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8. REFLECTING TELESCOPES       8.1.2. Newtonian collimation

8.1.1. Newtonian off-axis aberrations

Dominant off-axis aberration in the Newtonian is coma; astigmatism is low in comparison. Image surface deformation is field curvature, with distortion being zero with the stop at mirror surface. All three aberration are independent of mirror conic, which means that they are equally affecting both types of the Newtonian telescope, those with spherical and those with paraboloidal primary. Following are relations specifying the aberrations.

lower-order coma, from Eq. 12-13 and 15.1, after substitutions, as the P-V and RMS wavefront error at diffraction focus for object at infinity is given as:

Wc= αD/48F2  = h/48F3    and    ω = Wc/321/2 = h/272F3        (70)

respectively, with α being the field angle (α=h/, h being the linear height in the image plane, and the mirror focal length). Note that this is double the error given by Eq. 12, which expresses only the peak aberration.

In units of the λ=0.00055mm wavelength, the coma wavefront errors are, slightly rounded off:

Wcλ = 38αD/F2 =  38h/F3    and    ωλ = 6.7αD/F2 =  6.7h/F3       (70.1)

for D and h in mm (or with the numerical factor 38 and 6.7 replaced by 965 and 170, respectively, for D in inches).

For the 0.80 Strehl ratio, ωλ=1/180, which corresponds to the field angle α=F2/90D in radians, for D in mm (α=F2/2286D for D in inches), or the linear height in the image plane

hDL = αDL = F3/90            (70.2)

which represents the radius of diffraction limited field. Actual Strehl ratio at this field point is likely to be slightly lower than 0.80, due to the presence of astigmatism. Solving Eq.70.2 for α gives angular diffraction limited field radius as:

αDL = F2/90D          (70.2.1)

in radians, or α'=38F2/D in arc minutes, for D in mm (α'=1.5F2/D for D in inches). Evidently, unlike the linear quality field, which only changes in proportion to F3, quality angular field changes in proportion to F2 and in inverse proportion to the aperture diameter D.

As Eq. 70 implies, the wavefront error of coma in the Newtonian is inversely proportional to the square of its -ratio for given angular field radius, and to the third power of it for given linear field radius. Thus it decreases exponentially with the increase in focal ratio, as illustrated on a simplified scheme below.


FIGURE 113: The geometry of coma ray spot formation in a fast (black) vs. slow mirror (blue) for given field angle. The slower mirror has about twice smaller relative aperture,  thus its ray spot should be four times smaller (inversely proportional to the square of -ratio for given linear field radius), but it is made larger for visual clarity.

Also, Eq. 70 shows, coma of the concave mirror is independent of its conic. It, however, changes with the stop position and object distance, as described in 2.2. Coma (it also contains specifics on geometric, or ray aberration). In terms of object distance O and mirror radius of curvature R, the coma wavefront error changes in proportion to (O+R)/O, with R being, according to the sign convention, negative. Thus, mirror coma diminishes with the object distance, falling to zero for object at the center of curvature (O=-R).

What is usually seen of the coma in the eyepiece, with sufficiently bright stars, roughly corresponds to the dense 1/3 of its geometric blur (sagittal coma). Its angular size needs to be ~5 arc minutes for its shape to be recognized by an average eye. Since, from Eq. 17, (from Sa=S/e) angular sagittal coma in the eyepiece (neglecting eyepiece distortion) is 215tanε/F2 arc minutes, with ε being the apparent viewing field angle in the eyepiece, in degrees, the eyepiece viewing angle at which the geometric sagittal coma would become obvious to most people is given by tanε~F2/43. This means that one could see it at the edge of a 40 AFOV (w/AFOV=2ε) eyepiece at /4, regardless of its focal length.

