TO: File
FROM: Robert A. Arnold
SUBJECT: Secondary Mirror Motions Versus Actuator Steps
DATE: 11-July-1995 Rev. A
=============== Table of Contents ===============
I. Introduction
II. Actuator Steps Required For Each Mirror Motion
III. Mirror Motion For Single Actuator Steps
IV. Orthogonality of Mirror Motions
V. Telescope Image Plane Optical Aberrations
VI. Optical Consequences of a Single Actuator Step
VII. Optical Consequences of Random Actuator Errors
VIII. Conclusions
Figure 1. Secondary Mirror Nomenclature and Sign Conventions
Table 1. Actuator Steps Versus Secondary Mirror Motions
Table 2. Actuator Steps For One Unit of Secondary Mirror Motion
Table 3. Inherent Motion Increment For Each Secondary Mirror Motion
Table 4. Secondary Mirror Motion for a Single Actuator Step
Tilt about Vertex and Decentration of Vertex
Table 5. Secondary Mirror Motion for a Single Actuator Step
Tilts about Neutral Point and Center of Curvature
Table 6. Secondary Mirror Motion for a Single Actuator Step
Tilt about Mid Plane and Decentration of Vertex
Table 7. Orthogonality of the Secondary Mirror Motions
Tilt about Vertex and Decentration of Vertex
Table 8. Orthogonality of the Secondary Mirror Motions
Tilts about Neutral Point and Center of Curvature
Table 9. Orthogonality of the Secondary Mirror Motions
Tilt about Mid Plane and Decentration of Vertex
Table 10. Secondary Mirror Motion Induced Image Aberrations
Table 11. Wavefront Aberration Changes Induced by
Tilt of the Secondary Mirror About the Neutral Point
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A computer simulation study has been performed in order to analyze the motions of the secondary mirror as a function of the individual step size of the mirror positioning actuators. This study was performed for the purposes of: documenting the number of actuator steps required to produce a given mirror motion, quantifying the motions of the mirror which result from a single step of any one of the actuators, and to determine the influence of these steps upon the optical performance of the telescope. This information may also be used for estimating the anomalous mirror motions that would result from measured positioning errors for each of the actuators.
This study was performed in four parts. The first part was to use the secondary mirror actuator control software to calculate the number of actuator steps that would be required to produce each of a number of selected motions of the secondary mirror. The second part was to cast these numbers of actuator steps for each given motion as a six-by-six matrix of the number of steps versus six degrees of freedom of mechanical motion. This matrix was then matrix inverted so as to produce the mechanical motion in each of the six degrees of freedom that would result from a single step of any single actuator. The third part was to evaluate the telescope image plane optical aberrations that would result from various motions of the secondary mirror so as to quantify the optical consequences of these motions. The fourth part was to combine the known actuator positioning errors with the optical consequences of the secondary mirror motions so as to estimate the impact of the actuator errors on the telescope optical performance.
In the use of the secondary mirror actuator control software, two important factors were taken into consideration. The first is that the calculated number of actuator steps is a nearest integer value. In order to minimize the consequences of this (which in effect quantizes the number of actuator steps), comparatively large secondary mirror motions were used. The second factor is that (especially for these large motions) the required number of actuator steps is a non-linear function of the secondary mirror motion. This factor was minimized by performing the motions in both direction from the centered position and averaging the results. This in effect employed the chord line of the non-linear curve which is a close approximation to the slope of the curve in the vicinity of the centered position. It should be noted that this analysis is applicable to the center of the travel region of the secondary mirror motion and is only an approximation for motions that are well away from this center of the travel region. The coordinate system conventions and secondary mirror actuator numbering conventions followed in this study are presented in Figure 1.
II. Actuator Steps Required For Each Mirror Motion
The number of actuator steps required for each of a selection of secondary mirror motions are presented in Table 1. The focus and decentration travel values are in units of microns and the roll and tilt travel values are in units of arc-seconds. It is noted that in defining six independent degrees of freedom of mechanical motion, any pair of two motions may be selected from the group of motions labelled as: tilt about the vertex, tilt about the neutral point, tilt about the center of curvature, and decenter. The other motions within this group can be achieved by a linear combination of the selected two motions and therefore are not independent.
