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8.1.1. Newtonian off-axis aberrations       8.1.3. Newtonian reflector diagonal flat

8.1.2. Newtonian collimation

The difficulty in aligning optical elements of the Newtonian reflector arises from its diagonally oriented flat mirror, causing apparent shift of the projection of its surface away from the primary, in the view of the focuser opening. Due to this effect, in a perfectly aligned instrument (FIG. 116, top), with the center of the flat coinciding with the point of intersection of primary's and focuser axes - positioned at right angle - observing with the eye at the focal point, the flat appears off the focuser axis, shifted away from the primary. Also, image of the primary mirror in the reflected view of the flat appears decentered with respect to the flat, while in fact their respective centers do lie on the optical axis, and are perfectly aligned with the focuser axis.

There are two drawbacks of this type of alignment: (1) uneven illumination of the outer field, and (2) difficulty of determining the correct visual alignment of the three components. The former is pretty much insignificant in visual use, since the uneven illumination falloff at the two opposite sides of the field are unlikely to be noticed. The latter does present practical difficulty, although it is possible to determine and produce an image of the proper visual alignment for a particular telescope, and use it as a guide.

Much more common alternatives are those that are based on near-concentric visual alignment of the focuser, diagonal, primary mirror in the diagonal's view and diagonal's reflection in the primary. One of them achieves this by offsetting the diagonal by sliding it down along its surface plane, until it becomes visually centered in the focuser's view. It also evens up field illumination, without introducing any form of mechanical.

The other is a conventional 3-step collimation procedure, in which the diagonal is centered in the focuser by shifting it toward primary. It also achieves near-concentric visual alignment and even field illumination, but at a price of introducing some forms of misalignment, some generally negligible, some others possibly not (FIG. 116, bottom). This collimation mode is sometimes referred to as partial offset, as opposed to offset, or full offset when the flat shifts diagonally along its surface. A better terminology, suggested by Don Pensack, is unidirectional and bidirectional diagonal offset, respectively.

FIGURE 116: TOP: Visual alignment of the focuser, diagonal and primary mirror in a perfectly aligned Newtonian reflector with unoffset diagonal flat. Since the diagonal is not centered in the axial cone, so is not reflection of the primary in it. Diagonal itself, being on primary's axis, is projected to the focus centered around it, as indicated by a ray from the top diagonal point. BOTTOM: Conventional 3-step collimation procedure, with the forms of misalignment induced at each step, as well as those remaining after it is completed.

The starting assumption is that all three elements are in near-perfect mechanical alignment, with the diagonal center nearly coinciding with the focuser and tube axis point of intersection, and the primary nearly centered around tube axis as well. Focuser axis is perpendicular to the tube axis. Mirrors are likely to be deviating somewhat from their perfect orientation but, for the sake of illustration, we'll assume they are close to it (diagonal deviations from 45-degree angle - or in the plane orthogonal to it - will affect the position of primary's reflection in it, while primary's deviation from orthogonal to the axis position will affect position of diagonal's reflection in it, which is centered around primary's optical axis).

Following calculation, assuming surface rotation around their center point (except when acknowledged otherwise), and omitting negligible factors (such as depth of the primary) is not strict in mathematical sense, but should give close approximations practically as good as exact values for the range of amateur Newtonian telescopes.

The first step of this procedure (FIG. 116 bottom, 1) is centering the diagonal in the focuser's view by shifting it axially toward the primary. The needed axial shift a (to distinguish it from the offsetting shift ) to equalize apparent visual angles α and β for the top and bottom half of the diagonal, respectively, resulting in the apparent and actual center of the diagonal coincide, requires the equality (a-∆a)/(H-a+∆a)=(a+∆a)/(H+a+∆a), which solves to:

where A=2a is the diagonal's minor axis, and H the height of focus above the diagonal's center. By neglecting a in brackets with three parameters (i.e. neglecting that the focus-to-diagonal separation will increase by a after primary's tilt), it simplifies to a close approximation ∆a~a2/H = A2/4H.

