$$\require{cancel}$$

# Optics

### The Ray Model of Light

What is the nature of light? Light is a thing that has the ability to travel from position to another. Light is absolutely required when trying to form an image with our eyes or other optic tools. When identifying the color of a leaf, we identify it using our very own perception. Just because we see a leaf we immediately assume that the color of the leaf is green.

Today, photography provides the simplest experimental evidence that nothing has to be emitted from your eye and hit the leaf in order to make it green. A camera can take a picture of a leaf even if there are no eyes anywhere nearby. Since the leaf appears green regardless of whether it is being sensed by a camera, your eye, or an insect's eye, it seems to make more sense to say that the leaf's greenness is the cause, and something happening in the camera or eye is the effect.

Unlike water and air, light can travel without a medium thus making it able to travel through space which is essentially a vacuum. The fundamental distinction between sound and light is that sound is an oscillation in air pressure, so it requires air or any other medium like water in which it can move. With today's technology we know that space is a vacuum and we still have the ability to see stars billions of miles away proving light can travel in a vacuum.

The ripples show how water acts as a medium when it comes to vibration and this cannot be done with light.

### Light and Matter

Light interacts with matter in 4 ways:

1.  Emission e.g. hot filament in a light bulb emits light or electron jumps orbital shell

2.  Absorption e.g. your hand absorbs sunlight and feels warm or like a hot iron

3.  Transmission e.g we get to see through air and glass

4.  Reflection or Scattering mirrors reflect and create image (Discussed in ch.2)

### Images by Reflection

Images by reflection explains how we see images when we are seeing through a reflection. This is seen most easily in a mirror, as this is the most common way that people see and acknowledge reflected images.

In the figure to the left, a man is shown looking at a flat mirror. As rays are created from his nose they travel in all directions. If these rays hit the mirror they are reflected, and create a image to all those who look at it. While this Image isn't real, it still appears to be real and thus is called a Virtual Image. This can once again be seen in the image to the left. As the rays are created from the mans face they follow the path of 1->2, starting at 1 and ending at the end of 2 .However these rays are perceived by all those who look at it as traveling from 3 -> 2

However, the above information is only true when applying to flat  mirrors. When a curved mirror is placed into view, the images change. Just as with the flat mirror, rays that are produced from the man's nose will still reflect off the mirror, however this time the rays will reflect themselves at an angle. This causes the virtual image that is created to appear as bigger than the original image. The figure above shows this occurrence. As the rays are created from the man's nose, they hit the mirror and are reflected outward, thus creating a virtual image that appears to be larger than the original.

'The above figure in grey, shows what happens when a curved mirror is placed far enough away from an object (the man's face) and creates a something called a real image. (While this image isn't actual real material, it is called a real image because it can now be displayed an anything in its path.) This real image is created due to the reflections of the mirror actually aiming inward due to the angles created when the rays hit the mirror. In turn the rays combine and reconstruct another visable image, however it is reversed and upside down. The cause of this can be seen in the figure above, where the rays from the face clearly land in a position that causes them to be reversed and upside down.

From the above real image created by curved mirrors, one can now place another mirror where the real image is created in order to reflect that image. This can be seen in the figure to the left. In this figure a real image is created via a curved mirror, then the real image is reflected by another mirror to send the image to a more easily visible spot. This is one of the most basic ways to create a telescope. If you imagine A to be the position of someone's eyes, then they can then see the reflected real image created by the curved mirror. This can be useful to create a real image which mirrors an object that is very far away, and thus not clearly visible to the naked eye.

Images Quantitatively

Despite the numerous possible projections of images when dealing with lenses and curved mirrors, there is only one equation for the location and one equation for the magnification of the image.

Location of an image

C=location of center of mirror

heavily curved mirror means smaller C

lightly curved mirror means larger C

O=location of the object

I = point where image will be formed

To derive the equation:

First, Pick a singular point on the mirror

Second, Draw two lines, one from the image's point on the horizontal axis to the point on the mirror; the other from the object's point on the hor. axis to the point on the mirror.

First, we find that: θi + θo = θf

f= the focus point or the halfway point between the centre of a mirror and the sides

Equations that follow:

1 f=1/di+1/do

d0=-di

θi - θo = θf

Important Lens/Mirror Equation for Location

1 f=1/di-1/do

Aberrations:

Aberrations are imperfections on a mirror or lens or on the resulting images.

In reality, it is nearly impossible to construct a mirror or lens without some degree of aberration

This image illustrates some examples of aberrations:

A spherical mirror is great for images up close. However, at a great distance, the images will appear blurry with a spherical mirror.

So, astronomers use parabolic mirrors to offset this.

Another way to prevent aberrations is to only allow light near the axis to go through,

Magnification of the image:

Magnification of the image is the ratio between the size of the resulting image vs. the size of the object. We can calculate this magnification by measuring the distance from the resulting image to the lens/glass (i) and measuring the distance between the lens/glass to the object (o). The sign of your 'I" value depends on whether your image is on the same side as the object or on the opposite side (if the lens is the barrier).

Your "o" value is usually positive. With this information you can calculate magnification.

M = -i/o

If the M value is negative, the image is flipped upside-down from the object's orientation. Furthermore, the resulting image is real because, as shown in Chapter 2, the rays of light meet.

If the M value is positive, the image is oriented upright and the image is virtual (the rays of light never actually meet up).

Wave Optics

Why is visible light not optimal for microscopes or computer chips? Short Answer: Visible Light has wave properties and the interactions of these waves are not always ideal. We can no longer use the ray model of light - we must use the wave model.

Diffraction: the behavior of a wave when it encounters an obstacle in its medium. Diffraction, in general causes a wave to bend around obstacles and make patterns of strong and weak waves. Diffraction can be useful (such as using x-rays to find bones) or unhelpful (such as light entering telescopes)

Incoherent light: When multiple parts of the light wave are out of step from with each other and have different phase constants (ex. light from the sun)

Coherent light: When all parts of the wave are in step and have the same phase constant (light from a laser beam)

Diffraction angles depend only on the unit-less ratio of  λ/d in which λ = wavelength and d = center to center spacing between the slits.

Correspondence Principle:

When λ/d is small (< 10-4), the ray model of light and the wave model of light must give approximately the same result.

Huygen's Principle

Huygen's principle is used to describe the constructing and destructing interference that occurs between waves in a double slit experiment

It states that "any wavefront can be broken down into many small side-by-side wave peaks, which then spread out as circular ripples, and by the principle of superposition, the result of adding up these sets of ripples must give the same result as allowing the wave to propagate forward

.

### References

Crowell, Benjamin. Optics. Fullerton, CA: Light and Matter, 2008. Online.