Calculation workspace
Select a mode, enter measured values, and review assumptions.
Live formula preview
The displayed equation changes with the selected mode.
Included tools and options
The page combines common Michelson calculations with advanced corrections, visualization, education, and export functions.
Optical path difference
Single-pass, double-pass, signed, absolute, refractive-index, and angle-aware calculations.
Fringe shift
Whole and fractional fringe counts from mirror movement or sample changes.
Wavelength measurement
Recover wavelength from calibrated displacement and counted fringes.
Phase analysis
Radians, degrees, wrapped phase, fringe order, and bright-dark classification.
Intensity and visibility
Unequal beams, coherence factors, maximum, minimum, and Michelson contrast.
Refractive index
Gas-cell and sample-index calculations using observed fringe shifts.
Transparent plate
Normal and oblique plate models with Snell-law correction.
Coherence check
Gaussian, exponential, Lorentzian, and hard-cutoff visibility models.
Thermal drift
Linear expansion, optical path change, phase drift, and fringe drift.
Gas correction
Approximate pressure-temperature refractivity scaling for dilute gases.
Synthetic wavelength
Two-wavelength beat length, frequency difference, and range.
Uncertainty propagation
First-order root-sum-square propagation for key inputs.
Data export
Copy, CSV, PDF, print, and shareable URL tools.
Visual output
Dynamic interference, coherence, sensitivity, and calibration graphs.
Browser history
Local history and reusable examples without database storage.
Responsive access
Keyboard-friendly controls, mobile layout, labels, and theme support.
Michelson formula library
These equations use consistent SI units internally. The symbol P equals two for a standard round-trip Michelson arm and one for a single pass.
Optical path from mirror motion
At normal incidence in air, a mirror movement d normally produces approximately 2d optical path change.
Fringe count
One full fringe corresponds to one wavelength of optical path change.
Mirror displacement
This rearranged form converts counted fringes into calibrated mirror travel.
Measured wavelength
Use a traceable displacement and reliable fringe count for wavelength metrology.
Phase difference
Constructive conditions occur near φ = 2πm. Destructive conditions occur near φ = (2m+1)π.
Interference intensity
The factor V represents coherence, polarization, alignment, and detector averaging losses.
Refractive-index change
The sample must occupy a known optical length L in one arm.
Transparent plate at normal incidence
The plate replaces ambient medium of the same geometric thickness.
Two unequal arms
This general form includes independent lengths, indices, and effective angles.
Thermal expansion
Environmental compensation may also require refractive-index and mount-gradient corrections.
Synthetic wavelength
Close wavelengths create a long effective period for extended unambiguous measurements.
Independent uncertainty propagation
This first-order method assumes small, uncorrelated input uncertainties.
How to use this calculator
1. Choose the measurement objective
Select optical path difference, fringe count, mirror displacement, wavelength, phase, intensity, refractive index, plate, arm comparison, coherence, thermal, pressure, synthetic wavelength, or uncertainty mode.
2. Select the light source
Choose a preset laser or enter a custom wavelength. Confirm whether your wavelength is specified in vacuum, air, or another medium.
3. Define the optical geometry
Use double pass for a standard Michelson arm. Enter the refractive index and any meaningful cosine-angle correction.
4. Enter measured quantities
Provide mirror displacement, fringe count, path difference, sample length, beam intensities, pressure, temperature, or uncertainty values as required.
5. Calculate and inspect
Review the main result, converted units, phase, fringe order, visibility, classifications, warnings, and plotted sensitivity.
6. Export and document
Copy the table, download CSV, generate a PDF, print the report, or create a link containing the current form values.
Example data table
These examples provide quick reasonableness checks. Exact experimental values depend on medium, geometry, wavelength convention, and calibration.
| Scenario | Wavelength | Input | Expected scale |
|---|---|---|---|
| Mirror motion | 632.8 nm | 0.3164 µm | 1 fringe |
| Ten fringe count | 632.8 nm | 3.164 µm | 10 fringes |
| Gas index cell | 632.8 nm | 10 cm cell | n−1 = 3.164×10⁻⁵ for 10 fringes |
| Glass plate | 632.8 nm | 1 mm, n=1.5 | ≈1580 fringes, double pass |
| Phase example | 632.8 nm | 158.2 nm OPD | π/2 radians |
| Synthetic wavelength | 632.8 and 633.0 nm | 0.2 nm spacing | ≈2.002 m synthetic wavelength |
Michelson interferometer theory and measurement practice
How the interferometer creates fringes
A Michelson interferometer divides one optical field at a beam splitter. The reflected and transmitted portions travel through separate arms. Mirrors return both beams to the beam splitter, where portions recombine at the detector or observation screen.
The recombined intensity depends on relative phase. Relative phase depends primarily on optical path difference. A changing mirror position, refractive index, sample thickness, pressure, temperature, or wavelength can therefore move the fringe pattern.
Geometric distance versus optical distance
Geometric length is the physical distance measured with a ruler, encoder, or displacement stage. Optical path length weights each segment by refractive index. A ten-centimeter cell filled with gas contributes slightly more optical path than the same cell evacuated.
For nonuniform media, optical path length is an integral of refractive index along the ray. This calculator uses lumped uniform segments. Precision work with gradients, turbulence, dispersion, or curved rays may need numerical integration or ray tracing.
Bright and dark conditions
Equal coherent beams are brightest when their phase difference is an integer multiple of two pi. They are darkest when the phase difference is an odd multiple of pi. Unequal beam intensities prevent perfectly dark minima.