The shift has, however, separated reflected (imaginary) axis of the primary from the focuser axis, inducing decenter error. At this point, primary's axis is shifted toward the primary by a linearly, which brings to the field center off-axis field point of identical height, with the P-V wavefront error of coma, according to Eq. 73.1, given by ω=6.7∆a/F3. For 300mm f/5 Newtonian with the focus height H=250mm, diagonal's minor semi-axis a=35mm and corresponding axial shift ∆a=4.9mm, this gives ω=0.26 wave RMS.

The next step, with the diagonal centered in the focuser's view, is to bring reflection of the primary in the diagonal's view (i.e. on the surface of the diagonal) to the diagonal's center, making the two concentric. Since the axial shift resulted in nearly equalized apparent angles α and β, the primary reflection is also nearly centered in the secondary, and only slightly shifted toward tube opening due to focuser axis being displaced upward from primary's axis. Centering of the primary reflection is accomplished by tilting the diagonal clockwise, by an angle given as:

γ = a/2(f-H)           (74.1)

in radians (for this small angles, practically identical to tanγ), f being the mirror focal length. Obviously, an imaginary ray coinciding with the focuser axis will be now, after reflection from the diagonal, directed toward the center of the primary.

However, the diagonal tilt directs the "reflected" primary's axis - with the best image point on it - further away from the field center (FIG. 116, bottom 2). At this point, with both, diagonal and primary in its view centered in the focuser's view, the total RMS wavefront error of coma in the field center is that of decenter and tilt combined, or ω=(6.7∆a/F3)+13.4γH/F3 (the latter from Eq. 73, substituting γ for τ and multiplying with 57.3x60 for the conversion from arc minutes to radians, with the result reduced by a factor H/f for the focuser). For the same 300mm f/5 Newtonian, that comes to γ=0.00196 and the combined RMS wavefront error of coma ω=0.31 wave.

The final step is bringing the deflected optical axis of the primary - i.e. the focal point - back to the field center, in the middle of the focuser. This is accomplished by tilting the primary clockwise, so that its optical axis tilts upwards until it nearly coincides with the focuser axis "reflected" from the diagonal. The required angle of tilt equals 2γ. Since diagonal's reflection in the primary is centered around primary's axis (since the diagonal is axially centered itself), bringing primary's axis to the center of the focuser is, in terms of the view in the focuser, bringing diagonal's reflection in the primary to the center of the view, that is, making it concentric with both, diagonal and primary's reflection in it.

 As a result of primary's axis now coinciding with the focuser axis "reflected" from the diagonal, the two axes hit nearly identical point at the diagonal, and nearly coincide in the focuser. In other words, the point of best image is brought to nearly coincide with the focuser axis, that is, with the field center (note that the focal point is now slightly higher, effectively by the amount of diagonal's axial shift).

However, the consequence of partial offset is that the primary's axis and tube axis are at an angle. This angle equals 2γ, hence it is given by:

δ = a2/(f-H)H = A2/4(f-H)H        (74.2)

in radians, with a being, as before, the diagonal's semi-axis, f the mirror focal length, H the focus-to-diagonal axial separation, and A the diagonal's minor axis, A=2a (if we designate the diagonal-to-primary separation as S, it simplifies to δ=a2/HS=A2/4HS). For the above 300mm f/5 Newtonian, it would come to 0.0039, or 0.22 degrees.

 Axial disparity between primary's and tube axis - assuming tube axis aligned with mechanical axes of the mount - causes pointing error. Pointing error induces tracking error, and negatively affects efficiency of go-to guiding.

