◄ 7.2. Spider obstruction and apodization   ▐    8.1.1. Newtonian reflector diagonal flat ►
 

As one would think, reflecting telescopes use mirror objectives to form the image. They come in a variety of arrangements, from a single mirror objective to those consisting of several mirrors. Most of designs are obstructed, with the secondary mirror in the light path - such are Newtonian and Cassegrain varieties. Unobstructed designs, either with tilted or off-axis elements, enjoy small but steady popularity. Inherent optical quality varies from one design to another, and so do other characteristics, including needed properties of optical surfaces, and sensitivity to miscollimation. 

8.1. The Newton reflector

The most popular telescope design among amateurs - Newtonian reflector - consists from a single concave primary mirror and a reflecting flat. It is a single optical surface of the primary mirror that creates the image, while the flat merely directs light out to the side, to an accessible observing location. Ideal shape of the primary mirror is paraboloid, which is free from spherical aberration when imaging distant objects. Smaller, longer focus mirrors can be left spherical - which is the easiest to make and test mirror shape - but will have residual amounts of spherical aberration (FIG. 55). Concave mirror also produces off-axis aberrations which are, for distant object and stop at the surface, independent of its conic constant.

Omitting distortion, which is negligible for the typical field angles, a quick look at the aberration coefficients gives following wavefront errors for the concave mirror:

- lower-order spherical aberration, from Eq. 7,9.1-2 is given for object at infinity by

                                                              Ws=(K+1)D/2048F3                                       (66)

as the P-V wavefront error at best focus (1/4 of the error at paraxial focus), with K being the mirror conic, D the aperture diameter and F the focal ratio number. Evidently, the only surface form free from spherical aberration for object at infinity is a paraboloid (K=-1). The RMS wavefront error for primary spherical aberration is smaller than the P-V error by a constant ratio; it is given by: 

                                                                 ω=Ws/√11.25.                                            (67)

In units of the λ=0.00055mm wavelength, the P-V wavefront error is:

                                                            Wsl=0.888(K+1)D/F3                                         (68)   

for the aperture D in mm, and for the aperture D in inches:

                                                 Wsl=22.55(K+1)D/F3                                       (68.1)

Longitudinal and transverse aberrations are as given in 2.1 Spherical aberration. The RMS wavefront error, in units of the wavelength, can be used to calculate the appropriate Strehl ratio from Mahajan's close approximation (Eq. 56). For spherical mirror, it can also be expressed as:

                                      S ~ e-(1.66D/F3)2 ~ 1/e(1.66D/F3)2                            (69)

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 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-(2πω)2, needed F# for desired Strehl with spherical mirror is given by F=3.5D1/3/(1-S)1/6, for the aperture diameter D in inches, and F=1.18D1/3/(1-S)1/6 for D in mm.

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 best focus, after substituting for m in terms of the object distance o and mirror focal length ƒ, is W's=-[K+(1-2ψ)2]D/2048F3, with ψ=ƒ/o being the primary focal length in units of the object distance (which is 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 the object distance, for primary spherical aberration, as K=-(1-2ψ)2.

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

                                                           Wc=αD/48F2  = h/48F3                                     (70)

with α being the field angle (α=h/ƒ, h being the height in the image plane, and ƒ the mirror focal length). Note that this is double the error given by Eq. 12, which expresses only the peak aberration. The RMS wavefront error is related to the P-V error as:

                                                  ω=Wc/√32 = h/272F3                                     (71)

In units of the l=0.00055mm wavelength, the P-V error of coma is:

                                                      Wcl=38αD/F2 =  38h/F3                                       (72)

(for D in mm, or the same metric used for the wavelength), and the RMS error ωλ, slightly rounded off,

                                                ωλ=6.7αD/F2 =  6.7h/F3                                  (73)

For the 0.80 Strehl ratio, ωλ=1/√180, which corresponds to the field angle α=F2/90D in radians, for D in mm, or the height in the image plane h=ƒα=F3/90, also in mm (for the aperture D in inches, simply divide last two expressions by 25.4). Actual Strehl ratio is slightly smaller, due to the presence of astigmatism.

