PHYS 1008 Geometrical Optics

Advanced Technology Solar Telescope (ATST)

Objectives: by the end of this you will be able to explain mirrors understand Snell's law and refraction find what happens to light when it hits a transparent medium Draw ray diagrams Make calculations with simple lenses

Reflection

Off a plane surface : Note direction of propagation gets reversed

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• If we have an extended object, this will create an image. To find out where the image appears to be, extend the line of sight

• To get the sensation of depth, we need binocular vision

This is based on angle of incidence = angle of reflection

θ₁ = θ₂

• E.g. concave mirrors: different bits of the mirror reflect the wave according to the local angle of incidence

• The effect in this case is to focus the wave

This is reversible: if we have a source at the centre of a curved mirror, we have a plane wave (well almost) coming out

Convex mirrors cause waves to diverge Note that these behave as if there is a focus behind the mirror

Refraction

• • light going from air to glass
  • light going from air to water .
  • light going from glass to air

• Velocity in medium 1 =  v₁ = n₂ 
  Velocity in medium 2    v₂   n₁
  

• n is the refractive index of the medium: for light, the refractive index of the vacuum is defined to be 1, so n > 1
• e.g. in crown glass, c ~ 2x10⁸ ms^-1 so n = 1.5

• Note that the refracted wave is bent since the wavelength is decreased. This gives rise to Snell's law, when the wave hits the interface at an angle n₁ sin(θ₁) = n₂ sin(θ₂)

• a simulation

1. λ = 520 nm
2. λ = 780 nm
3. λ = 347 nm

• What is the refracted angle?
  1. 59.4⁰
  2. 52.5⁰
  3. 23.3⁰
  4. 22.5⁰

We have already seen how a single surface refracts. All optical instruments have at least 2 surfaces. A prism deflects light via two successive refractions sin(θ₁) = n sin(θ₂) etc

Total Internal Reflection

Light can go from a dense medium to a less dense one at an "impossible" angle: e.g in crown glass, what would happen to a ray whose angle of incidence was θ = 60o?

• e.g. lying on the bottom of a swimming pool looking up what do you see?

• A prism can be used to show total internal reflection

1. 60^o
2. 90^o
3. 42^o
4. 48^o

• 
• If you want to carry a large amount of signal on one carrier, need a very high frequency (roughly, a voice channel needs 10 kHz, so to carry N voice channels, need 10N kHz.
• How many voice channels can you carry on a 1 MHz radio wave?
• How many could you carry on red light?

Lenses

How does a lens form images?.

• We can build up a lens from a series of prisms

• We could add a 2nd. prism, to deviate light more, so that two rays go through the same place

• 
• 

There are a variety of lens, but essentially they are converging (usually convex) diverging (usually concave)

• Convex lenses create a real image (i.e. one that can be cast on a screen)

  The most important quantity for a lens is the focal length f: i.e. how far from the lens do parallel rays get focussed.

• Concave lenses cause light to diverge, but the rays can be traced back to an (imaginary) focus.

• Images are formed as either real or virtual: only a convex lens (positive focal length) can form a real image

• Rules for "ray-tracing" diagrams:
• a ray which goes through the centre of the lens is not deflected.
• a ray that is parallel to the axis must go through the focus.
• a ray that goes through a focus must go parallel to the axis.
• We can simulate lens on an optical bench

The Thin Lens Formula

This is the derivation of the "thin-lens" formula. We can use this to find the relation between the distance to the object, the image and the focal length

• The magnification \color{red}{ M = \frac{{{\rm{image height}}}}{{{\rm{object height}}}} = \frac{{h_i }}{{h_0 }}} (M can be < 1)

We have two sets of similar triangles: \color{red}{ \left| {\frac{{h_i }}{{h_0 }}} \right| = \frac{{d_i }}{{d_0 }} = \frac{{d_i - f}}{f}} so \color{red}{ \frac{{d_i - f}}{f} = \frac{{d_i }}{{d_0 }} \Rightarrow \frac{1}{f} = \frac{1}{{d_0 }} + \frac{1}{{d_i }}}

The thin lens equation

• In words

       1      =    1      +      1       
  focal length object dist.   image dist.
  
  

• The surprising thing about this formula is that it always works provided we remember certain conventions:
• real objects have d_o > 0
• real images have d_i> 0
• virtual ones have d_i < 0
• objects above the axis have h_o > 0
• images below the axis have h_i < 0
• convex lenses have f > 0
• concave lenses have f <0
• convex mirrors have f < 0
• concave mirrors have f >0
• mirrors have real images on the same side as the object
• lenses must be "thin": the approx. that is used is sin(θ)= θ

• We can simulate lens on an optical bench

The lens-maker's equation:

\color{red}{ \frac{1}{f} = \left( {n - 1} \right)\left( {\frac{1}{{R_1 }} + \frac{1}{{R_2 }}} \right)}

• applies if the lens is "thin" (means that all angles are small enough so that θ = sin(θ))
• R₁,R₂ are the radii of curvature of the lens surface:
• convex surfaces have positive curvature,

1. 16.6 cm
2. 30.0 cm
3. 12 cm
4. 0.060cm

f = R/2 because if you place a source at the centre the light must be reflected back there. 1 = 1 + 1 f R R

• We can simulate a mirror on an optical bench

e.g. a spoon What happens if you look at the front of the spoon? What is the focal length? What happens if you put an object inside the focal length?

• What happens if you look at the front of the spoon? Image is real and inverted

• Image is virtual and upright. Note we are drawing dotted lines to extend the rays through the foci.

1. -20 cm?
2. 10 cm?
3. 5 cm?
4. -5 cm?

1. real and inverted orientation
2. real and same orientation
3. virtual and inverted orientation
4. virtual and same orientation