Chapter 3

Refraction and Lenses

Light always travels through something, be it air, water, glass, or even empty space. The stuff that light travels through is called a medium. If a light ray stays in the same medium, it travels in a straight line 1, but when a light ray travels from one medium to a different medium, it changes direction at the surface between the two media. This is called refraction (Figure 1).

Figure 1. A ray of light travelling from air (coloured grey) into glass (coloured blue) bends when it strikes the glass surface. You can click and drag the incoming ray on the left hand side to see how much the passage from air to glass changes the direction of the ray. (Note that at extreme glancing angles, there is also a small reflected ray.)

What angle of the light ray causes no bending?

Snell’s Law.

The ray in Figure 1 bends as it passes from air into glass. The amount of bending is described by Snell's Law, named after the Dutch scientist Willebrord Snellius, who developed the formula in 1621. (The amount of bending was first described by Ibn Sahl, a Persian mathematician who lived in Baghdad, around 984CE.)

To describe and use Snell's Law we need some definitions:

The surface normal and the angles of incidence (red) and reflection (green) are shown in Figure 2.

Figure 2. The surface normal is an imaginary line at right angles to the refracting surface. Here the surface normal is drawn as a dotted line. The angle of incidence is the angle between the incident ray and the surface normal. Here it is drawn as a red wedge. The angle of refraction is the angle between the refracted ray and the surface normal. Here it is drawn as a green wedge.

You can click and drag the incoming ray to see how the angles of incidence and refraction change. What can you say, in general terms, about the relationship between the angle of incidence and the angle of refraction?

When a ray of light travels from air into glass (which is what is being simulated in Figure 2), the relation between the angle of incidence and the angle of refraction is

\[\sin(\text{angle of incidence}) = 1.52 \times \sin(\text{angle of refraction})\tag{1}\]

(The notation \( \sin( x ) \) means the sine of the angle \( x \). If you don't know what a sine is, read the trigonometry appendix.) The number 1.52 is called the index of refraction or the refractive index. Using this formula, we can work out the angle of refraction if we know the angle of incidence, or vice versa. For example, if the angle of incidence is \( 45^\circ \), the angle of refraction is \( 28^\circ \), because \( \sin(45)=1.52\times\sin(28) \)

If a ray of light travels from air into water, the relationship between the angle of incidence and the angle of refraction is

\[\sin( \text{angle of incidence}) = 1.33 \times \sin(\text{angle of refraction})\tag{2}\]

The only thing that has changed is the number on the right hand side: \( 1.33 \) is the refractive index for water. Each substance has its own refractive index, and some indices of refraction are given in the table below. There is no simple reason why a substance has a particular index of refraction, but denser substances tend to have higher indices. The index of refraction is never less than \( 1 \), and only rarely more than \( 2 \).

Medium Index of Refraction
Empty Space 1.0
Air 1.0003
Water 1.333
CR39 1.498
Crown Glass 1.52
Polycarbonate 1.585
Sapphire Glass 1.77
Diamond 2.42

The index of refraction for air is \( 1 \) (or nearly so), so Equation (2) for a ray of light passing from air into water can also be written as

\[1\times \sin(\text{angle of incidence}) = 1.33 \times \sin(\text{angle of refraction})\]

That is, each \( \sin( ) \) is multiplied by the refractive index of the medium that the light ray is in, air on the left and water on the right. Generally, for a ray travelling from one medium to another medium, the angles of incidence and refraction are related by the formula

\[n_{in} \times \sin(\text{angle of incidence}) = n_{out} \times \sin(\text{angle of refraction})\tag{3}\]

where \( n_{in} \) is the refractive index of the medium that the incident ray is travelling in, and \( n_{out} \) is the refractive index of the medium that the refracted ray is in, as it travels out from the surface. For example, if a ray of light travels from water (refractive index \( 1.33 \) ) to glass (index \( 1.52 \)), equation (3) says

\[1.33 \times \sin(\text{angle of incidence}) = 1.52 \times \sin(\text{angle of refraction})\]

Equation (3) is called Snell's Law.

Critical Angle and Total Internal Reflection.

