Re: [ATM] Lens Wedge

["vladimir sacek" <vla@toast.net>]

Hi Bob,

>Vlad, that makes your calculations produce about 1/4 of what the
>previous answer (.030").  In any case, it indicates that wedge
>appears to be a lot less critical than the .001" or less that I,
>and others, are shooting for with our constructions.  Glad to
>hear from both of you that I'm merely making good pieces of glass
>perfect!

I'm sure your lenses are much better pieces of work than my posts :)
The numbers I threw in are fairly random. For instance, the 50% red/blue
blur separation in an F=3D" achromat would be 1.5 Airy disc diameters.
Since both, separation and blur size practically go with the square of the
wavelength differential, at half the separation from the optimized, say, 
570nm
wavelength they would be about 4 times smaller at 528 and 612nm than at
either the F or C line. The blur at those wavelengths would be about 0.75
the AD, and the separation nearly 0.4 AD (~0.2AD each), which should
just keep them within the Airy disc.

Considering spherochromatism, other errors enlarging blurs, approximative
character of the consideration and desirabilty of having somewhat more of
the spectral range within the Airy disc, it is probably better idea to go 
with less
than the Airy disc F/C line separation as the max acceptable. What amount of
wedge it would mean depends on the refactive index. Front of rear, doesn't
matter. While the resulting front-surface wavefront tilt/refraction for 
given wedge
is smaller while in-glass, the advancing end of the wavefront gains more 
since
hitting the air sooner, and it equalizes the effect vs. rear surface wedge. 
This
comes as a correction to the statement in my previous post.

So, for either surface, the linear separation in the focal plane is given by 
d(n)TF,
"dn" being the index differential, "T" the max. linear wedge and "F" the 
scope's F#.
For a common crown, like BK7, with d(n) for the C/F lines of 0.008, the
F/C separation reaches the AD diameter for T~0.17mm for an f/12 system.

For a typical flint, like F2, with dn=0.017 for the F/C lines, this 
separation level
would be reached with ~0.08mm wedge, also at f/12.

One thing that still puzzles me is coma and astigmatism caused by the wedge.
With a concave mirror, it is clear where do they come from. For oblique 
incoming
pencils, wavefront points hitting one half of a mirror have longer paths vs. 
those
hitting the other half. This slightly flattens one side of the wavefront, 
while curving
more the other one. The result is coma, given by Da/48F^2 as the max p-v 
error
(the RMS is smaller by a factor of 1/sq.rt.32), with "D" being the aperture 
diameter,
"a" the field angle in radians and F=f.l./D.

Astigmatism results from the oblique wavefront hitting a mirror being 
extended
in the vertical (sagittal) direction by a factor 1/cos(a), while its 
effective diameter
projected toward the focus is reduced by a cos(a) factor. Since wavefront 
radius
(in effect, the f.l.) for given wavefront depth changes with the square of 
diameter,
the longitudinal astigmatism (3rd order) comes to [1-cos^4(a)]f, with its 
p-v
wavefront error being Wa=[1/cos^2(a) -cos^2(a)]D/16F (the RMS error is
smaller by a factor of 1/sq.rt.24).

Since the mechanism of aberration forming seems to be identical in the case 
of a
lens wedge, with the coma forming as a result of surface tilt, and the 
astigmatism
resulting from wavefront tilt, I expected that the same equations would 
apply, after
substituting "a" with T/D  and R/(n-1)D for "f" with coma, and (n-1)T/D for 
"a"
with the astigmatism. However, it gives much smaller wavefront errors for 
both,
than what the raytrace shows.

Sorry for the long post - got little carried away. There is something 
intriguing about
the mystique surrounding the wedge effect. It woud be nice to know why the
wavefront errors are as large as they are, but doesn't really matter. As 
already
mentioned, at the level of max acceptable color separation they are still 
negligible.

Vlad

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