For a photo-induced process to proceed, light must be absorbed. Consequently, the light intensity in photosensitive media decreases with depth. This interactive graph demonstrates the relationship between key variables.
Critical note: Assumes monochromatic light source.

Vertical lines denote key attenuation marks.
    For homogeneous sample cure, attenuation less than 10% is advised (90% mark); though 20% is preferred by some users.
    For vat polymerization, depth of cure \(D_p\) is a typical characterization metric.

Default values are for diphenyl(2,4,6-trimethylbenzoyl)phosphine oxide (TPO), a common radical photoinitator, at 405 nm.

Additional features are available including:
1) Calculating Napierian absorptivity from UV-Vis data
    Napierian absorptivity is in base e, rather than base 10
    Sigma catalog
    Ciba catalog
2) Converting wt% to mM
3) Calculating initiator half-life
4) Incorperation of an additional (inert) absorber

Interested in the impact of multiple wavelengths?

Plot details
$$ \frac{dI}{dz} = -I \, \varepsilon \, [C] $$ $$ \frac{I}{I_0} = \exp(-\varepsilon \, [C] \, z) $$ \( I \) – intensity of light at sample depth \( z \) \((\text{mW/cm}^2)\)
\( I_0 \) – initial intensity of light \((\text{mW/cm}^2)\)
\( \varepsilon \) – Napierian molar extinction coefficient (base \(e\)) \((\text{L/mol·cm})\)
\( [C] \) – concentration of chromophore (i.e., photoinitiator) \((\text{mol/L})\)
\( z \) – sample thickness \((\mu\text{m})\)

Napierian absorptivity details
$$ A = \frac{\varepsilon \, [C] \, l}{\ln(10)} $$ \( A \) – Absorbance (base 10)
\( \varepsilon \) – Napierian absorptivity \((\text{L/mol·cm})\)
\( l \) – path length \((\text{cm})\)
\( [C] \) – concentration of absorber \((\text{mM})\)

Initiator half-life
The energy of a photon is inversely proportional to the wavelength \(\lambda\). This relationship is useful for converting light intensity from \((\text{mW/cm}^2)\) to \((\text{Einstein/s·cm}^2)\), where \(1 \ \text{Einstein} = 1 \ \text{mol photons}\). $$ E = \frac{h \, c \, N_A}{\lambda} $$ \( E \) – photon energy \((\text{J/mol})\)
\( h \) – Planck's constant \((6.6261 \times 10^{-34} \ \text{J·s})\)
\( c \) – speed of light \((2.9979 \times 10^{8} \ \text{m/s})\)
\( N_A \) – Avogadro's number \((6.022 \times 10^{23} \ \text{mol}^{-1})\)
\( \lambda \) – wavelength of light \((\text{nm})\)

Half-life (assuming no diffusion, photobleaching, or photodarkening) is then: $$ t_{1/2} = \frac{\ln(2) \, E}{\phi \, \varepsilon \, I} $$ \( \phi \) – quantum yield \((\text{mol/Einstein})\)
\( \varepsilon \) – Napierian molar extinction coefficient (base \(e\)) \((\text{L/mol·cm})\)
\( I \) – light intensity \((\text{mW/cm}^2)\)

Inert absorber contribution
The total attenuation now accounts for both the active chromophore and the inert absorber. Note that this impacts intensity for half-life, but only the active species progresses. $$ \frac{dI}{dz} = -I \, (\varepsilon [C] + \varepsilon_\text{inert} [C]_\text{inert}) $$ $$ \frac{I}{I_0} = \exp\big(-(\varepsilon [C] + \varepsilon_\text{inert} [C]_\text{inert}) \, z\big) $$ \( \varepsilon_\text{inert} \) – Napierian molar extinction coefficient of inert absorber \((\text{L/mol·cm})\)
\( [C]_\text{inert} \) – concentration of inert absorber \((\text{mol/L})\)

This simulation was created in the Department of Chemical and Biological Engineering at University of Colorado Boulder for LearnChemE.com by Alexander Osterbaan under the direction of Professor John L. Falconer and Professor Christopher N. Bowman and with the assistance of Drew Smith and Meagan Arguien. Address any questions or comments to LearnChemE@gmail.com.