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Optics Simulator

Fresnel Equations Simulator — Polarization-Dependent Reflectance & Brewster Angle

Compute s polarization reflectance R_s, p polarization reflectance R_p, unpolarized reflectance and Brewster angle in real time from refractive indices n_1, n_2, incidence angle theta and s polarization fraction beta. Ray geometry and angle-resolved reflectance curves are visualized side by side.

Parameters
incident-side index n_1
—
transmitted-side index n_2
—
incidence angle theta_i
°
s polarization fraction beta
—

Defaults are n_1 = 1.00 (air), n_2 = 1.50 (glass), theta_i = 30 deg, beta = 0.5 (natural light). The s polarization fraction beta is the share of s polarized light in the input: beta = 1 is pure s, beta = 0 is pure p, beta = 0.5 is unpolarized natural light.

Results
—
s reflectance R_s
—
p reflectance R_p
—
unpolarized R
—
Brewster angle theta_B
Ray geometry diagram

Blue band = medium 1 (n_1) / Green band = medium 2 (n_2) / Yellow = incident ray / Red = reflected ray (angle = theta_i) / Cyan = refracted ray (theta_t, Snell's law) / White dashed = normal

Reflectance vs incidence angle

x-axis = theta_i (0 to 90 deg) / y-axis = R (0 to 1) / Cyan = R_s / Orange = R_p / Green = R_unpol / Green dashed line = Brewster angle / Yellow line = current theta_i

Theory & Key Formulas

The Fresnel equations give the reflection coefficients for s and p polarization at the interface between two media. Snell's law $n_1 \sin\theta_i = n_2 \sin\theta_t$ provides the transmitted angle $\theta_t$.

Reflection coefficients for s (electric field perpendicular to the plane of incidence) and p (parallel) waves:

$$r_s = \frac{n_1\cos\theta_i - n_2\cos\theta_t}{n_1\cos\theta_i + n_2\cos\theta_t},\quad r_p = \frac{n_2\cos\theta_i - n_1\cos\theta_t}{n_2\cos\theta_i + n_1\cos\theta_t}$$

Reflectance and unpolarized reflectance (beta is the s polarization fraction):

$$R_s = r_s^2,\ R_p = r_p^2,\ R_{\text{unpol}} = \beta R_s + (1-\beta) R_p$$

Brewster angle (where $r_p = 0$) and critical angle (when $n_1 > n_2$):

$$\tan\theta_B = \frac{n_2}{n_1},\qquad \sin\theta_c = \frac{n_2}{n_1}$$

$n_1, n_2$ are the refractive indices (dimensionless) and $\theta_i, \theta_t$ are the incidence and transmitted angles. There is no total internal reflection when $n_1 < n_2$.

What is the Fresnel Equations Simulator

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I already know Snell's law, but how is the Fresnel equation different? Aren't they both about how light bends?
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Good question. Snell's law fixes the direction of the refracted ray, while Fresnel tells you what fraction of the light is reflected and what fraction is transmitted, separately for each polarization. Augustin-Jean Fresnel derived these in 1823 from elastic-wave theory and they sit at the heart of optical design. With the defaults here (n_1 = 1.00, n_2 = 1.50, theta_i = 30 deg, beta = 0.5) the Results card shows R_s about 0.058, R_p about 0.025, unpolarized R about 0.041 and Brewster angle 56.31 deg. So even at 30 deg from air to glass only about 4 percent reflects.
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Why are the s and p reflectances so different? It's the same light, that feels strange.
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It comes down to how the electric field of the wave is oriented relative to the surface. The s wave (German senkrecht, perpendicular) has E perpendicular to the plane of incidence, while p (parallel) has E in that plane. The boundary conditions of Maxwell's equations then give different reflection coefficients. Sweep theta_i from 0 to 89 deg in the tool: R_s grows monotonically, but R_p drops to 0 at one specific angle and then rises. That null is the Brewster angle, discovered empirically by David Brewster in 1812.
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Brewster angle! I have heard about it from polarized sunglasses. How is it actually used?
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Polarized sunglasses and camera polarizing filters are the most everyday application. Light reflected off water, wet roads or windshields near the Brewster angle has nearly zero p component and a strong s component, so a polarizer that blocks the horizontal direction kills the glare. Set n_1 = 1.00 and n_2 = 1.33 (water) in the tool and theta_B becomes about 53.06 deg, the optimum angle to suppress water glare. Photographers rotate a circular polarizer to match this same physics.
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What happens when n_1 is larger than n_2? The graph must change a lot.
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Sharp catch. Set n_1 = 1.50 (glass) and n_2 = 1.00 (air). The critical angle theta_c = arcsin(1/1.5) is about 41.81 deg. Above it both R_s and R_p saturate at 1, the regime of total internal reflection (TIR). Optical fibers exploit TIR: light bounces inside the core (n about 1.47) against the cladding (n about 1.46) with essentially no loss for kilometers. The tool draws a dashed line at theta_c and the curves stick to the ceiling beyond it. The Brewster angle now sits at about 33.69 deg, where R_p still touches zero. In the ray diagram the refracted ray disappears above theta_c and the label "Total Internal Reflection (TIR)" appears.

