Fresnel Reflection Calculator

How reflection changes when light crosses from one medium to another

When a beam of light meets a boundary between two transparent materials, it almost never behaves in a perfectly all-or-nothing way. Part of the light can reflect, part can transmit into the second medium, and the proportions depend on both the refractive indices and the angle of incidence. That is the job of the Fresnel equations: they quantify how much of the incoming intensity is reflected for different polarizations. This calculator gives you a fast way to estimate that boundary behavior without working through the trigonometry by hand every time.

The tool is useful in practical optics because interfaces appear everywhere. A bare glass window reflects some light. Fiber-optic links depend on what happens at dielectric boundaries. Camera lenses rely on coatings because even a few percent of unwanted reflection at each surface can add up to glare and lost throughput. Laser setups, microscopes, optical sensors, solar covers, and lab prisms all involve the same basic question: for this pair of materials and this angle, how much light comes back, how much continues forward, and does the answer depend on polarization?

Three inputs control the calculation. The first is the refractive index of the incident medium, n₁. This is the medium the light starts in before it reaches the interface. The second is the refractive index of the transmitted medium, n₂. This is the medium on the far side of the boundary. Refractive index is unitless, so there is no length or time unit to convert, but you still need to enter physically sensible values. 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 input is the angle of incidence, θ_i, in degrees. The most common source of confusion is how that angle is measured. In optics, the angle is measured from the normal, which is the imaginary line perpendicular to the surface, not from the surface itself. Straight-on incidence is 0°, not 90°. A very shallow, grazing ray is near 90°. If you measure from the surface by mistake, the calculator will still produce a number, but it will answer a different physical question than the one you intended.

Once you enter those values, the first step is Snell's law, which determines the transmission angle in the second medium. If the geometry allows a real refracted ray, the calculator then applies the Fresnel formulas separately for s-polarized and p-polarized light. S-polarization means the electric field is perpendicular to the plane of incidence. P-polarization means the electric field lies in that plane. The two cases matter because the interface does not reflect them equally except at special angles, and in many optical systems that difference is exactly what you care about.

Like many scientific calculators, the Fresnel model can be described first as a general function of its inputs. The result depends on the refractive indices, the incident angle, and the rules that connect those quantities through geometry and boundary conditions. The generic notation below is broad, but it is still worth keeping because it reminds you that every output is produced by a specific mapping from measured inputs to a physical result.

Some scientific workflows also combine weighted contributions, which is what the next expression represents. In the present calculator the final answer is not literally a weighted sum of arbitrary inputs, but the notation is still a familiar way to think about how models turn several ingredients into one reported quantity. Here the relevant point is that the calculator is not guessing; it is applying a consistent mathematical rule set to a defined group of variables.

For this specific problem, the physically relevant relationship begins with Snell's law. It links the incident angle to the transmission angle. If the sine term required on the transmitted side would have magnitude greater than 1, the refracted angle is no longer real and the interface has entered total internal reflection instead.

The amplitude reflection coefficients for the two linear polarizations are then

The calculator reports reflectance, which is intensity reflection rather than amplitude reflection, so it squares those coefficients:

That average is a simple arithmetic mean of the two polarization reflectances, which is a common approximation for unpolarized light at an ideal interface. It is helpful when you want one quick number for comparison, but the separate s and p values are often the more informative outputs. If you are designing a polarization-sensitive system, the average can hide the behavior you actually care about.

Worked example: air to glass at 45°

Suppose light travels from air into ordinary glass. A useful first-pass set of values is n₁ = 1.00, n₂ = 1.50, and θ_i = 45°. Snell's law gives a transmitted angle of about 28.13°. From there, the Fresnel equations show that the interface reflects the two polarizations very differently: the s-polarized reflectance is about 9.20%, while the p-polarized reflectance is only about 0.85%. The calculator then reports an average reflectance of about 5.03%.

That example is valuable because the numbers are physically intuitive once you know what to look for. First, going from air into glass bends the ray toward the normal, so the transmission angle should be smaller than the incident angle; 28° passes that sanity check. Second, reflection is not the same for both polarizations away from normal incidence. The difference between roughly 9% and less than 1% tells you why polarization matters at a surface. Third, the average reflectance is only a few percent, which matches the everyday observation that a clean window is mostly transmissive but still visibly reflective.

How to read the result panel and what the special cases mean

The result panel reports four things. The transmission angle tells you the direction of the refracted ray in the second medium. The s-polarized reflectance tells you what fraction of incident s-polarized intensity is reflected. The p-polarized reflectance does the same for p-polarized light. The average reflectance is simply the mean of those two fractions, expressed as a percentage. If you are evaluating a system for glare, loss, or polarization response, it is usually best to look at all four numbers together rather than isolating a single percentage.

At normal incidence, where the incoming ray is perpendicular to the surface, the distinction between s and p disappears. Both polarizations reflect by the same amount because the geometry is symmetric. As the angle increases, the two curves separate. S-polarized reflectance usually rises more steadily, while p-polarized reflectance can dip dramatically. That dip reaches a special point known as the Brewster angle, where the p-polarized reflectance goes to zero for ideal non-absorbing media. For many readers, this is the first memorable landmark in Fresnel optics because it has such a clear experimental consequence: one polarization can stop reflecting at a particular angle even though the other still does.

The Brewster-angle condition is often written as

For air to glass, that angle is about 56.3°. Near that value, the p-polarized result from the calculator should approach 0%, while the s-polarized result remains distinctly nonzero. If your goal is to minimize reflected p-polarized light, that part of the curve matters more than the average.

The other major special case is total internal reflection. This occurs only when light tries to go from a higher index medium into a lower index medium, such as from glass into air, and only when the incident angle exceeds the critical angle. The critical angle satisfies

For glass to air with n₁ = 1.50 and n₂ = 1.00, the critical angle is about 41.8°. Above that, the calculator correctly reports total internal reflection and does not give a real transmission angle into the second medium. This is not an error state; it is a real physical regime, and it is the reason optical fibers can guide light efficiently.

If you are using the calculator for design work, a good habit is to vary only one input at a time. Keep the indices fixed and sweep the angle to see where the response changes quickly. Then swap the media or modify one refractive index and repeat. That approach makes it easier to build intuition. You begin to see, for example, that low-angle incidence often produces modest reflection, that grazing angles can become far more reflective, and that reversing the direction of travel across the same boundary can produce qualitatively different behavior because total internal reflection becomes possible in only one direction.

Assumptions, limits, and sensible interpretation

This calculator models an ideal, flat interface between two homogeneous media with real refractive indices. That means it is a clean starting point for many textbook and engineering estimates, but it does not include every effect that may matter in a real optical stack. It does not model surface roughness, thin-film interference, multiple internal reflections in a layered coating, absorption from complex refractive indices, anisotropic crystals, or wavefront distortion. If your application depends on those effects, the Fresnel result here is still informative, but it should be treated as a baseline rather than the final word.

A result is most useful when you can interpret it in context. If the average reflectance comes out around 4% to 5% for a bare air-glass interface at ordinary angles, that is plausible. If you accidentally enter an angle measured from the surface instead of the normal, you may get a surprisingly high or low answer that looks mathematically neat but is physically misframed. If the transmission angle seems larger when the ray enters a higher-index medium, that is a clue to revisit the inputs because the ray should bend toward the normal in that case. In short, use the number as part of a physical story, not as a detached percentage.

The form below includes example values so you can explore immediately. Try the air-to-glass example first, then change only the angle. After that, reverse the indices to see how the possibility of total internal reflection appears. Those small experiments turn the calculator from a one-time lookup tool into a compact optics lab for boundary behavior.