Why can't flat lenses focus light?
Published: September 30, 2013
Actually, flat lenses can focus light if they are built properly. Let us look at the basics of lenses and then at different types of flat lenses.
A traditional lens is essentially a curved piece of glass that bends light through a process known as "refraction". Whenever a ray of light travels from one transparent material, such as air, into another transparent material, such as glass, it changes direction (except in the special case of the light's direction being perfectly perpendicular to the interface). This change of direction occurs right at the interface between the two materials, and not deep in one material. The light bends because one material has electrons that interact more strongly with the light passing through it than the other material, causing the light to slow down and bend. The ability of a material to bend light is quantified by its "refractive index". A material with a higher refractive index, such as diamond, will bend light more. When light goes from a standard low-index material (such as air) into a standard high-index material (such as glass), it always bends towards the normal to the surface. In contrast, when the light goes from a standard high-index material (such as glass) into a low-index material (such as air), it always bends away from the normal. This is illustrated in the diagram.
If you look closely at the diagrams above, you will realize that if light enters one surface of glass and bends, and then exits a surface that is parallel to the first surface, it will bend back by the same amount and end up not bent at all. If we are using simple, standard, smooth, uniform pieces of glass, we have to curve one side of the glass so that the light remains somewhat bent upon exiting the piece of glass. This is how traditional lens are built: with one side (or both) curved. A concave lens is a lens that is thinner at the center than at the edges. If you follow the rays of light through a concave lens as I have done in the diagrams below, you find that a concave lens spreads out light. In contrast, a convex lens has a thicker center than its edges and tends to focus the light. A standard, flat, uniform piece of glass does indeed fail to focus or disperse the light, as can be seen in the rightmost diagram below. That is why we can look out of flat glass windows and see the outside world as it really is. But there are more things in the world than standard, flat, uniform pieces of glass.
A Fresnel lens is an interesting object that starts with a convex shape and then has useless material removed until you have an object that is mostly flat but still retains the essential properties of a convex lens. Look closely at the middle diagram above of a standard convex lens. The bending of light happens entirely at the curved exit surface. Because the entrance surface is normal to all the rays, none of the light bends at this surface. We can therefore remove most of the glass material as long as we keep the entrance surface normal to the rays entering. This is accomplished by dividing the lens into sections and removing as much glass as possible from each section while still allowing the entrance surface to be normal to the entering rays. This process is shown in the diagram below. Lastly, all the different sections can be aligned in a flat plane to make the material savings worthwhile without affecting the focusing ability much. The result is an object that is virtually flat (although it is a little rough in an essential way), but is able to focus light just like a convex lens.
The great advantage of a Fresnel lens is that you can make a huge lens that is still thin on average. Fresnel lenses are used in lighthouses to focus the light forwards, giving lighthouses their distinctive look. Fresnel lenses are also used in overhead projects, car headlamps, and emergency signal lights.
You may complain that a Fresnel lens is not really flat, so it does not count. Fair enough. Let us move on to another type of lens that is really flat: the graded-index lens. If you remember, we discussed the fact that a higher-index material will bend light more. If we vary the index of refraction of the lens from its center to its edge instead of varying the normal direction by curving the surface, we can get the same kind of focusing effect. To make the refractive index of the material change as you go from its center to edge, you just have to gradually change the material. For instance, a graded-index lens can be made by stacking thin layers of slightly different types of glasses in a controlled way. The effect is shown in the diagram below. Graded-index lenses are used in scanners and photocopiers where the flat lens shape makes them easy to mount.
You may still complain that a graded-index lens is indeed flat but not uniformly one material. A graded-index lens is, in a sense, many little lens made out of different material stuck together. Is there a lens that is both flat and uniform, and yet is still able to focus light? Amazingly the answer is yes, but you have to use non-standard materials.
There is a special kind of man-made optical material known as "meta-material", "left-handed material", or "negative-index material". The interesting thing about meta-materials is that they not only refract light, they also flip light to the other side of the normal upon refraction. The situation is illustrated in the diagram below.
This flip may seem trivial, but this behavior can lead to interesting results. For one thing, this property can be used to make cloaking devices. More to our point, though: a flat, uniform piece of meta-material can act just like a convex lens and is able to focus light. We can show this by tracing the light rays through a flat piece of meta-material, being careful to flip the rays across the normal as happens in real life. The result is shown below.
Meta-materials do not occur in nature and they are hard to fabricate, so there are not yet many commercial applications using flat plates of meta-materials to focus light.