{ "id": "epn9s2rd0c", "slug": "phase-sensitive-amplification-spectroscopy", "title": "Phase-sensitive amplification (spectroscopy)", "updated": "2026-09-30", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.physics", "label": "Physics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.physics" }, { "id": "physical.physics.practice", "label": "Physics methods, practice, and community", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.physics.practice" } ], "geo": [ { "id": "geo.nongeo.t1946.physical.physics.practice", "label": "Non-geographic · 1946 to 2000: Physics methods, practice, and community", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.physics.practice", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1946", "label": "Non-geographic · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946" }, { "id": "geo.nongeo.t1946.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical" }, { "id": "geo.nongeo.t1946.physical.physics", "label": "Physics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.physics" }, { "id": "geo.nongeo.t1946.physical.physics.practice", "label": "Physics methods, practice, and community", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.physics.practice" } ] } ], "excerpt": "Phase-sensitive amplification, also called lock-in amplification, is a signal-recovery technique that measures weak signals by detecting the component matching a known reference frequency, standard in spectroscopy for roughly a century.", "snippet": "Phase-sensitive amplification, also called lock-in amplification, is a signal-recovery technique that measures weak signals by detecting the component matching a known reference frequency, standard in spectroscopy for roughly a century.", "node": "physical.physics.practice", "markdown": "# Phase-sensitive amplification (spectroscopy)\n\nLock-in amplification, a form of phase-sensitive detection, is a signal-recovery technique that measures a weak signal by modulating it at a known reference frequency and detecting only the component whose phase matches that reference.[17](https://publications.lib.chalmers.se/records/fulltext/local_165060.pdf) It extracts AC signals down to a few nanovolts even when noise sources thousands of times larger obscure them[1](https://thinksrs.com/downloads/pdfs/applicationnotes/AboutLIAs.pdf), which is why it has been a standard tool in experimental spectroscopy and related fields for roughly a century.[2](https://pubs.aip.org/aip/rsi/article/97/9/094705/3404068/Exact-computation-of-lock-in-amplifier-outputs-for)\n\n| Key fact | Value |\n|---|---|\n| Output quantities | In-phase X and quadrature Y; amplitude R and phase θ derived from them[3](https://www.zhinst.com/sites/default/files/li_primer/zi_whitepaper_principles_of_lock-in_detection.pdf) |\n| Minimum detectable signal | A few nanovolts, in noise thousands of times larger[1](https://thinksrs.com/downloads/pdfs/applicationnotes/AboutLIAs.pdf) |\n| Dynamic reserve | 120 dB (best modern instruments)[3](https://www.zhinst.com/sites/default/files/li_primer/zi_whitepaper_principles_of_lock-in_detection.pdf); up to 130 dB commercially available[4](https://www.ameteksi.com/-/media/ameteksi/download_links/documentations/7210/tn1001_specifying_lock-in_amplifiers.pdf?revision=07cddc64-e72e-4478-8df2-59252950444b) |\n| Noise rejection | Proportional to the square root of the detection bandwidth, set by the time constant[5](https://wp.optics.arizona.edu/jpalmer/wp-content/uploads/sites/65/2018/11/Gesamttext.pdf) |\n| Modulation choice | Sine-wave deviation of 0.507 × FWHM for a Gaussian peak, 0.353 × FWHM for a Lorentzian[2](https://pubs.aip.org/aip/rsi/article/97/9/094705/3404068/Exact-computation-of-lock-in-amplifier-outputs-for) |\n| History | Invented in the 1930s; commercialized mid-20th century[3](https://www.zhinst.com/sites/default/files/li_primer/zi_whitepaper_principles_of_lock-in_detection.pdf) |\n\n## How it works\n\nThe experiment modulates the quantity of interest at a fixed, known frequency and phase. The detector output is multiplied (mixed) by a reference signal at that same frequency in a phase-sensitive detector (PSD). For an input of amplitude \\( A \\) and a reference of amplitude \\( B \\) with phase difference θ, the PSD output is \\( \\tfrac{1}{2}AB\\cos\\theta \\) plus a component at \\( 2\\omega t \\); a low-pass filter removes the doubled-frequency term, leaving a DC signal proportional to \\( A \\) and to \\( \\cos\\theta \\).