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5.2. Low-level turbulence, tube currents...   ▐    6. EFFECTS OF WAVEFRONT ABERRATIONS

5.3. Optical misalignment: tilt, decenter and despace

For best performance, optical surfaces of a telescope have to be in their optimum alignment. Any deviation from the optimum position - be it tilt, decenter or despace - will negatively affect wavefront accuracy (FIG. 90). Degree of sensitivity to misalignment vary with the design and system properties. In general, the more optical surfaces and the more strongly curved they are, the greater misalignment sensitivity. More specific consequences of misalignment are addressed with each particular telescope type.

FIGURE 90: Misaligned optical surface can be tilted, decentered, despaced, or any combination of the three in respect to its proper position. Tilt is expressed in angular form, while decenter and despace are linear quantities. Tilt and decenter result in one side of the optical surface becoming closer to the wavefront than the other one - a recipe for coma, which is by far the dominant resulting aberration. Astigmatism, which results from wavefront's inclination relative to the surface is small in comparison. Despace causes mainly spherical aberration with converging or diverging wavefronts - as shown to the right, result of the wavefront-to-surface respective curvatures at the location of reflection/refraction being different from those at the optimum position.

Probably the most common miscollimation error is coma induced by misaligned primary and secondary mirror. With a flat secondary mirror (Newtonian) it simply shifts the axis of the primary away from the eyepiece axis, bringing a portion of the image affected by (existing) off-axis aberrations to the field center. In a two-mirror system it creates coma where there was none before.

As can be grasped from the illustrations (FIG. 53, left and center), the misaligned surface induces the same amount of error to the wavefront regardless of its inclination. Hence this coma is invariant to field angle; in a coma-free system, it affects the entire field equally, while in a system with existing (inherent) coma, it is intact only in the field center, while either lessening - or adding to - the existing coma in the rest of the field.

Taking as an example a common 100mm f/10 achromat, its two lenses can be decentered, tilted and/or despaced. Despace induces only spherical aberration (widening the gap undercorrection, and vice versa). Tilt and decenter will induce all-field coma, but will also affect lateral color error (larger tilt/decenter values will also induce image tilt, but they are not possible in a functional lens cell).

Decentering either lens alone induces more than 1/2 wave P-V of all-field coma, as well as all-field lateral color error, with F and C line separation nearly equaling e-line Airy disc diameter. Tilting either lens by 0.05° induces near 0.6 waves P-V of all-field coma, and about half Airy disc F/C separation. If both lenses are tilted, the effect is negligible. If the rear lens is tilted as much but in opposite direction, the coma nearly doubles, but lateral color error is cut in half. The magnitude of aberration induced is commensurate to the nominal misalignment, and its sign changes with the sign of misalignment (same sign of tilt and decenter of the negative lens induce coma of opposite sign to that of the positive lens).

This exercise shows that even the common f/10 achromat requires fairly tightly aligned lenses in order to avoid significant aberrations induced. This is raised to another level with apo doublets and triplets. For instance, decentering one lens by 0.25mm in the Takahashi FC100DZ f/8 doublet induces 2.7 waves P-V of coma, and with faster doublets it can be up to 2-3 times as much. That same decenter in a 100mm f/6.5 BSC7/FCD100/BSC7 triplet (middle element) would induce 4.2 waves P-V of coma, while 0.05° tilt of that same element would induce 1.2 waves P-V of coma.

      5.4. Force-induced surface errors: thermal, pinching, gravitational pull

The enormously high surface accuracy requirements for optical surfaces of a telescope results in high sensitivity to even slight changes induced by external forces. Such forces commonly are: (1) thermal expansion and contraction, (2) pressure by the mounting elements, and (3) force of gravity.

Thermal expansion and contraction causes surface deformities due to their uneven rate within the body of an optical element. Given material homogeneity and thermal properties, it becomes more of a problem as the volume of an element increases, and as the mass distribution gets more uneven. Relatively small differences in the temperature can cause significant surface deformations and resulting wavefront error. The dominant aberration induced is usually spherical and/or edge defect error, but other, more or less random forms of wavefront deformation are frequent. The only cure to it is to get optical elements to a thermal near-equilibrium with the surrounding air.

In general, mirrors are significantly more affected by thermal expansion/contraction than lenses. This is due to the fact that deviation at the reflective surface causes change in the wavefront larger by a factor of (n-1)/2 than the same nominal deviation at the lens surface, n being the lens refractive index (also due to mirrors having typically significantly thicker edge for given diameter, both nominally and relative to the center thickness). The usual scenario is a telescope - including its optical components - warmer than surrounding air for more or less extended period of time (depends of telescope size, thermal characteristics of the optics and mechanics, and passive/natural vs. active/fan cooling).

