Lock-in amplifier

A lock-in amplifier is an instrument that recovers very small signals (often well below the noise floor of a direct measurement) by exploiting prior knowledge of the signal's frequency. The input signal is multiplied by a reference oscillation at the known signal frequency, and the result is low-pass filtered. Only signal components at the reference frequency (and within the filter bandwidth) survive; everything else is averaged out.

Operating principle. Consider an input signal $V_\text{in}(t) = V_S \cos(\omega_r t + \phi) + V_N(t)$, where $V_S$ is the signal amplitude at the reference angular frequency $\omega_r$, $\phi$ is a phase offset, and $V_N(t)$ is broadband noise.

Multiplying by the reference $\cos(\omega_r t)$:

$$V_\text{in}(t) \cdot \cos(\omega_r t) \;=\; \frac{V_S}{2} \cos\phi + \frac{V_S}{2} \cos(2\omega_r t + \phi) + V_N(t) \cos(\omega_r t).$$

Low-pass filtering removes the high-frequency $2\omega_r$ term and the noise terms (which spread across a broad spectrum and average to near-zero after filtering). What remains is the DC term $(V_S/2)\cos\phi$, proportional to the signal amplitude with a known phase factor.

A second multiplication by the quadrature reference $\sin(\omega_r t)$ yields the orthogonal component $(V_S/2)\sin\phi$. The two components are conventionally called $X$ and $Y$; their quadrature combination gives:

$$R \;=\; \sqrt{X^2 + Y^2} \;=\; \frac{V_S}{2}, \qquad \phi \;=\; \arctan(Y/X).$$

Thus the lock-in measures both the amplitude and phase of the signal at the reference frequency.

Why it works. Two key insights:

1. Frequency selectivity: only signal components within the lock-in's narrow output bandwidth (set by the low-pass filter) reach the output. Effective input bandwidth: $\sim 1$ Hz or less.

2. Phase sensitivity: noise without a defined phase relationship to the reference averages to zero. Signal components at $\omega_r$ with a definite phase (driven by the same source as the reference) survive.

The signal-to-noise improvement from a lock-in is roughly:

$$\frac{\text{SNR}_\text{out}}{\text{SNR}_\text{in}} \;=\; \sqrt{\frac{BW_\text{input}}{BW_\text{output}}}.$$

For typical lock-in measurements (input bandwidth 1 MHz, output bandwidth 0.1 Hz): SNR improvement $\sim 3000\times = +35$ dB.

Standard implementation.

Modern lock-in amplifiers (Stanford Research Systems SR860/SR865, Zurich Instruments MFLI/UHFLI) are digital: the input signal is digitized and all multiplication/filtering is done in firmware. Bandwidth ranges from DC to 600 MHz for state-of-the-art commercial units.

Standard applications in optics.

• Photoluminescence (PL) measurement: chop a CW pump laser at $\omega_r$ (e.g., 800 Hz) using a mechanical or electro-optic chopper; detect the PL signal at the same frequency through the lock-in. Allows PL detection at $\sim 10$ fW signal levels.

• Pump-probe spectroscopy: chop one beam at $\omega_r$, detect transmission of the other beam at $\omega_r$. Recovers the transmission change as small as $10^{-7}$, far below any direct measurement of intensity.

• Photodetector responsivity calibration: chop a known light source at $\omega_r$; measure detector output at $\omega_r$ via lock-in. Eliminates background light and detector dark current.

• Modulation-doped spectroscopy: modulate the sample (electrical bias, magnetic field, temperature) at $\omega_r$; detect optical response at $\omega_r$ to isolate the modulated component from the static background.

• Beam-deflection / position sensing: chop or modulate the beam at $\omega_r$; use lock-in to extract position-sensitive-detector signals at very low light levels.

• Wavemeter-locked laser stabilization: modulate the laser wavelength at a small dither frequency; use lock-in detection of the transmission through a reference cavity to lock the laser to a cavity peak.

Time constants and filter slopes. Lock-in output is set by:

• Time constant $\tau$: $1/e$ exponential settling time of the low-pass filter
• Filter slope: 6, 12, 18, or 24 dB/octave (set by filter order; sometimes higher)

Longer $\tau$ → narrower bandwidth → better SNR but slower response. Typical operation: $\tau = 30$ ms to 1 s for general bench measurements; $\tau = 0.1 - 10$ s for high-precision measurements.

Dynamic range and overload. Lock-in inputs have:

• Full-scale sensitivity: 1 mV to 1 V typically
• Overload threshold: the maximum input the front-end can accept without saturating
• Noise floor: $\sim 5$ nV/Hz^0.5 for low-noise inputs

For modern digital lock-ins (Zurich UHFLI etc.), dynamic range exceeds 100 dB.

Heterodyne detection (frequency-down-conversion). A lock-in can be seen as a single-channel coherent (or "homodyne") detector for electrical signals. The same idea applied to optical signals, multiplying with an optical reference (local oscillator) before photodetection, is the basis of coherent optical detection.

Modern alternatives.

• Boxcar averaging: time-gated integration around expected signal arrival; complementary to lock-in for pulsed signals
• Digital signal averaging: triggered acquisition repeated many times with averaging
• FFT analyzers: full spectrum at once, but with worse sensitivity per frequency bin

Lock-in amplification remains the gold standard for narrowband, low-frequency-modulated measurements.

References: Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019), Ch. 18 (optical detection noise considerations); Horowitz & Hill, The Art of Electronics (3rd ed., 2015), Ch. 8 for the comprehensive electronic engineering treatment; Stanford Research Systems "About Lock-in Amplifiers" technical note.