To understand what is depth of field in photography, you first need to understand what is **circle of confusion**.

### CIRCLE OF CONFUSION

In physics, **circle of confusion is the effect created by an image point, in relation to the blur on the sensor plane**.

Assuming you have a point placed at a certain distance from a single lens, the light rays starting from the point, pass through the lens and are focused on a plane that passes through the vertex of the cone projected to the other side of the lens. This plane, perpendicular to the optical axis of the lens, is called **focal plane** and corresponds to the plane of the sensor or film.

**The focal plane is the only plane on which it is possible to obtain a perfectly clear image** because it is the only plane on which the vertex of the cone of the light rays lies.

Moving forward or backwards in respect to the focal plane, the projected image will no longer be a point but a circle due to cone of light rays being cut from a point other than its vertex.

This circle is what is called the **circle of confusion**.

*The figure explains the principle of the formation of circles of confusion (also called discs), using the example of focusing an image point captured on the optical axis: when the focus of the image point is correct (example center in the figure), it will be clearly represented as a point. As the focus is moved forward or backward, the image point will increase its size more and more, forming and widening the so-called circle of confusion that represents it (in the figure, the image point is moved, but in practice the principle does not change).*

*The image points are refracted onto the*

**focal plane**(film or sensor) positioned at a precise distance on the**optical axis**, also called the**focal point**. On this plane, the focused image appears clear because it is reconstructed as points, or rather as point-like circles.Since the observational effect is closely related to human binocular visual acuity (average acuity is 10/10), the size of the circles of confusion can reach values such that the eye will no longer be able to distinguish them as circles: at this point the best focusing of the image points has been achieved.

Circles of confusion are round in shape like discs (as the lenses are circular) and are measured according to the diameter they depict on the image, but are subject to take on the shape (silhouette) of the profiles of the lamellae, when the diaphragm closes (hexagonal, pentagonal, etc).

Certain evaluation procedures, based on a large number of observers who evaluated contact prints with 8×10 inches (20×25 centimeters) frames from a distance equal to the diagonal (32 cm), found a value of the diameter of the circles of confusion equal to 0.2 millimeters. This very bland value (5 lines per millimeter from a distance of 32 cm, or 6.4 l / mm from 25 cm) corresponds to a rather poor visual acuity and is well below average, equal to 4.6 / 10 of the relative scale.

Assuming the value of 0.2 mm, as a reference of the **maximum acceptable diameter** of the CoC (Circle of Confusion) for the 8×10 inch frame, the respective values can then be calculated for each other film / sensor format, considering the ratio of the diagonals. For the 35mm format, having to be enlarged 7.5 times to be equivalent to the 8×10 inch format taken as a reference, the circle of confusion must be calculated as 0.2mm / 7.5 = 0.027mm approximately.

In the photographic field, the manufacturers, adapting to this convention, began to conform the production of the lenses, adopting indexes on the barrel that indicated an ideal depth of field (PdC) possible, based on the focus distance and the selected aperture ( read the Hyperfocal Distance section).

However, if you want to make the image even sharper in all cases (with more HP), it will naturally be necessary to decrease the value of the CdC in a separate calculation and, usually in the procedure, it will also be possible to establish an arbitrary, but more useful correct value to specific needs, capacity and possibility (It is important to remember that the values found during the tests represent a conventional value that is not always adequate for all situations).

#### CIRCLE OF CONFUSION DIAMETER IN RELATION TO THE FORMAT

Format | Frame size | CoC |
---|---|---|

SMALL FORMAT | ||

APS-C | 22,5 mm x 15,0 mm | 0,016 mm |

35mm | 36 mm x 24 mm | 0,027 mm |

MEDIUM FORMAT | ||

6x5 | 56 mm x 42 mm | 0,043 mm |

6x6 | 56 mm x 56 mm | 0,049 mm |

6x7 | 56 mm x 69 mm | 0,055 mm |

6x9 | 56 mm x 84 mm | 0,062 mm |

6x12 | 56 mm x 112 mm | 0,077 mm |

6x17 | 56 mm x 168 mm | 0,109 mm |

LARGE FORMAT | ||

4x5 | 90 mm x 127 mm | 0,100 mm |

5x7 | 127 mm x 178 mm | 0,135 mm |

8x10 | 203 mm x 254 mm | 0,200 mm |

*Source: Wikipedia*

#### Example of calculating circle of confusion for different formats

Based on 8×10 inch format values (CoC: **0,2 mm** – Diagonal (203 x 254 mm): **325,1538 mm)**

