Michelson Interferometer Simulator — Path Difference and Fringes
Vary path difference, wavelength and mirror travel and watch detected intensity and fringe order respond in real time. The optical layout and fringe pattern are shown side by side.
Parameters
Path difference ΔL
μm
Wavelength λ
nm
Mirror travel d
nm
Input intensity I0
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Total path difference Δ_total = ΔL + d. Detected intensity I = I0 cos²(2π·Δ_total/λ).
Results
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Detected intensity I/I0
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Phase difference (mod 2π)
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Fringe order m
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Mirror travel per fringe (λ/2)
Optical layout
Laser to beam splitter to mirror M1 (right) / M2 (top), then back to detector (bottom). Arm length difference ΔL is highlighted.
Fringe pattern I(d) and current point
Horizontal axis = mirror travel d (0 to λ) / vertical axis = intensity I/I0, yellow marker = current d.
Theory & Key Formulas
Given a total path difference Δ_total and input intensity I0, the detector intensity follows the two-beam interference law.
One fringe per λ/2 of mirror travel arises because each arm carries the light on a round trip, doubling the optical path difference.
What is the Michelson interferometer simulator?
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My optics class introduced the Michelson interferometer, but what exactly is it for in practical terms?
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In one line, it turns the wavelength of light into a yardstick for length. A laser is split into two arms, each reflects from its mirror, and the beams recombine on a detector. The intensity follows $I = I_0 \cos^2(2\pi \Delta_\text{total}/\lambda)$, so as you increase the mirror travel d in the simulator, the intensity oscillates cleanly.
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How far does the mirror have to move for one fringe?
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From the mirror's point of view, λ/2. For a He-Ne laser at 633 nm, that's 316.5 nm. The key fact is that moving the mirror by d adds 2d to the optical path because the beam goes round trip. The simulator shows fringe order m = 2·Δ_total/λ, so each integer step of m is one fringe.
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So how is this used to measure wavelength itself?
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It's the inverse problem. Move the mirror by a known distance D and count N fringes; then λ = 2D/N. Industrial laser interferometers use a stabilized He-Ne reference and measure stage positions with sub-nm resolution. They are everywhere in semiconductor lithography, ultra-precision machining, and engine-component inspection.
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I read LIGO uses a Michelson interferometer too. Is that the same idea?
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Yes. LIGO's arms are 4 km long and are enhanced by Fabry-Perot cavities. A passing gravitational wave changes each arm length by about 10⁻¹⁸ m — far smaller than a proton — and is detected by sitting near a dark fringe where $I = I_0 \cos^2(2\pi \Delta/\lambda)$ is most sensitive to small Δ changes. You can feel the same effect in this simulator: between bright and dark, a tiny d change produces a large I change.
Physical model and key formulas
The Michelson interferometer splits a coherent beam at a beam splitter, sends each half down a separate arm to a mirror, and recombines the returning beams. The detected intensity follows the two-beam interference formula $I = I_0 \cos^2(2\pi \Delta_\text{total}/\lambda)$, where Δ_total is the total optical path difference between the two arms. The simulator separates this into a static arm length difference ΔL and an additional mirror travel d that the user controls.
An important factor of two appears because each arm carries the light on a round trip: if the mirror moves by d, the optical path changes by 2d. The phase difference can equivalently be written as $\varphi = 4\pi \Delta_\text{total}/\lambda$, giving cos²(φ/2). One full bright-dark-bright cycle corresponds to a 2π change in φ, which is just λ/2 of mirror travel. For a He-Ne laser (λ = 633 nm) this is 316.5 nm; for a frequency-doubled Nd:YAG (532 nm) it is 266 nm.
The fringe order is defined as $m = 2\Delta_\text{total}/\lambda$, with integer m corresponding to bright fringes and half-integer m to dark fringes. With ΔL = 1 μm and λ = 633 nm one finds m ≈ 3.16, locating the operating point between the third and fourth bright fringe from centre.
Real-world applications
Precision length metrology: The stages of semiconductor steppers and ultra-precision machine tools are tracked by laser interferometers (typically He-Ne based two-beam systems) with sub-nm resolution. The principle is identical to this simulator: each mirror displacement modulates detector intensity, and the resulting fringe count translates into displacement.
Spectroscopy (FTIR): Fourier transform infrared spectrometers continuously sweep one mirror while recording the interferogram, which is then Fourier-transformed into a spectrum. They are workhorses of analytical chemistry, gas analysis and thin-film characterization.
Gravitational-wave detection (LIGO/KAGRA): Kilometre-scale Michelson interferometers use Fabry-Perot cavities and power recycling to achieve strain sensitivities below 10⁻²¹. The first direct detection of gravitational waves in 2015 was made possible by exactly this technique.
White-light interferometry (WLI): Using broadband light, the short coherence length lets the system locate the zero-path-difference point with high accuracy. Surface step heights and thin-film thickness can be measured with sub-10 nm precision, widely used in semiconductor and MEMS metrology.
Common misconceptions and pitfalls
The most frequent mistake is to identify the mirror displacement with the optical path difference. Because the light makes a round trip, moving a mirror by d changes the optical path by 2d. This simulator takes Δ_total directly as input for clarity, but in real instruments you must always remember the factor of two. The "one fringe per λ/2 of mirror travel" rule is a direct consequence.
Another pitfall is to assume that any light source produces fringes. Lasers can show interference at path differences up to metres or kilometres, but white light or LEDs have coherence lengths of just a few μm to a few hundred μm. Beyond that, contributions from different wavelengths randomize the phase and the fringes wash out. This monochromatic cos² model does not capture coherence-length effects.
Finally, when the cos² minimum reaches zero, students sometimes conclude that the light has been "destroyed". In reality, one output port shows cos² and the other (in a real beam splitter, the unused port) shows sin²; energy is conserved and their sum is always I0. Some LIGO configurations actually use the "other" port as the main signal output.
Frequently asked questions
A Michelson is a "folded" geometry where the light passes through the same beam splitter twice, doubling the path-difference sensitivity. A Mach-Zehnder is a "linear" geometry with two separate beam splitters, in which the sample beam and reference beam are physically separated; this is favoured in fibre sensors and quantum-optics experiments. This simulator handles only the Michelson case.
A real beam splitter is a glass slab; one beam passes through the glass once and the other passes through three times, creating an asymmetric optical path. A "compensator plate" of identical thickness is inserted in the other arm so that the glass paths balance. This matters most in white-light interferometry, where neglecting it kills fringe visibility. The simulator implicitly assumes such compensation.
Common causes are: (1) the arm-length difference exceeds the coherence length of the source, (2) the mirrors are tilted so the two beams do not overlap on the detector, (3) the beam splitter asymmetry is uncompensated, (4) vibration or thermal drift averages out the fringes. Alignment (case 2) is usually the hardest; experimenters first locate a coarse fringe with a highly coherent laser and then refine.
Yes. Place a gas cell or thin film in one arm: changing density or refractive index changes the optical path nL, shifting the fringes. Counting N fringe shifts for a known sample length L gives Δn = Nλ/(2L). This technique is used to measure pressure and temperature dependence of gas indices and the optical thickness of thin films, and was the basis of the original Michelson-Morley experiment.