Played with WinSpot - a great little tilted systems
program (does axial systems too) - in order to make sure
how much of wavefront error is induced by various shapes
of deformed Newtonian diagonals, when the surface error is smoothly distributed over the entire surface. In my previous post on the subject I came up with some results, but was drawing analogies for some shapes, instead of calculating (plain lazy), so there will be also some corrections.
The results are for an 8" f/5 Newtonian, with 0.25f diagonal to focus distance. In general, for given surface error, there is no appreciable change in the size of induced astigmatism with the change in aperture or focal ratio, nor relative size of the diagonal. Important assumption is that the diagonal is just as wide (minor axis) as the converging cone hitting it. If the diagonal is larger (as it usually is) the error diminishes with the square of relative cone width at the diagonal (that is, if the cone is 0.8 of the diagonal width, both surface p-v error and that induced to the wavefront are smaller by a factor of 0.64).
Since p-v errors for different aberrations have different effect contrast-wise, I'll have the errors for coma and astigmatism standardized to a comparable error of spherical aberration. So, "x" waves of, say, coma, will be effectivelly same as "x" waves of spherical aberration.
According to the WinSpot, the diagonal deformation inducing the least amount of error is one where the edges are flat, and the inside surface (either convex or concave) smoothly curved. This is a toroidal form in which the radii of minor vs. major axis relate as 1:2. Here, any given surface
p-v error "x" results in x/16 waves of coma, combined with some residual astigmatism. Obviously, tolerances for this form of deviation are much greater than what usual ideas are. It would take nearly 4 waves p-v surface error to cause 1/4 wave of coma. Moreover, this is an exception in that the error diminishes with smaller relative diagonal size.
Now, by flattening either of the axes in the above shape, so that ends of one axis are twice as high or deep as ends on the other one, we come to the next surafce deformation shape, in which radius of curvature are identical for both,
minor and major axis. At this point, astigmatism induced to the wavefront has become much larger and entirely dominant over coma. Any given surface p-v error "x" induces about x/2 waves of astigmatism to the wavefront (this agrees pretty well with my calculation - nice).
Further flattening of either of the two axes, until it has zero curvature (symmetrial cylindrical surface), results in doubling of the induced astigmatism, while coma further diminishes. Any surface p-v error "x" here results in "x" wavefront error of astigmatism.
Finally, bending either of the two axes so that their depth is identical, but the radii are of opposite sign, results in
"saddle shape" inducing "x" wave of nearly pure astigmatism
for every "x" p-v surface error. Apparently, size of induced error chages little going from the cylindrical towards various saddle shapes.
Conclusion is pretty much the same: error induced to the wavefront by a diagonal suffering from p-v deformation evenly distributed over the surface varies, depending on the shape of surface deformation, from ~1/16 of to equal to the surface error. However, local surface smoothness is even more important in the case of a diagonal than the primary, because any local surface deformation causes 2.8 times greater local wavefront deformation.
Vlad (in need of a break ')
program (does axial systems too) - in order to make sure
how much of wavefront error is induced by various shapes
of deformed Newtonian diagonals, when the surface error is smoothly distributed over the entire surface. In my previous post on the subject I came up with some results, but was drawing analogies for some shapes, instead of calculating (plain lazy), so there will be also some corrections.
The results are for an 8" f/5 Newtonian, with 0.25f diagonal to focus distance. In general, for given surface error, there is no appreciable change in the size of induced astigmatism with the change in aperture or focal ratio, nor relative size of the diagonal. Important assumption is that the diagonal is just as wide (minor axis) as the converging cone hitting it. If the diagonal is larger (as it usually is) the error diminishes with the square of relative cone width at the diagonal (that is, if the cone is 0.8 of the diagonal width, both surface p-v error and that induced to the wavefront are smaller by a factor of 0.64).
Since p-v errors for different aberrations have different effect contrast-wise, I'll have the errors for coma and astigmatism standardized to a comparable error of spherical aberration. So, "x" waves of, say, coma, will be effectivelly same as "x" waves of spherical aberration.
According to the WinSpot, the diagonal deformation inducing the least amount of error is one where the edges are flat, and the inside surface (either convex or concave) smoothly curved. This is a toroidal form in which the radii of minor vs. major axis relate as 1:2. Here, any given surface
p-v error "x" results in x/16 waves of coma, combined with some residual astigmatism. Obviously, tolerances for this form of deviation are much greater than what usual ideas are. It would take nearly 4 waves p-v surface error to cause 1/4 wave of coma. Moreover, this is an exception in that the error diminishes with smaller relative diagonal size.
Now, by flattening either of the axes in the above shape, so that ends of one axis are twice as high or deep as ends on the other one, we come to the next surafce deformation shape, in which radius of curvature are identical for both,
minor and major axis. At this point, astigmatism induced to the wavefront has become much larger and entirely dominant over coma. Any given surface p-v error "x" induces about x/2 waves of astigmatism to the wavefront (this agrees pretty well with my calculation - nice).
Further flattening of either of the two axes, until it has zero curvature (symmetrial cylindrical surface), results in doubling of the induced astigmatism, while coma further diminishes. Any surface p-v error "x" here results in "x" wavefront error of astigmatism.
Finally, bending either of the two axes so that their depth is identical, but the radii are of opposite sign, results in
"saddle shape" inducing "x" wave of nearly pure astigmatism
for every "x" p-v surface error. Apparently, size of induced error chages little going from the cylindrical towards various saddle shapes.
Conclusion is pretty much the same: error induced to the wavefront by a diagonal suffering from p-v deformation evenly distributed over the surface varies, depending on the shape of surface deformation, from ~1/16 of to equal to the surface error. However, local surface smoothness is even more important in the case of a diagonal than the primary, because any local surface deformation causes 2.8 times greater local wavefront deformation.
Vlad (in need of a break ')
