Snell's law

Snell’s Law Derivation

The diagram above shows the path of a light ray that moves from a medium with refractive index \( n_1 \) to one with refractive index \( n_2 \), where \( n_2 > n_1 \), and so the ray bends towards the normal. You might have already come across the equation that relates the speed of light to the wavelength and frequency of a wave:

$$c=v\lambda,$$

where \( c \) and \( v \) represent the same quantities as in the first equation, and \( \lambda \) is the wavelength of the light measured in meters.

When a light ray moves from one material to another, its frequency does not change, but its speed does, so this means that its wavelength must change. In the situation shown above, as light moves from the first medium to the next, its speed decreases, and so the wavelength must also have decreased. The diagram shows the wavefronts for the incoming and outgoing light rays. You can see that they are more widely spaced after crossing the boundary. We can use trigonometry along with the equations previously stated to prove Snell's Law.

Some of the useful distances and angles are labeled on the diagram. \( A \) and \( B \) represent distances that the incoming and outgoing rays travel in a certain time period (they could represent the wavelengths, and this will lead to the same result). They are related to the speeds of light in the different mediums by:

\[ A = v_1 T \]

\[ B = v_2 T\]

In these equations, \( v_1 \) and \( v_2 \) represent the speeds of light in mediums \( 1 \) and \( 2 \), respectively, and \( T \) is an arbitrary time period measured in seconds.

We can identify two right triangles in the boundary. Then, using trigonometry, we can relate the angles \( \theta_1 \) and \( \theta_2 \) to the known sides.

$$\sin(\theta_1)=\frac AC$$

$$\sin(\theta_2)=\frac BC$$

Hence:

$$\frac{\sin(\theta_1)}{\sin(\theta_2)}=\frac AB$$

$$\frac{\sin(\theta_1)}{\sin(\theta_2)}=\frac{v_1\mathrm T}{v_2\mathrm T}$$

$$\frac{\sin(\theta_1)}{\sin(\theta_2)}=\frac{v_1}{v_2}$$

Then, we can express the velocities in each medium in terms of their refractive index by using the definition of the refractive index.

$$ v=\frac cn \Rightarrow \frac{v_1}{v_2}=\frac{\displaystyle\frac c{n_1}}{\displaystyle\frac c{n_2}}=\frac{n_2}{n_1}$$

This leads to the expression:

$$\frac{\sin(\theta_1)}{\sin(\theta_2)}=\frac{n_2}{n_1},$$

which can be rearranged to give Snell's Law:

$$n_1\sin(\theta_1)=n_2\sin(\theta_2).$$

Snell's Law Equation

As derived above, Snell's Law is

$$n_1\sin(\theta_1)=n_2\sin(\theta_2),$$

where \( n_1 \) is the refractive index of the material on the side of the boundary where the light ray starts and \( n_2 \) is the refractive index of the second material. The diagram shown below illustrates what the angles in the equation represent. Snell's Law can be used to find these quantities when light is incident on different boundaries. The procedure for doing this is explained in the next section.

The diagram above is called a ray diagram, and it has several key features to remember:

• The line perpendicular to the surface of the new medium is called the normal. If the incident ray is along the normal, no refraction will occur, and the refracted ray will also be along the normal.

• The angle between the incoming light ray and the normal is known as the angle of incidence.

• The angle between the outgoing ray and the normal on the opposite side of the boundary is known as the angle of refraction.

Snell’s Law Procedure

Snell’s Law can be used in many problems to work out how a light ray will behave upon striking a boundary between two mediums of different refractive indices. There is a useful way of checking your calculations when working with Snell’s Law. Consider a car driving from a smooth road across a boundary into a muddy field. The car will move slower in the field than on the road. If the car were to move into the field at an angle, the side of the car entering the field first would move slower than the side still on the road, which would cause the car to turn in the direction of the slower-moving side until both sides move at the same speed. We can think of a light ray in the same way:

• If a light ray moves from a medium with a lower refractive index to a higher one, then the path of the ray will bend towards the normal (the light ray moves slower in the higher refractive index medium, just like how the car moves slower in the field).

• If a light ray moves from a medium with a higher refractive index to a lower one, then the path of the ray will bend away from the normal.

Snell's Law Example

There is a very particular and interesting case of the refracting angle that we can explore using Snell's Law. If we increase the incident angle of a light ray striking a boundary, the angle of refraction will increase as well, until it is greater than \( 90^\circ \). Then, all of the light is reflected instead of moving out of the first medium —this is called total internal reflection. This can only occur when the light ray bends away from the normal —when it moves from a lower refractive index material to a higher refractive index material.

Snell's Law is

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

It can be rearranged to

$$\frac{\sin(\theta_1)}{\sin(\theta_2)}=\frac{n_2}{n_1}$$

We want to find the angle that the light is refracted along the boundary, called the critical angle.

When light is reflected along the boundary, the refracted angle is \( 90^\circ \), so

$$\sin(\theta_2)=\sin(90)=1$$

and our equation becomes

$$\sin(\theta_c)=\frac{n_2}{n_1},$$

where \( \theta_c \) is the critical angle. This can be rearranged to find the critical angle:

$$\theta_c=\sin^{-1}(\frac{n_2}{n_1}).$$

Total internal reflection occurs when the angle of incidence is greater than the critical angle. The phenomenon of total internal reflection is used in many different applications, such as sending information down optical fibers and observing minuscule objects through microscopes!

Fig. 6. Light is sent long distances along optical fibers through the use of total internal reflection, image by mammothmemory.

Sending information using optical fibers is more efficient than traditional copper wires, as optical fibers can carry more information with less loss in the signal. These fibers allow transferring around 1 Gigabyte of data per second! It is amazing what we can achieve by understanding Snell's Law!