PHYS 208 Lecture 26

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Interferencee

In order to have interference, there must be multiple coherent sources of light combining in a region of space.

constructive interference

when two waves are in phase (peaks add, valleys add; double everything on wave)

destructive interference

when waves are out of phase by $\lambda /2$ (peaks to valleys, valleys to peaks; add to 0)

• Phase change of π ($\phi =\pi$) when $n_{2}>n_{1}$
• No phase change ($\phi =0$) when $n_{2}<n_{1}$

$m\pi =\left({\frac {2d}{\lambda }}\right)2\pi +\phi$

For two sources with a screen:

• $R_{1}$ is distance traveled by first ray
• $R_{2}$ is distance traveled by second ray
• $R_{2}-R_{1}$ is the optical path length difference
• When the optical path length difference
  • is a multiple of λ (i.e. $m\lambda$), there is constructive interference
  • is $(m+{\frac {1}{2}})\lambda$, there is destructive interference

Example

Phase change between rays that bounce off a thin layer of air between two plates of glass: (note the π since $n_{ain}<n_{glass}$)

$m\pi =\left({\frac {2d}{\lambda }}\right)2\pi +\pi ={\mbox{total phase change between 2 rays}}$

There will be bright spots at $d={\frac {\lambda (m\pi -\pi )}{2(2\pi )}}={\frac {\lambda }{2}}(m-1)$

Diffraction

Hugyen's principle: any point on a wave front can be considered a new point source of the wave propagating with the same velocity as the original wave.

For a single slit of length $a$ through which a single source of light shines

• For an angle θ from the normal between the slit and screen, the dark fringes will be when $\sin {\theta }={\frac {m\lambda }{a}}$, where $m\in \mathbb {Z}$