 However, what we see in the eyepiece is not a ray spot, but an actual diffraction image deformed by the aberration. The above "geometric" angle at which sagittal coma becomes visible is noticeably reduced in conventional eyepieces, due to significant additional blurring caused by eyepiece astigmatism. Also, in order for the comatic deformation to be seen in the actual star image, this image has to be noticeably deformed by the aberration. For coma, that occurs as the aberration decidedly exceeds ~1 wave P-V on the wavefront (or, as the full size blur exceeds four times the Airy disc diameter, and the sagittal coma is nearly 1.6 times the Airy disc diameter). At this aberration level, and beyond, the actual visible blur of relatively bright stars roughly resembles the form and size of geometric sagittal coma. As the error diminishes below ~1 wave P-V, the effect gradually transforms into one-sided intensity distribution asymmetry visible in the diffraction rings, but with the spurious disc still well defined. 

From Eq. 70, the field height at which the coma error becomes large enough (~1.25 wave P-V) for the central maxima deformation to begin to resemble sagittal coma in the actual star image is about three times the diffraction limit, or approximately h~F3/30, in mm. For an /4 mirror, that corresponds to ~2.1mm (FIG. 114, top right). Substituting h~F3/30 for h in the relation for angular sagittal coma, Sa=S/e, gives Sa=F/480e in radians, or Sa=7.2F/e in arc minutes.

From Sa=5, the eyepiece focal length needed to enlarge the actual comatic image of a bright star with this amount of coma to 5 arc minutes apparent size, is approximately e~1.4F. Hence an average observer, with a 5.6mm eyepiece and an /4 mirror, can expect to notice coma on bright stars from ~2mm off-axis out. This corresponds to larger than 20 field angles, as in the above consideration; the difference is that that the implication here is that the shorter f.l. eyepieces would not show edge coma due to the diminishing actual deformation of the central maxima at the image height they reach.

With an /6 mirror, it will take an 8.4mm eyepiece to show the coma becoming apparent from some 7.2mm off-axis (which, from ε~57.3h/e, e being the ep f.l., would require 80+ degrees distortion-free AFOV); higher magnification will reveal little of the characteristic comatic deformation inside 7mm off-axis, due to the deformation becoming less apparent with the decreasing wavefront error. Compromised image sharpness, however, will be noticeable well within the 7mm-radius field circle (FIG 115, bottom).



FIGURE 114
: Ray spots and diffraction images of 6"
/8.15 paraboloid and sphere (A). For better resemblance to their real-life appearance, diffraction images are reduced by a factor of 3; also, they include 0.2D (linear) central obstruction effect. At 0.28 off-axis, the coma RMS wavefront error of paraboloid - 0.075 wave (0.80 Strehl) is identical to the RMS wavefront error of spherical aberration of the sphere. The combined error of the sphere at 0.28 degrees off-axis is 0.12 wave RMS (0.56 Strehl). (B) Paraboloid's coma is changing with its focal ratio F. Linear blur size varies inversely to F2, while the Airy disc changes in proportion to F, resulting in the wavefront error for given field height to vary inversely to F3. Thus, quality field size changes inversely to F2 angular, and to F3 linear. (C) Monochromatic (e-line) ray spots of a paraboloid at /4.5 and /9, modified by typical level of astigmatism and spherical aberration induced by conventional eyepieces (astigmatic blur is 2.6 times smaller than coma for given RMS error).


 

Due to eyepiece astigmatism, which increases with the square of eyepiece field angle, comatic-like deformation in the eyepiece grows rapidly toward the field edge. For given linear field radius in the image plane, eyepiece astigmatism increases in inverse proportion to the square of the focal ratio as transverse aberration (i.e. ray spot size) and inversely to the cube of it as wavefront error (lens astigmatism, given by Eq. 22, increases with the square of entrance beam width, which is inversely proportional to mirror focal ratio, and inversely to lens' focal length, which is proportional to the mirror focal ratio for given aperture and nominal magnification). Therefore, its proportion to mirror coma at given linear field radius remains constant regardless of mirror focal ratio.

