Michelson interferometer

Michelson interferometer

1. What It Is

Michelson interferometer = amplitude-division interferometer with a beam splitter and two perpendicular mirrors ($M1$ fixed, $M2$ movable) giving two coherent beams that recombine to give circular fringes (or straight if misaligned).

Invented by A.A. Michelson for aether experiment; now used for precision length, $λ$ measurement, refractive index, and FTIR.

2. Construction (diagram description)

Source → Beam Splitter (45° half-silvered) →  [reflected → M1] and [transmitted → M2] (compensator plate C ensures equal glass)
                ← recombined beams ←          ← both return through BS → interference at Telescope/Eye
M2 is on movable carriage with micrometer screw.

The compensator plate (equal thickness to BS) equalizes optical paths in glass for white-light operation.

3. Path Difference and Fringes

Optical path difference between arms:

where $d_1,d_2$ are mirror distances from BS, $θ$ is viewing angle (near normal $cosθ≈1$).

• Bright:

• Dark:

If $M1$ and $M2$ are slightly tilted, fringes are concentric circles centred at normal; if tilted one mirror, straight parallel fringes.

Moving $M2$ by $Δd = λ/2$ shifts fringe pattern by one fringe (path changes $2Δd$). So counting fringes measures displacement.

4. Uses and Measurements

• Wavelength: move $M2$ by known $Δd$, count $N$ fringes:

• Refractive index: insert thin glass slab thickness $t$ in one arm ⇒ path increases $2(μ-1)t$ ⇒ shift $N =2(μ-1)t/λ$ ⇒

• Distance metrology: count $N$ for small $Δd$ (nanometer sensitivity with laser).
• Fourier Transform Spectroscopy: scan $M2$ and record interferogram, FFT gives spectrum.

5. Worked Examples

(a) M2 moves 0.147 mm, 500 fringes counted, λ? ⇒ λ=2·0.147e-3/500=588 nm ✓

(b) Glass slab t=0.01 mm inserted, 40 fringes shift, λ=600 nm ⇒ μ=1+40·6e-7/(2·1e-5)=1+0.012/2e-5? Actually 40·6e-7=2.4e-5 ⇒ /2e-5=1.2 ⇒ μ=2.2? Check: N=2(μ-1)t/λ ⇒ μ=1+ Nλ/2t =1+40·6e-7/2e-5=1+1.2=2.2 ✓ (typical glass 1.5 so maybe 20 fringes would be 1.6)

(c) Why compensator? Without it white-light central fringe not at zero delay ⇒ no white-light fringes.

(d) Circular vs localized fringes: equal inclination (Haidinger) vs equal thickness (Fizeau) — Michelson gives Haidinger type.

6. Exam Points

• Diagram and working of Michelson — 5-7 marks, very frequent.
• Derive $Δ =2(d_1-d_2)cosθ$ and fringe condition — 5 marks, frequent.
• Show $λ =2Δd/N$ and $μ =1+ Nλ/(2t)$ — 5-7 marks, very frequent.
• Use of compensator plate — 3 marks.