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Light Transport in Tissue Examples of Delta Functions Collimated light is often modelled with a Dirac delta function, In particular, if light is incident from the direction $\hat\mathbf{s}'$ then the radiant intensity is written $$I(\mathbf{r},\hat\mathbf{s})=E_0(\mathbf{r})\delta(1-\hat\mathbf{s}\cdot\hat\mathbf{s}')$$ (C26) where I is the radiant intensity [W/cm2/sr], E0 is the irradiance [W/cm2], and the Dirac delta function has units of 1/sr. The integral of the radiance over all angles is the irradiance E0 since $$\int_{4\pi}\delta(1-\hat\mathbf{s}\cdot\hat\mathbf{z})\, d\o... ...pi}^\pi\, d\varphi\int_{-1}^1{1\over2\pi}\delta(\mu-1)\,d\mu=1$$ (C27) The integral of the delta-Eddington phase function $$P_{delta-E}(\cos\theta)={1\over4\pi}\{2f\delta(1-\cos\theta)+(1-f)(1+3g'\cos\theta)\}$$ (C28) over all angles is unity since $$\int_{4\pi}2f\delta(1-\cos\theta)\,d\omega= 2f\int_{-\pi}^\p... ...phi')\, d\varphi \int_{-1}^1 2\pi\delta(\mu-\mu')\,d\mu=4\pi f$$ (C29) and $$\int_{4\pi}=(1-f)(1+3g'\cos\theta)\,d\omega=4\pi(1-f)$$ (C30)

S. A. Prahl."Light Transport in Tissue," PhD thesis, University of Texas at Austin, 1988.