So far, we've only considered light travelling from a low index medium (e.g. air) to a high-index medium (e.g. glass). In these cases, the angle of refraction is always smaller than the angle of incidence, as you can see from Figure 2. However, when light travels from a high-index medium to a low-index medium, the angle of refraction is always bigger than the angle of incidence (Figure 3 ).

This introduces a new phenomenon. Consider a ray of light travelling from water (index= \( 1.33 \)) into air, with an angle of incidence equal to \( 65^\circ \). Snell's law then says

\[1.33\sin{(65)}=\sin{(\text{angle of refraction})}\]

from which you can work out that \( \sin{(\text{angle of refraction})} = 1.205 \). However, if we try and work out the angle of refraction on a calculator by typing \( \sin^{-1}(1.205) \) we get an error, because the sine of any angle cannot be bigger than \( 1 \). Thus, Snell's Law doesn't work in this case.

What happens instead is that, when the angle of incidence is big enough, the light ray does not escape the denser medium and instead gets reflected. This is called total internal reflection. You can see total internal reflection in action by dragging the incident ray in Figure 3. At high enough angles, the ray is reflected instead of being refracted. The "total" in total internal reflection is because we always get some reflection at a surface; total indicates that all the light is reflected.

Figure 3. Here a ray of light travels from a medium with high refractive index (water in this case) to one with lower refractive index (air). If you click and slowly drag the incident ray, you can see the amount of reflection increases until the ray goes past the critical angle of about \( 49^\circ \). At that point, no light escapes into the air and all the light is reflected. This is called total internal reflection. You do get some partial reflection at angles less than the critical angle. It's just that past the critical angle, all the light is reflected, hence the word total.

The critical angle is the biggest angle of incidence that allows refraction to occur. The biggest possible angle of refraction is \( 90^\circ \), so the critical angle is the angle of incidence when the angle of refraction is \( 90 \); that is

\[n_{in}\sin{(\text{critical angle})}=n_{out}\sin{(90)}\]

Because \( \sin{(90)}=1 \), this simplifies to

\[n_{in}\sin{(\text{critical angle})}=n_{out}\]

For example, when a ray travels from water (\( n_{in}=1.33 \) ) to air \( (n_{out}=1) \) as in Figure 3, the critical angle is given by

\[1.33\sin{(\text{critical angle})} = 1\]

Solving this gives a critical angle of \( \sin^{-1}{(1/1.33)}=48.8^o \). You can confirm this using Figure 3, which simulates light travelling from water into air: angles of incidence less than \( 48.8^\circ \) yield refraction (and some reflection); angles greater than this yield only reflection.

Lenses Work By Refracting Light.

Light changes direction when it passes through a refracting surface. This means we can use refracting surfaces to control the direction of light rays. In Figures 1 to 3 above, the refracting surface has been fixed vertically and we have moved the light ray. In Figure 4, however, the light ray is fixed, but you can change the angle of the refracting surface. By moving the refracting surface, you can redirect the light ray almost anywhere you want to.

Figure 4. If you click and drag on this figure, you can tilt the refracting surface. When you do, the light ray is bent by refraction to travel in a different direction. You can choose the direction of the light by moving the surface.

Fresnel Lenses.

We can bend rays of light in any direction simply by changing the angle between the ray and the refracting surface it hits. Now suppose that we have a set of parallel rays, as in Figure 5 . We can bend all these rays to travel towards a single point (i.e. converge them) if we have a glass surface which has a different orientation for each rays. No flat glass block can do this of course, but if the glass block is faceted, as in Figure 5, and each facet can have a different orientation, then we can make the light rays converge to a point. A lens with this sort of design is called a Fresnel lens (pronounced “fraynel” or, less correctly, “freznel”).

Figure 5. A Fresnel Lens Builder. The refracting surface is broken up into 5 facets which can be independently tilted by the sliders on the right. Use them to alter the angle of each facet so each ray passes through the point shown by the yellow dot. What shape is the final Fresnel surface?

