Lock-in amplifier

A lock-in amplifier measures the amplitude and phase of a signal at one known frequency, the frequency at which the experiment is modulated, and rejects everything else. It multiplies the input by a reference at that frequency and averages the product with a low-pass filter, so that only the part of the input that stays in step with the reference survives, within a bandwidth that can be a fraction of a hertz. A signal buried in noise hundreds or thousands of times larger can be read this way, provided the experiment can be modulated: light chopped before it reaches a detector, a pump beam switched on and off, a laser whose current or wavelength is dithered. The price is speed, since a narrow bandwidth means a long averaging time.

How it works

Take an input $V_{\text{in}}(t)=V_S\cos ({\omega }_{r}t+\varphi )+V_N(t)$, where $V_S$ is the amplitude of the signal at the reference angular frequency ${\omega }_r$, $\varphi$ its phase relative to the reference and $V_N(t)$ the noise. Multiplying by the reference $\cos ({\omega }_{r}t)$ gives

$$\begin{array}{rl}{\displaystyle V_{\text{in}}(t)\cos ({\omega }_{r}t)}& {\displaystyle =\frac{V_S}{2}\cos \varphi +\frac{V_S}{2}\cos (2{\omega }_{r}t+\varphi )}\\ {\displaystyle }& {\displaystyle +V_N(t)\cos ({\omega }_{r}t)\mathrm{.}}\end{array}$$

The low-pass filter removes the term at $2{\omega }_r$, and the noise contributes only through its components within the filter's bandwidth of ${\omega }_r$, which the multiplication shifts down to near zero frequency. What remains is $X=(V_S\mathrm{/}2)\cos \varphi$. A second multiplier, driven by the reference shifted by 90°, gives $Y=(V_S\mathrm{/}2)\sin \varphi$, and together they give the amplitude and phase,

$$R=\sqrt{X^2+Y^2},\theta =\arctan (Y\mathrm{/}X)\mathrm{.}$$

Commercial instruments scale the outputs so that $R$ reads the root-mean-square amplitude of the signal component, $V_S\mathrm{/}\sqrt{2}$. Seen in the frequency domain, the lock-in is a band-pass filter centered on the reference frequency whose width is set by the output filter and which follows the reference if its frequency drifts (Scofield); a fixed filter at 1 kHz with a 0.1 Hz bandwidth would need a quality factor of 10,000 and could not follow a drifting chopper. Modern lock-ins digitize the input and do the multiplication and filtering numerically, and lock an internal oscillator to an external reference signal with a phase-locked loop.

Time constant, filter slope and noise bandwidth

The output filter is set by its time constant $\tau$ and its slope, 6, 12, 18 or 24 dB per octave for one to four cascaded stages. The noise that reaches the output is set by the filter's equivalent noise bandwidth (ENBW), and the time the output takes to follow a change by the settling time; for $n$ identical stages both follow from $\tau$ (SRS):

With $\tau$ = 100 ms and a slope of 24 dB/octave, the noise bandwidth is 0.78 Hz and each reading needs about 1 s to settle. The noise in the output is the input noise density at the reference frequency times the square root of the noise bandwidth: an input noise of 10 nV/√Hz leaves 8.8 nV rms in 0.78 Hz. Against a measurement with 1 MHz of bandwidth, narrowing to 0.1 Hz improves the amplitude signal-to-noise ratio by $\sqrt{{10}^7}$, about 3,200 or 70 dB, when the noise is white. A sweep must wait the settling time at each point, so halving the noise costs four times the measurement time.

The input can tolerate noise and interference much larger than the signal it is set to read, a margin called the dynamic reserve, but only up to the point where the front end overloads. An overloaded input distorts the reading without always being obvious, so the input range is set for the total signal, noise included, and the output sensitivity for the signal.

Setting up a chopped optical measurement

• Modulation frequency. The frequency is chosen away from the mains frequency and its harmonics, above the 1/f noise corner of the detector and preamplifier, and well within their bandwidth. Mechanical choppers reach up to several kilohertz; higher frequencies need a modulated source, such as a laser whose drive current is modulated, or an acousto-optic modulator.
• Detector and preamplifier. A photodiode's current goes to the lock-in's current input or through a transimpedance amplifier; the gain is chosen so that the input, with its background light, stays below overload.
• Reference and phase. The chopper controller's reference output drives the lock-in's reference input. With the signal present, the phase is adjusted, often automatically, until the signal lies in $X$ and $Y$ is near zero, and $X$ is then read. At low signal-to-noise ratio $R$ is biased upward, because the noise in $X$ and $Y$ adds in quadrature and does not average to zero, while $X$ alone averages to the signal.
• Square-wave modulation. A chopper switches the light between zero and a peak value $V_p$, and the lock-in reads only the fundamental of that square wave, whose rms amplitude is $\sqrt{2}{V}_p\mathrm{/}\pi =0.450V_p$. The reading is divided by 0.450 to recover the peak-to-peak signal.

