Re: [ATM] Lens Wedge

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

James Gort wrote:

>Yes, lateral chromatic aberration is the problem, but I'd l>ike to
>quantify it.  How much is 'too much' to achieve 90% >Strehl Ratio (at all
>wavelengths), say?  Or what is the lateral chromatic >aberration due to
>wedge as a percentage of lateral spherical aberration?

For 0.9 degradation factor, lateral color would have to pull out ~10% more
(effective) energy from the Airy disc. With most achromats, most of the 
red/blue
is already out, so it would be the wavelengths in the 500-600nm range.
For better corrected refractors, wider spectral range would need to be taken
into account.

The wedge causes wavefront tilt, resulting in image shift. The tilt angle
(tangent) is given by (n-1)T/D, "n" being the glass refractive index, T the 
max.
linear wedge and D the lens diameter. Consequently, the image shift in the 
focal
plane is given by (n-1)TF, with F=f.l./D.
Color separation in the focal plane is given by dnT/D as a tangens, and dnTF 
as
the linear shift in the focal plane ("dn" is the refractive index 
differential). I am not
aware of tolerances here but, roughly, keeping 600nm and 500nm wavelengths
shift within 10% of the Airy disc diameter seems to be a good idea
for an average achromat (color shift in Airy disc diameters is 745dnT, for 
the
550nm wavelength).

Lens surface tilt caused by wedge results in some coma. It is created due to 
the
wavefront now having longer in-glass path one half, than in the other one, 
with the
mid section path nearly unchanged. The deviation on both sides increases
to about 60% of the radius, diminishing to zero toward the edges. This 
creates
characteristic "S" shaped comatic wavefront profile.

Simplifying the geometry to triangles (which should be good enough 
considering
small angles), leads to the wavefront advance/retardatation (approx. 
maximum)
as +/-(n-1)TD^2/(40R^2), with "R" being the surface r.o.c. However, this 
doesn't
seem to be the main source of coma. Coma, as well as astigmatism seem to 
result
also from the wavefront deformation caused by the wedge alone: coma, as one 
half of the
tilted wavefront gets slightly flattened (stretched out), while the other 
half gets slightly
"compressed" (more curved). Asigmatism, as the tilted wavefront's 
orientation projects
slightly shorter tangential radius toward the new focus and, probably, 
slight streaching
of the tangential wavefront diameter.

At least, it appears that way on a sketch. Of course, means little without 
actual calculation,
which doesn't seem to be too complicated (maybe easier if done through 
raytracing, but
that would require some 3-D geometry). Coma/astigmatism related to wavefront 
tilt would
make the rear surface more sensitive to wedge error than the front one. Due 
to difference
in air-to-glass vs. glass-to-air light path, same linear wedge will result 
in (n-1)T/nD and
(n-1)T/D tilt angle, respectively. With the usual crown/flint indici, the 
rear lens would be
nearly twice as sensitive, more so due to the stronger exiting wavefront 
curvature.

In any instance, seems that wavefront errors are comparatively small. With 
even lose color
separation standards - such as the red and blue blurs separated by no more 
than 50% the
the blur radius - the wavefront aberrations induced are negligible. With an 
approximate
index differential between the red and blue of 0.01, for an average F#~5D 
achromat
it would allow as much as 0.2mm wedge.

Vlad





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