One of the most memorable experiences in a physics lab was from a graduate level module of Geometrical Optics. The tutor, Prof. Papathanasoglou -perhaps one of the most sweet and gentle men I have ever met- was an Emeritus professor in his late 70s (some said early 80s!), who ran the Optics module with devotion and love. During one of our lab sessions, he had set up a telescope on an optical bank, directed at a small projector slide on the other side of the room, holding a miniature photo of a Swiss chalet in the Alps (you might not be familiar with what a projector slide is if you were born after the year 2000; think of it as a miniature photo about 1.5cm x 2cm). The slide was lit by a miniature lamp behind it so that it could be seen when the lights in the room were turned off and was meant to simulate a small object for observation, having roughly the apparent size of the Moon.
The tutor invited us to look through the telescope and tell the class what we saw. I happened to go first. Placing my eye behind the eyepiece lens, I adjusted the scope until the image was brought into focus. I declared that I could see a beautiful chalet. Then the tutor asked that I should continue describing any changes I see on the image, without looking at what they were doing: “do not look away, keep your eye on the image and tell us what you see”.
The tutor did something and asked, “what about now?”. The chalet was still there, pretty much the same as before, which was my reply. Something shuffled again and they asked, “what about now?”. Giggles could be heard coming from my classmates. “It’s the same image“, I replied, “only a little darker”. Whispers intensified among my classmates. Another mysterious shuffle followed by gasps of surprise and the same question: “what about now?”. “Oh, now the image is the same, only considerably less bright”. Then the tutor said, “what if I told you I am now covering over two-thirds of the objective lens with a book?”. I remember thinking to myself “no way!!”. At that point I just had to see for myself: I looked away from the scope and lo and behold, most of the telescope lens had been covered by a book, held by the tutor, who seemed to be most amused by the look of genuine surprise on my face and the effect that the demonstration had on his pupils. We all had that look on our faces when one is deceived by a clever “magic” trick but were at a loss as to how the trick was performed!
I was shocked! How on Earth did that happen? The only effect that covering the objective lens with a book had on my view through the telescope, was a progressive reduction in the brightness of the image, and nothing else! No sign of a huge black area covering the corresponding part of the field of view as the objective lens was being progressively covered up by a book, which is what most of us would expect.
The explanation was elegantly given in a single sentence, which, as is so often the case in Physics, had a meaning whose extent and consequence was not immediately understood the first time one heard it: “every point on the lens receives the entire image”.
Having had the good fortune of learning about this during my training, I could not resist but do the same demo while teaching the Astronomy module at GCSE and A-level. But how to explain it to the students, who shared the same surprise and disbelief as I did after the demonstration?
The answer turned out to be simple, and familiar: the pinhole camera.
The pinhole camera is a simple and inexpensive experiment that is meant to demonstrate how photographic cameras work and is an activity frequently done in Y7 or Y8 in Science lessons in the UK. The students make a rectangular box using black card. The box is just like a regular rectangle only that its two smaller opposing sides are modified thus: one side has a square cut and is fitted with a semi-transparent piece of baking paper or tracing paper used in architectural drawings. The opposite side is pricked with a pin until to make a tiny hole in the centre, of an area smaller or equal to 1mm2.
Then, there needs to be some sort of bright source that the students can observe, such as a candle. The teacher puts the blinds down to darken the room and the students direct their boxes so that the hole is pointing toward the candle. Once they have the correct alignment, an inverted image of the candle appears, projected on the semi-transparent paper.
The pinhole camera is a case in point (pun intended) for the phrase “every point on the lens receives the entire image”. Indeed, the pinhole camera practically only has but a single point to receive light from the target, and yet the entire image is formed on the back, not just a bright spot.
So, what about those ray diagrams that you might have seen in science textbooks?
Traditional ray diagrams are two-dimentional; their purpose is to demonstrate how an image is formed from one or more lenses. One or two points are chosen from the object and two or three rays are drawn following the rules of geometrical optics, in order to show the location of the image and its properties. One can thus see whether the image is inverted or upright, magnified or diminished, real or virtual.
To shed light on our conundrum, we will need to look at a ray diagram in 3D. In the diagrams below, I have drawn the projector slide, the telescope and the image formed at the focal plane. Two points have been chosen on the image, A and B, and their respective bundle of rays. Light rays are emanating from each point on the image in all possible directions.
To illustrate our point, only that bundle of rays that reaches the lens has been drawn. Each bundle of rays reaching the lens is contained in a cone. The second diagram shows two such cones, for points A and B, as well as the position of those points on the image, after the converging lens has focussed the light rays on the focal plane.
The first step in untangling the mystery of the covered telescope, is to realise the following upshot from these diagrams: since each and every point on the image emits rays that are incident on the lens, each point on the lens correspondingly receives rays originating from every single point of the image.
This is a logical corollary that emerges from the statement above. It is best seen in a diagram that is almost never seen in schools (or even optics textbooks!), which we reproduce below.
The diagrams use two points on the lens, A and B, as an example to illustrate that the entire image is received by every single point on the lens. We can now begin to see how the unexpected result can be explained. If we imagine the lens being made up of a number of points, each point produces its own image on the focal plane, using the photons/rays that happen to pass through it.
Extending this principle to every point on the objective lens, one realizes that the final image is a cumulative effect; it is, in essence, composed of a stack of images, each originating from a unique point on the objective lens. Therefore, by covering up a portion of the lens, we are simply reducing the number of points, and therefore, the number of images that are superposed to create the final image. It is evident that, the fewer the number of images superposing to produce the image, the fewer the number of photons incident on the focal plane, and therefore the less bright the image will be.
Another way to look at the situation is this: our initial expectation was that by covering up the telescope, part of the image would be obscured and we would get a completely dark portion covering our field of view. However, darkness is the absence of light. As the diagrams clearly demonstrate, the effect of covering up part of the objective lens is manifestly not the absence of light. Light rays coming from the entire image will invariably fall on the lens, no matter how much of it we happen to cover (barring covering all of it). Even if we left a tiny hole, an image would still be formed, as it indeed does in the case of the pinhole camera.
As such, concepts such as the light collecting power of a telescope being directly proportional to the surface area become immediately manifest, as opposed to simply sounding plausible. Indeed, if one doubles the diameter of the objective element of a telescope (lens or mirror), one quadruples the light collecting capability of their telescope, which means that they can image very faint objects more easily.
And thus, the phenomenon of the darkening of the image when one covers the objective lens of a telescope is explained!
Our diagrams are, of course, simplified, since we have omitted what happens once the rays carry on after they converge on the focal plane and are incident on the eyepiece lens. They would then go through that lens and enter our eye, or end up stimulating any camera sensor that we happen to have attached to our instrument.
In a photography camera the rays go through several lenses, since there are several elements (lenses) one after the other, suitably placed to correct for various image distortions (called aberrations) that lenses normally produce. The number of elements and the elaborate mechanisms that move them by just the right amount when one zooms in or out justifies the high price tag that they carry, often in the several hundreds.
All in all, this makes for a memorable experiment that is worthwhile to demonstrate when teaching about Optics or telescopes in Astronomy, which I wholeheartedly recommend to anyone teaching those subjects, at any level.