GRIN Systems

GRIN stands for GRadient INdex, and refers to variations in the index of refraction in the lens. Even using a flat lens, you can still have parallel rays from a source focus at a common point. It turns out that the center of the lens has the highest index of refraction, and as you move out from the center, the index decreases. A front view of a lens would resemble the lens on the left. The image of the right is a side view of how the light rays bend to the common focus, f, through a lens of thickness, d.

It turns out that parallel light waves coming in from the left hand side are refracted into a spherical pattern when they exit. This allows them to come together at a common focus, similar with other curved lenses that were introduced previously.  The lens in our eye has a maximum index of refraction of approximately 1.406 in the middle, and 1.386 towards the outside. This gives our eye more "focusing power" because even slight variations in thickness also effect the way in which the index variation is distributed.

In this case, we will take an in-depth look into how a a flat lens can focus as in the above pictures. First off, let's begin with the necessary labels:

The major goal is to have all the light sources pass through the lens and meet at the focus at the same time. Here, we have selected two different rays. Ray 2 travels right through the center of the lens, and Ray 1 travels slightly outside of the center. As you can see from the picture, the Ray 1 has to travel a longer distance than Ray 2 does. By making the index of refraction higher in the middle, it slows down Ray 2 so that Ray 1 can travel the additional distance required. In the following equation, we relate the two by measuring the time it takes them to travel through the lens, and then to the focus.

Ray 2 travels through the lens, and then goes from Point B to the Focus through (presumably) air. The time it takes is:

Total Time = (Time through Lens) + (Time After Lens to Focus)

= (NMAX) d + (NAIR) f

d is the thickness of the lens
f is the focal length
NMAX is the index at the center (which is where is reaches a maximum)
NAIR is the index of refraction in wait

Ray 1 does something very similar. It travels through the lens at a point with a smaller different index of refraction, but a longer length from the lens to the focus.

Total Time = (Time through Lens) + (Time After Lens to Focus)

= (NR) d + (NAIR) (Distance from A to F)

d,f, and NAIR are the same as above
r is the distance of the ray from the central axis

NR is the index of refraction at a distance r

Since AB and the Focus form a right triangle, we can use Pythagoras' Theorem. We can replace the (Distance from A to F) with what Pythagoras' Theorem gives us:

(NR) d + (NAIR) (Distance from A to F) = (NR) d + (NAIR) [ sqrt ( r2 + f2 )]

At this point, we can compare the two equations, and try to solve to NR as a function of the other variables. In order to make it more viewable, the two sides of the equation have been  coloured.

Ray 2 = Ray 1

(NMAX) d + (NAIR) f  =(NR) d + (NAIR) [ sqrt ( r2 + f2 )]
(NMAX) d + (NAIR) f - (NAIR) [ sqrt ( r2 + f2 )] = (NR) d
(NMAX) d + (NAIR) [  f - { sqrt ( r2 + f2 ) } ] = (NR) d
NR = (NMAX) + (NAIR/d) [  f - { sqrt ( r2 + f2 ) } ]

In order to simplify the look of the equation, we will assume that NAIR = 1. This is a fair assumption, since the actual value of NAIR is around 1.00029 for light with a wavelength of 589 nm. Also, since term involving the square root [ f - { sqrt ( r2 + f2 ) } ] is a negative quantity, it is easier to reverse the order, and add a negative sign.

NR = (NMAX) - (1/d) [ { sqrt ( r2 + f2 ) }  - f ]

There is one last change that can be made; the rather confusing portion of the right hand side can be simplified. By using Newton's Binomial Theorem, we can make an approximation that the part in the right-hand square bracket is equal to a much simpler expression:

NR =(NMAX) - (r2/2fd)

In doing so, we can see that the index of refraction decrease with respect to the square of the radius. If you graphed the index of refraction versus the radius, you would see that it is a parabola. This is the second factor that allows the lens to focus. These slight variations in the make-up of the lens greatly increase its ability to focus images.

Colour Vision
Colour Math
Focal Lengths and Distances
GRIN Systems
Human Vision
Vision Problems