective index is 1.512; the value of the substrate index. The mode is no longer confined to the metal layer and begins radiating power into the substrate.
Due to the implementation of the frequency sweep, color coding of the modes becomes mixed up at crossing points. This problem usually does not occur with the "track selected mode" option, as we will see in the loss section below. However, the "track selected mode" option only allows for one mode to be tracked at a time. Therefore three frequency sweeps would be required.
Loss vs wavelength
Still in the Analysis - Frequency analysis tab, select Set calculation parameters from the options menu. Select the TM waveguide mode (mode 3). Select "track selected mode", then start the frequency sweep. Once the sweep is finished, select "Loss" from the list of Plot options below the figure. The following figure will be displayed.
As expected, the mode has much higher loss at approximately 550nm because it can couple to the surface plasmon mode. This result matches Figure 3 from Lavers and Wilkinson.
C.R. Lavers and J.S. Wilkinson, "A waveguide-coupled surface-plasmon sensor for an aqueous environment", Sensors and Actuators, B 22, 75-81, (1994)