This is further to converting the exposure recommendation of a flash exposure meter (incident mode) to the luminous energy density, denominated in lux•seconds. A later stage will be to convert that to radiant energy density in J/m^2.
The following derivation has been done "on the fly", and has not been audited. I will of course check it out as soon as I can. Please consider the intent of this note to show the train of thought and the procedure needed, not to report a reliable result. All work is done in SI units (I do not always take the time to mention all the units for each equation).
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We begin by assuming that the flash meter used practices this "exposure strategy":
When the exposure index setting of the meter (the "ISO setting") is set to the
ISO speed of the presumed camera (not the
ISO SOS rating), then with the f-number of the camera set to the value recommended by the meter after observing a test flash burst, an actual exposure of a test card with 100% reflectance, illuminated by an identical flash burst, will result in a photometric exposure, H, on the camera sensor of 1/2 stop below the "saturation" photometric exposure, Hsat. (
Photometric exposure is the illuminance-time product on the sensor.)
Let's do our work assuming that the exposure index on the meter is set to ISO 100.
Based on interpretation of the ISO standard for the ISO speed of a digital camera, we may presume that the Hsat of the camera is given by:
Hsat = 78/S******(1)
where S is the ISO speed setting and Hsat is in lux•seconds.
In this next part (for conciseness) I will work with luminance and illuminance, not their time products. We can easily switch to the latter domain when we are ready. (This work is for the moment generic, not pertaining to flash exposure.)
Let E1 be the illuminance upon a Lambertian test object of reflectance R (R running from 0-1). Then the luminance, L1, exhibited by the object is given by:
L1=(E1*R)/pi (2)
When that test object is regarded by a camera with f number N, the resulting illuminance on the focal plane, E2, is given approximately by:
E2=(pi/4)*(1/N^2)*L1 (3)
Combining equations 2 and 3 we get:
E2=(1/4*N^2)*E1*R1 (4)
Now, for the assumption of an ISO speed of ISO 100 for the camera, Hsat is given by:
Hsat = 78/100 (5)
If we assume some arbitrary exposure time (remember, we are not yet actually working with flash bursts), then the saturation illuminance on the sensor, Esat, is given by:
Esat=Hsat/t (6)
Thus:
Esat=0.78/t (7)
Working back through equation 4, and now assuming a target with R=1, we find the illuminance, E3, needed on the target to produce Esat on the sensor (assuming ISO 100) is given by:
E3 = (0.78/t)*4*N^2 (8)
Now to convert illuminance to illuminance-time product (in which we are interested for a flash burst), we just multiply both sides by t, thus (I will use the symbol "H" for the illuminance-time product, H3 in this case):
H3=0.78*4*N^2 (9)
or,
H3=3.12*N^2 (10)
That is:
With the incident light exposure meter set to an exposure index of ISO 100, the illuminance-time product* of the flash burst (in lux•seconds) at the meter location is given by 3.12*N^2, where N is the f-number recommended by the meter for exposure under that flash burst.
*This corresponds to the luminous energy density
(I don't know at the moment if that is even credible - I'll do a credibility test as soon as possible).
Well, I'll actually wake up now, have our nourishing Sunday breakfast (three kinds of fresh fruit, bacon, scrambled eggs, and hashed-brown potatoes, along with 12 IU of fast-acting insulin to support its processing), accompanied by a digest of world news from the New York Times and two local dailies (expecting a pic of mine in one of them today), and then carefully audit my result above.
Best regards,
Doug