Wave Optics

• Superposition of two sinusoidal waves

  𝑥1(𝑡) = 𝑎1 cos(𝜔𝑡 + 𝜃1)

  𝑥2(𝑡) = 𝑎2 cos(𝜔𝑡 + 𝜃2)
• The two displacements of disturbances have same frequency but different amplitudes and different initial phases
• Resultant displacement


  𝑥(𝑡) = 𝑥1(𝑡) + 𝑥2(𝑡)

  = cos (𝜔𝑡)(𝑎1 cos𝜃1 + 𝑎2 cos 𝜃2) - sin (𝜔𝑡)(𝑎1 sin𝜃1 + 𝑎2 sin 𝜃2)

  𝑥(𝑡) = 𝑎 cos(𝜔𝑡 + 𝜃)
• Resultant disturbance


  • It is also simple harmonic with different amplitudes and initial phases
• Calculating resultant amplitude and phase

  𝑎 cos 𝜃 = 𝑎1 cos𝜃1 + 𝑎2 cos 𝜃2

  𝑎 sin 𝜃 = 𝑎1 sin𝜃1 + 𝑎2 sin 𝜃2
• Conditions for


  • Constructive interference: 𝜃1 - 𝜃2 = 0, 2𝜋, 4𝜋, ...

  Destructive interference: 𝜃1 - 𝜃2 = 𝜋, 3𝜋, 5𝜋, ...
• Resultant displacement with n displacements

  𝑥 = 𝑥1 + 𝑥2 + ... + 𝑥𝑛 = 𝑎 cos(𝜔𝑡 + 𝜃)

  𝑎 cos 𝜃 = 𝑎1 cos𝜃1 + ... + 𝑎𝑛 cos 𝜃𝑛

  𝑎 sin 𝜃 = 𝑎1 sin𝜃1 + ... + 𝑎𝑛 sin 𝜃𝑛
• At a point B where 𝑆2𝐵 - 𝑆1𝐵 = 𝜆/2, the disturbance from 𝑆1 will always be out of phase with disturbance from 𝑆2
• At a point C where 𝑆2𝐶 - 𝑆1𝐶 = 𝜆, the phase of vibration are exactly the same as the point A
• Conditions for intensity maxima and minima

  Maxima: 𝑆2𝑃 - 𝑆1𝑃 = 𝑛𝜆 (𝑛 = 0,1,2,3,...)

  Minima: 𝑆2𝑃 - 𝑆1𝑃 = (𝑛 + 1/2)𝜆 (𝑛 = 0,1,2,3,...)
• Coherence


  Two sources vibrating at a constant phase difference
• If the phase difference varies rapidly, no stationary interference pattern would occur
• Resultant displacement and intensity for coherent sources


  𝑦 = 𝑦1 + 𝑦2 = 2𝑎 cos(𝜑/2) cos(𝜔𝑡 + 𝜑/2)

  𝐼 = 4𝐼0 cos^2(𝜑/2)
• Intensity for coherent sources


  • Minima: 𝜑 = ±𝜋, ±3𝜋, ±5𝜋, ...

  Maxima: 𝜑 = 0, ±2𝜋, ±4𝜋, ...

  
• For incoherent sources, 𝐼 = 2��0 (no stationary interference pattern)
• Young's double slit experiment

  Division of a single wavefront into two, which act as if they emanated from two coherent sources
• Determining positions of maxima and minima in Young's double slit experiment

  𝑦𝑛 = 𝑛𝜆𝐷/𝑑

  Distance between consecutive fringes: 𝛽 = 𝜆𝐷/𝑑
• Intensity distribution in Young's double slit experiment


  𝐼 = 𝐾(𝐸1 + 𝐸2)^2 = 2𝐼0(1 + cos 𝛿)

  𝛿 = 2𝜋(𝑆2𝑃 - 𝑆1𝑃)/𝜆
• Conditions for maxima and minima in Young's double slit experiment

  • Maxima: 𝑆2𝑃 - 𝑆1𝑃 = 𝑛𝜆

  Minima: 𝑆2𝑃 - 𝑆1𝑃 = (𝑛 + 1/2)𝜆
• For incoherent sources in Young's double slit experiment, 𝐼 = 𝐼1 + 𝐼2 (no interference pattern)
• Intensity distribution for coherent sources in Young's double slit experiment

  𝐼 = 4𝐼0 cos^2(𝛿/2)
• Constructive interference occurs at 𝑥 = 𝜆/4, 3𝜆/4, 5𝜆/4, ...

  Destructive interference occurs at 𝑥 = 0, 𝜆/2, 𝜆, 3𝜆/2, ...
• Reflection of wave at 𝑥 = 0 (by rigid end) results in a phase shift of 𝜋

  
• Diffraction


  • Spreading-out of a wave when it passes through a narrow opening

  Closely related to interference
• Fresnel diffraction: Source and screen at finite distances from the diffracting aperture

  Fraunhofer diffraction: Source and screen at infinite distances from the diffracting aperture

  
• Single-slit diffraction pattern


  Treat slit as a continuous distribution of point sources

  Intensity 𝐼 = 𝐼0 (sin ��/𝛽)^2, where 𝛽 = 𝜋𝑏 sin 𝜃/𝜆
• Positions of minima in single-slit diffraction: 𝑚 = ±1, ±2, ±3, ...
• Positions of maxima in single-slit diffraction: 𝛽 = 0, 1.43𝜋, 2.46𝜋, ...