Fresnel Reflection Calculator

How Fresnel reflection changes at a transparent material boundary

When light reaches the boundary between two transparent materials, the incident intensity is generally divided between reflected and transmitted rays rather than behaving in an all-or-nothing way. The reflected share depends on the two refractive indices, the incident angle, and polarization. The Fresnel equations quantify that division, and this calculator applies them to estimate the reflectance of an ideal optical interface without requiring you to work through the trigonometry by hand.

Fresnel reflection matters throughout practical optics. A bare window surface reflects some incident light, fiber systems rely on behavior at dielectric boundaries, and camera lenses use coatings because small reflections from multiple surfaces can become glare or lost throughput. Laser setups, microscopes, optical sensors, solar covers, and laboratory prisms all raise the same interface question: for these media and this angle, how much light returns, how much propagates into the second medium, and how does polarization change the result?

Three inputs define the Fresnel calculation. The first is the refractive index of the incident medium, n₁, meaning the material the light occupies before reaching the boundary. The second is the refractive index of the transmitted medium, n₂, on the far side of that boundary. Refractive index is unitless, but the entries still need to represent plausible materials. Air is close to 1.00, water is about 1.33, many common glasses are near 1.5, and denser optical materials can be noticeably higher.

The third Fresnel input is the angle of incidence, θ_i, in degrees. In optics, this angle is measured from the normal: an imaginary line perpendicular to the surface, not from the surface itself. Straight-on incidence is 0°, not 90°, and a shallow grazing ray approaches 90°. Entering an angle measured from the surface will still produce a number, but it describes a different optical geometry.

After you supply the indices and incident angle, the calculator first uses Snell's law to determine the transmission angle in the second medium. When a real refracted ray is possible, it then evaluates separate Fresnel formulas for s-polarized and p-polarized light. For s-polarization the electric field is perpendicular to the plane of incidence; for p-polarization it lies in that plane. An interface need not reflect those polarizations equally, and that distinction is often the important part of the result.

For Fresnel reflection, the physically relevant relationship starts with Snell's law, which connects the incident and transmission angles. When the required sine on the transmitted side would have magnitude greater than 1, no real refracted angle exists and the interface is in the total-internal-reflection regime.

For a Fresnel interface with a real transmitted angle, the amplitude reflection coefficients for the two linear polarizations are

The Fresnel outputs are reflectances, meaning intensity reflection rather than amplitude reflection. The calculator therefore squares the amplitude coefficients and reports their simple average:

For unpolarized light at an ideal interface, the displayed average is the arithmetic mean of the two polarization reflectances. It offers a convenient comparison value, but the separate s and p outputs are usually more useful when an optical system is polarization-sensitive. In particular, an average can conceal a strong p-polarized dip near Brewster's angle.

Worked example: Fresnel reflectance from air into glass at 45°

Consider light traveling from air into ordinary glass, with n₁ = 1.00, n₂ = 1.50, and θ_i = 45°. Snell's law gives a transmission angle of about 28.13°. The Fresnel equations then give about 9.20% s-polarized reflectance and about 0.85% p-polarized reflectance. Their average is about 5.03%.

This air-to-glass Fresnel example provides useful physical checks. Because the ray enters a higher-index medium, it bends toward the normal, so its transmission angle should be smaller than 45°; 28° meets that expectation. The large difference between the two reflectances also shows why polarization matters away from normal incidence. Finally, an average reflection of only a few percent is consistent with a clean window being mainly transmissive while still producing a visible reflection.

How to read Fresnel reflectance results and special interface cases

The Fresnel result panel reports four linked quantities for the selected interface. The transmission angle gives the direction of the refracted ray in the second medium. S-polarized reflectance is the fraction of incident s-polarized intensity reflected at the boundary, and p-polarized reflectance is the corresponding p-polarized fraction. Average reflectance is the mean of those two fractions, expressed as a percentage. For glare, throughput loss, or polarization-response work, read all four values together rather than relying on one percentage alone.

At normal incidence, where the ray is perpendicular to the interface, the s and p labels do not produce different reflectances because the geometry is symmetric. As the incident angle rises, the two Fresnel curves separate. S-polarized reflectance generally increases steadily, while p-polarized reflectance can fall sharply. Its minimum occurs at the Brewster angle, where p-polarized reflectance becomes zero for ideal non-absorbing media. This is a defining feature of Fresnel optics: one polarization can cease reflecting at a particular angle even though the other remains reflected.

The Fresnel Brewster-angle condition is

For an air-to-glass Fresnel interface, this angle is about 56.3°. Near it, the calculator's p-polarized result should approach 0%, while the s-polarized result remains nonzero. When minimizing reflected p-polarized light is the goal, this region matters more than the average reflectance.

The other major Fresnel special case is total internal reflection. It occurs only when light travels from a higher-index medium into a lower-index medium, such as glass into air, and the incident angle is above the critical angle. The critical-angle relationship is

For glass to air with n₁ = 1.50 and n₂ = 1.00, the critical angle is about 41.8°. Above that angle, the calculator reports total internal reflection rather than a real transmission angle in the second medium. This is a physical result, not an error condition, and it underlies optical-fiber guidance.

When exploring a Fresnel boundary for design work, vary one input at a time. Hold both refractive indices fixed while sweeping the incident angle to identify regions where the response changes rapidly. Then alter an index or reverse the media and repeat. This makes the direction-dependent behavior clear: low-angle reflection is often modest, grazing incidence can be much more reflective, and total internal reflection is possible only for high-to-low index travel.

Fresnel-interface assumptions, limits, and sensible interpretation

This Fresnel reflection calculator models an ideal flat boundary between homogeneous media with real refractive indices. It is a useful starting point for textbook and engineering estimates, but it does not include every effect in a practical optical stack. Surface roughness, thin-film interference, multiple internal reflections in coatings, absorption represented by complex refractive indices, anisotropic crystals, and wavefront distortion are outside this model. For applications where those effects matter, use this interface result as a baseline rather than a final prediction.

A Fresnel result is most valuable when interpreted with the optical geometry. An average reflectance around 4% to 5% for a bare air-glass interface at ordinary angles is plausible. An angle measured from the surface instead of the normal can yield a numerically tidy but physically misframed answer. Likewise, if a ray entering a higher-index material appears to have a larger transmission angle than its incident angle, recheck the input order and angle convention: the refracted ray should bend toward the normal in that case.

The form begins with an air-to-glass Fresnel test case so you can inspect the outputs immediately. Change only the incident angle first, then reverse the indices to observe when total internal reflection becomes available. These small comparisons make the calculator useful not just for a single lookup, but for building intuition about reflection at transparent boundaries.