Aperture size and depth of field (Alberte 2017)

The effect of aperture size on depth of field

Introduction

As one of the main ways of regulating the brightness of a photo, aperture size is a term widely used in the world of photography. However, altering the aperture size not only changes the light, it also affects the depth of field, which in itself may be a desired effect.

In this investigation the relationship between the aperture size and depth of field will be explored by placing a stationary camera in front of a ruler, lifted and slightly angled so as to see the ruler well but still keeping the distances to points mainly horizontal, and taking photos at different aperture sizes. These photos will then be processed to find the depth of field of each aperture and a relationship between the two will be sought.

Research question

How does the aperture size affect the depth of field in a camera?

Independent variable: aperture size

Dependent variable: size of depth of focus

Constant variables:

• Angle of camera

• Height of camera

• Focal length

• Position of focus

• Lighting

• ISO

Theory

Defining focus and depth of field

Before it is possible to find the relationship between aperture size and depth of focus, it is necessary to define what is meant by focus and depth of field.

Firstly, if an object is considered as a point emitting light in every direction, while most of the light rays will not hit the lens and hence are unimportant in this context, some of them will indeed go into the lens. These rays will get bent by the converging lens of the camera and will at a later point intersect (diagram 1, left).

To make the diagram more comprehensible the rays may be simplified so that only the maxima of the rays hitting the lens are shown, as only the outer extremes will be of significance (diagram 1, right).

Diagram 1: (left) an object sending out light rays, going into camera; (right) simplified

The point where the light rays intersect, called the focal point, is the point at which the object is in focus, as this is where the object point would be seen as a point again. In order to take a focused photo of the object, the sensor, be it film or photodiodes, must be here. If the sensor is placed elsewhere, the resultant image will not be of a point but instead of a circle (diagram 2). This circle is called the circle of confusion and will give a photo that is out of focus. In order to reach focus, the distance between lens and sensor can be adjusted on cameras.

Diagram 2: the object is in focus where the lines intersect (green point) and out of focus everywhere else (exemplified by golden circles as circles of confusion)

However, defining focus this strictly doesn’t allow for anything but points at a very specific distance to be in focus. As it can be seen by looking at any photo, this is not how it works in practice. This is due to the properties of the camera sensor: in a digital camera for example the smallest point registerable is the size of one photodiode and if the circle of confusion is smaller than this, it becomes insignificant as it is recorded in the same way as if it had been exactly in focus.

It is then not necessary to have the object completely in focus for the photo to be sharp; it is only needed for the focus to be good enough. The distance in which the circle of confusion is small enough for the photo to be sharp is called the depth of field.

For the purpose of this investigation the depth of field will be measured as the distance in which the millimeter lines on the ruler are seen as lines and not blurred together.

Changing the aperture

In the previous diagrams the outer extremes of the light rays were always the same. When the aperture size is changed, the light rays allowed into the lens changes as well.

The size of the aperture is described by f-stops. f-stops are the ratio between the diameter of the aperture and the focal length, given by:

f-stop=focal lengthdiameter of aperture

Since focal lengths vary by lens and zoom, f-stops are a practical way of describing the light let into the camera without having to go into detail with this. f-stops will be used to describe the aperture size in this investigation.

By decreasing the size of the aperture, the hole through to the lens is made smaller and vice versa (diagram 3). As it is seen from the drawings, a small aperture increases the range in which the circle of confusion (referred to as CoC) is small enough so as to create a sharp photo due to the smaller angle at which it will hit the photo sensor.

Diagram 3: (left) large aperture giving a small CoC; (right) small aperture giving a large CoC

Hypothesis

As a smaller aperture size, i.e. a higher f-stop, will result in a smaller angle of light passing through the lens, and this giving a smaller circle of confusion upon hitting the camera sensor, it is hypothesized that a small aperture will give a large depth of field and vice versa.

Instruments

A DSLR camera, specifically Canons EOS50D with a Canon EF 50mm f/1.8 II lens; a ruler; a stand; Playdoh

Method

The camera was placed on a stand at a height of 6 cm and the back end lifted 1 cm up with Playdoh to create an angle and height at which it was possible to see all of the ruler while still keeping the camera reasonably horizontal to the lines.  The ruler itself was placed in front of the camera at a distance of 25 cm (photo 1).

