Why is it that RF waves travel faster than c/√(εμ) in a coplanar waveguide (CPW) electrode?

In electro-optic modulators, one important task is matching the propagating RF and optical wave velocities. This begins a discussion on modulator electrode design.

The primary method of matching the RF and optical velocities is using a slow wave electrode, that is, a capacitively loaded electrode.

Before capacitive loading, we need to determine the initial velocity of RF waves travelling in the coplanar waveguide (CPW). The CPW electrode is as follows. The optical waveguides are positioned between the signal and ground electrodes.

Let’s think about the formula for wave propagation velocity for RF waves:

V_RF = c/n,

where c is the speed of light (3×10^8 m/s) and n is the microwave index. n is also equal to:

n = √(εμ),

where ε and μ are the relative permittivity and permeability of the medium that the RF waves are propagating in.

We might think, if we know what material the modulator is made of, then we can calculate the microwave index based on the relative permittivity and permeability of the material, and calculate it. Not so fast…

As shown in the diagram above, the propagating electromagnetic wave’s mode is not confined within the substrate. For this reason, we must determine a weighted average of the material properties based on the electromagnetic waves’ mediums, namely air and the substrate. I use Ansys HFSS to perform this calculation easily and accurately.

In summary, the propagating electromagnetic waves along the coplanar waveguide electrodes are present both in the semiconductor substrate and in air surrounding the device. The velocity of the propagating electromagnetic wave is therefore a weighted average of the electric field propagation in air and the semiconductor. Since the index of air is less than the semiconductor, the field propagates faster than if it were entirely propagating in semiconductor.