Snell's Law, Fresnel Equations & Brewster's Angle Calculator

Comprehensive Physics Guide to Snell's Law and Fresnel Equations

The propagation of electromagnetic radiation across the interface separating two media with different optical dielectric constants is governed by Maxwell's boundary equations. Formulated analytically by French civil engineer and physicist Augustin-Jean Fresnel in 1823 and building upon Willebrord Snellius's 1621 empirical law of refraction, **Fresnel optics** provides the exact quantitative framework for anti-reflective lens coatings, polarizing filters, fiber-optic waveguiding, and solar cell glass texturing.

1. Snell's Law of Refraction

According to Fermat's principle of least time, light traversing from medium 1 (phase velocity $v_1 = c/n_1$) to medium 2 ($v_2 = c/n_2$) bends at the planar interface such that:

where $theta_i$ is the angle of incidence and $theta_t$ is the angle of transmission (refraction), both measured relative to the surface normal vector $hat{n}$.

• Refraction toward the Normal ($n_1 < n_2$): When light enters a denser medium (e.g. air to glass), the ray decelerates and refracts toward the normal ($theta_t < theta_i$). A refracted ray exists for all incident angles from $0^circ$ to $90^circ$.
• Refraction away from Normal ($n_1 > n_2$): When propagating from a denser into a rarer medium (e.g. glass to air), $theta_t > theta_i$.

2. Total Internal Reflection (TIR) & The Critical Angle ($theta_c$)

When $n_1 > n_2$, increasing the incident angle eventually causes the refracted angle to reach $90^circ$ ($sintheta_t = 1.0$), grazing parallel to the surface. The incident angle producing this condition is the **Critical Angle** $theta_c$:

For any incident angle $theta_i ge theta_c$, no real refracted transmitted ray can propagate into medium 2. All incident optical power is reflected back into medium 1 with $100%$ efficiency ($R_s = R_p = 1.0$). This phenomenon of **Total Internal Reflection** forms the physical foundation of modern fiber-optic telecommunications (trapping laser pulses within high-index silica cores) and right-angle prism binoculars.

3. The Fresnel Equations & Polarization

Matching tangential electric and magnetic fields across the boundary yields the **Fresnel amplitude reflection coefficients** ($r$) and **transmission coefficients** ($t$):

A. s-Polarization (TE Mode, Senkrecht / Perpendicular)

The electric field vector $vec{E}$ oscillates strictly perpendicular to the plane of incidence (parallel to the interface plane):

B. p-Polarization (TM Mode, Parallel)

The electric field vector $vec{E}$ oscillates parallel to the plane of incidence:

C. Power Reflectance ($R$) & Transmittance ($T$)

Because optical intensity is proportional to the square of field amplitude, the power reflectances are:

By conservation of energy across lossless non-absorbing dielectric interfaces:

4. Brewster's Angle ($theta_B$) and Natural Polarization

In the expression for $r_p$, when the denominator approaches infinity ($theta_i + theta_t = 90^circ$), $tan(theta_i + theta_t) to infty$, causing the p-wave reflection coefficient to drop to zero ($r_p = 0$). Substituting $theta_t = 90^circ - theta_i$ into Snell's law:

At **Brewster's Angle** $theta_B$, reflected light is $100%$ linearly s-polarized because the dipoles excited in medium 2 oscillate parallel to the reflected direction, and dipoles cannot radiate electromagnetic energy along their oscillation axis. Polarized sunglasses exploit this principle to eliminate blinding horizontal road and water glare.

5. Normal Incidence Limit ($R_0$)

When light strikes perpendicular to the interface ($theta_i = 0^circ, theta_t = 0^circ$):

For an uncoated glass window ($n = 1.50$) in air ($n = 1.00$), $R_0 = (0.5 / 2.5)^2 = 0.04 = 4.0%$ per face (totaling $sim 8%$ two-face reflection loss).

6. Step-by-Step Worked Fresnel Calculation

Calculate reflection and refraction for unpolarized laser light entering crown glass:

• Incident medium: Air ($n_1 = 1.000$)
• Transmitted medium: Schott BK7 Crown Glass ($n_2 = 1.517$)
• Angle of incidence: $theta_i = 45.0^circ$

Step 1: Calculate Refraction Angle using Snell's Law

Step 2: Calculate Brewster's Angle and Normal Reflectance

Step 3: Calculate Fresnel s and p Reflectance at $theta_i = 45^circ$

At $45^circ$ incidence, s-polarized light reflects over $10times$ more strongly than p-polarized light ($9.6%$ vs $0.92%$), demonstrating partial polarization by reflection.

7. Evanescent Waves, TIR Phase Shifts, and Anti-Reflective Coatings

When total internal reflection occurs ($theta_i > theta_c$ when $n_1 > n_2$), electromagnetic theory dictates that the field does not terminate abruptly at the boundary. Instead, an evanescent wave is established in medium 2. The evanescent wave propagates along the boundary plane while decaying exponentially with distance $z$ into the rarer medium:

Here, $d_p$ represents the penetration depth, typically on the order of a few hundred nanometers for visible optical wavelengths. If a third medium with high refractive index is brought within a distance less than several penetration depths $d_p$, photons can tunnel across the gap into the third medium via Frustrated Total Internal Reflection (FTIR), which is the optical equivalent of quantum mechanical barrier tunneling. FTIR is utilized in optical beam splitters and fingerprint scanners.

Furthermore, upon total internal reflection, while the power reflectance $R = 1$, the reflected waves acquire polarization-dependent phase shifts $delta_s$ and $delta_p$. The differential phase shift $Delta = delta_p - delta_s$ between the two orthogonal polarizations is given by:

This phase retardance is exploited in the Fresnel rhomb, which converts linearly polarized light into circularly polarized light without the spectral dispersion inherent in birefringent waveplates.

In optical engineering and solar photovoltaics, minimizing surface reflection losses is critical. By applying a single-layer dielectric coating of refractive index $n_c = sqrt{n_1 n_2}$ and optical thickness equal to a quarter wavelength ($d = lambda_0 / (4 n_c)$), reflections from the top and bottom interfaces interfere destructively, reducing reflectance to near zero ($R approx 0%$) at the design wavelength.