For a long time, it wasn’t clear to me why a stick inside water appears bent- a phenomenon we all witness in our day to day lives and about which we have read in high school Physics textbooks.
A high school Physics textbook uses a schematic as below (Figure 1) and offers the following explanation: Light (ambient) reflected by the stick get bent when it traverses from water to air due to refraction. Our eyes can’t follow the bent path of rays, backtrace those rays as shown in the schematic (dotted lines) and hence we see the stick at a different position from where it really is.
Figure 1- Schematic used in high school physics textbooks to explain why a stick inside water appears bent. Light from point A on the stick bends at the surface of water, our eye can’t follow the bent path and so we see image of A at A’. Using the picture of an eye and back-tracing the light rays to a point basically involve one layer of abstraction, which we don’t use in the subsequent ray diagrams.
In high school, I took this explanation for granted, reproduced it on answer scripts of examinations and even solved numerical problems related to it. But I never really understood this phenomenon until I got into graduate school, where my lack of understanding of this phenomenon eventually made me conclude that I do not understand how science works in general. Then after a phase of “soul searching” and of course reading up on several things, a much more satisfactory explanation of the phenomenon dawned on me, which I shall describe here in details.
The fact that this is a blog post gives me the liberty to not only write about a field of science in which I am not an expert but also state something which may already have been published before or has been proven wrong. I simply may not be aware of it despite talking to several friends, pursuing research in the sciences, and spending a lot of time on the internet browsing on the topic. I don’t have this luxury while writing research articles in peer reviewed journals for my professional career.
Another reason to write this essay is that my explanation starts from physics, that governs all phenomena in the physical world, but eventually delves into the mental world and becomes a neuroscience problem, true to the theme of my blog. In my opinion, the neuroscience aspect is key to understanding the phenomenon, but has largely been ignored in high school textbooks, which led to a gap in my mind between what I read in science textbooks and what I witnessed in the real world.
So let’s first get back to the explanation provided in high school physics text books. Light from the stick travels in a straight line inside water, but when it crosses the water surface it bends since air has a refractive index different from water. Then light again travels in straight line in air to reach our eye. The fact that light travels in straight line in a medium and that it bends at the intersection of two media are consistent with the laws of physics. But these facts alone don’t solve the puzzle. The last part of the explanation is that our eyes cannot follow the bent path and backtrace the incoming rays in a straight line path to form an image of the stick at some other position. But there are no details on why this is so in high school textbooks. Similar issue arises with explanation of how magnifying glass works, why we see our reflection on the mirror or occurrence of mirages- basically any case where a “virtual image” is formed.
First let’s take the case of a magnifying glass and analyze it in more details. In order to solve the last part of the puzzle, we have considered the eye as a combination of a convex lens in front and a screen (retina) behind it in the ray diagram below (Figure 2). High school textbooks instead show the picture of an eye and backtrace the rays, which is basically a layer of abstraction which was the root cause of my confusion.
Figure 2- To our retina, there is no difference between an object at position x with lens at position z, and a larger object at position y with no lens. But our brain always thinks that it is the second case and that is what we “see”.
Actually, if an object AB is placed within the focal length of the convex lens (magnifying glass) rays from object AB go through the lens, diverge and then hit the lens of our eye only to converge again at the retina. There is absolutely no difference in the spatial intensity pattern formed on the retina between the case in figure 2 (object AB at position x and lens at position z leading to formation of virtual image ab at position y) and a simpler case of a larger object ab at position y with no lens at position z. However, our brain only considers the second case and hence we “see” a magnified object at position y. No matter how much we train our brain through physics textbooks, we can never instead “see” a much smaller object at position x even though we know that is the case physically. Thus there is a subtle difference between the intensity pattern/ image formed at the retina of the eye and what we “see”. This subtle difference is probably created by some extremely complicated signal processing in the brain. Instead of looking at the magnifying glass with our eye if we took a snapshot with our camera then also we will end up “see”-ing the same thing. This is because the lens of the camera acts like the lens of our eye leading to the same intensity pattern on the film/CMOS sensor as the retina. Then we interpret that intensity pattern with our brains the same way we do in the case of looking at the magnifying glass with our eyes.
Next let’s discuss why we see the reflection of an object on the mirror the way we see it. In Figure 3 below, we consider two cases: Case I (an object AB at position x and a mirror at position z) and Case II (an object AB at position x, another identical object CD at position y and no mirror)
Figure 3- To our retina, there is no difference between case I and case II, but our brain thinks that it can be only be case II. It is to be noted that A’B’ and C’D’/ a’b’ are formed on the same region of the screen. They have just been drawn slightly away from each other for the sake of clarity here.
Again, in either case, the intensity distribution on the retina is the same- a focussed image of object AB and a slightly defocussed image of object CD, or ab (light rays from object AB get reflected off the mirror and converge near the retina). However just like in the example of magnifying glass, our brain only considers case II and hence we “see” an object at position x and another identical object at position y. No matter how much we try we cannot “see” an object at x and a mirror at z which is reflecting off the light from the object at x.
At this point, I guess it is obvious what happens in the case of a stick immersed in water. Rays of light (ambient) reflected by the stick cross the surface, bend, hit our eyes and converge to form an image on our retina which is identical to an image of a bent stick in the air. Just like the previous cases, we end up “see”-ing a bent stick in air (yes we still see the water in all practical cases but that is for other reasons like presence of the vessel, water droplets, water reflecting off ambient light etc.) as opposed to a straight stick in water with light bending off as it comes towards our eyes.
The subtle point I am trying to make here through all the examples above is that light can travel through a bent path on its way from the object to our eyes if it passes from one medium to another with different refractive index. The image formed on our retina will be identical to an object being displaced from its actual position and light traveling from it to our eyes through vacuum/ air following a straight line path. However our brain can only conceive of light traveling straight through vacuum/ air and hence we “see” the object at a position different from where it actually is. This particular behavior of the brain may arise out of evolution because we and our ancestors have grown up in a planet with air of a nearly constant refractive index and our visual perception is hence calibrated to that. Essentially, the laptop/ computer on which the reader is reading this article, the table on which it is placed, the window in your room, etc. are present where they “see” it to be present simply because light is traveling through a medium of fixed refractive index on its way from the object to their eyes. If the refractive index of the medium changed along the trajectory of light, they will see the objects at different spots from where they actually are. If we could do an experiment where we could have brought aliens from a planet where the refractive index of the medium varies much more as a function of height from the surface of the planet than it does in the case of our earth and ask them where they locate different objects on earth, then my hypothesis could have been tested. My guess would be that they would locate all objects on earth wrongly because their brains are calibrated to how light travels in their planet, which is not usually in a straight line unlike our planet.
At this point, the really imaginative readers may be wondering if what we see around us indeed exist or not. Probably they have asked this question to themselves before. My humble opinion in this regard is that there is no absolute reality, or at least we can’t perceive it. We can only be more convinced of the existence of something we see through other senses like smell, touch, etc. but can never be convinced of the absolute existence of something. A subjective aspect of consciousness always accompanies our perception of reality, which is essentially a calibration of the current signal we are receiving from the external physical world to some previously received signal, which we may have received in our own lifetime or inherited from our predecessors through evolution, as in the case of all the optical phenomena discussed in this essay.