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## Mathematics of depth of field

Current time:0:00Total duration:2:33

# Thin lens approximation

## Video transcript

- 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.