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Basic principles of the ray model (2013)

Page history last edited by Joe Redish 1 year ago

Working Content > Three models of light > The ray model

 

Prerequisites:

 

We have a lot of experience with light. One of a newborn infant's first tasks is to learn to interpret the light its eyes receive and to parse the world into objects. We live with light everyday and interpret it naturally, automatically, and without thinking about it. But understanding what's going on and what light "is", is not so trivial. For centuries, some people treated light as if it were a kind of touch -- as if we were reaching out with our eyes to get information rather than collecting information that came to our eyes naturally. We still have some of this in common speech: the sense that you can "feel" when someone is looking at you even if you can't see them. Careful experiments show that this is not the case. (There are animals that send out signals and probe the world through the signals' reflection -- dolphins and bats, for example -- but they use sound, not light.)

 

To build a model of what light "is" we need to develop an idea of the nature of the phenomenon: what kind of thing light is, and how it behaves. Although many of the principles of the ray model of light were known to the Greeks 2000 years ago and were improved and refined by the Arabs 1000 years ago, we will describe the ray approach to light in terms of a model developed by Newton in the 1600's since he was the first to build a theoretical foundation for what kind of thing light might be -- not just a description of how it behaved.

 

Everyday experience: Phenomenological footholds

In figuring out what our experience tells us about light we start with the basic assumption

 

Light is some kind of substance that travels and carries information.

 

The "traveling" is not obvious. Light moves so fast that the technology to measure its speed directly was not available until the middle of the 1800's. (Though Roemer inferred a value for the speed of light from small deviations from the prediction that Newton's laws of motion made about the orbits of the moons of Jupiter in the late 1600's.)

 

From our experience with light, dark, and shadows, we can infer some basic foothold ideas:

 

  • Some objects (the sun, fires, light bulbs) give off light.
  • Other objects scatter that light, typically in all directions.
  • We only see something when light that has come from it enters our eyes.
  • Light typically consists of a mixture of different kinds of light -- colors -- that can be separated by passing the light through a prism (triangle of glass).
  • In a uniform medium -- air, glass, water -- light travels in straight lines.

 

A bit of light traveling in a straight line is called a light ray.

 

This gives us a simple picture of ordinary vision. Light from one of the sources of light goes off in all directions traveling in straight lines. When it hits some object, it scatters from every point in all directions. Some of those rays come into our eyes and permit us to see the object. We can infer that the light is scattered in all directions since many people in different places can see the same spot.

 

There is an interesting implication. The light that comes to our eyes from the classroom wall doesn't interfere with a student looking across that light to see the projection screen at the front of the room.

 

From this we can infer

 

  • Light is invisible (transparent) and doesn't interfere with other light passing through it.

 

This is a peculiar statement! How can light be invisible if it's what we see? This just means we can't see it unless it comes into our eyes. Light doesn't scatter off of light; rays of light just pass through each other without interfering with their travel. (This should remind you of our wave superposition principle.) This is a well-known principle for teasing cats -- since they can only see the spot of light when it scatters off the wall, not when it is traveling from the laser to the wall. An interesting experiment to try is to shine a beam of laser light on a wall. The spot is visible but the beam (ray) from the laser to the wall is not -- unless you sprinkle dust in its path!

 

These foothold ideas lead to the analysis of shadows and light passing through holes in screens. This sounds trivial, but it isn't! (See the very nice lesson "Light and Shadow" in Tutorials in Introductory Physics, by the University of Washington Physics Education Group.)

 

Two more phenomenological principles are needed to complete the ray model: what happens when a ray hits a mirror (reflection) and what happens when it crosses a boundary between two transparent media (refraction).

 

  • Reflection: When a light ray hits a smooth polished surface (mirror) it bounces off not at random but in a regular way. The incoming angle with the normal (perpendicular) to the surface is equal to the outgoing angle with the normal; or briefly, at a mirror the angle of incidence (θi) equals the angle of reflection (θr).

 

  • Refraction: Each transparent medium has a dimensionless constant associated with it called its index of refraction, n. When a light ray passes from one transparent medium to another it bends towards or away from the normal to the interface according to Snell's law. For vacuum, we define n = 1. (We can choose n to be one for something since all the experiments measuring n are relative to each other.)

