Re: [ATM] wedge in lenses

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

vladimir sacek wrote:
> 
> For any Fraunhofer doublet,[...]. 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), and t~(14.6wF^2)/D. For "w" in units of 
> the 555nm wavelength, it would come to t~(wF^2)/123D.
>

Not quite. I took another look at it, and found a couple of 
things to add and correct. The good news is that the tolerance
is more forgiving, by about a factor of 2. For the wedge
angle "t" in degrees it is approximated by:

t~(wF^2)/64D

with "w" being the RMS wavefront error induced (in units 
of the 555nm wavelength), "F" the objective F# and D the 
aperture diameter in inches. The difference vs. initial result 
is on the account of using (proper) balanced coma RMS 
(50%), and the rest an error in calculating aberration 
coefficient.

For the linear wedge "T" in inches, it is approximated by:

T~(wF^2)/3667

For any f/15 Fraunhofer doublet and 1/28 wave RMS error
(0.95 Strehl/MTF degradation factor) limit, the lens wedge
can't exceed 0.0022". For an f/10, it comes to 0.001".

Strictly talking, this is the tolerance for the two inner surfaces,
for which the sensitivity is highest and nearly leveled. For the 
front surface it is nearly 10 times loser, and for the last one
nearly 5 times loser. 

Since this is based on a particular Fraunhofer configuration,
(radii proportion 1 : 0.58 : 0.59 : 2.43), variations in radii 
proportion will result in the tolerance level variation. While
it is not likely to be more than 15%-20%, it can be easily
adjusted by applying a correction factor, as given by

c=(1.03fR1/R3^2)^2,

where "f" is the objective f.l., R1 the front surface radius and
R3 the third surface radius (assumed the one about equally or
slightly more sensitive than the second surface). So, better
approximation of the limiting wedge "T" in inches is:

T~(wF^2)/3667c

In my checkups, it comes within a few percent from 
raytracing results. Normal glass indici variations can be 
neglected. Other than Fraunhofer doublet configurations will 
likely have different wedge sensitivity. The Steinheil, which 
for all else equal has the radii roughly 1/3 shorter, has 
roughly twice the sensitivity of the Fraunhofer. 

A quick look at the wedge caused color separation in the focal
plane. For the linear wedge "t" in inches, it is given by 

s=18,900dt

in Airy disc diameters, with "d" being the index differential.
With commonly used glasses, the F-C differential is ~0.008
for the crown, and ~0.017 for the flint element. For an f/15 
objective with 0.0022" wedge at the 3rd surface, it would 
result in 0.7AD F/C separation (blur centers). That should 
be acceptable for achromats, in general, although I can 
imagine one could prefer somewhat less of spectral
separation.

Vlad

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