The mechanical motions of decenter and tilt about the vertex are natural motions in which to visualize the motion of the secondary mirror. However, the mechanical motions of tilt about the neutral point and tilt about the center of curvature are natural motions in which to evaluate the optical effects which result from these motions. The tilt about the neutral point results in changing the line of sight of the telescope (along with changing the tilt of the telescope image plane and the centration of the astigmatism pattern in the image plane of the Ritchey-Chretien telescope) while not affecting the field independent coma produced across the telescope image plane. In contrast, the tilt about the center of curvature results in changing the field independent coma while not affecting the line of sight of the telescope. The mechanical motions of decenter and tilt about the mid plane (that plane which is mid way between the upper and lower actuator attachment points) are special in that they produce a set of actuator step vectors that are orthogonal for the six degrees of freedom represented by these motions.
Table 1. Actuator Steps Versus Secondary Mirror Motions
Motion Travel Act 1 Act 2 Act 3 Act 4 Act 5 Act 6
Focus +4000 -1401 -1401 -1401 -1401 -1401 -1401
-4000 1428 1428 1428 1428 1428 1428
Roll +7000 1349 -1335 1349 -1335 1349 -1335
-7000 -1335 1349 -1335 1349 -1335 1349
X Tilt +4000 -205 -889 1117 1117 -889 -205
Vertex -4000 234 893 -1104 -1104 893 234
X Tilt +4000 -664 -3 666 666 -3 -664
Mid Plane -4000 666 -3 -664 -664 -3 666
X Tilt +1000 -549 790 -215 -215 790 -549
Neutral -1000 567 -777 236 236 -777 567
X Tilt +400 -497 887 -363 -363 887 -497
Cen Curv -400 517 -874 384 384 -874 517
X Dec +3000 -908 15 927 -908 15 927
-3000 927 15 -908 927 15 -908
Y Tilt +4000 -1155 -755 396 -379 781 1156
Vertex -4000 1156 781 -379 396 -755 -1155
Y Tilt +4000 -386 -766 -386 382 770 382
Mid Plane -4000 382 770 382 -386 -766 -386
Y Tilt +1000 591 -181 -768 781 203 -574
Neutral -1000 -574 203 781 -768 -181 591
Y Tilt +400 732 -65 -794 808 88 -716
Cen Curv -400 -716 88 808 -794 -65 732
Y Dec +3000 -517 1068 -517 -517 1068 -517
-3000 543 -1052 543 543 -1052 543
The data presented in Table 1 was analyzed so as to determine the number of actuator steps (including fractional parts of a step) required to produce one unit of motion for each of the motion cases considered. The results from this analysis are presented in Table 2.
Table 2. Actuator Steps For One Unit of Secondary Mirror Motion
Motion Act 1 Act 2 Act 3 Act 4 Act 5 Act 6
Focus -0.354 -0.354 -0.354 -0.354 -0.354 -0.354
Roll 0.192 -0.192 0.192 -0.192 0.192 -0.192
X Tilt - Vertex -0.055 -0.223 0.278 0.278 -0.223 -0.055
X Tilt - Mid Plane -0.166 0.000 0.166 0.166 0.000 -0.166
X Tilt - Neutral -0.558 0.784 -0.225 -0.225 0.784 -0.558
X Tilt - Cen Curv -1.268 2.201 -0.934 -0.934 2.201 -1.268
X Dec -0.306 0.000 0.306 -0.306 0.000 0.306
Y Tilt - Vertex -0.289 -0.192 0.097 -0.097 0.192 0.289
Y Tilt - Mid Plane -0.096 -0.192 -0.096 0.096 0.192 0.096
Y Tilt - Neutral 0.583 -0.192 -0.774 0.774 0.192 -0.583
Y Tilt - Cen Curv 1.810 -0.191 -2.002 2.002 0.191 -1.810
Y Dec -0.177 0.353 -0.177 -0.177 0.353 -0.177
It is noted that since the actuators can be moved only in integer numbers of steps, there is an inherent increment size applicable to each individual motion. These inherent increments and the associated numbers of actuator steps are presented in Table 3. The presented inherent increment sizes and the associated numbers of actuator steps are only approximate values and are intended for providing insight rather than precise values. While seventy of the seventy-two values for the numbers of actuator steps are within one percent of the exact values, two of the values (which are for one step each) are in error by about eight percent of the exact value.
