M1 Microroughness and Dust Contamination

Transcription

1 Project Documentation Technical Note TN-0013 Rev. D M1 Microroughness and Dust Contamination Robert Hubbard Systems Engineering November 2013 Advanced Technology Solar Telescope 950 N. Cherry Avenue Tucson, AZ Phone atst@nso.edu Fax

2 REVISION SUMMARY: 1. Date: 6 September 2002 Revision: A Changes: Initial release 2. Date: 22 October 2002 Revision: B Changes: Correct gray area in figures. Add additional information about the cause of small-angle scatter by dust. 3. Date: 28 October 2002 Revision: C Changes: Corrected figure 12 to include better low-angle data. Corrected cleanliness level from 230 to 240. Added additional clarification of dust discussion in section 5.2. Added van de Hulst reference. 4. Date: 6 November 2013 Revision: D Changes: Updated scattered light requirement per most recent SRD version. Updated format to conform with modern document template. TN-0013, Rev. D Page i

3 Table of Contents 1. INTRODUCTION SCIENCE REQUIREMENTS AND SKY BRIGHTNESS EXPECTATIONS THE ASAP MODEL THE OPTICAL SYSTEM MODEL THE SOURCE MODEL SCATTER MODELS AND IMPORTANT AREA SAMPLING MIRROR MICROROUGHNESS SCATTER DUE TO MICROROUGHNESS BSDF AND THE HARVEY MODEL RMS MICROROUGHNESS REASONABLE ASSUMPTIONS, AND MICROROUGHNESS RESULTS DUST CONTAMINATION THE MIE MODEL FOR SCATTER BY DUST PARTICLES RESULTS FOR A CLEAN MIRROR CONCLUSIONS AND RECOMMENDATIONS REFERENCES TN-0013, Rev. D Page ii

4 1. INTRODUCTION The optical design of the Advanced Technology Solar Telescope (ATST) is an all reflecting, off-axis Gregorian telescope. Some of the considerations that drove this solution stem from the need to perform coronagraphic observations close to the sun s limb. Even with this configuration, scattered light in the telescope optics has the potential to be the limiting factor on the quality of such observations at a site with low sky brightness. Care must be taken in the design, fabrication, and maintenance of the telescope to prevent this. If occulting is performed at prime focus, then the dominant instrumental stray-light contributor during coronal observations will be the microroughness of the primary mirror, and dust accumulating on that same surface. The is the result primary mirror being large, fully illuminated by the sun, and because of the relatively low scattering angles necessary to couple this unwanted light into the system. These straylight sources are the subject of this report. Mirror microroughness is a static condition that will persist throughout the lifetime of the telescope, or at least during the lifetime of the primary mirror. It is a function of mirror polish applied during manufacture, so the project s only opportunity to mitigate its effects comes in the specification and acceptance testing of this major telescope component. On the other hand, an aggressive roughness spec has the potential to significantly impact project cost, schedule, and risk management. The effects of mirror microroughness need to be well understood so that the project can make good decisions. Dust control, on the other hand, has a larger systems impact. It will be affected by site selection, enclosure design, ventilation, mirror-cell design, telescope support structure design, and a variety of operational issues such as mirror washing, cleaning, and recoating schedules. Once again we need to thoroughly understand the degradation of telescope performance due to dust contamination. In this case good data and good analysis needs to be in hand early to guide the design process in many areas. Buffington and Jackson (UCSD Center for Astrophysics and Space Sciences) have already done work on these and other stray-light topics relevant to ATST 1. On the subject of primary mirror microroughness they note that something like one percent of the Total Integrated Scatter (TIS) from microroughness will enter the coronagraphic field of view, and warn that, Specifying, manufacturing, testing and certifying [the primary mirror] could prove a significant challenge for ATST. This report attempts to further quantify Buffington and Jackson s microroughness results over a range of realizable mirror-polish parameters. Results will also show how scatter levels due to primary mirror microroughness will vary with limb distance. On the subject of scattering by dust particles, Buffington and Jackson note that there are modeling challenges associated with the absence of small-angle observational results in the visible. The power-law relationship between scattered light and scatter angle cannot continue indefinitely if TIS is to remain finite, and must roll off at small angles. But since Buffington and Jackson were uncertain where, exactly, this roll-off begins, and because the details of the angular dependence in this unobserved region are critical to the performance of ATST, they were only able to establish a relatively broad range of dustcontamination predictions. In their worst-case scenario, there is little chance of ever being able to do state-of-the-art coronagraphic observations without resorting to heroic measurements. Even at the more optimistic end of their parameter space, dust control of ATST would appear to be challenging. This report attempts to extend that work, and further clarify the impact of dust on the primary mirror. It will support the view that their optimistic parameters are probably the correct ones, and further explore the issue of dust accumulation rates in this context. This will lead to recommendations for mirrorcleaning strategies. TN-0013, Rev. D Page 1 of 18

