Path:  physics insights > basics > simple optics > image brightness > Brightness of a Virtual Image

On this page we will consider a virtual image produced by either a positive or negative lens, with the scene and the lens image both on the same side of the lens, and the camera on the other side "looking through" the lens (see figure 1).  We'll show that the brightness of the camera image is unchanged by the introduction of the lens ("Lens 1" in the figure).

Please see the proof that image brightness is unaffected by a positive lens producing a real image, here, for a discussion of the terminology we're using and a general idea of how we're going to proceed in this case.  We'll start with a geometric argument, then give a symbolic proof.

Figure 1: Camera focusing virtual image, formed by lens

Start by setting l₁ equal to l₂.  When we do that, Lens 1 is reduced to a simple piece of glass, and the brightness of the camera image is clearly unchanged by its presence.

Now let l₁ vary.  As it varies, the overall magnification of the image will vary inversely with l₁; the virtual image is magnified by l₂/l₁.  Consequently, the magnification of the area of the scene goes as (1/l₁)².  But the amount of light gathered from the also depends on the ratio of p₁ to l₁; it, also, scales as  (1/l₁)².  Since the brightness is the light received divided by the area of the camera image, the factors of 1/l₁ cancel, and the image brightness does not vary.

A Symbolic Proof

In case that went by too fast, or it didn't seem rigorous enough, here's a symbolic proof that the image brightness in this case is the same as it is when the camera views the scene directly.  (This proof follows the proof that a positive lens forming a real image doesn't affect camera image brightness, given here.  In fact, this is exactly the same proof, line for line; it refers to a different illustration but the labels are the same.)

Let's consider a tiny piece of the scene, right at the arrow's base, which has area δv.  The total light emitted from δv will be δE.  We'll assume that the light is radiated evenly over a full hemisphere.  This is the same assumption we made on the camera image brightness page.  With that assumption, and with Lens 1 in place, we're going to compute the brightness of the camera image and see if it's the same as the image brightness for the camera alone.

Light gathered by the camera will be the solid angle subtended by the yellow triangle shown in figure 1, divided by the solid angle of a full hemisphere, times the light emitted by the image.  We'll call that amount δE_e.  We're going to assume that the scene is distant relative to the lens diameter, so we can use small-angle approximations.  Then we have:

(1)    

The area of the camera image of δv will be the square of the magnification due to Lens 1 times the magnification due to the camera's lens, or

(2)    

From our page on ideal lenses, case (2) and case (4) (or just by inspecting figure 1), we see that we can write the magnifications as

(3)    

Looking at figure 1, we can see, by similar triangles, that we can write p₁ as

(4)    

Now we divide (1) by (2),  and plug in the values from (3) and (4), to obtain,

(5)    

Almost everything cancels; after taking the limit in small δv, we're left with:

(6)    

which is the same formula we found in the simple camera brightness page, equation (SC-6), which was to be shown.




Page created on 09/25/2007