**FULL FRAME**

Diagonal(24 x 36mm): 43,2666 mm

Ratio: 325,1538 mm / 43,2666 mm = 7,5151

CdC = 0,2 mm / 7,5151 = **0,0266 mm**

**HASSELBLAD X2D**

Diagonal(32,9 x 43,8 mm): 54,7600 mm

Ratio: 325,1538 mm / 54,7600 mm = 5,9377

CoC = 0,2mm / 5,9377 = **0,0337 mm**

**PHASE OBE IQ4**

Diagonal (40 x 53,4 mm): 66,7200 mm

Ratio: 325,1538 mm / 66,7200 mm = 4,8734

CoC = 0,2 mm / 4,8734 = **0,0410 mm**

### DEPTH OF FIELD

In general,* depth of field in photography is the distance between the closest object and the farthest object in an image that still appear sharp and sufficiently in focus, despite the fact that the plane in focus is one and only one*. In other words,

*.*

**depth of field is the portion of space in front of and behind the point on which the focus is made, bounded by two planes, within which all the points have a diameter smaller than that of the circle of confusion and , for this reason, they appear sharp and in focus even if really they are not**Depth of field depends on several factors: **focal length**, **distance to the subject**, **aperture** and **diameter of the circle of confusion**.

It can be roughly calculated with the following formula:

DoF\approx\dfrac{2u^2NC}{f^2}

where

* DoF *= depth of field

*= distance from subject*

**u***= f-stop number*

**N***= circle of confusion diameter*

**c***= focal length*

**f**Analyzing the equation it is evident that the depth of field is influenced linearly by the f-stops number and circle of confusion diameter and exponentially by the focal length and distance from the subject..

Depth of field is directly proportional to the square of focal distance, the f-stops number, the circle of confusion diameter , and is inversely proportional to the square of the focal length of the lens.

#### CONTROLLING DEPTH OF FIELD

The depth of field can be controlled by acting on three of the four parameters in the formula described above. The parameter that cannot be changed is the circle of confusion as it is bound by the format used.

**Focal length**: to the denominator in the equation with squared incidence. Leaving the other parameters unchanged, the greater the focal length is, the shorter the depth of field. To obtain a shallower depth of field, and therefore more blurred effect in the photograph, it is necessary to use longer focal lengths. Remember that focal length has an exponential impact on the depth of field. Even small variations in focal length will have a major impact on the part of the scene that will be in focus.

**Distance from the subject**: to the numerator in the equation with squared incidence. Leaving the other parameters unchanged, to obtain a smaller depth of field it is necessary to decrease the distance between the subject and the camera. While, if we want to increase, it is necessary to increase the distance between the subject and the camera. Remember that distance to the subject has a exponential impact on the depth of field. Even small variations in distance between subject and camera will have a major impact on the part of the scene that will be in focus.

**Aperture ( f-stops number)**: to the numerator in the equation with linear incidence. Leaving the other parameters unchanged, to obtain a smaller depth of field it is necessary to open the diaphragm(set a smaller f-stop) while to obtain a greater depth of field it is necessary to close the diaphragm (set a larger f-stop).

From the point of view of physics, by reducing the diameter of the aperture through which light rays can pass, by increasing the number of f-stops, the depth of field increases because only rays with narrower angles are able to pass (having a narrower angle) and enter in the diameter of the circle of confusion for a greater distance.

It is demonstrated very well in the drawing below. The point in focus indicated with the number 2 projects a point on the focal plane but points 1 and 3 which are at a different distance project a blurred image. By reducing the size of the aperture, the size of these blurred images is reduced, reaching the point where the size is such that it can be perceived as in focus.

### HYPERFOCAL DISTANCE

**The hyperfocal distance is a particular focusing distance, which can be calculated for a given focal length and a given aperture and which allows one to obtain the maximum extension of the depth of field. It is the closest distance to which a lens can focus while maintaining acceptably sharp objects to infinity**.

When the lens focuses at this distance (H), all objects at a distance ranging from half the hyperfocal distance (H / 2) to infinity will be acceptably sharp. It is the distance that allows the greatest possible depth of field, in relation to the aperture of the diaphragm and the focal length of the lens.