Table 3. Inherent Motion Increment For Each Secondary Mirror Motion
Motion Increment actuator steps
Focus 2.83 microns -1, -1, -1, -1, -1, -1,
Roll 5.21 arc-secs 1, -1, 1, -1, 1, -1,
X Tilt - Vertex 18.0 arc-secs -1, -4, 5, 5, -4, -1,
X Tilt - Mid Plane 6.00 arc-secs -1, 0, 1, 1, 0, -1,
X Tilt - Neutral 8.89 arc-secs -5, 7, -2, -2, 7, -5,
X Tilt - Cen Curv 8.56 arc-secs -11, 19, -8, -8, 19, -11,
X Dec 3.27 microns -1, 0, 1, -1, 0, 1,
Y Tilt - Vertex 10.4 arc-secs -3, -2, 1, -1, 2, 3,
Y Tilt - Mid Plane 10.4 arc-secs -1, -2, -1, 1, 2, 1,
Y Tilt - Neutral 5.21 arc-secs 3, -1, -4, 4, 1, -3,
Y Tilt - Cen Curv 10.5 arc-secs 19, -2, -21, 21, 2, -19,
Y Dec 5.65 microns -1, 2, -1, -1, 2, -1,
It is noted that a focus motion equal to one-half of the given increment can accomplished by combining a focus motion of 1.4 microns with a roll motion of 2.6 arc-seconds. This would result in every other actuator moving just one step and the resulting roll component would have no optical effect in the telescope. While smaller increments of motion can be requested from the software and hardware, the resulting mechanical motions would be only rough approximations of the requested motions.
III. Mirror Motion For Single Actuator Steps
The data presented in Table 2. was further analyzed using matrix inversion so as to determine the secondary mirror motions that would result from a single step motion of each actuator. It is noted that a single step for an actuator is equal to a two micron motion for that actuator. The units for focus and decenter motions are in microns and the units for roll and tilt motions are in arc- seconds. For purposes of matrix processing, the data was organized into three groups of six motions. Each six-by-six matrix was then inverted to obtain the combinations of secondary mirror motions for that group of motions that would result from a single step of one actuator. The results from this analysis are presented in Tables 4, 5 and 6.
Table 4. Secondary Mirror Motion for a Single Actuator Step
Tilt about Vertex and Decentration of Vertex
Actuator Focus X Tilt Y Tilt X Decn Y Decn Roll
microns arc-sec arc-sec microns microns arc-sec
1 -0.471 -1.504 -0.868 -0.270 -1.420 0.869
2 -0.471 0.000 -1.736 1.095 0.943 -0.869
3 -0.471 1.504 -0.868 1.365 0.476 0.869
4 -0.471 1.504 0.868 -1.365 0.476 -0.869
5 -0.471 0.000 1.736 -1.095 0.943 0.869
6 -0.471 -1.504 0.868 0.270 -1.420 -0.869
Table 5. Secondary Mirror Motion for a Single Actuator Step
Tilts about Neutral Point and Center of Curvature
Actuator Focus XTiltN YTiltN XTiltC YTiltC Roll
microns arc-sec arc-sec arc-sec arc-sec arc-sec
1 -0.471 -2.221 -1.550 0.715 0.683 0.871
2 -0.471 -0.236 -2.693 0.236 0.959 -0.867
3 -0.471 2.458 -1.143 -0.951 0.276 0.870
4 -0.471 2.458 1.143 -0.951 -0.276 -0.870
5 -0.471 -0.236 2.693 0.236 -0.959 0.867
6 -0.471 -2.221 1.550 0.751 -0.683 -0.871
Table 6. Secondary Mirror Motion for a Single Actuator Step
Tilt about Mid Plane and Decentration of Vertex
Actuator Focus X Tilt Y Tilt X Decn Y Decn Roll
microns arc-sec arc-sec microns microns arc-sec
1 -0.471 -1.504 -0.868 -0.817 -0.472 0.869
2 -0.471 0.000 -1.736 0.000 0.943 -0.869
3 -0.471 1.504 -0.868 0.817 -0.472 0.869
4 -0.471 1.504 0.868 -0.817 -0.472 -0.869
5 -0.471 0.000 1.736 0.000 0.943 0.869
6 -0.471 -1.504 0.868 0.817 -0.472 -0.869
The results as presented in Tables 4, 5 and 6 were computed by selecting three sets of six combinations of motions and inverting each six-by-six matrix. Each value shown is an expression of the actual secondary mirror motion as represented by the specific mechanical motion. All six secondary mirror motions included within a given set of motions must be applied in order to realize the single actuator step that was assumed for the initial input. As stated previously, the various tilt and decenter motions can be expressed as linear combinations of each other and therefore are not all unique degrees of freedom.