The drawback with Fresnel lenses is that the sharpness of the image is limited by the width of the facets. A distant point has millions of parallel rays leaving it. All of the parallel rays that hit one facet are bent by exactly the same amount, so they all remain parallel, and thus they can't converge on a single point (Figure 6). Despite this problem, Fresnel lenses are still useful when the sharpness of the image doesn't matter too much, and they have the advantage that they can be made very flat and light. Fresnel lenses are thus commonly used when one needs a large lens but wants the lens to be thin and light, e.g. in overhead projectors, or large searchlights, or page-sized magnifiers.

fig6
Figure 6. The problem with Fresnel lenses is that the rays that hit a facet are not all bent towards the same point. Here, parallel rays (from the same distant point) hitting the centre three facets are all bent by the same amount, and so remain parallel. That means they can't all converge on the yellow point, and so the image of the point is blurred. The amount of blur can be reduced by making the facets smaller, but it can't be eliminated.

Refraction at Curved Surfaces.

If we want all of the parallel rays from a distant point to converge on a single focus, then each and every ray has to be bent by a different amount. The only way this can be done is by using a smoothly curved surface. To apply Snell’s law to a curved surface, we have to be able to know where the surface normal on a curved surface is (because all the angles in Snell’s law are measured from a surface normal). A surface normal on a flat surface is easy, it’s just at right angles to the surface. A surface normal on a curved surface is a little more subtle. To get the surface normal of a curve, we hold a flat surface against the curve so that it just touches the curve, and draw the normal at right angles to that flat surface ( Figure 7).

fig7
Figure 7. The surface normal to a curved surface (thick curve) is defined by first holding a flat surface against the curve so that it just touches, and then drawing the normal at right angles to the flat surface.

Figure 8 below allows you to design a curved surface which focuses light. For simplicity, this figure only lets you adjust the curve at five places (rather like Figure 5 ). If we were really designing a curved surface to focus light, we would have more control over its shape.

Figure 8. The Curved Surface Builder. The surface shape is controlled by 5 points which can be independently moved. Use the sliders on the right to reshape the surface so the rays pass through the focus point shown by the yellow dot. The surface normals are drawn as short dotted lines. Roughly what shape is the final curved surface?

Spherical Surfaces.

Any curved surface could potentially have a very complicated shape, and may be difficult or expensive to make. However, spherical curved surfaces are fairly easy and cheap to manufacture, and for that reason we tend to use spherical surfaces in optics. Another advantage is they are easier to analyze mathematically. Spherical surfaces are quite good at focussing light to a point, although not perfect. Figure 9 allows you to converge or diverge light rays from a single distant point by adjusting the radius of the spherical surface in the middle.

Figure 9. Spherical Surfaces. Use the slider at the bottom to change the sphere's radius and look at how the rays converge or diverge. Moving the slider right gives a positive radius; moving it left gives a negative radius. The yellow arrow (marked \( r \)) points towards the centre of the sphere. The centre only comes on screen at extreme values of the slider. The refractive indices \( n_{in} \) and \( n_{out} \) refer to Equation (4).

In Figure 9 the parallel rays can be made to converge to a single point (or appear to diverge from a point). That is, the spherical surface has a power. The power depends on the radius of the sphere and the refractive indices. The formula for the power of a spherical surface is

\[F_{surface} = \frac{n_{out} - n_{in}}{r}\tag{4}\]

where \( r \) is the radius of the surface in metres, and \( n_{in} \) and \( n_{out} \) are the refractive indices for the incident and refracted rays.

In Figure 9, \( n_{in} \) is the refractive index to the left of the spherical surface; \( n_{out} \) is the refractive index to the right of the surface; and \( r \) is the distance measured from the surface to the centre of the sphere. The distance \( r \) follows the sign convention, so in Figure 9, radii pointing to the right are positive, and radii pointing to the left are negative, because the light in the figure travels from right to left.

Equation (4) gives rise to four different situations, depending on whether \( \ n_{out} - n_{in} \) is positive or negative, and whether \( r \) is positive or negative. These four situations are shown in Figure 10.

fig10
Figure 10. Spherical surfaces. The shading shows which side of the sphere has the higher refractive index. The light is always travelling from left to right. The refractive indices \( n_{in} \) and \( n_{out} \) refer to the refractive indices to the left and the right of the surface respectively. The surfaces on the left both have positive power; those on the right both have negative power. The top two cases can be created in Figure 9.

