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## Brightness of an Image Viewed Through Lenses

Wouldn't it be great if it were possible to look through a large telescope, and see, directly, those fabulous images of galaxies which one sees in astrophotographs?  I certainly think it would be wonderful.  I was disappointed when I found out it's not just impractical -- it's impossible to see such things by looking through a telescope directly.  We can only ever see them in photographs, or on computer displays fed from CCDs.  We can't see them through an eyepiece.  On this page we'll say a little more about that, and then we'll prove it.

The term "night glasses" -- a name sometimes given to 7x50 binoculars -- seems to suggest that when we look through them at night, they should reveal things that are too dim to see with the unaided eye.   In a word, things should look brighter through them.  And, in fact, anyone who's used them can attest that in dim light, you can indeed typically see more through them.  However, it turns out they don't actually make the scene any brighter.   Through binoculars, everything is bigger, and hence easier to see -- particularly in dim light -- but that's the only advantage.

More generally, it is impossible to build an optical system out of simple lenses and mirrors which makes things brighter.  The brightest image you can see through a pair of binoculars is exactly as bright as the image you see with unaided eyes.  And that rules out ever seeing distant galaxies directly, just looking through a telescope eyepiece.

Taking this as a "given" for a moment, let's think about why we can't buy 7x60 or 7x80 binoculars.  Why are 7x50 binoculars the largest 7 power glasses available?

Since magnifying an image spreads out its light over a larger area, if binoculars magnify a scene, they must also gather more light than your unaided eye if you are to see the scene as brightly as you do with bare eyes.  Human pupils are generally taken to be about 7 mm in diameter when dilated.  If you're using 7 power binoculars, then, in order to provide a scene of the same brightness you see without them, the front lenses must be about 7 times the diameter of the pupils of your eyes -- or about 49 mm.  Hence, 7x50 binoculars provide a "full brightness" scene.  Making them bigger wouldn't help.

Stars, on the other hand, are an odd exception to all we've been saying.  A star is a point source -- its image doesn't spread with higher magnification.  Consequently, if you look at the stars through binoculars (or a telescope), the stars will look brighter, and you'll see more of them.  (The galaxies and nebulae, however, will look bigger, but no brighter.)  We're still stuck with an inability to get more light into our eyes without increasing the magnification, however (as we'll see later), which means 7x binoculars larger than 50 mm wouldn't help.  However, "giant binoculars" -- 10x80 or larger -- are useful for star gazing.  (You may need a tripod to support them -- they're not light.)

Now let's get on with the proof, and then we'll talk about a way around it.

There is more than one way to approach this, and it appears that there should be at least a couple of clever ways of showing, elegantly and clearly, why this is true independent of the optical system used.

A general argument involving magnification and the shape of the cone of light from a particular spot on an image, independent of the agent providing the magnification, seemed promising but I haven't gotten anywhere with it as yet.

I've been told that it's possible to show that image brightness is conserved (so to speak!) using thermodynamics.  But my grasp of thermodynamics proved too weak to get anywhere with this approach, either.

So, since flying over the mountains in a balloon proved infeasible, I pulled on the hip boots and slogged through the swamp.

### Case by Case Proof that No Single Simple Lens can Change Brightness

By treating each case in turn, I'll first show that any scene, viewed through a sufficiently large simple lens, will have the same visual brightness as the same scene viewed directly.  There are actually twelve cases:  Three orientations each of a positive and negative lens, and the same three orientations each of a positive and negative mirror.  However, with regard to the issues we're interested in, these reduce to just four cases. First, of course, mirrors and lenses are "mirror images" of each other, and if we understand how image brightness works with lenses, then we understand it for mirrors as well; that reduces it to six cases for lenses.  But then two pairs of cases for positive and negative lenses turn out to be geometrically identical, and can be treated as single cases; that gets us down to four.

We'll start by treating each of those four cases in turn.  They have been broken out into separate pages.

For each of the four distinct configurations, we're going to show that we can't increase the image brightness of a simple box camera by using a simple lens in that configuration.  Once we've done that, we'll show by induction that no cascade of simple lenses can increase the image brightness, either.
Positive lens, real image -- We show that using a single positive lens to produce a real image of a scene doesn't change the perceived brightness.