It is observed in Tables 4, 5 and 6 that the magnitude of the resulting tilts and decenters are variable depending upon which set of motions the results are expressed in. This relationship is a consequence of some of the motions not being orthogonal in the first and second groups. As a result, there is not a unique set of values for the tilt and decenter that are produced by a given actuator motion.
IV. Orthogonality of Mirror Motions
In addition to the linear dependencies between the various tilt and decenter motions cited above, there also is the relationship that a number of these motions are not orthogonal when expressed in terms of the actuator steps required to produce each motion. In order to explore and effectively present this relationship, further analysis was performed using the data from Table 2, grouped as in Tables 4, 5 and 6. In each case the number of actuator steps required for a single unit of motion was renormalized so that the sum of the squares of the actuator steps was equal to unity. The resulting matrix was then multiplied by its own transpose so as to produce a matrix which is the term by term vector dot product of the vector of actuator steps for one motion multiplied by the vector of actuator steps for another motion. When the two vectors are orthogonal, the dot product is zero. When the two vectors are collinear, the dot product is plus or minus one. The results from this analysis are presented in Tables 7, 8 and 9.
Table 7. Orthogonality of the Secondary Mirror Motions
Tilt about Vertex and Decentration of Vertex
Motion Focus X Tilt Y Tilt X Decn Y Decn Roll
Focus 1.000 0.0 0.0 0.0 0.0 0.0
X Tilt 0.0 1.000 0.0 0.0 -0.758 0.0
Y Tilt 0.0 0.0 1.000 0.757 0.0 0.0
X Dec 0.0 0.0 0.757 1.000 0.0 0.0
Y Dec 0.0 -0.758 0.0 0.0 1.000 0.0
Roll 0.0 0.0 0.0 0.0 0.0 1.000
Table 8. Orthogonality of the Secondary Mirror Motions
Tilts about Neutral Point and Center of Curvature
Motion Focus X Tilt Y Tilt X Tilt Y Tilt Roll
Neutral Neutral Center Center
Focus 1.000 0.0 0.0 0.0 0.0 0.0
X Tilt 0.0 1.000 0.0 0.988 0.0 0.0
Neutral
Y Tilt 0.0 0.0 1.000 0.0 0.988 0.0
Neutral
X Tilt 0.0 0.988 0.0 1.000 0.0 0.0
Center
Y Tilt 0.0 0.0 0.988 0.0 1.000 -0.0003
Center
Roll 0.0 0.0 0.0 0.0 -0.0003 1.000
Table 9. Orthogonality of the Secondary Mirror Motions
Tilt about Mid Plane and Decentration of Vertex
Motion Focus X Tilt Y Tilt X Decn Y Decn Roll
Focus 1.000 0.0 0.0 0.0 0.0 0.0
X Tilt 0.0 1.000 0.0 0.0 0.0 0.0
Y Tilt 0.0 0.0 1.000 0.0 0.0 0.0
X Dec 0.0 0.0 0.0 1.000 0.0 0.0
Y Dec 0.0 0.0 0.0 0.0 1.000 0.0
Roll 0.0 0.0 0.0 0.0 0.0 1.000
As can be seen in Tables 7, 8 and 9, the vectors of numbers of actuator steps have been normalized as indicated by the unity values along the diagonal of each matrix. In Table 7, the two pairs of vectors which are not orthogonal are the X Tilt and Y Decenter pair and the Y Tilt and X Decenter pair. In Table 8, the two pairs of vectors which are not orthogonal are the X Tilt about the Neutral Point and the X Tilt about the Center of Curvature pair and the Y Tilt about the Neutral Point and the Y Tilt about the Center of Curvature pair. In Table 9, all six motions are orthogonal. It is noted that the selected tilt and decenter motions as used in Tables 6 and 9 are not a unique set of orthogonal motions. There is an infinite number of alternative orthogonal motion pairs which can be formed by suitable linear combinations of these motion pairs.
