In the previous discussion (Topic 29. Camera Lenses, Part 1. Focusing on the Object), we showed how the position of the lens had to be moved relative to the film on the image plane in order to bring objects at different distances in front of the lens into sharp focus. However, simply changing the position of the lens is not adequate for real-life situations, since anything more complicated than the very simplest of objects will have components that are at different distances from the lens. In principle, all of these different components cannot be brought into sharp focus on the film plane simultaneously. The following figure illustrates the problem.
If the lens is adjusted so that the image of the red object is in sharp focus (that is, the lens is adjusted so that the film plane is at distance Sr behind the lens) the rays from the blue object strike the image plane in an extended region rather than at a single point. Depending on the goal, the lens can be moved to bring either of the two objects into sharp focus (but not both simultaneously) or it can be adjusted to a point midway between the two images so that both objects are about equally blurred.
The magnitude of this problem is determined by two independent factors. The first is the angle at which rays from the lens converge to the point of sharpest focus. This angle determines how much blurring will result from a given difference in the positions of the images from the different parts of the object. (The rays diverge on the far side of the point of sharpest focus at the same angle, so that the blurring is equally large at some point in front of the actual focus or at a point an equal distance behind it.) This angle is illustrated in the following figure.
The size of this angle is determined by the ratio of the diameter of the lens to the distance between the lens and the image. Since objects are usually many focal lengths in front of the lens, the distance between the lens and the image is approximately equal to the focal length of the lens (see the previous topic). The ratio of the diameter of the lens to its focal length has appeared before as the parameter that governs the applicability of the paraxial ray approximation, and its inverse is the f-number.
The magnitude of this angle is clearly determined by the extreme rays – the rays that strike the lens at its edges, so that the problem can be reduced either by increasing the focal length of the lens (leaving everything else unchanged) or by inserting a stop or diaphragm that blocks the edges of the lens and prevents these extreme rays from reaching the image. Since the f-number is the ratio of the focal length to the diameter, reducing the effective diameter of the lens leaving the focal length unchanged increases the effective f-number of the lens. (Increasing the focal length leaving the diameter unchanged obviously has the same effect.) This increase in f-number reduces the angle at which the rays converge to (and diverge from) the point of sharpest focus and reduces the blurring for any given configuration.
The second factor that governs the magnitude of this blurring is the relationship between D, the distance between the two points in the object space, and d, the distance between the corresponding images. Decreasing d, keeping everything else the same, will decrease the size of the problem because the same angular divergence of the various rays produce a smaller divergence of the rays on the image plane.
Therefore: the blurring due to depth of field can be reduced by using a larger f-number and leaving everything else unchanged.
For a given distance D between two points on the object, the corresponding distance between the two images decreases as the objects move further away from the lens. This point is illustrated in the following table, which gives the variation in the ratio d/D as the object moves further away from the lens.
Distance of the object from the lens d/D
in units of the focal length
For example, if an object is 5 m (about 16 feet) in front of a lens whose focal length is 50 mm (a distance of 100 focal lengths), the ratio of d/D is 0.0001 – the images of two objects are spaced only 0.0001 times the distance between the two objects themselves. As you can see from the table, the fraction d/D decreases as the square of the distance from the object to the lens.
Therefore: the blurring due to depth of field will be reduced as the object moves further away from the camera, leaving everything else unchanged.
The final factor in estimating the depth of field of a real lens is the fact that no lens can focus the rays from a single point on an object to a single image point. Even the best lens produces a somewhat blurry image because of residual uncorrected aberrations, because of scattering of light in the lens, and because of dispersion (the fact that the ratio of the wavelength of the light to the size of the aperture is not exactly zero so that light always has some residual wave-like character).
The practical depth of field of a lens is then set by the balance between the imperfections in the size of the image point due to all of these problems and the additional blurring caused by depth of field. That is, the practical depth of field of a lens is set as the point at which the blurring due to the depth of the real object is not greater than the blurring that is inherent in the performance of the lens itself when it is imaging a perfectly flat object. For any given lens and object, the two factors above play important roles – the depth of field can be increased by increasing the f-number of the lens or by moving the camera further away from the object.
The following table shows the actual depth of field for a typical lens. That is, the table shows the range of object distances over which the image is essentially in “perfect” focus because the blurring due to the differences in the object distances is not greater than the blurring due to the other imperfections in the optical system. This lens has a focal length of 50 mm and a maximum aperture of f/1.7 (In other words, the maximum aperture is 50/1.7=29.4 mm). These parameters are typical of the lens that is used on an ordinary 35 mm camera.
1.7 4 8 11 16
5 4.5-5.5 4-6.5 3.4-9.6 3-15 2.5-175
10 8.3-12.6 6.8-20 5-500 4-infinity 3.4-infinity
100 47-infinity 20-infinity 10-infinity 7-infinity 5-infinity
Although the details of this table will vary from lens to lens, the general features are universal. In particular, when a lens is used at a small aperture opening corresponding to f/11 or f/16 and is focused on an object 10 m (about 30 feet) away, almost all objects (except for those that are quite close to the camera) will be in focus. This is how most cheap cameras are configured – they operate at a fixed lens aperture of f/11 or f/16 with a fixed focus lens adjusted so that objects about 5-10 m away are in sharpest focus. The depth of field at this small aperture is so large that almost everything else is acceptably in focus without the need for any adjustments.