Note that this calculation is nearly accurate for the focuser perpendicular to the tube/structure axis. If the focuser is at an angle, it will either add to, or deduct from the axial disparity caused by collimating with only partially offset diagonal. The change in pointing error due to the focuser tilt angle τ in radians (i.e. as tanτ) is given by Δδ=Lτ/2S, with L being the separation from geometric center of diagonal's surface to its effective center of rotation (approximately the point of intersection of diagonal's center axis and the plane of position adjustment), and τ being positive for the tilt away, and negative for focuser tilt toward primary (hence, increasing pointing error for the former, and reducing it for the latter). Change in the pointing error is a consequence of the effective decenter of the diagonal vs. tube/primary, given by Lτ/2 (since the diagonal tilts by about τ/2 in order to compensate for the focuser tilt and bring primary's axis to nearly coincide with the focuser axis).

For the above 300mm f/5 Newtonian, assuming L=1.5A=105mm, a 2° focuser tilt away from the primary (τ=2) would add 0.08° (Δδ=0.0015) to its pointing error, raising it to ~0.3°. A 2° focuser tilt toward primary would result in Δδ=-0.0015, thus ~0.14° pointing error.

Focuser tilt in the plane perpendicular to the vertical plane containing tube axis also adds decenter error to the focuser's axis "reflected" toward primary. As a result, the sideways tilt of the diagonal is needed to bring it to the center of the primary (it also requires to decenter diagonal sideways by τH with respect to the tube axis, in order to have the diagonal centered in the focuser's view; this means that the diagonal's reflection in the primary is no more centered). Now the diagonal tilt has two components: (1) the upward angle resulting from the axial shift toward primary alone, and (2) the sideways angle resulting from the diagonal decenter due to focuser tilt. Hence, the effective disparity between the axes of the tube and the primary - pointing error - is given by:

 δ'=(δ2+τ'2)0.5         (74.3)

with τ' - the angle of diagonal's sideways tilt needed to bring "reflected" axis of the focuser to primary's center - given by τ'=τH/2S. For the above 300mm f/5 Newtonian and sideways focuser tilt of 2 degrees (τ=0.035), τ'=0.0035 and the effective pointing error is 0.0052 in radians, or 0.3°.

For a given focuser tilt in the plane in between the one containing tube axis and that perpendicular to it, the value of resulting pointing error is somewhere between δ and δ'.

All considered, offsetting diagonal bidirectionally is preferable to the partial offset, i.e. with it being axially centered without offsetting vertically. While the apparent alignment of the reflective surfaces in the focuser is not entirely concentric with the diagonal being offset as well - reflection of the diagonal in the primary is shifted slightly toward primary, due to it being lowered with respect to primary's axis (its relative shift is still considerably less than in optimally aligned system without offsetting) - no other tradeoffs are involved. That said, pointing error inherent to the conventional (axially offset) collimation is small enough to be generally insignificant for visual use without computerized guiding.

These considerations are valid for collimating from the proximity of the focal plane. It is the most logical vantage point for visual use, since it is where the eyepiece is positioned. For other vantage points, the above relation for the needed shift centering the diagonal in the focuser's view, given as A2/4H is still valid, assuming that the actual point height is substituted for H. Using higher vantage points reduces the needed shift, and by the same amount the extent of slightly decentered field of full illumination. The field is exactly centered when the vantage point coincides with the tip of the cone formed by extending the lines connecting edges of the primary and diagonal (FIG. 74.1); from this point, the apparent circles of the diagonal and primary coincide. Since this point is roughly ~10cm above the focal plane in Newtonians - specific value is closely approximated by {S/[(D/A)-1]}-H - it does make collimation more complicated, and potentially less accurate. With knowledge and experience in collimating Newtonian, however, it should not be a significant concern. Publication that details this collimation mode - including more general information on Newtonian collimation and use - is Vic Menard's "New Perspectives on Newtonian Collimation".

Also, collimating without centering the diagonal in the focuser is not as difficult as it may seem. The only non-concentric element is the diagonal, and the proportion of primary reflection's distance from the diagonal's inner (toward the primary) and outer edge is given by {[af/(H-a)]-d}/{[af/(H+a)]-d}, d being the aperture radius.

8.1.1. Newtonian off-axis aberrations       8.1.3. Newtonian reflector diagonal flat

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