5 2. SCIENCE REQUIREMENTS AND SKY BRIGHTNESS EXPECTATIONS The science requirement for stray light in the ATST dictate that scattered light must be controlled to less than of the solar disk irradiance at R/R sun = 1.1 and = 1 m. This is derived from a desire to keep instrumental scatter comfortably below scatter due to the earth s atmosphere, presuming the best day at a good coronal site. While we have been unable to find direct sky-limited irradiance measurements at 1.1 R sun, estimates abound. LaBonte reports median values for Haleakala of , with some observations as low as one or two millionths 2. This, however, corresponds to measurements in an annular ring spanning a range between 1.6 and 4.4 R sun at 0.53 m. Jacques Beckers uses a value of at 1.1 R sun at 0.5 m in the Clear studies 3. David Elmore recently reported values of order at 1.5 R sun at 0.7 m (for Mauna Loa Solar Observatory) and estimated that we could expect values closer to at 1.1 R sun at 1 m. Many other observations of the sun obtained with telescopes, instruments, and sites not optimized for coronal observations show values very much higher than a part in Analysis by Barducci et al. attempts to separate out instrumental scatter from that due to the atmosphere 4. Their results for the Donate Solar Tower in Arcetri suggest values of 250 to 1000 millionths for the sky-brightness component at that admittedly non-coronal site. It is not surprising that there is considerable variation in these estimates because of the difficulty of the measurement and the wide variation in site-to-site and day-to-day sky brightness. In any event, the instrumental scatter predictions that follow should be viewed in the context of possible sky-brightness values at 1.1 R sun in a range between a few millionths to few tens of millionths of the solar irradiance. 3. THE ASAP MODEL The analysis that follows was performed using version 7.1 of the Advanced Systems Analysis Program (ASAP) sold by the Breault Research Organization. Because the scatter models used in this analysis all have analytical representations, it is possible in principle to perform these calculations by just doing the integrals of the scatter functions over the appropriate solid angles. ASAP obtains its results by performing a Monte-Carlo style simulation involving a large number of geometrical rays. ASAP is fundamentally a non-sequential ray-tracing engine. The term non-sequential in this context means that rays proceed through even complex three-dimensional system models making no assumptions about the order in which objects will be encountered. This allows us to model scattered light where the power carried initially by each source ray can be split into multiple child rays with the appropriate distribution of power as a function of scatter angle. These child rays are then traced independently through the system to determine the stray-light impact at any point or surface in the system. 3.1 THE OPTICAL SYSTEM MODEL The system component definitions are entered into ASAP in terms of their optical parameters: halfdiameters, surface radii of curvature, and conic constants. The only system geometry that needs to be modeled in this study is the primary mirror, and a detector to collect the rays located at the position of the telescope s heat stop. The primary mirror is modeled after the current f/2 off-axis Gregorian design. It is a four-meter diameter off-axis parabola with its center located four meters out from the axis of the twelve-meter parent parabola. The focal length at prime focus is eight meters. The detector is a circular plane located along the axis of the parent parabola at the prime focus. It is tipped toward the primary mirror at an angle of 28.2 degrees to the axis of the parent parabola for optimum image quality. Its diameter is 1.3 mm, which is about one tenth the size of the actual heat-stop aperture in TN-0013, Rev. D Page 2 of 18

