Why does a larger aperture cause more blur of the background?

Question

Why is it that a larger lens aperture produces a shorter depth of field, causing the background of a photo to be more blurred?

Answer

Why the wide aperture blurs the background more

Let me start with Wikipedia figure:

Above we have a wide open aperture. Only point 2 is in focus. Points 1 and 3 are out of focus. Due to wide aperture, the rays coming from them through different parts of the lens intersect the screen 5 (a film or a digital sensor) in different points. We may also tell that these rays form a point (intersect) before (red) or beyond (green) the screen. The corresponding cones of light intersect with the screen and form an ellipse-like image on the screen. Wider aperture allows for wider cone of light (so it allows to collect more light and blurs more).

Effectively, an out-of-focus point produces a circle of confusion. This is what we can call blur or bokeh.

For smaller aperture below, the rays too far from the center are cut off, so the circle of out-of-focus point is smaller.

If the circle of confusion is smaller than film grain or sensor subpixel, we cannot tell if it is out of focus at all, and then the point appears as in focus even if it is not. So with finite aperture, there is a range of distances which all appear as in focus. The depth of this range is called the depth of field (DoF). It is bigger for smaller apertures.

If the aperture is really, really small, then only the central rays can pass, and we have an infinite depth of field no matter what. Every point, close or far away, is represented as a point on the image. This is how pinhole camera works. Adjustable aperture allows to have anything in between.

How it looks like

At smaller aperture f/32:

At larger aperture f/5, an out-of-focus background is blured more:

(images are again from Wikipedia)

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