Thin lens & depth of field: circle of confusion, interactive demo

The thin lens

A pinhole has to be tiny to give a sharp image, so it collects very little light. A lens can have a large aperture of diameter $A$ and still be sharp, because it bends all rays from a scene point that pass through it onto one image point. For an ideal thin lens with focal length $f$ , a point at distance $z_o$ in front of the lens is imaged at distance $z_i$ behind it, with

\frac{1}{f} = \frac{1}{z_o} + \frac{1}{z_i}

A point at infinity is imaged at $z_i = f$ , closer points a bit farther behind the lens. The sensor is at a fixed distance $z_S$ behind the lens, so only points at one distance, the focus distance $S$ , are imaged exactly on it. Focusing a camera means moving the lens to change $z_S$ . The aperture is usually given as the f-number $N = f / A$ , written as f/1.8, f/8 and so on.

Circle of confusion

A point at another distance $D$ comes into focus at $z_D \ne z_S$ : in front of the sensor if it is farther away than $S$ , behind the sensor if it is closer. Its light forms a cone with the aperture as the base and the tip at $z_D$ , and the sensor cuts this cone in a disk. By similar triangles, the disk's diameter is $A \, |z_S - z_D| / z_D$ . With the thin lens equation this becomes

c(D) = \frac{A \, f \, |D - S|}{D \, (S - f)}

It is zero at $D = S$ and grows on both sides, faster in front of the focus distance than behind it.
It is proportional to the aperture $A$ : half the aperture, half the blur (but also a quarter of the light).
For a very distant background it approaches $A f / (S - f)$ , so background blur has an upper limit.

Depth of field

No real image is perfectly sharp, so a small blur is acceptable. A common choice for the largest acceptable circle of confusion is $c = \text{sensor diagonal} / 1500$ , about 0.03 mm for a full-frame sensor. The range of distances where $c(D) \le c$ is the depth of field (also called depth of focus). Solving $c(D) = c$ gives its limits:

D_\text{near} = \frac{A f S}{A f + c \, (S - f)}, \qquad D_\text{far} = \frac{A f S}{A f - c \, (S - f)}

When the denominator of $D_\text{far}$ is zero or negative, everything behind the focus distance is sharp enough. This first happens at the hyperfocal distance $H = f + A f / c$ : focused there, the depth of field reaches from $H / 2$ to infinity. A smaller aperture (larger f-number) increases the depth of field, and a larger one makes it shallower.

Smartphone vs large sensor

A smartphone camera and a full-frame portrait lens can have the same f-number, f/1.8, and thus collect the same light per area. But the phone's lens has $f = 4.3$ mm, so its aperture is only $A = 2.4$ mm across, while an 85 mm lens at f/1.8 has $A = 47$ mm. Focused at 2 m, the phone keeps everything from about 1 m to 21 m sharp, and the portrait lens only about 5 cm around the subject:

Smartphone: small aperture, large depth of field. Near and far objects appear in focus at the same time.
Large-sensor camera (DSLR) with a wide aperture: shallow depth of field. The subject is isolated from a blurred background (bokeh).

The portrait modes of phones imitate the second look in software: they estimate a depth map and blur each pixel according to its depth, much like the layers in this demo.

About the simulation. The scene is made of three flat layers, each at a single distance. Each layer is blurred with a disk the size of its circle of confusion, with 240 pixels across the sensor width, and the layers are then combined from back to front in linear light. Real scenes have continuously varying depth, and the light that a blurred foreground edge would let through from the hidden part of the background is missing. The view stays the same for every lens and sensor, although a real camera would see more or less of the scene (see focal length & field of view).

Try this

Start at 50 mm, f/2, focused on the subject: its circle of confusion is zero, while the foreground and the background are blurred.
Stop down from f/2 to f/8. The aperture shrinks 4×, and so do all circles of confusion, while the depth of field grows from 18 cm to 74 cm.
Focus on the background at 10 m. The foreground is now much blurrier than the background was before: the curve rises faster in front of $S$ .
Press the smartphone preset: everything is sharp. Then press the full-frame portrait preset. Both are f/1.8.
At f/8, press “Focus at the hyperfocal distance” and check that $D_\text{far} = \infty$ and $D_\text{near} = H / 2$ .
Move the background from 10 m to 100 m. Its blur hardly changes: it approaches $A f / (S - f)$ .
Keep the lens and switch to a smaller sensor. The acceptable $c$ shrinks with the diagonal, and so does the depth of field.