Engineering
Fresnel Optical Reflectance, Transmittance & Brewster's Angle formulas
Calculate uncoated glass or crystal surface reflection loss R = ((n1 - n2) / (n1 + n2))^2, single-interface transmittance 1 - R, and Brewster's polarizing angle ATAN(n2 / n1).
Model a single lossless dielectric interface
These formulas assume a planar uncoated interface between isotropic, nonmagnetic, nonabsorbing media with positive real refractive indices at the chosen wavelength. They do not describe metals, absorbing media, coatings or a complete lens or glass plate.
At normal incidence the power reflectance is symmetric in the two indices. Power transmittance is1-R for this lossless interface; it is not the electric-field amplitude transmission coefficient. A finite slab also has another surface and potentially multiple reflections.
The Brewster angle is measured from the normal in the incident medium. It gives zero reflection for the p-polarized component under these ideal assumptions; the s component can still reflect. Polarizing sunglasses suppress reflected polarization rather than making all glare vanish. Equal indices give zero reflection at all angles, so there is no unique Brewster angle.
| Optical Interface Metric | Formula | Air to Crown Glass (1.00 -> 1.50) | Air to Diamond (1.00 -> 2.42) |
|---|---|---|---|
| Normal Power Reflectance R | ((B2 - B3) / (B2 + B3)) ^ 2 | 4.00% (0.0400) | 17.24% (0.1724) |
| Single-Surface Transmittance T | 1 - (((B2 - B3)/(B2 + B3)) ^ 2) | 96.00% (0.9600) | 82.76% (0.8276) |
| Brewster Polarizing Angle (deg) | DEGREES(ATAN(B3 / B2)) | 56.31 degrees | 67.55 degrees |
Normal-Incidence Fresnel Power Reflectance R = ((n1 - n2) / (n1 + n2))^2
Symmetric in n1 and n2: light exiting glass into air experiences the exact same 4.00% reflection loss as light entering glass from air.
=IF(COUNT(B2:B3)<>2,"Invalid inputs",IF(OR(B2<=0,B3<=0),"Invalid inputs",ROUND(((B2 - B3) / (B2 + B3)) ^ 2, 4)))Brewster angle for the p-polarized component
Evaluates arctangent ATAN(n2 / n1) converted from radians to degrees.
=IF(COUNT(B2:B3)<>2,"Invalid inputs",IF(OR(B2<=0,B3<=0),"Invalid inputs",IF(B2=B3,"No unique angle",ROUND(DEGREES(ATAN(B3/B2)),2))))Normal-incidence power transmittance
For the single lossless interface, subtract the unrounded reflectance from1 before rounding the displayed result.
=IF(COUNT(B2:B3)<>2,"Invalid inputs",IF(OR(B2<=0,B3<=0),"Invalid inputs",ROUND(1 - (((B2 - B3) / (B2 + B3)) ^ 2), 4)))Try the example
Use the original values or change them to test your understanding.