Precision displacement measurement with a Michelson interferometer Created 2026-10-18 Updated 2026-10-20
A Michelson interferometer turns displacement into a count of interference fringes: a moving mirror changes both the outward and return paths. This lets a wavelength act as a ruler whose divisions need no engraving. Assume normal incidence, a fixed reference arm, stable refractive index along the moving arm, and fixed vacuum wavelength . Write for the wavelength in that medium and take positive away from the beam splitter. The change in optical path difference, the accumulated phase difference, and the signed number of complete interference fringe cycles areThe factor two comes from propagation geometry: moving the mirror by adds before reflection and afterward. One bright-to-bright cycle means mirror travel ; bright-to-dark means , half a cycle. All lengths must use compatible units, and is dimensionless. A constant reflection phase difference changes the initial fringe position but cancels from this change formula.
Mirror displacement, doubled optical path difference, and counted interference fringes for a hypothetical Michelson interferometer
. Details. For an original hypothetical example, suppose in the actual surrounding medium and a continuous scan gives cycles, including the final fractional cycle. Then . With the wavelength treated as exact, a fringe-reading measurement uncertainty of cycle contributes to the displacement measurement uncertainty. An uncertainty of in a local phase difference instead corresponds to . This illustrates measurement resolution, not a complete claim of accuracy: wavelength calibration, air changes, mirror tilt and missed cycles can dominate.
To separate the oscillating readout from its optical coherence envelope, use the local modelHere is detected light intensity, is the mean level, is a fixed bias wave phase, and is interferometric visibility. A fine scan of measures the carrier phase difference; a coarse scan can map the temporal optical coherence envelope through . The envelope maximum locates matched delays only under the source and apparatus assumptions; it is not a bright fringe identifying an arbitrary mirror position. Even with known , one intensity reading leaves the cosine sign and whole-cycle ambiguities. Continuous motion of known direction, or an independently obtained pair of signals in phase quadrature, is needed to recover signed travel through reversals.
Proposed Michelson interferometer readout and hypothetical optical coherence and refractive index controls
. Details. The figure's original control example stipulates equal arm light intensities, matching linear polarizations, and a Lorentzian optical spectrum giving with . In vacuum, arm mismatches and give and . Matching the arms improves the phase difference readout without changing its travel per cycle. A fine mirror scan of one cycle changes by only for the stipulated source, much less than . These are calculated values, not historical data.
A proposed measurement sequence follows from this model. Overlap the returning beams, match their linear polarizations, find a high-contrast delay and use a small calibrated mirror scan to determine the sign and fringe period. Read a fixed spatial sample of the interference fringes with a photodiode; record each arm separately and the dark signal, avoid detector saturation, and retain enough time resolution to count every cycle. Approach mechanical settings consistently and repeat the return scan to expose lost motion or drift. For very small changes, bias near phase quadrature, where is largest; an intensity extremum has zero first-order sensitivity. A known drive voltage alone does not provide measurement calibration for mirror travel: compare the actual travel with the fringe count using an independently known wavelength.
Control temperature, pressure, vibration and air currents because the instrument measures optical path difference, not mirror motion alone. For a fixed reference arm and fixed , the first-order change of the moving arm's round-trip path is . Thus an unrecognized mimics . A hypothetical , , and give apparent travel. Holding the mirror still while varying or monitoring the air conditions tests this confounder. Separate checks of alignment, wavelength stability and thermal expansion are necessary before sub-fringe measurement resolution becomes a defensible measurement uncertainty.
Pulse energy measurement with a photodiode 2026-11-05
Pulse energy measurement with a photodiode estimates incident laser pulse energy from collected signal electric charge. At fixed wavelength, suppose a photodiode has constant calibrated spectral responsivity in amperes per watt and remains in its linear operating range. If is the incident radiant flux, the signal photocurrent is . Integrating over the complete response givesThe units agree because amperes per watt equal coulombs per joule. Spectral responsivity must apply at the measurement wavelength; a broadband pulse requires accounting for its spectrum.
