Employing applied mathematics to expand the bandwidth of heterodyne carrier signals with a small phase modulation index

Frequency modulation (fm) theory states that heterodyne carrier signals cannot be frequency modulated with modulation frequencies fm higher than the carrier frequency fc subtracted by the maximum frequency deviation fd,max. Thus, fm ⩽ fc − fd,max is considered as maximum limit to obtain the correct modulation signal from the carrier signal by fm demodulation. This paper proves mathematically that this limit can be breached for small phase modulation indices. The result is advanced to a new algorithm to demodulate heterodyne carrier signals with a modulation frequency of up to twice the carrier frequency if a small phase modulation index M can be assumed. This assumption is valid in heterodyne laser interferometers for ultrasonic testing where the phase modulation of a detector signal is originated by vibration amplitudes much smaller than the wavelength of the laser. Simulations of the demodulation demonstrate the functioning of the algorithm. The employment of the presented results in an interferometric system demonstrates the impact in metrology instruments for vibration measurements at ultra-high frequencies. Therefore, the influence of the presented algorithm to the measurement uncertainty of interferometric systems is also derived in this paper.