Penetration Depth Calculator
Optical penetration depth from the complex index n + ik at any angle of incidence: d_p = λ₀/(4π Im √(N² − n₀² sin²θ)), covering absorbing media, metals (skin depth) and total internal reflection.
Absorption Coefficient α
157.1 cm⁻¹
4πk/λ₀ = 1.571e+4 m⁻¹
Intensity Depth d_p at θ
63.66 µm
1/e of intensity, normal to the surface; refracted wave, absorbed
Field 1/e Depth 2d_p
127.3 µm
skin depth λ₀/(2πk) at normal incidence
Depth at Normal Incidence 1/α
63.66 µm
λ in medium λ₀/n = 533.3 nm
Normal incidence: α = 4πk/λ₀, intensity I(z) = I₀ e^(−αz), d_p = 1/α.
Oblique incidence from index n₀: k_z = (2π/λ₀) √(N² − n₀² sin²θ), N = n + ik, so d_p = λ₀ / (4π · Im √(N² − n₀² sin²θ)).
This one formula gives 1/α at normal incidence and the evanescent TIR depth for k = 0 above the critical angle. The intensity is the depth-dependent factor of the transmitted field; reflection losses at the surface are not included.
Model and references
Exact. The equations are exact within the stated assumptions, or are a standard's tabulated values. The results are as good as the inputs. About the tiers
Exact plane-wave result for one planar interface from a lossless medium: the depth is that of the transmitted (or evanescent) field; reflection at the surface is not included.
- Born M., Wolf E. (1999). Principles of Optics, 7th ed., ch. 14 (optics of metals). Cambridge University Press.
- Axelrod D. (2001). Total internal reflection fluorescence microscopy in cell biology. Traffic 2, 764–774 (eq. 2). doi:10.1034/j.1600-0854.2001.21104.x