How do you determine focal length/wideness of fisheye conversion lenses?

I currently use a 0.7x adapter on a 28mm lens, but I want to go wider. I've seen a selection of different 0.*x lenses but with no indication to the focal length it provides. How do you determine the focal length/wideness of fisheye adapter lenses?

As Michael notes, screw-on wide angle adapters are usually marked as an area multiplier (unlike teleconverters which screw in behind the lens). So, you multiply your lens's focal length by the square root of the given multiplier, so for 0.7×, something like 0.84 × 28mm = (approximately) 23 or 24mm.

However, when you throw fisheye into the mix, it becomes more complicated. Our normal understanding of the relationship between field of view and focal length assumes a rectilinear projection — the regular, non-fisheye view of the world. If your add-on lens does more than make your normal lens wide angle but also converts the projection to fisheye, it's hard to say exactly what the effect will be.

• This is incorrect. Don't forget, screw on adapters use areal rather that linear magnification factors. Mar 12, 2019 at 0:01
• @MichaelC Huh. That makes total sense but I hadn't encountered that before. Mar 12, 2019 at 0:03
• @dodds — don't be so quick to accept answers :) Mar 12, 2019 at 0:03
• You must have forgotten that you had encountered this before. Mar 12, 2019 at 0:12
• @MichaelC Very likely! I learned, like, twenty years ago, to dismiss these things pretty much entirely as gimmicks. Mar 12, 2019 at 0:13

One thing to keep in mind about screw on lens adapters is that they base the magnification numbers on areal instead of linear measurement. This is unlike, for example, teleconverters or "crop factors" when we compare cameras with differently sized sensors that both use linear magnification ratios.

Compare this theoretical scenario:

We have a 100mm lens, a 1.4X teleconverter and a 1.4X screw-on adapter.

• With the 1.4X TC, our effective focal length will be 140mm (actually it should be 141mm, since a 1.4X TC is theoretically a √2X or 1.414213562...X TC)
• With the 1.4X screw-on adapter, our effective focal length will be 118mm. This is because the square root of 1.4 is 1.18. (If we use √2 instead of 1.4, it figures out to 118.9mm, or 119mm.)
• The same holds true with screw-on "wide angle" adapters. A 0.7X adapter increases the areal coverage of the lens by the reciprocal of 0.7, or 1.42X. This equates to a linear increase of 1.195X. So our effective focal length with a 100mm lens would be 84mm (83.666mm), rather than the intuitively expected 70mm! (Again, if we use the reciprocal of √2 instead of 0.7, it figures out to almost exactly 84mm.)

To apply this to any screw-on adapter, take the square root of the "X" number (i.e. 0.7X) and multiply it by the focal length of the lens to which you are attaching it.

For your 28mm lens (assuming it is exactly 28mm, which may not be the case, even when focused at infinity), a 0.7X "wide angle adapter" will result in an effective focal length of 23.43mm. If we use the reciprocal of √2 instead of 0.7, it figures out to 23.55mm. Keep in mind that none of these measurements are precise enough to worry about the difference.

• Your 28mm lens could be anywhere from about 26mm to 31mm in actual focal length when focused at infinity. When focused at closer distances, the actual focal length usually changes. You can test this by looking through the camera's viewfinder while changing the focus distance and observing whether the field of view gets smaller, stays the same, or gets larger. One reason "cinema" lenses cost so much is that they are designed so that the field of view does not change as the focus distance is changed.
• The screw-on 0.7X "wide angle adapter" is probably not exactly either 0.7X or 1/√2X. Even if it gives that exact magnification in the center of the image field, it probably varies one way or the other on the edges. This is what we call geometric distortion - a lens magnifies by a slightly different ratio at the edges as compared to the center.
• Although many screw-on "wide angle adapters" appear to give a fisheye projection, this is more than likely a result of barrel distortion than an actual fisheye projection. If you screw an adapter onto a rectilinear lens, you're getting a distorted rectilinear projection rather than an actual fisheye projection.

Rectilinear and fisheye projections are different ways in which a lens projects an image of a three dimensional world onto a flat, two-dimensional imaging plane. It's more than just the angle of view rendered by a lens. Canon, for example, makes rectilinear lenses for full frame cameras as wide as 11mm in the EF 11-24mm f/4 L ultra-wide angle zoom lens. They also make an EF 8-15mm f/4 L fisheye lens. The projection of the EF 11-24mm f/4 L set to 15mm looks very different than the projection of the EF 8-15mm f/4 L Fisheye set to 15mm when both are used on the same camera.

• When you say 144mm don’t you really mean 141mm. Square root of two is 1.414 not 1.44. Mar 12, 2019 at 13:52
• @EricShain Yeah, it was a slip of the fingers. Mar 12, 2019 at 17:16