Beer-Lambert Optical Depth From Transmittance Calculator

Beer-Lambert attenuation for a uniform medium is commonly written as: T = I / I0 = e-τ

From a measured transmittance T, the optical depth is: τ = -ln(T)

Absorbance (base-10) is: A = -log10(T) = τ / ln(10)

If a path length L is provided, the average attenuation coefficient is: μ = τ / L (reported in 1/m).

1. Select whether your transmittance is a fraction or percent.
2. Enter the transmittance value, or provide both I0 and It.
3. Optionally enter a path length to compute μ (attenuation per meter).
4. Press Calculate to display results above the form.
5. Use Download CSV or Download PDF after calculation.

Optical depth as a practical metric

Optical depth (τ) compresses transmission behavior into a single, unitless number. Because transmittance follows an exponential law, τ scales linearly with absorber amount and path length, making it easier to compare measurements across instruments and setups. It is widely used in spectroscopy, imaging, and radiative‑transfer calculations.

From transmittance to τ

For a measured transmittance T, the conversion is τ = −ln(T). A value of T = 0.90 gives τ ≈ 0.105, while T = 0.50 gives τ ≈ 0.693. Small changes in T near 1.0 correspond to small τ, but the same absolute T change at low T can represent a large τ jump.

Link to absorbance and decibels

Many labs report absorbance A on a base‑10 scale: A = −log10(T) = τ/ln(10). In optics and telecom, attenuation is often expressed in decibels: dB = 10·log10(I0/It) = 4.343·τ. This calculator outputs all three so you can match your reporting standard.

Using I0 and It measurements

If you measure incident intensity I0 and transmitted intensity It, transmittance is T = It/I0. Good practice is to keep I0 stable, avoid detector saturation, and subtract dark current/background. Averaging repeated reads reduces random noise before computing τ.

Including path length and μ

When you provide a physical path length L, the calculator also estimates an average attenuation coefficient μ = τ/L in 1/m. For example, τ = 0.35 across L = 0.10 m implies μ = 3.5 m⁻¹. This is useful for filters, cuvettes, optical windows, fog chambers, and tissue phantoms.

Where the model applies

The Beer–Lambert form assumes a uniform medium, negligible scattering into the detector, and no fluorescence or gain. In real samples, strong scattering, turbidity, or inhomogeneity can make apparent τ deviate from true absorption, especially at short wavelengths.

Interpreting typical ranges

In atmospheric and remote‑sensing contexts, aerosol optical depth is often reported at a reference wavelength (e.g., 500–550 nm). For many clean conditions τ may be below ~0.2, while hazy or dusty episodes can push τ higher. Always pair τ with wavelength and geometry. As a quick guide, τ = 1 means T ≈ 0.37 and τ = 2 means T ≈ 0.14.

Uncertainty and sensitivity

Because τ depends on ln(T), relative error grows as T becomes very small. If T has an uncertainty ΔT, a first‑order estimate is Δτ ≈ ΔT/T. Measuring T = 0.05 with ±0.005 uncertainty yields Δτ ≈ 0.1, which can dominate downstream μ estimates.

What is the difference between optical depth and absorbance?

Optical depth τ uses the natural logarithm: τ = −ln(T). Absorbance A uses base‑10: A = −log10(T). They are proportional: A = τ/ln(10) ≈ 0.4343·τ.

Can transmittance be greater than 1?

Physically, ideal transmittance is between 0 and 1. Values above 1 usually indicate calibration issues, background subtraction errors, detector nonlinearity, or gain changes between I0 and It measurements.

Why does the calculator ask for path length?

Path length L lets you compute an average attenuation coefficient μ = τ/L in 1/m. This normalizes results so materials or samples with different thicknesses can be compared more directly.

How do I enter percent transmittance?

Choose the “Percent” option, then enter values like 75 for 75%. The calculator converts it to a fraction internally (0.75) before computing τ, A, dB, and μ (if L is provided).

What happens when transmittance is very small?

As T approaches zero, τ grows rapidly and uncertainty increases. A small absolute error in T can create a large τ change because Δτ ≈ ΔT/T, so low‑T measurements need careful calibration.

Is Beer–Lambert always valid for real samples?

Not always. Strong scattering, turbidity, fluorescence, or non‑uniform concentration can break the simple exponential model. In such cases, τ from T is an “effective” optical depth for the measurement geometry.

How can I reduce measurement error in τ?

Stabilize the source, avoid detector saturation, subtract dark/background signals, and average repeated readings. Keep alignment fixed so I0 and It are measured with the same optics and detector settings.