V. Telescope Image Plane Optical Aberrations
The telescope optical form was modeled using the Lens Design computer program ZEMAX as published by Focus Software, Inc., Tucson, Arizona. This model was evaluated by geometric ray trace and Zernike polynomial wavefront analysis in order to determine the optical aberrations present in the image plane for various motions of the telescope secondary mirror. The telescope prescription used for this analysis was as follows:
Primary Mirror Radius of Curvature = -6,605.995 mm.
Primary Mirror Conic Constant = -1.007514
Primary Mirror Thickness Following = -2,703.614 mm.
Secondary Mirror Radius of Curvature = -1,415.170 mm.
Secondary Mirror Conic Constant = -1.920796
Secondary Mirror Thickness Following = 3,919.677 mm.
Effective Focal Length = 21,600.000 mm.
Angular Magnification = 4.81952
This prescription yields a telescope design which is very nearly a
Ritchey-Chretien design (it is noted that a small amount of field
dependent third order coma remains). The images formed by the
telescope were evaluated for object space field angles of +0.08,
+0.04, 0.0, -0.04 and -0.08 degrees. These field angles correspond
to image heights from approximately +30 to -30 millimeters in the
telescope image plane. The optical wavefronts which form the images
were analyzed using Zernike Polynomials and the image shifts were
determined by geometric raytrace intersections with the image
plane. The results for this analysis are presented in Table 10. The
secondary mirror motion values are as shown and the field angles
are in units of degrees. The image decenter values are in units of
millimeters and the focus, astigmatism and coma values are in units
of waves RMS of optical wavefront deformation at a wavelength of
0.5 microns.
Table 10. Secondary Mirror Motion Induced Image Aberrations
Motion Image Field Focus Astigmatism Coma
Decenter Angle (3) (4) (7)
None 0.0 0.08 0.2798 -0.0822 -0.0294
0.04 0.0721 -0.0206 -0.0147
0.0 0.0029 0.0 0.0
-0.04 0.0721 -0.0206 0.0147
-0.08 0.2798 -0.0822 0.0294
Focus 0.0 0.08 0.2324 -0.0822 -0.0294
2.83 0.04 0.0247 -0.0206 -0.0147
microns 0.0 -0.0445 0.0 0.0
-0.04 0.0247 -0.0206 0.0147
-0.08 0.2324 -0.0822 0.0294
Focus 0.0 0.08 0.1850 -0.0822 -0.0294
5.66 0.04 -0.0227 -0.0206 -0.0147
microns 0.0 -0.0919 0.0 0.0
-0.04 -0.0227 -0.0206 0.0147
-0.08 0.1850 -0.0822 0.0294
Tilt about 3.4*E-7 0.08 0.2953 -0.0977 0.1132
center of 0.04 0.0801 -0.0286 0.1279
curvature 0.0 0.0036 -0.0005 0.1426
0.01 -0.04 0.0651 -0.0136 0.1574
degrees -0.08 0.2653 -0.0677 0.1721
Tilt about 4.5*E-7 0.08 0.3117 -0.1142 0.2559
center of 0.04 0.0890 -0.0375 0.2706
curvature 0.0 0.0048 -0.0020 0.2853
0.02 -0.04 0.0591 -0.0075 0.3000
degrees -0.08 0.2518 -0.0542 0.3147
Tilt about -4.004 0.08 0.2686 -0.1480 -0.0293
neutral point 0.04 0.0668 -0.0534 -0.0146
-587 (0.415) 0.0 0.0036 0.0001 0.0001
0.05 -0.04 0.0787 0.0124 0.0149
degrees -0.08 0.2924 -0.0163 0.0296
Tilt about -8.007 0.08 0.2588 -0.2136 -0.0292
neutral point 0.04 0.0630 -0.0861 -0.0145
-587 (0.415) 0.0 0.0056 0.0003 0.0003
0.10 -0.04 0.0867 0.0456 0.0150
degrees -0.08 0.3063 0.0497 0.0297
Tilt about 0.0 0.08 0.4190 -0.2980 -0.0369
neutral point 0.04 0.1497 -0.1487 -0.0223
-587 (0.415) 0.0 0.0188 -0.0405 -0.0076
0.10 degrees -0.04 0.0264 0.0267 0.0072
field bias 0.021239 -0.08 0.1725 0.0526 0.0219
Tilt about 0.0 0.08 0.5902 -0.5947 -0.0444
neutral point 0.04 0.2592 -0.3577 -0.0298
-587 (0.415) 0.0 0.0667 -0.1618 -0.0151
0.20 degrees -0.04 0.0127 -0.0070 -0.0004
field bias 0.042478 -0.08 0.0971 0.1066 0.0144
