Geometrical optics

1. Ray Optics

2. The Nature of Light
The question of the nature of light is an old one. Throughout the history of
physics, it has been argued back and forth as to whether light is made up of
particles or if it is some sort of wave.
Newton took the side of the particle, or corpuscles, as they were called.
However, others, such as Young, showed experimentally that light exhibits
wave-like features. For example, light can produce interference patterns.
So light is a wave, spread out over space.
But that’s not the whole story. Certain effects connected with the absorption
and emission of light reveal that it also has a particle aspect. For example, the
energy carried by light is packaged in discrete bundles called photons or
quanta.
So light is a particle, localized in space.
This unusual and seemingly contradictory nature is referred to as
wave-particle duality.

3. The Ray Model of Light
Given the wave-particle duality of light, should we treat it as a wave or as a
particle when it comes to optics?
The answer is neither! That is, neither if we are considering geometric optics.
For geometric optics we use the ray model of light.
A light ray is a line in the direction along which light energy is flowing. A light
ray is an abstract idea, not a physical entity or a “thing.”

4. Light rays travel in straight lines.
Light rays can cross without either being affected.
A light ray travels forever unless it interacts with
matter, which can occur in a number of ways:
Light can be reflected, refracted, scattered, or absorbed.
An object is a source of light rays.
Rays originate from every point on the object, and
each point sends rays in all directions.
Objects may be self-luminous or reflective.

5. In order for our eye to see an
object, rays from that object
must enter the eye.

6. Reflection
There are two varieties of reflection—specular and diffuse.
Specular (regular) reflection is what we observe from a flat, smooth
mirrored surface.
In this case we observe a law of reflection:
Specular reflection
The angle of incidence equals the
angle of reflection.
Note that the angles are taken
from the normal.

7. The Law of Reflection
The angle of incidence is equal to the angle of reflection.
θi = θr
Both angles are measured from the normal to the surface.

8. Reflection
There are two varieties of reflection—specular and diffuse.
Diffuse reflection occurs when light hits a rough surface.

10. Scattering by Diffuse Reflection

11. Refraction
When light hits a surface, it does not always reflect. Sometimes it enters into the
second medium. If the second medium differs from the first in its index of
refraction, then the light will be bent, or refracted as it enters the second
material.
The transmission of light from one medium to another, but with a change in
direction, is called refraction.

12. Refraction
Refraction depends on the index of refraction of the two media. We’ve noted
that light in vacuum travels with the speed c = 3 × 108 m/s. But in other
materials, light slows down to some speed v. The ratio of these speeds is called
the index of refraction:
Index of refraction = n = c/v

14. Refraction ray diagrams are “reversible.”

15. When light travels into a medium of higher index, it slows down. Recall the
equation
v = λ f.
So what gets smaller—the wavelength or the frequency, or both?
Important Note
When light passes from one material to another, the frequency f does not change.
It is the wavelength λ that is changed.
The reason that the frequency doesn’t change is that each crest hitting the
interface must be a crest as it passes into the second medium. Each time a crest
comes in, that crest is transmitted. So the frequency of crests on one side of the
boundary must be that on the other side of the boundary.
We can write the index of refraction in this way:
n = c/v = λ vac fvac/λmat fvac = λvac /λmat
Since n is greater than 1,
the wavelength in the material is less than the wavelength in vacuum.

17. Snell’s Law
Index of refraction = n = c/v
Suppose medium 1 has an index of refraction n1, and medium 2 has an index of
refraction n2.
If a ray refracts between medium 1 and 2, the ray angles θ1 and θ2 are related by
the following:
n1 sinθ1 = n2 sinθ2.
This relation is known as Snell’s law.
There are three situations:
A ray entering a material of larger index bends toward the normal.
A ray entering a material of smaller index bends away from the normal.
A ray oriented perpendicular to a surface does not bend, regardless of the
materials.

18. Total Internal Reflection
With θ1 = θc , and θ2 = 90º, Snell’s law gives us sinθc = n2/n1.

19. Fiber Optics

20. Color and Dispersion
One of the most obvious visual aspects of light is the phenomenon of color. Yet
color is a perception, and not something inherent in the light itself.
Our perception of color is based on the wavelength of the light. But the fact that
we see a wavelength of 650 nm as “red” tells how our visual system responds to
the electromagnetic waves. There is no “redness” associated with the light itself.
Newton ran a series of experiments and showed that white light is a mixture of
all colors. He also showed that a prism does not somehow add the color to the
light, as was previously thought.

21. Color and Dispersion
If a prism does not alter the light or add anything to it, why does the incoming
white light emerge as a multitude of colors?
The reason is that the prism causes the different color components of the white
light to take slightly different paths through the material.
In other words, different colors—that is,
different wavelengths—are bent at the
interface to slightly different angles.
As we’ve seen, this bending depends on
the index of refraction, n. So the index of
refraction of a material must vary
slightly with wavelength.
For example, glass has a slightly larger
index for violet light than for green or
red light. Thus violet light refracts
more than red light.

24. Mirrors and Lenses

25. Thin Lenses
To create a bright, well-focused image, we often use a lens. A lens is a
transparent material that uses refraction of light rays at curved surfaces to
form an image.
A converging lens causes the rays to
refract toward the optical axis. The point
where the rays meet is called the focal
point, and the distance from the mirror to
the focal point is the focal length.
A diverging lens causes parallel rays to
refract away from the optical axis. The
point where the rays, if traced back,
appear to diverge from is called the focal
point, and the distance from the mirror to
the focal point is the focal length.

26. The focal length is the distance from the lens at which rays parallel to the optical
axis converge or from which they diverge.

27. Note that a lens actually has two focal points.

28. The image here is a real image, since the light rays actually converge at, and pass
through, a particular point.

30. Ray tracing for a converging lens

31. 1 1 1
'
s s f
 
This equation holds for both converging and diverging lenses.
It also holds equally well for mirrors.

32. Virtual Images
We’ve seen that for a converging lens with the object at a distance s > f, we
have a real image on the side of the lens opposite from where the object is.
Furthermore, the image was inverted.
This isn’t always the case for a converging lens.
When s < f, we have a virtual
image that is upright.

34. Diverging Lenses
Diverging lenses always make virtual images and, for this reason they are
rarely used alone. However, they are often used in combination. For example,
cameras and eyeglasses use diverging lenses.
Object Image

36. The Thin-Lens Equation
The thin-lens equation relates the object distance s, the image distance s’, the
index of refraction n of the lens material, and the radii R1 and R2 and the lens
surfaces.
1 2
1 1 1 1 1
( 1)
'
n
s s f R R
 
    
 
 