# Diffuse reflection of a laser beam

I'm trying to calculate how much power is detected from a laser beam after it reflects off a diffuse material. The setup is the following (image source: https://pe2bz.philpem.me.uk/Lights/-%20Laser/Info-999-LaserCourse/C03-M14-LaserSafety-HazardEvaluation/mod03_14.htm):

I know:

• the laser transmitted power $$P_T$$
• the reflectance of the surface $$\rho$$
• the angle between the detector and the surface normal $$\phi$$
• the distance from the surface to the detector $$r$$ and
• the area of the detector $$A_D$$.

I'm looking for the power incident on the detector $$P_D$$.

My issue is with how or even whether to include Lambert's cosine law into the calculation and I'm thoroughly confused with irradiance, radiance, radiant intensity, ...

The image source gives an equation (Equation 8), which, after transforming it to use my variables, is:

$$E(r, \phi) = \frac{\rho P_T cos\phi}{\pi r^2}$$

However with no derivation, especially for the factors of $$1/\pi$$ and $$cos\phi$$.

In https://gll.urk.edu.pl/zasoby/74/1-3-2014.pdf I found equation (1), which is (without irrelevant variables):

$$P_D = \frac{\rho A_D P_T}{2\pi r^2}$$

Apart from the differences due to it being for power instead of irradiance, it does not have a dependence on the cosine but instead has a factor of $$\frac{1}{2}$$. I was able to derive this equation by assuming that the reflected power is distributed evenly over the whole hemisphere.

Is the assumption of reflected power being reflected evenly in all directions valid? And how does Lambert's cosine law come into this? Is it even relevant in the case of a single small laser spot that is fully in the detector's field of view?

I'd be very happy about pointers to a credible source with good explanations of diffuse reflection in this scenario.