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8.1.1. Newtonian
reflector diagonal flat
Wavefront error at the diagonal
is of particular interest with the
Newtonian design, the diagonal flat being its constitutive element. Of
course, perfectly flat diagonal won't induce any aberrations. In
reality, every diagonal is less than perfectly flat, and the result is
some form of wavefront deformation. Magnitude of this deformation
depends directly on the size and type of surface error at the diagonal.
In general, there are two types of diagonal surface error: (1) local
error, covering relatively small portion of its surface, and (2) error
smoothly distributed over the most, or all of the surface.
Any local surface error, be it a
single defect, any number of local errors scattered over the surface,
turned edge, or zone, will multiply in the wavefront by a factor of
√2,
or ~1.4. This is the consequence of its ~45° surface inclination, making
the diagonal about 30% less sensitive to local surface errors and
roughness than a "regular" mirror (FIG.
57).
FIGURE
57: Left: When the direction of light after reflection is
nearly opposite to its incoming direction, local surface error
resulting in an air path difference of the thickness t
results in the wavefront P-V error of ~2t. This applies to
both, flat and curved surface mirrors orthogonal to the optical
axis.
Right: For the diagonal surface, the wavefront points after
reflection move in nearly orthogonal direction to that of wavefront
points coming onto the surface. Consequently, there is no
added compounding of the surface error in the wavefront. A surface error
creating an air path difference t will result in the
wavefront P-V error of ~1.4t, due to the actual surface error (i.e.
optical path difference) being enlarged by the ~45° position angle. Neither surface angles nor
angles of convergence in amateur telescopes result in appreciable
difference in added path difference vs. strict "orthogonal"
scenario. In general, Newtonian diagonal flat is less sensitive to
surface errors than main mirror.
For errors smoothly
distributed over diagonal's surface it is somewhat more complicated. The reason is that they, in general, change
the curvature of the wavefront so that part of the error induced can be
compensated by refocusing. As long as radii depth along the minor and
major axis are equal, there is no astigmatism induced, and the coma
wavefront error is very small in comparison to the surface error. It is
the difference in depth along the two radii that induces astigmatism,
not the surface P-V error itself (FIG.
58). Due to the
inclination angle, the final P-V wavefront error of astigmatism is ~1.4
times the difference in radii depth (when the radii are of the opposite
sign, as they are for the saddle surface form, the error is 1.4 times
the sum of their depths). This conveniently allows for obtaining good
approximations of the size of wavefront errors induced by these types of
surface error using quite simple calculations.
FIGURE
58: Four major types of smooth diagonal surface error, shown in
the side view along the major axis. The toroid with the minor
axis radius of curvature shorter by a factor of 2
than the major axis radius, has equal depths along both axes and
flat edges, resulting in near-zero astigmatism and P-V wavefront error
(WPV) of coma ~1/16 of the surface P-V
error (SPV). As the radii change toward equalization, depths
along the two radii grow uneven, and astigmatism induced to the
wavefront increases. Spherical surface has equal radii along
both axes, resulting in uneven edge with the center depth along the minor axis being
half of that along the major axis. The wavefront error induced - mostly
astigmatism, with traces of coma - comes from the difference in
radii depth, multiplied by a
√2
factor, due to the 45° angle of surface inclination. Thus, the
wavefront error is ~0.7 of the surface P-V
error. Surface change toward cylindrical and saddle form result in
further increase of the induced wavefront error of astigmatism: it
becomes equal to the
surface error multiplied by ~1.4 factor (cylindrical form has the
best focus RMS error lower by nearly 20%). Concavity vs. convexity of the surface deformation is
not a factor in the surface/wavefront P-V error relationship, and
neither is the axis - major or minor - along which are oriented
cylindrical and saddle form surface deformation. Also, the wavefront
error is independent of focal ratio and aperture. Note that the
diagonal surface error is for the area on the diagonal transmitting
the wavefront, normally somewhat smaller than the entire diagonal.
The dependence of the size of
wavefront error caused by the diagonal flat on both, size and form of
the diagonal's surface error makes its actual performance level
uncertain, even if the nominal surface RMS error is known (the P-V error
is, as usual, pretty much meaningless). Safe policy is to go with the
worst-case scenario, expecting the surface error to multiply in the
wavefront by a factor of ~1.4.
With star diagonals, converging cone
uses much smaller portion of the optical surface. Thus, even much larger
errors smoothly distributed over most of the surface have no appreciable
effect on wavefront quality. However, local surface errors,
especially in the mid-area, transmitting the light for the central
portion of the field, can have significantly greater effect.
◄
8.1. Newton reflector
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8.2. Two-mirror telescopes
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