Lock-in amplifier

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Lock-in Amplifier is a technique for extracting a signal from noise ... and/or a measuring instrument that exploits this technique. It is useful in situations where the signal of interest has a relatively narrow bandwidth, and the noise level depends significantly on frequency; i.e. the noise is not white noise. A common example of such a situation is a low-frequency signal in the presence of flicker noise.

In the uncommon situation where the signal fortuitously sits in a low-noise region of the noise spectrum, it might suffice to use a narrowband detector. Much more commonly, the signal (in its natural form) sits in a high-noise region of the noise spectrum. A key part of the lock-in technique is to modulate the signal, shifting it to a part of the spectrum where there is significantly less noise. The shifted signal can be measured using a narrowband detector. Furthermore, if desired, the shifted signal can be measured using a synchronous detector, aka phase sensitive detector, where the detector is phase-locked to the modulation applied to the signal.

Famed Princeton University physicist Robert H. Dicke invented the technique, and also founded the company Princeton Applied Research (PAR) to market a line of general-purpose lock-in amplifier instruments.

[edit] Application to signal measurements in noisy environment

Let us suppose that we want to measure the (linear) response of a system to a stimulus. Unfortunately, the environment is noisy, so we observe not just the response to our stimulus but also additive non-white noise. We choose the stimulus to be a periodic function of time with frequency f. The response to our excitation will also be oscillatory, with the same frequency f. We choose f to be in a relatively low-noise region of the noise spectrum.

To recover the signal, it sometimes suffices to use a band-pass filter at the output of the system, centered at frequency f. More commonly it is advantageous to use a synchronous detector. A simple way (certainly not the only way) to implement a synchronous detector is to multiply the received signal by a reference signal at the same frequency f, and then pass the output of the multiplier through a narrow baseband filter. [recall that multiplying a signal by exp(iω₀t) in the time domain shifts its (Fourier) transform by ω₀ in the frequency domain.]

For example, in order for an Atomic force microscope to achieve nanometer and piconewton resolution, the cantilever position is modulated at a high frequency and the signal is detected synchronously. Without lock-in techniques, the signal would likely be swamped by low-frequency noise.

Lock-in techniques can also be applied to nonlinear systems. In such a case, if the applied stimulus is at frequency f, it may be advantageous to tune the detector to some harmonic of f, perhaps 2f or 3f.

[edit] External links

• Explanation of lock in amplifiers from Boston Electronics, which sells lock-in amplifiers. Other significant lock-in vendors include Scitec Instruments, SRS and Ametek, which is the direct successor to PAR.
• Online Lock-In Amplifier Simulation Applets - a series of online virtual 'experiments' in which the well-understood underlying mathematics is used to model and predict the behavior of actual Lock-in amplifiers, which is less well-understood. The introductory pages provide a reasonably complete overview and demonstration of basic behavior. The full series of experiments take a recommended 6-8 hours to complete.

[edit] Articles about Lock-in Amplifiers

• A. Restelli, R. Abbiati, and A. Geraci (2005). "Digital field programmable gate array-based lock-in amplifier for high-performance photon counting applications". Review of Scientific Instruments 76: 093112. DOI:10.1063/1.2008991.

• Maximiliano Osvaldo Sonnaillon and Fabián Jose Bonetto (2005). "A low-cost, high-performance, digital signal processor-based lock-in amplifier capable of measuring multiple frequency sweeps simultaneously". Review of Scientific Instruments 76: 024703. DOI:10.1063/1.1854196.

• Libbrecht KG, Black ED, Hirata CM (Nov 2003). "A basic lock-in amplifier experiment for the undergraduate laboratory". American Journal of Physics 71 (11): 1208-1213. DOI:10.1119/1.1579497.

• Barragán, L. A. Artigas, J. I. Alonso, R. Villuendas, F. (Jan 2001). "A modular, low-cost, digital signal processor-based lock-in card for measuring optical attenuation". Review of Scientific Instruments 72 (1): 247. DOI:10.1063/1.1333046.

• John H. Scofield (Feb 1994). "Frequency-domain description of a lock-in amplifier". American Journal of Physics 62 (2): 129-133. DOI:10.1119/1.17629.

• Probst, Pierre-Alain Jaquier, Alain (Mar 1994). "Multiple-channel digital lock-in amplifier with PPM resolution". Review of Scientific Instruments 65 (3): 747. DOI:10.1063/1.1145096.

• Wang, Xiaoyi (1990). "Sensitive digital lock-in amplifier using a personal computer". Review of Scientific Instruments 61 (70): 1999. DOI:10.1063/1.1141413.

• Richard Wolfson (Jun 1991). "The lock-in amplifier: A student experiment". American Journal of Physics 59 (6): 569-572. DOI:10.1119/1.16824.

• Dixon, Paul K. Wu, Lei (Oct 1989). "Broadband digital lock-in amplifier techniques". Review of Scientific Instruments 60 (10): 3329. DOI:10.1063/1.1140523.

• Vanexter M, Lagendijk A (Mar 1986). "Converting an AM radio into a high-frequency lock-in amplifier in a stimulated Raman experiment". Review of Scientific Instruments 57 (3): 390. DOI:10.1063/1.1138952.

• Probst, P. A. Collet, B (Mar 1985). "Low-frequency digital lock-in amplifier". Review of Scientific Instruments 56 (3): 466. DOI:10.1063/1.1138324.

• Paul A. Temple (1975). "An introduction to phase-sensitive amplifiers: An inexpensive student instrument". American Journal of Physics 43 (9): 801-807. DOI:10.1119/1.9690.