§ 42. Michelson Interferometer & Phase Shift of Beams (Grade 11 · Uni)
Division of a coherent laser beam by a beam splitter into two orthogonal arms L₁, L₂ and fringe shift under mirror displacement Δd. Displacing a mirror by merely half a wavelength of light reverses the central interference fringe from bright to dark.
Theory Summary & Scientific Mechanism
Invented by Albert A. Michelson in 1881, the Michelson interferometer splits a coherent monochromatic light beam of wavelength λ at a 45^° beam splitter (BS) into two orthogonal arms of lengths L₁ and L₂. After reflecting off mirrors M₁ (micrometer-adjustable) and M₂ (fixed), the beams recombine at BS and interfere on the detector screen.
Because light traverses arm 1 twice (out and back), translating mirror M₁ by distance Δ d changes the round-trip optical path length by Δ = 2Δ d, introducing a phase shift Δφ = (4π Δ d)/(λ). Central fringe intensity varies as I(Δ d) = Iₘaxcos ² ≤ft((2π Δ d)/(λ) ).
Consequently, moving mirror M₁ by just a quarter-wavelength (Δ d = λ/4) flips a bright maximum into a dark minimum, and every Δ d = λ/2 shifts the pattern by one full fringe (N = (2Δ d)/(λ)). This design powered the historic 1887 Michelson–Morley experiment and forms the optical core of the LIGO gravitational-wave observatories.
Everyday Analogy
The most precise ruler in the Universe! A laser beam splits in two, races down perpendicular arms to mirrors, and recombines. Shift one mirror by just half a wavelength of light and the bullseye fringes shift by a full ring, how LIGO detects gravitational waves!
Interactive Question & Derivation
By what distance Δ d must the movable mirror M₁ of a Michelson interferometer be translated for the interference pattern on the screen to shift by exactly one full fringe (Δ m = 1)?
- A: By half a wavelength Δ d = λ / 2, because the beam traverses the adjustable arm twice (out and back), making the optical path change Δ = 2Δ d = λ
- B: By one full wavelength Δ d = λ
- C: By two wavelengths Δ d = 2λ
Study Hint: Because light travels to the movable mirror and back, shifting the mirror by Δ d changes the optical path by Δ = 2Δ d. What mirror shift Δ d produces a shift of 1 full fringe (Δ = λ)?
Proof & Explanation: In the arm with movable mirror M₁, the light beam travels from the beam splitter to the mirror and back again. Moving the mirror by Δ d therefore changes the round-trip optical path difference by Δ = 2Δ d. Shifting by one full fringe requires 2Δ d = λ, or Δ d = λ/2.
Key Equations & 3D Simulation Parameters
Δ = 2 Δ d = mλ, I = Iₘaxcos ² ≤ft((2π Δ d)/(λ) )
- Laser wavelength λ: 420 … 680 nm (default: 550 nm)
- Micrometer mirror displacement Δd: 0 … 700 nm (default: 0 nm)
- Laser beam power P₀: 2 … 10 mW (default: 5 mW)