6 the current design. The collection aperture was undersized so that the scatter from the primary could be probed as close as 1.1 R sun in the focal plane. The full five-arcmin aperture would otherwise have included a section of the solar limb. Thus, in the calculations that follow the values represent the normalized irradiance in a small area at the center of the telescope s field of view, not the integrated scatter over the full 5-arcmin aperture. 3.2 THE SOURCE MODEL For the analysis to follow, we need a set of rays that models the sun, which is an extended circular source located at infinity and subtending a 32-arcmin full angle. We must be able to move this source over a range of small angles relative to the optical axis of the telescope so we can collect scatted light at sample positions on the sky near the sun s limb. We begin with a parallel grid of rays representing a point at the sun s center. The angle that these rays make with the optical axis is the parameter that will eventually control the position of the sun relative to the detector. The extended source is created by introducing these rays onto a dummy plane just in front of the primary mirror (Figure 1) which randomly scatters each parallel ray into a cone of 16 arcmin half angle centered around the original ray direction (Figure 2). Each parent ray is converted into 100 random child rays distributed within this cone. The scatter function is Lambertian, but at such small angles the resulting extended source has nearly constant irradiance from center to limb. Dummy surface to create an extended source. Primary mirror Figure 1. A dummy scatter surface is created just in front of the primary mirror to create a set of rays appropriate for an extended source. Figure 2. Each of the original source rays is spread randomly into a cone centered on the original ray direction. Both the angle of the parallel rays and the size of the cone have been exaggerated in the figure for clarity. Many more parent rays were also used in practice to give more uniform surface coverage that what is shown here. Using these methods, ten ray sets were created at the position of the dummy surface corresponding to limb positions from 1.1 to 2.0 R sun in increments of 0.1 R sun. Each source contained approximately 800,000 rays, though in the end this was more than was needed to converge on a stable result. All rays This presumes a plate scale of mm/degree, which was determined by measuring the positions of rays traced onto the tipped detector plane. TN-0013, Rev. D Page 3 of 18

7 were assigned a wavelength of 1 m, which will be used in the scatter calculations that are eventually performed during the ray trace. These ten sources were then saved for future use so that the rays would not have to be recreated each time the primary mirror s scatter model was changed. The initial source flux after creating the rays was arbitrary. One additional centered ray set was created and used to obtain a flux measurement at sun center. This value was then used to normalize the total ray flux in the other sources, obtaining what the scatter community calls normalized detector irradiance values. Hence, all results that follow are unitless, and refer to the ratio to on-disk measurements. Figure 3 shows the position of the model sun relative to the collector surface for each of these sources. To check the source model, one of the sources (1.1 R sun ) was traced to the prime focus location with no scattering applied to the primary mirror, and a spot diagram was created in a view that included the small scattered-light collecting surface (Figure 4). The detector size, location, solar image size, and irradiance as a function of position were then carefully measured to confirm the source model. Scattering properties are added to ASAP surfaces as object modifiers. If scatter properties have been added to a surface, power will be scattered according to a user-defined prescription or model. Even though many light-scattering mechanisms are diffractive (and hence wave-optical phenomena) geometrical rays can still be used in stray-light analysis. This is possible because once the scatter distribution is characterized (by either theory or measurement) it can be treated as a statistical rather than a deterministic phenomenon. In ASAP, each incident ray is split into a set of child rays in a pseudorandom way. While the directions given to the child rays are random, the scatter model dictates the probability of a ray being created with a given direction. By tracing many rays, the scatter distribution function can be adequately sampled to obtain accurate predictions Figure 3. The sun s disk (right) is shown to scale next to the ten sample positions (left) spaced 0.1 solar radii apart from 1.1 to 2.0 solar radii. TN-0013, Rev. D Page 4 of 18