The two surfaces on the left are convex surfaces and the two surfaces on the right are concave.

The Paraxial Equation.

The thin lens equation (given in Chapter 1) is

\[V_{in}+F=V_{out}\]

There are no refractive indices in this equation because the vergences \( V_{in} \) and \( V_{out} \) are for light in air, which has a refractive index of \( 1 \). If we put in the refractive indices for air, the thin lens equation would look like

\[1\times V_{in}+F=1\times V_{out}\]

There is a similar formula which is used to work out the change in vergence as light travels across a spherical surface, called the the paraxial equation:

\[\left( n_{in} \times V_{in} \right) + F_{surface} = (n_{out}{\times V}_{out})\tag{5}\]

In this equation \( n_{in} \) and \( n_{out} \) are the refractive indices to the left and right of the surface (assuming the light travels from left to right as in Figure 6) and \( F_{surface} \) is the surface power, given by Equation (4). We can use the paraxial equation in the same way as we used the thin lens equation, as shown in the example below.

Example.

Let's use the paraxial equation to work out where a refracting surface forms an image. An object in air has been placed 0.8m to the left of a refracting surface made of glass. The glass surface has a radius of 25cm, and the centre of the surface is to the right. The glass has a refractive index of 1.5 and, as always, air has a refractive index of 1. Where does the image form?

Answer.

The reason this equation is called the paraxial equation is because it is only valid for light rays that are near to ( par-) the optic axis (-axial) of a refracting surface. The optic axis of a refracting surface or lens is the axis of rotational symmetry. A paraxial ray has to do two things:

  1. It must hit the surface close to the optic axis
  2. It must make a small angle with the optic axis

These two rules ensure that the angle of incidenceof the ray is small. Figure 11 shows examples of two paraxial rays (in green) and two non-paraxial rays (in red). The paraxial equation applies to bundles of converging or diverging paraxial rays; if the rays aren't paraxial, the paraxial equation becomes inaccurate. Luckily, we will mostly be dealing with paraxial rays in optics.

fig11
Figure 11. Paraxial and non-paraxial rays. The two rays in green are paraxial because they hit the refracting surface close to the optic axis (dashed line) and there is only a small angle between the optic axis and the ray. The two rays in red are non-paraxial. The one at the top hits the surface far from the optic axis, and the one coming from the bottom has a large angle between the ray and the optic axis. The paraxial equation can only be used on rays like those in green.

Focal Lengths of Surfaces.

The image focal length and object focal length can be defined exactly as they were for a lens. The image focal length is the distance from surface to image when parallel light ( \( V_{in}=0 \)) strikes the surface, and can be worked out from the paraxial equation as follows:

Step Statement Reason
1 \( n_{in}V_{in}+F_{surface}=n_{out}V_{out} \) The paraxial equation
2 \( F_{surface}=n_{out}V_{out} \) Set \( V_{in}=0 \)
3
\[ \frac{F_{surface}}{n_{out}} = V_{out} \]
 Divide both sides by \( n_{out} \)
4
\[ \frac{n_{out}}{F_{surface}} = \frac{1}{V_{out}} \]
 \( 1/V_{out} \) is the image distance, so take the reciprocal of both sides
5
\[ \frac{n_{out}}{F_{surface}} =\text{image distance} \]
 \( 1/V_{out} \) is the image distance
6
\[ \frac{n_{out}}{F_{surface}} =\text{focal length} \]
 The image distance is the focal length when \( V_{in}=0 \)

Thus, the image focal length \( f_{img} \) is

\[f_{\text{img}} = \frac{n_{out}}{F_{surface}}\]

The object focal length is the distance from the surface to an object which causes parallel light to leave the surface (\( V_{out}=0 \)). We can show that the object focal length is

\[f_{\text{obj}} = \frac{-n_{in}}{F_{surface}}\]

The difference between these focal lengths and the ones given in Chapter 1 is the involvement of the refractive indices to the left and right of the surface. It means that the two focal lengths for a refracting surface are different, unlike the two focal lengths of a thin lens which are the same.

