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:

$$I(L) \;=\; I_0 \, e^{-\varepsilon c L},$$

where $c$ is concentration (mol/L) and $\varepsilon$ is the molar absorption coefficient (L/(mol·cm)).

Decibel form for engineering use:

$$\text{Loss [dB]} \;=\; 10 \log_{10}(I_0/I) \;=\; 4.343 \, \alpha L.$$

The factor 4.343 converts between nepers ($\ln$) and 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:

$$\alpha_\text{total} \;=\; \sum_i \alpha_i.$$

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

Applications:

Domain	Use of Beer–Lambert
UV-vis spectroscopy	Concentration measurement from transmission
Optical fiber attenuation	Propagation loss in dB/km
Semiconductor absorption	Bandgap measurement from absorption edge
Gas sensing	Concentration of CO$_2$, CH$_4$, etc. from absorption depth
Laser saturable absorption	Linear regime before saturation

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).