Tool · Optical materials

Snell’s Law Calculator

Light crossing a surface bends by Snell’s law, n₁ sin θ₁ = n₂ sin θ₂, and from the denser side it is totally reflected beyond the critical angle. From the two refractive indices and the angle of incidence the calculator gives the refracted angle, the critical and Brewster angles, and the reflected and transmitted power. Background: Snell’s law, critical angle, Brewster angle, total internal reflection, and Fresnel equations.

Interface

Index of the first medium n₁

Index of the second medium n₂

Angle of incidence θ₁ (°)

Wavelength (nm), for the evanescent depth

Light travels from the first medium into the second. Refractive indices depend on wavelength; the refractive index calculator gives them for common glasses and crystals.

Presets

Index values are typical: water and diamond near 589 nm, the fiber core and cladding of a standard single-mode fiber and silicon near 1550 nm.

Readouts

Rays at the interface

Reflectance against angle of incidence

s-polarizedp-polarizedunpolarized

Learn with it

Three short experiments. Each one sets the inputs, says where to look, and asks for a prediction before it shows the result.

Checked against

These checks run in your browser on every load. Angles and reflectances are compared with values worked out by hand, energy conservation is checked with the transmission coefficients, and the behavior at the critical and grazing angles is checked for continuity.

Check	Expected	Computed	Tolerance

The expected values follow Snell’s law and the Fresnel equations as given in Hecht, Optics, and Born and Wolf, Principles of Optics, evaluated by hand for the stated cases. The tolerance is the largest difference from Expected that still passes, relative to Expected or to 1, whichever is larger.

The model

At a flat interface between transparent media of refractive indices $n_{1}$ and $n_{2}$ , light arriving at angle $θ_{1}$ from the normal is refracted to $θ_{2}$ given by

n_{1} \sin θ_{1} = n_{2} \sin θ_{2}, θ_{c} = \arcsin \frac{n_{2}}{n_{1}}, θ_{B} = \arctan \frac{n_{2}}{n_{1}}

Beyond the critical angle $θ_{c}$ , which exists only when $n_{1} > n_{2}$ , no refracted beam exists and all the light is reflected. At Brewster’s angle $θ_{B}$ the p-polarized reflection vanishes. The reflected power follows the Fresnel equations,

r_{s} = \frac{n_{1} \cos θ_{1} - n_{2} \cos θ_{2}}{n_{1} \cos θ_{1} + n_{2} \cos θ_{2}}, r_{p} = \frac{n_{2} \cos θ_{1} - n_{1} \cos θ_{2}}{n_{2} \cos θ_{1} + n_{1} \cos θ_{2}}

with $R = ∣ r ∣^{2}$ and, for lossless media, $T = 1 - R$ ; unpolarized light has the average of the two. Under total internal reflection the field in the second medium decays over a depth $λ / (2 π \sqrt{n_{1}^{2} \sin^{2} θ_{1} - n_{2}^{2}})$ , the basis of evanescent coupling. The intensity decays twice as fast, so the penetration depth quoted in TIRF microscopy, which refers to intensity, is half this value. The model takes both media as transparent and isotropic and the surface as flat and uncoated; for the phase on reflection and absorbing surfaces see the Fresnel equations calculator, and for coatings the thin-film reflectance calculator.

Worked example

Light passing from air into glass of index 1.5 at 30° is refracted to 19.471°, and 4.152 % of unpolarized light is reflected. In the other direction, from glass into air, the critical angle is 41.81°; at 45° all the light is reflected, and at 633 nm the evanescent field decays to 1/e within 285 nm of the surface. Brewster’s angle for air to glass is 56.31°.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017), ch. 4. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), ch. 1.