September 2024
Point source scaling is a way to proportionally decrease the divergence of light by virtually reducing the size of its source. It allows one to study light from a light source as if the light source were smaller or further away. It also allows one to scale models of the solar system in order to study or observe solar illumination as it might appear at distances greater than one astronomical unit.
Concave mirrors cause parallel light rays to converge and convex mirrors cause parallel light rays to diverge. When parallel light rays from the sun strike the surface of a concave mirror, the light rays are focused to form a real image of the sun. When parallel light rays from the sun strike the surface of a convex mirror, the light rays diverge and form a virtual image of the sun. Using the rays from a virtual image of the sun, one can approximate the appearance of the sun’s rays on other planets in terms of intensity, divergence, shadows, diffraction and interference.
Practically speaking, concave mirrors can magnify objects by forming virtual images. A makeup mirror is one example of this. Some solar devices use concave mirrors for their ability to form a real image of the sun. Convex mirrors, on the other hand, can provide a panoramic reflection of surrounding objects. The panoramic reflections, which are imaginary rather than real images, are often used as vehicle mirrors, security devices, and in any situation where a broadened view of one’s surroundings is required.
So now for the interesting stuff - the sun’s rays are, practically speaking, quite parallel. On earth, the sun’s rays spread about ½ degree angle, or 9.3 milliradians. In fact, the sun is probably the brightest source of well collimated light that we encounter on an everyday basis. The divergence of sunlight is probably lower than that of most common sources of collimated light such as flashlights and spotlights, while at the same time being higher than that of some artificial sources - most notably lasers.
It’s the good collimation of the sun’s rays that not only make shadows possible, but also make it possible for sunlight to be focused by mirrors and lenses to an intensity that is adequate to heat or burn things. There’s a sort of paradox that is common to light sources whereby tight focusing is required for high intensity, good collimation is required for tight focusing and a point source is required for good collimation. There seems to be no way around this limitation considering that most of the light from a source must be discarded in order to produce or obtain a well collimated portion of it. The gains of focusing are offset by the losses of collimation, but in the case of the sun those losses are compensated for by the overwhelming optical power. Nevertheless the more light retained from a source, the less collimation that is possible. The game changer came with the advent of the laser - for the first time in history it was possible to have light with maximum intensity and collimation.
Here’s where the confusing part comes in - a convex mirror causes light rays to diverge, or spread out. So then how is it possible to make sunlight less divergent (more spatially coherent) by spreading it out with a convex mirror? While light from the sun, at 9.3 mR divergence on earth, becomes more divergent after being reflected by a convex mirror, the reflected light also appears to originate from a much smaller source. The key to understanding this is in knowing why sunlight has a divergence of approximately 9.3mR - unlike parallel rays of light from a laser or a device that uses optics to collimate light, the collimated light from the sun is a result of the diameter of the sun and the distance between the sun and the earth. The diameter of the sun and the distance between the sun and earth are directly proportional so that if the sun were larger or closer to the earth, the divergence of its rays would be greater. If the sun were smaller or further from the earth, the divergence of its rays on earth would be lower. Regardless of whether an observer is standing at the bottom or the top of Mount Everest, the divergence of the sun’s rays are going to be practically the same because the distance between the sun and earth is so great by comparison to the mere height of Mount Everest. If however we are able to make the sun smaller, then our distance from the sun will also need to be smaller in order to obtain the same divergence of 9.3mR. And shrinking the sun is exactly what the convex mirror does, because sunlight that is reflected from a convex mirror appears to originate from a virtual image of the sun that is smaller than the sun.
We cannot change the size of the sun or its distance from the earth, but we can use a convex mirror to produce a scaled model of the sun and its output. Let’s look at an example to clarify how this process works.
| Magnification Equation |
|---|
| Height of Image/Height of Object |
| = |
| Distance of Image/Distance of Object |
| Hi/Ho = Di/Do |
| Hi = 4"/5.89e 10" x 5.5e 10" |
| Hi = 3.74 inches |
| Angula Diameter |
|---|
| Angular Diameter (in radians) = diameter/Distance |
| Solving for Distance |
| 3.74"/0.0093 radians = Distance |
| 402 inches or 33 feet 6 inches |
Our convex mirror with a focal length of 4 inches has reflected light from the sun, and in so doing, has produced a virtual image of the sun that is 3.74 inches in diameter. Given the ratio of diameter to distance that results in an approximate angular divergence in radians, we can solve for distance using the diameter of the image to determine the distance at which sunlight reflected by the mirror will have a divergence of 9.3mR. Given that the virtual image is located 4 inches behind the mirror, we can basically say that at 33 feet in front of the mirror, the reflected sunlight has again attained its original divergence of 9.3mR.
