Depth of hearth

Note:

It is highly advised to also consult the following articles:

* Diaphragm

* Performances of the eye

* Iris and diaphragm

* orthoscopic Distance

* Fuzzy, clearness and contrast

* Depth of hearth

* Depth of field

which forms a coherent unit and is necessary to the comprehension of this last article.

Note: This article is initially resulting from that entitled Depth of field which was the subject of a cutting to form or supplement the articles mentioned above.

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One calls depth of hearth (not to be confused with the Depth of field) the interval in which must be the plan of a film or of a sensor so that the image of a luminous point on which one made the development is regarded as clear, for a given use.

Assumptions

We will make the following assumptions:

the objective will be regarded as a simple lens; the calculation which follows would be exactly the

even by using either the optical center O of the lens, but the nodal points of one centered system.

the objective will be regarded as perfect, able to give a specific image of a point

luminous; calculation will thus apply as much better than the objective will be of high-quality.

the spot-image limits will be seen, since the optical center O, under the angle ε

defined higher.

one will suppose that the possible later operations, like the enlarging on paper or projection, do not cause any degradation of the image.

Formulas

The traditional formulas of the simple lenses are in the article relating to the geometrical Optique, point out simply the two following ones:

\ frac {1} {p} + \ frac {1} {p'} = \ frac {1} {F}

and

g= \ frac {p'} {p}

which gives

p'=f \, (g+1)

These formulas are given here, as in all the remainder of the article, in their arithmetic form: it is indeed impossible to assign a negative or positive sign to the enlarging of an image by data-processing ways…

The luminous rays resulting from P converge in P' by forming a all the more open cone as it diameter D of the diaphragm is more important. If the plan of the film or the sensor is not placed exactly in P', the recorded image will be not a point, but a small circular spot whose diameter will not have to exceed the value:

$\ delta \ approx \ epsilon p' = \ epsilon F (g+1)$

It is now necessary to calculate the maximum acceptable shift X of the plan of the receiver by report/ratio at the P' point:

$\ frac {X} {\ delta} = \ frac {p'} {D}$ (similar triangles) from where $x= \ frac {\ delta
p'} {D}$ By replacing δ by the computed value previously and p' according to G , one obtains:

$x= \ frac {D}$

One sees appearing here the relative opening of the objective $n= \ frac {F} {D}$ .

Finally:

The interval in which the plan of the film or the sensor must be so that the image that is to say regarded as clear is all the more large as focal is longer to it, it more important growth and the diaphragm more closed.

Example: one wants to photograph an object located ad infinitum or very far (g=0) with an objective from focal distance 50 mm open to F: 2 and one limit of blur tolerated of 1/1500 radian:

$x= \ frac {2.50} {1500} = {0,066 \, mm}$

Under these conditions, it is necessary that the apparatus either built with a high degree of accuracy, in particular if it comprises an aiming reflex camera, but it is necessary also that the film remains perfectly planes. All things being equal, an adjustment of the diaphragm with F: 22 would give

X = 0,7 mm, which is obviously much less constraining.

For much of scientific uses, the limit of clearness of 1/1500 can be regarded as very insufficient. The requirements of precision are obviously reinforced and nothing is possible without using an objective of high-quality very. According to the same principle, one can calculate a tolerance of development but, except in some particular cases, this concept hardly has practical interest.

See too

wikilivre of photography - chapter 14: Clearness of the photographic images