Derivative measurements on a laser

A lock-in reads the derivatives of a laser diode's characteristics directly. A small sinusoidal current is added to the DC drive, through a bias tee or the modulation input of the laser driver, and the lock-in, referenced to the modulation, reads the resulting modulation of the optical power, which gives $dL\mathrm{/}dI$, or of the voltage, which gives $dV\mathrm{/}dI$. For the example laser of the LIV curve entry, with a slope efficiency of 0.25 mW/mA, a modulation of 0.1 mA rms produces 25 µW rms of optical modulation, or 25 µA rms from a photodiode with a responsivity of 1.00 A/W. Kinks in the L-I curve appear as dips or steps in $dL\mathrm{/}dI$ (Why Laser Diode L-I Curves Kink), and the saturation of the junction voltage at threshold appears as a step in $dV\mathrm{/}dI$ and a sharp feature in $d^{2}V\mathrm{/}d{I}^2$, which Paoli observed at threshold. The modulation must be small compared with the width of the features it resolves, and at low modulation frequencies the chip temperature follows the modulation, so the measured derivative includes the thermal response, as a slow DC sweep does.

Other uses in optics

• Photoluminescence. The pump laser is chopped and the emission detected at the chopping frequency, which removes the detector's dark current and the room's background light.
• Pump-probe spectroscopy. The pump is modulated and the lock-in reads the change it causes in the transmitted probe. Shot noise sets the floor: with 1 mA of probe photocurrent the relative shot noise is $1.8\times {10}^{-8}$ per √Hz, and the laser's intensity noise, usually larger at low frequencies, is why the pump is often modulated at high frequencies.
• Detector calibration. Chopped light from a calibrated source measures a detector's responsivity without error from its dark current or from ambient light.
• Wavelength-modulation spectroscopy. The wavelength of a diode laser is dithered across an absorption line and the lock-in reads the transmitted power at twice the dither frequency. For a small dither that signal is proportional to the second derivative of the line and peaks at its center, and it is the basis of sensitive trace-gas detection.
• Laser locking. Dithering a laser's frequency and demodulating the transmission through a reference cavity or an absorption line at the dither frequency gives the derivative of the transmission, which passes through zero at the peak and serves as the error signal of a servo that holds the laser there (ring resonator thermal tuning and locking uses the same idea on a chip).

A boxcar averager, which integrates the signal in a time window after each pulse, suits low-duty-cycle pulsed signals better, since a lock-in spends most of each period averaging the noise between pulses. An FFT spectrum analyzer shows every frequency at once but, for the same measurement time, with less rejection at any one of them. The lock-in is also the electrical counterpart of heterodyne detection, in which the multiplication by a reference happens optically, on the photodetector.

Common questions

What does a lock-in amplifier do?

It measures the amplitude and phase of the part of a signal that is at a known reference frequency, by multiplying the signal by the reference and low-pass filtering the result. Noise at other frequencies averages away, so a modulated signal far smaller than the noise can be measured.

What time constant should be used?

The longest the measurement can afford. The noise falls as the square root of the equivalent noise bandwidth, which is $5\mathrm{/}(64\tau )$ for a 24 dB/octave filter, but each reading then needs about $10\tau$ to settle, so a sweep of many points at $\tau$ = 1 s takes about 10 s per point.

Why read X instead of R?

At low signal-to-noise ratio, $R=\sqrt{X^2+Y^2}$ contains the noise of both channels added in quadrature, which never averages to zero and biases the reading upward. Once the phase is set so that the signal lies in $X$, reading $X$ gives an unbiased value.

How is a chopped signal converted to optical power?

The lock-in reads the rms value of the fundamental of the square wave, 0.450 times the peak-to-peak signal for light switched fully on and off with equal on and off times, so the reading is divided by 0.450 and then by the detector's responsivity and gain.

References: J. H. Scofield, "Frequency-domain description of a lock-in amplifier," American Journal of Physics 62, 129 (1994). Stanford Research Systems, About Lock-In Amplifiers, Application Note 3. M. L. Meade, Lock-in Amplifiers: Principles and Applications (Peter Peregrinus, 1983). P. Horowitz and W. Hill, The Art of Electronics, 3rd ed. (Cambridge University Press, 2015). T. L. Paoli, "Observation of second derivatives of the electrical characteristics of double-heterostructure junction lasers," IEEE Transactions on Electron Devices 23, 1333 (1976).