From the data presented in Tables 3 and 10, the suitability for this application of the secondary mirror positioning system can be determined with respect to minimizing the optical aberrations present in the images formed. This data has been combined so as to determine the minimum changes of wavefront aberration that can be achieved for the motions of focus, tilt about the center of curvature and tilt about the neutral point.
The change in wavefront defocus aberration which would result from a secondary mirror inherent focus motion step size of 2.8 microns is approximately 0.0474 waves RMS and this wavefront aberration scales linearly with the focus motion of the secondary mirror. By suitably combining a small amount of roll motion with the focus motion, a focus motion of 1.4 microns can be obtained. This motion would produce a wavefront focus aberration change of approximately 0.0237 waves RMS. This change is approximately one-third of the wavefront aberration that corresponds to the Raleigh limit for image degradation which is about 0.071 waves RMS. The conclusion is therefore drawn that the focus motion is able to provide sufficient positioning resolution for this application.
The telescope optical image aberrations that are induced by the tilt of the secondary mirror about the center of curvature of that mirror are strongly dominated by field independent coma. The induced aberration is approximately 0.1426 waves RMS for a tilt angle of 0.01 degrees (36 arc-seconds) and scales linearly with the tilt angle. It is noted that the telescope line-of-sight is negligibly affected by this tilt motion. The inherent motion step size for tilt about the center of curvature in approximately 10 arc-seconds which corresponds to a wavefront aberration change of about 0.0396 waves RMS of coma. This change is equal to approximately one-half of the wavefront aberration that would correspond to the Raleigh limit for image degradation. It is noted that substantially smaller tilt motions can be achieved, but not with the same motion purity as can be produced by the inherent motion step size. The conclusion is therefore drawn that the tilt about the center of curvature motion is also able to provide sufficient positioning resolution for the application.
The telescope optical image aberrations that are induced by the tilt of the secondary mirror about the neutral point are a rich mixture of wavefront aberrations. These aberrations include: telescope pointing direction changes, axial shifting and tilting of the image surface, and changes to the astigmatism across the image surface. The changes in wavefront aberrations due to the tilt of the secondary mirror about the neutral point are presented in Table 11.
Table 11. Wavefront Aberration Changes Induced by
Tilt of the Secondary Mirror About the Neutral Point
Wavefront Induced Tilt Angle Field Angle
Aberration Change Dependence Dependence
Pointing dir. 76.464 linear with angle independent
Focus Shift 0.0159 square of angle independent
Image Tilt 0.1233 linear with angle linear with field
Astigmatism 0.0405 square of angle independent
Astigmatism 0.1754 linear with angle linear with field
Notes: Assumed tilt of secondary mirror about neutral point
= 0.1 degrees (360 arc-seconds)
Assumed field angle for field dependent aberrations
= 0.08 degrees (288 arc-seconds, 30.15 mm image height)
The units for Pointing Direction change are arc-seconds of
telescope pointing direction change for a 360 arc-second
tilt of the secondary mirror about the neutral point.
The units for Focus Shift, Image Plane Tilt and Astigmatism
are waves RMS of wavefront aberration for an evaluation
wavelength of 0.5 microns.