8 Figure 4. This is a spot diagram made by tracing rays from the source model to a tipped plane at prime focus. The small, blue circle on the left is the detector. The lack of sharpness in the image on the far right is caused by coma at the edge of a field of view that here exceeds a degree. 3.3 SCATTER MODELS AND IMPORTANT AREA SAMPLING The collecting area near prime focus subtends only a very small solid angle in the ATST model. Given that small size, and given that scatter due to both microroughness and particle contamination is strongly peaked in the specular direction, the probability of a ray being scattering into the tiny target is very small. A bruit-force scatter analysis by the methods described above would require tracing many millions of rays to get just a handful onto the collecting surface. ASAP allows the user to define important areas when generating child rays. We are able to declare the direction and solid angle subtended by our detector to be such an important area. The program will only create child rays that scatter toward this important area, though their flux will be scaled as if power had been scattered into the entire hemisphere above the scatter surface. In this way, every ray created and traced contributes to the final result. 4. MIRROR MICROROUGHNESS 4.1 SCATTER DUE TO MICROROUGHNESS Because of its importance in optical instrumentation, the scattering of light by polished glass surfaces has been the subject of considerable experimental and theoretical study. John Stover has captured much of this progress in his book entitled Optical Scattering Measurement and Analysis 5, which also includes an extensive bibliography. The observed behavior of light scattered from polished glass is best understood in terms of a diffractive model. Any reflecting surface that is rough at the microscopic level can be viewed as a two-dimensional superposition of sinusoidal height variations with a range of frequencies. Each individual component will behave as a diffraction grating, obeying the familiar grating equation: power incident at an angle 0 will be diffracted at an angle such that sin sin nf, (Equation 1) 0 g TN-0013, Rev. D Page 5 of 18

9 where n is an integer representing the diffractive order, f g is the frequency of the sinusoid, and is the wavelength of light. For an isotropic scattering surface (i.e., one whose grooves have no preferred direction), one can infer the spectrum of frequencies by measuring the roughness in just one dimension using a profilometer. In its simplest form this is just a stylus similar to that used to play a phonograph record, which is dragged across the polished glass surface. The Fourier transform of the resulting surface profile yields the Power Spectral Density (PSD) function of the surface. If this is plotted on a log-log scale, it is generally well approximated by a straight-line fit (Figure 5). This power-law relationship implies that mirror-polishing methods impart roughness with a fractal character over a broad range of spatial dimensions. Stover shows that there is a direct relationship between the PSD function and scattered power as a function of scatter angle. They are, in fact, proportional once the horizontal scale is converted to angle, and adjusted to include the effects of the second dimension. In Figure 5 the horizontal frequency scale is labeled in units of inverse length. We can use the grating equation to convert these to scatter angles (assuming normal incidence and diffraction order 1), wherein we find that the power law for the fused silica sample in figure 5 persists for scatter angles from about six degrees down to angles around two arcmin, easily covering the angles of interest in the ATST near-limb analysis Frequency (m -1 ) Figure 5. The one-dimensional power-spectral density (PSD) of polished fused silica derived from profilometer data. The figure is taken from Church 6. The dashed red line shows a linear fit to the data. TN-0013, Rev. D Page 6 of 18

10 4.2 BSDF AND THE HARVEY MODEL The relationship between the scattered power and scatter angle is usually expressed as a Bidirectional Scatter Distribution Function or BSDF. This function of incident angle and scatter direction can be obtained experimentally by illuminating a sample with a bright, collimated source (often a laser), and measuring the power as a function of angle in a spherical coordinate system with a sensitive moving detector. By convention, the BSDF is normalized to remove all experimental factors like detector size, source power, illumination spot size, projection effects, and detector distance. When this is done, BSDF becomes the ratio of radiance (power per unit angle per steradian) on the scatterometer s detector to the irradiance (power per unit area) of the sample spot. The resulting BSDF has units of steradian -1. Figure 6a and 6b show typical plots of BSDF versus scatter angle on both a linear and a logarithmic scale for polished glass. These normal-incidence scatter curves show how strongly peaked the function is in the specular direction. The power-law relationship between scattered power and scatter angle inferred from surface profiles is Figure 6a. A typical BSDF for polished glass is sharply peaked around the spectral direction. This plot represents scatter for light at normal incidence. Figure 6b. The same BSDF shown in Figure 6a plotted on a log scale to show the large dynamic range of the BSDF function. borne out by direct scatter measurements. Figure 7 shows actual data from a polished quartz surface obtained for a 45-degree incident angle. If we plot log BSDF versus log (sin) at normal incidence, we get a straight line. For other angles of incidence 0, if we plot log (sin sin 0 ) on the horizontal axis, the exact same line is obtained for the same sample. This property of polished glass is called shift invariance, and further simplifies the modeling problem at hand. The absolute value taken of this difference causes the forward scatter points (where (sin sin 0 ) > 0) to overlap the backscatter points (where (sin sin 0 ) < 0), both lying along the same line. Out of the plane of incidence, the scatter function remains symmetric about the specular direction when plotted in direction cosine space, which is effectively the projection of the direction vectors down onto the plane of the scatter surface. The simplifications that occur when working in this coordinate system (effectively direction-cosine space) once again point up the relevance of diffraction and the grating equation in scatter from polished-glass surfaces. As the figure shows, this remains true for all angles until we approach grazing incidence. This function is sometimes called a BRDF when it applies to reflected scatter, and BTDF in transmission. TN-0013, Rev. D Page 7 of 18