 

n1 sin θ1 = n2 sin θ2

 

 

Note that in both the case of reflection and refraction, the angles are measured to the normal, NOT to the surface!

 

When these principles are added to our psychological interpretation of light, a huge number of valuable results can be obtained. See the follow-ons for explicit examples.

 

Newton's particle model: A theory of rays

Newton wasn't satisfied with these foothold principles, most of which were already known when he started thinking about light as a young man while fleeing London during the plague. (He did discover the color principle.) He was always interested in mechanism -- what were things -- and in mathematizing -- what were the mathematical rules they followed; and of course, how did it all fit together?

 

He proposed the idea that light consisted of small particles moving very, very fast. As a result, they seemed to move in straight lines. Although gravity would make them fall, because of the high speed of the light, this would be a tiny and negligible effect. Reflection came out simply assuming elastic bounce and no friction at the surface so that the component of velocity parallel to the surface stayed the same and the one perpendicular to it flipped.

 

To get Snell's law, keeping the idea that there was no friction at the surface so the parallel to the surface component of the velocity didn't change, he had to assume that the particle of light went faster in the more dense medium. As the component of velocity perpendicular to the surface got longer, it bent the velocity vector towards the normal -- exactly as observed.


 

Newton's model turns out to be wrong. Light actually goes slower in dense media (in contrast to sound). But Newton's model held sway over Huygens' competing wave model for 100 years, in part because Huygens' model was a lot harder to think about, in part because of Newton's prestige, and in part because it gave excellent results for the design of optical instruments -- telescopes, microscopes, and eyeglasses.  The modern photon model is a complex weaving together of a particle and wave model. Newton's particle model is still a useful way to think about most ray situations.

 

The psychology: How we see by looking

To complete the ray model we have to add some additional principles -- but psychological ones, not physical ones. We are interested in what the ray model tells us that things look like, but despite our tendency to assume "what we see is what there is", our brain plays a huge interpretive role in what we see. Every retina has a hole -- the optic disk where the cable of nerves that carries the signals received by the eye back to the brain. The brain compensates for this bit of bad design by "filling in" with plausible continuations. (This leads to some nice "blind spot" demonstrations.)

 

A lovely demonstration of the fact that the brain interprets the signals from vision rather than just "seeing" them directly is Ed Adelson's Checkershadow demonstration shown at the right.  Surprisingly, the two squares labeled A and B are the identical shade of gray. Check it yourself by printing it out and cutting, or by using a clipping program to cut out images of the two squares on your screen and putting them next to each other. Some computers come with a digital color meter that lets you read the color details for each pixel on your screen. (On a Mac look for DigitalColorMeter in Applications/Utilities.) The brain here is not really making an error; it is interpreting the information using the context. If two regions have the same color and one is in light and the other in shadow, then "really" (when displayed in the same light) the one in shadow will be lighter -- and your brain actually handles that.
 

 

It's really quite amazing when you think about how the brain puts together the information that it receives from its two two-dimensional retinas into a physical three-dimensional world, parsing what are only dots of color into distinct objects. The process is only partly understood as of this writing. The question we want to ask is: How do we know where something is by looking? Though there are many distinct clues the brain uses to place objects in a three-dimensional field using its two-dimensional information -- motion when we move our eyes or head, fuzziness (for very distant objects), and expectations (we know how big it really is and interpret how far it is by how big it appears) -- we will concentrate on binocular vision: the use of two perspectives.

 

The lens of the eye and the particular pixel (rod or cone) that a bit of light hits determines a direction. If both eyes detect a bit of light coming from the same point on an object the two rays will trace back to the same point. (Leave the question of how the brain manages to identify which bits of light go together to the neuroscientists for now!) The angles between the two directions allow us to tell where it came from. So we propose the following foothold psychological principle.

 

  • Our brain registers the images from the two eyes against each other, matching points by context. The brain then identifies the distance of a point on an object by tracing back rays and assuming the point is at the convergence point of the traced back rays.

With these few principles an extraordinary amount of optics can be done, giving us a powerful set of tools for improving our vision by "fooling" our brains through the bending of light.

 

Follow-ons:

 

Joe Redish 4/10/12

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