A lock-in amplifier is a type of amplifier that can extract a signal with
a known carrier wave from an extremely noisy environment. Depending on the
dynamic reserve of the instrument, signals up to 1 million times smaller
than noise components, potentially fairly close by in frequency, can still
be reliably detected. It is essentially a homodyne detector followed by low pass
filter that is often adjustable in cut off frequency and filter order. Whereas
traditional lock-in amplifiers use analog frequency mixers and RC filters
for the demodulation, state of the art instruments have both steps implemented
by fast digital signal processing for example on an FPGA. Usually sine and
cosine demodulation is performed simultaneously, which is sometimes also
referred to as dual phase demodulation. This allows the extraction of the
in-phase and the quadrature component that can then be transferred into polar
coordinates, i.e. amplitude and phase, or further processed as real and
imaginary part of a complex number. 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 is commonly
believed to be invented by Princeton University physicist Robert H. Dicke who
founded the company Princeton Applied Research to market the product. However,
in an interview with Martin Harwit, Dicke claims that even though he is
often credited with the invention of the device, he believes he read about it in
a review of scientific equipment written by Walter C Michels, a professor at Bryn
Mawr College. This could have been a 1941 paper by Michels and Curtis, which
in turn cites a 1934 paper by C. R. Cosens. Another timeless paper was
written by C.A. Stutt in 1949. Basic principles
Operation of a lock-in amplifier relies on the orthogonality of sinusoidal
functions. Specifically, when a sinusoidal function of frequency f1 is
multiplied by another sinusoidal function of frequency f2 not equal to f1
and integrated over a time much longer than the period of the two functions,
the result is zero. Instead, when f1 is equal to f2 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, and integrates it over a specified time, usually on the order
of milliseconds to a few seconds. The resulting signal is a DC signal, where
the contribution from any signal that is not at the same frequency as the
reference signal is attenuated close to zero. The out-of-phase component of the
signal that has the same frequency as the reference signal is also attenuated,
making a lock-in a phase-sensitive detector.
For a sine reference signal and an input waveform , the DC output signal can be
calculated for an analog lock-in amplifier by:
where φ is a phase that can be set on the lock-in.
If the averaging time T is large enough to suppress all unwanted parts like
noise and the variations at twice the reference frequency, the output is
where is the signal amplitude at the reference frequency and is the phase
difference between the signal and reference.
Many applications of the lock-in only require recovering the signal amplitude
rather than relative phase to the reference signal. For a simple so called
single phase lock-in-amplifier the phase difference is adjusted to zero to get
the full signal. More advanced, so called two phase
lock-in-amplifiers have a second detector, doing the same calculation as
before, but with an additional 90 degree phase shift. Thus one has two outputs:
is called the 'in-phase' component and the 'quadrature' component. These two
quantities represent the signal as a vector relative to the lock-in reference
oscillator. By computing the magnitude of the signal vector, the phase
dependency is removed: The phase can be calculated from
Signal measurement in noisy environments 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 103 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, a bandpass filter
with width much narrower than 100 Hz would accomplish this. The averaging
time of the lock-in-amplifier determines the bandwidth, and allows very narrow
filters, less than 1 Hz if needed. However this comes at the price of a
slow response to changes in the signal. 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, 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
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 nonlinear systems, higher harmonics of the modulation-frequency
appear. A simple example is the light of a conventional light bulb being
modulated at twice the line frequency. Some lock-in-Amplifiers also allow
separate measurements of these higher harmonics.
Furthermore, the response width of detected signal depends on the amplitude
of the modulation. Generally, linewidth/modulation function has a
monotonically increasing, non-linear behavior.
References Publications
Scofield, John H.. "Frequency-domain description of a lock-in amplifier".
American Journal of Physics 62: 129–133. Bibcode:1994AmJPh..62..129S.
doi:10.1119/1.17629. Jaquier, Pierre-Alain; Jaquier, Alain.
"Multiple-channel digital lock-in amplifier with PPM resolution". Review
of Scientific Instruments 65: 747. Bibcode:1994RScI...65..747P.
doi:10.1063/1.1145096. Wang, Xiaoyi. "Sensitive digital lock-in
amplifier using a personal computer". Review of Scientific Instruments 61:
1999. Bibcode:1990RScI...61.1999W. doi:10.1063/1.1141413.
Wolfson, Richard. "The lock-in amplifier: A student experiment".
American Journal of Physics 59: 569–572. Bibcode:1991AmJPh..59..569W.
doi:10.1119/1.16824. Dixon, Paul K.; Wu, Lei. "Broadband
digital lock-in amplifier techniques". Review of Scientific Instruments 60:
3329. Bibcode:1989RScI...60.3329D. doi:10.1063/1.1140523.
van Exter, Martin; Lagendijk, Ad. "Converting an AM radio into a
high-frequency lock-in amplifier in a stimulated Raman experiment". Review of
Scientific Instruments 57: 390. Bibcode:1986RScI...57..390V.
doi:10.1063/1.1138952. Probst, P. A.; Collet, B..
"Low-frequency digital lock-in amplifier". Review of Scientific
Instruments 56: 466. Bibcode:1985RScI...56..466P.
doi:10.1063/1.1138324. Temple, Paul A.. "An introduction to
phase-sensitive amplifiers: An inexpensive student instrument".
American Journal of Physics 43: 801–807. Bibcode:1975AmJPh..43..801T.
doi:10.1119/1.9690. Burdett, Richard. "Amplitude Modulated
Signals - The Lock-in Amplifier". Handbook of Measuring System Design.
ISBN 978-0-470-02143-9. External links
About LIAs from Stanford Research Systems. Application note detailing how
lock-in amplifiers work. Calculation of a modulated time
dependent Lorentz signal and its line broadening due to the finite modulation
by Lock-In Technique. Lock-in amplifier tutorial from Bentham
Instruments. Comprehensive tutorial about the why and how of lock-in
amplifiers. 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.