If we express the coma blur length (Eq. 17) in arc minutes, it comes to 645h/DF3. Angular size of this blur on the retina is multiplied by telescope magnification M, to 645hM/DF3. Setting 645hM/DF3=3, since it is the angular size at which an object begins to appear non-point like to the average eye (although shape recognition requires at least 5 arc minutes size), gives the field radius h at which angular size of the coma blur is three arc minutes as

h=DF3/215M.

Average P-V wavefront error of astigmatism in a conventional eyepiece is approximated by WA~0.02e(α/F)2 in units of 550nm wavelength, where e is the ep focal length, and α is the ep field angle (AFOV/2). Substituting α~57.3h/e gives WA~66h2/eF2, with the astigmatism wavefront error relating to that of coma (WC=37.9h/F3, also in units of the wavelength) as 1.7hF/e.

The combined visible blur size can be, at least roughly, approximated as larger than coma-alone blur by a factor of 1+(WA/WC). If we assume that only about the bright 1/3 of the coma blur (sagittal coma) is visible, which is most often the case (the exception are bright stars, especially in large apertures) then the combined blur is roughly three times smaller.

Graph below (FIG. 115, top) illustrates how the quality visual field in the Newtonian telescope varies due to coma alone, but it also outlines how the magnitude of astigmatism in conventional eyepieces affects the actual field. Note that D/M in the above equations is, for convenience, replaced by the pupil diameter P. FIG. 115 bottom illustrates quality of the field within which comatic deformation of stellar images is not detectable, but the aberration is still large enough to affect image contrast.


FIGURE 115: TOP - Quality of the visual field in Newtonian is mainly determined by two factors: mirror coma and eyepiece astigmatism. Diffraction-limited field radius hL for mirror coma (blue plot) is based on the wavefront error (WFE), but says little about the actual star appearance. Whether coma will be noticed, or not, depends primarily on the angular size of the coma/astigmatism-bloated central diffraction maxima in the eyepiece. Since it changes with the relative magnification of a telescope, it can be expressed in terms of the eyepiece exit pupil diameter P. For all but very bright telescopic stars it is the bright 1/3 tip of the coma ray spot (sagittal coma) that approximates the visible aberration, in which case the coma-free visual field is the largest (green plots). Assuming the entire coma spot (tangential coma) visible, the actual visual diffraction-limited field radius (in a sense that within it there won't be appreciable difference in the appearance of a star vs. aberration-free image), is smaller than the nominal WFE-based field for exit pupils of about 2mm and smaller, but larger for larger exit pupils (red plots). Assuming only sagittal portion of the coma ray spot visible, the actual visual diffraction-limited field is larger than the nominal WFE-based field already at around 1mm exit pupil diameter, and more so for larger exit pupils. But, as mentioned, mirror coma is only one factor causing off-axis bloating of star images. At larger relative apertures (~/5 and larger with conventional eyepieces) eyepiece astigmatism becomes significant and/or dominant (w/conventional eyepieces). Average WFE of eyepiece astigmatism (WA) can be also roughly estimated, as well as its effect on the size of combined coma/astigmatism blur. The inset with diffraction pattern illustrates how eyepiece astigmatism combines with coma in producing the actual combined blur (ray spot on white background corresponds to 1 wave P-V of coma, next at right to it; the bottom 1/3 of the spot is sagittal coma). With an average conventional eyepiece, the combined blur is approximated by 1+(1.7h/P) in units of the coma blur (either sagittal or tangential), thus increasing exponentially with the field radius h. With corrected eyepiece, the combined blur is roughly twice the comatic blur alone. Neither is plotted, being only rough approximations, but it is safe to conclude that the actual diffraction limited visual field for mirror/eyepiece combinations is between the red and green plot for given exit pupil: closer to the green for corrected eyepieces, and closer to the red for the average conventional eyepiece.
BOTTOM - Graph above describes the size of field in the Newtonian in terms of a noticeable deformation of the stellar image. However, within a portion of the field w/o noticeable deformation of star images, coma is still large enough to affect performance. Starting at the field height where the coma deforms the central maxima just enough to show the typical deformation under sufficiently high magnification - F
3/30 in mm, as mentioned before - will turn to MTF for the indication of performance level toward field center (tangential MTF is for the blur axis perpendicular to MTF bars, and sagittal MTF is for the blur and bars in the same orientation; the average is approximately midway between the two). It is not the level of aberration present by itself, rather magnification used that determines when the effect of aberration will become noticeable.

