Geometric optics

## Big Picture

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.

## Key Terms

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.

## Reflection and Refraction

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:

• Specular Reflection: Rays of light reflect off a smooth surface parallel to each other. This means it is possible to see the original image that hit the surface. All mirrors employ specular reflection to display an image.
• Diffuse Reflection: When rays of light reflect off a rough surface, light reflects in all directions due to surface irregularities. 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.

• How much light changes direction is determined by the medium’s index of refraction, the ratio of the speed of light in a vacuum to the speed of light in the medium, and the incident angle of the light ray. This relationship is described by Snell’s law.
• To a lesser degree, the angle of refraction also depends on the wavelength of light being refracted.
• All mirrors use specular reflection to create an image. Lenses are special shapes made of transparent material that refract light into an image.

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

Ray diagrams illustrate the path of light when it interacts with mirrors and lenses.

• If the light is interacting with a mirror, it will reflect off the surface.
• If the light is interacting with a lens, it will refract through the lens.

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.

## Physics

Geometric optics cont.

## Ray Diagrams (cont.)

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.

• Real images can be projected onto a surface. You can see this by putting a white piece of paper on one side of a magnifying glass and then pointing it at a bright window. Move the paper back and forth relative to the magnifying glass until you see the window on the paper.
• Although the actual path of the light rays never intersect, when the diverging light rays hit your eyes, your brain will work backwards and construct an image at the point where the rays seem to intersect. Virtual images cannot be projected onto a surface. Flat mirrors always create virtual images.

There are three main characteristics of the image produced by the arrangements below that we need to keep track of:

• type of image: real vs. virtual
• inverted vs. upright
• magnification: magnified vs. reduced

## Concave Mirrors

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.

### Object Between Focus and Mirror

Resulting image: virtual, upright, magnified ### Object at 2F

Resulting image: real, inverted, same size ### Object at Focus

Rays reflect parallely so an image never forms ### Object Beyond 2F

Resulting image: real, inverted, reduced ### Object Between Focus and 2F

Resulting image: real, inverted, magnified Image Credit: All mirror images by Cronholm144, CC-BY-SA 2.5

## Physics

Geometric optics cont.

## Convex Mirror

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

## Concave Lens

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

## Convex Lens

Also called converging lens because light rays will come together at the focus. There are a couple different configurations that are important.

### Object Between Focus and Lens

Resulting image: real, inverted, reduced Image Credit: DrBob, GNU-FDL 1.2

### Object Beyond Focus

Resulting image: virtual, upright, magnified Image Credit: DrBob, GNU-FDL 1.2

## Important Equations

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