8.1.2. Newtonian reflector's diagonal flat

The role of Newtonian diagonal flat mirror is to divert light converging from the primary to the side, so that the image forms outside the incoming light, and can be observed without obstructing it (except for the relatively small obstruction by the diagonal flat). Perfectly made diagonal flat doesn't induce optical aberrations; its only effects are: (1) usually small reduction in light transmission, equal to (1-υ2)r, with r being the diagonal reflectivity ratio, and υ its relative diameter in units of the aperture, and (2) worsening of the diffraction induced imaging error due to the effect of central obstruction.

Fast Newtonian telescopes usually have the flat "offset", or slightly shifted from strictly centered position away from the focuser and toward the mirror, both directions by ∆~(D-H)/4(F-1/16F)D, in units of the diagonal flat's minor axis, where D and H are the aperture diameter and fully illuminated image field diameter, respectively, F-(1/16F) is the actual mirror focal ratio (when depth of sagitta is deducted from the focal length), and F is the conventional mirror focal ratio number F=ƒ/D (offsetting amounts to sliding the surface downward along the inclination angle by √2∆). For all practical purposes, a simplified form ∆~(D-H)/4FD, or (D-H)/4ƒ of the minor axis (ƒ being the mirror focal length), is just as accurate.

The fully illuminated field diameter H depends on the flat's minor axis Ami and flat-to-focus separation s, as H=[(F-1/16F)Ami-s]/[(F-1/16F)-(s/D)]. Again, the difference is practically negligible when the relation is simplified to H~(FAmi-s)/[F-(s/D)]~[Ami-(s/F)]/[1-(s/ƒ)]. Note that numerically negative result for H indicates too small flat size for its location. Substituting H into the above relation for the offset ∆ gives, after consolidation, more useable offset formula with all the factors known, ∆=(D-Ami)/4[(F-1/16F)D-s], in units of flat's minor axis Ami. Replacing the actual mirror focal ratio F-(1/16F) with the conventional focal ratio number F doesn't make appreciable difference in the result. It leads to the final flat's offset formula, as an actual shift away from the focuser and toward the mirror (as illustrated on FIG. 55):

Offsetting Newtonian diagonal flat doesn't influence image quality, but evens up field illumination and makes collimation easier, by centering the flat in the focuser opening (for near-minimum size diagonal flat, offsetting also prevents small light loss at the field center).

Minimum diagonal size (minor axis) to transmit the entire axial cone is given by Amin=s/F, s being the diagonal-to-focus separation (note that, due to the conical form of the converging cone, the major axis needs to be slightly larger than √2Amin, specifically [1+1/(4F2-1)]√2Amin; this is however, of no significance in regular Newtonian systems). At the minimum secondary size, field illumination, normalized to 1 at the field center, is

with cos(β)=c/2 and c=[1-(s/ƒ)]FH/s, where H is the image field diameter, ƒ is the mirror focal length, and c is the axial center separation of the cone cross section at the diagonal, in units of diagonal's minor semi-axis. Quantitatively, c is determined by the overlap of the marginal ray of an off-axis point away from diagonal's edge in units of diagonal's minimum minor semi-axis. The actual nominal overlap value for the marginal ray is given by Omin=[1-(s/f)]H/2 for the minimum diagonal size Amin, and for a larger diagonal AA by OA=[(1-H/D)s/2F]-(AA-H)/2.

Following graph (FIG. 74) shows illumination drop as a function of c, for a range of relative diagonal sizes.

FIGURE 74: Illumination I of off-axis image points in the Newtonian as a function of the relative overlap of the marginal ray away from the diagonal's edge (c), for the actual minor axis AA equal to the cone width at the diagonal (minimum diagonal size), 1.5 and 2 times the minimum size. The illumination plot for the minimum size diagonal is identical to the MTF plot, since with c=2ν Eq. 75.1 is identical to Eq. 57. For c~1 and smaller, the plot changes linearly, with I=1-0.61c in that range of c values. Drop in illumination is somewhat slower for the larger minor axis AA sizes, but not significantly. Linear diameter of the fully illuminated field is given by I100=[AA-(s/F)]/[1-(s/f)]. The lowest acceptable limit to visual edge illumination is for c~1, with ~1 magnitude loss. The relative overlap ratio c in units of the minimum diagonal's semi-axis is given by c=1-[AA-(1-s/ƒ)H]F/s.

EXAMPLE: D=400mm f/4.5 Newtonian with diagonal-to-focus separation s=300mm, minimum diagonal minor axis Amin=66.7mm and actual minor axis AA=80mm. Diameter of fully illuminated field is I100=16mm (0.5° angularly), relative overlap of marginal ray for (H/2)=22mm off-axis point is c=1-[AA-(1-s/ƒ)H]F/s=0.35, in units of the minimum diagonal's semi-axis. Corresponding illumination for the minimum diagonal size is I=(β/90)-csinβ/π=0.78; from FIG. 74, gain in illumination for AA/Amin=1.2 for c=0.35 is nearly 2%, with the appropriate illumination at 22mm off-axis I~0.8, or nearly 80%.

For computerized aid to this and other mechanical design aspects of a Newtonian, see Dave Keller's NEWT freeware.

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. 75.1).

FIGURE 75.1: 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. 75.2). 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 75.2: 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 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.

◄ 8.1.1. Newtonian off-axis aberrations ▐ 8.2. Two-mirror telescopes ►

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