focal points, image formatio

Physics II

Class Notes, 2/22/99

The figure below depicts parallel rays traveling through air and striking a circular interface with a material whose index of refraction is 1.33.

Rays close to the center ray (the center rate is the one which strikes the circular interface at a perpendicular and hence passes through undeflected) will all converge at or very near a single point, called the focal point.
Points further away from the center ray will not converge as close to the focal point.

The figure below shows parallel rays being reflected from a mirror.

Rays close to the center ray (the center ray is the one which strikes the mirror at a perpendicular and is therefore reflected straight back) will all converge at or very near a single point, called the focal point.
As with a circular lens, points further away from the center ray will not converge as close to the focal point.

The reflection from a curved surface has the property that the angle thetaI of the incident ray, as measured from the normal, is equal to the angle thetaR of the reflected ray.

Video Clip #1

A spherical reflector of radius R reflects all parallel rays close to the center ray to a focal point which lies a distance R/2 from the reflector.

This result can be fairly easily derived from the geometry of a circle.

The figure below shows a central ray and a parallel ray at a small distance `dx from the central ray entering a clear, thin spherical container full of water (index of refraction about 1.33).

The distance `dx is small in the sense that it is much less than the radius R of the circle.
The parallel ray strikes the surface at an angle `thetaI with the normal to the surface.
The normal line is a radial line from the center of the sphere.
By Snell's Law the ray will be refracted to an angle thetaR such that sin(`thetaI) / sin(`thetaR) = 1.33 / 1.

Video Clip #2

The triangle formed by the dotted lines within the circle extends from the center of the circle to the point at which the parallel ray strikes, then back to the central ray and from their to the center of the circle.

This triangle shows that the sine of `thetaI is `dx / R.
It follows that the sine of the angle of refraction `thetaR is `dx / (1.33 R).
Since for angles measured in radians, the sine of the small angle is approximately equal to the angle, we see that when `dx is much smaller than R, `thetaI = `dx / R (in radians) and `thetaR = `dx / (1.33 R) (in radians).

Video Clip #3

The point where the refracted ray joins the center ray will be the focal point of the system, provided it lies within the circle or sphere.

By the geometry of the triangles, the refracted ray joins the center ray at an angle of `thetaI - `thetaR.
The refracted ray, the central ray and the side of length `dx perpendicular to the central ray form the right triangle indicated on the right-hand side of the figure below.
If we let f stand for the focal distance--i.e., the distance from the spherical surface to the focal point along the path of the central ray-- then we see that `dx / f = tan(`thetaI - `thetaR), so focal distance = f = `dx / tan(`thetaI - `thetaR) (provided the focal point is inside the circle).
As we saw before, `thetaI = `dx / R and `thetaR = `dx / (1.33 R), so `thetaI - `thetaR = `dx / R - `dx / (1.33 R).
For small angles, the tangent of the angle in radians is very nearly equal to the angle, so the denominator tan(`thetaI = `thetaR) is therefore very close to `dx / R - `dx / (1.33 R) .

Algebraic simplification of the resulting expression for f yields focal distance f = 4 R.

We note that this focal distance is not within the circular or spherical lens we postulated at the outset.
The parallel rays therefore reach the other side of the circle and are again refracted toward the central ray, as we will see shortly.
If the far side of the lens was lengthened to permit focus within the medium for which n = 1.33, the rays would in fact converge at this focal distance.
Note that `dx does not appear in the expression for focal distance, indicating that all parallel rays will converge at this point (to the extent that `dx is small--otherwise the approximations used in this analysis do not hold and the conclusion that the ray converges with the central ray at f = 4 R is not valid).

Video Clip #4

If we generalize this analysis to a general index of refraction n, rather than the specific 1.33, we obtain the expressions indicated in the figure below.

The algebraic simplification of the resulting expression for the focal distance f yields focal distance f = n / (n-1) * R.

We note that if n = 2, f = 2 R and the rays converge at the far side of the circle.
If n > 2, f < 2 R and the rays converge at a point inside the circle.
Materials for which n > 2 are very uncommon and usually quite valuable (diamond being an example).

Video Clip #5

For an actual circular or spherical lens (e.g., a soft drink bottle or a water-filled clear Christmas tree ornament), the refraction of the rays at the far side of the lens results in a focal point at a distance (2 - n) / (2n - 2) * R from that side.

This provides a convenient way to measure the index of refraction of a liquid--just put it in a cylindrical container, shine a laser pointer through the material at various points (as in the experiments) and determine the distance of the focal point from the far side.

Set the distance equal to (2 - n) / (2n - 2) * R and solve for n (better measure R first, of course).

The refraction at the far side of the lens is as indicated in the figure below. note that in this and the above figure the ray leaving the circle should be deflected toward the central ray; this is not clearly shown and, in fact, appears to be just the opposite

The details of the analysis are straightforward enough to be done using high school geometry, but they are complicated enough to provide a real challenge.

Video Clip #6

The spherical, circular and cylindrical lenses analyzed above are called thick lenses, because the rays are displaced a significant distance perpendicular to their original direction as they pass through the lens.

The effect of a thick lens is therefore to both change the direction of light rays striking them, and to move the rays over a significant distance as they pass through.

Most lenses used in optics are thin lenses, in which rays are displaced only an insignificant distance perpendicular to the original direction while passing through the lens.

The only significant effect of a thin lens is to change the direction of rays passing through.

In the figure below we depict a thin lens, perhaps made of a broken piece of a clear Christmas ornament glued to a flat piece of clear plastic and filled with water or another material.

As with thick lenses, a well-made thin lens is constructed so that all parallel rays focus at the same point.
As with thick lenses, spherical or circular lenses work well as long as incoming parallel rays are close to the central ray; for rays further from the central ray, the shape of the lens must be adjusted.

The distance at which parallel rays focuses called the focal distance f.

In the figure below we see how the light reflected from the top of an individual's head might converge at some point on the other side of a lens.

A light ray entering the lens parallel to the central ray must pass through the focus.
A light ray passing through the center of the lens will end up passing through the lens effectively undeflected.
If the distance of the individual from the lens is greater than the focal distance (as assumed in the figure below, though the sketch doesn't really look that way), then the rays from the top of the individual's head must converge at some point.

For a well-made lens, every other ray reflected from the top of the individual's head which passes through the lens will also converge at that point.

Light reflected off another part of the individual and passing through the lens will also converge in a similar manner.

This convergence of reflected rays will be at the same distance beyond the lens for every point on the individual located at the same distance in front of the lens.

As a result an image of the individual will form at this distance.

If some parts of the individual are significantly closer to the lens than others, then parts of the image will be blurry while other parts are sharply resolved.

Video Clip #7

Video Clip #8

The figure below shows the convergence of two rays from the individual's head, and of two rays from the individual's arm.

If the individual is illuminated brightly enough, if the lens is large enough that a significant amount of the light reflected from the individual passes through it, if there is a screen at the position where the image forms, and if it is dark enough around the screen, then the image of the individual can be seen projected onto the screen.
We therefore say that the image formed by this lens is a real image. It really is there--all we need to see it are the right circumstances.

Video Clip #9