Module 10

Optics

Last Updated: Tue Nov 5 03:11:21 PM CST 2024

Geometric Optics

We model light as rays that propagate through space.
A light source emits rays of light.
Surfaces reflect or refract light rays (or absorb) according to the law of reflection and refraction.
Optics problems basically become geometry problems.

Reflection and Refraction

Convention

Incident ray
Reflected ray
Refracted ray
Angle of incidence, reflection, and refraction are all measured with respect to the surface normal.

Reflection:

Refraction (Snell’s Law):

Light Ray Box: Reflection and Refraction

Laser beams travel along straight lines, just like rays. This gives us a nice way to “see” ray optics.

Law of reflection
Law of refraction

Refraction vs. Reflection

Refraction vs. reflection is not either-or. Some light reflects, and some light refracts.

The amount of each depends on the angle.

https://phet.colorado.edu/en/simulations/bending-light

Light Ray Box: Total Internal Reflection

If the refracted angle is greater than 90 degrees, then we get complete reflection.

This is called “total internal reflection”.

Examples: Retro Reflector

Consider a ray of light incident on two mirrors that make a 90 degree angle.

How far away is the moon?

Examples: Depth Perception

If you collect two rays from the same source, you can work out where the source is at.

This is an example of triangulation

Examples: Apparent Position and Mirrors

We can determine where an image of an object in a mirror will appear to be by considering two rays from the object reflected off of the mirror.

Examples: Apparent Position and Refraction

We can do the same thing for refraction.

How deep does a fish look if it actually sits a distance \(d\) below the surface of the water?

Small Angle Approximation

We will often find that the small angle approximation greatly simplifies our algebra.

Small Angle Approximation

When angles are small, we can approximate sine, cosine, and tangent in terms of the angle directly.

\(\sin\theta \approx \theta\) (in radians!)
\(\cos\theta \approx 1 - \frac{1}{2}\theta^2\)
\(\tan\theta \approx \sin\theta \approx \theta\)

Small Angle Approximation

All approximations will have less than a 1% error for angles less than 0.17 radian, which is about 10 degrees.

Examples: Apparent Position and Refraction

Some objects are not where they appear…

Examples: Apparent Position and Refraction

We can do the same thing for refraction.

How deep does a fish look if it actually sits a distance \(d\) below the surface of the water?

Examples: Spherical Refractive Surface

A spherical refractive surface will bend light toward a single point, its focal point.

Example: What is the focal length of a convex spherical surface made of a material with a higher refractive index than the surrounding media?

Applications: Lenses

Take two spherical refractive surfaces and put them together.
Both the front and back surfaces will refract light.

Lensmaker’s Equation

Sign Convention:

The radius of curvature for each surface may be positive or negative.
The RoC is positive if the center of the sphere is on the “back” side of the lens (where the light rays are going).
The RoC is negative if the center of the sphere is on the “front” side of the lens (away from where the light rays are going).

Applications: Optical Fiber

Total internal reflection is used to transmit light through flexible glass rods (light wires)

For example, consider a glass rod with a refractive index of 1.5. What is the “Acceptance Angle”?

Dispersion

Different colors refract different amounts.

This is called “dispersion”.

Dispersion

Light is a wave, so it has a wavelength.
Our brain interprets different wavelengths of light as different colors

The refractive index of most materials depends on its wavelength, i.e. color

Applications: Prism

What is the angular separation of red and blue light emerging from a prism?

Is red above blue?

Applications: Rainbows

Water droplets can refract and reflect light rays back toward the Sun

Applications: Higher-order Rainbows

More than one rainbow can appear. These are called Higher-order.

Applications: Spherical Mirrors

Mirrors reflect light rays.
Mirrors can be shaped to reflect light rays to (or away from) a single point.

Collimating Light

The principle of ray reversal says that for any path that a ray can take through an optical system, the reverse path is also possible.

For Example

A lens will bend parallel rays to the focal point.
A lens will bend rays coming from the focal point to parallel.

Fresnel Lens

What if we needed a lens with a big diameter?

Aberrations

Not all lenses will bring all rays to the same points.
Since the refractive index depends on color, different colors will not be sent to the same points.
This imperfect ability to focus all rays is called “aberration”.
There are various types:
- Spherical
- Chromatic
- Etc.
Some of these can be removed, some cannot.

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