Michelson Interferometer

A Michelson interferometer measures the intensity \(I(x)\) at the detector as a function of the optical path difference \(x = 2d\), where \(d\) is how far the moving mirror has been displaced. For two monochromatic lines, each one contributes its own cosine fringe, and together they beat with period \(1/\Delta\sigma\) (with \(\sigma = 1/\lambda\)). The spectrum is the Fourier transform of the measured part of \(I(x)\). Scanning a length \(L\) gives a resolution of about \(\delta\sigma \approx 1/L\), i.e. \(\delta\lambda \approx \lambda^2/L\). To separate the two lines you therefore have to scan at least one beat period.

Michelson interferometer: from I(x) to the spectrum Press Scan, or click and drag on I(x) to move the mirror

Beam 1

Wavelength λ₁

Intensity I₁

Intensity noise

Beam 2

Separation Δλ = λ₂ − λ₁ nm

Intensity I₂

Intensity noise

Measurement

Scan start (OPD)

Scan end (OPD)

Sampling step Δx

Detector noise

Scan duration

▶ Scan Measure whole range New noise Apodise (Hann window) Log spectrum Spectrum: zoom on the lines Spectrum: full range (shows aliasing)

(a) The interferometer. Mirror M₂ moves by d = x/2 (displacement exaggerated).

(b) I(x) around the mirror position: beam 1, beam 2, their sum, and the measured samples (dots)

(c) The interferogram I(x) over the measurement range. Click or drag to move the mirror.

(d) Spectrum = |Fourier transform| of the measured part of I(x). The dashed lines mark the true wavelengths.

Things to try

1. Watch the resolution build up. With the sodium doublet (589.0 / 589.6 nm), press Scan. At first the spectrum is a single broad peak. The two lines separate once the scan has covered about one beat period of \(I(x)\), \(L \approx 1/\Delta\sigma \approx 0.58\) mm. Click on \(I(x)\) to rewind and step through it.
2. Closer lines need longer scans. Make \(\Delta\lambda\) smaller. The beats in \(I(x)\) get longer, and you need a longer scan before the readout turns green. With the full 4 mm the limit is \(\delta\lambda \approx \lambda^2/L \approx 0.09\) nm.
3. Only the scan length matters. Move the scan start away from zero. The resolution depends on \(L = x_{\text{end}} - x_{\text{start}}\), not on where you measure.
4. Noise. Increase the intensity noise on one beam or the detector noise. The fringes in (b) scatter and the spectrum gets a noise floor (turn on Log spectrum). Scanning longer helps: the line height grows with the number of samples \(N\), the noise only with \(\sqrt N\).
5. A weak line next to a strong one. Set \(I_2 = 0.1\) and choose \(\Delta\lambda\) so the lines are just resolved. The weak line hides in the side lobes of the strong one. Apodise suppresses the side lobes, but each line becomes twice as broad.
6. Sampling and aliasing. Switch to full range and increase \(\Delta x\) beyond \(\lambda/2 \approx 295\) nm. Panel (b) shows fewer than two samples per fringe, and the lines jump below the Nyquist frequency to the wrong wavelength.