Photonic Doppler Velocimetry

What is modelled

Light reflected from a surface moving at velocity v along the beam is Doppler shifted by 2v/lambda. A coherent receiver mixes it with a reference that is offset by f_off, so the detector sees a beat at f_off + 2v/lambda, and the optical phase 4 pi x / lambda advances one full cycle (one fringe) for every lambda/2 of travel. The surface motion (harmonic, constant speed or ring-down) is computed with the 3D page module; the page builds the beat waveform, counts fringes and takes a windowed spectrum.

Equations

• Doppler shift fD = 2 v / λ.
• Optical phase φ = 4 π x / λ; one fringe per λ / 2 of travel.
• Detector signal cos(2 π foff t + φ(t)), beat frequency foff + fD.
• Harmonic motion x = A sin(2 π f t), peak speed 2 π f A.
• Largest unambiguous speed for sampling rate f_s: v = λ fs / 4 (Nyquist).

Checks run in the page

• Phase derivative against 2v over lambda and the 3D module dopplerHz.
• Zero crossings of the phase over a quarter stroke against travel over lambda/2.
• Hann-window DFT peak of a constant-speed pass against offset plus 2v over lambda.
• Module velocity against the central difference of its position, and wavelength scaling.

Honest simplifications

Ideal narrowband laser, perfect quadrature, no speckle, noise, dispersion or fibre-delay effects, and a surface that stays perpendicular to the beam. The spectrum uses a short 8 ms window, so its peak is wide when the speed changes quickly.

• Press Green laser 532 nm and see the Doppler shift per mm/s nearly triple.
• Press Constant speed and watch the spectrum collapse into one sharp line at the offset plus 2v/lambda.
• Raise the stroke and read how many fringes the mirror sweeps: every 775 nm of motion is one fringe at 1550 nm.