More Light and Optics

CS185c

Chris Pollett

Mar 11, 2019

Outline

Remember your first project progress report is due this Wednesday.
Last week, we began talking about light and optics and their role in VR.
We talked about three different ways to view light: as photons, as waves, and as rays.
We described how the intensity of light fall off as `1/r^2` and how waves from a distant sources will tend to hit a plane in a more and more parallel way.
We then described different processes by which light might interact with a material: specular reflection, diffuse reflection, absorption, and transmission.
We said that the above is a good starting viewpoint for looking at idealized lenses, but misses some details that become important when studying lens aberrations.
Today, we begin by briefly discussing coherent light and lasers.

One way in which our picture so far is too simple is that light sources don't typically emit coherent light.
Coherent Light is light with perfectly aligned wavefronts in both time and space.
An example of a coherent light source is a laser.
It emits parallel waves of a constant wavelength such that all the peaks in the waves align as they propagate.
Common light sources such as a light bulb and the sun emit a jumble of waves that have various wavelengths that do not have aligned peaks.

Which of the following statements is true?

The function navigator.getVRDisplays() returns a Promise from which an array of VR Display objects can be gotten.
The only Javascript object allowed to have a requestAnimationFrame() method is Window.
It is impossible to do screen casting using the Android debugger (adb).

The first picture above shows the wavelengths of light visible to humans.
Each wavelength corresponds to a spectral color that we would see it as if it came from a coherent source.
Wavelength between 700 and 1000nm are called infrared, longer wavelengths correspond to microwaves (between a third of a meter and 3 mm) and radio waves (fractions of a meter to kilometer lengths).
Wavelength between 100 and 400nm are called ultraviolet. Although we can't see these waves, some birds, insects, and fish can.
The second picture above shows what various wavelengths make up typical light sources.
If all light sources were equally present one would see white light; if no light was present, we would perceive black.

We don't usually perceive light straight from light sources.
Instead, we see the reflections of light off of various surfaces.
Each surface has its own distribution of wavelengths it reflects based on the material it is made of. See figure above.
For us to perceive an object as red, the red wavelengths must be included in the light source, and the surface must strougly reflect red wavelengths.
For example, a white light has all wavelengths of light and so if we shine it on a tomato, we can see that the main color that would be reflected is red. The other colors although present in the light source would not be reflected so we would see them.

Often it is useful to discuss the frequency of light rather than its wavelength. This is how many wavefronts pass a given location in a given unit of time.
One can convert between wavelength and frequency using the formula:
`f = s/lambda`
whee `f` is the frequency `s` is the speed of light in the medium of interest, and `lambda` is the wavelength.
In the case of a vacuum `s` is a constant `c = 3 times 10^8` m/s.
Light travels 0.03 percent faster in a vacuum than in air, which can be neglected in most engineering situations.
Using the formula above, one can calculate that visible light has a frequency between 400 THz (red) and 800 THz (blue).
Light propagates through denser media such as water or a lens in a significantly slower fashion.
This difference in speed is the basis for most optical systems.

Lenses are materials that bend rays of light so that a focused image is formed.
The left hand figure above shows an early lens from Nimrud in Assyria that dates to 700BC.
Over the centuries, lenses have been used in many devices such as eye-glasses (right hand figure above), telescopes, magnifying glasses, binoculars, cameras, and microscopes.
Over this time period, many design patterns have been developed to optimize the making of these well-understood devices.
VR headsets are unlike classical optical devices, and give rise to many new challenges that are outside the standard patterns that have existed to date.
To understand these challenges we first need to understand how simple lenses work.

Snell's Law, which we have mentioned before, describes how much light rays bend when entering or exiting a transparent material.
Recall that speed of light is slower in a non-vacuum medium, `s`, then it is in a vacuum, `c`.
Define the refractive index to be the ratio:
`n = c/s`.
For example, `n=2` means that light takes twice as long to traverse the medium than through a vacuum.
Here are the values of `n` for some common materials: `n= 1.000293` for air, `n=1.33` for water, `n=1.523` for crown glass.
The figure above shows parallel incident wave peaks hitting a surface at an angle for light switching from a material with index `n_1` to one with index `n_2` where `n_1 < n_2`.
Once the waves start passing through the surface the distance they can travel in the same amount of time is less. This causes the bending in the direction of the wavefronts.
Snell's law tells us:
`n_1 sin theta_1 = n_2 sin theta_2`.

Solving Snell's Equation for `theta_2`, yields:
`theta_2 = sin^{-1}((n_1sin theta_1)/(n_2))`.
If `n_1 < n_2` the quantity inside the parenthesis will be smaller that `sin theta_1` and so the angle will be closer to the perpendicular to the surface.
If `n_1 > n_2` the opposite occurs, the angle will be further from the perpendicular.
If `n_1` is so much different than `n_2` that the quantity inside the parentheses is out of the range of `sin^{-1}`, i.e., 0 to 1. then the light rays reflect from the surface.
For example, a swimmer might see a reflection of themselves if they look up at the surface of the water from underwater depending on the viewing angle.

Suppose we shine light through a prism as in the above illustrations.
Snell's Law can be applied to calculate how the light ray bends after it enters and exits the prism.
Note for the upright prism, a ray pointing slightly upward now points slight downwards; for the upside down prism the light gets bent in the opposite way.
Once the refractive index of the prism is fixed, the bending only depends on the angles at which the rays enter and exit the surface rather than the thickness of the medium.
We will exploit this principle when constructing lenses.

The above figure on the left shows a simple convex lens.
Instead, of making a diamond shape, the lens surface is spherically curved so that incoming, parallel, horizontal rays of light converge to a point on the other side of the lens.
This special place of convergence is called the focal point.
Its distance from the lens center is called the focal depth or focal length.
The incoming rays on the left above are special in two ways:
1. They are parallel, as from a source infinitely far away.
2. They are perpendicular to the lens plane.
The figure on the right shows what happens if the rays remain parallel but are no longer perpendicular to the lens plane.
Using this concept, light from different parts of an object would form a real image on the image plane as it were projected on a screen.
Next day, we'll talk about what happens if the rays are not parallel.