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.

Component	Function
Reference input (electrical or optical)	Provides the frequency reference
Phase-locked loop (PLL)	Tracks reference frequency drift
Input preamplifier	Boosts signal before mixer
Mixer (X and Y)	Multiply signal × reference (and quadrature reference)
Low-pass filter	Sets output bandwidth (often 24 dB/octave with adjustable time constant)
Output stage	DC analog or digital output of X, Y, R, $\phi$

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.