i want to do analytic ray tracing by using ABCD matrices and retrieve the wavefront at the detector via Zernike coefficients.

$ \begin{pmatrix}x_{out} \\ \theta_{out} \end{pmatrix} = \begin{pmatrix}A & B \\ C & D \end{pmatrix} \begin{pmatrix}x_{in} \\ \theta_{in} \end{pmatrix} $

where $x$ and $\theta$ are position and angle.

A simple setup would be to misalign a lens in an optical system (tilt/decenter). Apparently this induces a coma-like aberration as seen in the wikipedia article for coma.

When i use extended ABCD matrices from literature (Siegmann 3x3 or Shaomin 4x4), tilting a lens would have no effect on the wavefront. However, this is not true in reality.

I found two papers (Link1, Link2) that just add the offset $\triangle \theta$ and $\triangle x$ to the tilt and decenter, respectively.

The only effects i can reproduce are tip/tilt and defocus. However, i cannot produce the effects stated in the paper.

I know this is really specific but is anyone able to help me out here?


The transfer matrix of an optical system, by definition, is the Jacobi matrix at a single point in optical phase space of the general transformation wrought by the system. It therefore describes the system's behavior linearized about that point. In other words, it holds only for rays that travel paths through the system that are near to that of the "chief ray", i.e. the ray that defines the linearization point.

Thus the transfer matrix can only "see" linearized behavior. All lenses described by a $2\times 2$ transfer matrix are therefore perfect: rays converge perfectly to their focusses when described by the transfer matrix. A $4\times 4$ transfer matrix, tracing the $(x,\,y)$ positions and $(\theta_x,\,\theta_y)$ angles of rays, can account for an astigmatic lens: different focusses for the meridional and sagittal planes. By definition, it cannot "see" higher order aberrations, so astigmatism is the only aberration that be modelled.

Even tilts are awkward to model, because a tilt is an inhomogeneous transformation - a constant angle added to ray angles and matrices model homogeneous linear transformations. You can model a tilt by adding the offset to the transformation and then taking it away again. If you do this for a thin lens, the procedure has no effect, reflecting the approximation that a ray through a thin lens's optical center is undeviated. If you do this procedure with a thick lens, i.e. one with a nonzero distance between its principal planes, it will correctly model the deviation caused by the rotation of the "lever arm" between the two principal planes.

Aberrations must be calculated either by full ray tracing or, as you have seen, various ad-hoc correction terms and procedures added to the transfer matrix method. Transfer matrices are most useful in calculating imaging geometry and magnification. They are also symplectic matrices, which means that they also model the important principle of conservation of the optical (Helmholtz) invariant and conservation of étendue and the limitations imposed by these principles on imaging geometry and magnification.

  • $\begingroup$ Hi Rod, thanks for the elaborate answer, i appreciate that alot. I also noticed that adding tilts before and substracting them after a a thin lens does cancel out the effects. If i do the same thing with a thick lens it does not cancel out as you already mentioned. However, this still does only induce tip/tilt (at the detector) and not any coma aberration. I know this might be asked a lot, but can you take a look at one of the two links to two publications i mentioned in the original post. $\endgroup$ – C. Schindlbeck Sep 5 '17 at 7:51
  • $\begingroup$ Therein, they add (without subtracting) lens decentering/tilting and state that this leads to coma aberration (change in the respective zernike coefficient). How is this possible? I am not able to reproduce the effects, neither do i get a reply from the autors. It might not be the highest ranked journal out there, but it does seem to be peer-reviewed. Are they flat out lying, making up results, have an error in their implementation or faking the experimental validation? Or am i missing something? $\endgroup$ – C. Schindlbeck Sep 5 '17 at 7:51
  • $\begingroup$ @C.Schindlbeck I'll have a look at the papers. I'm certain they're using a different/ augmented technique for the aberration. $\endgroup$ – WetSavannaAnimal Sep 5 '17 at 14:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.