Demodulation

SNOM relies heavily on lock-in amplifier demodulation of various \(z_{tip}\) -dependent signals at different harmonics, \(n\), of an AFM tip’s tapping frequency. For this reason snompy features code to simulate lock-in measurements of arbitrary functions. In most cases, you can ignore this, and rely on versions of the snompy functions that have this functionality built-in. For example snompy.fdm.eff_pol_n() is a modulated and demodulated version of snompy.fdm.eff_pol(), so you don’t need to worry about simulating the demodulation in your own code.

This section describes how lock-in measurements are simulated in snompy, and also how you can use the demodulation functionality on arbitrary functions that aren’t included in the package.

Lock-in amplifier simulation

A lock-in amplifier extracts a periodic signal with a known frequency from a noisy background. It works by multiplying the input signal by a sinusoidal reference signal of the desired frequency, then using a low-pass filter to remove unwanted high-frequency components. This effectively suppresses all frequency components except for the component which is at the same frequency and in-phase-with the reference signal. In modern lock-ins, two orthogonal reference signals are usually used in order to measure both the amplitude and phase of the desired signal.

To simulate this, we first need to convert a non-periodic function \(g(x)\) into a periodic function \(f(\theta)\). We apply a modulation as:

\[f(\theta) = g\left(x_0 + A_x \cos(\theta)\right)\]

We can then simulate demodulation of \(f(\theta)\) at different harmonics, \(n\), by multiplying it by a complex sinusoid (\(e^{-i \theta n}\), whose real and imaginary components are orthogonal) then integrating over a single period of the oscillation as:

\[\frac{1}{2 \pi} \int_{-\pi}^{\pi} f(\theta) e^{-i \theta n} d{\theta}\]

In snompy this process is implemented by the function snompy.demodulate.demod().

Adapting the process for AFM

For AFM demodulation we apply a slight variation of the general approach above. To simulate a periodic modulation of \(z_{tip}\) with tapping amplitude \(A_{tip}\), we apply a modulation like:

\[f(\theta) = g\left(z_{tip} + A_{tip} \left(1+\cos(\theta)\right)\right)\]

Note that the cosine term here has been offset so that the oscillation is centred on \(z_{tip}+A_{tip}\), rather than on \(z_{tip}\) itself. This is to ensure that when \(z_{tip}=0\) the lowest part of the oscillation never becomes negative, which would mean intersecting with the sample surface.