The principles of geometric optics describe the interactions of light on a macroscopic scale. Geometric optics is mainly used to determine how light will change direction and form images through reflection or refraction. The path of light is approximated by rays that travel in straight lines. All angles regarding reflection and refraction are measured from the normal, which is a line perpendicular to the surface the light is interacting with.
Reflection: When light travels away from an object at the same angle that it hits the surface.
Normal: Line perpendicular to the surface of interest.
Refraction: Occurs when light changes the medium it is moving through.
Snell’s Law: Determines the angle at which a ray of light refracts.
Total Internal Reflection: Occurs when the light ray’s incident angle is greater than the critical angle, causing all of the light to be reflected back into the original medium.
Image: An image forms where rays of light coming from an object seem to converge. There are two kinds:
Real Image: Forms where the path of the light rays actually do converge.
Virtual Image: Forms where the light rays actually converge.
Focal Length (Focus): The point where originally parallel light rays converge when they reflect off a curved mirror or refract through a lens.
Lensmaker’s Equation: Allows us to calculate the focal length of a lens based on the material its made of and the curvature of the lens.
When light hits a surface, it always reflects away from the object at the same angle that it hits the surface. The angle is measured relative the normal. Depending on the surface, two types of reflection are possible:
Image Credit: Johan Arvelius, CC-BY-SA 2.5 | Image Credit: Marcelo Reis, CC-BY-SA 3.0
Light refracts when it travels from one medium to another (examples: from air to water, from glass to air). Light will change direction when entering a new medium because it will travel at a different speed in the new medium.
Depending on the angle, an interesting phenomenon called total internal reflection can occur. Whenever light is moving from a medium of a higher index of refraction to a medium of lower index of refraction, some of the light is reflected back into the original medium. If the angle is greater than the critical angle, all the light is reflected. The critical angle is the angle at which light is refracted at 90° from the normal (along the interface between the two mediums).
Ray diagrams illustrate the path of light when it interacts with mirrors and lenses.
Light from an object hitting a mirror or lens is approximated by two or three arrows that go towards the top, center, and bottom of the mirror or lens. Since ray diagrams are approximations, we will not need to calculate any specific angles of reflection or refraction.
In a ray diagram, the solid lines indicate the actual path of light rays. Real images form where the rays converge. On the other hand, dotted lines indicate the path of virtual rays. They form virtual images when they converge.
There are three main characteristics of the image produced by the arrangements below that we need to keep track of:
Concave mirrors are shaped like a parabola and curve towards the object. One key factor of a concave mirror is that light rays coming in parallel (from an infinite distance) will all come together at the focus. There are a few different arrangements that are associated with concave mirrors. Each one produces a different sort of image.
Resulting image: virtual, upright, magnified
Resulting image: real, inverted, same size
Rays reflect parallely so an image never forms
Resulting image: real, inverted, reduced
Resulting image: real, inverted, magnified
Image Credit: All mirror images by Cronholm144, CC-BY-SA 2.5
Convex mirrors are mirrors that curve away from the object. Convex mirrors always produce the same kind of image.
Image Credit: Cronholm144, CC-BY-SA 2.5
Resulting image: virtual, upright, reduced
Also called a diverging lens because the light rays will diverge after passing through the lens. Diverging lenses always form a virtual, upright, diminished image.
Image Credit: Cronholm144, CC-BY-SA 2.5
Also called converging lens because light rays will come together at the focus. There are a couple different configurations that are important.
Resulting image: real, inverted, reduced
Image Credit: DrBob, GNU-FDL 1.2
Resulting image: virtual, upright, magnified
Image Credit: DrBob, GNU-FDL 1.2
Note: The Lensmaker’s equation shown here is actually an approximation that assumes the lens is thin and is in air.
Snell’s law:
\(n_1 \sin \theta_2 = n_2 \sin \theta_2\)
n - index of refraction
θ - angle from normal
Lensmaker’s equation:
\(\frac{1}{f} = (n - 1) (\frac{1}{R_1} - \frac{1}{R_2})\)
f - focus/focal length
R - radius of lens curvature
\(\frac{1}{f} = \frac{1}{S_\circ} + \frac{1}{S_i}\)
\(S_o\) - distance to object
\(S_i\) - distance to image