Once the entire objective cools down to a new temperature, the only consequence is slight reduction in focal length, caused by the slight reduction in its size and proportional to it. With the linear coefficient of thermal expansion for standard optical glasses of ~0.000006/�C, the change in focal length for as much as 10�C change in the temperature of the objective is still only ~/17,000, being the focal length (it is usually smaller than reduction of tube/structure length caused by cooling). In the process, however, surface deformations caused by thermally caused shrinkage or expansion can create wavefront aberrations, especially when portions of the optical element (front vs. back, top vs. bottom, center vs. edges) are cooling or warming up at different rates (FIG. 91).

FIGURE 91: Effect of thermal contraction on optical surface. Once the optical element has cooled down entirely, it is slightly smaller, and the only consequence is slightly shorter focal length (A). During the cooling process, surfaces are stretched outward, due to the outer portion cooling (and shrinking) at a faster rate than the inner mass (B). If one side is cooling at a faster rate, it will also be more deformed (C). The deformation weakens curvature of a concave surface toward the edge, while strengthening the outer curvature of a convex surface. The inset to the right gives a simplistic illustration of the mechanism causing the outer surface points to be pulled inward relative to the surface points closer to the center. 

The general form of surface deformation is opposite for contraction vs. expansion. In a concave mirror, contraction will induce overcorrection. Quantifying the error involves extensive calculation with material, dimensions, temperatures and present factors other than the optical element itself influencing the process of thermal transfer specified. For a rough approximation of the error level in the scenario with uneven rate of cool down for the front and rear, an arbitrary assumption can be made that contraction of the cooler, front side, and partial contraction of the circumference wall area, have brought the edge point E of the wormer shape to the point H of the cooler shape, with the length GH being 1/2 of the full contraction along the edge (width-wise parallel to axis).

For a mirror, with relatively small axial vs. edge thickness difference, arbitrarily assuming axial contraction 1/2 of the edge contraction, the edge contraction versus center is by ~τT/2 greater, with τ being the glass thermal expansion coefficient (linear) and T the edge thickness. This contraction differential equals the peak surface deviation from its original form. With the thermal expansion coefficient being for the linear expansion per 1�C, this simple relation gives an idea of the level of peak (edge) surface deviation for every 1�C differential between the front and back of the mirror.

With the coefficient of expansion for plate glass τ~0.0000085/C�, the edge vs. center surface deviation is ~1/5 wave (for 550nm wavelength) edge surface error for every inch of mirror thickness, doubling in the wavefront. Pyrex, with about 2.5 times smaller thermal expansion coefficient, will have proportionally smaller deformation. Due to the deviation increasing from center toward the edge, it produces wavefront deviation generally resembling spherical aberration and/or defocus. For an actual surface error, the wavefront error is probably somewhat smaller than double as much - possibly significantly - due to refocusing (i.e. better fit of the deformed wavefront to a reference sphere of slightly longer radius). The RMS error may be lower than for ordinary spherical aberration even before refocusing, since it seems possible that the wavefront could be comparatively less affected away from the edge area, hence more resembling the higher-order terms, affecting relatively smaller wavefront area. In the scenario where both, front and rear of a mirror are similarly exposed to cooler air, the error is likely to be greater, due to more rapid cooldown of the entire circumference area.

Pressure from mounting elements usually causes some form of astigmatism, due to the typically radially symmetric distribution of the points of support and/or retaining. Typical pinching pressure, for instance, induces trefoil (under "pinching") - a three-winged form of astigmatism, quickly revealing itself in the appearance of diffraction pattern. Mounting pressure can result from thermal expansion of optical elements and/or mechanical structure, which is one more reason why optical elements should be left slightly loose within mechanical structures holding them.

Gravitational force tends to deform larger pieces of glass, especially if they are relatively thin. The form of deformation depends on the position angle, as well as on the support points distribution and level. While the error induced by gravitational force is usually low to very low, it can become significant in larger diameters, especially large thin mirrors. Proper support structure here can be critical. Popular free software, PLOP, evaluates the error induced by gravitational force to a mirror on mirror cell support points.

Common characteristic of induced telescope aberrations is that they do not have pre-determined level. Unlike the aberrations inherent to the optical set, they vary with the user, telescope and the circumstance. The effect on image quality is directly related to the RMS wavefront error they cause, which is often times hard to determine. Partly due to this elusiveness, they are, in general, less well known of, and taken less seriously than intrinsic telescope aberrations. However, there is no difference in the effect of aberration, regardless of its origin. Aberrations induced to a near-perfect optics can make it perform as a third-grade system. Thus, knowledge and control of induced telescope aberrations are unavoidable part of the proper routine of using a telescope.

5.2. Low-level turbulence, tube currents...   ▐    6. EFFECTS OF WAVEFRONT ABERRATIONS

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