Lock-in amplifier

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 extremely noisy environment (S/N ratio can be as low as -60 dB or even lessFact|date=December 2007). 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&mdash;actually a time-varying low-frequency&mdash;voltage signal.

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. The PAR brandname is now used for electrochemical instruments, but their successors, SIGNAL RECOVERY, continue to design and produce lock-in amplifiers.

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 $U_mathrm\left\{in\right\}\left(t\right)$, the DC output signal $U_mathrm\left\{out\right\}\left(t\right)$ can be calculated for an analog lock-in amplifier by: :$U_\left\{mathrm\left\{out\left(t\right)= frac\left\{1\right\}\left\{T\right\} int_\left\{t-T\right\}^t \left\{sinleft \left[2pi f_\left\{mathrm\left\{refcdot s + phi ight\right] U_\left\{mathrm\left\{in\left(s\right)\right\};mathrm\left\{d\right\}s$

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; a lock-in usually measures both in-phase ("X") and out-of-phase ("Y") components of the signal and can calculate the magnitude ("R") from that.

More basic principles

Lock-in amplifiers are used to measure the amplitude and phase of signals buried in noise. They achieve this by acting as a narrow bandpass filter which removes much of the unwanted noise while allowing through the signal which is to be measured.

The frequency of the signal to be measured and hence the passband region of the filter is set by a reference signal, which has to be supplied to the lock-in amplifier along with the unknown signal. The reference signal must be at the same frequency as the modulation of the signal to be measured.

A basic lock-in amplifier can be split into 4 stages: an input gain stage, the reference circuit, a demodulator and a low pass filter.

* Input Gain Stage: The variable gain input stage pre-processes the signal by amplifying it to a level suitable for the demodulator. Nothing complicated here, but high performance amplifiers are required.

* Reference Circuit: The reference circuit allows the reference signal to be phase shifted.

* Demodulator: The demodulator is a multiplier. It takes the input signal and the reference and multiplies them together. When you multiply two waveforms together you get the sum and difference frequencies as the result. As the input signal to be measured and the reference signal are of the same frequency, the difference frequency is zero and you get a DC output which is proportional to the amplitude of the input signal and the cosine of the phase difference between the signals. By adjusting the phase of the reference signal using the reference circuit, the phase difference between the input signal and the reference can be brought to zero and hence the DC output level from the multiplier is proportional to the input signal. The noise signals will still be present at the output of the demodulator and may have amplitudes 1000 times as large as the DC offset.
* Low Pass Filter: As the various noise components on the input signal are at different frequencies to the reference signal, the sum and difference frequencies will be non zero and will not contribute to the DC level of the output signal. This DC level (which is proportional to the input signal) can now be recovered by passing the output from the demodulator through a low pass filter.

The above gives an idea of how a basic lock-in amplifier works. Actual lock-in amplifiers are more complicated, as there are instrument offsets that need to be removed, but the basic principle of operation is the same.

Application to signal measurements in a noisy environment

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 is indistinguishable in time domain, if 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, in the case of detection of 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, in order to achieve nanometer and piconewton resolution, the cantilever position is modulated at a high frequency, to which lock-in amplifier is again referenced.

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

* [http://www.boselec.com/products/siglimwhat.shtml Explanation of lock in amplifiers] from Boston Electronics, which sells lock-in amplifiers. Other significant lock-in vendors include [http://www.scitec.uk.com Scitec Instruments] and [http://www.thinksrs.com/products/sci.htm SRS]
* [http://www.bentham.co.uk/pdf/F225.pdf Lock-in amplifier tutorial from Bentham Instruments] Comprehensive tutorial about the why and how of lock-in amplifiers.

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