In this lecture the following are introduced:
• Total internal reflection
• Optical fibres
• Image formation by convex lenses
• Image formation by concave lenses

Total Internal Reflection

When a light beam is directed along the axis of a cylindrical glass fibre, it reflects repeatedly off the edges of the fibre, without appreciable loss.

This kind of reflection is called Total Internal Reflection. To explain how it arises, we need to look at the refraction which occurs when light, which is travelling in a medium with higher refractive index, hits an interface with a medium of lower refractive index.

Example:
A light ray is travelling in glass of refractive index, 1.5. It meets an interface with air (refractive index = 1.0) at an incident angle of 30⁰. Find the angle of refraction.

In general, when a light ray from a higher refractive index medium enters a lower refractive index medium, then the ray bends away from the normal.

Critical Angle

When a refracted angle equals 90⁰ then a critical situation arises where the refracted ray travels along the interface.

For typical glass with refractive index 1.5, this occurs when:

For angles of incidence greater than the critical angle, the refracted ray does not exist, and the ray reflects off the interface at the same angle as it was incident. This reflection is called total internal reflection because there is no energy loss on reflection. The Silver reflecting surface in most domestic mirrors will absorb some of the incident light, so for them, multiple reflections will cause the image to fade away.

Even though there is no energy loss in total internal reflection, the electric and magnetic fields of the light do penetrate a few wavelengths into the lower refractive medium. This wavelength penetration can be used to make variable-reflectance mirrors by placing another glass surface in near contact. This is known as frustrated total internal reflection.

Optical fibres have a protective envelope around them so that wavelength penetration does not produce any energy loss.

To couple light into optic fibres, the incoming light should be within a light cone that produces total internal reflection.

This can be calculated by considering the geometry of the fibre core/envelope and the definition of the critical angle.

Example:

Calculate the largest angle of incidence available for light to be transmitted down an optic fibre which has core refractive index 1.50, and envelope refractive index 1.49.

[What would happen if there was no cladding around the core (n₁=1) and you tried to find the incident light cone for transmission by total internal reflection?]

Optical Lenses

Optical lenses work by refracting light rays. There are two main types of lens, "convex" and "concave". A convex lens will bring rays together. Its glass-in-air shape is (), i.e. it fatter at the middle than the edge. A concave lens has a "cave" or hollow. Its glass-in-air shape is )(, i.e. it is thinner at the middle than the edges.

These are the four rays which you should be able to draw for a convex lens.

1 • Any ray through the centre of the lens will pass through undeviated.

2 • Any ray diverging from the primary focal point will emerge parallel to the axis

3 • Any ray parallel to the axis will converge to the secondary focal point.

4 • Any ray will converge to that point in the secondary focal plane, where a parallel ray through the centre intersects with the focal plane.

These are the four rays which you should be able to draw for a concave lens.

5 • Any ray through the centre of the lens will pass through undeviated.

6 • Any ray parallel to the axis will diverge as if it came from the primary focal point.

7 • Any ray converging towards the secondary focal point will emerge parallel to the axis.

8 • Any ray will diverge from that point in the primary focal plane, where a parallel ray through the centre intersects with the focal plane.

Image formation by a convex lens

1 • Object inside the focal length.
the image is virtual, upright & enlarged.

2 • Object between the focal length and twice the focal length.
The image is real, inverted & enlarged

3 • Object greater than twice focal length.
The image is real, inverted & diminished

Image formation by a concave lens

4 • Real object
The image is virtual, upright & diminished

5 • Virtual object
The image is real, upright & enlarged

The lens formula

From similar triangles:
    

Also from different similar triangles:    

Example
An object 30mm high is placed 300mm in front of a 200mm convex lens. Find the position, size and nature of the image.

The image is real (v is positive), inverted (m is negative), and enlarged (m is greater than 1). It is 600mm behind the lens and 30 x 2.0 = 60mm high.

Example
An object 50mm high is placed 100mm in front of a 200mm convex lens. Find the position, size and nature of the image.

The image is virtual (v is negative), upright (m is positive) and enlarged (m is greater than 1). It is 200mm in front of the lens and 50 x 2.0 = 100mm high.

Example
An object is placed in front of a 120mm concave lens. The image is upright, 25mm high and -80mm from the lens. Find the position and size of the object.

The object is real (u is positive), 240mm in front of the lens. Its size is 3x the image, i.e. 75mm high.

Summarising:

When light changes from a higher refractive index medium to a lower refractive index it bends away from the normal
At the critical angle the refracted ray is at 90⁰, i.e. parallel to the surface.
and
At incidence greater than the critical angle there is total internal reflection.
Optical fibres have a protective envelope around them so that wavelength penetration does not produce any energy loss.
Convex lenses are thicker at the middle than the edges.
Concave lenses are thinner at the middle than the edges.
There are four standard light paths for convex and concave lenses.
The lens equations use the convention: real is positive

(where minus means inverted).