Magnification acts as a frequency filter, with increasing magnification generally expanding frequency range toward higher frequencies, ultimately reaching system's cutoff frequency. Assuming that the threshold detection at cutoff frequency λ/D (in radians; 1.89/D in arc minutes, for λ=0.00055mm and aperture diameter D in mm) requires the separation between two consecutive bright lines to be 5 arc minutes, 1.89M/D=5, with M being the nominal telescope magnification; taking relative magnification per mm of aperture m=M/D, it becomes 1.89m=5, and m=5/1.89=2.64 (67x per inch). Since the inverse of relative magnification gives the corresponding eyepiece exit pupil diameter, the "cutoff pupil" is about P=0.37mm. At h=F
3/30 mm linear field radius (red), angular size of the sagittal coma is, 215h/DF3 arc minutes which, after substituting for h, gives 7.2/D arc minutes. The relative magnification m=M/D that will magnify it to 5 arc minutes is obtained from 7.2m=5, thus m=0.7, and P=1.4. This magnification is smaller than the system cutoff magnification by a factor of 0.26, thus it is effectively a low-pass filter for frequencies of ~0.26 and lower, with the higher frequencies attenuated. Still, as the graph indicates, the contrast drop in this range is still quite significant (since this field radius can be considered a median between the inner field with no coma deformation detectable, and outer field where it is obvious). The field graph top right shows that the linear field radius with this much coma ranges from below 6mm at /4 to over 17mm at /8, with the corresponding eyepiece apparent field radius needed to reach this portion of mirror image from 22 to nearly 70, respectively, which means it would be seen near edge of 50 AFOV eyepiece (and more inside the field with wider-angle eyepieces) at /4, but no eyepiece would reach that far out at /8 (recall that the ep f.l. needed to enlarge the coma to 5 arc minutes is 1.4F mm). For comparison, MTF graph show contrast drop at a three times larger field radius (hence three times lower magnification, and corresponding exit pupil P=4.2mm), where the cutoff frequency is below 0.09 (black). The magnitude of contrast drop toward the effective cutoff frequency for the magnification is similar, but the resolution limit is three times more coarse. Contrast is significantly lower roughly in the higher half of the range of resolvable frequencies, becoming similar to that at three times smaller field radius - and field center - inside the lower half. Effective cutoff magnification is achievable at the very edge of widest-field (~100 AFOV) eyepieces at /4, and somewhat farther from the edge with faster mirrors. At the field radius twice smaller than the one at which the deformation of stellar image due to coma becomes apparent, the effective cutoff is at 0.52 frequency, with the corresponding exit pupil P=0.7mm (orange). There is no stellar deformation detectable this far off axis, but the contrast drop is still significant. The radius is only about 1mm at /4, increasing to 8.5mm at /8, and the effect near cutoff frequency is seen at about 11 apparent field radius at /4, and 40 at /8. At yet twice smaller field radius (F3/120 mm radius, blue), the entire range of frequencies is covered, but contrast drop is less than the diffraction-limited maximum (F3/90 mm radius).

 The green plots on FIG. 115 top, for the field radius with the angular sagittal coma within 3 arc minutes, represent the field quality with zero eyepiece astigmatism. The red plots, for the entire coma blur (tangential coma) are probably more representative with respect to the actual blur size, at least as a rough guide, considering that eyepiece astigmatism commonly enlarges the final blur. Somewhat larger blur, 3-5 arc minutes is still acceptable, and toward field edge, as much as 6-9 arc minutes is tolerable.

