I. Concept of Depth of Field

Before taking a picture, the camera must be focused. Theoretically, only the part of the image that is accurately focused is sharp, while objects in front of and behind the focal plane gradually become blurred as they deviate from the plane of focus. Influenced by factors such as the lens and shooting distance, there is still a range of sharp objects in front of and behind the focal point; this range is the depth of field. Simply put, the depth of field is the distance range in front of and behind the focal point where the image remains acceptably sharp.

In photography, depth of field effects are conventionally categorized as “deep” or “shallow,” divided into shallow depth of field and deep depth of field.

What is a shallow depth of field?
Take an intuitive shooting example: when photographing a cluster of flowers and focusing on the front petals, if we shoot with a large aperture, only a very small range of the petals will be sharp, while the foreground and background will be significantly blurred. This effect with a narrow range of sharpness is a shallow depth of field, as shown in Figure 1. A shallow depth of field refers to a small range of sharp imaging, where only objects near the focal point remain sharp. It is commonly used to emphasize the subject and minimize background distractions.

What is a deep (large) depth of field?
Similarly, for example, when photographing a landscape, stopping down the aperture and placing the focal point at the 1/3 position from the front of the frame will result in sharp imaging from the foreground to the background. This is a deep depth of field, as shown in Figure 2. A deep depth of field refers to a large range of sharp imaging, where objects over a large distance range remain sharp. It is commonly used in architectural aerial photography, landscape photography, etc.

II. Influencing Factors
Depth of field is determined by four factors: focal length, subject distance, permissible circle of confusion diameter, and aperture. The formula for calculating depth of field is as follows: $$\text{DOF} \approx \frac{2 u^2 N c}{f^2}$$

1. Permissible Circle of Confusion Diameter
The permissible circle of confusion diameter is an artificially set maximum acceptable blur spot size in photographic optics, used to define the boundary between “sharp” and “blurred” on the image plane: when the actual circle of confusion diameter is smaller than this threshold, the corresponding object point is considered to be within the depth of field. Under the condition that the focal length, f-number, and focus distance are all kept constant, the permissible circle of confusion diameter is directly proportional to the depth of field—the larger this value, the deeper the calculated depth of field, and vice versa.

2. Aperture ( f-number )
The f-number (abbreviated as N, also called f-ratio or f-stop) is the core parameter that directly determines the amount of light entering the image and the range of sharpness. Its mathematical definition is the ratio of the focal length $f$ of the imaging system to the effective aperture diameter D, with the formula:
$$N=\frac{f}{D}$$

Under the premise that the focal length $f$, object distance $u$, and permissible circle of confusion $c$ are all fixed, the depth of field is directly proportional to the f-number. The larger the f-number (N), the smaller the aperture (referred to as a “small aperture” in photography), and the greater the depth of field; conversely, the smaller the f-number, the larger the aperture (referred to as a “large aperture” in photography), and the shallower the depth of field.

3. Focal Length
Under the premise that the aperture $N$, object distance $u$, and permissible circle of confusion $c$ are all fixed, the longer the focal length, the shallower the depth of field; the shorter the focal length, the deeper the depth of field.

4. Shooting Distance / Object Distance
Under the premise that the aperture $N$, focal length $f$, and permissible circle of confusion $c$ are all fixed, the farther the camera is from the subject, the greater the depth of field; the closer the camera is to the subject, the shallower the depth of field.

III. Principle of Depth of Field

Before understanding the principle of depth of field, it is necessary to clarify the optical characteristics of focusing: when a lens is focused, theoretically only one plane parallel to the image plane can achieve precise focus, namely the focal plane. Object points in front of or behind this plane are in an out-of-focus state. Light rays emitted from object points on the focal plane converge into a sharp point on the image plane after passing through the lens; whereas light rays emitted from object points not on the focal plane cannot converge perfectly on the image plane, but instead form a spread-out circular spot, which is called the circle of confusion (as shown in Figure 4).

The size of the circle of confusion is closely related to the distance from the object point to the focal plane. As shown in Figure 5, out-of-focus plane 1 is farther from the focal plane, and its object points form a larger circle of confusion on the image plane; whereas out-of-focus plane 2 is closer to the focal plane, forming a smaller circle of confusion. That is, the closer the object point is to the focal plane, the smaller the circle of confusion.

If the circle of confusion is so small that it is invisible to the human eye, the blur circle can be considered as the imaging of a point, appearing just as sharp as an in-focus object. Such a circle of confusion is called the permissible circle of confusion. Based on this principle, there is a distance range in front of and behind the subject (focal plane) within which the circles of confusion formed by object points on the image plane are all smaller than the permissible circle of confusion, and their images appear visually sharp, with the naked eye unable to distinguish any blur. This spatial distance over which relatively sharp images can be obtained is the depth of field, as shown in Figure 6.

IV. Calculation of Depth of Field

As shown in Figure 6, the distance from the foreground to the focal plane is the near depth of field, and its calculation formula is as follows:
$$\Delta L_1 = \frac{F \delta L (L - f)}{f^2 + F \delta (L - f)}$$

The distance from the background to the focal plane is the far depth of field, and its formula is:
$$\Delta L_2 = \frac{F \delta L (L - f)}{f^2 - F \delta (L - f)}$$

The sum of the near depth of field and the far depth of field is the depth of field of the subject, and its calculation formula is as follows:
$$\text{DOF} = \Delta L_1 + \Delta L_2 = \frac{2f^2 F \delta L (L - f)}{f^4 - F^2 \delta^2 (L - f)^2}$$ where L is the object distance, i.e., the shooting distance; $\delta$ is the permissible circle of confusion diameter; F is the lens f-number; and f is the lens focal length. It can be seen from the above formulas that the near depth of field < the far depth of field.