# Fresnel Equations Calculator

r_s, r_p, R_s, R_p, T = 1−R, and the Brewster angle for any dielectric interface, with the 4% normal-incidence anchor.

> Canonical page: https://elysiatools.com/en/tools/fresnel-equations

- **Category:** Science & Education

- **Keywords:** fresnel equations, reflectance, transmittance, s polarization, p polarization, brewster angle, optics

## Overview

The Fresnel Equations Calculator computes reflection and transmission coefficients at the planar interface between two dielectric media. It provides exact values for amplitude coefficients (r_s, r_p), power reflectance (R_s, R_p), transmittance (T_s, T_p), Snell's refracted angle, and the Brewster angle based on incident and transmitting refractive indices.

## Inputs

- **Refractive index n₁ (incident)** (number): e.g. 1.0
- **Refractive index n₂ (transmitting)** (number): e.g. 1.5
- **Incidence angle θ₁ (degrees)** (number): e.g. 30

## When to use

- Evaluating surface reflection losses and transmission through uncoated optical windows, prisms, or lenses.
- Analyzing polarization-dependent reflectance differences between s-polarized and p-polarized light at non-normal incidence.
- Determining the Brewster angle for laser cavity windows or polarizing beam splitters to minimize reflection losses.

## How it works

- Enter the refractive index of the incident medium (n₁) and the transmitting medium (n₂).
- Specify the angle of incidence (θ₁) in degrees measured from the surface normal.
- The tool computes the refracted angle (θ₂) via Snell's law, solves the Fresnel equations for s and p polarizations, and determines the Brewster angle.

## Use cases

- Optical system engineering: Quantifying transmission throughput and ghost-reflection power across multiple optical interfaces.
- Laser and polarization optics: Aligning Brewster windows to eliminate insertion loss for linearly polarized laser beams.
- Physics education: Demonstrating electromagnetic wave boundary conditions, polarization extinction, and Snell's law behavior.

## Frequently asked questions

### What do the Fresnel equations calculate?

They calculate the amplitude and power ratios of reflected and transmitted electromagnetic waves across a boundary between two dielectric materials.

### What is the distinction between s-polarization and p-polarization?

s-polarization has the electric field vector perpendicular to the plane of incidence, whereas p-polarization has the electric field vector parallel to the plane of incidence.

### What occurs at the Brewster angle?

At the Brewster angle θ_B = arctan(n₂/n₁), the reflectance for p-polarized light (R_p) becomes zero, resulting in 100% transmission of the p-polarized component.

### What is the normal incidence reflection formula?

At 0° incidence, both polarizations share the same reflectance formula: R = ((n₁ - n₂) / (n₁ + n₂))², which equals 4% at an air-to-glass (1.0 to 1.5) boundary.

### How is transmittance derived?

For non-absorbing lossless dielectric interfaces, conservation of energy applies such that transmittance is computed directly as T = 1 - R for each polarization state.

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