Parabolic Reflectors and the Ideal Flashlight
by J. Schattman
Sir John A. Macdonald Secondary School
Waterloo, Ontario, Canada
Introduction
A flashlight projects light in one general direction because of the reflecting surface behind the lightbulb. This redirection of light is called collimation, a fine SAT word if I've ever heard one. A good flashlight collimates more of the bulb's light than a poor one. How well a flashlight collimates depends on the shape of the reflecting surface and on the distance of the bulb from the surface.
Which shape is the collimation champion? This application compares the light-reflecting properties of four different surfaces:
Shape of the reflecting surface
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Equation of the cross-section
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Quality of collimation
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Parabolic
|
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Perfect
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Spherical
|
|
Not half bad
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Quartic
|
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Half bad
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Conical
|
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All bad
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Experiment with the four flashlight simulators below, and you'll see why the parabola is the perennial favorite of flashlight engineers. (See my final note about modern advances in flashlight research.)
Initializations
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(2.1) |
The perfect parabola: 
Parabolas were born to reflect. In the simulation below, the red ball is the light bulb, and the blue region is a cross-section of the reflecting surface. The yellow lines are light beams. (Only the reflected beams are shown; beams that miss the surface are not shown.)
To move the bulb, click on the graphic and drag the slider in the animation toolbar. Try setting the bulb distance to .24694, which is the approximate focal distance of the parabola 
. At this distance, the reflected beams are in perfect focus.
It turns out that the paraboloid is the only surface that collimates perfectly, whether the stuff being collimated happens to be light, heat, gamma radiation or television signals. That's why satellite dishes and telescope mirrors are also parabolic!
Cheaper to make and almost as good: 
The hemisphere is the first surface most students suggest when I ask them to guess the ideal reflecting surface. For beams reflected close to the origin, it's not half bad, but that's only because a circle approximates a parabola near the origin. No matter where you place the bulb, the collimation aberrates (worsens) as you move towards the edge of the surface. Try it!
Why not 
Don't get me wrong, the curve 
is a sublime shape that has rightfully earned its place in the annals of polynomial lore. But it makes a terrible flashlight. Try it.
What about other even powers 
? Sorry to say, but after 
, it's all down hill. The larger n gets, the more box-like the curve becomes, and thus, the worse the collimation.
And whatever you do, don't use a cone: 
Another common suggestion from students is to use a cone. Well, they're wrong. Here's why.
Want to learn more?
Of course, no reflecting surface can control the forward-directed beams. To see how modern top-of-the-line flashlights cope with this, see the eloquently titled article Automated optimization advances software for illumination design. (I would have just named it "Advances in flashlight design", but then again, I never was cut out for marketing.)
For an even more stunning demonstration of wave reflection from parabolas and a myriad of other shapes, try Paul Falstad's marvelous ripple-tank simulation applet at http://www.falstad.com/ripple/ex-parabola.html.
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