for the natural logarithm base e~2.718, and the aperture diameter **D** in
mm. Taking conventional 0.80 Strehl, or the RMS wavefront error in units
of the wavelength ω=1/**√**180
as the maximum acceptable amount of wavefront degradation, sets the
appropriate F# limit for spherical mirror at F=(3.55D)1/3
or larger
for **D** in mm, and

F=(90.17D)1/3
for **D** in inches. Counting in the effect of central obstruction, the
criterion becomes more demanding, as described in
5. Obstruction effects.

From S~1-(2π
ω)2
,
needed F# for a desired Strehl **S** with spherical mirror
is given by F~3.5D1/3
/(1-S)1/6
,
for the aperture diameter **D** in inches, and F~1.18D
1/3
/(1-S)1/6
for **D** in mm (since based on the approximation, it is accurate for
Strehl ratios of ~0.9 and higher; for the ratios ~0.8 and lower, it
gives increasingly higher F# than the actual value).

For objects close enough that the
primary magnification **m**, defined as one of
Eq. 9 parameters,
appreciably differs from zero, the P-V wavefront error at the best focus,
after substituting for **m** in terms of the object distance **o**
and mirror focal length **f**, is W'**s**
=-[K+(1-2ψ)2]D/2048F3
(the minus sign indicating the aberration is numerically negative at the
best focus location; consequently, it is positive, or overcorrected at
paraxial focus), with ψ=f/o being the primary focal length in units of
the object distance (the reciprocal of the object distance in units of
the mirror focal length). Obviously, the wavefront error is zero if the
expression in the brackets is zero, which defines the zero-aberration
conic in terms of object distance, for primary spherical aberration, as
K=-(1-2ψ)2
.

Inversely, object distance in terms of wavefront
error induced can be written as (1-2ψ)2=[2048WF3/D]-K.
For the wavefront error in units of 550nm wavelength, it is
(1-2ψ)2
=-[1.126WF3/D]-K.
For a selected wavefront error, the right side gives the value of (1-2ψ)
squared, which makes finding **ψ**
easy. For instance, for W=0.05 wave in units of 550nm wavelength,
D=400mm f/4 paraboloidal mirror
(thus F=4 and K=-1), the value of (1-2ψ)2
is 0.99099, thus (1-2ψ)=0.99549 and
ψ=(1-0.99549)/2=0.00226. The corresponding distance is
f/0.00226=443f=708.9 meters,
nearly identical to the value obtained with somewhat rounded off
relation given in the section on
star testing.

Most Newtonian telescopes nowadays use paraboloidal
primary. If well made, its spherical aberration is practically
cancelled. However, off-axis aberrations are present, and can be
significant - particularly coma. Before addressing full aperture
off axis aberrations of a Newtonian, a quick look at the effect
of off-axis mask, in a fairly common use with larger instruments
of this kind.

EFFECTS OF OFF AXIS MASK

With off axis mask on, light effectively uses off-axis section of a
paraboloid. This causes: (1)reduction in the level of off axis
aberration, which is also from dominant coma transformed to dominant
- although still somewhat coma-like in appearance - astigmatism, and
(2) tilted best image surface. In the case shown, a large 560mm
f/3.6 mirror, with a 200mm mask
opening, tilt is significant enough to cause deformation of the
diffraction image farther off axis (1.5mm of tilt-caused defocus at
17.5mm radius corresponds to 100mm field curvature radius, thus
requires significant eye accommodation). The field is also
asymmetrical, with the astigmatism along the tilted (vertical)
radius having different form than along the horizontal field radius.

Mask distance has a minor influence on the magnitude of off axis
aberration. Since this astigmatism originates in the mirror coma,
it also changes with the field radius. Placing mask farther away
increases the chief ray height at the mirror, increasing by it off
axis error. As a relative aberration, in units of the aberration
with the mask at the mirror surface, the increase is approximated
as 1+αS/[29(D-M)], where **α** is the off axis angle in degrees,
**S** is the mirror-to-mask separation, **D** is the mirror
diameter and **M** is the mask opening diameter. In this case,
α=0.5, S=1600mm, D=560mm, M=200mm, and the relative aberration is
1.077 times larger than with the mask at the mirror.