[6](https://www.ameteksi.com/-/media/ameteksi/download_links/documentations/7210/tn1000_what_is_a_lock-in_amplifier.pdf?revision=8c096882-f88b-463a-9a3e-f242d44c0c6b) The PSD produces a DC output only when its two inputs share a frequency, with polarity set by their phase difference.[7](https://cpb-us-e1.wpmucdn.com/sites.psu.edu/dist/1/37364/files/2015/12/traficante_detection_1990.pdf)\n\nNoise at frequencies other than the reference is rejected.[8](https://thinksrs.com/downloads/pdfs/applicationnotes/Lock-In%20Basics.pdf) After detection, the desired signal is a DC level while noise appears as an AC fluctuation, so a simple output low-pass filter separates them.[6](https://www.ameteksi.com/-/media/ameteksi/download_links/documentations/7210/tn1000_what_is_a_lock-in_amplifier.pdf?revision=8c096882-f88b-463a-9a3e-f242d44c0c6b) By the [Wiener–Khinchin theorem](https://www.edgepedia.org/wiener-khinchin-theorem), the noise voltage reduction is proportional to the square root of the filter bandwidth, and the lock-in behaves like a bandpass filter at the reference frequency whose bandwidth is set by the integration time; SNR improvements of 60 dB or more are possible.[5](https://wp.optics.arizona.edu/jpalmer/wp-content/uploads/sites/65/2018/11/Gesamttext.pdf) Choosing a modulation frequency above roughly 1 kHz avoids the 1/f noise (noise whose power grows as frequency drops) that dominates lower-frequency backgrounds.[2](https://pubs.aip.org/aip/rsi/article/97/9/094705/3404068/Exact-computation-of-lock-in-amplifier-outputs-for)\n\nDual-phase demodulation multiplies the input with the reference and a 90°-shifted copy, giving \\( X = R\\cos\\theta \\) and \\( Y = R\\sin\\theta \\), with \\( \\Theta = \\mathrm{atan2}(Y, X) \\) covering all four quadrants.[3](https://www.zhinst.com/sites/default/files/li_primer/zi_whitepaper_principles_of_lock-in_detection.pdf) The magnitude \\( |Z| = R \\) equals the root-mean-square amplitude of the signal and \\( \\arg(Z) = \\Theta \\) its phase relative to the reference.[3](https://www.zhinst.com/sites/default/files/li_primer/zi_whitepaper_principles_of_lock-in_detection.pdf) When the modulation is small, the fundamental output is proportional to the first derivative of the system function and the \\( 2 \\cdot f \\) output to its second derivative (curvature)[5](https://wp.optics.arizona.edu/jpalmer/wp-content/uploads/sites/65/2018/11/Gesamttext.pdf), which is how the output maps to [spectral line shape](https://www.edgepedia.org/spectral-line-shape).\n\n## How it is done\n\n1. **Modulate the signal** at a frequency above the 1/f noise region. For a spectral peak, the optimal sine-wave modulation deviation maximizing zero-crossing slope is 0.507 × FWHM for a Gaussian line and 0.353 × FWHM for a Lorentzian.[2](https://pubs.aip.org/aip/rsi/article/97/9/094705/3404068/Exact-computation-of-lock-in-amplifier-outputs-for) Square-wave modulation reduces the signal by a factor of 1/4 (half lost in modulation, half in filtering the frequency-doubled component) but is nevertheless preferable when technically possible.[2](https://pubs.aip.org/aip/rsi/article/97/9/094705/3404068/Exact-computation-of-lock-in-amplifier-outputs-for)\n2. **Supply the reference** from the modulator drive so the instrument knows frequency and phase directly.\n3. **Choose the output filter.** Filters are specified by time constant, inversely proportional to the −3 dB frequency, and stacked for 12, 18, or 24 dB/octave roll-off; roll-off above 6 dB/octave should not be used in feedback applications because of instability.