Light Refraction: Bending of light

By: Elisa Mercando

The Main Idea

Light refraction is the wavelengths of light entering different materials, where the speed is different. The refraction of light when it passes through a fast medium to a slow medium causing the light ray to bend between the two mediums.

The wavelength, [math]\displaystyle{ \gamma }[/math], of the radiation can either be shorter or longer (depending on the material). The frequency and period of the oscillation do not change, so the speed of the ray changes ([math]\displaystyle{ speed = \frac{\gamma}{T} }[/math]).

A Mathematical Model

The refraction of light can be explained by Snell's law. Snell's law describes a relationship between the bending of the light rays at the medium and speed of the ray in the two medium.

The rays can be represented as straight lines, and the direction of the incoming and outgoing rays can be expressed as angles between the ray and the normal vector (perpendicular to the surface). We can assign [math]\displaystyle{ \theta_1 }[/math] for the incoming angle of the ray and [math]\displaystyle{ \theta_2 }[/math] for the outgoing angle of the ray. The speed of the ray in the air is [math]\displaystyle{ v_1 }[/math] and the speed in the material is [math]\displaystyle{ v_2 }[/math]. In the time T, the ray travels [math]\displaystyle{ v_1*T }[/math] in the air and [math]\displaystyle{ v_2*T }[/math] in the material. The two triangles formed by the angles share a side, d.

The incoming angle is expressed as [math]\displaystyle{ \sin(\theta_1) = \frac{v_1*T}{d} }[/math] and the outgoing angle is expressed as [math]\displaystyle{ \sin(\theta_2) = \frac{v_2*T}{d} }[/math]. [math]\displaystyle{ \frac{T}{d} }[/math] is the same for both triangles. So, [math]\displaystyle{ \frac{\sin(\theta_1}{v_1} = \frac{\sin(\theta_2}{v_2} }[/math].

The index of refraction is expressed as [math]\displaystyle{ n = c/v }[/math].

Therefore, [math]\displaystyle{ \frac{\sin(\theta_1}{c/n_1} = \frac{\sin(\theta_2}{c/n_2} }[/math]. Then we cancel the common factor, c.

Snell's Law: [math]\displaystyle{ n_1\sin(\theta_1) = n_2\sin(\theta_2) }[/math]

A Computational Model

How do we visualize or predict using this topic. Consider embedding some vpython code here Teach hands-on with GlowScript

Examples

Ray Travels from Water into Glass

A beam of light travels through water and hits a glass surface. The angle [math]\displaystyle{ \theta_1 }[/math] between the incident beam and the normal to the glass surface is 23 degrees. The index of refraction of water is 1.33 and the index of refraction of this type of glass is 1.65. What is the angle [math]\displaystyle{ \theta_2 }[/math]?

Solution:

[math]\displaystyle{ n_1\sin(\theta_1) = n_2\sin(\theta_2) }[/math]

[math]\displaystyle{ \sin(\theta_2) = \frac{n_1\sin(\theta_1}{n_2} }[/math]

[math]\displaystyle{ \theta_2 = arcsin\frac{1.33*\sin(23)}{1.65} = }[/math]18.4 degrees

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