In an original example, let and let the baseline-corrected collected charge be . Then . An ideal charge amplifier with feedback capacitance would produce an output change of magnitude . The amplifier's polarity depends on its circuit convention.
A practical apparatus needs a calibrated detector, a charge-sensitive readout and a measurement window containing the full response. Detector baseline subtraction removes the estimated background, including dark current. A residual constant current of1microampere integrated over10microseconds contributes10picocoulombs; with the example responsivity this is a25picojoule energy error. Extending the window can capture a missing signal tail but also accumulate more background error.
Photodiode saturation invalidates the assumed constant response. Robert E. Vest and collaborators measured pulsed silicon-detector behavior and reported dependencies on applied reverse bias and illuminated area.[ref]
Measuring Pulse Energy With Solid-State Photodiodes by Robert E. Vest and others. Abstract, nonlinear pulsed response and bias/area dependence
A low average optical power therefore does not by itself establish linear response to individual pulses.Rodney W. Leonhardt's calibration report describes a complementary reference method: a slow thermal standard measures accumulated energy, while pulse count and a calibrated beam splitter ratio determine average energy in the test beam.[ref]
Calibration Service for Low-level Pulsed-Laser Radiometers at 1.06 µm: Pulse Energy and Peak Power by Rodney W. Leonhardt. §4, printed pp9–10
This connects a fast photodiode readout to an energy reference without requiring the thermal standard to resolve every pulse.Collected charge measures energy under the stated assumptions; it does not alone determine peak power. A flat30nanojoule pulse lasting100nanoseconds has power0.30watts, whereas a flat pulse of the same energy lasting1microsecond has power0.030watts. Those stipulated rectangular profiles illustrate why pulse shape and pulse duration are additional information.
Radiant exposure 2026-11-05
Radiant exposure measures the radiant energy received per unit surface area during a specified interval. At a fixed surface point , it is the time integral of incident irradiance :The area belongs to the receiving surface: its orientation and location matter. An irradiance in specifies a rate, while radiant exposure in specifies the accumulated arrival of energy.[ref]
International Lighting Vocabulary by CIE. CIE S 017:2020, entry 17-21-071, definition, symbols and note 3
For a uniform rectangular pulse, , with constant irradiance and duration . Stipulate one pulse at for , and another at for . Both deliver , but their peak irradiances differ by a factor of four. The two radiant exposures agree only after each complete pulse has arrived. A shorter observation window can intercept different fractions of them, and a nonlinear receiver can respond differently even to their complete pulses.
Original rectangular-pulse example: different irradiance histories accumulate the same radiant exposure.
Details. Integrating over a receiving patch gives its incident radiant energy:Here is the spatially averaged radiant exposure, rather than necessarily the local maximum. Uniform illumination is the special case where this spatially averaged radiant exposure equals the local value everywhere on the patch. In the rectangular-pulse example, a uniformly illuminated patch receives .
For an original spatial example, consider a circular Gaussian beam at a fixed transverse plane, with constant Gaussian beam radius throughout the pulse. Let be its total radiant power and its laser pulse energy. Integrating its transverse irradiance in time gives the radiant exposure of a Gaussian beam:The normalization follows by integrating from to . The disk of radius contains only , so its actual spatially averaged radiant exposure is , not . With stipulated and , the peak is , while the average inside that disk is . State the Gaussian beam radius convention and the receiving area with any reported value.