[4](https://www.ameteksi.com/-/media/ameteksi/download_links/documentations/7210/tn1001_specifying_lock-in_amplifiers.pdf?revision=07cddc64-e72e-4478-8df2-59252950444b)\n4. **Set the phase** so the signal appears in X, or record both channels.\n\n## Origin\n\nLock-in amplifiers are instruments that extract signal amplitudes and phases in extremely noisy environments using homodyne detection and low-pass filtering.[3](https://www.zhinst.com/sites/default/files/li_primer/zi_whitepaper_principles_of_lock-in_detection.pdf) By 1965 a National Bureau of Standards paper could state that the phase-sensitive detector plays an important role in instrumentation for spectroscopy and radio astronomy and in atomic frequency standards.[9](https://tf.nist.gov/general/pdf/269.pdf) Over the last century the technique became a standard measurement tool across experimental physics.[2](https://pubs.aip.org/aip/rsi/article/97/9/094705/3404068/Exact-computation-of-lock-in-amplifier-outputs-for) Specific attributions of the invention to individual authors and papers circulate in secondary accounts but are not established by the primary technical literature cited here.\n\n## Variants\n\n**Single-phase versus dual-phase.** A single-phase lock-in output is proportional to \\( \\sin(\\Delta\\phi) \\) rather than \\( \\Delta\\phi \\), limiting the linear phase range to less than \\( \\pi \\); dual-phase demodulation extends it to \\( 2 \\pi \\).[10](https://liquidinstruments.com/application-notes/phase-detection/) A dual-phase lock-in uses two PSDs with references 90° apart and measures X, Y, and R directly, with \\( \\theta = \\tan^{-1}(Y/X) \\).[1](https://thinksrs.com/downloads/pdfs/applicationnotes/AboutLIAs.pdf) Quadrature output is needed when signals in quadrature must be measured simultaneously.[4](https://www.ameteksi.com/-/media/ameteksi/download_links/documentations/7210/tn1001_specifying_lock-in_amplifiers.pdf?revision=07cddc64-e72e-4478-8df2-59252950444b)\n\n**Digital versus analog.** Digital lock-ins convert the input immediately with an ADC and perform all subsequent steps by digital signal processing, which is less prone to pathway mismatch, cross-talk, and temperature drift than analog mixing with voltage-controlled oscillators and RC filters; FPGAs provide the real-time processing.[3](https://www.zhinst.com/sites/default/files/li_primer/zi_whitepaper_principles_of_lock-in_detection.pdf) Digital implementations have outperformed analog instruments in bandwidth, adjustable parameter range, accuracy, and versatility.[11](https://re.public.polimi.it/bitstream/11311/988417/1/1.4941721.pdf) Digitally computed 20-bit reference sine waves have harmonics at the −120 dB level, making such instruments insensitive to signals at harmonics of the reference, whereas square-wave multiplying lock-ins detect at all odd harmonics and analog PSDs suffer harmonic rejection problems, output offsets, limited dynamic reserve, and gain error.[1](https://thinksrs.com/downloads/pdfs/applicationnotes/AboutLIAs.pdf) Because digitizing a signal buried in 100 dB of noise to 15-bit accuracy would require a 32-bit converter, digital instruments recover resolution by averaging many samples at the expense of response time.[6](https://www.ameteksi.com/-/media/ameteksi/download_links/documentations/7210/tn1000_what_is_a_lock-in_amplifier.pdf?revision=8c096882-f88b-463a-9a3e-f242d44c0c6b)\n\n## Applications\n\nLock-in detection underpins photoacoustic spectroscopy, [Raman scattering](https://www.edgepedia.org/raman-scattering) microscopy, atomic force and scanning probe microscopy, impedance measurements, single-ion quantum lock-in amplification, quantum sensors, and biomedical sensing.[2](https://pubs.aip.org/aip/rsi/article/97/9/094705/3404068/Exact-computation-of-lock-in-amplifier-outputs-for) Multi-channel lock-ins accelerate scanning-probe measurements such as nonlinear tip-sample force mapping and conductive atomic force microscopy.[12](https://hoffman.physics.harvard.edu/publications/Multi_channel_lock_in_amplifier.pdf) Beyond spectroscopy, phase-sensitive amplification is applied in quantum-state detection, optical communications, enhanced quantum metrology, and interferometers.