Lens math? Image circle ⭕️ calculation

[pchaplo]

Do you love optics math calculation problems? Here’s one!

Givens: 150mm APO-Sironar-S at f/22
Infinity focus
4x5 film in typical plastic film holder
Edit: no shifts!

At what distance behind the lens would the image (on the 4x5 film) begin vignetted by a metal disc with an circular opening of 80mm diameter?

Application: For example. If I use an 80mm I.D. tube to make an extension (with the lens mounted at one end), how long (maximum length) can the tube be before it vignettes the image?

For extra points: the increments of length to build the tube are sections of he following lengths: 20mm, 30mm, and 50mm. Multiples of any length are allowed. What would be the optimal combination?

Do you know the math? I’m thinking this is a slice of a cone at a certain distance problem, with the corners of the 5x4 inch (diagonal) the limit for the cone diameter. Anyone brushed-up on their calc?

[Graham Patterson]

Simplify it further. It's a triangle. Base of the triangle is half the image circle diameter. The base of a sub-triangle is 80mm/2, with a height of the tube length. Solve for the angle, then use that angle and the base to get the new distance to the film. The triangles have the same angles, and the sides are proportional.

Now, if you want some real fun, try placing the lens axis off-center for the film!

[Pere Casals]

At 78mm from rear nodal point, then if you want you can add the nodal point vs flange distance. The critical projection section is the sheet's diagonal, as vigneting starts in the corners. With that triangle you can calculate any case. This is because we have an small apeture... if it was a large aperture we should use pupils, I guess.

[Nodda Duma]

Here’s the more accurate answer


First part of problem. The f/# defines the cone angle in the image space. You can back out the answer that the marginal ray height at the 150mm f/22 lens is 3.41mm (paraxial approximation fine for this f/#).

See first pic showing calculation



From there the marginal ray travels to the corner of the film, 3.05” or 77.47mm off-axis for 4x5.

Similar triangles means the distance from stop to the end of the 80mm diameter cylinder closest to the film plane is readily calculated. See second pic. You should measure from the film plane since you don’t know the actual stop position relative to the lens mount.

That distance is 75.89mm from film to the back of the tube. See second pic. Pere’s calculation is close, but small errors can be critical when designing system baffling to control stray light (one of the few optical design lessons I’ve learned the easy way), so it’s worth being accurate.



The calculations don’t account for shifts nor for closer focus than infinity ... that can be accommodated by incorporating magnification factor to determine marginal ray height and back focal distance.

[pchaplo]

Nodda,

Many thanks for the accurate calculations. I appreciate you taking the time to share this! Indeed the focus will be infinity with no shifts, therefore this info is immediately and practically useful to me.

Wishing you great light,
Paul

Here’s the more accurate answer


First part of problem. The f/# defines the cone angle in the image space. You can back out the answer that the marginal ray height at the 150mm f/22 lens is 3.41mm (paraxial approximation fine for this f/#).

See first pic showing calculation



From there the marginal ray travels to the corner of the film, 3.05” or 77.47mm off-axis for 4x5.

Similar triangles means the distance from stop to the end of the 80mm diameter cylinder closest to the film plane is readily calculated. See second pic. You should measure from the film plane since you don’t know the actual stop position relative to the lens mount.

That distance is 75.89mm from film to the back of the tube. See second pic. Pere’s calculation is close, but small errors can be critical when designing system baffling to control stray light (one of the few optical design lessons I’ve learned the easy way), so it’s worth being accurate.



The calculations don’t account for shifts nor for closer focus than infinity ... that can be accommodated by incorporating magnification factor to determine marginal ray height and back focal distance.