The Intermodulation Lockin Analyzer

Abstract: Nonlinear systems can be probed by driving them with two or more pure tones while measuring the intermodulation products of the drive tones in the response. We describe a digital lock-in analyzer which is designed explicitly for this purpose. The analyzer is implemented on a field-programmable gate array, providing speed in analysis, real-time feedback and stability in operation. The use of the analyzer is demonstrated for Intermodulation Atomic Force Microscopy. A generalization of the intermodulation spectral technique to arbitrary drive waveforms is discussed.

• The paper introduces an FPGA-based analyzer that performs rapid, parallel lock-in analysis of intermodulation products in multi-tone nonlinear systems.
• It employs digital synchronization and advanced signal processing to mitigate Fourier leakage and achieve sub-noise-floor detection.
• The analyzer demonstrates high-resolution applications in AFM for mapping nonlinear tip-sample interactions, with potential extensions to RF and photonic systems.

The Intermodulation Lockin Analyzer: Architecture, Capabilities, and Applications

Introduction

The paper "The Intermodulation Lockin Analyzer" (1008.2722) presents the design, implementation, and application of a digital instrument specifically devised for real-time measurement of intermodulation products in nonlinear two-port systems under multi-tone excitation. Emphasizing both the hardware and signal processing aspects, the work details the deployment of a Field-Programmable Gate Array (FPGA) platform, enabling rapid, parallelized lock-in analysis across multiple frequencies. The system's utility is manifested through its application to Intermodulation Atomic Force Microscopy (ImAFM), where it serves as a crucial diagnostic and analytic component for resolving nonlinear tip-surface interactions at the nanoscale.

Intermodulation Analysis and Lock-in Techniques

Traditional lock-in amplifiers excel in isolating linear responses at a drive frequency, but their utility diminishes in the investigation of nonlinearities generating frequency mixing beyond simple harmonics. The presented ImLA extends lock-in techniques to the efficient simultaneous quantification of both harmonics and intermodulation products from multiple drive tones. This is achieved through a digitally synchronized reference and sampled measurement protocol, mitigating Fourier leakage and enabling sub-noise-floor detection. The intermodulation frequencies are constructed as integer linear combinations of base drive frequencies, leveraging the digital paradigm to ensure all relevant measured frequencies remain commensurate with the base clock $\Delta\omega$.

The FPGA-based analyzer is architectured for high-throughput: 12-bit A/D input at 61.4 MSa/s, 16-bit D/A output, and parallel calculation of real and imaginary output quadratures at 32 user-selected frequencies. A CORDIC module allows real-time extraction of amplitude and phase for feedback, critical for resonance tracking or high-speed dynamic measurements such as in AFM.

Instrument Architecture and Performance

The implementation emphasizes a fully digital, FPGA-based pipeline with robust clock and triggering architecture. Networked communication via Ethernet ensures broad integration capability with standard laboratory software platforms. The analyzer can operate in both frequency-selective lock-in mode and broadband time-domain streaming (at 3.9 MSa/s), offering versatility in measurement strategies, from efficient real-time feedback to comprehensive post-facto spectral analysis.

Experimental characterization reports an internal Total Harmonic Distortion (THD) and intermodulation distortion (IMD) of less than $-75$ dB under maximum drive amplitude loopback, with a dynamic reserve of 80 dB at 1 kHz measurement bandwidth. These metrics establish a performance floor commensurate with the requirements for high fidelity extraction of nonlinear response, with the limiting factor being the DUT rather than the analyzer's intrinsic artifacts.

Application: Intermodulation Atomic Force Microscopy

A primary application, Intermodulation AFM, leverages the ImLA's fast, multichannel capabilities. By exciting the cantilever at two closely spaced frequencies near the resonance, the ImLA resolves both direct and nonlinear response components. As the AFM tip approaches the surface, nonlinear tip-sample forces manifest as a proliferation of intermodulation products; these are directly measured, providing a compressed and information-rich descriptor of the interaction force landscape at each scanned pixel.

The capacity to extract key amplitude and phase data at intermodulation frequencies, while disregarding out-of-band noise, enables real-time high-resolution mapping of material properties. This approach is particularly advantageous for high-throughput or high-resolution imaging, where computational and storage constraints preclude exhaustive spectral recording.

Generalization to Multi-Frequency and Arbitrary Waveform Excitation

The described analyzer architecture is not restricted to two-tone excitation. The digital design permits arbitrary superpositions of tones at frequencies sampled from integer multiples of the base clock, supporting advanced probing schemes such as chirplets or band-excitation protocols. While the resulting spectrum becomes densely populated with mixing products, theoretical frameworks based on polynomial mappings (such as the X-parameter or poly-harmonic distortion models) can still enable rigorous interpretation of the spectrum and nonlinear device identification. Such flexibility opens new directions for the extraction of device or material physics in complex, multitone-driven nonlinear systems.

Implications and Future Directions

This development shifts the paradigm for experimental nonlinear system identification, compressing the acquisition and analysis process via parallel, synchronized frequency channel monitoring. The capacity for real-time feedback and integration into complex measurement workflows—particularly in the context of scanning probe microscopy and resonant sensors—enables direct access to spectral fingerprints of nonlinear interactions. Beyond SPM applications, the instrument is positioned for use in RF/microwave device characterization, photonic components, and any context where multi-frequency nonlinear responses are of interest.

While the demonstrated application focuses on dynamic AFM, future work could exploit arbitrary waveform excitation for tailored probing of specific nonlinearities, adaptive feedback stabilization in quantum or low-dimensional systems, or synchronized hybrid operation with other timetagged acquisition subsystems. The possibility of direct firmware-level modification provides an open framework for future instrument augmentation and community-driven application development.

Conclusion

The Intermodulation Lockin Analyzer constitutes a significant contribution to the measurement and analysis ecosystem for nonlinear systems, coupling high sensitivity with versatile, multi-frequency capability. By extending digital lock-in methodology beyond harmonics to arbitrary intermodulation spectra, it enables the efficient extraction and interpretation of nonlinear response features, with demonstrated impact in AFM and broad potential across the physical sciences.