Expectedly, apparent size of the comatic image is in inverse proportion to the exit pupil size, i.e. relative telescope magnification. But it doesn't translate into inferior field performance. For instance, at the exit pupil diameter P=1 and /5, coma blur reaches 3 arc minutes at nearly 0.6mm off axis. However, the corresponding error is still more than twice smaller than "diffraction limited". Since 1mm exit pupil at /5 requires 5mm f.l. eyepiece (i.e. f.l.=F), its field radius at a standard ~45 AFOV is ~2.1mm, implying that roughly the inner half of the field area (~3/4 of the field radius) will be better than diffraction limited. At the very edge, coma is only about 1/3 larger than diffraction limited, or about 0.55 wave P-V, with the angular size of about 10 arc minutes.

On the other hand, at three times lower magnification (P=3), the 3-minute coma blur field radius also triples, to 1.8mm, with the coma this far off now somewhat worse than diffraction limited. At the field edge (assuming the same ~45 AFOV), it is as much as 1.9 wave P-V, or 10 arc minutes, same as at 1mm exit pupil. The difference is that both, comatic pattern deformation and energy lost to the rings are significantly greater at 1.9 than at 0.55 wave P-V. In other words, coma alone is more noticeable and detrimental at the lower magnification.

Plugging in the eyepiece astigmatism factor, with the astigmatism P-V wavefront error approximately  WA~1.7hWC/P, where h is the field radius and WC the P-V wavefront error of coma, makes the actual blur larger. At 1mm exit pupil (P=1), edge coma at h=2.1mm is about 0.55 wave P-V, implying about 2 waves P-V of eyepiece astigmatism; at 3mm exit pupil (P=3), h=6.2 and WC=2, implying as much as 7 waves of astigmatism. In both cases eyepiece astigmatism dominates mirror coma by roughly the same ratio, about 3.5 (eyepiece astigmatism for given field angle increases with the focal length, and coma increases with the image field radius), but both coma and astigmatism are significantly larger at lower magnification. Using

Bcomb=[1+(WA/WC)]BC

approximation for the combined geometric blur, with BC being the coma blur, gives that the combined edge blur for for both, 1mm and 3mm exit pupil is over four times greater than that for coma alone. That comes to over 40 arc-minute blur, more than the naked-eye full Moon. Of course, not all of it is visible, even with bright stars; roughly, only about 1/3 of it, or less, can be seen. That is still nearly half the full Moon diameter; since in both cases the eyepiece astigmatism is strong, and by nearly the same ratio so than coma, degree of deformation of the actual pattern is similar in both cases, resembling the right most pattern above, with somewhat flatter base and a fainter triangular tail.

The difference is that the blur size relative to the Airy disc is three times smaller at 1mm exit pupil, since at 1mm exit pupil everything is three times more magnified than at 3mm. So, while the apparent size of stellar images is similar to that at 3mm exit pupil, energy concentration vs. image scale at the higher magnification is significantly better, and the bloated outer-field stars are roughly three times smaller relative to near-perfect mid-field stars.

The 1 wave P-V wavefront error of coma is characterized by Sidgwick as limit to tolerable, visually. Obviously, it may not be so at high enough magnifications, and/or with significant eyepiece astigmatism present. In a 200mm /5 Newtonian, coma is that strong at 3.3mm off axis. In the 10mm f.l. eyepiece, tangential coma is already 8.5 arc minutes, and sagittal near to 3. Average eyepiece astigmatism P-V wavefront error at this field height (close to 20 field angle) is nearly three times stronger than coma's, enlarging sagittal coma to roughly 11 arc minutes - quite obvious for most observers.

Summing it up, stellar deformation due to coma is not apparent at any magnification within field radius less than three times the diffraction-limited coma field. The field radius at which coma becomes apparent, if sufficiently magnified, is given by F3/30 in mm, or tanα=F2/39 angularly, with the magnification needed to make it apparent to the eye (~5 arc minutes sagittal coma) being provided by an eyepiece of 1.4F mm focal length. This assumes negligible eyepiece astigmatism. With conventional eyepieces, stellar deformation in faster mirrors will progressively more deteriorate toward the outer field due to eyepiece astigmatism. For instance, linear field radius at which stellar deformation becomes apparent is more than three times larger at /6 than /4, but the corresponding angular radius in the eyepiece magnifying it to 5 arc minutes is only about twice larger at /6 (FIG. 115 top right). Thus the astigmatism at /6 is effectively somewhat more than doubled at twice the angular radius (quadrupled due to doubled apparent angle, and larger by a factor 1.5 due to larger e.p. f.l. but more than halved due to the slower focal ratio), while at /4 it is quadrupled, and combined with twice larger coma.