OPTICAL WINDOW EFFECT

While not widespread among amateurs, and pretty much ignored by the
commercial makers, sealing off a tube of the Newtonian by placing
optical window on its front end seems to keep its appeal for at
least a few. The benefits are avoiding spider vanes
diffraction and protecting the tube inside and primary from dust
and impurities contained in the air. Does such window produce
optical effects, and how accurately made it needs to be in order
not to induce significant aberrations?

To find the answers, will
raytrace a 200mm f/5 Newtonian (below). Configuration is more or less
standard, and small differences will not significantly effect results.
Unavoidable effect, which is not even caused by the window itself,
rather by placing aperture stop at a distance from the primary
mirror, is change in off axis astigmatism, and wit it in the field
curvature. As raytrace for arrangements without (top) and with
optical window (below it) show, stop shift significantly reduced
astigmatism, making best image surface somewhat more relaxed,
and with the curvature of opposite sign than that with the stop at the
mirror. While in the latter best image radius (absolute value) equals mirror's focal
length and is concave toward mirror, with the window on best image
radius is 1370mm, convex toward mirror (if astigmatism would go to
zero, image field radius would equal Petzval curvature, i.e. one
half of the mirror surface curvature radius. With the new stop
location, longitudinal astigmatism at 0.5° is reduced from
nearly 0.08mm to 0.012mm (since the P-V error is - same as for
defocus - smaller by 8F^{2}, from 0.7 to 0.1 wave P-V,
respectively, for
550nm wavelength). Despite the significant reduction in astigmatism,
off axis blur at 0.5° is little changed, due to the coma
accounting for most of the blurring (about 2.4 waves P-V).
If the window clear opening equals that of the mirror, some
vignetting is induced: less than 6% at 0.5° and twice as much
at 1° off (for zero vignetting, mirror radius needs to be
7.5mm/15mm larger).

As for the effects of the window itself, first about
what don't cause any: its thickness, and possible tilt
(without going to extremes, of course).
The window is very forgiving with respect to
near-spherical deformation of its surfaces. As much as 10 waves
deep deformation would induce as little as 1/25 wave P-V in the
blue (F) and red (C) lines, with entirely negligible spherical
aberration in the central line (bottom left). Twice as deep
deformation would induce twice more of each, and so would
the same magnitude deformation on both window surfaces.
This means that for approaching the "true apo" level the
window would need five times deeper sphericity (50 waves,
or 0.0275mm) on its one surface, or half as much on each of
the two. No lateral color, or other aberrations are induced.

It becomes much less forgiving when it comes to wedge error.
As little as 1/20 of a degree tilt of one of the surfaces
(in this case the front surface, tilted counterclockwise,
with the top point 0.175mm farther out than the bottom)
would induce all-field lateral color separating F and C
lines at a distance about equaling the Airy disc diameter,
amounting to 0.9 wave P-V in the F, and half as much in the C
(bottom right). The center field coma induced is still very
low, causing 0.975 Strehl degradation. But the lateral color
error causes polychromatic Strehl (430-670nm, photopic sensitivity)
to drop to 0.53. Half as strong wedge - causing half as much of
lateral color error - would still limit the
poly-Strehl to 0.8. For reaching 0.95 poly-Strehl, i.e. 1/8 wave
P-V of spherical aberration level - wedge error would have
to be yet another 2.5 times smaller, or 1/100 degree (0.035mm).

Internal glass strains can induce significant large-scale
wavefront deformations, and impurities/inhomogeneity will induce
small-scale wavefront deformation and scattering. To avoid this,
the window should be made from a quality optical glass.

◄
7.3. Apodizing mask
▐
8.1.1. Newtonian off-axis
aberrations
►

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