What's the actual physics behind the optical apodization?

Question

Canon seems to plan to introduce a 50mm with apodization for a more natural bokeh [1], but it is not the first in photography [2]. Wikipedia article is only an introduction [2], and does not develop the physics behind.

1. What is the quantitative physics (mathematics) behind ?
2. Which physical quantity can we build to quantify this esthetical quality manifested in the bokeh ?
3. How do we achieve this in terms of engineering with a piece of glass ?

[1] https://www.canonrumors.com/patent-canon-ef-50mm-f-1-4-with-apodization-filter/

[2] https://en.wikipedia.org/wiki/Apodization

Answer 1

Not physics, but the math of filtering, i.e. filter theory in the frequency domain. Specifically you would need (the textbook result of) a 2D Laplace transform.

If you want a nicely blurred background with little disturbing detail then you'll want the lens to act as a good low-pass filter. Normally the 2D impulse response for out-of-focus objects is a circle with a flat top in the brightness direction. The corresponding frequency response is a 2D sin(r)/r function, which is a bad low-pass filter. In particular the stop band is not deep enough, which indeed we see in the form of "busy" bokeh circles with sharp edges.

A better filter with a deeper stop band is a Gaussian filter, where both the impulse response and its frequency transform have a 2D Gaussian shape, like exp(-r^2). High frequencies will be completely suppressed, no sharp circles will be seen.

The way to realize a Gaussian aperture is by painting the edges of the aperture darker and darker. The expensive edges of a large aperture lens are thus hardly used ! This can be done with ink on a glass plate near the aperture, or with a concave (negative) lens made from dark glass close placed to the aperture, or by opening and closing an iris = aperture or central shutter as a function of time so that the center passes more light than the edges. (Or try aperture stacking !)

It is important to approach a Gaussian profile because other shapes like a linear profile (tipi tent shaped) do not nearly yield as good a low-pass filter with a deep stop-band.

See also How does an apodization filter improve bokeh? . You asked abouth the math.

Answer 2

Apodization filters have been around photography for a long time. Often they are called "soft focus" filters. Lenses that include such filter are sometimes called "soft focus" lenses.

Lower quality, non-apodizing "soft focus" filters are basically diffusion filters that also reduce contrast and result in softening the overall image. Higher end "soft focus" filters, such as the Zeiss Softar line, let center light through undiffused and only diffuse the light entering the edges of the lens.

This is the basic concept of apodization: Allow all of the more collimated light to pass through the center of the lens while allowing some, but not all, of the less collimated light from the edge of the lense to pass. Stopping down a normal lens would only allow the light through the center, but not allow a portion of the more diffused light from the edges of the lens.

Rather than use material that diffuses light, such as some soft focus filters do, apodizing lenses use either a ring of neutral density material that is more dense near the edges than at the boundary nearer the center where all light is allowed to pass or they use plates with holes of various sizes and patterns near the edges.

The classic early soft focus lens was the Rodenstock Imagon made to fit in the lens board of a Large Format view camera. Introduced in 1930/31, it was in production until the 1990s. It had a set of different "sink strainer" aperture diaphragms, a/k/a apodization filters, that included a main opening in the middle that produced the relatively sharp image. Each "sink strainer" also had smaller holes of varying sizes and patterns around the edges of the various diaphragms that could "dial in" the amount of soft focus that was blended with the sharp image projected through the center hole in the diaphragm.

Combined with uncorrected spherical aberration/field curvature that made the focus distance at the edges closer to the camera than the focus distance at the center of the lens' optical axis, this allowed the proportions of 'in focus' versus 'soft focus' light to be controlled.

Fuji also produced such a lens for 135 format (35mm) cameras way back when, although the pattern was fixed and inside the lens, rather than on the front of a Large Format/Medium Format lens such as the Rodenstock.

Other soft focus lenses, such as the discontinued Canon EF 135mm f/2.8 Soft Focus (with Softfocus mechanism) used different mechanics to do the same thing: combine in-focus light coming through the center of a lens that had uncorrected/undercorrected spherical aberration/field curvature with softer light entering through the edges of the lens in a way which could be more variably controlled than by a traditional lens design with a single opening in the aperture diaphragm.

In addition, one can simulate the look digitally by placing layers with varying levels of simulated spherical aberration applied to them in various amounts of transparency/opacity on top of the in-focus base image.