How do graduated ND filters actually work?

Question

My question stems from the fact that the surface of the filter is completely out of focus.

My understanding is that light from any one point of the subject hits the lens all over... some of it passing through the darker part and some of it passing through the lighter part of the filter. Assuming the subject is in focus, all this light goes back to a single point on the sensor.

So it would seem logical to me that each point on the sensor will receive an equal amount of light that had passed through each of the halves (for simplicity sake) of the graduated filter.... and therefore I derive that logically the filter will not function differently from a uniform ND filter.

Obviously I am wrong, as evident by many successful photos of sunsets. Do lenses not use all of the light that hits the front element? If not, where does it all go?

Answer 1

Your intuition is essentially correct but there are a few important points.

1. When the lens is stopped right down, only light heading for the centre of the front element will make it into the picture, so the whole front element isn't used for every point of light hitting the sensor (though all of it is used for some point of light).

2. Even when the aperture fully open and light is passing through the whole front element for every pixel, the filter will be a small but significant distance in front of the lens.

Time for some crudely handdrawn diagrams. For the first case, it's easy to see that if the filter is placed anywhere except dead level with the iris, then it's going to block light coming from the top of the scene but not the bottom:

Now consider the second case. The lens is wide open, and bar the effects of vignetting, each point of light from an object in focus spreads out, hitting the whole front element before being focussed down to a dot. This suggests that light from the top of the object and bottom of the object both pass through the same amount of ND filter, meaning there is no graduated effect. However the filter wont be mounted flush with the front element:

When slightly in front there will be a angle at which light just skirts under the filter when coming from the bottom of the scene, whilst light from the top is partially filtered out, leading to a difference in brightness between top and bottom.

However if the lens is very fast, and the filter is mounted quite close to the lens then the grad ND really wont work, as suggested by your question. To prove this I've taped a black card over a 50mm f/1.4 lens so it's just above the centre, covering almost half of the front element:

That's wide open, pointing at the sky (focus distance 3m). There is a slight vignetting visible at the top, but nowhere near the effect you'd expect from what is essentially a graduated filter that goes abruptly from 0 to 100% filtration! Here's the same shot at f/11:

Answer 2

Not every pixel will use light from all light reaching the lens.

Your assumption about all light going to a single point is correct only if you take a single point from your object. If you take another adjacent point from your subject, that point will be projected next to the previous point. In the end you have a focus plane (projection), not just a single dot.

In other words, central pixel of your image will use mainly light coming from the center of a lens as border pixels will use light from the edges of a lens. In that way part of the image that covered by grad-ND will be darker.