What’s your Q?

Another way of looking at the minimum resolvable distance between two points is to turn it on its head and look at spatial frequencies. When we do that, we can restate one of the conclusions of this post as: the upper cutoff spatial frequency of a diffraction–limited lens is one over the Sparrow distance, or

Fclens = 1 / lambda * N, where lambda is the wavelength of the light, and N is the f-stop.

Now let’s turn our attention to the upper spatial frequency a sensor can resolve. A consequence of the Nyquist criterion is that the spatial frequency cutoff of a sensor is

Fcsensor = 1 / (2 * pitch) where pitch is the sensel pitch.

In the ground imaging world (satellites, aerial photography) a digital optical capture device is “balanced” when the two frequencies are equal. There’s another, less binary way of looking at the question of lens and sensor balance; the Q of an imaging system is defined as:

Q = 2 * Fcsensor / Fclens

So

Q = 2 * (lambda * N) / (2 * pitch) = lamda * N / pitch

I’ve included some references at the end of this post. For systems that image a range of frequencies, like normal photographic digital cameras, the convention is to set lambda equal to the average of the highest and lowest imaged wavelength, or, for systems imaging visible light,

lambda = (380 + 720) / 2 = 550 nanometers.

Since we will be working in micrometers, let’s call it half a micrometer.

At a Q of 2, the system is balanced; the diffraction of the lens is just enough to provide the proper AA filter to the sensor. For Q’s of greater than that, the sensor can resolve more detail than the lens can supply, and for Q’s of less than that, the lens can supply more detail than the sensor can resolve, and aliasing can result.

High-resolution full frame cameras like the Nikon D800e and the Sony alpha 7R have pixel pitches of slightly finer than 5 micrometers.

The Q of a diffraction-limited f/11 system with a sensor pitch of 5 micrometers is:

Q = (0.5 * 11) / * 5 = 5.5/5 = 1.1

The sensor can’t resolve all of the lens’s detail.

How fine a pitch would we need to have a balanced system with a diffraction-limited f/11 lens?

pitch = lambda * N /Q = 0.5 * 11 / 2 = 2.75 micrometers.

As you can see, using Q = 2 as a definition of balance between the lens and the sensor yields desirable pixel pitches that are much finer than what are commonly thought to be adequate.

Next, a discussion of the consequences of removing some of the assumptions.

Reference:

http://kiss.caltech.edu/workshops/gazing/presentations/green.pdf