Lenses and Optics


Curvature of waves

When light is emitted by an object the radiation travels out in a wave like way. It's kind out like the waves seen after a pebble in dropped in a pond. The only difference is that the waves travelling out on the pond are confined to one direction and travel out circularly, whereas the waves given off by an object travel out in shells, in spherical wavefronts. The wave front is what connects wave peaks that are at the same distance away from the source. This is one way of depicting the way in which light travels away from an object. Another way is by drawing a ray. This shows the direction of a single wave, and will always be perpendicular to the wave front, assuming it too originated from the center of curvature. The idea of light travelling in straight lines can be thought of as rectilinear propagation. 

These fronts propagate (or travel) away from the source in the same way as a balloon filling with air. Because the waves propagate from the same source, the object from which they are given off serves as their center of curvature. Curvature  refers to how flat a wave front is. These spherical waves expand as they move away from the source, and therefore their curvature can be thought of  as directly related. The further away from a source a wave front is, the less it's curvature is. In the same way, the closer the wave front is to a source, the higher the value of its curvature. In air, which has a refractive index of 1, curvature of waves can be thought of as:

Curvature = 1 / distance from source

Curvature is therefore inversely proportional to the distance from the source. 

For example, the rays of light leaving a book and travelling 0.4m meters to the eyes (average reading distance) The curvature at the eyes is 1/-0.4 = -2.50 D. As the value of the distance from the source increases, The curvature of the wave tends to zero, therefore, objects infinitely far away will have a curvature of zero, and wavefronts are therefore drawn parallel to each other. 

It is also important to note that converging waves will spread out from a single point, such as the waves delineated above, and converging waves will focus on a single point, having started off being 'spread out'. Cartesian convention states that diverging waves are considered negative, whereas converging waves are generally considered positive. What this means for AS level is the waves before they hit a lens are given a negative value, and the waves after they hit the lens are given a positive value. It's almost like drawing a graph, with the y axis being straight through the lens. anything to the left of this axis will take on a negative value, and anything to the right of the line will take on a positive value. 


Power of waves

The function of a lens is to either increase or decrease the curvature of a wavefront. Lenses have a power, which is simply a measure of how much curvature they add to the wave. The power of a lens is given by the equation:

Power = 1/focal length

The focal length of a wave is the point at which rays to the right of the lens will intersect with one another, if the wavefronts entering the lens are parallel to each other. It is where the light from an object infinitely far away will be focused. 

The wave equation

The wave equation take the power of the lens (how much curvature the lens adds to the wave), and the curvature of the wave entering the lens, and then calculates the curvature of the new wave. From the curvature of the new wave, the point at which the light will be focused can be derived. 

1/v = 1/u + 1/f

Because of cartesian convention, the value for u will be a negative number. 

Drawing ray diagrams 

There are three rules when drawing ray diagrams:

1. A ray parallel to the principle axis when entering the lens will be refracted to the focal point. 
2. A ray going through the centre of the lens will continue; experiencing no change in direction.
3. A ray going through the focal point on its way to the lens will be refracted to be parallel to the principle axis. 

The first image depicts what happens normally, the object is further away from the focal length of the lens. The second image gets a little more confusing. This is what happens when the object is between the focal length and the lens. The waves disperse (due to the three rules above) and when they hit the eye, the brain assumes that the rays have travelled in a straight line on their way to the eye, (it can't really comprehend reflection or refraction) and so creates a virtual image behind the lens, which is where all the rays would meet behind the lens. The third image above is what would happen if the object is on the focal length of the lens, the refracted rays would always be parallel to each other, and so an in focus image would be seen an infinite distance away.