What is the theoretical relation between zoom length and guide number for a flash?

The point of a guide number is to specify the relation between aperture number and distance. For zoomable flashes there are still extensive tables considering ISO number and zoom length. The relation with ISO is straightforward (the square root of the ISO number factors into the guide number). The relation with zoom length tends to be inscrutable. At wider zoom settings, zoom length and light angle are not fully inverse (should be 2 arctan (18mm/f) horizontally and 2 arctan (12mm/f) vertically with f being the 35mm equivalent). But at least at longer zoom levels, one would imagine a somewhat predictable relationship.

Assuming that the zoom head makes best use of the available light output (and for the sake of simplicity we are zoomed far enough that arctan(x)=x is a reasonable approximation) what would be the expected effect on the guide number? Whatever I come up with seems so far from the actual tables that I seriously doubt my assumptions.

• How do you come to these asumptions? and why do you need the guide number so exact? I think your look on photography is a more scientific look than an artistic look, which is ok but the numbers are just numbers and they will only give a course guidance and are not dependend on the situation in my opinion... Mar 4, 2019 at 13:04
• @LuZel: what assumptions? The formulae for the angles are simple trigonometry. A constant zoom length makes for a constant spherical angle and makes for a straightforward relation between aperture and distance: the same amount of light reaches the scene at different distances, but the returned amount diminishes with inverse square law. All that (and factoring the square root of ISO values in) are straightforward consequences of the assumptions needed to specify a "guide number" in the first place. They are not controversial. But the guide numbers for different zoom lengths elude me. Mar 4, 2019 at 13:32
• ok the formulas are clear when you think about them but why is the exact guide number so important for you? or do you just want to know them and the assumptions for yourself. because I personally have not needed them in my photography life. this would make it easier to find a answer which would satisfy you. Mar 4, 2019 at 13:36
• "Assuming that the zoom head makes best use of the available light output" should make an answer independent of my motivation, shouldn't it? Diffuse reflection is already being assumed for the normal guide number arithmetic. There is a large variety between tables: a theoretic optimum would give a hint at which zoom lengths the actually usable light yield could be improved for different flashes and how much they fall short. I have a 35mm specified flash that loses 2 aperture steps with a 24mm diffusor. Is that good, bad, to be expected? If it is bad, it may imply heat problems. Mar 4, 2019 at 14:19
• "Assuming that the zoom head makes best use of the available light output... " is a hopelessly idealistic assumption. That would require the reflector behind the flash lamp to change shape for each distance. That doesn't happen. Most speedlights don't change the shape of the reflector at all, they just move it forward or backwards. At best, the flash will make the best use of available light at a single distance. Mar 4, 2019 at 20:08

Imagining theoretically, twice the speedlight zoom mm would compute half size field dimensions with 1/4 the area coverage. Same flash power concentrated into 1/4 the area theoretically would be 4x brightness, which would be 2 EV increase at twice the zoom mm.

However, that simply does not happen (not even close).

The speedlight zoom is very simple, the reflector simply moves back and forth, and does not change curvature to focus. The front fresnel lens is the same constant at any zoom, it does not change focus. Plus the design includes excess safety factor to ensure complete coverage regardless of slight mis-aiming in hot shoe.

Real world, speedlight 4x zoom (like 24 mm to 105 mm) is typically near 2 EV increase. Which is also near 2x Guide Number (because Guide Number is lens fstop number x distance). 4x zoom might be 1.9x GN instead of 2, but 2x zoom might be 1.5x GN instead of 1.414x (not linear). I imagine every speedlight design varies a bit. It would be great if zoom could be computed with Guide Number, but that is simply not the real world. Zoom is Not a precise mechanism. Still a big plus even if a bit crude.

• GN for 24mm to 105mm (where applicable): Cullman CX35: 25 to 45. Regula Variant 740: 20 to 53. Metz MZ40-3: 28 to 50. The Regula is irregular in that it has separate heads both for wide angle and for zoom. But if you look only at the zoom head settings, it's at 70mm, 100mm, 135mm, 200mm (more or less a factor of 1.4 from one setting to the other) GN 50, 53, 58, 70. What's with the large jump at the end? Is just the light distribution subpar for the wider settings? Maybe I need to take a look at the coverage patterns to understand how zoom settings are getting labeled. Mar 4, 2019 at 16:04
• The lens field size can be easily computed for zoom focal lengths, but it's a stretch of the imagination to assume the flash field is the same, or necessarily even very similar. Flash coverage will be minimal at shortest specified zoom, and excessively wide at long zoom. The flash head is designed its one way, and then (in best case) various zoom guide numbers are actually measured (in controlled setting, i.e., in wide open space) to determine what they actually are. Mar 4, 2019 at 18:08
• In summary, this answer would be "theory should consider about 2EV win for doubling of focal length possible, in practice it's more like 1EV", correct? Mar 4, 2019 at 21:56
• Not sure we have theory to account for the actual design. :) 1 EV might be a rough estimate in the general case. IMO, maybe a bit more in first half of the zoom range, and a bit less at the long end. It's a shame speedlight zoom result cannot be computed for guide number and exposure, but it is not done.The flash manuals typically show how to exactly compute for ISO, and also aperture or power level can all be computed too (see my Guide Number calculator at scantips.com/lights/flashbasics1c.html ). But for zoom, the manuals only offer a chart of actual measured GN results. Mar 4, 2019 at 23:19

Imagine the rectangular area visible to the sensor to be lit uniformly with outside areas being unlit. Now you move the camera away doing a "Hitchcock zoom", keeping the same area filling the same amount of image and the zoom follows suit. The same amount of light (namely everything the flash gun produces) will emit from the same area and, given the same aperture number and assuming fully diffuse reflection, the same image will result.

In other words: guide number should be proportional to focal length since focal length grows proportionally with object distance if you keep the same image size.

This is so far from actual guide number tables that maintaining comparable efficiency across zoom ranges does not appear to usefully approximate common flash zoom mechanism performance.