An energy readout alone therefore does not specify a surface's radiant exposure. In an ideal linear model for pulse energy measurement with a photodiode, dark-subtracted photocurrent has constant spectral responsivity at the chosen wavelength. Integrating the electric current gives signal electric charge . If a known patch of area is uniformly illuminated and all of its incident energy is collected, thenAs a separate stipulated example, and correspond to . Uniform illumination over gives . Background subtraction, collection geometry, detector linearity and an integration window covering the full response are part of this measurement model. Continuous-wave responsivity alone does not establish linearity for a short pulse: silicon-photodiode experiments have observed pulse-dependent saturation.[ref]
Measuring Pulse Energy With Solid-State Photodiodes by Robert E. Vest and others. Abstract: pulsed nonlinearity and dependence on reverse bias and illuminated area
A historical NIST measurement calibration separates these operations experimentally. A modulator produces pulses from a continuous-wave laser; a calibrated beam splitter sends one beam to a slow thermal reference and another to the pulse detector. The reference records the accumulated energy, while the pulse repetition rate and injection duration determine the mean energy per pulse. The detector's integrating electronics provide a pulse-energy readout. Converting such a readout to radiant exposure additionally requires the illuminated area and its spatial distribution.[ref]
The beam-use term fluence denotes the same time integral for a directed beam normal to the chosen surface. For light arriving from many directions, specify the angular geometry: planar irradiance and spherical irradiance are different quantities, so a value defined using one cannot simply be substituted for the other.[ref]
International Lighting Vocabulary by CIE. Entry 17-21-054, notes 1–2: directional radiance and the small-sphere construction
In 1977, H. Jeff Kimble, Mario Dagenais, and Leonard Mandel excited a dilute sodium atomic beam with a resonant dye laser. Apertures restricted the observed atoms; a microscope objective collected fluorescence perpendicular to both beams. A beam splitter sent the collected light to two photomultipliers. Shaped pulses drove the start and stop inputs of a time-to-digital converter; its delay histogram rose away from zero delay, indicating photon antibunching. Turning the atomic beam off measured scattered-laser backgrounds. More than one contributing atom and scattered light limited the comparison with an ideal isolated two-level system; the paper explicitly treated corrections.[ref]
Photon Antibunching in Resonance Fluorescence by Harry Jeffrey Kimble and others. 1977, p. 693, figure 2, equation (6), apparatus and atomic-beam-off/on control; MIT mirror
A simple reason for suppressed immediate re-emission is visible in an ideal two-level system. After spontaneous emission, the atom is in . Its emission operator obeys , so two instantaneous emissions are impossible. Excitation must rebuild the excited-state population before another emission. This argument explains the zero-delay idealization; the actual photon counting record additionally depends on collection, background, timing, and the number of contributing atoms.
Rodney W. Leonhardt 2026-11-05
Rodney W. Leonhardt is the author of a2004 National Institute of Standards and Technology report on measurement calibration of low-level pulsed-laser radiometers at a wavelength of1.06micrometres. It treats laser pulse energy and peak power as separate measurement quantities and describes the apparatus and measurement uncertainty needed to transfer a laboratory reference to a test instrument.[ref]
Calibration Service for Low-level Pulsed-Laser Radiometers at 1.06 µm: Pulse Energy and Peak Power by Rodney W. Leonhardt. NIST SP250-64, title/byline and §§3–5
The pulse-energy procedure uses a modulated laser, a calibrated beam splitter arrangement, a relatively slow thermal reference and a fast test radiometer. The reference accumulates energy over many pulses. Pulse repetition rate and the illumination interval determine the pulse count; the splitter ratio transfers the reference energy to the lower-energy test beam. The thermal detector need not resolve individual pulses.[ref]
For an original idealized transfer calculation, let the reference receive a total over equal pulses. Define as the reference-to-test beam energy ratio. The average energy per test pulse isIf the calibrated radiometer's selected output measure is0.20volts, the corresponding energy-per-output coefficient would be50picojoules per volt. These invented values explain the transfer arithmetic and do not reproduce the report's calibration results.
The output measure must be specified together with its coefficient. Leonhardt's report discusses offsets in the integrating-amplifier waveform and use of peak-to-peak electric potential difference as a readout. Its peak-power procedure separately compares continuous and pulsed conditions.[ref]
Calibration Service for Low-level Pulsed-Laser Radiometers at 1.06 µm: Pulse Energy and Peak Power by Rodney W. Leonhardt. §4, waveform/offset discussion; §5.1, peak-power comparison
The work illustrates how a measurement becomes reproducible by recording the optical transfer and electrical readout conventions, rather than only stating a detector sensitivity. Sagnac effect 2026-10-18
The Sagnac effect is a difference in return times, or an associated interference fringe displacement, for light sent in opposite directions around a closed optical path in a rotating apparatus. The light leaves one beam splitter, travels around the same loop in opposite senses, and returns to the rotating beam splitter and readout. For a slowly rotating apparatus in flat spacetime, the leading laboratory arrival-time difference isHere is the vacuum speed of light, is the apparatus's angular velocity relative to an inertial frame, and is the vector area oriented by the chosen positive direction around the closed curve . With positive rotation about the upward normal vector, the counterclockwise beam is labelled and arrives later. Reversing rotation, or reversing the orientation convention alone, changes the sign of .