[13](https://iopscience.iop.org/article/10.1088/1367-2630/ad8778)\n\n## Limitations and alternatives\n\n**Input overload bounds the dynamic reserve.** Peak input is limited to the linear range of the input amplifier, typically a few volts, so 60 dB of dynamic reserve is impossible at 1 V full-scale sensitivity, since that would imply a 1 kV input capability.[4](https://www.ameteksi.com/-/media/ameteksi/download_links/documentations/7210/tn1001_specifying_lock-in_amplifiers.pdf?revision=07cddc64-e72e-4478-8df2-59252950444b) Dynamic reserve is also frequency-dependent: it is 0 dB at the reference frequency and increases at the rate at which the output low-pass filter rolls off.[1](https://thinksrs.com/downloads/pdfs/applicationnotes/AboutLIAs.pdf)\n\n**Drift and phase stability.** Analog lock-in DC drift is on the order of 1000 ppm/°C at 60 dB dynamic reserve, moving the zero 1% of full scale over a 10 °C change[1](https://thinksrs.com/downloads/pdfs/applicationnotes/AboutLIAs.pdf); modern digital instruments specify analog output drift below 5 ppm/°C.[14](https://www.salukitec.com/wp-content/uploads/2021/06/SE1022-Signal-Channel-Lock-in-Amplifier-Datasheet.pdf) [Phase noise](https://www.edgepedia.org/phase-noise), the random variation in phase difference between signal and reference inputs, becomes a problem when the quadrature signal is much smaller than the in-phase signal.[4](https://www.ameteksi.com/-/media/ameteksi/download_links/documentations/7210/tn1001_specifying_lock-in_amplifiers.pdf?revision=07cddc64-e72e-4478-8df2-59252950444b) The lock-in tolerates long measurements because it is insensitive to DC offset drift, but drift in device resistance or amplifier gain still affects long measurements, making stable temperature crucial[3](https://www.zhinst.com/sites/default/files/li_primer/zi_whitepaper_principles_of_lock-in_detection.pdf), and slow fluctuations limit the resolution of digital lock-ins, motivating differential configurations.[11](https://re.public.polimi.it/bitstream/11311/988417/1/1.4941721.pdf) When the studied system responds non-instantaneously to the pump field, cross-talk between the [I and Q components](https://www.edgepedia.org/in-phase-and-quadrature-components) prevents their complete separation.[15](https://pmc.ncbi.nlm.nih.gov/articles/PMC5695695/)\n\n**When to use something else.** Purely sinusoidal signals, as in tunable diode absorption spectroscopy and photoluminescence, are best analyzed with lock-in amplifiers, while ultra-low duty-cycle signals, as in low-repetition-rate pump–probe, Raman, and terahertz spectroscopy, are best analyzed with boxcar averagers, because for such signals the power spreads across harmonics with a span inversely proportional to the duty cycle and requires a detection bandwidth wider than a single harmonic.[16](https://www.zhinst.com/americas/en/blogs/lock-in-amplifier-or-boxcar-averager-comparing-two-measurement-approaches-periodic-signal/) A boxcar averager multiplies the signal with a square wave of arbitrary duty cycle, integrates while the pulse is high, and filters with a moving average.[16](https://www.zhinst.com/americas/en/blogs/lock-in-amplifier-or-boxcar-averager-comparing-two-measurement-approaches-periodic-signal/)\n\n## References\n\n1. [About Lock-In Amplifiers (Stanford Research Systems application note)](https://thinksrs.com/downloads/pdfs/applicationnotes/AboutLIAs.pdf)\n2. [Exact computation of lock-in amplifier outputs for arbitrary frequency modulations using Gauss–Chebyshev quadrature (Review of Scientific Instruments, 2025)](https://pubs.aip.org/aip/rsi/article/97/9/094705/3404068/Exact-computation-of-lock-in-amplifier-outputs-for)\n3. [Principles of lock-in detection and the state of the art (Zurich Instruments whitepaper)](https://www.zhinst.com/sites/default/files/li_primer/zi_whitepaper_principles_of_lock-in_detection.pdf)\n4. [Specifying Lock-in Amplifiers (AMETEK Signal Recovery