Lock-in amplifier

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A lock-in amplifier from Stanford Research Systems.

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One from SIGNAL RECOVERY.

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A PC-controlled model from Zurich Instruments.

A lock-in amplifier (also known as a phase-sensitive detector) is a type of amplifier that can extract a signal with a known carrier wave from an extremely noisy environment (the signal-to-noise ratio can be -60 dB or even less). It is essentially a homodyne with an extremely low pass filter (making it very narrow band). Lock-in amplifiers use mixing, through a frequency mixer, to convert the signal's phase and amplitude to a DC—actually a time-varying low-frequency—voltage signal.

The device is often used to measure phase shift, even when the signals are large and of high signal-to-noise ratio, and do not need further improvement.

Recovering signals at low signal-to-noise ratios requires a strong, clean reference signal the same frequency as the received signal. This is not the case in many experiments, so the instrument can recover signals buried in the noise only in a limited set of circumstances.

The lock-in amplifier was invented by Princeton University physicist Robert H. Dicke who founded the company Princeton Applied Research (PAR) to market the product.

Basic principles

Operation of a lock-in amplifier relies on the orthogonality of sinusoidal functions. Specifically, when a sinusoidal function of frequency ν is multiplied by another sinusoidal function of frequency μ not equal to ν and integrated over a time much longer than the period of the two functions, the result is zero. In the case when μ is equal to ν, and the two functions are in phase, the average value is equal to half of the product of the amplitudes.

In essence, a lock-in amplifier takes the input signal, multiplies it by the reference signal (either provided from the internal oscillator or an external source), and integrates it over a specified time, usually on the order of milliseconds to a few seconds. The resulting signal is an essentially DC signal, where the contribution from any signal that is not at the same frequency as the reference signal is attenuated essentially to zero, as well as the out-of-phase component of the signal that has the same frequency as the reference signal (because sine functions are orthogonal to the cosine functions of the same frequency), and this is also why a lock-in is a phase sensitive detector.

For a sine reference signal and an input waveform <math>U_\mathrm{in}(t)</math>, the DC output signal <math>U_\mathrm{out}(t)</math> can be calculated for an analog lock-in amplifier by:

<math>U_{\mathrm{out}}(t)= \frac{1}{T} \int_{t-T}^t {\sin\left[2\pi f_{\mathrm{ref}}\cdot s + \varphi\right] U_{\mathrm{in}}(s)}\;\mathrm{d}s</math>

where φ is a phase that can be set on the lock-in (set to zero by default).

Practically, many applications of the lock-in only require recovering the signal amplitude rather than relative phase to the reference signal. For these purposes, lock-in usually measures two components and there are two outputs: <math>X=V_{sig}\cos\theta</math> and <math>Y=V_{sig}\sin\theta</math>, where <math>\theta</math> - phase difference between the signal and reference. These two quantities represent the signal as a vector relative to the lock-in reference oscillator. X is called the 'in-phase' component and Y the 'quadrature' component. By computing the magnitude (R) of the signal vector, the phase dependency is removed:

<math>R=\sqrt{X^2+Y^2}=V_{sig}</math>.

Signal measurement in noisy environments

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Typical experimental setup

The essential idea in signal recovery is that noise tends to be spread over a wider spectrum, often much wider than the signal. In the simplest case of white noise, even if the root mean square of noise is 10⁶ times as large as the signal to be recovered, if the bandwidth of the measurement instrument can be reduced by a factor much greater than 10⁶ around the signal frequency, then the equipment can be relatively insensitive to the noise. In a typical 100 MHz bandwidth (e.g. an oscilloscope), a bandpass filter with width much narrower than 100 Hz would accomplish this.

In summary, even when noise and signal are indistinguishable in the time domain, if the signal has a definite frequency band and there is no large noise peak within that band, noise and signal can be separated sufficiently in the frequency domain.

If the signal is either slowly varying or otherwise constant (essentially a DC signal), then 1/f noise typically overwhelms the signal. It may then be necessary to use external means to modulate the signal. For example, when detecting a small light signal against a bright background, the signal can be modulated either by a chopper wheel, acousto-optical modulator, photoelastic modulator at a large enough frequency so that 1/f noise drops off significantly, and the lock-in amplifier is referenced to the operating frequency of the modulator. In the case of an atomic force microscope, to achieve nanometer and piconewton resolution, the cantilever position is modulated at a high frequency, to which the lock-in amplifier is again referenced.

When the lock-in technique is applied, care must be taken to calibrate the signal, because lock-in amplifiers generally detect only the root-mean-square signal of the operating frequency. For a sinusoidal modulation, this would introduce a factor of <math>\sqrt{2}</math> between the lock-in amplifier output and the peak amplitude of the signal, and a different factor for non-sinusoidal modulation. In the case of extremely nonlinear systems, it may in fact be advantageous to use a higher harmonic for the reference frequency, because of frequency-doubling that takes place in a nonlinear medium.

Furthermore, the response width (effective bandwidth) of detected signal depends on the amplitude of the modulation. Generally, linewidth/modulation function has a monotonically increasing, non-linear behavior.

References

• Wang, Xiaoyi (1990). "Sensitive digital lock-in amplifier using a personal computer". Review of Scientific Instruments. AIP. 61 (70): 1999. Bibcode:1990RScI...61.1999W. doi:10.1063/1.1141413. 

• Temple, Paul A. (1975). "An introduction to phase-sensitive amplifiers: An inexpensive student instrument". American Journal of Physics. AAPT. 43 (9): 801–807. Bibcode:1975AmJPh..43..801T. doi:10.1119/1.9690. 

• Burdett, Richard (2005). "Amplitude Modulated Signals - The Lock-in Amplifier". Handbook of Measuring System Design. Wiley. ISBN 978-0-470-02143-9. 

External links

• Calculation of a modulated time dependent Lorentz signal and its line broadening due to the finite modulation by Lock-In Technique.
• Explanation of lock in amplifiers from Boston Electronics (USA), which sells lock-in amplifiers.
• Lock-in amplifier tutorial from Bentham Instruments. Comprehensive tutorial about the why and how of lock-in amplifiers.
• Lock-ins, Boxcars, and Photon Counting from Stanford Research Systems. Application note detailing how lock-in amplifiers fit into the spectrum of signal recovery techniques.
• Lock-in Technical Notes Range of Technical and Applications notes describing the design of digital and analog lock-ins, and guide to their specifications from SIGNAL RECOVERY.
• PCSC-Lock-in Tool for data acquisition on acoustic chopping frequency using a computer sound card.
• An example of a model of Lock-in Amplifierde:Lock-in-Verstärker