Angle of Transmission - Snell's Law

The Angle of Transmission calculator computes  the angle of refraction based on the refractive indices of two media and the angle of incidence based on Snell's Law.

INSTRUCTIONS: Choose units and enter the following:

• (`theta_i`) Angle of Incidence
• (n₂) Refractive index of medium 2
• (n₁) Refractive index of medium 1

Angle of Transmission(`theta_t`): The calculator returns the angle in degrees.  However this can be automatically converted to compatible units via the pull-down menu.

The Math / Science

Snell's law (also known as Snell-Descartes law and the law of refraction) is a formula used to describe the relationship between the angles of incidence and transmission, when referring to light or other waves passing through a boundary between two different isotropic media, such as water, glass, or air.  

In optics, the law is used in ray tracing to compute the angles of incidence or transmission, and in experimental optics to find the refractive index of a material. The law is also satisfied in metamaterials, which allow light to be bent "backward" at a negative angle of refraction with a negative refractive index.

Snell's law states that the ratio of the sines of the angles of incidence and refraction is equivalent to the ratio of phase velocities in the two media, or equivalent to the reciprocal of the ratio of the indices of refraction.  The relationship can be seen in the following formula:

`(sin theta_t)/(sin theta_i) = v_2 / v_1 = n_1 / n_2`

where:

• `theta_t` = angle of transmission
• `theta_i` = angle of incidence 
• n₂ = refractive index of medium 2
• n₁ = refractive index of medium 1
• v₂ = velocity of light in medium 2
• v₁ = velocity of light in medium 1

The largest possible angle where of incidence that still result in refracted light is called the Critical Angle.   The formula for Critical Angle between refraction and reflection is:

    `theta_c = sin^(-1) (n_2/n_1)`

where:

• `theta_c` = Critical Angle
• n₂ = refractive index of medium 2
• n₁ = refractive index of medium 1

Reflection

The Law of Reflection is fairly straightforward: `theta_i = theta_r`¹.  As you can see, the angle of reflection is entirely independent of the indices of refraction of the two materials.   Both `theta_i` and `theta_r` are measured from the normal, but they're on opposite sides of the normal.

Refraction

The Law of Refraction, commonly known as Snell's Law², is `n_1 sin(theta_i) = n_2 sin(theta_t)`.  Both `theta_i` and `theta_t` are measured from the normal, but they're on opposite sides of the normal and interface.

Total Internal Reflection

If `n_2 < n_1`, there's an interesting phenomena termed Total Internal Reflection (TIR)³.  As the name suggests,  TIR is when all of the incident is reflected, so no light transmits into the second material.  To see why, or at least when, this happens, let's look at Snell's Law rearranged to solve for `theta_t`.

• `theta_t = sin^(-1) (n_1/n_2 sin(theta_i) ) `

Since `n_2 < n_1`, there an angle, called the Critical Angle⁴, that is the largest incident angle that will still result in a transmitted wave.  In other words, it's the largest possible value of `theta_i` such that `theta_t = sin^(-1) (n_1/n_2 sin(theta_i) ) ` evaluates to an answer.  The critical angle is given by the formula:

• `theta_c =sin^(-1) ( n_2/n_1)`

It's worth mentioning that the critical angle is also where the angle of transmission is 90 degrees.   This means that the transmitted wave won't travel into the second material so much as along the interface between the two materials.  Any incident angle greater than the critical angle won't result in any transmission at all.  To be clear, TIR and critical angles are only relevant when `n_2 < n_1`, i.e. when the wave travels from a material with a higher index of refraction to a material with a lower index of refraction.

Snell's Law Calculators

• Angle of Transmission
• Angle of Incidence
• Refractive Index of Medium 1 (n₁)
• Refractive Index of Medium 2 (n₂)
• Critical Angle