Depth of Field and the F-Stop

In Chapter 14 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I briefly discuss depth of field: the distance between the nearest and the furthest objects that are in focus in an image captured with a lens. However, we don’t go into much detail. Today, I want to explain depth of field in more—ahem—depth, and explore its relationship to other concepts like the f-stop. Rather than examine these ideas quantitatively using lots of math, I’ll explain them qualitatively using pictures.

Consider a simple optical system consisting of a converging lens, an aperture, and a screen to detect the image. This configuration looks like what you might find in a camera with the screen being film (oh, how 20^th century) or an array of light detectors. Yet, it also could be the eye, with the aperture representing the pupil and the screen being the retina. We’ll consider a generic object positioned to the left of the focal point of the lens. 

 

To determine where the image is formed, we can draw three light rays. The first leaves the object horizontally and is refracted by the lens so it passes through the focal point on the right. The second passes through the center of the lens and is not refracted. The third passes through the focal point on the left and after it is refracted by the lens it travels horizontally. Where these three rays meet is where the image forms. Ideally, you would put your screen at this location and record a nice crisp image. 

Suppose you are really interested in another object (not shown) to the right of the one in the picture above. Its image would be to the right of the image shown, so that is where we place our screen. In that case, the image of our first object would not be in focus. Instead, it would form a blur where the three rays hit the screen. The questions for today are: how bad is this blurring and what can we do to minimize it?

So far, we haven’t talked about the aperture. All three of our rays drawn in red pass through the aperture. Yet, these aren’t the only three rays coming from the object. There are many more, shown in blue below. Ones that hit the lens near its top or bottom never reach the screen because they are blocked by the aperture. The size of the blurry spot on the screen is specified by a dimensionless number called the f-stop: the ratio of the aperture diameter to the focal length of the lens. It is usually written f/#, where # is the numerical value of the f-stop. In the picture below, the aperture diameter is twice the focal length, so the f-stop is f/0.5.

 

We can reduce the blurriness of the out-of-focus object by partially closing the aperture. In the illustration below, the aperture is narrower and now has a diameter equal to the focal length, so the f-stop is f/1. More rays are blocked from reaching the screen, and the size of the blur is decreased. In other words, our image looks closer to being in focus than it did before. The blurring of an out-of-focus image is reduced. 

 

It seems like we got something for nothing. Our image is crisper and better just by narrowing the aperture. Why not narrow it further? We can, and the figure below has an f-stop of f/2. The blurring is reduced even more. But we have paid a price. The narrower the aperture, the less light reaches the screen. Your image is dimmer. And this is a bigger effect than you might think from my illustration, because the amount of light goes as the square of the aperture diameter (think in three dimensions). To make up for the lack of light, you could detect the light for a longer time. In a camera, the shutter speed indicates how long the aperture is open and light reaches the screen. Usually as the f-stop is increased (the aperture is narrowed), the shutter speed is changed so the light hits the screen for a longer time. If you are taking a picture of a stationary object, this is not a problem. If the object is moving, you will get a blurry image not because the image is out of focus on the screen, but because the image is moving across the screen. So, there are tradeoffs. If you want a large depth of focus and you don’t mind using a slow shutter speed, use a narrow aperture (a large f-stop). If you want to get a picture of a fast moving object using a fast shutter speed, your image may be too dim unless you use a wide aperture (small f-stop), and you will have to sacrifice depth of field. 

With your eye, there is no shutter speed. The eye is open all the time, and your pupil adjusts its radius to let in the proper amount of light. If you are looking at objects in dim light, your pupil will open up (have a larger radius) and you will have problems with depth-of-focus. In bright light the pupil will narrow down and images will appear crisper. If you are like me and you want to read some fine print but you forgot where you put your reading glasses, the next best thing is to try reading under a bright light.

Most photojournalists use fairly large f-stops, like f/8 or f/16, and a shutter speed of perhaps 5 ms. The human eye has an f-stop between f/2 (dim light) and f/8 (bright light). So, my illustrations above aren’t really typical; the aperture is generally much narrower.