No, in general the rays that pass an aberrated lens at different angles are not in phase on the optical axis. If they all were, the lens would have no aberrations. Some paths will arrive almost in phase (e.g. paths close to the optical axis) so that if you put an aperture stop in front of the lens you reduce the effect of the spherical aberration.

I'm not actually considering all rays, just two of them. Let's look at an ideal lens to start with. Here all the rays are in phase when they intersect in the ideal image point since an ideal lens turns a diverging spherical wave into a converging one. I'm always assuming that the object point gives off a spherical wave. Now to the lens with spherical aberration. As you state, all rays close to the optical axis will actually intersect close to the Gaussian image plane almost in phase. But what about just two rays with very different angles? In some point they will intersect. Shouldn't they also be in phase there since that is where they intersect?

Now I am not sure what you meanby "two rays with very different angles" - Two rays at the same but opposite angle (theta and - theta) will intersect at the axis and be in phase with each other there. But they will not be in phase with the ray at the optical axis (theta=0) or with rays at most other values of theta. You can visualize this nicely by putting a ring aperture in front of an aberrated lens - you will get a sharp image in a slightly different plane. This is a nice experiment in our undergraduate optics lab, all you really need is a badly aberrated lens, e.g. a singlet, some cardboard apertures and a lamp.

With a disk aperture of the same size your picture will not be sharp.

Thanks for your help. I've realized now that I was wrong. There was too much wishful thinking since I needed a starting point for a derivation. I've come up with a little thought experiment that shows that they cannot be in phase in general. I start with an ideal lens where all rays are in phase in the image point. Then I deform the lens

everywhere apart from where the rays at angle theta pass through it so that these rays still intersect in the same point. The deformation is such that it leads to a lens with spherical surfaces. Since I've changed the thickness of the lens close to the optical axis thes rays at theta cannot in general be in phase with the theta=0 ray anymore.

@Philip: In your initial question you state that the ray inclined to the optical axis hits the optical axis closer to the lens as the Gaussian image plane. All my intuition agrees with that. However, in the famous textbook by Born&Wolf: Principles of Optics, I found the sentence 'The rays from an axial object point which makes an appreciable angle with the axis will intersect the axis in points which lie in front of or behind the Gaussian focus'. . Along the lines of your nice thought experiment, you easily find artificial lens shapes which let the 'outer rays' hit behind: let the outer parts of the lens approach the profile oh a plano-parallel plate. By the way, do you know that the shape of your 'ideal lens' is hyperbolic (conjectured by Kepler, and proved by Descartes)?

@Ulrich: I always forget whether it's hyper- or parabolic. Thanks for the information. A lens where the outer rays intersect the optical axis behind the Gaussian image plane should have a negative spherical aberration coefficient, I suppose. Is that right? The reason why I brought up this topic is that I'm looking for a derivation for the phase shift due to spherical aberration in the objective lens of a transmission electron microscope (TEM). I thought I had my own but it seems I over-used Fermat's principle. According to literature the correct expression is exp( -i C theta4). Here C is a constant containing the spherical aberration coefficient Cs and the wavelength of the electrons. "theta4" stands for the angle of the ray coming from the object taken to the power of 4. To get this simple formula it might be necessary to make many approximations specific for the TEM, for example the wave length of electrons is about 0.002 nm in this case, the focal length of the lens is about 2mm and so is the value for Cs. The maximum value for theta that needs to be considered is about 0.005 radians. The phase shift should be measured in the Gaussian image plane using a converging spherical wave as a reference. There are derivations in several books about electron microscopy but they don't make complete sense to me. I wonder if somebody could point me to a clear and simple derivation.

Dear Nilotpal,

I agree that that is one of the approximations made in this treatment of lens aberrations. If you think about the dimensions of the optical system compared to the wavelength of the electrons this is, however, an approximation that's easy to defend.

The wavelength is about 0,002 nm, the focal length of the objective lens 2 mm and the diameter of the objective aperture between 10 and 200 micrometers.

I agree that this forces me to talk about this question in a strange way. The aberration is described in terms of geometric optics as deviations from the ideal rays. However, since I want a formula for phase differences I have to "attach" a wave length to each ray. Physics seems to force me into conceptual somersaults like that all the time.