The surprise is that a small loop can reveal its rotation by comparing two beams locally, even though each local vacuum speed of light measurement gives . A rotating reference frame is not a single inertial frame: a beam must catch the moving receiver after a full circuit. The Sagnac area law makes this difference a practical way to measure angular velocity, while the Relativistic Sagnac delay on a circular ring distinguishes laboratory coordinate time from the receiver's proper time.
Sagnac interferometer 2026-10-18
A Sagnac interferometer uses a beam splitter to send light with optical coherence around a common closed optical path in opposite directions, then recombines the beams to read their phase difference. Mirrors or a guiding path define the loop; the source, beam splitter, loop and readout rotate together in the basic experiment. A small relative beam tilt makes spatial interference fringes whose displacement reveals a change of phase difference. At one fixed output, a photodetector can instead record the corresponding change in light intensity.
Sagnac rotating-ring experiment 2026-10-18
In 1913 Georges Sagnac mounted a lamp, collimator, beam splitter, four mirrors and photographic plate on a horizontal rotating platform. The mirrors defined a closed polygon; the two beams traversed it in opposite senses. A slight beam splitter adjustment produced visible interference fringes. A motor brought the platform to steady rotation, and switching the lamp exposed the plate. Comparing opposite senses of rotation revealed a reversing displacement. Changing the beam splitter adjustment supplied a further check against optical deformation. The report compared measured and calculated displacements. Georges Sagnac interpreted this as evidence for a luminiferous aether; the relativistic derivation below reproduces the delay without that hypothesis.[ref]
Regarding the Proof for the Existence of a Luminiferous Ether Using a Rotating Inteferometer Experiment by Georges Sagnac. Georges Sagnac, 1913 second report, inspected in William Lonc's 2008 English translation, pp. 77–80, §§1–3: rotating apparatus, photographic procedure, reversal and deformation controls, comparison table, historical interpretation
The readout measures a relative wave phase, not two stopwatch readings separated by an optical transit time. Reversal isolates a contribution odd in angular velocity: if a signed interference fringe position is to leading order, then cancels the fixed offset . Changes of alignment or temperature correlated with reversal can survive this subtraction. Thus the experimental inference concerns a reproducible rotating-loop phase difference, with controls for disturbances; it is not by itself a unique test of an underlying propagation hypothesis.
Arthur Leonard Schawlow's 1981 lecture describes an external-cell arrangement: a beam splitter makes opposed beams, an optical chopper interrupts the pump, and a lock-in amplifier reads the probe detector with the chopper reference. Pump-on reduces absorption and raises the detector signal; scanning the laser finds the narrow resonance. Its first application used iodine; pressure and laser jitter limited the resolution.[ref]
Spectroscopy in a New Light by Arthur Leonard Schawlow. Schawlow, Spectroscopy in a New Light, pp. 39–40, Fig. 2
Counterpropagating pump and probe with synchronous transmission readout
. Original functional drawing of the cited external-cell principle. Beam separation is exaggerated; this is not a reconstruction of the pulsed sodium apparatus. Details.For an original detector model, let be the probe power reaching a photodetector without sample attenuation and let be the sample's probe optical depths. After subtracting a stable detector background, the transmitted powers areA positive synchronous change means less probe photon absorption, provided incident probe power is stable and scattered pump light is excluded. For small optical-depth changes the fractional signal is approximately . A lock-in amplifier's voltage also depends on gain, reference phase and chopping waveform; it is not automatically this dimensionless ratio.
