BIT 1002 Physical Optics

Objectives: by the end of this you will be able to Understand Young's slits experiment and the single slit Understand the diffraction grating Understand dispersion of colours Understand polarisation

Diffraction and Interference

A point source of waves produces spherical waves. If we see them a long distance from the source, they look like plane waves.

• When we have two sources, we can get interference between them

• At the green points, we will get constructive interference (crests add) At the yellow points, we get destructive interference (crest and trough)

• This means we get nodal lines: (this shows the lines of maxima)

Young's slits

is this interference experiment done for light.

• Distance between sources is small (d) Distance to detector is large (D) so waves can be treated as plane waves, giving constructive interference

• If we have an extra ½ λ, will get destructive interference.

• 
• Must have D₁ - D₂ = nλ for constructive interference n = 0,±1,±2,±3.....................
• We can derive

  θ = nλ
      d
  

• For water waves, this can be done by two sources. For light, there is an extra problem: sources don't stay in phase...

Huyghen's Principle

Every point on a wave acts as a source of new (spherical) wavelets

When these add together (interfere constructively) one gets a new wavefront

• Huyghen's principle allows us to understand a variety of phenomena: e.g refraction.

• We can use this to understand how waves pass through narrow gaps

  Each successive wave is "in phase with itself"

Can replace two sources by one source and two slits: Young's slits experiment

e.g. a light produces Na light (λ = 589 nm) and it passes through two slits separated by 50μ (5x10-5 m). How will it appear on a screen at a distance of 1 m? What is the distance from the central max to the 2nd max?

• Waves can go straight through. Wow! However, this isn't very interesting!
• But they can also add up in different directions
• What makes diffraction gratings interesting is that bands are very narrow:
• even if one slit and the next produce waves that are only 1⁰ out of phase, the 180^th slit will be completely out of phase, so they only add up at one angle.

• λ = d sin(θ) so if white light is incident on diff. grating, it will be split up into constituent wavelengths

• We may also be able to arrange a second maximum 2λ = d sin(θ₂)

e.g. Na light consists of two wavelengths very close to 589 nm. It is incident on a diffraction grating with 8000 lines/cm. What will you see? At what angle will the first maximum be found?

• Note that a diffraction grating is really an interference grating...

Single Slit

• Geometry

  To have destructive interference between light from centre and edges, need D₁-D₂ = λ/2 difference in paths

• i.e.

  sin(θ) = λ/2   =  λ 
           a/2      a
  

• A simulation

• The intensity falls of very rapidly away from the centre line.
• Minima are given by

  sin(θ) =  mλ m = 1,2,3.. (but not m=0)
             a

• Centre (bright) is given by m=0
• Maxima are given by

  sin(θ) =  (m+½)λ m = 1,2,3.. (but not m=0)
                 a
  

Thin Film interference

• e.g. a wedge:

• if the light from the bottom level travels λ/2 further, there will be destructive interference between the two reflections. Since the path will depend on the separation, there will be a pattern of lines

• e.g. suppose we are using hydrogen light (λ = 643 nm) and the wedge has an angle of .001 radians, what will the separation between the lines be? d₂-d₁ = λ

We can use this constructively to reduce the reflection from lenses: e.g. a surface will reflect

• By coating the lens, we can arrange for destructive interference in the reflected wave

Dispersion

• Glass has (slightly) different refractive indices n for different λ, Hence a prism can be used to produce a spectrum.

• Note that this mechanism is quite different from a diffraction grating (where different wavelengths interfere constructively at different angles).
• With a prism, the red light is deviated least (i.e. the velocity is largest), whereas for a grating it is deviated most.

• Mirages: Hot air is less dense than cold air

• Green Flash

  Atmosphere bends light, but blue is bent most. Hence image of the "blue" sun is sometimes visible just after sunset, or just before sunrise. You see this as a momentary "green flash" lasting for about 2 s. Need a distant horizon (i.e. mountains or sea) to give an adequate distance through the atmosphere

combination of refraction and total internal reflection in raindrop.

1. red
2. blue
3. green
4. yellow

• .

• Why is the sky darker outside the rainbow than inside?

Polarization

• We have introduced polarization of transverse waves: this of course applies to light

• Polaroid acts as a "slot" to let through radiation in only one poln. state

• Hence two successive sheets of polaroid at right angles will eliminate all light

• if a ray with some polarization at an angle θ passes through a filter with a vertical polarization axis then the projection of E is E₀cos(θ). However, \color{red}{I \propto E_y ^2 = E_0^2 \cos ^2 \left( \theta \right)} so the intensity is reduced by cos²(θ).

• If the original light is unpolarized, the intensity is given by the average value
  \color{red}{ \left\langle {\cos ^2 \left( \theta \right)} \right\rangle = \frac{1}{2}\left\langle {1 + \cos \left( {2\theta } \right)} \right\rangle = \frac{1}{2}}

• Hence if light passes through a succession of filters, it will lose a factor cos²(θ) for each one.

• Note: this assumes that the filter is perfect: in fact some absorption will always occur.

1. 3/4
2. 3/8
3. 1/2
4. 1/8

• Depends on angle of incidence if tan(θ) = n₂/n₁ = n (if outer medium is air) then reflected ray is 100% polarized

• What is this for water (n = 1.33)?
• Now we understand physical optics and geometric optics, we can look at how optical instruments work