Tool · Units and conversions

Optical Density Calculator

How much light does a filter pass, what OD brings a beam down to a safe working power, and how much does a sample absorb? The calculator converts between optical density, transmission, decibels and attenuation factor, adds up a stack of filters, and applies the Beer-Lambert law to an absorbing sample. Background: optical density, transmittance, neutral-density filter, Beer-Lambert law, and decibel.

Convert

Enter the attenuation as

Optical density

Photographic ND filters labeled ND2, ND4 and ND8 give their attenuation factor; enter it as a factor.

Filter stack

Filter 1 OD

Filter 2 OD

Filter 3 OD

Filter 4 OD

Power into the stack (mW)

Power you want out (mW)

An OD of 0 leaves a slot empty. 1 µW is 0.001 mW.

Absorbing sample

Absorption given as

Absorption coefficient α (cm⁻¹)

Molar absorptivity ε (L mol⁻¹ cm⁻¹)

Concentration c (mol/L)

Thickness or path length (mm)

Refractive index, for the surface reflections

α is the natural-log coefficient, T_i = e^−αℓ; ε is base 10, A = εcℓ. Set the index to 1 to leave out the reflections, as a spectrophotometer does when it measures against a blank cuvette.

Presets

Readouts

Power through the filter stack

Each filter lowers the power by 10 dB per unit of OD. The dashed orange line is the power you want out.

Transmission against optical density

On a logarithmic transmission axis the relation is a straight line: each unit of OD is a factor of ten. The dots mark the conversion, the stack and the sample.

Learn with it

Three short experiments. Each one sets the inputs, says where to look, and asks for a prediction before it shows the result.

Checked against

These checks run in your browser on every load. The conversions between OD, transmission, decibels and attenuation factor, the filter stack, the Beer-Lambert absorbance and the slab transmittance are compared with values worked out by hand and with the worked numbers in the site’s entries.

Check	Expected	Computed	Tolerance

The expected values follow the definitions OD = −log₁₀T and dB = 10 OD, the Beer-Lambert law and the incoherent slab formula with Fresnel reflectance at normal incidence, evaluated by hand for the stated cases. The tolerance is the largest difference from Expected that still passes, relative to Expected or to 1, whichever is larger.

The model

Optical density is the base-10 logarithm of the ratio of incident to transmitted power, and the loss in decibels is ten times it:

\text{OD} = -\log_{10} T = \log_{10}\frac{P_0}{P}, \qquad \text{loss (dB)} = 10\,\text{OD}

Filters in series multiply their transmittances, so their optical densities add, provided the reflections between them do not send light back through. In an absorbing sample of thickness $\ell$ the internal optical density is $\varepsilon c \ell$ in the chemist’s form of the Beer-Lambert law, or $\alpha \ell / \ln 10$ with the natural-log absorption coefficient $\alpha$ . A slab with surface reflectance $R = \left(\frac{n-1}{n+1}\right)^2$ at each face and internal transmittance $T_i$ transmits

T = \frac{(1-R)^2\, T_i}{1 - R^2 T_i^2}

counting the multiple reflections between its faces incoherently, as for any sample much thicker than the coherence length of the light. The model takes the surfaces as uncoated and the light as arriving at normal incidence. It does not compute the optical density that laser safety eyewear needs, which comes from the maximum permissible exposures in ANSI Z136.1 or IEC 60825-1.

Worked example

A 1 W beam through an OD 1 filter and an OD 2 filter meets a total of OD 3 and leaves at 1 mW, 0 dBm. A further OD 3 brings it to 1 µW. An uncoated window of index 1.5 reflects 4 % at each face and transmits 92.31 %, an optical density of 0.03476 before it absorbs anything.

References: IUPAC, Compendium of Chemical Terminology (the Gold Book), entries “absorbance” and “attenuance”. E. Hecht, Optics, 5th ed. (Pearson, 2017), ch. 4. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), ch. 1.