Thin lens approximation

- Now, it's time to add a lens to our pinhole camera. In the previous lesson, we explored how lenses bend or refract incoming parallel rays of light and focus them at a single point known as a focal point. We called the distance from the lens to the focal point the focal length of the lens. Now, if an object is really far from a lens, all the light rays leaving it are effectively parallel like this, and these rays focus at this point, which is distance f from the lens. We saw this relationship in lesson one, but now let's think about nearby objects. Where do they focus? That is, where would you have to put the image plane to make this image sharp? To get started, we need to introduce some new variables. As we say in lesson one, f is the focal length of the lens. Let o be the distance to the object we want to focus on, and let i represent the distance from the lens and when the object comes into focus. Let's call this the focus plane. So, we need to develop a formula that lets us compute i from f and o. Before we can do this, we need to look more closely at how lenses refract light. Refraction is actually a little complicated, and it's described by an equation called Snell's law, but we won't need it here because we'll make an important simplifying assumption called the thin lens approximation, which makes the math easier to solve. The thin lens approximation says that any ray that passes through the center of the lens doesn't get bent at all. It remains straight. These are called medial rays. And we know that parallel rays will refract as they pass through the lens and pass through the focal point of the lens. That point is at distance f from the lens on the x-axis. So, the focal length efines where a parallel ray crosses the x-axis. So, now we have a definition of two rays which leave our object, medial and parallel, and where the two rays intersect defines the distance i at which the object comes into focus. And the really cool part, the thin lens approximation says that all rays leaving this point will come into focus at the same place. And similarly, all points on the plane parallel to the lens and at a distance o will come into focus on the same plane on the other side, which is why we call it our focus plane. If we can find the intersection of those two rays, then we'll know where all rays from this object will come into focus. Pretty cool. Okay, let's pause here so you can build your intuition about this diagram using the next exercise.