As mentioned in all the other posts, you shine two lights of different wavelengths, measure the difference in absorption of the lights, and then apply some simple maths to find the ratio of the absorption.
I just want to add that this is then compared to a graph of absorption vs oxygen saturation to spit out the number.
This graph was initially obtained by healthy individuals breathing a low O2 mixture of gas, having a pulse oximeter connected, and then samples are taken to determine the saturation at various absorption ratios. This was done down to a SpO2 of 70% and further extrapolated down to 50%
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