Positive or negative lens, virtual image -- We show that using either a positive or negative lens to produce a virtual image, which is then photographed with a simple camera looking "through" the lens, doesn't change the image brightness in the camera.

Projected scene, forming a real image -- Suppose we project a real image through a lens (the "projected scene").  Suppose further that the lens, which may be positive or negative, forms a real image from the scene.  The image so formed may fall on either side of the original "projected scene".   We show that, if we view this new image with a simple camera, the image brightness in the camera will be the same as if we viewed the projected scene directly.

Projected scene, forming a virtual image -- Suppose we project a real image through a negative lens (the "projected scene").  Suppose further that the lens forms a virtual image from the scene; the virtual image will lie on the other side of the lens from the original "projected scene".  We show that, if we view this virtual image with a simple camera by "looking through" the lens with it, the image brightness in the camera will be the same as if we viewed the projected scene directly.

### Induction Proof for a General System

We've shown that no single, simple lens can increase visual image brightness.  But how do we know some exotic combination of lenses can't increase visual image brightness?

Suppose it could.  Now, take the smallest possible number of lenses which can combine to increase visual brightness of a scene.  Since no single lens can do this, we know the system must have at least two lenses in it.  Call the lens closest to the eye -- the one we look through -- the last lens.

Now, take the last lens away.  Call the remaining system the reduced system.  The last lens was forming an image from an image -- i.e., the rest of the reduced system produces an image which was, in turn, acted on by the last lens.  Look directly at that image, produced by the reduced system.  Observe the brightness of that image.

Since the original system used the smallest possible number of lenses to produce brightness enhancement, and the reduced system has fewer lenses, we know that the apparent brightness of the image produced by the reduced system must be no brighter than what you see with the unaided eye.

Now put the last lens back.

The last lens is a simple lens, and it must be acting on the image produced by the reduced system via one of the four configurations discussed above.  So, the last lens can't increase the apparent brightness of the image beyond what the reduced system produced on its own.

And so the entire system must also not be able to produce an apparent image brightness increase.

### Holes in the Proof

I'm aware of two major holes in the proof.
• I assumed without proof that the brightness distribution in the cone of light emitted by a lens is the same as the distribution in the cone entering the lens.  Without this assumption, the induction step doesn't work.
I believe this assertion is true; a geometric argument to prove it seems reasonably straightforward but some of the details may be tricky.
• I used small angle approximations everywhere.  This makes sense for a single lens used to view a distant scene.  It doesn't make much sense for the internal lenses in a complex optical system, however.
I haven't worked the proofs through with exact values, but I think the result would be to show that it's possible to reduce the apparent image intensity, but it is still not possible to increase it.  However, without working out all the details one cannot be sure.

### Improvements to the Proof

The cases of real and virtual images are very similar, save that distance from the lens to the image may be viewed as negative in one case, positive in the other.

Furthermore, all lenses have the same formula for magnification:  If l1 is the distance from the lens to the scene, and l2 is the distance from the lens to the image, then the magnification is l2/l1.
Given these two facts, it seems likely that, with a little fiddling, the proof could be reduced to just two cases:  A lens used to focus a scene which is in front of the lens, and a lens used to focus a scene which is behind the lens (i.e., a scene projected through the lens).

I haven't done that, however, and probably won't do so at this point, as I've already spent too much time on this.

### A Way Around It

Ah, so now we've proved visual image intensity can't be increased by simple lenses .... but as a matter of fact it can. You just need to think outside the box.

The given constraints are pupil radius and camera length.  We can't change those.  And, if we can only do things outside the camera, we can't affect the image brightness.  However, if we are allowed to insert an extra lens inside the camera, then we can, indeed, increase the brightness of the final image!

An eyeball has a pupil which is about 1/4" in diameter.  The eyeball itself is about 1" in diameter, or four times the diameter of the pupil.  That gives us a lot of extra room to play with.   Supposed we inserted a positive lens 3/4" in diameter in the middle of the eyeball, and then placed a negative contact lens on the front of the eye in order to bring a distant scene into focus at the retina.  This "back-end telecompressor" would result in a far smaller view than normal, but it would result in a brighter view than normal, as well.  If we combined it with a telescope of large light gathering power and large magnification, the result would be a brighter view of distant galaxies.

Page created on 09/23/2007