Employing applied mathematics to expand the bandwidth of heterodyne carrier signals with a small phase modulation index Christian Rembe Polytec GmbH, Research and Development, Polytec Platz 1-7, 76337 Waldbronn, Germany article info Keywords: Frequency demodulation Phase demodulation Unitary operator Heterodyne carrier signal Heterodyne interferometry Laser-Doppler vibrometry Ultrasonic testing abstract Frequency modulation (fm) theory states that heterodyne carrier signals cannot be fre- quency modulated with modulation frequencies f m higher than the carrier frequency f c subtracted by the maximum frequency deviation f d,max . Thus, f m 6 f c  f d,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 het- erodyne 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 hetero- dyne 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. Ó 2010 Elsevier Inc. All rights reserved. 1. Introduction Heterodyne interferometers can measure accurately broad-bandwidth vibrations without contact and with high resolu- tion. Nowadays heterodyne interferometers have a photo-detector signal which is only limited by the quantum nature of the returning measurement light (shot-noise limited detection). Recently a novel optical sensor was presented to measure vibra- tion frequencies up to 1.2 GHz [1]. First measurements with the sensor showed that the technique employed in this sensor makes it possible to obtain accurate vibration amplitudes for the first time [2] and measurements above 1 GHz have been presented [3]. The optical sensor has been developed in an industrial environment and, therefore, the core ideas for the sys- tem together with its mathematical procedures have been patented [4,5]. So far, heterodyne interferometers were not capa- ble to measure at frequencies higher as the carrier frequency. This paper high-lights the mathematical procedure to expand the demodulation bandwidth of the heterodyne carrier by a factor of two if only small vibration amplitudes are detected what is typically the case for frequencies of 10 MHz or more. The core idea of the procedure is as follows: the components of the spectrum of the phase-modulated carrier at frequencies lower than the carrier frequency have a well-defined phase relation to the components at the higher frequencies if small modula- tion indices can be assumed. It is demonstrated for the case of a digitized carrier signal that the lower frequency components can be computed with an unitary operation from the higher frequency components by taking the phase of the carrier into account. Therefore, an algorithm is developed which needs only the carrier and the components higher than the carrier 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.04.006 E-mail address: chrembe@aol.com Applied Mathematics and Computation 217 (2010) 1202–1212 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc