Cornell University CCAT
Memo:
/CCAT-TM-010
/Date:
/29-Apr-05
Subject:
/CCAT Optical Performance Requirements
/Version:
/1.0
Distribution:
/CCAT Project
/Orig. Date:
/29-Apr-05
Author:
/Terry Herter, German Cortes
/Posted:
/09-May-05
This note considers the optical performance requirements (delivered image quality) for CCAT. We require that when operating at 350 mm the increase in integration time due to degradation in PSF shape and throughput (due to surface accuracy errors) is less than 50% relative to the ideal case. The goal is less than 25% for the integration time increase. When translated in to one-half wavefront error (or rms surface error), WFE0.5, the integration time requirement and goal correspond to WFE0.5 = 12.6 and 9.3 mm respectively. [Note: Changing the performance goal and requirement to less than 30% and 15% increase in integration time results in WFE0.5 = 10 and 7.5 mm respectively. Likewise decreasing the goal and requirement to 100% and 50% give WFE0.5 = 16.5 and 12.6 mm respectively.]
If there are also aberrations that affect the shape of the PSF then WFE0.5 can incorporate and constrain these as well. However, we will take WFE0.5 to apply to the nominal Ruze criteria and characterize geometric-like (PSF shape changing) aberrations by a broadening of the PSF. This separates “large scale” wavefront errors (such as spherical aberration and coma) from “small scale” wavefront errors. Assuming a Ruze factor of one (WFE0.5 = 0) and that the ideal PSF is broadened by convolution with a Gaussian of FWHM equal to g*1.02l/D, then the maximum allowable “blurring” of the diffraction pattern is g = 0.68 and 0.48 for an integration time increase of less than 50% and 25% relative to the ideal case, respectively. The detailed trade-off between PSF broadening/blurring and WFE0.5 is shown in Figure 4.
Performance Requirement:
The point spread function (PSF) has a direct impact on the limiting point source sensitivity of CCAT in three ways: throughput, background flux, and confusion. Small scale errors in the wavefront will scatter light out of the central regions of the diffraction pattern reducing throughput. Aberrations always increase the size (shape/width) of the PSF. A wider PSF will mean that a larger solid angle must be used for extracting point sources. This increases the background flux contamination from the atmosphere and telescope, and reduces sensitivity. Finally confusion is a function of the beam area. Since CCAT is expected to reach the confusion limit in less than one hour for all but the shortest and longest wavelengths, keeping the beam as small as possible is important. Confusion will be dealt with in another tech memo.
We chose the performance requirement such that the throughput losses due to surface errors do not increase the integration time required to reach a background limited observation by more than 50% over the “idealized” diffraction limited observation. The goal is to not increase this time by more than 25%.
If the surface errors are random then there is little change in shape of the PSF. Power is simply scattered out of the main beam decreasing the power received by the Ruze factor.
(1)
WFE0.5 is the 1/2 Wavefront Error and srms is the root-mean-square surface roughness. This last equality should take into account projection effects for the surfaces. Table 1 and Figure 1 show how integration time changes versus surface rms (or (WFE0.5). (See technical memo CCAT_TM-006 for more details on the sensitivity calculation.) This surface rms can cover all wavefront errors including those introduced during manufacturing mirror segments, alignment and positioning of optics, geometric aberrations, etc. but the shape of the PSF will depend on the detailed aberrations present.
The integration time requirement and goal correspond to a one-half wavefront error (WFE0.5) of 12.6 and 9.3 microns when operating at 350 mm. This wavelength places the most stringent requirements on telescope performance. Reasonable operation at 200 mm is a goal. Figure 1 and Table 1 show the variation in integration time with WFE0.5 for l = 200 mm.
Table 1: Relative Integration Time vs. WFE0.5*
WFE0.5
(mm) / Ruze Factor / Relative Integration Time / Ruze Factor / Relative Integration Time
4 / 0.98 / 1.04 / 0.94 / 1.13
6 / 0.95 / 1.10 / 0.87 / 1.33
8 / 0.92 / 1.18 / 0.78 / 1.66
10 / 0.88 / 1.29 / 0.67 / 2.20
12 / 0.83 / 1.45 / 0.57 / 3.12
14 / 0.78 / 1.66 / 0.46 / 4.70
16 / 0.72 / 1.93 / 0.36 / 7.55
18 / 0.66 / 2.31 / 0.28 / 12.9
20 / 0.60 / 2.80 / 0.21 / 23.5
22 / 0.54 / 3.48 / 0.15 / 45.7
24 / 0.48 / 4.42 / 0.10 / 94.4
26 / 0.42 / 5.71 / 0.069 / 208
28 / 0.36 / 7.55 / 0.045 / 488
30 / 0.31 / 10.2 / 0.029 / 1219
For other wavelengths scale WFE0.5 by 350/l where l is in microns.
