Laws of Geometry in Optics

Geometric optics considers beams of light as rays that spread in straight lines within a homogenous sphere. Geometric optics is bases on four axioms.

• Rays of light propagate independently of each other. The total intensity of two beams is equal to the sum of the intensities of each individual beam in the absence of other beams (the principle of superposition). This axiom does not hold in the presence of a phenomenon known as interference.

• Within homogenous media, rays of light propagate in direct lines. The fact that light bends around obstacles that it encounters was discovered in the beginning of XIX century. It is known as diffraction.

• The law of luminous reflection states that the angle of a light beam is equal to the angle of its reflection. Incident and reflected rays - as well as the perpendicular constructed at the incident point - all lie in one plane.

• Snell's law refers to the notion that the ratio of the sine of angle of incidence to the sine of angle of refraction is a constant value for a given pair of materials.

 sin φ1 = ν1 = n2 sin φ2 ν2 n1

The incident and refracted rays, as well as the perpendicular constructed in incident point, all lie within the same plane.

Geometrical optics is not an exact science. Approximation is essential for the existence of paraxial optics, and approximation necessarily leads to a certain amount of impreciseness. Paraxial optics considers only those rays that undergo insubstantial deviations from the optical axis of the system along their route. The angle between the optical axis and the incident beam is so small that paraxial optics considers sin φ ? tg φ ? φ. Deviations from the laws of paraxial optics in optical instruments result in a variety of aberrations.

Parallel beams of lights that propagate along the optical axis concentrate at a point that is known as the spherical mirror focus. This phenomenon transpires after the beams of light are reflected from the spherical mirror; it occurs when paraxial optic approximation is used. The spherical mirror focus deviates from center of the sphere by F = R/2. A relationship between the distance to the object and the distance to the image is described by formula:

 1 1 = 1 = 2 a b F R

Both the mirror and the lens gather the parallel beams in one point. If the distance to the lens focus is equal to F, then lens formula is

 1 + 1 = 1 a b F

This model illustrates the laws of reflection and refraction. A thin beam of light (shown in yellow) falls on a glass hemisphere with a silver-coated upper edge. Input the position of the light source (determined by circumference degrees from the vertical) in the upper input window. The second input window determines the glass refraction index.

As a beam of light falls on the center of the hemisphere from above, it is partially reflected (at the same angle φ), and partially passes through the glass and undergoes refraction upon exiting it. The angle at which the beam leaves the glass (ψ) appears in the message window.

Press the top radio button to watch the beam move towards the center of the hemisphere from below. As the beam reaches the upper surface of the hemisphere, a part of it will travel outside at the angle ψ, and another part will be reflected by the silver-coated surface. The diagram also displays the total reflection angle φ0. When a beam comes into contact with the upper edge at this angle, it will be completely reflected, and no part of it will travel outside.