Impedance generalization for plasmonic waveguides beyond the lumped circuit model

We analytically derive a rigorous expression for the relative impedance ratio between two photonic structures based on their electromagnetic interaction. Our approach generalizes the physical meaning of the impedance to a measure for the reciprocity-based overlap of eigenmodes. The consistency with known cases in the radio-frequency and optical domain is shown. The analysis reveals where the applicability of simple circuit parameters ends and how the impedance can be interpreted beyond this point. We illustrate our approach by successfully describing a Bragg reflector that terminates an insulator-metal-insulator plasmonic waveguide in the near infrared by our impedance concept.

Figure 1

(a) Sketch of the test geometry considered in this paper. The fundamental mode of a plasmonic insulator-metal-insulator waveguide is reflected by a Bragg reflector. (b) The situation is characterized in terms of the relative impedance between the two structures as in an equivalent circuit model.

Figure 2

Reflection of the fundamental mode of a section with ${\epsilon }_f=1$ in a rectangular microwave waveguide ($a=1$ mm, $b=0.5$ mm) at a section of different dielectric filling ${\epsilon }_f\ne 1$. Lines represent the rigorous result obtained by comsol Multiphysics whereas circles show the results using the impedance framework. The inset shows the geometry and the fundamental mode profile.

Figure 3

Rigorous results (lines) obtained by the aperiodic Fourier modal method and results of the impedance framework (crosses) for the problem sketched in Fig. 1 for different filling factors of the Bragg reflector. The free space wavelength was ${\lambda }_0=1550$ nm and the corrugation depth 50$\%$. A very good agreement is obtained.

Figure 4

Magnitude and phase of the relative impedance for the results of Fig. 3. Depending on the filling factor (FF), the reflection peak has its origin either in a strong mismatch in the former or the latter.

Figure 5

Map of the reflection amplitude for different filling factors and periods calculated by the impedance framework. The optimal parameters were found to be a 33$\%$ filling factor at a period of 528 nm.