Blur size and bokeh character
The size of a distant blur disc follows the entrance pupil: focal length divided by the f-number. An ideal 50mm f/1 and the Canon 50mm f/1.0 L land at 50.0 mm and 50.0 mm. The diameters match. The discs do not.
A thin lens bends every ray at one plane. A real fast prime sends the beam through a stack of curved elements. The discs here are ray traces of that stack. They show three things a thin lens never has: a corner clipped by the barrel, an iris that turns into a polygon as it stops down, and a brightness profile from spherical aberration.
| Optical trait | Ideal thin lens | Traced patent lens |
|---|---|---|
| Corner, wide open | A circle. Center and corner match. | Clipped into a cat’s eye by the barrel. |
| Stopped down | Stays a circle. | Takes the polygon of the blades. |
| Inside the disc | Even from center to rim. | Soft, even, or bright at the rim. |
| Surface | Smooth. | Onion rings are not drawn. An apodized disc feathers at the rim. |
Cat’s eye and swirl
Wide open, a highlight in the center sends a round beam through the barrel. Toward the corner the beam enters at an angle. The front rim cuts one side and the rear rim cuts the other. What remains is an almond, a lemon, or a D. That is cat’s-eye bokeh.
The almond is shorter toward the middle of the picture and longer around it. A background full of those highlights rings the subject. That is the swirl of a Biotar such as the Helios 44-2 58mm f/2. Each drawing here is one highlight: center, 60% of the way out, and the 90% corner.
In this trace the Helios 44-2 58mm f/2 is 61% round at the corner at f/2. The Nikon NIKKOR Z 135mm f/1.8 S Plena is 100% round at f/1.8. Stop the Helios to f/4 and the corner comes back to 95%, because the iris is then smaller than the rims that were clipping the beam. A fast modern prime is not automatically round there: the Nikon NIKKOR Z 58mm f/0.95 S Noct is 55% at f/0.95, and the Sony FE 50mm F1.2 GM is 43% at f/1.2.
Shape from the aperture blades
Wide open, the blades sit out at the rim and the barrel sets the shape. Stop down, and the highlight takes the silhouette of the iris.
The Nikon 58mm f/1.2 Noct-NIKKOR has 7 straight blades, so the center disc becomes a heptagon. The Nikon NIKKOR Z 58mm f/0.95 S Noct has 11 rounded blades, so the same f-number stays nearer a circle. An even blade count draws that many long diffraction spikes on a street lamp. An odd count splits them into twice as many. This simulator draws the iris. It does not draw the spikes.
Compare the Z 58mm f/0.95 Noct with the 58mm f/1.2 Noct-NIKKOR
Creamy fade and bright edges
In an ideal lens, rays through the rim of the glass focus at the same distance as rays through the center. Spherical aberration shifts the rim rays. Behind the subject, that changes where the light sits inside the disc.
Under-corrected, the light gathers in the middle and feathers out. The Canon 50mm f/1.0 L traces as soft-edged at f/1. Even correction fills the disc and leaves a clean edge. The Nikon NIKKOR Z 135mm f/1.8 S Plena traces as even. Over-corrected, the light piles up on the rim, the soap-bubble edge. The Helios 44-2 58mm f/2 traces as bright-rimmed at f/2. The gradient is that class from the ray trace, not a photograph of a bulb.
Onion rings, aspheres and apodization
Aspherical elements let a fast lens stay smaller. The patent records the aspheric curve, and the trace follows that curve. It does not follow the faint concentric ridges a mold can leave on the glass. Those ridges, onion rings, are a manufacturing texture. They are outside this simulation, together with coatings and sample-to-sample tolerances.
The Nikon NIKKOR Z 58mm f/0.95 S Noct patent has 3 aspherical surfaces. The Sony FE 50mm F1.2 GM has 2. The discs still show the pupil and the spherical-aberration class, not grooves.
Apodization is a radial filter at the stop. It darkens the rim of the pupil, so a real disc fades instead of ending on a hard line. These discs use that weight, the same Gaussian as the 3D portrait. The Canon RF 85mm F1.2 L USM DS and the Canon RF 85mm F1.2 L USM share a 48% corner outline, and the DS filter feathers the rim of every disc. On the Canon RF 85mm F1.2 L USM DS the filter is about T2.2 at f/1.2. On the Fujifilm XF 56mm F1.2 R APD it is about T1.7. On the Minolta 135mm f/2.8 [T4.5] STF it is T4.5 at f/2.8.
Diagrams of the idea, not ray traces.
Also on this site: Lens compression, Crop factor, Depth of field, How the simulator works and Lens comparisons.
Questions
What is the difference between blur and bokeh?
Blur is how large the out-of-focus disc is. It follows the entrance pupil, focal length divided by the f-number. Bokeh is the shape of that disc and the light inside it: the corner cut, the aperture blades, and the brightness from spherical aberration.
What causes cat’s-eye bokeh?
Wide open, an oblique beam is clipped by the front and rear rims of the barrel. The surviving bundle is the overlap of two circles. Stopping down until the iris is smaller than those rims brings the corner back toward a circle. At f/2 the traced Helios 44-2 is 61% round in the 90% corner; at f/4 it is 95% round. The Nikon Z 135mm f/1.8 S Plena stays 100% round at f/1.8.
Why do background lights become polygons?
Stopped down, the highlight takes the shape of the iris. Straight blades make a crisp polygon. Rounded blades, and a higher count, stay nearer a circle. The Nikon 58mm f/1.2 Noct-NIKKOR has 7 straight blades. The Nikon Z 58mm f/0.95 S Noct has 11 rounded blades.
What makes a bokeh disc creamy or bright-edged?
Spherical aberration shifts the focus of the rim rays. Under-correction piles light in the center, so the rim feathers off. Over-correction piles light on the rim, the soap-bubble edge. In this trace the Canon EF 50mm f/1.0L is soft-edged at f/1, the Plena is even, and the Helios 44-2 is bright-rimmed at f/2.
Does this page show onion rings?
No. Onion rings are microscopic ridges left by molding or grinding an asphere. The trace follows the aspheric curve in the patent and does not add that texture. The radial fade of an apodization filter is drawn on the Canon RF 85mm F1.2 DS disc, using the same Gaussian as the 3D portrait.