Apparent Distances.

If you've ever looked into a swimming pool, you might have noticed that it appears shallower than it really is. That is, the apparent depth of the pool is less than the true depth. This is a consequence of the paraxial equation that was introduced just above.

Figure 12. This diagram shows what happens to the vergence of light as it leaves a medium of high refractive index to one of low (here, from water to air). The rays leaving the surface are diverged more. If followed backwards, they create a virtual image. The distance of the virtual image from the surface is less than the distance of the object from the surface. However, if you move the object to the other side, in the less dense medium, the situation is reversed: the virtual image is now further away than the object.

Take a flat surface, with a high refractive index to the left, and air to the right, as in Figure 12. The surface power of the flat surface is simply zero. There is an object somewhere to the left of the flat surface, in the higher index medium. By the time the rays of light from the object strike the flat surface, they have some vergence \( V_{in} \). According to the paraxial equation,

\[\left( n_{in} \times V_{in} \right) + F_{surface} = (n_{out}{\times V}_{out}) \]

But since \( F_{surface} = 0 \), this becomes

\[n_{in} \times V_{in} = n_{out}{\times V}_{out}\]

or, rearranging,

\[V_{out}=\frac{n_{in}}{n_{out}}V_{in} \tag{6}\]

so the vergence has changed because of the different refractive indices. However, unlike a lens which adds vergence, the flat refracting surface has multiplied the vergence. In the case of Figure 11, the light leaving the surface is diverging more than the light entering it, and this is always the case when light goes from a more dense to a less dense medium.

If the increase in divergence reminds you a little of a negative lens, it should, because the divergent light leaving the surface creates a virtual image of the object which happens to be closer than the object. The distance from the surface to the virtual image is just \( 1/V_{out} \). If we looked at the diverging light leaving the surface, we would see the object at the location of the virtual image, and not at its true location.

Example.

Let's figure out how deep a \( 2 \text{m} \) pool appears to be.

So if you're \( 1.75\text{m} \) tall, the pool looks shallow enough to stand in, but in fact you'll go over your head if you try.

Self-Test

Now that you've read this chapter, you can do a self-test

Ibn Sahl's Law.

Although we call the law of refraction "Snell's Law", it was actually discovered by Persian mathematician Ibn Sahl who lived in Baghdad from roughly 940-1000 CE (CE means Current Era and is the same as AD). Here we're going to have a quick look at this.

The figure below is a short slideshow setting up Ibn Sahl's law. Click through the slideshow and read all the captions carefully, then continue.

a
Figure 13(a) A page from Ibn Sahl's optics treatise showing his law of refraction. The two labels \( L_1 \) and \( L_2 \) (in blue) have been added to the diagram, and indicate the lengths of the two lines. Ibn Sahl said that
\[\dfrac{L_1}{L_2} = \dfrac{n_{out}}{n_{in}}. \]

(The curved scribbles to the right are part of another diagram, and we don't care about them.) In the next few slides we will see how this is the same as Snell's Law.

Click the arrow on the right to continue.

a
Figure 13(b) Here, we've added a few more things to Ibn Sahl's drawing.
  • The line marked \( L_2 \) is actually the incident ray going into a refracting surface.
  • The refracting medium has been added as a light blue region.
  • The surface normal has been added as a dotted black line.
  • The refracted ray has been added, as have the angles of incidence and refraction \( \theta_{in} \) and \( \theta_{out} \) from Snell's Law.

Click the right arrow to continue.

a
Figure 13(c) Here we've added a dotted line which traces the refracted ray backwards. The label \( L_1 \) is the length of this imaginary line.

Click the right arrow to continue.

a
Figure 13(d) This diagram shows that the angle of incidence \( \theta_{in} \) on the surface normal is the same as the angle between the line \( L_2 \) and the horizontal line in Ibn Sahl's drawing, because these are complementary angles (the surface normal is parallel to the line that Ibn Sahl drew at the top of his diagram)

Click the right arrow to continue.

a
Figure 13(e) This diagram simply shows that the angle of refraction \( \theta_{out} \) on the surface normal is the same as the angle between the line \( L_1 \) and the horizontal line in Ibn Sahl's drawing, because again these are complementary angles.

