The guide number of a flash is determined as the F-number multiplied by the distance that gives a proper exposure. The guide number is different for different ISO sensitivities, but they are typically given for ISO 100, and when calculating a guide number for e.g. ISO 3200, you calculate:
GN3200 = sqrt(3200/100)*GN100
and similarly for other ISO sensitivities.
Smartphone flash guide numbers are peculiar in that they are not only affected by the ISO number but also affected by the exposure time. For example if you have GN_{100,10} for ISO 100 and 1/10 second exposure time, and want to know GN_{400,5} for ISO 400 and 1/5 second exposure time, you calculate:
GN_{400,5} = sqrt(10/5)*sqrt(400/100)*GN_{100,10}
Let us determine GN_{100,10} first and then other exposure times and ISO sensitivities can be calculated from that.
Let us fix the subject to be 1 meter away from the smartphone camera and let us consider an f/1 lens (fairly unusual for smartphones but F-number will be accounted for later), which would give a guide number of 1 meter. A field of view calculator gives image dimension of 1.2857 x 0.85714 units for a subject 1 unit away if the focal length is 28mm and the image format is 3:2 (usually smartphones might have a differently shaped image sensor, but this shouldn't affect the calculations much because the crop factor is based on comparing the diagonal).
We also need to know about exposure value. For f/1 lens and 1/10 second exposure time, the exposure value is
log(1.0^2 / 0.1)/log(2) = 3.3219
Also we know that exposure value of 0 corresponds to 2.5 lux (from that Wikipedia article), so exposure value of 3.3219 corresponds to 25 lux.
If the smartphone manufacturer has modified the LED optics to produce an even light field for the field of view of the camera lens, the smartphone flash produces an illuminance of
50 lm / (1.2857 m * 0.85714 m) = 45.371 lm/m^2 = 45.371 lux
So we have
45.371 lux / 25 lux = 1.8184
times as much light as we would need. This means guide number is not 1 meter as we initially assumed but rather
sqrt(1.8148) * 1 m = 1.3471 m
If the smartphone has a different ISO sensitivity (let's say 400), a different 35mm equivalent focal length (let's say 32 mm), a different exposure tame (let's say 1/5 s) and a different brightness LED (let's say 70 lm) then we can take into account these corrections to the guide number:
GN_{400,5,32mm,70lm} = 35mm/28mm * sqrt(70lm/50lm) * sqrt(400/100) * sqrt(10/5) * GN_{100,10,28mm,50lm}
= 4.1833 * GN_{100,10,28mm,50lm}
= 4.1833 * 1.3471 m
= 5.6353 m
Some comparisons for the guide number:
- A professional Canon EL-1 flash has an ISO100 guide number at 24mm of 19 m (this is measured, advertised is 27 m) and at 35mm of 24 m (this is measured, advertised is 32 m)
- To convert 24mm guide number from 24mm to 28mm (assuming a perfect light distribution), you would calculate
(28/24)^2*19 m = 25.9 m
- To convert 35mm gide number from 35mm to 28mm (assuming a perfect light distribution), you would calculate
(28/35)^2*24 m = 15.4 m
- A very small Canon flash EL-100 has an ISO100 guide number at 24mm of 21 m (advertised; measured would probably be 19/27 * 21 m = 14.8 m)
- To convert 24mm guide number from 24mm to 28mm (assuming a perfect light distribution), you would calculate
(28/24)^2*14.8 m = 20.1 m
- A typical DSLR built-in flash has an ISO100 guide number of 12 m; not sure if this is measured or advertised, but if it's advertised the measured probably would be 19/27 * 12 m = 8.4 m as advertised guide numbers are typically much larger than measured guide numbers
Compared to these, the smartphone guide number of 1.3471 m is very poor and also to achieve that 1.3471 m, you need 1/10 second exposure time which means a shaken picture unless the smartphone is equipped with an image stabilizer, and also may mean you get motion blur unless the subject is perfectly still. So the smartphone flash is only useful at very near ranges in a very dark environment such as levels of light during streetlight periods, or in a very dark room. Also in macro photography in general home light levels the LED of a smartphone may be helpful.
