Jack said, "The correct question is, how small a spot can you make? The
applicable physics is related to diffraction."
Stefan said, "Just a comment... If I recall my fourier optics class
correctly, no real lens system can direct all "rays" to the focal point,
even in the case of perfect planes waves emanating from a source an
infinite distance away (for which the sun mostly qualifies)."
I agree with Jack's first sentence. Getting all the sunlight entering
the lens to hit the fuel within the smallest spot is the goal. So the
question is how small of a spot can you get from the sun, and/or how do
you achieve the smallest spot from the sun?
I disagree with the Jack's second statement about the diffraction limit.
Stefan's statements also appear to invoke the diffraction limit. In
addition, Stefan indicates the sun mostly qualifies as a source from an
infinite distance. These ideas are not correct for answering the
question of how small of a spot can you get from a lens that is focusing
the sun.
It is indeed true that diffraction places a limit on the resolution of
the lens. By this we mean that light leaving a point on the object does
not all arrive at the same point in the image. Indeed we do discuss the
diffraction limit for optics, especially telescope optics where the
diffraction limit keeps us from imaging a star as a point, and it keeps
us from imaging the individual stars in clusters or galaxies that are
too far away.
As stars go, our sun is an exception. Its distance does not qualify as
infinite because its diameter is too big compared to its distance.
Rayleigh's criterion for resolving images is that the angular separation
must be greater than inversesine(1.22lambda over lensdiameter). For
example, with a 90-mm diameter lens, the Raleigh diffraction limit for
550 nm light is about 4.27E-4 degrees or 2.56E-2 arcminutes or 1.54
arcseconds.
The angular diameter of the sun's disk is a whopping 32.5 arcminutes =
1950 arcseconds which is more than a thousand times larger than the
diffraction limit of a 90-mm telescope lens. It is the sun's angular
diameter coupled with the focal length of the lens that limits the
smallness of the spot, and diffraction has essentially nothing to do
with it for any lens large enough to hold in your fingers.
Remember when doing thin-lens ray tracing that rays headed for the
center of the lens continue straight on through. Rays headed from the
edges of the sun's disk toward the center of a lens will continue
through, and diverge from the lens center with the same angle as they
hit the lens. The angular spread of these rays after the lens is
therefore 32.5 arcminutes, the same as the angular diameter of the sun.
If we extend these beyond the lens to the focal plane, that will give us
the diameter of the spot that is in focus at the focal plane.
For the 90-mm diameter 1000-mm focal length lens I tested, this
calculation yields a spot diameter of 9.5 mm. In my previous post you
will see that I estimated the solar spot size as about 10 mm when using
this lens. I comfortably held my hand at that position.
On the other hand, if we used a 90-mm lens with focal length of 100 mm,
typical of a magnifying glass, the spot would be just under 1-mm
diameter. That's a decrease in spot area of 100-fold. I will not hold
my hand at that position.
Michael D. Edmiston, Ph.D.
Professor of Physics and Chemistry
Bluffton University
Bluffton, OH 45817
(419)-358-3270
edmiston@bluffton.edu