Lambert’s Cosine Law

Lambert’s cosine law states that radiation from a black body in thermal equilibrium is emitted equally in all directions but appears weaker at an angle due to projection effects. This law indicates that the observed radiant intensity from an ideal diffuse radiator is directly proportional to the cosine of the angle between the infrared sensor’s line of sight and the surface normal. The radiation flux from a small surface area decreases with the cosine of the emission angle θ relative to the surface normal. This means that maximum radiation is emitted perpendicular to the surface (θ = 0), while at larger angles, the apparent emitting area decreases in proportion to cos(θ). Since a black body emits radiation uniformly in all directions, its spectral radiance remains constant, but the observed intensity varies based on projection effects. The total emitted power per unit area can be described where the spectral radiance accounts for the projected emitting area.  

[math]\frac{d\Phi (dA,\ \theta ,d\Omega ,d\lambda )}{d\Omega }\ =\ B_{\lambda }(T)\ dA\ d\lambda \ \cos (\Theta )[/math]

In the equation, the differential solid angle [math]d\Omega \ [/math] stands for a small portion of the solid angle into which radiation is emitted or measured. It describes the directionality of radiation and is measured in steradians sr. Differential Surface Area [math]dA\ [/math] denotes an infinitesimally small area of the emitting surface. It represents the tiny portion of a black body or any radiating surface contributing to the radiation emitted. Differential Wavelength Bandwidth [math]d\lambda [/math] refers to a small spectral interval within the emitted radiation spectrum. 

Most nonconducting surfaces can be considered to be diffuse or “Lambertian”. This means that for a Lambertian surface, the spectral radiance [math]B_{\lambda }(T)\ [/math] and the spectral radiant intensity [math]I_{\lambda }(T)[/math] are numerically equal. Spectral emissive power [math]M^{\circ}\left(T\right)[/math]  represents the total power emitted per unit area of a blackbody per unit wavelength, integrated over all directions in a hemisphere. 

[math]I_{\lambda}=B_{\lambda}\left(T\right)=M°\left(T\right)/[/math][math]\pi[/math]

In terms of radiant intensity [math]\ I[/math], it follows equations, where [math]I_{\theta }[/math] is the radiant intensity in [math]W/sr[/math] from a surface viewed at the angle [math]\theta [/math] normal to the surface and [math]I_{\bot }[/math] is the maximum emitted intensity in [math]W/sr[/math] perpendicular to the surface. 

[math]I_{\theta }=\ \ I_{\bot }\cos \left(\theta \right)[/math]

The law is also known as the cosine emission law. Figure 1 illustrates this relationship for a black body. 

The rule is based on theory and empirical observation and assumes that the surface is not specular. As wavelength increases, a given material surface is less likely to be considered Lambertian. This is because most surfaces are considered to be “rough” at the scale of the wavelength of the light if the material is not polished. Therefore, when a sensor is viewing a surface at an angle, the radiant intensity is decreased by the cosine of the angle between the target and the sensor. Additionally, it is to note that mirrors do not follow this rule but Snell’s law of reflection for geometrical optics.