Spheric, Aspheric and Freeform lenses
- Subhabrata Bhattacharya
- May 17, 2022
- 7 min read
Updated: Mar 21
The objective of this article is to understand different types of lenses and their effect on spherical aberration correction. We understand that spherical aberration occurs when marginal rays of light are over refracted (or under refracted) than the paraxial rays of light. The amount of spherical aberration that occurs while marginal rays pass through the lens, will depend on the lens shape in itself. For those of us who are connected to ophthalmology, a spherical lens design almost always connects to a biconvex shape. This is because all IOLs are shaped biconvex. But in effect, a spherical lens could be a plano convex, a convex plano, or even a meniscus shaped lens.
Thus all spherical lenses, be it a biconvex or a plano convex, or otherwise would give rise to spherical aberration. Spherical aberration cannot be totally eliminated by spherical lenses, although its effect can be minimized with a spherical lens of an appropriate bending of light. This is also called the Coddington Shape Factor. Another factor that also determines the amount of spherical aberration that happens through a particular spherical lens is the position of the image and object distances, also called the Coddington Position Factor. Combining the two, would help us to achieve the least spherical aberration with a spherical lens.
C (coddington shape factor) = r2 +r1/r2 - r1 ----- (1)
( wherein the r2 and r1 are the two radius of curvature of the anterior and posterior of the lens)
P (coddington position factor) = i + o/i- o ------- (2)
( where i and o are image and object distances)
Considering Fig 1, the minimum spherical aberration that could be achieved through IOLs, appear to be a shape factor between 1 and 0 when the object is at infiniti.
However, one other conclusion that we can draw from Fig 1 is that regardless of the design of the spherical lens, that is the shape factor, the spherical aberration cannot be altogether eliminated with such lenses. And herein comes the importance of aspheric lenses. But before we describe aspheric lenses, let us look into the spherical lens surface in Fig 2, so that it would be easier then to understand different aspheric lenses when we further go ahead.
Spherical lenses - Spherical lenses are a cross section of a circle. A circle can be defined as a set of points from the same plane which are having an equal distance from a centre point. Thus a line that joins the centre of the circle to the edge of lens is the radius (r). The radius of curvature (R) is constant across the surface of the sphere, therefore the surface contour of the lens follows a regular arc where the lens surface is always the same distance away from the centre of curvature (centre point).
The Z(y) represents the deviation of the surface of the sphere from the tangent drawn as a straight line on the vertex of the lens. The 'y' is the distance from the vertex of the lens to the saggital (sag) point being measured. By finding the Z(y), that is the sag, we can understand the surface of the sphere or construct the sphere.
The equation therefore for a spherical surface is :
It is not about memorizing this formulae, but understanding the three components of the formulae, that is the 'y' is the distance from the vertex where the consecutive sagittal, Z(y) is being measured as a function of the R (radius of curvature of circle). With this formulae therefore you can calculate the full optical surface of the sphere.
Conic section
Like the circle, there are other shapes like elipse, parabola, hyperbola, etc. The earth's orbit around the sun is an elipse. On the other hand the shape of a parabola and hyperbola is described in Fig 3. In a parabola, all rays of light passing through the lens surface will come and meet at the same point. Thus, the mirror of a torch light, or a dish of the satellite is shaped like a parabola. Since a parabola has the property to focus all rays in a single point, therefore satellite dish is made in the shape of parabola. Thus all rays that reflect off the dish of the satellite are focussed on the receiver. A human cornea that is shaped like a parabola has no spherical aberration. All such shapes come under one category and is called conics as they can be cut out from a cone, that is they are all part of cross section of a cone. That is why they are called conic section.