TN1001)](https://www.ameteksi.com/-/media/ameteksi/download_links/documentations/7210/tn1001_specifying_lock-in_amplifiers.pdf?revision=07cddc64-e72e-4478-8df2-59252950444b)\n5. [The Lock-In: Noise Reduction and Phase Sensitive Detection (University of Arizona course text)](https://wp.optics.arizona.edu/jpalmer/wp-content/uploads/sites/65/2018/11/Gesamttext.pdf)\n6. [What is a Lock-in Amplifier? (AMETEK Signal Recovery technical note TN1000)](https://www.ameteksi.com/-/media/ameteksi/download_links/documentations/7210/tn1000_what_is_a_lock-in_amplifier.pdf?revision=8c096882-f88b-463a-9a3e-f242d44c0c6b)\n7. [Phase-Sensitive Detection. Part I: Phase, Gates, Phase-Sensitive Detectors, Mixers, and the Rotating Frame (Traficante, 1990)](https://cpb-us-e1.wpmucdn.com/sites.psu.edu/dist/1/37364/files/2015/12/traficante_detection_1990.pdf)\n8. [SRS Tech Note: Lock-In Basics](https://thinksrs.com/downloads/pdfs/applicationnotes/Lock-In%20Basics.pdf)\n9. [The reference voltage is contra' (Beers, National Bureau of Standards, 1965), phase-sensitive detector in spectroscopy instrumentation](https://tf.nist.gov/general/pdf/269.pdf)\n10. [Phase detection with the Moku Lock-in Amplifier and Phasemeter (Liquid Instruments)](https://liquidinstruments.com/application-notes/phase-detection/)\n11. [Note: Differential configurations for the mitigation of slow fluctuations limiting the resolution of digital lock-in amplifiers (Review of Scientific Instruments, DOI 10.1063/1.4941721)](https://re.public.polimi.it/bitstream/11311/988417/1/1.4941721.pdf)\n12. [Open-source platform for an efficient multi-channel lock-in amplifier](https://hoffman.physics.harvard.edu/publications/Multi_channel_lock_in_amplifier.pdf)\n13. [Enhancement of single-photon level signal detection based on phase sensitive amplification strategy (New Journal of Physics)](https://iopscience.iop.org/article/10.1088/1367-2630/ad8778)\n14. [SE1022 Digital Lock-in Amplifier Datasheet (Saluki)](https://www.salukitec.com/wp-content/uploads/2021/06/SE1022-Signal-Channel-Lock-in-Amplifier-Datasheet.pdf)\n15. [Quadrature Demodulation of a Quantum Dot Optical Response to Faint Light Fields (PMC)](https://pmc.ncbi.nlm.nih.gov/articles/PMC5695695/)\n16. [Lock-in Amplifier or Boxcar Averager? Comparing Two Measurement Approaches for Periodic Signal Analysis (Zurich Instruments blog)](https://www.zhinst.com/americas/en/blogs/lock-in-amplifier-or-boxcar-averager-comparing-two-measurement-approaches-periodic-signal/)\n17. [Local 165060 (publications.lib.chalmers.se)](https://publications.lib.chalmers.se/records/fulltext/local_165060.pdf)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Physics › Physics methods, practice, and community*\n\n*Initially written Sep 29, 2026 · Reviewed: Sep 30, 2026 · Edited: Sep 30, 2026 · Last review: Sep 30, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [], "url": "https://www.edgepedia.org/phase-sensitive-amplification-spectroscopy", "markdown_url": "https://www.edgepedia.org/phase-sensitive-amplification-spectroscopy.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Phase-sensitive amplification (spectroscopy)\", Edgepedia (EdgeChat), https://www.edgepedia.org/phase-sensitive-amplification-spectroscopy. Edgepedia Community License 1.0.", "credit_md": "\"[Phase-sensitive amplification (spectroscopy)](https://www.edgepedia.org/phase-sensitive-amplification-spectroscopy)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/phase-sensitive-amplification-spectroscopy](https://www.edgepedia.org/phase-sensitive-amplification-spectroscopy). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Phase-sensitive amplification (spectroscopy)\", Edgepedia (EdgeChat), https://www.edgepedia.org/phase-sensitive-amplification-spectroscopy. Edgepedia Community License 1.0.", "speakable": "Phase-sensitive amplification, also called lock-in amplification, is a signal-recovery technique that measures weak signals by detecting the component matching a known reference frequency, standard in spectroscopy for roughly a century." }