Reconstructing Nonlinearities with Intermodulation Spectroscopy

Abstract

We describe a method of analysis which allows for reconstructing the nonlinear disturbance of a high Q harmonic oscillator. When the oscillator is driven with two or more frequencies, the nonlinearity causes intermodulation of the drives, resulting in a complicated spectral response. Analysis of this spectrum allows one to approximate the nonlinearity. The method, which is generally applicable to measurements based on resonant detection, increases the information content of the measurement without requiring a large detection bandwidth, and optimally uses the enhanced sensitivity near resonance to extract information and minimize error due to detector noise.

Nonlinear motion and mechanical mixing in as-grown GaAs nanowires

“…(1) by taking the cubic term to be a perturbation to the driving force and solving the equation iteratively. 21,30 One then obtains new terms in the response at the intermodulation frequencies f 1 À nDf and f 2 þ nDf (where n is an integer) for each iteration. The amplitudes of these new intermodulation terms have coefficients proportional to P n ðf…”

Noninvasive Protein Structural Flexibility Mapping by Bimodal Dynamic Force Microscopy

“…Atomic force microscopy (AFM) [1] has enabled molecular resolution images of packed arrays of biomolecules [2-5], sub-2-nm images of individual biomolecules [6], and studying biomolecular interactions in liquid [7]. Single force spectroscopy measurements are already an established tool to measure intermolecular and intrabiomolecule forces [8]; however, those measurements are not compatible with high resolution imaging.Recently, several multifrequency AFM schemes have been proposed to improve high resolution imaging, contrast, and quantitative mapping of material properties [9][10][11][12][13][14][15][16][17][18][19][20][21]. Generically, those schemes exploit the nonlinear character of the tip-surface forces to either activate or detect higher eigenmodes or harmonics and to open new channels to improve imaging and composition sensitivity.…”

“…Recently, several multifrequency AFM schemes have been proposed to improve high resolution imaging, contrast, and quantitative mapping of material properties [9][10][11][12][13][14][15][16][17][18][19][20][21]. Generically, those schemes exploit the nonlinear character of the tip-surface forces to either activate or detect higher eigenmodes or harmonics and to open new channels to improve imaging and composition sensitivity.…”

Effect of Material Stiffness on Intermodulation Response in Dynamic Atomic Force Microscopy

“…All intermodulation products appear at frequencies which are integer multiples of ∆Ω, so the response can be expressed as a Fourier series in ∆Ω [2]. In the simulations one drive was placed slightly below resonance, f 1 = 276.5 kHz and the other drive on resonance f 2 = 277 kHz, so that ∆Ω/2π = f 2 − f 1 = 500 Hz.…”

“…Thus, in contrast to traditional dynamic AFM, where response amplitude and phase are measured only at the drive frequency, intermodulation AFM acquires many (typically of order 30) amplitude and phase quantities, which together contain information about the nonlinear tip-surface force that created the intermodulation response. By analysis of the intermodulation spectrum one can, in principal, reconstruct a polynomial approximation to a conservative tip-surface force F ts (z) as a function of the cantilever displacement z [2]. Reconstruction methods which include the possibility of non-conservative tip-surface forces are under development.…”