Click the right arrow to continue.

a
Figure 13(f) Almost there. All the Snell's law additions have been removed, leaving only the lines \( L_1 \) and \( L_2 \), and the angles \( \theta_{in} \) and \( \theta_{out} \) .

One another line has been added to the drawing. The length of this line is O, and it's been named O because it's opposite the two angles \( \theta_{in} \) and \( \theta_{out} \) . \( L_1 \) and \( L_2 \) are hypotenueses of two triangles.

Now read on:

In Figure 13(f), we have two angles \( \theta_{in} \) and \( \theta_{out} \), and three lines with lengths \( L_1 \), \( L_2 \) and \( \mathbf{O} \). From what we have in Figure 13(f), and Ibn Sahl's Law, we can get Snell's Law, as follows:

Step Statement Reason
1 \( \dfrac{L_1}{L_2} = \dfrac{n_{out}}{n_{in}} \) Ibn Sahl's Law from Figure 13(a)
2 \( \dfrac{n_{in}}{L_2} = \dfrac{n_{out}}{L_1} \) Cross multiply
3 \( \sin{\theta_{in}} = \dfrac{\mathbf{O}}{L_2} \) From Figure 13(f), sides of triangle
holding \( \theta_{in} \)
4 \( L_2 = \dfrac{\mathbf{O}}{\sin{\theta_{in}}} \) Cross multiply statement 3
5 \( \sin{\theta_{out}} = \dfrac{\mathbf{O}}{L_1} \) From Figure 13(f), sides of triangle
holding \( \theta_{out} \)
6 \( L_1 = \dfrac{\mathbf{O}}{\sin{\theta_{out}}} \) Cross multiply statement 5
7 \( \dfrac{n_{in}}{\mathbf{O}/\sin{\theta_{in}}} = \dfrac{n_{out}}{L_1} \) Substitute statement 4 for \( L_2 \)
8 \( \dfrac{n_{in}}{\mathbf{O}/\sin{\theta_{in}}} = \dfrac{n_{out}}{\mathbf{O}/\sin{\theta_{out}}} \) Substitute statement 6 for \( L_1 \)
9 \( \dfrac{n_{in}}{1/\sin{\theta_{in}}} = \dfrac{n_{out}}{1/\sin{\theta_{out}}} \) Multiply both sides by mathbf{O}
10 \( \dfrac{n_{in}\sin{\theta_{in}}}{1} = \dfrac{n_{out}}{1/\sin{\theta_{out}}} \) Multiply top and bottom of left hand side by \( \sin{\theta_{in}} \)
11 \( \dfrac{n_{in}\sin{\theta_{in}}}{1} = \dfrac{n_{out}\sin{\theta_{out}}}{1} \) Multiply top and bottom of right hand side by \( \sin{\theta_{out}} \)
12 \( n_{in}\sin{\theta_{in}} = n_{out}\sin{\theta_{out}} \) Simplify to get Snell's Law

Thus Snell's Law can be derived from Ibn Sahl's law (and vice versa, of course).

Why is it called Snell's Law then if Ibn Sahl was first?

Who knows. Racism perhaps? There's a good chance Willebrord Snell knew the work of some of the Persian mathematicians from Ibn Sahl's time, but it's hard to prove he read Ibn Sahl's book. Even if he had, he might not have bothered crediting Ibn Sahl (academic credit was even less common back then than it is now.)

The only thing in favour of calling it Snell's Law rather than Ibn Sahl's Law is that Snell provided us with a formula we could use for calculation, but Ibn Sahl didn't. He could have, easily, because trigonometry was well-known in his time, and he would have been able to do the steps I took above, but he chose not to. We may never know why, but if I were to guess, I'd say it was because Ibn Sahl, like most mathematicians of his time, had enormous respect for ancient Greek geometry, and the Greek style of geometry did not seem much interested in formulas.

  1. Actually, light is bent by gravity too, but we don't usually care about this because the effect of gravity is very weak.  ↩