Beer–Lambert law

For light propagating through a homogeneous absorbing medium, intensity decays exponentially with path length:

$$I(L)=I_0{e}^{-\alpha L},$$

where $I_0$ is the input intensity, $L$ is the path length, and $\alpha$ is the absorption coefficient with units of 1/length (typically cm${}^{-1}$ or m${}^{-1}$).

Alternative formulations:

Concentration form for dilute solutions or low-density gases:

$$I(L)=I_0{10}^{-\epsilon cL},$$

where $c$ is concentration (mol/L) and $\epsilon$ is the decadic molar absorption coefficient (L/(mol·cm)); the absorbance $A=\epsilon cL$ is the base-10 optical density. The Optical Density Calculator computes either form, with the surface reflections of the sample.

Decibel form for engineering use:

$$\text{Loss [dB]}=10{\log }_{10}(I_0\mathrm{/}I)=4.343\alpha L\mathrm{.}$$

The factor 4.343, which is $10{\log }_{10}e$, converts the natural-log coefficient into decibels ($10{\log }_{10}$); see propagation loss for the conversion between dB/cm and cm${}^{-1}$.

Multi-mechanism form. When multiple absorption mechanisms (electronic, vibrational, scattering, Auger) act independently:

$${\alpha }_{\text{total}}=\sum _i{\alpha }_i\mathrm{.}$$

This is the basis for cumulative loss models in optical fibers (Rayleigh scattering + OH absorption + UV/IR tail absorption + impurity absorption).

Applications:

Limitations. Beer–Lambert assumes:

• Monochromatic light (or absorption coefficient constant over bandwidth)
• Homogeneous medium
• Independent absorption events (no saturation)
• Negligible scattering loss into other directions (or scattering treated separately)
• Linear absorption (intensity-independent $\alpha$)

For high-intensity light, saturation reduces absorption (saturable absorbers); for non-uniform media, spatial integration is required; for resonant scattering or guided-mode geometries, the simple exponential breaks down. In single-mode fibers, the Beer–Lambert relation holds extremely well, which is why fiber attenuation is reported as a single $\alpha$ value (typically 0.20 dB/km at 1550 nm).