It should be possible to calculate all of this theoretically, but it would be difficult. The main jewel of this tutorial is the 3.13 magic number, I'd call it the Warske constant or Kw
With this value he's letting us bypass all the complicated stuff and get straight to the point - relating a change in temperature to power output. Kw rolls into one number multiple values of emissivity and reflectance for several difference surfaces, as well as probably picking up the measurement error in the IR thermometer device. Kw might be true only for aluminum foil targets with one side painted of a size relatively close to 0.25 in^2, so I wouldnt know how this would scale to a patch of 1 m^2...
That said, I have done some online reading and I think I can shed some mathematical light on this. I would really appreciate if some of the smarter people here confirm or deny everything im about to say
By shining a laser energy on a target we heat it up. Black paint helps absorb the most energy, it looks black because it does not reflect any colors (wavelengths). A white surface is the opposite, it reflects back all of the light at all wavelengths. I think this property is called "reflectance". Any energy from the light not reflected back still has to go somewhere, so it manifests itself in the form of heat, increasing the temperature of the target. All objects at any temperature emit IR radiation, i think this is called "emissivity". There is a concept of a theoretical object called "blackbody", it emits all of its energy in the form of radiation, and has emissivity e=1. Flat black paint has a lower e, and i dont know what it is (maybe ~0.9), but this difference to e=1 is accounted for with Kw!
Now let's think about our lasers, targets and thermometers. The target is black on one side, very shiny on the other, and the target's temperature is being measured by the thermometer. The black side has a low reflectance, it absorbs most of the energy from the laser. This energy converts into heat and then gets emitted into the thermometer as IR. First, we start with no laser light, just the target floating infront of the sensor - it's at some temperature (room's) T1, and therefore emitts some IR (naturally) at power output P1 (watts). Second, we shoot the laser on the target and raise its temperature to T2 - the target now has much more energy going into it and start emitting at a higher output of P2. We now make an assumption that all of the energy going into the target also gets emitted out from it. This is not 100% true because there is no such thing as a true "blackbody", so in reality we should only get some fraction (80-90% prolly a good guess). We can now relate the power of IR radiated from the target by using the Stefan–Boltzmann law (
Stefan, see 3rd eq from top):
P = A * e * s * T^4
where
P is power of the IR output from the target,
e is emissivity of the target's surfaces,
s is the Stefan–Boltzmann constant (0.0000000567) (it's not 's' on Wiki, but i dont know how to type the real lowercase Sigma
),
T is temperature of the target in K (ex, 63.5F = 17.9C = 290.7K)
Our laser's output power is
P(laser) = P2 - P1.
A,
e,
s remain constant, the only change is in the T. Let's do the calculations for the 10F temp change Warske observed with a
P(laser) = 31mW =
0.031 W. From the pictures i can see that
T2 = 70.3F =
294.43K, so
T1 = 60.3F =
288.87K. The square target has side of 0.5in = 1.27cm = 0.0127 m, and area is 0.00016129 m^2; we have TWO sides of the target emitting IR, so
A =
0.00032258 m^2. Plug it all in:
P1 = 0.00032258 * 1 * 0.0000000567 * 288.87^4 =
0.1274 W
P2 = 0.00032258 * 1 * 0.0000000567 * 294.43^4 =
0.1374 W
P(laser) = P2 - P1 = 0.01 W =
10 mW
Hmm, something is not right, Warske got
31 mW here... Oh, that's right, we made a ton of assumptions along the way and pretended the target is a "blackbody"! Multiplying by
3.13 * 10 =
31.3 mW, that's why we need Warske's constant
So the real equation should be:
P(laser) = Kw * A * e * s * (T2^4 - T1^4)