Using another convex mirror with a focal length of 27 ½ inches, we find that our virtual image of the sun is considerably smaller than it was with the previous mirror:
| Hi = 27.5 in/5.89e12 in x 5.5e10 in |
| 0.25685 in = 6 ½ mm |
With a focal length of 27.5 inches, the virtual image of the sun is only 6.5 mm in diameter.
Given a helium neon laser with a divergence of 1.6mR, we can solve for the distance from the mirror at which the reflected sunlight will have the same divergence as the beam from the helium neon laser:
| Given: diameter = 6.5 mm, |
| θ = 1.6 mR | diameter/Distance = θ |
| Distance = diameter/θ |
| 6.5 mm / 0.0016 Rad |
| D = 4062 mm = 160 in |
| 160 in - 27.5 in (distance from virtual image of sun to surface of mirror) |
| 123.5 in = 11 feet |
Aside from obtaining a better understanding of the behavior of light from common sources at significant distances from those sources, source point scaling also enables one to model sunlight as it might appear on distant planets. For example, Neptune is approximately 4.5 billion kilometers from the sun, or 30 times further than Earth is from the sun. By dividing the diameter of the sun by this distance, we obtain an impressively low divergence of only 0.31 mR (light that is more parallel than the light from most lasers). If we want to see what sunlight would look like on Neptune, we can simply divide 6.5 mm from our previous example, by 0.00031 Radians, to determine that sunlight reflected from the convex mirror (with the 27.5” focal length) would appear similar to sunlight on Neptune at only 67 feet from the mirror.
In this experiment, the 27.5 focal length convex mirror and a plane mirror were used to compare the effects of reflected sunlight over a distance. At a distance of approximately 28 feet from each mirror respectively, an opaque surface with a small opening was located so that it intercepted the rays that were reflected from the mirrors. The hole in this surface was made using a hole punch, and it had a diameter roughly equal to the sides of the individual squares that formed the grids on a piece of graph paper. A graph paper target was located about 16.5 feet from the hole, opposite each mirror. When reflected sunlight from the plane mirror passed through the small opening, it produced a rectangle shaped pattern on the graph paper that was approximately 5 by 7 squares on the grid. It was effectively an image of the plane mirror. Occasional shadows appeared within this pattern of rectangular light that was presumed to be silhouette-like images of tree limbs that blocked part of the path between the setting sun and the mirror.
When sunlight from the convex mirror passed through the hole and fell on the graph paper, it produced a round spot of light that was only slightly larger than a single grid in diameter - an impressively low divergence of 0.7 milliradians at 28 feet from the mirror. This explains why the sunlight, when passed through the opening of approximaly 0.25 inches in diameter, spread very little beyond its original diameter at 16.5 feet past the hole. The puzzling part is when looking at sunlight reflected from both mirrors before it had passed through the hole: the sunlight from the plane mirror was limited only by the quality of the mirror, and was otherwise presumed to retain its original divergence of 9.3 milliradians. From the frame of reference of an observer who is standing behind one of the mirrors, the reflected beam of sunlight from the plane mirror is more parallel than the reflected beam of sunlight from the convex mirror. From the frame of reference of an observer who is on the side of the hole opposite the mirror, light from the convex mirror that has passed through the hole is more parallel than light from the plane mirror that has passed through the hole.
From the limited perspective of an observer on earth, sunlight reflected from a convex mirror is more divergent than sunlight reflected from a plane mirror. From a relative perspective however, it is the more divergent light beam that wins the race for low divergence despite how counterintuitive this might seem. One way to understand this is to remember that one must be further from a large light source to obtain the same divergence that one would obtain at a shorter distance from a small light source. On the scale that would exist using artificial light sources on earth, very short distances from a light source would result in dramatic changes in both the intensity and divergence of the light. On the scale that exist with the sun and the earth, one would be required to leave the earth in a spacecraft and travel to a point in space that is significantly further from from the sun before substantial changes in the intensity and divergence of the sunlight would be noticed.
In terms of non-isotropic sources of light such as flashlights, spotlights, car headlamps or searchlights, the collimating apparatuses collect and direct the light so that intensity and useful illumination is preserved over distance. Ironically, light from such light sources would exceed the divergence of light from an isotropic point source of light, such as a tiny filament or even a candle, over a surprisingly short distance. Isotropic sources obey the inverse square law and the intensity would thus be infinitesimally small - a large price for spatial coherence. Over enough distance however, light from both collimated and isotropic light sources should obey the inverse square law.
Last updated on Monday December 30, 2024