Calculator
Formula Used
In a Michelson interferometer, moving one mirror changes the round-trip optical distance. For a medium with refractive index n and beam angle θ from the mirror normal:
- OPD = 2 · n · Δd · cos(θ)
- N = OPD / λ (fringe count)
- φ = 2π · OPD / λ (phase difference)
If θ = 0° and n ≈ 1, then OPD ≈ 2Δd, and each fringe corresponds to a mirror shift of λ/2.
How to Use This Calculator
- Select what you want to solve for (OPD, Δd, N, λ, or φ).
- Enter the known values and choose appropriate units.
- Use n for your medium and θ for the beam angle.
- Press Calculate to show results above the form.
- Use the CSV or PDF buttons to export the computed table.
Example Data Table
| Δd (mm) | n | θ (deg) | λ (nm) | OPD (mm) | N (fringes) |
|---|---|---|---|---|---|
| 0.50 | 1.000 | 0 | 632.8 | 1.00 | 1580.29 |
| 0.25 | 1.333 | 10 | 532 | 0.66 | 1238.93 |
| 1.00 | 1.000 | 5 | 1064 | 1.99 | 1868.50 |
Examples assume the calculator relations and rounding in display.
Michelson Path Difference in Practice
Interferometer geometry and round trips
In a Michelson setup, each arm is traversed twice, so mirror motion creates a doubled optical change. With normal incidence in air, a 1.0 mm mirror shift produces about 2.0 mm optical path difference. This doubling is why interferometers achieve high displacement sensitivity with simple mechanics.
Optical path difference is the effective distance mismatch between arms, scaled by refractive index. It sets the interference condition and determines whether fringes brighten or darken. When OPD changes by one wavelength, the intensity pattern repeats, making OPD a direct bridge between motion and optical phase.
Fringe counting links motion to wavelength
Fringe count N is the number of intensity cycles observed as OPD changes. For θ ≈ 0° and n ≈ 1, each fringe corresponds to a mirror displacement of λ/2. For a 632.8 nm source, one fringe is about 316.4 nm of mirror travel.
Phase difference adds sub-fringe resolution
Counting fringes gives integer cycles, while phase φ captures fractional cycles. A phase shift of π indicates half a wavelength of OPD change. Modern phase-tracking methods can resolve small fractions of a fringe, enabling nanometer and sometimes sub-nanometer displacement measurement under stable conditions.
Refractive index and environmental corrections
The factor n matters whenever light propagates in air, glass, or liquids. Air index is near 1, but it varies with temperature, pressure, humidity, and CO₂. In precision metrology, a small change in n can bias OPD and the inferred displacement, so environmental monitoring improves accuracy.
Angle term and alignment tolerance
When the beam hits the mirror at angle θ, the effective component of motion along the beam is reduced by cos(θ). Even a modest 10° angle yields about a 1.5% reduction. This calculator includes angle to help compare ideal alignment against real optical layouts and mounting constraints.
Choosing wavelength and coherence considerations
Shorter wavelengths increase fringe density, improving displacement sensitivity per millimeter of travel. However, the usable OPD range is limited by coherence length and spectral width. Single-frequency lasers support long OPD ranges, while broadband sources wash out fringes quickly. Selecting a source balances range, stability, and measurement goals.
Recommended workflow for reliable results
Start by setting n and θ to match your setup, then enter mirror travel or measured fringes. Use consistent units and keep the wavelength in the same medium. After calculating, export CSV or PDF to document the assumptions, inputs, and computed quantities for lab notes or reports.
FAQs
1) Why is there a factor of 2 in OPD?
The beam travels to the moving mirror and back. A mirror displacement changes the round-trip distance twice, so the optical path difference becomes 2·Δd, then scaled by n and cos(θ) when applicable.
2) How many fringes appear for a given mirror shift?
Use N = OPD/λ. For normal incidence in air, OPD ≈ 2Δd, so N ≈ 2Δd/λ. A 10 µm shift at 632.8 nm yields roughly 31.6 fringes.
3) What does the beam angle input mean?
It is the angle between the beam and the mirror normal. The effective OPD contribution from mirror travel is reduced by cos(θ). If your setup is close to normal incidence, set θ near 0° for best agreement.
4) Should I use refractive index as 1.000?
For many lab estimates in air, n ≈ 1.000 is acceptable. For precision work, use a corrected air index or the medium index in the optical path. Errors in n directly scale OPD and inferred displacement.
5) Can this calculator estimate wavelength from fringes?
Yes. Choose “Wavelength (λ)” and enter mirror displacement, refractive index, beam angle, and fringe count. The calculator returns λ = OPD/N, which is useful when a stable source is measured via fringe counting.
6) What is phase difference used for?
Phase converts OPD into an angular measure: φ = 2π·OPD/λ. It is helpful for tracking fractional fringes and analyzing interference signals from photodiodes or demodulation electronics, especially when the intensity varies continuously.
7) Why do fringes disappear at large OPD?
Visibility drops when OPD exceeds the coherence length of the source. Broadband sources have short coherence lengths, so fringes wash out quickly. Narrow-line lasers maintain coherence longer, allowing larger mirror scans before contrast becomes too low.