Mathematically, the hyperfocal can be calculated for any objective with the following formula:

H=\dfrac{f^2}{Nc}+f

where

* H* = hyperfocal distance

*= focal length*

**f***= aperture*

**N***= circle of confusion diameter*

**c**The use of hyperfocal is very useful for the landscape photographer who must have everything in focus, from the foreground to infinity.

Wanting to reduce the hyperfocal, it is necessary to decrease the focal length and close the diaphragm, which is why in landscape photography the wide-angle lenses are able to keep the whole landscape in focus.

With a wide-angle lens, carefully chosen focus distance and a reasonably small aperture (from f/8 to f/11), it is possible to achieve good focus from short distances to infinity.

With lenses with longer focal lengths it is more complicated. Just think that a 200mm at f/11 has a hyperfocal distance that is about 140 m and a depth of field that will extend from about 70 m to infinity.

On the barrel lens of many lenses, especially those with manual focus, there is the distance scale and the depth of field scale that allows you to set the hyperfocal very easily.

By setting an aperture with the **aperture ring** and observing the **depth of field scale**, you can read the indication of the aperture used twice, one on the left and one on the right.

The distance number aligned with the left aperture number on the **distance scale,** indicates the distance from which the focus area begins, where depth of field starts, while the one on the right indicates how far the focus area extends, and where depth of field ends.

Everything that will be included between the distances indicated by the two diaphragms on the distance scale will be in focus because it is contained within the depth of field.

The hyperfocal calculation is done in the reverse way, starting from the farthest distance from us, that is, infinity and inserting as much space as possible between us and infinity itself within the depth of field.

Quite simply, it’s enough to place the infinite value ∞ of the distance scale at the value to the right of the aperture used on the depth of field scale.

In this way, everything between infinity and the corresponding value on the distance scale aligned with the left aperture number will be in focus.

#### MATHEMATICAL INSIGHTS

#### Circle of confusion, depth of field and hyperfocal distance

A subject in front of a lens (for simplification consisting of a single lens) at a distance **s** is focused at a distance **d** behind it, according to the equation of the lens: **1/d = 1/f – 1/s** , where **f** is the focal length of the lens.

If the lens were perfect (without aberration or diffraction) a point in **s** would focus on an infinitely small point in **d**.

An object in **uf**, in front of **u**, focuses in **df**, behind **d**. On the sensor plane **d**, the object would be blurred; it would be reproduced as a circle whose diameter **cf** is called **circle of confusion**.

Similarly, an object in **ur**, behind **u**, focuses in **ur**, in front of **d**. Its circle of confusion in **d** has **c**r diameter.

The **depth of field (DoF)** is represented by the interval of distances between uf and **ur**, (**Dr + Df**), where the circles of confusion, **cf** and **cr**, are small enough to make the image appear in focus.

The depth of field formula comes from the Gaussian thin lenses equation:

\dfrac{1}{f} = \dfrac{1}{u} + \dfrac{1}{d}

*where*

* f = focal length*

**u**= distance from subject

**d**= diatance from imageTo calculate the near and far limits of the depth of field substitute in the Gaussian equation of the thin lens:

\dfrac{1}{F} + \dfrac{1}{[d(1-\dfrac{1}{\dfrac{c}{a+c}})]} = \dfrac{1}{f} \quad => \quad F = \dfrac{fua}{fa-c(u-f)}

\dfrac{1}{N} + \dfrac{1}{[d(1+\dfrac{1}{\dfrac{c}{a-c}})]} = \dfrac{1}{f} \quad => \quad N = \dfrac{fua}{fa+c(u-f)}

\dfrac{1}{F} + \dfrac{1}{[d(1-\dfrac{1}{\dfrac{c}{a+c}})]} = \dfrac{1}{f}

\dfrac{1}{N} + \dfrac{1}{[d(1+\dfrac{1}{\dfrac{c}{a-c}})]} = \dfrac{1}{f}

F = \dfrac{fua}{fa-c(u-f)}

N = \dfrac{fua}{fa+c(u-f)}

*where*

* F = ur is the distance within which the image is acceptably sharp (far limit of depth of field)*

**N = uf**is the distance from which the image is acceptably sharp (near depth of field limit)Depth of Field (PdC) is the difference between the far and near limits:

DoF = F-N = fua[\dfrac{1}{fa - c(u - f)} - \dfrac{1}{fa + c(u - f)}] = \dfrac{2fuac(u-f)}{(fa)^2-c^2(u-f)^2}