As 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 to be clearly recognized by an average eye. Since the angular size of sagittal coma in the eyepiece can be approximated with 4ε/F2, in arc minutes, with ε being the eyepiece apparent viewing semi-angle in degrees, the viewing angle in the eyepiece at which the geometric sagittal coma would become obvious to most people is given by ε~1.25F2. This means that one could see it at the edge of a 40° AFOV eyepiece at f/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 the 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 the 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 1.5 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 the geometric sagittal coma. As the error diminishes bellow ~1 wave P-V, the effect gradually transforms into one-sided intensity distribution asymmetry, 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) to begin to resemble sagittal coma in the actual star image can be approximated by h~F3/30, in mm. For an f/4 mirror, that corresponds to ~2.1mm (FIG. 56). For the magnification level neded to enlarge the actual comatic image of a bright star with this amount of coma to 5 arc minutes apparent size, the eyepiece focal length is approximately f~1.3F. Hence an average observer, with a 5mm eyepiece and an f/4 mirror, can expect to notice coma on bright stars from ~2mm off-axis out. With an f/6 mirror, it will take an 8mm eyepiece to show coma becoming apparent some 7mm off-axis; 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.

Due to eyepiece astigmatism, which increases with the square of off-axis distance, comatic-like deformation in the eyepiece grows rapidly toward the field edge. Eyepiece astigmatism increases with the square of the focal ratio; combining it with the coma wavefront error increasing with the cube of the focal ratio, it becomes understandable why off-axis performance of the paraboloidal mirror drops so quickly with the increase in relative aperture beyond ~f/6. Field curvature, if present to a significant degree, has similar effect, although it is unusual for it to be strong enough to affect visual observing. 

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

                                                               Wa=Dα2/8F = h2/8ƒF2                                   (74)

with ƒ being the mirror focal length. The RMS wavefront error is: 

                                                         ωa=Wa/√24                                          (75)

In units of the l=0.00055mm wavelength:

                            Wal=227Dα2/F       and      ωal=Wal/√24                             (76)

for D in mm. 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=αƒ. It bears no significance, with the coma error being absolutely dominant at this field angle. The two aberrations' RMS wavefront error equalizes 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 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 mirror's focal length:

                                                                          Rm=-R/2                                                (77)

Change in the aperture stop position causes change in the mirror astigmatism, which in turn causes changes in the median image 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, bringing in the field center certain amount of off-axis coma. From Eq. 73, coma induced by the primary mirror tilt can be expressed as the RMS error (in units of the 0.00055mm wavelength)  as:

                                                         ωt' = τD/514F2                                          (78)

for the aperture diameter D in mm and τ the tilt angle in arc minutes. Assigning Λ to the diagonal-to-focus separation, and Γ 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 2Λ/ƒ for the diagonal, and by a factor of Γ/ƒ for the focuser, respectively, with ƒ being the mirror focal length.

Sensitivity to decenter is identical for all three elements, and given by:

                                                          ωd'=6.7∆/F3                                             (79)

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

 ATM mirror primer: aberrations in the Foucault test setup

Not seldom, amateurs make their own primary mirror for the Newtonian, typically using Foucault test during the figuring. In it, the mirror is examined under the light coming from a point source placed at mirror's center of curvature (in practice, slightly off laterally, so that the image can be observed). This object position differs from one for which the mirror is intended - infinity - and so do its exerted optical aberrations.

The P-V wavefront error of lower-order spherical aberration, from Eq. 7 and 9, is given by Ws=KD4/256R3=KD/2048F3, with K being the mirror conic, and F being the mirror focal number for object at infinity. For prolate ellipsoids, paraboloid and hyperboloids, the negative sign determined by the conic indicates over-correction. Comparing it with Eq. 66 shows that paraboloid with object at the center of curvature exerts the same amount of spherical aberration as a comparable sphere for object at infinity, only of the opposite sign.

However, since the effective focal ratio number F has doubled, the geometric aberration has changed: longitudinal spherical is larger by a factor of four vs. that for the sphere and object at infinity, while the transverse spherical is now doubled (keeping the same proportion to the Airy disc).

The wavefront aberration, transverse and longitudinal aberrations change in proportion to the fourth, third and second power of the zonal height, respectively.

For the stop at the surface, coma is cancelled, regardless of the conic. Wavefront error of astigmatism, on the other hand, doesn't change with the object distance, remaining as given by Eq. 74. The geometric aberration, again, changes due to the doubled effective focal ratio number F, with the longitudinal aberrations larger by a factor of four, and the transverse aberration nominally doubled. The wavefront error changes in proportion to the square of the zonal height, transverse aberration with the zonal height, and the longitudinal is, as expected, constant.
 

◄ 7.2. Spider obstruction and apodization   ▐    8.1.1. Newtonian reflector diagonal flat ►

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