I am not a physicist or engineer, and that is why I turn to you for help. This is a new post from an old topic. This is a repost because important information was absent from my first post on this topic. I have an infrared LED that emits light at 940 nm, and the manufacturer claims it's about 1000 mW/sr. While the light is not visible to human eyes, light of this wavelength can damage human eyes if it is strong enough. The following equation can help me determine if this LED represents a hazard. I recognize that the LED is probably not a hazard but I need science to back that statement.
Equation One:
E_IR_only (W/m^2) = SUM (from 770 nm to 3000 nm) E_lambda x delta_lambda
Equation One, Definition of terms
E_IR_only = irradiance in W/m^2;
E_lambda = spectral irradiance (W * m^-2 * um^-1);
delta_lambda = spectral bandwidth (um);Note: manufacturer reported spectral bandwidth = 30 nm.
Step One:
I first calculated the energy of a single photon of light at 940 nm. I am not being careful with my significant figures…apologies to the community.
E940 = hc/lambda = (6.626e-34 Js)(2.998e8 m/s)/(9.4e-7 m) = 2.113e-19 J
Step Two:
From the spec sheet I believe the radiant power of this LED is 40 mW or 0.04 W, and from that I can calculate the flux, phi.
phi = radiant power/E940 = 0.04 W / 2.113e-19 J = 1.893e17 photons/m^2
Step Three:
The next step is where I get confused. Every source of information I can find says that spectral irradiance (E_lambda from Equation One) is defined as:
F_lambda (W * m^-2 * um^-1) = phi * E940 * (1/delta_lambda)
In my case that would simplify to F(940) = radiant power / delta_lambda.
F(940 nm) = 0.04 W / 3e-8 m ~ 1.33e6 W m^-3 or 1.3 W * m^-2 * um^-1.
Question:
I feel my calculation is incorrect because if I apply the equation I need, E_IR_only = E_lamba x delta_lambda = 1.3 x 30 nm I just end up with radiant power again, or 0.04 W. Clearly, 0.04 W is not SOME_NUMBER W/m^2. Am I supposed to used the spectral bandwidth (delta_lambda, 30 nm) in the calculation for spectral irradiance, or am I supposed to use the specified wavelength (940 nm)?
I've made an error somewhere, and I keep going in mathematical circles. Any help the community can provide is very welcome.
Best Answer
Well it's nice you have a link to the LED. You can see all the data you need on the second page. (The good thing about an IR led is you don't have to deal with lumens :^) You can see the total power is 40 mW. And the radiant intensity is only 170 mW/sr at 100 mA of current. (you've got to pulse it quickly to get more.)
You can also see that the beam (full) width is 20 degree's (+/-10 deg.)
One way to think of steradians is as square degrees. This is OK
So a radian is about 57 degrees, and a square radian (Steradian) is about (57)^2 degree's squared. So as a check with a total power of 40 mW over (20 deg)^2 is about 40 mW *(57/20)^2 = ~325 mW/sr. That's about twice the listed number... which is fine because the width is the half power point and the intensity is a measure of just the central maximum. (the 40 mW is not constant over the beam profile so my quick calculation over estimates the intensity.)