Interference

• one type of interference is diffraction

  • Diffraction is the bending or spreading out of a wave around obstacles

• More commonly with the wave equation \begin{align} E_{1}=E_{0}\cos(wt+\phi_{1})\\E_{2}=E_{0}\cos(wt+\phi_{2})\\\text{phase difference:}\Delta \phi=\phi_{2}-\phi_{1}\end{align}

  • Constructive

    • adding 2 waves amplify each other
      • $\Delta \varphi =0,2\pi ,4\pi \dots$

  • Destructive

    • adding 2 waves destroy each other
      • $\Delta \varphi =\pi ,3\pi ,5\mathbf{\pi }..$

  • They can be true at the same time! (partially destructive interference)

Huygens’ Principle

• Definition

  • Every point on a wave front acts as a point source

    • the wave front as it develops is tangent to all the wavelets
      • basically we can take a wave, an the subsect it into many tiny wavelets
      • 

interference (double-slit)

• If light is a wave there should be an interference pattern!

  • we are ignoring wave-particle duality rn

• The young’s double-slit experiment

  • the interference occurs because each point on the screen is not the same distance from both slits!

    • depending on the path length difference the wave can act differently
    • 

  • Constructive: $d\sin \theta =m\lambda ,m=0,1,2,\dots$

  • Destructive: $d\sin \theta =\left(m+\frac{1}{2}\right),m=0,1,2,\dots$

• Intensity in the double-slit

Interference in Thin films

• Another way path lengths can differ, and waves interfere

  • is if they travel through diffrent media. if there is a very thin film of material (a few wavelengths thick)
    • light will reflect both at the bottom and the top of the layer
    • 

• Interference in thin films

  • Wave length in medium = $\frac{\lambda }{n}$

• A similar effect is when a shallowly curved piece of glass is placed on a flat one

  • when viewed from above concentric circles appear that are called newton’s rings
  • 

  • When $n_2>n_1$, the phase changes by $180^o$ $\left(\frac{\lambda }{2}\right)$ upon reflection