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Factors Affecting Mirror Performance
Daniel Vukobratovich
CONTENTS
1.1 Introduction and Summary
1.2 Tolerances, Error Budgets, and Superposition
1.3 Gravity Deflections
1.4 Mirror Deformations from Shear Effects
1.5 Mirror Deformations from Mounting Effects
1.6 Mirror Deformations from Dynamic Effects
1.7 Fundamental Frequencies
1.7.1 Drumhead Frequency of a Vertical-Axis Mirror
1.7.2 Rigid Body Fundamental Frequencies of an Axisymmetric Mirror
1.7.3 Damping
1.8 Rigid Body Response to Dynamic Excitation
1.8.1 Displacement
1.8.2 Response to Random Vibration
1.8.3 Response to Mechanical Shock
1.9 Thermal Effects
1.9.1 Uniform Change in Temperature
1.9.2 Temperature Gradients
1.9.3 Anisotropy of Coefficient of Thermal Expansion
1.9.4 Bimetallic Bending Effects
References
1.1 Introduction and Summary
The most important performance specifications for a mirror are associated with the shape, that is, optical figure, and position of its reflecting surface at all operational orientations relative to gravity. Secondary specifications include cost, weight, material availability, maintainability, and technical risk. Design of a mirror and its mount involves balancing performance with these and other considerations, such as pertinent environmental influences. Simple approximate closed-form solutions are a useful means for rapidly evaluating performance during conceptual design. Following the selection of conceptual designs, more detailed analysis methods, such as the finite element method, can be used to predict performance more accurately.
In this chapter, we consider the establishment and application of tolerances on various possible errors, such as gravitational effects, mounting influences, dynamic influences, and thermal effects on mirror performance. The combined effect of these on mirror performance is represented by an error budget. Numerical examples are presented to illustrate the application of performance estimation techniques.
1.2 Tolerances, Error Budgets, and Superposition
Mirror optical performance is defined in terms of two tolerances: optical figure and position. The optical figure tolerance determines the allowable deviation from the ideal form of the mirror. The position tolerance determines the location and orientation of the mirror surface with respect to the optical axis of the optical system as well as to other components of that system. The optical figure of the mirror is influenced by the mirror material, its shape, and method of mounting, as well as environmental factors, including temperature. The mirror mounting controls the location of the mirror and is affected by dynamic effects such as mechanical shock and vibration as well as the specified service environment.
A variety of factors influence the optical figure of the mirror. Included are self-weight, excitation from vibration, temperature changes, temperature gradients, and dimensional instability. Normally, the deflections of a mirror from all such causes are small, on the order of 10ā6 of the mirror diameter. These deflections then are well within the linear part of the stressāstrain curve, so the mirror material functions within its linear range. It is, therefore, possible to deal with each source of deflection independently and to add all deflections together using the principle of superposition. It is also assumed that the deflections are not correlated. In this case, the overall deflection can be defined as the root-sum-square (rss) of all deflections.
If the mirror optical surface deflections are measured at a series of points, as is...