11 Figure 7. Actual measurements of polished aluminum-coated quartz show that the power law relationship apparent in the PSD persists in the scatter measurements as long as we work in direction cosine space. The symbol is frequently used to represent sin() and 0 for sin( 0 ), leading to the horizontal label here. Gary Peterson and the Breault Research Organization 10 provided these data, originally obtained by James Harvey. The Harvey model is included in ASAP for these and other scatter situations where a power-law relationship exists. In its simplest form the Harvey function can be written as sin( ) sin( 0 ) BSDF b 0.01 S (Equation 2) where S is the slope of the line on the log-log plot, and b is the value of the BSDF at = 0.01 radians. If we integrate this function over all angles on the surface of the hemisphere, the result diverges for some values of S. This is caused by the assumption that the fractal behavior extends all the way to zero spatial frequency, which is not physically possible for finite apertures and not likely in any case. ASAP allows the Harvey function to roll off to a constant value beyond a certain spatial frequency by introducing a third parameter, l, into the Harvey formula as follows: BSDF sin b 0 1 sin l 0 2 S 2. (Equation 3) TN-0013, Rev. D Page 8 of 18

12 Log(BS DF) M1 Microroughness and Dust Contamination 1 b l Log(Beta-Beta0) Figure 8. The l parameter in the Harvey function is used to roll off the BSDF at low angles. For this plot, l = 0.001, and this is the approximate angle at which the knee of the curve is located. The l parameter controls the location of the angle of the knee of the curve as shown in Figure 8. The other new parameter, b 0, is the constant value that the BSDF asymptotically approaches at small angles. It replaces b in the previous simpler Harvey expression (Equation 2) as the parameter that controls the total integrated scatter for a given slope value S. They are related by the expression, S b0 b 100l. (Equation 4) This new expression for BSDF can be integrated over the hemisphere with finite results in all cases. For scatter at normal incidence the total integrated scatter is as long as S 2 *. S2 S l 2 l 2 S 100 TIS 2 b (Equation 5) S 2 The Power Spectral Density function shown in Figure 5 suggests that the roll-off controlled by the l parameter occurs at angles at least no larger than radians for polished glass, and it is generally found to be much smaller. In ASAP, if l is not specified, the program assumes a value of , which is small enough to have little effect on the TIS for the range of S values we will use. For such small l values, we can write * A separate expression can be derived for the special case of S = 2, but since we will not be looking at slopes of this magnitude, it has not been reproduced here. See Reference 7 for the other form. TN-0013, Rev. D Page 9 of 18

