# 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.

# Calculating Bandwidth for RF/Photonic Components based on Velocity mismatch

The bandwidth of a device such as a modulator or photodetector is an important figure. When designing a modulator or photodetector for high frequencies, much attention is paid to matching the velocity of the optical waves and the RF waves.

By finding the propagation time difference between the optical and RF waves, we model this in the time domain as a rect function. Note that for the rect function, the difference in propagation time is the tau variable. Performing the Fourier transform on the rect function will give us a sinc function. The 3dB cutoff point of this sinc function in the frequency domain gives us the device bandwidth. Note the MATLAB algorithm used below. The 3dB bandwidth is calculated using a simple manipulation of the frequency vector indices.

v_optical = ; %simulated optical velocity [define]

v_RF = ; %simulated RF velocity [define]

l_device = ; %device length [define]

f_max = ; %max frequency of vector (should be higher than bandwidth) [define]

f_num = ; %number of frequencies in vector [define]

tau = abs((l_device/v_optical)-(l_device/v_RF)) ; %propagation time difference

W = linspace(0,f_max,f_num); %frequency vector

S = tau*sinc(W*tau/2); %sinc function in frequency domain

Qs = find(20*log10(S)<=(20*log10(S(1))-3)); %intermediate calculation for index of 3db cutoff

BW_3dB= f_max*(Qs(1))/f_num %This is the result