Figure 1: Relative integration time change vs. 1/2 wavefront error at 200, 350, 450 and 850 mm. This is the factor needed to get the equivalent signal-to-noise ratio on a source for background-limited performance for a change in surface rms while other parameters are kept constant. The integration time is normalized to unity at an rms of 12 mm. For instance, at l = 350 mm, it will take 3.5 times longer to get the same signal-to-noise ratio with a 25 mm surface rms than when the surface rms is 12 mm.
PSF Shape:
Assuming that light is scattered out of the beam (rather than distorting the PSF), then the Strehl ratio is simply given by equation 1. Thus the requirement and goals (at 350 mm) above correspond to Strehl ratios of 0.89 and 0.82 respectively and the encircled energy is that of a diffraction-limited encircled energy profile but with an overall scale factor given by the Strehl ratio.
This gives little information on the tolerances with respect to changes in the PSF shape. In principle, aberrations that affect the shape of the PSF can be incorporated into WFE0.5. However, the Ruze criterion is usually thought of as scattering light out of the beam rather than changing the PSF shape (since this shape change depends on the type of aberrations present). To examine the impact of the PSF shape change on sensitivity we convolve the diffraction-limited profile with a Gaussian blurring function. This is taken to be a first order representation of geometric aberrations or other contributions to the PSF. The results will later be combined with the Ruze calculation to investigate the total system tolerance. Figure 2 shows the encircled energy for different amounts of blurring (assuming a Ruze factor of 1).
Figure 2: Encircled energy vs. radius (in arcseconds) for a Gaussian blurring of diffraction limited PSF. From top to bottom the lines correspond to a Gaussian FWHM blurring of g = 0.0, 0.3, 0.5, 0.7, and 0.9 times the FWHM of the Airy pattern (= 1.02l/D = 2.945 arcseconds at 350 mm). The horizontal scale is for CCAT operating at 350 mm. To convert to units of l/D multiply by 0.346 and to get the scale for another wavelength multiply by 350/l(mm). Primary and secondary mirror diameters of 25 m and 3.2 m respectively have been adopted for CCAT.
The optimal extraction radius (maximizing the signal-to-noise ratio assuming no pixilation/sampling effects from the imaging array) gives an approximately constant flux fraction of 62% within the extracted beam independent of blurring. The optimal extraction radius, re, is given by:
(2)
where qg = 1.02l/D is the FWHM of the Airy function and g = fractional Gaussian blurring (FWHM = gqg). This has the correct limits for the Airy function (g = 0) and a Gaussian (large g). Since the flux fraction is constant, for background limited operation the signal-to-noise ratio relative to the peak signal-to-noise ratio (g = 0) is
(3)
Figure 3 shows the relative signal-to-noise ratio versus extraction radius. The optimal extraction radius and signal-to-noise ratio as predicted by equations 2 and 3 are within a few percent of the detailed numerical calculation.
Figure 3: Relative signal-to-noise (R-SNR) vs. extraction radius (in arcseconds) for a Gaussian blurring of diffraction limited PSF at 350 mm. From top to bottom the lines correspond to a Gaussian FWHM blurring of g = 0.0, 0.3, 0.5, 0.7, and 0.9 times the FWHM of the Airy pattern (= 1.02l/D = 2.945 arcseconds at 350 mm). The solid black line represents the predicted extraction radius and R-SNR from equations 2 and 3. The horizontal scale is for CCAT operating at 350 mm. To convert the x-axis to units of l/D multiply by 0.346 and to get the scale for another wavelength multiply by 350/l(mm).
