Refraction of Light

What Happens When Light Enters a New Medium?

You have already seen that when light hits a surface, part of it bounces back (reflection). But what about the part that goes through? When a beam of light travelling through one transparent medium meets the boundary of another transparent medium, the transmitted portion changes its direction of travel. This change in direction at the boundary is called refraction.

Refraction only happens when the light hits the boundary at an angle (not head-on). If a ray arrives perpendicular to the surface, it passes straight through without bending. For any oblique angle of incidence (between $0°$ and $90°$), the direction of the transmitted ray shifts as it crosses the interface.

Snell’s Laws of Refraction

Through careful experiments, the scientist Snell worked out two rules that govern how light refracts. These are known as Snell’s laws of refraction:

Law 1 (Coplanarity): The incident ray, the refracted ray, and the normal to the interface at the point of incidence all lie in the same plane. This is the same coplanarity condition you saw for reflection, now extended to refraction.

Law 2 (The Sine Rule): The ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant for a given pair of media:

$\frac{\sin i}{\sin r} = n_{21} \qquad \text{(9.10)}$

Here, $i$ is the angle between the incident ray and the normal, $r$ is the angle between the refracted ray and the normal, and $n_{21}$ is called the refractive index (a number that tells you how much the light bends) of the second medium with respect to the first medium.

Fig 9.8: Refraction and reflection of light at a boundary between two media

A few key points about the refractive index:

• It is a property of the pair of media involved. Change either medium and you get a different value.
• It also depends on the wavelength of the light. Different colours of light bend by slightly different amounts in the same medium.
• It does not depend on the angle of incidence. Whatever angle the light arrives at, $n_{21}$ stays the same.

Bending Towards or Away from the Normal

The value of $n_{21}$ tells you which way the refracted ray bends:

• When $n_{21} > 1$: From Equation (9.10), $\sin r < \sin i$, which gives $r < i$. The refracted ray makes a smaller angle with the normal than the incident ray does, so it bends towards the normal. This happens when light enters a medium where it travels more slowly. Such a medium is called optically denser than the first.

• When $n_{21} < 1$: Now $\sin r > \sin i$, so $r > i$. The refracted ray makes a larger angle with the normal, meaning it bends away from the normal. This is the case when light passes from a denser medium into a rarer one.

A Common Confusion: Optical Density vs Mass Density

Students often mix up optical density with mass density (mass per unit volume), but they are completely different things. Optical density is about how fast light travels through a material, while mass density is about how much matter is packed into a given space. The two do not always go hand in hand.

A good example is turpentine and water. Turpentine is lighter than water (lower mass density), yet light moves more slowly through turpentine (higher optical density). So turpentine is optically denser than water despite being physically lighter.

Connecting Refractive Indices: The Reciprocal and Chain Rules

There are two handy relationships that connect refractive indices across different media.

The Reciprocal Rule

If $n_{21}$ is the refractive index of medium 2 with respect to medium 1, and $n_{12}$ is the refractive index of medium 1 with respect to medium 2, then:

$n_{12} = \frac{1}{n_{21}} \qquad \text{(9.11)}$

This makes intuitive sense. If going from medium 1 to medium 2 bends light towards the normal (meaning $n_{21} > 1$), then going the other way, from medium 2 to medium 1, must bend light away from the normal (meaning $n_{12} < 1$). The two refractive indices are simply reciprocals of each other.

The Chain Rule

If you know the refractive index of medium 3 relative to medium 1 ($n_{31}$) and the refractive index of medium 1 relative to medium 2 ($n_{12}$), you can find the refractive index of medium 3 relative to medium 2 by multiplying them:

$n_{32} = n_{31} \times n_{12}$

This chain rule works because refractive indices are ratios. When you multiply two such ratios, the intermediate medium cancels out, leaving you with a direct ratio between the two end media.

What Happens Inside a Glass Slab: Lateral Shift

Consider a ray of light passing through a rectangular glass slab, like a thick pane of glass sitting in air. The ray meets two parallel surfaces: first the air-glass boundary, then the glass-air boundary.

Fig 9.9: Lateral shift of a ray refracted through a parallel-sided slab

At the first surface (air to glass), the ray bends towards the normal because glass is optically denser than air. It travels through the glass at this new angle. At the second surface (glass to air), the ray bends away from the normal by exactly the same amount. The geometry of the two parallel surfaces guarantees that $r_2 = i_1$, meaning the emergent ray comes out parallel to the original incident ray.

So a glass slab does not change the overall direction of the ray. There is no net angular deviation. However, the emergent ray is not on the same line as the incident ray. It has been shifted sideways. This sideways shift is called lateral displacement or lateral shift. The thicker the slab and the larger the angle of incidence, the greater this lateral shift.

Why Things Look Closer Than They Are: Apparent Depth

Place a coin at the bottom of a glass filled with water and look straight down at it. The coin appears to be closer to the surface than it really is. This is the phenomenon of apparent depth, and it is a direct consequence of refraction.

Fig 9.10: Apparent depth for (a) normal and (b) oblique viewing

Light rays leaving the coin travel upward through the water and bend away from the normal as they exit into air (going from a denser to a rarer medium). Your eyes receive these bent rays and, since your brain assumes light always travels in straight lines, it traces them backward in straight lines to find where they appear to come from. These traced-back lines meet at a point that is higher than where the coin actually sits. The coin therefore looks closer to you than it really is.

For viewing nearly straight down (close to the normal direction), the relationship is straightforward:

$h_1 = \frac{h_2}{n}$

where $h_1$ is the apparent depth (how deep the object looks), $h_2$ is the real depth (the actual distance of the object below the surface), and $n$ is the refractive index of the medium (water, in this case).

Since $n > 1$ for water (and for any medium denser than air), the apparent depth is always less than the real depth. A swimming pool that is 2 metres deep, for instance, looks noticeably shallower when you peer in from above. This is the same reason a stick partially dipped in water appears to be bent at the water surface.