Henyey–Greenstein phase function

Henyey–Greenstein phase function is a mathematical model used to approximate the angular distribution of light scattered by particles. First introduced by Louis Henyey and Jesse Greenstein in 1941 to simulate the scattering of light by interstellar dust,^[1] the function has since become a standard tool in radiative transfer, atmospheric optics, biomedical imaging, and computer graphics.

It is particularly valued for its ability to represent strongly forward-scattering media (such as biological tissue or clouds) using a single parameter, the asymmetry factor (${\displaystyle g}$), without requiring the computational complexity of full Mie theory.^[2]

Mathematical formulation

The phase function, denoted as ${\displaystyle p(\theta )}$, describes the probability density of a photon being scattered into a direction ${\displaystyle \theta }$ relative to its original trajectory. The standard form, normalized over the solid angle sphere ${\displaystyle 4\pi }$, is given by:^[2]

${\displaystyle p(\theta )={\frac {1}{4\pi }}{\frac {1-g^{2}}{(1+g^{2}-2g\cos \theta )^{3/2}}}}$

Where:

• ${\displaystyle \theta }$ is the scattering angle (the angle between the incident and scattered light directions).
• ${\displaystyle g}$ is the anisotropy factor (or asymmetry parameter), where ${\displaystyle g\in (-1,1)}$.

Normalization

The function is normalized such that the integral over all solid angles ${\displaystyle \Omega }$ equals unity:

${\displaystyle \int _{4\pi }p(\theta )d\Omega =\int _{0}^{2\pi }\int _{0}^{\pi }p(\theta )\sin \theta \,d\theta \,d\phi =1}$

The asymmetry parameter (g)

The shape of the scattering distribution is controlled entirely by the parameter ${\displaystyle g}$. Physically, ${\displaystyle g}$ represents the average cosine of the scattering angle:

${\displaystyle g=\langle \cos \theta \rangle =\int _{4\pi }p(\theta )\cos \theta \,d\Omega }$

The behavior of the function changes based on the value of ${\displaystyle g}$:

• ${\displaystyle g=0}$: Isotropic scattering (light is scattered equally in all directions). This approximates Rayleigh scattering for very small particles.
• ${\displaystyle g>0}$: Forward scattering. As ${\displaystyle g}$ approaches 1, the scattering becomes highly peaked in the forward direction (${\displaystyle \theta =0^{\circ }}$).
• ${\displaystyle g<0}$: Backward scattering. As ${\displaystyle g}$ approaches -1, the scattering is directed back toward the source (${\displaystyle \theta =180^{\circ }}$).

In most natural environments, such as aerosols, clouds, and biological tissues, scattering is predominantly forward, with ${\displaystyle g}$ typically ranging between 0.7 and 0.99.^[3]

Applications

Astrophysics

Originally developed to model diffuse galactic light scattered by interstellar dust grains, the function remains a staple in simulating reflection nebulae and planetary atmospheres.^[1]

Biomedical optics

In tissue optics, the Henyey–Greenstein function is widely used to model photon transport in biological tissues. Most soft tissues are strongly forward-scattering, with typical ${\displaystyle g}$ values between 0.8 and 0.95. It is a critical component of the Radiative Transport Equation (RTE) used in optical tomography and Monte Carlo simulations of light propagation in tissue.^[3]

Computer graphics

The function is a standard in physically based rendering (PBR) for simulating volumetric effects. It is used to render clouds, fog, subsurface scattering in skin, wax, and marble, and particulate matter in water.^[4]

Because the function is analytically invertible, it allows for efficient importance sampling in Monte Carlo ray tracing algorithms.^[4]

Limitations and extensions

While the Henyey–Greenstein function is mathematically convenient, it is an approximation. It often fails to capture complex scattering phenomena present in real-world Mie scattering, such as the glory effect (a brightening in the direct backscatter direction) or rainbows.^[5]

Double Henyey–Greenstein function

To better model media that exhibit both a strong forward peak and a significant backward peak (which a single HG function cannot do simultaneously), a linear combination of two HG functions is often used:^[6]

${\displaystyle p_{double}(\theta )=\alpha p(\theta ;g_{1})+(1-\alpha )p(\theta ;g_{2})}$

Where ${\displaystyle \alpha }$ is a weighting factor between 0 and 1, ${\displaystyle g_{1}}$ controls the forward lobe, and ${\displaystyle g_{2}}$ controls the backward lobe.

See also

• Mathematics portal

• Mie scattering
• Rayleigh scattering
• Radiative transfer
• Volume rendering