DoF = F-N

= fua[\dfrac{1}{fa - c(u - f)} - \dfrac{1}{fa + c(u - f)}]

DoF= \dfrac{2fuac(u-f)}{(fa)^2-c^2(u-f)^2}

Focusing at the hyperfocal distance, when u = H and the far limit of the depth of field extends to infinity and the denominator fa-c (H-f) = 0

\dfrac{fHa}{fa-c(H-f)} = ∞ \quad \quad fa-c(H-f) = 0

\dfrac{fHa}{fa-c(H-f)} = ∞

fa-c(H-f) = 0

The hyperfocal distance is:

H= \dfrac{fa}{c}+f

By replacing the variable a (aperture) with **f/f-stop** in the equation, the hyperfocal distance becomes:

H=\dfrac{f^2}{Nc}+f

In terms of hyperfocal distance, the depth of field becomes:

DoF=\dfrac{2Hu(u-f)}{H^2-(u-f)^2}+f

In most cases, in photography, the focus distance is much greater than the focal length of the lens (u >> f), so the simplified depth of field equation becomes:

DoF\approx \dfrac{2Hu^2}{H^-u^2}

When the hyperfocal distance is much greater than the focus distance (H >> u), the approximate depth of field is given by:

DoF \approx\dfrac{2u^2NC}{f^2}

In terms of hyperfocal distance H, the near and far limit distances of the depth of field, N and F, are given by:

N = \dfrac{Hu}{H+u} \quad \quad F = \dfrac{Hu}{H-u}

For an infinity-focused lens **u = ∞**, the distance from which the depth of field begins (near limit) **N = H**.

For a lens focused at the hyperfocal distance **u = H**, the distance from which the depth of field begins (near limit) **N = H / 2** and the distance within which the depth of field (far limit) extends **F = ∞** .

Also, the relationship between the front depth of field focus point (**u – N**) and the rear depth of field focus point (**F – u**) is:

\dfrac{Front DoF}{Rear DoF} \quad=\quad \dfrac{H-u}{H+u}

At the focus distance of u = H / 3, the front depth of field is half the rear depth of field, i.e. the depth of field extends 1/3 in front of and 2/3 behind the focus point ( called the 1/3 rule of the hyperfocal distance).

The distribution of the front and rear depths of field depends on the focus distance (for a given focal length and number of f-stops).

The rule above is accurate only to 1/3 of the hyperfocal distance. As the focus distance decreases, the total depth of field decreases and the front and rear depths of field (the front and rear depths of field are expressed as fractions of the hyperfocal distance) are divided equally.

FOCUS DISTANCE | FRONT DOF | REAR DOF |
---|---|---|

∞ | ∞ | 0 |

H | H/2 | ∞ |

H3 | H/12 | H6 |

H10 | H/110 | H90 |

For close-up and macro photography, magnification is the appropriate working variable.

Recalling that **u – f = f / m**, where **m** is the magnification, the relationship between the front and rear depth of field can be rewritten.

In terms of the parameter **K = (c f-stop) / f,** the ratio between the front and rear depth of field is given by:

\dfrac{Front DoF}{Rear DoF} \quad=\quad \dfrac{m-K}{m+K}

As the distance between the camera and the subject decreases (and the magnification increases), the ratio between the front and rear depth of field approaches unity.

Lastly, given **N** and **F**, the focus distance that positions the depth of field for an optimal effect is given by:

u=\dfrac{2NF}{N+F}

The relationship is useful, provided that it is possible to measure the distances of the near (N) and far (F) limits.

#### Thin lenses and linear magnification equations

To accurately determine the characteristics of the image of an object formed by a thin lens, two equations can be used: the** thin lens equation** and the **linear magnification equation**.

*where*

* f = lens focal length*

**u**= object distance from the lens

**d**= image distance from the lens

**h0**=object height

**h1**= image heightT1 and T2 are similar triangles so we can write:

\dfrac{h0}{f} = - \dfrac{h1}{d-f}

Also in the figure below the triangles T1 and T2 are similar triangles

\dfrac{h0}{u} = - \dfrac{h1}{d}

By combining the two equations it is possible to obtain the final equation for thin lenses:

\dfrac{1}{f} = \dfrac{1}{u} + \dfrac{1}{d} \quad => \quad f=\dfrac{ud}{u+d}

dfrac{1}{f} is called **dioptric power of the lens** and is measured in m^{-1} or diopters.