Photo 1: How the camera and ruler were set up during the experiment

Positioned like this the camera was used to take pictures at different apertures.

To keep track of the height and angle of the camera, the abovementioned procedure was followed and not moved throughout the experiment. The Playdoh was quite malleable but with care taken during the changing of settings, it was possible to avoid changing the positioning of the camera.
Keeping the focal length constant was achieved by using a non-zoom lens in which this could not be changed. In order for the positioning of the focus not to be changed, the focus ring was kept stable and not turned.
It was tried to keep all other settings of the camera constant and only the aperture changing, but due to the changes in light caused by this, it was decided to change the shutter speed in accordance to this. This was, however, the only setting that was changed.
In trying to keep the lighting constant, the overhead lamps of the physics lab were the only lighting used. It was found that at small apertures with short shutter speeds, the AC lighting would visibly change the lighting of the picture, but the effect found negligible. This will be elaborated upon under Raw data.

Range of independent variable

The range of apertures used were determined by the capabilities of the camera and the lens. These were the f-stops 1.8, 2, 2.5, 2.8, 3.5, 4, 4.5, 5.6, 6.7, 8, 9.5, 11, 13, 16, 19 and 22. For each aperture only one picture was taken since it was found that the depth of field would not change between different pictures of same settings, see Raw data.

Measurement of depth of field

The depth of field connected to each aperture size was not directly found by the experiment but was measured based on the photos taken. The measurement was done by looking at the pictures and finding the number of millimeter lines still distinguishable.

Raw data

Photo 2: Example of how the counting of visible millimeter lines was done, here f/1.8.

The number of visible millimeter lines given in the table below were determined by counting the number of distinguishable millimeter lines at the right side of the lines (photo 2).

Table 1: Raw data collected from photos

 f-stop Visible mm lines ± 1 lines f-stop Visible mm lines ± 1 lines 1.8 12 6.7 27 2 13 8 36 2.5 14 9.5 41 2.8 14 11 46 3.5 15 13 63 4 17 16 75 4.5 19 19 95 5.6 17 22 113

The uncertainty of the number of lines is determined to be ± 1 line due to the ruler used for measuring. This gave an uncertainty of ± 0.5 line in each end of the depth of field, giving a total of ± 1 line for each measurement.

Each counting of lines has only been done once since taking multiple pictures at the same settings should result in the same number of lines. However, to make sure of this, a sample was done at f/1.8. It was found that the short shutter speed resulted in the lighting of the pictures varying with the AC lighting in the physics lab (photo 3).

Photo 3: 6 photos taken with exactly the same settings at f/1.8

The visible lines were counted and found to be the same in the samples, as exemplified in photo 4, and it was found unnecessary to make multiple countings of lines at each aperture.

Photo 4: Two measurements at f/1.8 with varying light

Processing data

By plotting the found data in LoggerPro, graph 1 was obtained.

Graph 1: plot of raw data

As it is seen from the graph, there is an overall tendency for the increasing f-stop to give a larger number of visible lines. At f/5.6 there is a sudden diversion from this tendency which indicates a mistake in the measurements made here. This might for example have been counting the lines wrong or the lighting of the lab hitting a low where it was made more difficult to see the lines clearly. It is not clear from the plot nor the theory which type of curve might be suited for the graph.

Simulation

To find the curve suited for the above graph and hence the relationship between f-stop and the number of lines, the experiment was simulated in Algodoo (photo 5). An “object” sending out light was placed to the left of the lens with the changing aperture in front and to the right of the lens was placed a red box representing a photodiode. The object was moved to a distance where it was in focus with the red box in the center of the focus. The object was then moved to see how far back and forth it could be placed and still be in focus - i.e. still have all lines going through the lens passing within the red box (photo 6).

Photo 5: overview of simulation in Algodoo

Photo 6: Left is in focus, right is out of focus

The lengths, distances and sizes used in the simulation were not like those in the real experiment but this was not needed for the relationship to be found. However, the relative lengths were consistent, meaning that aperture diameter and focal length were constructed to give the ratio - the f-stop - used in the data table below. For example, for f/2 the opening to the lens was 0,5 m and the focal length 1 m.

Controlling these values allows for calculating the f-stop using the equation in Theory: Changing the Aperture.

