Wednesday, 27 November 2013

Relationship Between F-Stop Numbers and the Size of the Diaphragm Opening Explained

After we published this article Making Sense of the Odd Progression of the F-Stop Scale? I got a handful of e-mails mainly asking doubts like;

  • Do f-stop numbers represent a specific size of lens opening?
  • Do all lenses set to a specific aperture let in the same amount of light? etc

So in this post we will discuss the relation between f-stop numbers and the actual diaphragm openings.

What is a Diaphragm?

Iris Diaphragm
Iris Diaphragm animation courtesy: Wikipedia

In photography, a diaphragm is a device that controls effective diameter of the lens opening and is called the “aperture” or “iris” of the lens. Diaphragm is located within the lens and is made up of a series of metal blades that form a circle. The adjustable diameter of the aperture is how the camera regulates the amount of light that pass through the lens to the sensor.

How are F-Stop numbers and Aperture Diameters Related?

The formula to find f-stop is:

f-stop = focal length / diameter of lens opening

What are F-Stops?

F-stops are a measure of the aperture of a lens; they express the ratio of focal length to apparent lens aperture. F-stops tell us how wide the opening of the iris is.

What is Focal Length?

Focal Length
Focal Length

Focal length of a lens is the distance between the optical center of the lens and the sensor plane.

So when trying to figure out the actual diameter of the diaphragm opening we should also consider the focal length of the lens in addition to the f/number.

f/2 = focal length / 2 
f/4 = focal length / 4 
f/5.6 = focal length / 5.6 and so on.

Let us try the formula on two different lenses one a 50mm and another 100mm.

Canon 50mm f/1.8 and Canon 100mm f/2.8 both set to f/2.8
Canon 50mm f/1.8 and Canon 100mm f/2.8 both set to f/2.8

f/2 for a 50mm lens will be a lens opening of 25mm. (50/2). And
f/2 for a 100mm lens will be a lens opening of 50mm. (100/2).

It is clear that when the focal length increases so does the diameter of the diaphragm. In this case when focal length increased from 50mm to 100mm the size of the diaphragm doubled from 25mm to 50mm.

But to determine the amount of light reaching the sensor shouldn’t we be considering the area of the circle instead of its diameter.

Let us have a look our two lenses, the 50mm and the 100mm

The 50mm lens when set to f/2 has a diameter of 25mm or radius of 25/2 = 12.5mm and
The 100mm lens when set to f/2 has a diameter of 50mm or radius of 50/2 = 25mm.

Applying the formula A=πr^2 (read π r squared)

The area of our 50mm lens set to f/2 will be 3.14 x 12,5² = 490,6 mm² and
The area of our 100mm lens set to f/2 will be 3.14 x 25² = 1962,5 mm²

From the above example it is clear that at an aperture of f/2 the 100mm lens has a diaphragm opening which is 4 times bigger than the 50mm lens which is also set to f/2 (1962.5 / 490.6 = 4).

Why is the Aperture area 4 times bigger when the focal length is doubled?

The aperture is 4 times bigger because the area varies in relation to the square of the radius. So doubling the radius is the same as saying 2² which is 4.

How does two diaphragm sizes one 4 times bigger than the other let in the same amount of light?

It might sound confusing at first but the answer to this question lies partly in mathematics and partly in physics.  First focal length plays an important role in deter mining the amount of light and then there is the Inverse Square Law.

Inverse square law

Inverse Square Law image courtesy: Wikipedia

Inverse square law states that some physical quantity or strength is inversely proportional to the square of the distance from the source of that physical quantity.

To put matters simply as per Inverse square law, the power of the light will be inversely proportional to the square of the distance.

For example

When a distance of 2 is squared we get 4 the inverse of which is ¼. Meaning when light travels double the distance the power of light is reduced to ¼.

Summing Up

To conclude let us state some bare facts.

  1. F-stop numbers does not refer to any particular size of aperture it is the ratio of focal length to apparent lens aperture.
  2. Same f-stop number when set on different focal length lenses will have different diaphragm sizes.
  3. Same f-stop numbers set on different lenses will let in exactly the same amount of light irrespective of the size of their aperture opening.
  4. At any aperture when focal length is doubled the size of diaphragm increases 4 times.

Related Reading