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.
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.
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
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
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.
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.