Introduction

On Monday, we were talking about light and optics.
We talked about what coherent light was and the color spectrum
We looked at how to convert between the frequency and wavelength of light.
We then talked about Snell's law and gave an application of Snell's law to two prisms.
We generalized this to parallel light waves hitting a simple lens, first perpendicularly to the lens and then at an angle.
We start today, by discussing focusing non-parallel waves.

Image Focusing

The first image above shows a situation where rays of light emitted from a point in front of a lens will converge to a point behind the lens.
The equation which governs is:
`1/s_1 +1/s_2 = 1/f`,
where `s_1` is the distance in front of the lens, and `s_2` is the distance behind the lens.
Notice the plane of points at the same `s_1` distance from the lens will all converge on the plane at distance `s_2` behind the lens, forming an image a show.
The above was under the assumption that `s_1 > f`.
If `s_1=f` then light from a point source would have parallel rays on the other side of the lens.
If `s_1 < f`, then the light coming a distance `s_1` will appear as if it came from a source at distance `s_2` from in front the lens, and one would get a magnifying effect as shown in the lower image.

The Lensmaker's equation, which can be derived from Snell's Law, tells us how to compute the focal length, `f`, of a lens based on the radius of curvature of its front and back surfaces, `r_1` (a positive value) and `r_2` (a negative value), and the difference in the index of refraction outside and inside the lens, `n_1` and `n_2`:
`(n_2 - n_1)(1/r_1 - 1/r_2) = 1/f`.
The version above assumes the thin lens approximation. I.e., that the lens thickness is small compared to `r_1` and `r_2`.
Concave: A concave simple lens is one in which parallel rays are forced to diverge (see picture above). In this case, the focal length is negative. The focal length length is in front of the lens. The lensmaker's equations is modified in this case to use a negative value for `r_1` and a positive value for `r_2`.

Sometimes it is more convenient to use the quantity `D= 1/f`, called diopters, directly rather than the focal length.
A large positive diopter lens has greater converging power than a smaller one. Similarly, a more negative diopter lens (a concave lens) has a greater diverging power than a smaller diopter one.
In VR applications, we will combine several lenses.
The effective number of diopters of the result can be computed by adding the diopters of the component lens, as in the figure below:

Various imperfections called aberrations can cause lenses not to behave in the ideal way just described.
These aberrations need to be considered when designing and designing for VR headsets.
Chromatic aberrations occur when the focal length of a lens varies with the wavelength of light.
This is caused because the index of refraction of a lens is really a function of the wavelength of light, `n(\lambda)`.
The top left figure shows a lens with focal lengths varying depending on if the light is blue, green, or red. The top right figure shows how an image with a lot of chromatic aberration might look.
Spherical Aberration is caused when rays further away from the center of a lens are refracted more than those closer to the center of the lens.
This kind of aberration does not depend on the light wavelength and so is called a monochromatic aberration.
If spherical aberration occurs, there will be no single plane on which the image will be in focus. There will, however, be a curved surface known as a Petzval surface on which the image will still be in focus.
To prevent spherical aberrations one can use lens, called aspheric lenses, which do not have one single radius of curvature.

Even if the image projects onto the image plane, it may be distorted on the periphery.
Assuming the lens is radially symmetric, the distortion can be described as a stretching (causing a barrel distortion) or compression (causing a pincushion) of the image away from the optical axis.
The effects of these can be seen in the three images above which correspond to an original image with a spherical lens beneath it, a barrel distorted image, with a lens of greater than spherical curvature, and a pincushion distorted images with a lesser curvature lens beneath it.
As we want wide fields of view in our VR headsets, we often make use of a fish-eye lens for which barrel distortions are very strong.
This might yield images like the lower one seen above.
To make usable VR systems, it is crucial to be able to correct for these kinds of distortions.

It a lens is elongated with respect to one axis over another (not necessarily vertical and horizontal) then one gets astigmatism aberrations, as illustrated by the top diagram.
It can result in separate focus planes with respect to one axis versus the perpendicular axis.
The right image shows the effect of this on text.
A Coma aberration is caused when when image magnification varies dramatically when incoming rays are far from perpendicular. See lower left image above.
A flare aberration (lens flare) occurs when very bright light scatters when passing through a lens rather that being transmitted in a straight fashion (Lower right image above).
This is often seen in movies when the viewpoint passes by the sun or stars and is sometimes added artificially because it looks cool.

The human eye is important to understand to develop effective VR headsets.
Many of the lens concepts we have been talking about directly apply to it.
The above figure illustrates how a light ray might travel through the different materials of the eye, each with its own index of refraction.
The focal plane of a lens in the eye is replaced by a spherically curved surface called the retina which contains photoreceptors that convert the light into neural impulses.
We will finish talking about this Monday, and then start talking about the Oculus native interface.