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A New Light on Lenses

Jacob Mattie

Jul 3, 2023

What's really going on with your camera's optics?

Hello, my friends. I hope you are doing well. I am doing well too. We're all here because we like movies, or moving pictures, as we like to call them in the industry. Haha yes, I too know many things about film. Anyways, there's more to film than just knowing the lingo. A lot of terms get thrown around, and sure -- we know how to work with them, and we have a sense of what knobs and settings to fiddle with if we're looking for a certain effect, but it may be the case that we've not had the time to sit down and properly study what these things are, and what they're actually doing.

Well trouble yourselves no longer, my dear companions -- I've got you covered. I'll walk us through the world of lenses, building from the diffraction of light to what an aperture is doing when it affects your images. I will note that in these following paragraphs, I'll refer to the photography of still-frame images. If you've got your heart set on a film-specific article and are unwilling to read anything else, just print out a bunch of copies and reread them at approximately 24 times per second. Tomato, toe-mah-to, amirite?

Lenses are founded on a physical phenomenon called Snell's Law. In short, this law highlights a consequence of how the speed of light changes depending on the medium it is in. When moving from a low-density medium to a higher one, light slows down and changes its path to align more closely with the perpendicular to the surface barrier. The change in direction is proportional to difference in densities between the two mediums -- so light would bend more in a transition from air to glass in comparison to air to water (since glass is more dense than water).

It bears noting here that the speed of light constant (1) that is so often mentioned in scientific journalism refers to an upper speed limit -- the speed of light in a vacuum (e.g. outer space). A good chunk of research has been done into optics, and the refractive indices of a great many materials have been found. For a handy reference, a few mediums are listed here in order of increasing density/ refractive index: Vacuum < air < water < glass < diamond.

You can validate Snell's Law experimentally if you try to seize something out of a body of water. Chances are you won't be able to grab it on your first try (unless you're a heron, in which case -- hi cutie <3 call me) since the water bends the light to make things appear to be in a different position than they already are. Or, perhaps more relevant to our purposes, you can shape glass in a way that it magnifies the image of whatever you point it at. Incidentally, this shape looks like a lentil -- so much so that it has inspired the name of such pieces of glass. A lentil-shaped piece of glass? We call it a lens, after the Latin name for lentil; lens.

An ideal lens will refract any incoming light to a certain point (note that Snell's Law is applied here twice; once from the air to the lens, and again when the light exits the lens and re-enters air). This point at which all incoming light converges is called the lens' focal point. Since Snell's Law is dependent on the angle at which incoming light hits a surface, the focal point of a lens is also dependent on the distance of the light source.

1.) The speed of light in a vacuum, also known as c, is measured to be 299,792,458 meters per second (very fast).

Physicists have done some calculations (in its algebraic form. Snell's Law can be written as 2:n sin(01) = n2 * sin (02), where n1 and n2 are the refractive inidices of the mediums, and 01 and 02 are the angles between the light ray and a line perpendicular to the surface, and determined that the further an object is from the lens, the closer its focal point will be to the lens itself. You can sketch out some geometry and try the equations yourself if you'd like, or you can just take a look at these lovely pictures I drew for you and take my word for it.

In reality, there are a few issues with focal points. One of the biggest ones comes the fact that lenses are tricky little devils to manufacture. Our best technologies still fall short of perfect, and can be prohibitively expensive -- meaning that we the common are not likely to be using top-of-the-line lenses. The result of this being that the lenses we work with don't have an explicit focal point. The light rays don't all converge at the same place; some meet closer, and some meet further. To resolve this, we need to introduce the concept of the circle of confusion.

Take a look at this image of a realistic lens refracting light. At any point after the lens, you can sketch a line from the lowest light ray to the highest one. This is a 2-dimensional analogue to the circle of confusion. In our 3-dimensional world, just remember that the lens is a circle, and this line we're describing matches its shape -- also a circle.

The circle of confusion basically describes, for a certain subject, how much blur is in a lens' image at a given point. Too close to the lens? The circle of confusion is gonna be large. The same will happen if you go too far away. Somewhere in the middle, we can observe how the light rays, while not converging entirely, get pretty close.

While we can't find an explicit focal point, we can find a distance from the lens at which the circle of confusion is "small enough". And there we have it -- the industry gold standard: "good enough". Our eyes aren't perfect, and our cameras' photoreceptors have limited resolution. If we can limit our circle of confusion to something fairly small, it'll be lost amongst the other imperfections of the world, and our images will come out as if in perfect clarity.

We can pull a free term from here: Depth of field. If we give ourselves a certain bound on the allowable size of the circle of confusion, we can check out the light passing through the lens and notice that there's a range of distances from the lens that are considered valid. Slap a name on it. It's the depth of field. These are all the objects that are in focus (or close enough to be referred as such).

We've still got one more tool in our pockets that we haven't addressed, although now we have the knowledge and the tools to address it properly. It's our good friend, Aperture. Hello, Aperture.

Aperture is a restriction on the light that enters a lens. In practice, what this does is trim away the outer edges of an image -- that is, it limits the angles at which light rays enter a lens. Since the outer edges of an image -- that is, it limits the angles at which light rays enter a lens. Since the angles of light leaving a lens are dependent on the angles at which they came in, a smaller aperture means that light is closer to parallel within the camera. Given that we've defined "in focus" to be all light contained within a certain size of circle of confusion, by narrowing the aperture we can flatten our light rays and extend our field of focus.

Ain't that something neat?

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