Tool · Optical materials

Critical Angle Calculator

At what angle does light stop leaving a dense medium, and how far does the reflected wave reach past the surface? From the two refractive indices, the calculator gives the critical angle, the escape cone of an emitter inside the denser medium, and, at a chosen angle beyond it, the evanescent penetration depth and the phase shifts of s and p light on total internal reflection. Background: critical angle, total internal reflection and evanescent wave.

Interface

Index of the medium the light is in, n₁

Index beyond the surface, n₂

Total internal reflection needs n₁ > n₂: light inside glass, water or a semiconductor meeting a less dense medium.

Beam

Angle of incidence (° from the normal)

Vacuum wavelength (nm)

The penetration depth and the phase shifts apply beyond the critical angle.

Presets

Readouts

Penetration depth beyond the critical angle

The depth diverges at the critical angle and falls to λ/(2π√(n₁² − n₂²)) at grazing incidence. The dot is the current angle.

Phase shift on total internal reflection

Blue: s polarization. Orange: p. The difference between them is what a Fresnel rhomb uses to turn linear into circular polarization with two reflections of 45° each.

Learn with it

Two 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. The critical angle, penetration depth, phase shifts and escape cone are compared with closed-form values.

Check	Expected	Computed	Tolerance

The expected values follow Snell’s law, the evanescent decay constant k√(n₁²sin²θ − n₂²) and the Fresnel phase on total reflection (Born and Wolf, section 1.5.4). The tolerance is the largest difference from Expected that still passes, relative to Expected or to 1, whichever is larger.

The model

Light going from index $n_{1}$ into a lower index $n_{2}$ refracts away from the normal, and Snell’s law has no real solution beyond

θ_{c} = \arcsin \frac{n_{2}}{n_{1}}

Past that angle all the light is reflected, and the field beyond the surface decays as $e^{- z / d}$ with

d = \frac{λ}{2 π \sqrt{n_{1}^{2} \sin^{2} θ - n_{2}^{2}}}

for the field; the intensity falls with $d / 2$ . The reflected wave is phase-shifted, by different amounts for the two polarizations, with $n = n_{2} / n_{1}$ :

δ_{s} = 2 \arctan \frac{\sqrt{\sin^{2} θ - n^{2}}}{\cos θ}, δ_{p} = 2 \arctan \frac{\sqrt{\sin^{2} θ - n^{2}}}{n^{2} \cos θ}

An isotropic emitter inside the dense medium sends a fraction $(1 - \cos θ_{c}) / 2$ of its light into the escape cone of one flat face, before the Fresnel reflection inside the cone is counted; this is the starting point of LED light-extraction estimates. The model assumes lossless media and a flat surface; an absorbing second medium or a third layer within a few $d$ of the surface makes the reflection frustrated and no longer total.

Worked example

Glass with $n = 1.5$ against air has $θ_{c} = \arcsin (2 / 3) = {41.81}^{\circ}$ , so a ray inside a 45° prism is totally reflected at the hypotenuse. At 45° and 633 nm the field reaches 285 nm into the air. For total internal reflection fluorescence microscopy, 488 nm light in a 1.518 coverslip against water (1.333) has $θ_{c} = {61.42}^{\circ}$ ; at 65° the intensity decays in 114.1 nm, and at 62° the field depth grows to 555.4 nm. A GaN chip with $n = 2.4$ sends only 4.547 % of its light into the escape cone of its top face.

Related: the Snell’s law calculator covers refraction below the critical angle, the Fresnel equations calculator gives the reflectance at every angle, and the numerical aperture calculator applies the same physics to fibers. See also frustrated total internal reflection.

References: M. Born and E. Wolf, Principles of Optics, 7th ed., Cambridge University Press (1999), section 1.5.4. D. Axelrod, “Cell-substrate contacts illuminated by total internal reflection fluorescence,” J. Cell Biol. 89, 141–145 (1981). E. Hecht, Optics, 5th ed., Pearson (2017), section 4.7.