Law of Refraction

$n_2*sin(theta_2) = n_1*sin(theta_1)$
Calculator of the law of refraction also known as Snell's law.
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This tool is a calculator of the law of refraction (Snell's law) :

$n_2*sin(theta_2) = n_1*sin(theta_1)$

n₁ : Medium 1 refractive index (incident ray)
n₂ : Medium 2 refractive index (refracted ray)
$theta_1$ : angle of incidence
$theta_2$ : angle of refraction

This law describes the behaviour of a light ray when it changes mediums. Specifically, it calculates the angle of deviation of the light ray following a transition from a refractive medium of index n₁ to another medium with refractive index n₂.

Refraction ability and medium index

The refractive index of a medium is an indication of its ability to bend light. When a light ray travels from a medium into another one, the angle of refraction depends on the ratio between the two medium indices n₁ and n₂ because of,

$sin(theta_2) = (n_1/n_2)*sin(theta_1)$

Accordingly,

- if n₂ > n₁ : the light ray passes to a more refractive medium, then,

$sin(theta_2) < sin(theta_1)$

so the light ray will be closer to normal ( $theta_2 < theta_1$ ). This is the case in the scheme above.

- if n₂ < n₁: the light ray passes to a less refractive medium, then,

$sin(theta_2) > sin(theta_1)$

so the light ray will move further from the normal (

$theta_2 > theta_1$ ).

Critical Angle and Refraction

$sin(theta_2) = (n_1/n_2)*sin(theta_1)$

We know that $sin(theta_2) <= 1$ therefore,

$(n_1/n_2)*sin(theta_1) <= 1$

$sin(theta_1) <= n_2/n_1$

This equation is always satisfied when n₂ > n₁.

However, for n₁ > n₂, it is mathematiquelly possible to find values of $theta_1$ such that this inequality is not satisfied ! if we enter these values into the calculator then, it will display NaN as value of $theta_2$ (ie 'not a number').

In fact, for these values of $theta_1$ , there is no refraction but a total reflection of the light ray which doesn't go into the 2^d medium.

The angle $theta_1$ at which this happens is called the critical angle of refractive and is calculated as follows (only when n₂ < n₁),

$sin(theta_1) = n_2/n_1$

$theta_1 = Arcsin(n_2/n_1)$

In summary, we've noticed that if n₂ > n₁ (light moving to a more refractive medium) then the ray of light will always be refracted closer to normal.

If n₂ < n₁ (light moving to a less refractive medium) then the light ray will be refracted away from normal provided that the angle of incidence does not exceed a limit (critical angle of refraction). Beyond this limit, the light ray will be fully reflected (middle 2 will act like a mirror).

Law of Refraction

Refraction ability and medium index

Critical Angle and Refraction

See also