Re: [ATM] wedge in lenses

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

>
> The tolerance likely lessens exponentially with faster systems (depends on
> both,
> specific front surface r.o.c. and individual coma contributions at each
> surface).

To wrap it up, for a 4" f/15 Fraunhofer doublet, the
individual surface coma contribution does increase nearly
with the square of decrease in F#. It also increases in proportion
with the aperture diameter. On the other hand, astigmatism
increases in proportion to both, F# decrease and aperture
increase. Being already significantly smaller than coma,
it can be neglected as a factor in determining the lens
wedge tolerance.

The Fraunhofer doublet radii proportions are given, and
so are the relative individual surface coma contributions.
Assuming that a given angular surface tilt caused by lens
wedge will cause approximately as much of center-field
coma as it would be produced by this surface at an identical
field angle when properly oriented, individual surface wedge
tolerances can be determined based on the coma contribution
of the first surface alone.

For a 4" f/15 Fraunhofer doublet, center-field coma caused by
the wedge "t" (in degrees) at the first surface is approximated by
Wc~t/2470 as a p-v wavefront error, or w=t/14,000 as the RMS
wavefront error. For any other aperture and F#, it will be
smaller or greater approximately by a factor 56.25D/F^2,
"D" being the aperture diameter in inches, and F the lens
objective F#.

Coma resulting from the wedge caused tilt at the second, third
and fourth surface is approximately 16, 17 and 2 times greater,
respectively. If the wedge can be assigned to individual surfaces,
so it can the needed tolerance. In more likely scenario, when
the wedge is measured for the lens element, not an individual
surface, the tolerance needs to be determined according to
the more sensitive surface. That makes it 16 to 17 times greater
than that for the first surface. Taking 17 and rounding off, gives
the tolerance "meter" as Wc~t/145 p-v wavefront error
(w~t/823 the RMS) for the 4" f/15.

For any Fraunhofer doublet, this worst case scenario wedge
tolerance would be determined by applying the 56.25D/F^2
factor to the RMS error w~t/823. This gives the general RMS
wavefront (coma) error as w~tD/14.6F^2 (for "D" in inches,
F the objective F# - f.l./D - and the wedge angle "t" in degrees),
as t~(14.6wF^2)/D. For "w" in units if the 555nm wavelength,
it would come to t~(wF^2)/123D.

For a 6" f/15 Fraunhofer doublet and, say, w=1/20 wave RMS
wavefront error tolerance, angular wedge "t" would
have to be limited to 0.015 degrees, or 0.0016 inch. For an
8" f/12, the worst case scenario wedge limit for this error level
would be 0.0073 degrees, or 0.001 inch. Obviously, the limit
scales with the RMS wavefront error tolerance.

As far as I can see, this approach seems appropriate. It sacifices
total (3rd order) accuracy for the sake of simplicity and
practicality, but should be accurate enough for the purpose.


Vlad










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