6 September 2013
W. Traub
ThermallyInduced Wavefront Error of the AFTA Telescope (v.1) Introduction
This memo estimates the raw contrast of the AFTA telescope caused by thermallydriven deformations of the optics. The resulting raw contrast values can be reduced by wavefront sensing and control methods, addressed elsewhere.
Thermallydriven deformations
The thermallydriven deformations of the AFTA primary and secondary mirrors (PM, SM) were calculated for 6 cases by Chang and Kuan (2013), who tabulated the resulting wavefront error (WFE) of each mirror, as a function of time, throughout a 24hour orbit. For use in the current memo, these authors provided pointbypoint mirror surface maps for their Case 5, a typical example. The locations of the points where the surface deformation was calculated are shown in Fig. 1 for the PM and SM. The PM points are uniformly distributed across the surface, except for the areas around the three main support points, where the density of points is much greater. The distribution of points across the SM is uniform. In both cases, the boundary of the illuminated annulus (owing to baffling) is indicated by red circles. Points outside this annulus are excluded from the calculation in this memo.
Fig. 1. (left) Plan view of the AFTA PM, showing points (x,y) where the thermallydriven surface deformation (z) is calculated. The heavilysampled support points are apparent. Red circles indicate inner and outer boundaries of the illuminated annulus. (right) Same, for the SM.
Diffracted light from a warped mirror generates speckles in the focal plane. The intensity of each speckle is directly related to the amplitude of a sinusoidal component of the warped surface. The ratio of the intensity of a speckle to the intensity of a star in the center of the focal plane is called the contrast. If a sinusoidal component of the surface deformation across a mirror has an amplitude h_{0} (nm), then that ripple will generate two speckles, one each on opposite sides of the star image. The contrast of each speckle is
C = I(speckle)/I(star)
= (2h_{0}/λ)^{2}
where λ is the wavelength of light, and both h_{0} and λ are in the same units, e.g., nm. The factor of 2 converts mirror amplitude to reflected wavefront amplitude. If the sinusoidal ripple has M waves per mirror diameter, then the angular location of each speckle, with respect to the star image, is
= Mλ/D (radians).
(See Traub and Oppenheimer (2010), sec. 4.7, for more background.)
We can estimate the amplitude of the ripples that make up the overall surface deformation of a mirror by projecting the surface deformation onto sinusoids. Moving step by step, we solve the leastsquares equation of this projection, for a surface ripple that is a function of x only, finding
h_{0}(sine,M) = Σ z_{i}*sin(2Mx_{i}/D) / Σ sin^{2}(2Mx_{i}/D)
and
h_{0}(cosine,M) = Σ z_{i}*cos(2Mx_{i}/D) / Σ cos^{2}(2Mx_{i}/D)
where the sums are over the x_{i} component of the (x_{i}, y_{i}) coordinates of the tabulated points on the mirror. The net amplitude of this ripple is
h_{0}(M) = [h_{0}^{2}(sine,M) + h_{0}^{2}(cosine,M) ]^{1/2}
The ripples in other directions are found by rotating the mirror by a few degrees, reapplying this equation, and repeating, until a full 180 degree rotation has been achieved. Then the RMS amplitude is calculated over all speckles having a given value of M. This RMS value is plugged into the equation for contrast, giving C(M), the contrast as a function of the number of ripples of deformation across the diameter of the mirror.
Numerical Example
The resulting contrast C(M) is shown in Fig. 2, for the PM and SM separately, and for two times of day, in the current example (“Case 5”). We see that the contrast is relatively high for the curve labeled “PM, noon”, which occurs for about 6 hours per day. The green curve just below it shows the improved contrast that is expected if the spherical aberration is removed, for example by an adjustment of the focus of the PM. For the other 18 hours, the typical situation is the contrast labeled “PM, typical”, a much lower level of thermal deformation, giving a lower (and better) raw contrast. The SM curves (dashed) for these times both are well below the PM curves (solid), so the deformation of the SM is negligible compared to that of the PM; the reason for this lower deformation is likely that the SM is shielded from direct view of space, whereas the PM sees that view directly, and can therefore be heated or cooled to a greater degree.
Fig. 2. The contrast in the focal plane, caused by thermal deformation of the PM and SM, is shown. The abscissa is angular distance from the star image, in units of λ/D, where D is the clear aperture, and the effect of the central obscuration is ignored for the purpose of this memo. The curve “PM, noon” is the worst case during the 24hr orbit. The curve “w/o SA” shows the contrast after spherical aberration is removed, for example by changing the telescope focus. The curve labeled “PM, typical” shows that during most of the day (18 out of 24 hours), even including the SA term, as here, that the thermallydriven deformations are negligible, compared to the 10^{10} level.
Summary
These curves are good news, because they show that for about 18 hours per day the thermal deformation from impinging Earth radiation is quite small, and essentially negligible, certainly well below the 10^{10} level, in the angular range between a typical IWA of 3λ/D and OWA of 22λ/D. During the 6hour or so “noon” period, the worstcast deformation generates a contrast above the 10^{10} level in the range 310 λ/D, but this contrast can be reduced by a WFSC unit, as described in a separate memo.
In summary, the thermal deformation of the AFTA telescope mirrors is expected to be negligible (well below 10^{10}) during most of each day. During the short “noon” period each day, the contrast can be sensed and controlled (down to about 10^{10}) as discussed in a separate memo (Traub, 2013).
References
Zensheu Chang and Gary Kuan, “WFIRST 2.4 Telescope Forward Optics Assembly Transient Thermal Elastic Analysis with WFI Optical Performance Analysis”, 30 May 2013.
W. Traub and B. Oppenheimer, “Direct Imaging”, in Exoplanets, Sara Seager, editor, 2010.
W. Traub, “LowOrder Wavefront Sensing for AFTA”, 6 Sept. 2013.
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