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 a 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. The device is often used to measure phase shift, even when the signals are large, have a 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 with 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 have been invented by Princeton University physicist Robert H. Dicke who founded the company Princeton Applied Research (PAR) 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 that 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 article by Michels and Curtis, which in turn cites a 1934 article by C. R. Cosens, while another timeless article was written by C. A. Stutt in 1949. 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 (e.g. for complex FFT analysis).
Basic principles The operation of a lock-in amplifier relies on the orthogonality of sinusoidal functions. Specifically, when a sinusoidal function of frequency f1 is multiplied by a sinusoidal function of another frequency f2 and integrated over a time much longer than the period of the two functions, the result is close to zero. If instead 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 (either provided from the internal oscillator or an external source, and can be sinusoidal or square wave), 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 (because sine functions are orthogonal to the cosine functions of the same frequency), making a lock-in a phase-sensitive detector. For a sine reference signal and an input waveform U in ( t ) {\displaystyle U_{\text{in}}(t)} , the DC output signal U out ( t ) {\displaystyle U_{\text{out}}(t)} can be calculated for an analog lock-in amplifier as
U out ( t ) = 1 T ∫ t − T t sin [ 2 π f ref ⋅ s + φ ] U in ( s ) d s , {\displaystyle U_{\text{out}}(t)={\frac {1}{T}}\int _{t-T}^{t}\sin \left[2\pi f_{\text{ref}}\cdot s+\varphi \right]U_{\text{in}}(s)\,ds,}
where φ is a phase that can be set on the lock-in (set to zero by default). If the averaging time T is large enough (i.e. much larger than the signal period) to suppress all unwanted parts like noise and the variations at twice the reference frequency, the output is
U out = 1 2 V sig cos θ , {\displaystyle U_{\text{out}}={\frac {1}{2}}V_{\text{sig}}\cos \theta ,}
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