Based on yours and @realChem@beehaw.org’s replies, as well as the links that you two provided, I think that I understood 1) why there’s a cursed unit there, 2) how to uncurse it, and 3) why it’s better to just leave it cursed.
In the formula Δτ = D√L (from Wikipedia), that “√L” is just a practical hack. The time differential doesn’t depend on the square root of the length itself, but of the number of events causing the light to disperse. Number of events is unitless, so is its square root.
So physics-wise a more accurate formula would be Δτ = k√E; where
k = some constant; probably universal??? measured in an unit of time (ps is fine)
E = number of events causing the diffraction
In turn, you can measure the number of events as the product of a parameter and length, like E=NL:
N = a parameter of the fibre, proportional to the concentration of junk and irregularities forcing diffraction, in km⁻¹ (or any the inverse of any other unit of length)
L = the actual length of the fibre
Feeding this back into the equation, you get Δτ = k√E → Δτ = k√(NL). You can expand the square root there to get Δτ = k√N√L if you want; and then with k√N = D, you’re back to the old formula - including the cursed ps/√km unit. And that shows why the practical hack is there on first place: why would you keep track of a constant and a parameter, if just a parameter is enough?
(By the way, thanks to both of you for your explanations.)
Based on yours and @realChem@beehaw.org’s replies, as well as the links that you two provided, I think that I understood 1) why there’s a cursed unit there, 2) how to uncurse it, and 3) why it’s better to just leave it cursed.
In the formula Δτ = D√L (from Wikipedia), that “√L” is just a practical hack. The time differential doesn’t depend on the square root of the length itself, but of the number of events causing the light to disperse. Number of events is unitless, so is its square root.
So physics-wise a more accurate formula would be Δτ = k√E; where
In turn, you can measure the number of events as the product of a parameter and length, like E=NL:
Feeding this back into the equation, you get Δτ = k√E → Δτ = k√(NL). You can expand the square root there to get Δτ = k√N√L if you want; and then with k√N = D, you’re back to the old formula - including the cursed ps/√km unit. And that shows why the practical hack is there on first place: why would you keep track of a constant and a parameter, if just a parameter is enough?
(By the way, thanks to both of you for your explanations.)