Equivalence in Pictures: Focal Length, f-number, diffraction

Equivalence – as we’ve discussed one of the fairest ways to compare the performance of two cameras of different physical formats, characteristics and specifications – essentially boils down to two simple realizations for digital photographers:

  1. metrics need to be expressed in units of picture height (or diagonal where the aspect ratio is significantly different) in order to easily compare performance with images displayed at the same size; and
  2. focal length changes proportionally to sensor size in order to capture identical scene content on a given sensor, all other things being equal.

The first realization should be intuitive (future post).  The second one is the subject of this post: I will deal with it through a couple of geometrical diagrams.

Equivalence asks a simple question: which final photograph would look better if two cameras were set up in the same spot, captured identical scene content and their output were then displayed and viewed at the same size?

Recall that final displayed images are considered to be Equivalent for the purpose of this series if and only if (the conditions in black are relevant to this particular post):

  1. They measure the same length diagonally (same display size)
  2. They are viewed from the same distance by a person of average visual acuity (same Circle of Confusion)
  3. They represent exactly the same scene (same lighting, subject and field of view)
  4. The perspective in the captured scene is the same (same distance to subject)
  5. They are subject to the same motion blur constraints (same exposure time)
  6. They show the same depth of field (DOF)

To make things simpler, let’s assume that we have a single camera with a thin lens fixed in space.  The camera has the ability to  mount sensors of different formats to its sensing plane and to move it backwards and forwards with respect to the lens to adjust focal length.  The lens’ aperture is constant.

We set up our camera with the movable back to capture a single tree at the center of a distant scene.  We position the non-movable lens a fixed distance U from the scene ensuring that the tree fills the field of view when the camera back is in the Full Frame focal length position.  Scene to the left of the lens, camera+sensor to the right:

Equivalence FF

This somewhat simplified – but for the purposes of field of view accurate – diagram shows the tree barely fitting within the field of view of our FF sensor, as desired.   In the field we call it a ‘field of view’ or FOV but speaking in geometrical terms it is really an angle of view, AOV.  The angle of view is obviously the same on both the sensor  and scene side of the diagram and it is determined solely by the size of the subject and its distance from the center of the thin lens.

If we replaced the FF sensor with one of a different size and we wanted the tree to just fit within it as in the case above in order to fulfil the requirement for equal Field of View for Equivalence – all we would need to do is move the back of our flexible camera backward or forward so that the sensor in question just fit within the lines defined by the AOV, as shown below

Equivalence FL

When moving the back forward all we are doing is creating similar isosceles triangles with a smaller base.  We know that if the base (sensor) of one triangle is half the size of a second similar one, the height (its distance to the lens) of the first triangle will also be half of the second one.  Half the sensor height means half the distance to the lens (approximately the focal length for our purposes) if field of view is to be maintained.  Base 1 is to base 2 as height 1 is to height 2, or in our case for example:


We are taking a couple of shortcuts but this in essence proves realization 2) above: all other things being equal two differently sized sensors will capture  identical scene content when the ratio of their focal lengths is varied proportionally to the ratio of their size.

Equivalent Focal Length: Proportional to Sensor Size

Using the diagonal of the sensor instead of its height for the calculations in order to mitigate aspect ratio issues, one therefore gets the focal lengths that would produce an equivalent diagonal field of view for different format sizes all else staying equal:

Diagonals Equivalence

That’s why to maintain equivalent scene content the focal length in different formats needs to be adjusted proportionately to sensor height (or its diagonal).  No magic, just geometry.

Once one realizes this simple fact, many other parameters used in photogarphy fall into Equivalent line.

Equivalent f-number: Proportional to Sensor Size

f-number (N) in photographic applications is usually approximated by the ratio of focal length (f) to effective lens aperture (D)

N = \frac{f}{D}

In our movable back camera example the lens has the same aperture for all sensors (refer to diagram 2 above).  It follows that with a constant D if f changes so does N, proportionately.

So when maintaining an equivalent field of view all other things being equal the f-number varies proportionately to sensor diagonal.   For example with an effective aperture D equal to 7.7mm:

Equivalence N

Equivalent Diffraction: Proportional to Sensor Size

The same applies to diffraction, which spreads out a beam of light passing through a circular aperture into an (Airy) disk whose first zero has an approximate diameter of

Diameter_{Airy} =2.44 \cdot \lambda \cdot N

in the same units as lambda.  With \lambda on average around 0.55 microns this is how much the light beam would be spread out at the various format’s focal lengths (sensor locations, for our flexible camera)

Airy Disk Diameter


Wait (!), you say.  You mean that in an Equivalent situation the Airy disk is larger on an FF sensor than in a smaller format’s?  Of course, but realize (item 2 in the intro) that what counts when viewed at an equal display size is its size in units of picture (or sensor) diagonals.  If we compute the number of Airy disks in the diagonal of the displayed image of each format we realize that diffraction affects equally all formats set up Equivalently when viewed at the same size and distance.


Airy Disks in Diagonal

The direct relationship with focal length breaks down with more complex parameters, as shown earlier in the case of defocus for instance.

Let’s next take a pictorial look at Equivalence and ‘sharpness’.