Given that the inherent step size for tilt of the secondary mirror about the neutral point is on the order of 8.9 arc-seconds, the changes in the image aberrations which are induced by this step size are approximately: 1.9 arc-seconds of pointing direction change, 0.003 waves RMS of tilt of the image plane, and 0.004 waves RMS of astigmatism change across the assumed 60 millimeter diameter image area. These wavefront aberration changes are extremely small compared to the wavefront aberration that would correspond to the Raleigh limit for image degradation (about 0.071 waves RMS). The conclusion is therefore drawn that the tilt about the neutral point is able to provide sufficient positioning resolution for control of the telescope image aberrations. However, the pointing direction of the telescope will be very substantially affected by this step size.
It is noted that the field locations evaluated for this analysis are all positioned along a line in the image plane which passes through the center of the image and is perpendicular to the axis about which the secondary mirror is rotated. For field locations not along this line, the wavefront aberration values will be split between the two Zernike polynomial terms which describe the applicable wavefront aberration.
VI. Optical Consequences of a Single Actuator Step
From Table 5, it can be seen that a single step of any one actuator would produce, among other motions, a tilt change about the neutral point of about 2.7 arc-seconds. The direction of this tilt depends upon which actuator is moved the one step. Combining this motion with the data from Table 11., this tilt change would produce a change in the telescope pointing direction of about 0.57 arc- seconds, which is equivalent to an image shift of approximately 60 microns in the telescope image plane. The change in the tilt of the telescope image plane and the change in the astigmatism across a 60 millimeter diameter centered on the telescope image plane which would result from a single actuator step would be negligible at about 0.001 waves RMS change in wavefront for each aberration.
In addition to the tilt about the neutral point, a single step of one actuator would also produce a tilt about the center of curvature of the secondary mirror of about 1.0 arc-second and an axial shift of the secondary mirror of about 0.47 microns. This tilt of the secondary mirror about it's center of curvature would induce about 0.004 waves RMS of field independent third order coma and the axial shift of the secondary mirror would induce about 0.008 waves RMS of focus change. Both of these wavefront changes are negligible with respect to their effect upon the quality of the optical images formed.
From the above discussion, it is apparent that the only optically significant consequence of a two micron step of any one actuator is the perturbation of the telescope pointing direction.
It should be noted that this effect upon the telescope pointing direction is not a cumulative effect. As the full complement of individual actuator steps are executed in order to perform an integer multiple of an inherent increment of motion of the secondary mirror for the motions of focus, roll or tilt about the center of curvature, the telescope line of sight is restored back to the direction in effect prior to the motion. However, any tilt about the neutral point that is performed will of necessity alter the telescope line of sight. In addition, any focus, roll or tilt about the center of curvature that is a fraction of an inherent increment of motion will also result in a perturbation of the telescope line of sight.
VII. Optical Consequences of Random Actuator Errors
The optical pointing direction of the telescope is, as discussed above, affected by the motions of each of the six actuators which position the secondary mirror. To the degree that the actuator motions deviate from the desired ideal motions, the pointing direction of the telescope will be perturbed. The magnitude of this perturbation is dependent upon the magnitude of the motion errors associated with each of the actuators. There are four principal expressions of the secondary mirror actuator motion related errors. These are:
1. departure of the actual hardware geometry from
the analytic model
2. quantization of the actuator motion to the nearest
two micron step
3. peak variation of the actuator motion per step from
the ideal motion per step
4. peak departure of the actuator motion from the ideal
linear motion
The first listed source of secondary mirror actuator errors,
departure of the actual hardware geometry from the analytic model,
would produce pointing direction changes that vary both slowly and
smoothly as a function of secondary mirror motions. This error
source includes both the dimensional errors in the secondary mirror
assembly and the scale factor errors relating actuator steps to
actuator motions for each of the six actuators. Given that the
perturbation of the telescope line of sight is significant only
over the range of secondary mirror motions that will be experienced
after the telescope is best aligned, these errors are expected to
be relatively benign. The magnitude of these errors and their
effect upon the telescope line of sight have not been estimated.