Assuming a Ruze factor of one (WFE0.5 = 0), then the maximum allowable blurring of the diffraction pattern is g = 0.68 and 0.48 for an integration time increase of less than 50% and 25% relative to the ideal case, respectively. However, the allowable blurring of the PSF must be played off against the Ruze factor. The total time increase, trel, to achieve a fixed sensitivity threshold including both of these effects is given by
(4)
For a given trel and l, equation 4 can be solved for g in terms of WFE0.5. Figure 4 displays the trade-off between blurring and rms wavefront errors at 350 mm. For instance for an allowable increase in integration time of 50% and for WFE0.5 = 10 mm, the fractional blurring allowed is g = 0.38. This corresponds to a Gaussian with a FWHM of 1.13 arcsecond blurring (convolved with) the diffraction-limited PSF. If the aberrations that cause this blurring are independent of wavelength then, of course, the system performs relatively better at longer wavelengths and worse at shorter wavelengths.
Figure 4: Trade-off in allowable fractional blurring vs. WFE0.5. The upper curve is for an increase in integration time of 50% over the ideal case while the lower curve is for a 25% increase. The plot is for l = 350 mm. Multiply the allowable fractional blur by 2.95 arcseconds to get the FWHM of the Gaussian blurring function.
PSF Shape: Strehl Ratio and Encircled Energy:
For the above assumptions we can now calculate the encircled energy and Strehl ratio criteria trading off Ruze factor and PSF shape. (Note: reflectivity losses are ignored.) Combining both effects, the Strehl ratio is given by:
(5)
Where again g = fractional Gaussian blurring (FWHM = gqg, qg = 1.02l/D) and WFE0.5 is the 1/2 wavefront error. Combining with equation (4) we have
(6)
trel and g must be chosen to be consistent with equation 4 (that is, g is a function of trel and WFE0.5). Figure 5 shows how the Strehl ratio varies according to equation 6 subject to the constraint give by equation 4. For instance to satisfy the criteria of less than a 50% increase in integration time, the Strehl ratio needs to be greater than 68 % when WFE0.5 = 0 and greater than 81% when WFE0.5 = 12.6 mm.
Figure 5: Trade-off in Strehl Ratio vs. WFE0.5 at l = 350 mm. The Strehl ratio must be greater than the lower curve for an increase in integration time of less than 50% over the ideal case. The upper curve is for a less than 25% increase in integration time. WFE0.5 and PSF degradation (Gaussian blurring) are constrained by equation 4 as shown in Figure 4.
The optimal extraction radius (equation 2) does not change with Ruze factor, but the encircled energy will change (a fact used to derive equation 4). The encircled energy will be given by:
(6)
The factor of 0.62 is the constant flux fraction found for optimal extraction with the extraction radius given by equation 2. Again, these equations are subject to the constraint imposed by equation 4 relating g, trel and WFE0.5. Figure 6 plots the optimal extraction radius vs. WFE0.5. The radius increases with smaller WFE0.5 since the width of the PSF can be larger because the Ruze factor is closer to unity. Figure 7 shows the encircled energy vs. WFE0.5. Here again the allowable PSF width is increases with decreasing WFE0.5 so that trel remains constant. The overall encircled energy decreases with larger WFE0.5 because the Ruze factor is getting smaller (equation 6).
Figure 6: Allowable extraction radius vs. WFE0.5 at l = 350 mm. The radius must be smaller than the lower curve for an increase in integration time of less than 50% over the ideal case while the upper curve is for a less than 25% increase. WFE0.5 and PSF degradation (Gaussian blurring) are constrained by equation 4 as shown in Figure 4.
Figure 7: Encircled Energy vs. WFE0.5 at l = 350 mm. The encircled energy must be greater this for an integration time of less than 50% and less than 25% over the ideal case. The extraction radius is a function of WFE0.5 and is shown in Figure 6. WFE0.5 and PSF degradation (Gaussian blurring) are constrained by equation 4 as shown in Figure 4.
Supporting Files
Version
/Comments/Changes
CCAT_Sensitivity(2005-04-13).xls
/Calculates sensitivity and survey speeds
CCAT_PSF_resp.mcd
/This MathCad program convolves the diffraction pattern with a Gaussian blurring function. It is used to produce encircled energy, SNR, and allowable fractional blurring plots.
Revision History
Version
/Date
/Comments/Changes
0.1
/29-Apr-05
/Initial version
1.0
/09-May-05
/Updated with discussion of Strehl ratio and Encircle Energy and posted.
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