Table 2: data obtained from simulation

 F-stop ± 0,05% Min. distance/m ± 0,05 m Max. distance/m ± 0,05 m Depth of Field/m ± 0,1 m 2 3,3 3,4 0,1 2.8 2,3 2,6 0,3 4 2,7 3,3 0,6 8 1,9 4,5 2,6 9.5 1,9 9,2 7,3

The uncertainties of distances are based on the measurements being taken with a ruler-like feature going to 0,1 m at the zoom used. The uncertainty of f-stop is based on the uncertainties in measuring focal length (±0,05 m) and diameter of aperture (±0,0005 m at f/2- this is much smaller due to the close zoom used for making the aperture boxes). Since the uncertainty of diameter is very small at all apertures it was deemed insignificant and the uncertainty constant for all f-stops.

Graph 2: an exponential curve fitted to the plot of data found in the simulation

Conclusion

As it was found from the simulation and its graph, the relationship between the f-stop and the depth of field is increasing exponentially. However, upon graphing an exponential curve based on the raw data plot (excluding the outlier at f/5.6), it does not look quite as certain.

Graph 3: raw data plot, excluding outlier, with an exponential curve

As it is seen here, the exponential curve does not fit well with the data plot despite this being found from the simulation. Yet this might be explained by expanding upon the uncertainties and errors in the experiment, see Evaluation.

Evaluation

The simulation showed an increasing f-stop to cause the depth of field to increase exponentially but this did not fit very nicely with the data plot obtained from the experiment. What is seen is that for the graph to fit better, the data obtained for the highest f-stops should have been greater and the general development more consistent.

As for the f/22 depth of field, the reason for the low number may be found in the conditions for counting the lines: as the outer lines to be counted are farther away, the lines will converge and become harder to distinguish. More so, the sheer distance to the counted lines will make them smaller in the photo and again make it more difficult. For these reasons the actual depth of field should be bigger than that which is found in the experiment.

For the inconsistent progress of the data found, this may for the higher f-stops be caused by the same reasons as for f/22 and in general may be caused simply by the difficulty in counting precisely. Another factor that may have had an impact is the lighting - not the AC lighting of the lab but rather the shutter speed of the photos. Where the photos have been exposed to give a nice-looking photo, it seems that more light would have given for easier and more precise counting of the lines.

With this kept in mind, the experiment in unison with the simulation does give an indication of the existing relationship between the aperture size and the depth of field and while the exact formula is not found, the exponential relation is.

Discussion

For improving the precision of the experiment, the photos might have been exposed for a longer time so as to make it easier to see exactly to where the lines are in focus. Another camera feature that might have been used to improve this would be to take the pictures in RAW-format to capture more details and be able to enhance each photo to be as clear as possible.

To help with the problem of the lines converging at higher f-stops, the experiment might have been conducted with something else than a ruler - it could for example have been made with a moveable object on a track which would then be moved back and forth, similarly to the simulation, to see where it would move in and out of focus.

Another approach that could have been taking in finding out the theory would be to use the lens equation, relating object distance, image distance and focal length, to look at it.

As a last thought for this investigation I would like to mention the camera with the infinite depth of field - the pinhole camera. This has a very small aperture, the size of a pinhole, and hence has basically no circle of confusion. It requires a very long shutter opening to take a picture, but everything should be clear.

Bibliography

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https://physicssoup.wordpress.com/2012/05/18/why-does-a-small-aperture-increase-depth-of-fie ld/. Lochocki, Ed. Why Does a Small Aperture Increase Depth of Field?, 18 May 2012.

https://physicsandphotography.com/2014/10/02/depth-of-field/.Ana. Depth of Field, 02 October 2014.
http://www.cambridgeincolour.com/forums/thread1357.htm.Humphries, Dave. Why does depth of field change with aperture size?, 18 April 2009.

http://www.fujifilm.com.au/blogs/digital-photography/circle-of-confusion-and-what-it-means-to-be -in-focus. Diprose, Leigh. Circle of Confusion explained and what it means to be in focus, 08 January 2015.
http://www.uscoles.com/fstop.htm. Cole, Matthew. A Tedious Explanation of the f-stop.

http://www.student.thinkib.net/physics/page/19676/mini-investigation-activity.Hamper, Chris. Mini investigation activity.

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