# Ideal surface for a perfect lens

in this physics lecture, on slide 15-16, it is found that the ideal surface for a perfect lens (which maps a plane wavefront into a perfect spherical wavefront, i.e. which makes focus all input parallel rays into one point) can be an hyperbola or an ellipsoid according to the refraction index ratio being higher or lower than 1: Now, I don't understand this result quite well. My doubts are:

• Imagine the rays start from right (being parallel). In case of an hyperbola, they are already in glass and then go into air. In case of an ellipsoid, they are in air and then go into glass. None of them are actually the common "thin lens" we usually study in basic optics (air - lens - air). How could we adapt these results to a thin lens? Should it have a hyperbolic/ellipsoid shape on both sides?

• I cannot visualize why a spherical surface is not ideal to map a plane wavefront into a spherical wavefront. A spherical surface slow down the input plane wavefront points in a spherical wave. I find strange that this does not occur, whilst the ideal surface are hyperbola and ellipsoid!

• Some books propose a different ideal surface for the perfect lens, precisely a cartesian oval. Other sources say the ideal surface should be parabolic, like for a mirror... which is the truth?

• You are assuming that the rays must be parallel inside an ideal lens. That is not a requirement. Rather that points S and P are fixed for all the rays that go through the lens. Sep 30, 2022 at 12:40

Spherical surfaces are used for lenses because they are much easier to manufacture precisely. The curvature is the same everywhere. You rub a spherical grinding surface all over the glass.

Lenses do have to be precise. An error of a wavelength of light matters. For visible light, that is about half a micron.

Fortunately, lenses typically use relatively large diameter spheres. In this case, a sphere is extremely close to the ideal shape.

For a small diameter lens, the approximation is close enough to perfect. For a larger diameter, the difference in shape is larger. The outer edge is a good match to an ideal shape with a different wavelength. The outer rays come to a focus in a different spot than the center rays. This is called spherical aberration, the aberration or error cause by using a spherical surface instead of an ideal surface.

To make an ideal lens from your left diagram, put a planar surface on the right side of the lens. Planar wavefronts will pass through a planar surface without changing direction.

For the right diagram, put a spherical surface centered on the focal point on the left surface. Spherical wavefronts will not change direction passing through it.

Of course this only works for waves parallel to the axis. There are other aberrations to deal with for off axis rays. And variations in wavelength also cause aberrations because the index of refraction varies with wavelength. Lens design gets complicated quickly. But for an application like focusing a laser, these simple lenses solve the problem.

In a Cartesian (Maxwell) oval the 2nd focal point is a finite distance from the first. These transform a spherical wavefront emanating from the 1st focus into another spherical wavefront concentrated into the 2nd focal point. When the 2nd focus is moved to infinity the oval becomes a conic section (hyperboloid or ellipsoid) and the 2nd wavefront becomes planar, a plane wave whose rays are parallel as your pictures show.

• So should the lens shape be different depending on if the input rays are in focus or not? Does it mean there is no ideal shape for all? Sep 30, 2022 at 15:11
• It depends on what you mean by "ideal". The cartesian oval maps a single point ideally (in a geometric sense, diffraction neglected) into another point, usually called stigmatic imaging, but it cannot map even two different points in that "ideal" sense, only one. It is possible to create a lens with several even four foci but these are complicated things used, as far as I know, only in RF antennas. Sep 30, 2022 at 15:28
• There is a famous theorem by Maxwell-Caratheodory that states the impossibility of perfect, i.e., stigmatic, imaging of an arbitrarily small but finite volume, and mapping a surface can only be done in a very limited sense, see Born-Wolf Chapter 4.2. Sep 30, 2022 at 15:29
• I mean: even if we could manufacture any arbitrary surface, it won't be perfect because the ideal surface to make a spherical wavefront converge is different from that designed to make a planar wavefront converge. Sep 30, 2022 at 17:46