Here are some numbers. Numbers are based on assumption, therefore, please read assumptions.
Assumptions:
- LED is a point source (i.e. its dimensions are negligible comparing to distance)
- LED is at the height H=14"=355 mm above water level at the center of a fishtank. Tank width W=24"=610 mm
- Eye pupil diameter D= 7 mm (fully dilated pupil - what's required by laser safety standard). This correspond to unrealistic case when you sit in completely dark room and LED is turned on all the sudden. Makes sense for lasers, but not much sense in our case. In reality, pupil diameter, when looking at the tank would be 4-5 mm
- Distance between eye and LED
- LED is a Lambertian source. Of course, most LED has front lens or other type of light concentrators. It results in more "useful" light for the tank and less leakage into the eye. So, Lambertian would be the worst case.
- Will use radiometric quantities as opposed to photometric, since energy is what matters
- Eye is right next to fishtank front glass, at its top.
- Eye is turned up, so it looks right at the LED. It makes thing worse (more energy). We can omit cosine
- LED electrical power P = 1W in blue part of spectrum. Results should be multiplied for real LED values
- Blue LED efficiency b=24%, i.e. for each 1W of electrical power, the blue LED produces 0.24W of light energy. Neglecting losses in power supply (based on Luminus PT-40)
- Radiation is not focused at the eye retina, i.e. this is not a laser source
Calculations:
- Angle between LED normal (vertical) and viewing direction: a=atan(W/(2H))=atan(305/355)=0.71 rad = 41 degrees.
- Distance between LED and the eye L=sqrt(H^2+(W/2)^2)=468 mm = 18.5"
- Pupil area: S=pi*D^2/4=39 mm2
- Solid angle subtended by the eye from LED: dA=S/L^2=1.75 E-4 steradian = (approximately) 0.0002 sr
- LED radiant flux: F=b*P=0.24W
- LED is lamebrtian, so its maximal radiant intensity (similar to photometric luminous intensity) is at normal direction: Imax=F/pi=0.24/3.14=0.076 W/sr
- Radiant intensity into direction at eye : I=Imax*cos(a)=0.076*cos(40)=0.058 W/sr
- Radiant flux into eye: dF=dA*I=0.058*1.75e-4=1.0e-5 W=10 uW (microwatt)
So, per each "blue" LED watt human eye receives 10 uW into pupil.
Estimating MPE (maximum permissible exposure - IEC-60825-1 standard, is a incredible mess, so we use guide:
http://onlinelibrary.wiley.com/doi/10.1002/9781118688977.app1/pdf, Table A4,
wavelength 0.4-0.5 um, exposure time = 3000 s, Photochemical damage to retina, retinal burn ): MPE=1.e-4 W/cm2 (power density).
Power density (W/cm2): U=dF/S (S - in cm2 now) = 1.e-5/0.39=2.64e-5 W/cm2)
So, we're U/MPE=0.26, i.e.
you can safely stare into 4W blue LED for 1 hr
For occidental short exposures (10s), MPE=1.8e-3*t^0.75 (J/cm2)= approximately 1e-2 J/cm2.
Our case Q=U*t=2.64e-5*10=2.64e-4 J/cm2
So, Q/MPE=2.64e-4/1.e-2=0.026, so
you can safely stare into 40W blue LED for 10s
Again, each case is different, the model is very simplified, however, it should give you some idea.