Beating beats mixing in heterodyne detection schemes

Abstract

Heterodyne detection schemes are widely used to detect and analyse high-frequency signals, which are unmeasurable with conventional techniques. It is the general conception that the heterodyne signal is generated only by mixing and that beating can be fully neglected, as it is a linear effect that, therefore, cannot produce a heterodyne signal. Deriving a general analytical theory, we show, in contrast, that both beating and mixing are crucial to explain the heterodyne signal generation. Beating even dominates the heterodyne signal, if the nonlinearity of the mixing element (mixer) is of higher order than quadratic. The specific characteristic of the mixer determines its sensitivity for beating. We confirm our results with both a full numerical simulation and an experiment using heterodyne force microscopy, which represents a model system with a highly non-quadratic mixer. As quadratic mixers are the exception, many results of previously reported heterodyne measurements may need to be reconsidered.

Interference and polarization beating of independent arbitrarily polarized polychromatic optical waves

“…This phenomenon is a result of interference of optical waves with the same polarizations, but different frequencies. It is applied, e.g., in heterodyne-type detection and sensing [4,21], optical velocimetry [22], optical frequency stabilization [23,24], and laser mode locking [6].…”

Subsurface contrast due to friction in heterodyne force microscopy

“…Considering the amplitude contrast ∆A dif f , the absolute values in the experiment are up to 100 times larger than we calculated the tip-sample interaction and, as a function of the applied contact force, the corresponding sample indentation as well as the amplitude A dif f and phase φ dif f of the heterodyne signal for different sample elasticities: 2 GPa (black), 3 GPa (red), 4 GPa (magenta), 5 GPa (green), and 6 GPa (blue). The inset in the lower left panel shows A dif f for 6 GPa plotted as a function of the height of the cantilever's base, z b , such that a comparison becomes possible with other calculations [21,22,26]. I: Comparison between experimentally determined and analytically predicted values: The obtained contrasts in the height, the amplitude A dif f , the normalized amplitude Ac (for which we also provide the background amplitude A b ), and the phase φ dif f for a contact force of 163 nN and 115 nN.…”

“…Figure 2 shows the excitation scheme and the vibration spectrum of the free hanging cantilever, of which we calibrated the spring constant to be 2.7 N/m using the thermal noise method [33]. Using the method described in [22,26], we determined the ultrasonic tip amplitude to be A t = 1.34 nm and the ultrasonic sample amplitude to be A s = 0.37 nm.…”

A subsurface add-on for standard atomic force microscopes

“…In comparison to the original holder, this breaks down to an improvement of a factor 4 in frequency range, which delivered an increase in subsurface resolution as well as in contrast with a factor of 16. 22 This allowed us to gather significant insight in the general technical aspects of HFMs [26][27][28] as well as in the determination of the physical mechanism that is responsible for the contrast formation! 29 By applying a similar piezo element mounted underneath the sample, we were also able to transmit sufficiently strong ultrasound waves into the sample such that we easily could measure the ultrasonic motion of ∼3 Å at the sample surface.…”

“…UFM and Waveguide-UFM work with a low frequency amplitude modulation, whereas HFM works with a nonlinear coupling between the two ultrasonic sound waves to generate a low frequency excitation at their difference frequency that is explicitly tuned on (or below) the first resonance of the cantilever. [26][27][28] It is the nonlinear interaction between the cantilever's tip and the sample that generates the low frequency signal in the cantilever's motion in all three ultrasound AFM techniques.…”