PHYS 208 Lecture 26

« previous | "Thursday", December 6, 2011 | next »

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda/2} (peaks to valleys, valleys to peaks; add to 0)

• Phase change of π (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_{ain} < n_{glass}} )

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \pi = \left( \frac{2d}{\lambda} \right) 2 \pi + \pi = \mbox{total phase change between 2 rays}}

There will be bright spots at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin{\theta} = \frac{m \lambda}{a}} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \in \mathbb{Z}}