Phase-sensitive detection of signals with non- sinusoidal modulation: The rectangular wave modulation case and its application to the photoacoustic technique

The phase-sensitive detection is a powerful mathematical tool in the recovering of signals, at low noise/signal ratio, in many areas of science and engineering. Specifically in Photothermal Science and Techniques (in all its variants) is essential the implementation of a stage of amplification and filtering (for example, by means of a lock-in amplifier) of the obtained photothermal signal from a measurement system, before to analyze the experimental data. Due to the general preference of sinusoidal modulation in the modelling of the generation of the photothermal signal, it is a common practice to neglect the information contained in superior harmonics, since sinusoidal functions are used as reference signals in the phase-sensitive detection stage. However, because the modulation of the optical excitation is often produced by means of a mechanical modulator (or directly by electronic control of a shutter integrated in the laser source, using a TTL signal) the photothermal signal is in fact modulated by a rectangular wave train, and therefore the neglected harmonics could be relevant, depending on the thermal regime in which the measurement is performed, and the thermal relaxation processes involved. In this paper, the phase-sensitive detection is applied to a square-integrable signal with non-sinusoidal modulation, analyzing the influence of the harmonics in the amplified output signal in frequency domain, and it is demonstrate that considering reference functions congruent to the modulations of the photothermal signal, the contribution of the superior harmonics do not vanishes.