Units of Transmittance: Defining & Explaining

[mikeph]

I don't get this. Everywhere I look, the units of transmittance of light are dimensionless.

I get that it's a ratio of incident to transmitted radiation, t(f) = I(f)/T(f), and these two have the same units.

However, light intensity is always properly defined as the power per metre squared of radiation in a given spectral range, f + df, then the intensity is ∫I(f)df. If you want an intensity at a single frequency, you'll never find it because you will never detect a photon at EXACTLY f.

This is the same as probability distributions. The pdf of a distribution is not a probability itself, it must be integrated over probability space. In the same way, the spectral intensity is not an intensity until it is also integrated.

So that leaves me thinking, what on Earth is the transmittance?

In my reasoning we should still define it in terms of a spectral range, and say the transmittance between f and df is t(f)df. Then the units are still inverse frequency because t is the ratio (dimensionless) and df has dimensions of frequency.


Anyone follow? Would love a yes/no answer!

 

[SteamKing]

Transmittance is defined as the fraction of light which passes thru a sample:
http://en.wikipedia.org/wiki/Transmittance

Same units in : same units out, therefore it is dimensionless.

Equivalently, what are the units of mechanical efficiency?

 

[Simon Bridge]

Anyone follow? Would love a yes/no answer!

"Anyone follow?" Is the only question you asked which can be answered "yes" or "no". The answer to that question is, "yes".

The other two questions have been answered above ;)
Note: if you don't know what something is, then it is not a good idea to try defining it.

 

[mikeph]

Thanks for the replies but neither helps address my confusion. Does the probability analogy not make sense?

Nobody I know talks about "fractions of light", they talk about spectral irradiance. I suppose if the transmittance is not a ratio of spectral irradiances, then what is? Because this is the ratio I am interested in.

 

[Simon Bridge]

The fraction of light through a medium would be the probability of an incident photon being transmitted or the ratio of the transmitted to incident intensity (and irradiance) of the light. These are equivalent statements.

You can do it just as well with spectral irradiance - in which case you have to specify the bandwidth as well - and the dimensions still cancel out. It looks to me like the dimensional analysis in post #1 is the confusion. Go through it again, step by step, slowly, and pedantically - careful to focus on what it is you are trying to find the dimensions of.

You clearly have the math - and are aware of the definitions of the terms - so I don't see what your difficulty is. Perhaps if you tell us what you are trying to use this for we'll see what is going on?
What's the context?