# Can a complex lens be represented by wavefront aberrations?

Usually complex lenses are described by their geometric design - usually a sequence of spherical cap sufraces and materials. The lens behavior, ie. how it transfers rays, can be evaluated by ray tracing. On the other hand wavefront aberrations can be computed for a particular lens. Are they enough to represent the ray transfer behavior of a lens?

Is it true that wavefront aberrations describe the way a beam of collinear rays are transferred though the lens? Is the list of wavefront aberrations valid only for a single direction of collinear rays or for any direction?

Thanks!

PS: I'm not a physicist but a computer graphics programmer, so excuse me in case the questions seem trivial for an optician :).

• What do you exactly mean by "wavefront aberrations"? If general relation of electric fields $E_{in}(x,y)\mapsto E_{out}(x,y)$ then yes. If sth like a photo of a grid projected by the lens - then no. – Piotr Migdal Feb 16 '12 at 18:18
• What's a "complex lens" anyway? Is a classic prism a complex lens? – MSalters Feb 17 '12 at 9:34
• The meaning of wavefront aberrations should be as in link. – Bohumir Zamecnik Apr 10 '12 at 6:25
• By a complex lens is meant a lens system composed of multiple elements such as refractive lenses made of glass or diaphraghms to limit the rays. We can assume the lens system is rotationally symmetric around the optical axis. – Bohumir Zamecnik Apr 10 '12 at 6:26

## 1 Answer

From your questions it follows that you understand all the problems already. The aberrations are computed independently for each direction of the beam incidence—and each shape of the beam. For a so called thin lens (the thickness may be ignored) the whole transformation of the field by the lens can be expressed as adding a fixed (for any beam) phase shift at each point of the lens and multiplying the result by the amplitude function describing the lens aperture:

$$E_\text{out}(x,y) = E_\text{in}(x,y) \exp(\Delta\phi(x,y))A(x,y),$$

where $A(x,y)$ equals 1 within the lens and 0 everywhere else. However this is an approximation which will give smaller or larger error for any real lens. In a real lens the equation should be written as

$$E_\text{out}(x',y') \approx \frac{\text{i}}{\lambda}\int_A E_\text{in}(x,y) \frac{\exp(-\text{i}kr(x,y,x',y'))}{r}\text{d}x\text{d}y,$$

where $r(x,y,x',y')$ is the distance between point $(x,y)$ on the left surface of the lens and point $(x',y')$ on the right surface. As you see, the ouput now depends not only on the lens shape (which is defined by $r$) but also on the input field $E_\text{in}$, which is different for each beam shape and each direction of incidence.

The more exact expression for the above integral you can find here.