# Formal mathematical definition of radiance and irradiance

Radiance is defined as $$L = \frac{\partial^2 \Phi}{\cos{\theta} \partial A \partial\Omega}$$ and irradiance as $$E = \frac{\partial \Phi}{\partial A},$$ where $$\Phi$$ is the radiant flux, $$\Omega$$ is the solid angle and $$\partial A$$ is some "infinitesimal area". In my understanding, $$\Phi$$ is a function defined on $$\mathbb{R}^3$$, i.e., $$\Phi = \Phi(x, y, z)$$. So what does it mean to "differentiate" wrt $$A$$ and $$\Omega$$? Many articles define these quantities by talking about "infinitesimal" measures, but those don't really mean anything meaningful to me and don't seem to be mathematically rigorous. How could one rigorously define radiance and irradiance?

## 2 Answers

It is true that we tend to play a bit fast and loose with the notation, and that's usually OK.

If we insist on being a bit more precise, suppose we have a small flat surface of area $$A$$. We may denote the flux through it with the function $$\Phi(\vec r,\hat n,A)$$, where $$\vec r$$ is the position of the surface and $$\hat n$$ is the unit normal vector that defines its orientation. Irradiance can be defined as the one-sided limit $$E(\vec r,\vec n)=\left.\frac{\partial}{\partial A}\Phi(\vec r,\hat n,A) \right|_{A=0}\equiv \lim_{A\to0^+}\frac{\Phi(\vec r,\vec n,A)}{A}.$$

Radiance is a directional quantity, so we need one additional auxiliary definition to specify the direction the power is going in. We have a small surface of area $$A$$ emitting flux. Like mmesser314 did, we define a cone whose vertex lies in the surface emitting flux. The axis of the cone is specified by the normal vector $$\hat m$$, pointing away from the surface. The solid angle subtended by the cone is $$\Omega=4\pi\sin^2(\alpha/4)$$, where $$\alpha$$ is the apex angle.

Letting $$\Phi(\hat m, \Omega, A)$$ be the flux passing through the base of the cone, we can now define radiance as $$L(\hat m) = \left.\frac{1}{\cos\theta}\frac{\partial^2}{\partial A~\partial\Omega}\Phi(\hat m, \Omega, A)\right|_{\substack{\Omega=0\\A=0}}\equiv\frac{1}{\cos\theta}\lim_{\substack{\Omega\to0^+\\A\to0^+}} \frac{\Phi(\hat m, \Omega, A)}{A\Omega}$$ where $$\theta$$ is the angle between $$\hat m$$ and the normal vector $$\hat n$$ of the emitting surface, with $$\cos \theta = \hat m \cdot \hat n$$.

These definitions could still be made more precise/rigorous, but it quickly gets cumbersome and that level of rigor is rarely needed in applications.

On rereding this post, I have the names hopelessly mixed up. I hope at least this makes the derivatives more clear.

There is radiance, irradiance, and intensity. These use the total flux. They deal with energy in the light.

Then there is luminance, illuminance, and intensity? These use the flux weighted by the sensitivity of the eye to the wavelengths in the light. They deal with how bright the light appears to be.

I have shown definitions for luminance and illuminance. I think I have names backwards?

You have a point receiving light from an extended source like the Sun. Light arrives from a variety of angles.

Take a cone with a vertex at your target point that extends to the source. The cone has a solid angle $$\Omega = Area/r$$, where $$Area$$ is the cross sectional area of the cone at distance $$r$$. There is a certain flux, $$\Phi$$ that arrives inside the cone. If all is uniform, luminance $$L = \Phi / \Omega$$.

Take a cone with a slightly larger solid angle $$\Omega + \Delta \Omega$$. The flux inside is $$\Phi + \Delta \Phi$$. The general form is

$$L = \frac{\partial \Phi}{\partial \Omega} = lim \frac{\Delta \Phi}{\Delta \Omega}$$

Now expand the receiving point to a patch with area $$A$$. The total flux arriving at the patch is $$\Phi$$. Again if all is uniform, intensity = $$I = \Phi/A$$.

More generally, take a slightly larger patch with area $$A + \Delta A$$. You now have a slightly larger $$\Phi + \Delta \Phi$$.

$$I = \frac{\partial \Phi}{\partial A}$$

For illuminance, take a patch A. Each point in it has a cone with the same $$\Omega$$. The total flux arriving at the patch from angles within the cones is $$\Phi$$. Illuminance $$= lx = \Phi/LA$$.

I hate these names. We need $$I$$, but we have already used it for intensity. The symbol for illuminance is $$lx$$.

More generally,

$$lx = \frac{\partial^2 \Phi}{\partial \Omega \partial A}$$

You can get at illuminance two ways.

$$lx = \partial I/ \partial \Omega$$

That is, if you take cones with a slightly larger $$\Omega + \Delta \Omega$$, you get a slightly larger $$\Phi + \Delta \Phi$$, and a slightly larger $$I + \Delta I$$.

$$lx = \partial L/\partial A$$

That is, if you take a slightly larger patch $$A + \Delta A$$, you get a slightly larger $$\Phi + \Delta \Phi$$. You can calculate L over the patch. You can calculate $$L + \Delta L$$ over the larger patch.

Cos factors appear when you have a beam illuminating a patch A at an angle $$\theta$$. This is the same as a beam at normal incidence illuminating a patch of area $$A cos(\theta)$$.

• Your $L$ is actually luminous intensity (or radiant intensity in radiometry). Your $I$ is illuminance (irradiance). Your $lx$ is luminance (radiance), which could be confusing because $\text{lx}$ is the unit of illuminance. Intensity is a vague term that usually but not always means illuminance (irradiance).
– Puk
Commented Dec 21, 2023 at 16:48
• The names for these optical quantities are hopelessly complicated. I've never understood why the intensity vector (Poynting vector with units W/m^2) isn't sufficient for all applications. Commented Dec 21, 2023 at 17:42