Fresnel Reflectance Calculator
The Fresnel equations describe the fraction of light reflected and transmitted at an interface between two media with different refractive indices. At normal incidence (0 degrees), the reflectance is R = ((n1 - n2)/(n1 + n2))^2 and transmittance is T = 1 - R. At oblique angles, the s-polarization and p-polarization components have different reflectances. This calculator computes the normal-incidence reflectance and, for oblique incidence, both Rs and Rp plus the average reflectance for unpolarized light. The Fresnel equations are fundamental to designing anti-reflection coatings, beam splitters, and understanding natural phenomena such as glare from water.
Fresnel reflectance formulas
Normal incidence: R = ((n1 - n2) / (n1 + n2))2
Rs = ((n1 cos theta_i - n2 cos theta_t) / (n1 cos theta_i + n2 cos theta_t))2
Rp = ((n2 cos theta_i - n1 cos theta_t) / (n2 cos theta_i + n1 cos theta_t))2
Where theta_t is the refraction angle from Snell's law: n1 sin(theta_i) = n2 sin(theta_t). Transmittance T = 1 - R for lossless media.
Practical notes on Fresnel reflectance
- Single uncoated glass surface (n=1.52 in air): approximately 4.3% reflectance at normal incidence.
- Water surface viewed at low sun elevation (grazing angle): reflectance approaches 100%.
- At Brewster's angle, Rp = 0 and the reflected light is completely s-polarized.
- Anti-reflection coating goal: quarter-wave layer with n_coating = sqrt(n1 x n2) gives R = 0 for a single wavelength.
Fresnel reflectance: frequently asked questions
What is Fresnel reflectance at normal incidence?
At normal incidence (light hitting a surface perpendicularly), the reflectance is R = ((n1 - n2)/(n1 + n2))^2, where n1 and n2 are the refractive indices of the two media. For an air-glass interface (n1=1, n2=1.52), R = ((1-1.52)/(1+1.52))^2 = (-0.52/2.52)^2 approximately 0.0426 or 4.26%.
What are the Fresnel equations for oblique incidence?
At angle theta_i (with refracted angle theta_t from Snell's law), the s-polarization reflectance is Rs = ((n1 cos theta_i - n2 cos theta_t)/(n1 cos theta_i + n2 cos theta_t))^2 and the p-polarization reflectance is Rp = ((n1 cos theta_t - n2 cos theta_i)/(n1 cos theta_t + n2 cos theta_i))^2. The average unpolarized reflectance is (Rs + Rp)/2.
Why is anti-reflection coating needed?
Each uncoated glass surface reflects about 4% of light. A camera lens with 10 elements and 20 surfaces would lose 1 - 0.96^20 = about 56% of light without coatings. Single-layer anti-reflection coatings reduce reflectance per surface to below 1%. Multi-layer coatings (MgF2, ZrO2) can reduce it to below 0.1%.
At what angle is reflectance maximum?
Reflectance increases toward 100% as the angle of incidence approaches 90 degrees (grazing incidence). For angles beyond the critical angle in a denser medium, total internal reflection gives 100% reflectance for both polarizations.
What is the difference between reflectance and reflectivity?
Reflectance (R) is the fraction of incident light power reflected by a specific surface, while reflectivity is an intrinsic property of the bulk material at normal incidence. In practice the terms are often used interchangeably, but reflectance refers to a surface while reflectivity refers to the material itself.
Official sources
- OpenStax University Physics Volume 3, Chapter 1: The Nature of Light. openstax.org.
- NIST, Physical Measurement Laboratory. physics.nist.gov.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.