Useful controls are blocking the pump, detuning far from the feature, reversing or adjusting beam alignment, and repeating the scan at several pump light intensities. Frequency calibration, sample pressure, transit through the beams and laser frequency fluctuations must be assessed separately. Nearly opposed beams leave a residual Doppler shift: for wave vectors , the simultaneous-resonance condition gives . Exact opposition removes this transverse sensitivity in the first-order model.
Tracy S. Clement 2026-11-05
Tracy S. Clement is a metrology researcher and coauthor, with Paul D. Hale and Dylan F. Williams, of NIST work on optical receiver frequency response. The 2001 paper describes combining two tunable lasers, a beam splitter, a polarizer and a single-mode material fibre for optical heterodyne detection.[ref]
The apparatus turns the difference between two optical frequencies into a measurable intensity oscillation. For beams in the same spatial mode and with matched polarization, an original derivation from interference givesHere and are the separate received powers and is their relative optical phase. The optical heterodyne modulation depth is thereforeEqual powers give . Powers in the ratio instead give , even though a clear beat remains. Dividing the measured photocurrent amplitude by the wrong assumed would give the wrong receiver response. Matching the beams' polarization and spatial modes is thus part of establishing a known stimulus, rather than merely improving the visual appearance of an interference pattern.
The equations explain the measurement principle; they do not assign individual apparatus tasks to Clement or claim a separate invention. The paper documents a joint contribution to making an optical stimulus useful for quantitative measurement calibration.
Ultraviolet Created 2026-10-05 Updated 2026-10-20
Ultraviolet is electromagnetic radiation with wavelengths shorter than the violet end of ordinarily visible light. The CIE convention uses –, subdivided into UV-A at –, UV-B at – and UV-C at –. These divisions name intervals; they do not mark universal changes in every material's response. The boundary with visible light is not sharp, and other fields use near-, far-, vacuum- and extreme-ultraviolet divisions. [ref]
https://cie.co.at/eilvterm/17-21-008. CIE, International Lighting Vocabulary (2020), ultraviolet radiation: ranges and boundary caveats
. For example, NASA's Solar Dynamics Observatory observes extreme-ultraviolet bands below ; the CIE interval is therefore not the full extent of astronomical ultraviolet usage. [ref]https://sdo.gsfc.nasa.gov/data/channels.php. NASA, Solar Dynamics Observatory HMI and AIA channels: the Atmospheric Imaging Assembly's ultraviolet and extreme-ultraviolet bands
.The CIE ultraviolet interval corresponds to frequencies about – and photon energies –, using vacuum wavelength and with Planck constant and speed of light . Shorter wavelength means larger energy per photon; larger irradiance means more incident power per area. These are different comparisons. At the same incident power, a ultraviolet beam has fewer photons per second than a infrared beam, even though each ultraviolet photon has more energy.
A common ultraviolet mechanism is photon absorption promoting an electron from a ground state to an excited state of a chemical atom or molecule. By contrast, many infrared absorption bands involve stretching or bending chemical bonds. For either mechanism, both the energy difference and the coupling to the incident electromagnetic field matter. One concrete example is the absorption discussed for butadiene: an electron moves between molecular orbitals, and the position of the absorption reveals their energy separation. Ultraviolet absorption can therefore probe molecular orbitals, while infrared spectroscopy probes many molecular motions. [ref]
https://openstax.org/books/organic-chemistry/pages/14-7-ultraviolet-spectroscopy. John McMurry, Organic Chemistry, §14.7: ultraviolet excitation and the 217 nm butadiene example
.Subsequent chemical reactions depend on the absorbing chemical substance and its environment; the label ultraviolet alone does not predict a reaction rate. At sufficiently short wavelength, a photon can cause photoionization if its energy exceeds the relevant ionization energy. There is no single wavelength that separates all ionizing from all nonionizing behavior for every chemical substance. Likewise, eventual heating after photon absorption is possible in both ultraviolet and infrared. The distinction is useful because different optical spectra expose different processes, not because the two ranges obey different laws of electromagnetism.
