Determining camera lens focus step size

[Chuck Perrygo]

If one is looking into motorizing their telescope focuser, determining the focus shift per motor step is easy on telescopes where the location of the focal plane is fixed and the focuser directly moves the camera. One simply measures how much the focuser tube moves for one turn of the focus knob. However, if one wants to motorize a camera lens, a different approach may be required. If the camera lens moves all of the internal optical components together as a single assembly when focusing, then one can directly measure the change in lens length if one has a dial micrometer of sufficient length. For other situations, the method presented below may be of help. Applicable to any camera lens with focus-distance markings, the approach uses a few simple measurements to determine how much a lens’ focus plane shifts per motor step.

The key principle is the simple-lens equation, which models a camera lens’ actual system of multiple lenses, necessary to produce a wide, flat and aberration-free image, with a single ideal thin lens. The simple lens equation relates the lens focal length f to object and image distances o and i as follows:

 1/f = 1/i + 1/o

The object distance o is the distance from the lens to the object being viewed. The image distance i is the distance from the lens to the image plane, or focus point, which for a focused image is coincident with the camera’s detector.

The simple lens equation can be rearranged to give:

 i = 1/(1/f – 1/o)

When focused on a star, the value of o is infinity and i equals the focal length f. When focused on an object closer than infinity, i is larger than the lens focal length, requiring the lens to be moved farther away from the detector to achieve focus. This shifting of the lens position is visible when the lens focus ring is turned, unless your particular lens uses internal focusing optics.

Lenses capable of true manual focusing (not to be confused with focus-by-wire lenses) have focus-distance marks engraved somewhere on the lens housing, typically next to the focus ring. Starting at infinity focus, let A be the value of the next adjacent focus mark, for example, the “20” m mark on the lens in the photo. Using the simple lens equation, the distance S the image plane shifts when refocusing the lens from infinity to A is i - f, where i is calculated for an object distance of A:

 S = i - f = 1/(1/f – 1/A) - f

Let D be the diameter of the lens housing with the focus distance marks and L be the distance between the infinity focus mark and focus mark A. Note that L is the actual distance on your lens that you measure using a ruler. One full rotation of the focus ring shifts the image plane by the ratio of the lens housing circumference to distance L, or 3.14 DS/L.

As an example, my old Zeiss Contax lens in the photo has the following characteristics:

f = 135 mm
A = 20 m
L = 6.9 mm
D = 60 mm

Calculating S yields:

 S  =  1/(1/f – 1/A) - f
    =  1/(1/135 mm - 1/20,000 mm)  - 135 mm
    =  0.92 mm

Being a lens design where all of the internal optical elements move together, I was able to measure the change in lens length directly. This measurement matched the above calculated value of 0.92 mm.

One full rotation of the focus ring shifts the image plane by:

  3.14 DS/L  =  3.14 x 60mm x 0.92 mm /6.9 mm  =  25.1 mm

As a check, I repeated the above calculation for the 10-m distance mark and obtained the following results:

 L = 14.3 mm
 S = 1.847 mm
 Focus shift = 24.4 mm/focus ring rev

This result differs by 3% from the value obtained using the 20-m mark – more than close enough for our purpose. Two such calculations can be expected to differ some due to errors in the as-manufactured location of the focus distance marks and errors in measuring L.

Assume now that I rotate the lens focus ring using a belt driven by a 15-mm diameter pulley mounted on a 400 step/rev stepper motor. One full rotation of the stepper motor shifts the image plane in proportion to the ratio of the drive pulley diameter to the focus ring diameter, which is 68 mm for this lens:

 25 mm x 15/68  =  5.5 mm/motor rev

Finally, one motor step shifts the focus by:

 (5.5 mm/motor rev)/(400 steps/motor rev) = 14 um/step

This focus step size can now be compared to the width of the lens’ critical focus zone to decide if one thinks it is acceptable or if more steps/rev are required.

[Scott Mitchell]

Chuck, this is super helpful. I've run the numbers for my 135mm lens and will be testing it out with an artificial star here in the next day or so. Hopefully, clear skies will return sometime soon as well. I'll report back on how it works.