# Radiance at a nearly perpendicular angle is almost infinite?

As far as I understand, radiance is a measure of the "intensity" of light in a direction defined as a differential cone . It's measure is Flux per solid angle per projected area. $$L = \frac{d\Phi}{d\omega dA^\perp} = \frac{d\Phi}{d\omega dAcos(\theta)}$$. Using this definition, it seems like as the $$\theta$$ angle increases, radiance increases, thus light is somehow more "intense" in a direction nearly perpendicular to the normal at point p. This contradicts my intuition.
Where am I interpreting this wrong?

• It should be $L = \frac{d\phi}{d\omega dAcos(\theta)}$ Feb 22 '17 at 13:54
• Yeah, sorry, fixed that now. Feb 22 '17 at 14:14
• I've posted an answer to this, but have been thinking about it and think I can see what you might be getting at. If there was some differential flux (even a teeny weeny bit) at a grazing angle and $\theta$ was very close to $90°$, then the radiance would be very large because of the very small denominator. Is this what you're getting at? If so, I'm not sure what the explanation would be and wonder "is there any surfaces that would have a significant $d\phi$ at a grazing angles?" Feb 22 '17 at 20:11

It's only more intense if $d\phi$ remains constant. However, for light source with constant luminance the flux is a function of $cos(\theta)$, i.e. depends on the light's angle of incidence with the surface and cancels out the term.

Edit: I feel the need to clarify this a bit because you say that:

thus light is somehow more "intense"

It's not that the "light is somehow more intense", but rather that the light has to be more intense to have same flux in grazing angles as when it's parallel to the surface. If you think of flux as particles hitting to this differential surface, the number of particles hitting the surface is smaller more perpendicular the light gets to the surface. So you have to throw more particles at the surface (i.e. increase the radiance) for the same flux until when the surface is completely perpendicular to the light, none of the particles hit the surface, thus the radiance goes to infinity.

• I know calculating irradiance, for example, cancels out the term, but this doesn't answer my question regarding radiance itself. More precisely, how can it approach infinity when theta is approaching 90. Feb 22 '17 at 17:53
• Because if the flux stays constant, you need infinite radiance that there's any flux through a surface that's perpendicular to the light source Feb 22 '17 at 18:05
• It makes sense that the light "has" to be more intense in this context, indeed. Feb 22 '17 at 21:25

Remember radiance corresponds to a particular direction. When the equation is considering directions quasi parallel to the surface normal, the differential flux (the numerator) will typically be a much greater quantity than for grazing angles. The numerator is not remaining constant.