Ritter's 1801 observation used a chemical readout rather than a thermometer. In Caltech/IPAC's account, sunlight passed through a glass optical prism and silver chloride was placed in different parts of the optical spectrum. Darkening was weak toward red, stronger toward violet, and pronounced beyond the visible violet edge. The measured evidence was a change in the chemical substance's appearance at a spatial location with no visible color, establishing radiation beyond the eye's ordinary range. It was not a calibrated measurement of a universal ultraviolet optical spectrum. [ref]
https://coolcosmos.ipac.caltech.edu/page/lesson_ritter_experiment. Caltech/IPAC Cool Cosmos, The Ritter Experiment (2022): the historical silver-chloride observation
.A controlled modern interpretation of such a chemical test would compare equal exposure durations and matched fresh samples, keep an unexposed sample in darkness, monitor temperature, and distinguish scattered visible light from the intended short-wavelength beam with characterized filtering. These are proposed controls, not an assertion that Ritter performed them. Unequal sample preparation or heating can change a reaction rate. The optical prism, terrestrial atmosphere and sample response restrict which wavelengths the test detects. Darkening is a detector-weighted response, so its spatial maximum is not automatically the maximum of incoming radiant flux or of photon energy.
Thomas Larason and Jeanne Houston's NIST report documents a quantitative ultraviolet detector comparison. An argon source feeds a double-diffraction grating spectrometer; a shutter and mirrors deliver the selected band to a working-standard or test photodiode, successively at the same position. A beam splitter supplies a monitor photodiode. Amplifiers and voltmeters record detector and monitor simultaneously; shutter-closed readings provide dark baselines. The – scan, typically with bandpass, measures spectral responsivity. Monitor normalization suppresses source drift; known spectral lines calibrate wavelength. The report assesses temperature effects and unwanted light outside the band, weighted by detector response. The latter need not cancel between different spectral responsivities. [ref].
This determines electrical response without visual perception. Bandpass and spatial nonuniformity limit the calibration; aging and changed temperature can affect later use. The scan includes visible light and ultraviolet; no response below follows from it.
For a narrow beam, the idealized substitution calculation isHere is spectral responsivity; and are dark-subtracted detector and monitor electric currents. This follows by writing for each detector and using the monitor to track changes in radiant flux. It assumes linear response, unchanged beam splitting and collection geometry, and a sufficiently narrow band. For an illustrative standard with , ratios and give . These invented arithmetic inputs illustrate the method; they are not NIST measurements.
An independent photon-counting model connects spectral responsivity to quantum efficiency. If is the number of collected electrons per incident photon, there is no internal multiplication, and is the elementary charge, thenAt illustrative constant and , this gives about at and at . The ultraviolet readout is smaller in this model because there are fewer incident photons, not because their individual energies are smaller. Real quantum efficiency depends on wavelength, windows, absorption and carrier collection; a detector's band gap and construction determine its usable range. Constant across these two wavelengths is an analytical comparison, not a claimed specification of a real photodiode.
Photon energy across conventional ultraviolet and infrared ranges, with a schematic detector-substitution experiment
. Details. The curve is the calculated relation , and the shaded ranges use the stated CIE conventions. The lower diagram abstracts the detector comparison rather than reproducing an instrument drawing. Together with the infrared heat-balance illustration, it shows why a calibrated detector response, rather than what looks bright to an observer, is the bridge between invisible radiation and a quantitative measurement.
Z. Y. Ou Created 2026-10-20 Updated 2026-11-05
An experimental optical physicist who coauthored the 1987 Hong-Ou-Mandel effect experiment. The team varied the relative arrival time of a photon pair at a beam splitter and counted coincidences. A narrow interference feature revealed subpicosecond timing information, converting a time scale too short for direct electronic timing into a path-delay measurement.[ref]
Measurement of Subpicosecond Time Intervals between Two Photons by Interference by C. K. Hong and others. Abstract and experimental arrangement: down-converted pairs and relative-delay interference





