Of Light

Rays

and geometric optics

Geometric optics

Reflection

Refraction

Total internal reflection

Lens

Geometric optics

The model called "Geometric optics" operates with the concept of "ray" and perfectly describes how light propagates through transparent media. As well as the operation of almost any optical devices (binoculars, telescopes, microscopes, camera lenses, etc.). Without geometric optics it would have been impossible to develop neither the first spyglass nor the modern radio telescope. Geometric optics is sufficient for solving many problems ... as long as the "examined" objects are not too small. What is amazing that only two basic laws lie at the basis of geometric optics - reflection and refraction of light at the boundary of two media. And these laws are consequences of just one (sic!) principle of the minimum time for ray propagation.

It should be noted that geometric optics does not respond to the question: how does the ray find the shortest path in time? - but it is an another story...

Reflection

Let's consider the reflection of a ray from a flat boundary. Since the ray propagates in one substance (with a constant speed V), then the propagation time T is proportional to the length of path L:

Let's find the minimum distance between two points A and B, in condition of reflection: the ray must pass through a certain point X on the "reflecting surface (line)".

The minimum distance L corresponds to the condition

Taking into account that

we have the following:

That means,

So, the angle of incidence is equal to the angle of reflection: β = α.

Note that the second solution: β = π - α corresponds to the ray АВ', passing through the "reflection" of point В.

Refraction

Let's consider ray crossing the boundary of two media (with different propagation speeds). What is the path taking the minimum time to go from A to B? Now the minimum is reached not along a straight line, but along an optimal broken trajectory. Small increase of time (and path length) in a medium with a higher velocity is compensated by a decreaseing of time (and the path length) in a medium with a lower velocity. Let's show it.

Let's find the path that requires the minimum time for the ray to propagate between points A and B.

Depending on the position of the point X on the "refracting line", the length of the path is:

The propagation velocities V₁ (in the "upper" medium where point A is located) and V₂ (in the lower medium where point B is located) are different from each other.

Then the time taken is:

The minimum time T corresponds to the condition

Taking into account that

we have the following:

So, that means

It turns out that the increase of the path in one substance (with a higher propagation velocity), which means an increase in the time Т₁ is more than compensated by the shortening of the path in another substance (with a lower propagation velocity) and by corresponding decrease in the time Т₂.

The ratio of propagation velocities is called the refractive index n = V1/V2.

Total internal reflection

When a ray passes from one medium to another, the angle of refraction depends on the angle of incidence (it is measured from the perpendicular to the interface between the media).

Under certain conditions -

transition from a medium with a lower speed of propagation of rays to a medium with a higher speed
sufficiently large incidence angles α

- the expression for the angle of refraction loses its meaning: the value of the sine function cannot be greater than one! To understand what this means, let's follow the angle of refraction with a gradual increase in the angle of incidence.

Everything goes ordinary until the angle of refraction reaches the maximum possible value of 90 degrees. In this case, the refracted ray does not pass into the second medium (with a higher propagation velocity), but goes along the boundary. Obviously, an increase in the angle of incidence will only lead to the fact that the ray is completely reflected from the interface and remains in the first medium (with a lower propagation velocity)! This phenomenon is called total internal reflection. An example of using this effect is light guides - thin fibers in which light propagates from one end to the other, reflecting from the side walls.

Lens

A remarkable optical device - a lens - is a transparent substance bounded by one or two spherical (or almost spherical) surfaces. Who was the first inventor of lens and when - it is unknown. Most likely, everything started with jewelry made of (semi-) transparent stones, and, suddenly, someone noticed the amazing optical properties...

Let a beam of rays parallel to each other falls on a flat/plane transparent (glass) plate perpendicular to the surface. After passing through the plate, the rays will remain parallel to each other and to the initial direction. After all, the angle of incidence of rays on the surface is equal to zero both "at the entrance" and "at the exit" ...

What if the second surface of the plate is not parallel to the first one? Then on the second surface (interface) the angle of incidence is not equal to zero, and the rays, being refracted, will change direction.

And if the second surface is not flat, then when a beam of parallel rays reaches it, the angles of incidence for every ray will depend points on the surface. Incident angles will be different from each other for different rays.

Thus the refraction angles will be different from each other too. So, rays that were parallel to each other "at entrance" will cross each other after passing non-flat glass surface.

BTW: glass thickness for every ray will be different. Which means that the propagation time in the glass will be different for different rays.

The spherical surface of the interface of medias with different permeability (the speed of light in the medium) generates amazing properties of lens. The lens can either converge parallel rays to a point or scatter them. For example, a converging lens operates in such a way that each beam of parallel rays is focused (converged) to a single point on the focal plane.

Due to the difference in the angles of incidence / refraction, each ray deviates from the initial direction in its own way. Moreover, the difference in lens thickness (for each ray) compensates the difference in path length from the lens to the point of convergence. And the rays come to the focal point at the same time. How important this is, we'll talk later...

The lens can be considered as an "angle-to-coordinate converter". The direction of the beams (the angle between the beam and the axis of lens) is converted into a point on the focal plane. Moreover, such a “converter” works in both directions: if a small lamp (light source) is placed at the focus of the lens, then the lens will form a parallel beam of light from the lamp's diverging rays.

By combining suitable lenses, one can build a telescope for observing distant stars and planets, and a microscope that allows to see the smallest objects, and many other useful optical devices: illuminators, camera lenses, etc.

This is how just one principle of the shortest time, coupled with geometry, makes it possible to describe the characteristics of any optical devices, develop new devices with desired properties, etc.

Note that beams of light can also be focused using mirrors. Only if the refractive surface of the lens is not flat, then the focusing mirror must have a non-flat reflecting surface.

Scheme of an illuminator that forms a parallel beam of rays. Used (for example) in the construction of lighthouses.

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