A Michelson interferometer splits a beam of light into two paths using a beamsplitter.  These paths are then reflected back and recombined, creating an interference pattern.  The intensity of this pattern depends on the path difference between the two beams

Superposition of Two Sine Waves: Resultant Intensity at the Michelson Interferometer Detector

Example: Two Sine Waves

Let's consider two sine waves representing the two beams:

• Wave 1: y1 = A * sin(kx)
• Wave 2: y2 = A * sin(kx + φ)

Where:

• A is the amplitude.
• k is the wave number.
• x is the position.
• φ is the phase difference, which is related to the path difference (Δ).

The path difference (Δ) and phase difference (φ) are related by: φ = (2π/λ) * Δ, where λ is the wavelength.

Specific Cases:

1. Path Difference Δ = 0 (φ = 0): Constructive Interference
   • y2 = A * sin(kx)
   • y_total = y1 + y2 = 2A * sin(kx)
   • The waves are in phase, and the resulting wave has double the amplitude (constructive interference). The intensity is maximum.
2. Path Difference Δ = λ/2 (φ = π): Destructive Interference
   • y2 = A * sin(kx + π) = -A * sin(kx)
   • y_total = y1 + y2 = A * sin(kx) - A * sin(kx) = 0
   • The waves are completely out of phase, and they cancel each other out (destructive interference). The intensity is zero.
3. Path Difference Δ = λ (φ = 2π): Constructive Interference
   • y2 = A * sin(kx + 2π) = A * sin(kx)
   • y_total = y1 + y2 = 2A * sin(kx)
   • The waves are in phase, and the resulting wave has double the amplitude (constructive interference). The intensity is maximum.

The detector output, representing the interference of the two waves, is visualized as an intensity variation. When you digitize the interference pattern of a monochromatic laser in a Michelson interferometer with a mirror moving at a constant velocity, here's what you can infer:

Key Relationship:

• The crucial relationship is between the frequency of the detected sinusoidal waveform, the velocity of the moving mirror, and the wavelength of the incident light. It's expressed by the equation:
  • f = 2v / λ
  • Where:
    • f is the frequency of the detector's output signal.
    • v is the velocity of the moving mirror.
    • λ is the wavelength of the incident light.

Detector Output:

• A sinusoidal waveform: The detector will capture a signal that oscillates like a sine wave. This is because the movement of the mirror at a constant velocity causes a linear change in the path difference between the two beams of light. This linear change in path difference results in a periodic variation of the interference pattern's intensity.
• Frequency proportional to velocity: The frequency of this sinusoidal waveform is directly proportional to the velocity of the moving mirror. A faster mirror movement will result in a higher frequency sine wave, and a slower movement will result in a lower frequency.

Inference of the Sine Wave (recorded at the detector):

• Measurement of small displacements: The ability to detect changes in the interference pattern allows for very precise measurements of small displacements. This principle is used in various applications, including laser Doppler velocimetry and precision metrology.
• Characterization of light source: In some cases, analyzing the sine wave can also provide information about the characteristics of the laser light source, such as its stability and coherence.
• Determination of Wavelength: Conversely, if the velocity of the mirror is accurately known, the wavelength of the incident light can be determined. By rearranging the formula (λ = 2v/f), and measuring the frequency of the sinewave, the wavelength can be calculated. 

So far, we've focused on one frequency of light. In the next section, we'll see how the interferometer reacts to light with a mix of frequencies