Field curvature in the Newtonian, with the radius equaling its focal length, is relatively weak, with no significant effect on visual observing. 
 

lower-order astigmatism, from Eq. 18, after substitutions (for θ=0), as the P-V and RMS wavefront error at the best focus, for object at infinity, are:

Wa = Dα2/8F = -h2/8fF2  = h2/8DF3   and    ωa = Wa/241/2         (71)

with being the mirror focal length.

In units of the λ=0.00055mm wavelength, the errors are:

Waλ = 227Dα2/F       and      ω = W/241/2         (71.1)

for D in mm (or with the numerical factor 227 replaced by 5766 for D in inches).

The field angle at which astigmatism reaches 0.80 Strehl is α=√F/622D in radians, or at the off-axis height in the image plane of h=α. It has no significance, with the coma error being absolutely dominant at this field angle. The two aberrations' RMS wavefront errors equalize for the field angle α=1/6F in radians, or for the h=D/6 height in the image plane, after which the astigmatism quickly becomes the dominant aberration.

The wavefront error doesn't change with object distance. However, it does change with the stop position. More details on this, as well as on the geometric (ray) aberration in 2.3. Astigmatism.

field curvature: Petzval curvature of a concave mirror is Rp=R/2, R being the mirror radius. Due to the presence of astigmatism, actual best image curvature is different from the Petzval. Also, it varies with the stop position. For the stop at the surface, best, or "median" image surface equals the negative of mirror's focal length:

Rm = -R/2          (72)

which makes it positive (concave toward mirror) for mirror oriented to the left, regardless of mirror conic. Position of the aperture stop influences mirror astigmatism, which in turn causes changes in the median image curvature. As Eq. 39 shows, it varies somewhat with the conic. For a paraboloid, best image surface is flat with the stop at half the focal length from mirror, and with the stop at the focal length away mirror astigmatism is cancelled, but image curvature equals R/2 (convex toward mirror). For a sphere, best astigmatic field is flat for the stop at (1-0.51/2) focal lengths from mirror, and astigmatism is cancelled for the stop at the center of curvature.

Miscollimation sensitivity of a Newtonian is determined by the diameter and F-number of the primary mirror. Sources of miscollimation are: (1) primary tilt/decenter, (2) flat tilt/decenter/despace, (3) focuser tilt/decenter and (4) tube/structural flow resulting in any of the former. As a result, axial image point is shifted away from the field center, and replaced with certain amount of off-axis coma.

Expressing the linear center field shift as 2τ/(60x57.3), for τ the mirror tilt angle in arc minutes and the focal length, and substituting it for h in Eq. 70.1 (right), gives the RMS wavefront error of coma shifted to the field center (in units of 0.00055mm wavelength) as:

ωt = τD/257F2         (73)

for the aperture diameter D in mm (for D in inches, ωt = τD/10F2).

Assigning H to the diagonal-to-focus separation, and H' to the focus height in the focuser (measured from the focuser base), the miscollimation sensitivity per arc minute of tilt is smaller by a factor H/ for the diagonal, and by a factor of H'/2 for the focuser, respectively, with being the mirror focal length.

Sensitivity to decenter - also in units of 550nm wavelength - is identical for all three, mirror, flat and focuser, and given by:

ωd = 6.7/F3         (73.1)

for the decenter in mm. It is also the despace sensitivity of the flat for despace in mm.

More details on collimating Newtonian on the following page.

8. REFLECTING TELESCOPES       8.1.2. Newtonian collimation

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