Real fringes also depend on polarization and spatial overlap. Orthogonal polarizations do not produce ordinary intensity fringes at a polarization-insensitive detector. Beam curvature, tilt, and aperture geometry determine whether fringes appear circular, straight, localized, or distorted.
Circular and straight fringes
Nearly parallel virtual mirrors can produce circular fringes of equal inclination. A slight relative tilt often produces approximately straight fringes. Their spacing and curvature contain alignment information beyond simple path difference.
This calculator focuses on scalar optical path and phase. It does not fully model wavefront curvature, beam divergence, aberrations, shear, detector pixel integration, or spatially varying phase maps.
Wavelength measurement
A calibrated translation stage moves one mirror through a measured distance. The experiment counts the number of fringes passing a reference point. For normal double-pass geometry, wavelength is approximately twice displacement divided by fringe count.
Reliable fringe counting may use a photodiode, comparator, phase-unwrapping algorithm, or image-processing system. Direction reversals can introduce stage backlash. Air-index changes can bias long or high-precision measurements.
Refractive-index measurement
A cell in one arm is evacuated, filled, pressurized, heated, or otherwise changed. The optical path change shifts fringes. Dividing the measured path change by the traversed sample length gives the refractive-index change.
The correct pass count matters. A cell positioned before the end mirror is usually traversed twice. A cell placed in a single-pass branch or external arrangement may require a different factor.
Coherence and spectral width
Interference requires a stable phase relationship over the arm delay. A narrow-line laser can tolerate a much larger path difference than a broadband source. White-light interference is localized near zero path difference.
Coherence length definitions vary by spectral line shape and visibility threshold. Gaussian and Lorentzian spectra produce different envelope forms. The calculator therefore offers multiple simplified models rather than claiming one universal relation.
Environmental stability
Temperature can expand mirror mounts and optical tables. Air temperature, pressure, humidity, and composition change refractive index. Acoustic noise and floor vibration change alignment and path length.
High-resolution systems often use enclosures, thermal control, vibration isolation, short exposed air paths, common-path designs, vacuum paths, environmental sensors, active stabilization, and differential measurement strategies.
Dispersion and wavelength convention
Refractive index varies with wavelength. Phase index governs monochromatic phase accumulation, while group index governs pulse-envelope delay. Broadband or multiwavelength systems may require wavelength-dependent index values.
Laser specifications may quote vacuum wavelength, air wavelength, nominal center wavelength, or a transition line. Use the correct convention and uncertainty for the required accuracy.
Uncertainty analysis guidance
A calculator result is not automatically a complete measurement result. A defensible report identifies the measurand, model equation, input estimates, standard uncertainties, probability distributions, sensitivity coefficients, correlations, combined uncertainty, coverage factor, and traceability.
Common random contributions
- Fringe-count repeatability and threshold noise.
- Phase-estimator noise and detector shot noise.
- Vibration and acoustic path fluctuations.
- Short-term laser frequency variation.
- Environmental sensor repeatability.
- Image-fitting and centroid uncertainty.
Common systematic contributions
- Translation-stage scale calibration and nonlinearity.
- Cosine error from beam-stage misalignment.
- Dead path and refractive-index compensation error.
- Beam splitter phase and polarization effects.
- Abbe offset, pitch, yaw, and parasitic rotations.
- Incorrect wavelength or pass-count convention.
Root-sum-square propagation is appropriate for small independent uncertainties in smooth models. Monte Carlo propagation is often better for nonlinear models, bounded distributions, significant asymmetry, phase wrapping, threshold fringe counts, and correlated inputs.
Frequently asked questions
Optical interferometry glossary
| Term | Meaning |
|---|---|
| Beam splitter | An optic that divides one incident beam into reflected and transmitted beams. |
| Coherence | The degree to which optical phase relationships remain predictable over time or space. |
| Coherence envelope | The visibility curve surrounding rapidly oscillating interference fringes. |
| Fringe | A bright or dark interference feature caused by phase-dependent intensity. |
| Fringe order | The optical path difference expressed in wavelength units. |
| Group index | An index governing pulse-envelope propagation, different from phase index in dispersive media. |
| Michelson interferometer | A two-arm interferometer using a beam splitter and retroreflected beams. |
| Optical path length | The integral of refractive index along a light path. |
| Optical path difference | The difference between two optical path lengths that recombine. |
| Phase difference | The angular offset between two sinusoidal optical fields. |
| Refractive index | The ratio of vacuum light speed to phase velocity in a medium. |
| Synthetic wavelength | A long effective wavelength formed from the difference between two optical frequencies. |
| Visibility | A normalized measure of fringe contrast. |
| Wavenumber | The spatial frequency of a wave, commonly 1/λ or 2π/λ depending on convention. |
| Zero path difference | The condition where both interferometer arms have equal optical path length. |
Laser safety
Never look directly into a laser beam or a specular reflection. Use the lowest practical power, proper beam stops, controlled beam height, suitable eyewear when required, and procedures appropriate to the laser classification.
Infrared beams may be invisible while remaining hazardous. Follow your institution’s laser safety program, equipment documentation, and applicable regulations.
Implementation notes for developers
This PHP file is intentionally self-contained and large. Core calculations run on the server. Interface behavior, live previews, charts, exports, share links, presets, and local history run in the browser.
No database is required. Browser history uses localStorage. External libraries load from CDNs, so offline deployments should host Bootstrap, Chart.js, jsPDF, and AutoTable locally.
For production metrology, add authenticated project storage, calibration records, Monte Carlo propagation, correlated uncertainty matrices, environmental sensor import, instrument APIs, wavelength-in-air models, humidity corrections, and traceable report signatures.