Frequently Asked Questions

The Fresnel equations describe how much light is reflected and transmitted at the boundary between two media of different refractive indices, separately for s and p polarization. Augustin-Jean Fresnel derived them in 1823 from the elastic theory of waves. The s wave (electric field perpendicular to the plane of incidence) and the p wave (parallel) have different reflection coefficients r_s and r_p, and the reflectance is R = r squared. With n_1 = 1.0, n_2 = 1.5, theta = 30 deg and beta = 0.5 (natural light) this tool reports R_s about 0.058, R_p about 0.025, unpolarized R about 0.041 and Brewster angle about 56.3 deg.
The Brewster angle theta_B is the special incidence angle where the p polarization reflectance is exactly zero, given by tan theta_B = n_2 / n_1. At this angle the reflected and refracted rays are perpendicular, so the dipoles oscillating in the p direction radiate along the surface normal of the interface and not back toward the reflected direction. For glass (n = 1.5) theta_B is about 56.3 deg, for water (n = 1.33) about 53.1 deg. Set theta to 56.31 deg in this tool and R_p drops to 0; the reflected light becomes purely s polarized. Polarized sunglasses and photographic anti-glare filters exploit exactly this effect.
When light moves from a denser medium (higher n_1) to a less dense one (n_2 below n_1), incidence angles above the critical angle theta_c = arcsin(n_2 / n_1) cause total internal reflection (TIR), giving R_s = R_p = 1. For glass (n_1 = 1.5) into air (n_2 = 1.0) theta_c is about 41.8 deg, beyond which 100 percent of the light is reflected. Set n_1 = 1.5, n_2 = 1.0 in this tool and sweep theta from 0 to 89 deg to watch the reflectance jump to unity at theta_c. Optical fiber telecommunication is built on this same total internal reflection.
The simulator only handles the real-valued Fresnel equations and ignores absorbing media (metals, which require complex refractive index n + ik), thin-film interference (constructive and destructive interference in multilayer coatings), wavelength dispersion, surface-roughness scattering and quantum effects. Real anti-reflection coatings and high-reflection mirrors are designed with thin-film software such as TFCalc or Essential Macleod, which include multilayer interference. This tool is sufficient for the basic understanding of polarized reflection at a single interface and for first-pass estimates; complex-index metal reflectance calculations need a dedicated tool.

Real-World Applications

Anti-reflection (AR) coatings: AR coatings on camera lenses, eyeglasses and smartphone screens use thin-film interference to push the single-interface reflectance predicted by Fresnel (about 4 percent for glass-to-air) close to zero. The classic single-layer optimum has a quarter-wave thickness with refractive index sqrt(n_substrate), often MgF_2; modern premium lenses stack ten or more layers to reach below 0.5 percent across the visible spectrum. Inspect R_unpol in this tool: without AR, 4 percent is reflected at normal incidence, so a 20-element zoom lens transmits only (0.96)^20 of about 44 percent of the input.

Polarizing sunglasses and filters: Light reflected off water, roads and car bodies near the Brewster angle has almost no p component and a strong s component. Polarized sunglasses absorb the horizontal (s) direction and cut glare by 70 to 90 percent, helping anglers see fish underwater and reducing oncoming headlight reflections at night. Move theta near theta_B in this tool: R_p drops to zero and the reflected light becomes fully polarized, exactly the principle that polarizing filters rely on.

Optical-fiber telecommunication: The core and cladding of a glass fiber are designed so that any incidence angle larger than the critical angle is reflected without loss. Setting n_1 = 1.47 (core) and n_2 = 1.46 (cladding) in this tool gives theta_c about 81.9 deg; only light entering within the corresponding acceptance cone stays trapped in the core. Single-mode fibers exploit this even more selectively, enabling long-haul links of 100 km or more with losses as low as 0.2 dB per kilometer.

Solar cells and architectural glass: Solar cells minimize surface reflection through texturing and AR coatings, dropping the reflectance from about 30 percent for bare silicon to under 1 percent. Low-emissivity (Low-E) building glass enhances infrared reflectance to improve heating and cooling efficiency. Set n_2 to about 3.5 (silicon) in this tool: bare Si shows R_unpol about 30 percent at normal incidence and about 35 percent at 60 deg, illustrating why AR engineering is critical for photovoltaic efficiency.

Common Misconceptions and Caveats

The single most common confusion is the idea that "reflection is zero at the Brewster angle". In reality only the p component R_p falls to zero there; R_s remains finite. For unpolarized natural light with beta = 0.5 the reflectance becomes R = 0.5 (R_s + 0) = R_s / 2, so on glass (n = 1.5) at theta_B you still get about 0.078, not zero. To suppress reflection completely you must first prepare the input as pure p polarization. Drag the beta slider down to 0.0 (pure p) and read R at theta_B in this tool: only then does it actually reach zero.

Another frequent mistake is to assume "reflectance grows monotonically with incidence angle". The s curve R_s does grow monotonically from 0 to 90 deg, but R_p is non-monotonic: it dips to zero at the Brewster angle and then climbs sharply. This behaviour follows directly from the boundary conditions of Maxwell's equations; Brewster found it empirically in 1812 and Fresnel proved it in 1823. Watch the orange R_p curve in the chart: it always touches zero at theta_B before rising. That non-monotonic dip is precisely the physical signature of the Brewster angle.

The third common pitfall is to think that "the Fresnel equations are real, so a complex refractive index is unnecessary". Reflectance from metals (silver, aluminium, gold) and semiconductors must be computed with the complex index n + ik (k is the extinction coefficient). This tool is only for transparent dielectrics with k = 0 (glass, water, plastics). At 550 nm aluminium has n about 0.96 and k about 6.69; plugging only the real n into Fresnel's formulas grossly underestimates its reflectance. Complex-valued Fresnel equations (with phase changes) require dedicated thin-film or ellipsometry software.

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