Gravitational Wave Detector: Michelson Interferometer Response (2D)

Wave Parameters

Strain amplitude h₀ 0.30

Frequency f (Hz) 0.50

Polarization mix ψ 0°

Fringe sensitivity L/λ 1.50

Live Readouts

Strain h₊(t)

0.000

Strain h₍ₓ₎(t)

0.000

Round-trip Δφ

0.00 rad

Fringe order n

0.000

Output intensity I(t)

50.0%

Period T

2.00 s

How it works (2D model)

The 3D companion shows what a gravitational wave does to free-falling test masses directly: a ring stretches along one axis while it squeezes the perpendicular one. This 2D simulator instead models the instrument that actually detects that effect — an L-shaped Michelson interferometer with arms fixed along its own x- and y-axes, exactly like LIGO. Projecting the same geodesic-deviation strain onto an arm at angle θ gives the detector's antenna-pattern response:

ΔL(θ)/L = ½[ h₊(t)·cos(2θ) + h₍ₓ₎(t)·sin(2θ) ]
h₊(t)  = h₀ cos(2ψ) cos(ωt)      θ = 0°  (x-arm)
h₍ₓ₎(t) = h₀ sin(2ψ) cos(ωt)      θ = 90° (y-arm)

For arms at 0° and 90°, cos(2θ) = +1 and −1 while sin(2θ) = 0 for both — so the differential arm signal ΔL_x − ΔL_y = L·h₊(t) depends only on h₊ measured in the detector's own frame. That is a real, well-known feature of interferometric detectors, not a simplification: an L-shaped detector aligned with the plus axis is exactly blind to a wave that is pure cross (ψ = 45°) in its own frame — you can watch h₍ₓ₎(t) read a nonzero value on the left while the interferometer signal stays flat.

The differential arm motion accumulates a real round-trip optical phase shift, which the photodetector converts into an interference intensity — the two-beam interference law:

Δφ(t) = 4π (L/λ) · h₊(t)          [round trip = 2·(ΔL_x − ΔL_y)]
I(t)/I₀ = ½[ 1 + cos(Δφ(t)) ]

Because Δφ is itself a cosine of a cosine, the output intensity trace is a genuinely different — and genuinely nonlinear — waveform from the strain that drives it, exactly as a real detector's photocurrent is not a scaled copy of h(t). Real strains are h₀ ~ 10⁻²¹ and L/λ ~ 10⁹, giving a slowly-drifting fringe; both are exaggerated here by many orders of magnitude, purely for visibility, in the same spirit as the 3D companion's exaggerated amplitude.

h₀ slider — peak strain amplitude (exaggerated for visibility).
f slider — wave frequency; sets angular frequency ω = 2πf.
ψ slider — polarization mix angle between pure plus (0°) and pure cross (45°) states, defined relative to the detector's own arms.
Fringe sensitivity L/λ — how many optical wavelengths fit the arm length; higher values pack more interference fringes into the same strain swing, just like a longer real arm or shorter laser wavelength.

Arm mirrors (x, y) Photodetector output Strain h₊(t) trace