The second listed source of secondary mirror actuator errors, the quantization of the actuator positions to the nearest two micron increment, would produce a uniformly distributed random error in a given actuator position which has a zero-to-peak value of one micron and an RMS value of 0.577 microns RMS. ( This RMS value is computed by assuming that the error values are a function of the form f(x) = x, integrating the square of this function from zero to one, and taking the square root of the result. ) This random actuator position error would result in a telescope pointing direction perturbation that would vary rapidly and erratically as a function of secondary mirror motion and the errors from each of the six actuators would be completely uncorrelated. As a result, the errors from the six individual actuators would combine as the root of the sum of the squares for an RMS value of 1.414 microns RMS. This random position error would induce a random tilt about the neutral point of about 1.91 arc-seconds RMS which would yield a random telescope pointing direction perturbation of about 0.41 arc-seconds RMS.
The third listed source of secondary mirror actuator errors, the peak variation of the actuator motion per step from the ideal motion per step, would also produce a random error in any given actuator position. This random error would be a function of both the magnitude of the variation in motion per step and the maximum travel from the reference position for the given actuator. An estimate of the likely maximum travel from the reference position can be made by assuming that the travel will be predominantly focus in nature and that it may be as large as perhaps ten times the Rayleigh limit for defocus at the telescope prime focus. The Rayleigh limit at the prime focus is nominally 7.56 microns ( for a wavelength of 0.5 microns ) and ten time this value would be about 76 microns. From the actuator calibration data as reported on 5/19/95, the peak variation in the actuator motion per step varies over the range of 0.03 to 0.06 as a fraction of the nominal distance traveled. The RMS of the seven peak variation values is 0.043. Multiplying the assumed travel of 76 microns by the RMS of the peak variation of 0.043 gives a peak position error for a typical actuator on the order of 3.27 microns. Assuming that this error has a uniform random probability over this peak range, the RMS error for a single actuator under these assumptions is on the order of 1.89 microns RMS. Combining six uncorrelated actuators yields a total error of about 4.6 microns RMS. This error would result in a tilt error about the neutral point of about 6.25 arc- seconds RMS which would produce a perturbation of the telescope line of sight of about 1.33 arc-seconds RMS.
The fourth listed source of secondary mirror actuator errors, the peak departure of the actuator motion from the ideal linear motion, establishes an upper limit on the consequences of the variation of the actuator motion per step from the ideal motion per step. Again, from the actuator calibration data as reported on 5/19/95, the zero-to-peak variation in the actuator motion from the nominal motion varies over the range of 1.1 microns to 3.4 microns with an RMS value of 2.27 microns. Assuming that this error has a uniform random probability over this peak range, the RMS error for a single actuator under these conditions is on the order of 1.31 microns RMS. Combining six uncorrelated actuators yields a total error of about 3.21 microns RMS. This error would result in a tilt error about the neutral point of about 4.3 arc-seconds RMS which would produce a perturbation of the telescope line of sight of about 0.92 arc-seconds RMS. This result suggests that the above assumption of ten times the Rayleigh limit produces an over estimate of the consequences of these errors.
Combining the results from the quantization of the actuator steps ( 0.41 arc-seconds RMS ) with the results from the nonlinearity of the actuator motions ( 0.92 arc-seconds RMS ), the expected random perturbation of the telescope line of sight which would result from commanding motions of the telescope secondary mirror is about 1.0 arc-second RMS.
This study has determined that the secondary mirror actuator system will provide sufficiently fine control of the mirror position for control of the telescope image aberrations. The secondary mirror actuators are able to adjust the position of the mirror in each of the necessary mechanical degrees of freedom with motion resolutions sufficiently fine so as to allow the residual image aberrations to be accurately controlled.
The primary challenge when aligning the telescope will be to quantitatively determine the amount of each image aberration present across the telescope image plane for a given position of the secondary mirror. Once this data is obtained, the corrective mirror motions can be readily calculated. The secondary mirror actuator control software and the actuators themselves can then accomplish the necessary mirror motions with suitable precision.
The pointing direction of the telescope will be substantially perturbed by any motion of any of the secondary mirror actuators. This pointing direction perturbation will require that active star image guiding be employed if any secondary mirror motions are to be executed during a science instrument exposure period.