Hi Geoffrey,

Some clarifications:

1.- I assume you are working with an off-axis object. Spherical lenses images a point object on curved surfaces for off-axis locations.

2.- Your lens has no astigmatism. If not, two astigmatic images are formed, the sagital and tangential image since the lens has also field curvature.

3.- Astigmatic lenses always has astigmatism, even for an on-axis point object. Therefore, we should work with those two astigmatic images.

Seidel theory stems from the rotational symmetry of the system, therefore you can work on a meridional plane. If not, Seidel theory extension to non-symmetrical systems must be done with caution.

Toroidal systems can be treated as spherical in two orthogonal meridional planes when the off-axis object lays on one of those planes. Therefore, the two astigmatic images would be in two curved surfaces, likely having different curvatures in those two orthogonal meridional planes. The problem arises when the object is not on those meridional planes, and it is needed to ascertain that the curvatures of the two astigmatic surfaces will be that of a toroidal surface. I guess that the two astigmatic images will not be on a corresponding toroidal surface, but a more complicated surface. Maybe, based on some approximations, you could be right. At this time, I cannot assured this.

The problem is not simple. However, I suggest you read the sections 11.1 and 11.2 of the chapter 11 in Aberration of Optical systems from W.T Welford, and the recent papers from Sheng Yuan & José Sasián in Applied Optics about anamorphic systems.

Oh, finally, as far I know, crossed cylinders are rarely used as astigmatic surfaces.

Hope this helps,

José Antonio

Hi Geoffrey,

Some clarifications:

1.- I assume you are working with an off-axis object. Spherical lenses images a point object on curved surfaces for off-axis locations.

2.- Your lens has no astigmatism. If not, two astigmatic images are formed, the sagital and tangential image since the lens has also field curvature.

3.- Astigmatic lenses always has astigmatism, even for an on-axis point object. Therefore, we should work with those two astigmatic images.

Seidel theory stems from the rotational symmetry of the system, therefore you can work on a meridional plane. If not, Seidel theory extension to non-symmetrical systems must be done with caution.

Toroidal systems can be treated as spherical in two orthogonal meridional planes when the off-axis object lays on one of those planes. Therefore, the two astigmatic images would be in two curved surfaces, likely having different curvatures in those two orthogonal meridional planes. The problem arises when the object is not on those meridional planes, and it is needed to ascertain that the curvatures of the two astigmatic surfaces will be that of a toroidal surface. I guess that the two astigmatic images will not be on a corresponding toroidal surface, but a more complicated surface. Maybe, based on some approximations, you could be right. At this time, I cannot assured this.

The problem is not simple. However, I suggest you read the sections 11.1 and 11.2 of the chapter 11 in Aberration of Optical systems from W.T Welford, and the recent papers from Sheng Yuan & José Sasián in Applied Optics about anamorphic systems.

Oh, finally, as far I know, crossed cylinders are rarely used as astigmatic surfaces.

Hope this helps,

José Antonio

Dear Jose,

Thank you for your reply and comments!!

I am curious about what will happen if the condition does not fit the assumptions of Seidel theory.

According to Yuan & Sasián, "in the paraxial region, an anamorphic system can be treated as two associated rotationally symmetrical optical systems (RSOS)". Does it hold true in the third-order optics sense?

Regards,

Geoffrey

Geoffrey,

Yes. However, while I have not read deeply the papers from Yuan & Sasian, they stablished third order theory for anamorphic systems, and this is a fully third order theory applicable. However, please, be advised that the point is to ascertain whether the two astigmatic image planes are toric surfaces.

The third order theory says that you can obtain the image plane curvature as it is a defocus dependence on the object position, when no astigmatism is present. However, if we apply the same reasoning for anamorphic systems, this is not the case. Additional coefficients are present since the wavefront is not simply the composition of the equivalent Seidel coefficients in the two orthogonal planes. I mean, additional coefficients appears from skew rays in anamorphic systems.

Therefore, It is not clear for me that astigmatic image planes will be toric. Nevertheless, this point deserves some study for sure.

Oh, I forgot it, please, be advised that the Petzval surface is parabolic having a Pertzval curvature in the optical axis. This could add complexity to the geometry of the astigmatic image planes.

Regards,

José Antonio

Dear Jose,

They also mentioned that anamorphic systems made from different surface types should be treated differently; for example, the cylindrical anamorphic system, more specificly, the cross cylindrical anamorphic system. Therefore, they developed a set of primary aberration coefficients to similate that of Seidel aberrations for rotationally symmetrical optical system. They said, "... The aberration coefficients include first-order quantities only" in their paper. What do they mean?

I am a bit confused. Should we treat the tangential and saittal surfaces as two independent surfaces or a continuous surface?

Regards,

Geoffrey

Hi Geoffrey,

I think they talk about a general anamorphic system in order to derive a particular one by setting approriate parameters. For example, a toric surface can be transformed to a cylindrical one by setting one curvature to zero value. HOwever, they note that the analytical coefficients are different, but numerically equal. therefore, the coefficients for toroidal systems can be used for cylindrical ones.

1.- They said, "... The aberration coefficients include first-order quantities only" in their paper. What do they mean?

They mean that the coefccicients can be calculated by using parameter quantities from paraxial raytracing (paraxial angles and heights) as in the case of Seidel theory.

2.- Should we treat the tangential and saittal surfaces as two independent surfaces or a continuous surface?

Please, be advised that two astigmatic images are formed. Therefore, one image plane correspond to sagital images and the other to the tangential plane. this means that you must work with those two different image (curved) planes.

Hope this helps.

José Antonio

Dear Jose,

Thank you!

Perhaps, could we compare the two forms of astigmatism as follows:

First, oblique astigmatism: in the RSOS, off-axis point object images on two rotationally symmetrical conic srfaces, tangential and sagittal surfaces. In this case, when one of the surfaces is in focus, the other will be out of focus.

Second, simple astigmatism: the astigmatism is due to uneven curvatures of an optical system (on-axis point object). In this case, when one of the focal lines is in focus, the other will be out of focus.

So, I guess when an off-axis point object is focused through a cylindrical optic system, two toroidal conic surfaces with four foci may possibly be formed. The foci are tengential meridian on tangential surface, sagittal meridian on tangential surface, tengential meridian on sagittal surface, and sagittal meridian on sagittal surface.

What do you think?

Regards,

Geoffrey

Geoffrey,

I am assuming that with "simple astigmatism" you are referring to "on-axis" astigmatism.

Regarding your last paragraph, please elaborate. Do you mean that the two astigmatic image surfaces (sagital and tangential) will be toric, each one having two curvatures?

I suggest you could use optical design software for a quick checking.

José Antonio

Dear Jose,

For illustration, a figure is attached.

I tried to modify the classic example, the spoked wheel, for ellaborating my idea.

Thank you for your suggestion, I will try to find a suitable optical design software.

Regards,

Geoffrey