13 S 100 TIS 2 b. (Equation 6) S 2 The simple power law relationship between BSDF and angle (and the resulting TIS expression) makes it relatively easy to model scatter from polished glass surfaces if we can only determine or infer the values of S and b. These parameters vary with glass type and mirror-polishing methods and quality. The slope is typically in the range 1 to 3, and b varies between and 1. It will be possible, however, to narrow these ranges considerably for the analysis that follows, based on relevant measurements. 4.3 RMS MICROROUGHNESS The discussion in the previous section was intended to establish that there is sound theoretical and experimental evidence to support a simple analytical expression for the scatter of polished glass surfaces. We only need to know two pieces of information to proceed: slope and intercept of the scatter function on the log-log BSDF plot. Unfortunately we only know one parameter at this point, the RMS microroughness requirement that will be placed on the mirror. The RMS microroughness of a polished surface is a statistical value derived from surface profile data, calculated as the root mean square deviation of the surface height z of a range L of positional values x, defined mathematically as follows: L 2 1 Lim zx z dx L L. (Equation 7) The limit as L goes to infinity is, of course, problematic, and will return to this point shortly. But if the microroughness defined in this way can be known or inferred, the total integrated scatter is given by a simple formula 5 : L TIS. (Equation 8) It would appear that we could now equate the TIS expression above to the TIS expression derived for the Harvey function in the previous section (Equations 5 and 6). Although only knowing RMS roughness is insufficient to specify both S and b in the Harvey expression, it does further constrain them, allowing us to explore a reasonable range of values. There are worse things lurking in these simple formulae, however. The presumption of an infinitely long measurement used in the expression for is, of course, incorrect. Whether the surface roughness measurements are made with a stylus-based profilometer or a non-invasive interferometer-based surface profiler, the measurements cover little more than a millimeter or two in practice. This limits the effective frequency bandwidth of the RMS calculation. The instrumental response, sampling frequency, and the detrending that is done to remove piston, tilt, and quadratic curvature from the raw profile data further corrupts the RMS roughness parameter by throwing away data within the measured bandwidth. As a result, trying to infer too much from this one parameter is a little treacherous. Still, the fundamental fractal nature of the surfaces involved has allowed investigators to estimate the errors introduced into RMS roughness values obtained over limited bandwidths 6,7. For our case, assuming a 1 mm sample spot on the mirror, we can safely neglect these effects without biasing our results by more than about 25%. Hence, in spite of the limited information currently available, we can now make some reasonable assumptions that will allow us to place some bounds on scatter due to microroughness for ATST. TN-0013, Rev. D Page 10 of 18

14 BSDF M1 Microroughness and Dust Contamination 4.4 REASONABLE ASSUMPTIONS, AND MICROROUGHNESS RESULTS Even with only an RMS roughness parameter to work from, an abundance of good measurements of actual polished glass surfaces gives us guidance in making pretty good estimates of the scattered-light performance we can expect from the telescope. Returning to the ASAP model, we begin by assuming a slope value of 1.5, the most likely value based on relevant measurements 7. By equating the right side of equation 6 to the right side of equation 8 we can solve for b for any value of microroughness. At the present time, mirror manufacturers are reluctant to bid on large mirror projects requiring RMS roughness values less than 20 Ångstroms. If we take this value for (which corresponds to a total integrated scatter value of percent) and derive a value for b as described above, we obtain the Harvey curve shown in Figure 9. We can now enter these values into the ASAP model and proceed with the analysis. The result for all ten of the sample sources for S = 1.5, presuming a wavelength of 1 m, is shown in Figure 10. This predicts scatter levels around at R/R sun = 1.1. What if the polish is better? The second curve in Figure 10 shows the results for a slope of 1.5, and a 12 Ångstrom finish. This value of is midway between what was actually achieved for the two eight-meter Gemini mirrors even though the specification called for only a 20 Ångstroms finish. It corresponds to a TIS of percent. For a mirror in that class we can expect performance close to at R/R sun = 1.1. S = 1.5, 20 Angstrom 1.E+02 1.E+01 1.E+00 1.E-01 1.E-02 1.E-03 1.E-04 1.E Sin (theta) Figure 9. BSDF for a 20-Ångstrom microroughness at 1 m for a slope of 1.5 is shown. This corresponds to a total integrated scatter of percent. The gray area shows the range of sun angles relevant to this analysis, with the left edge strongly favored. TN-0013, Rev. D Page 11 of 18

15 Ratio to On-Disk Irradiance M1 Microroughness and Dust Contamination Two Sample Polishes 1.2E E E E A A E E E Distance from Sun Center (solar radii) Figure 10. The figure shows the impact of scattered light due to microroughness for two different mirror polishes as a function of limb position. Both presume a slope of 1.5. Finally, how bad will things get if the more extreme slope value of 1.8 is used? For this case, the scatter at 1.1 R sun for12 and 20 Ångstroms will be about and respectively. The tabulated results appear below, with the most likely result (S = 1.5, = 12 Å) highlighted in red.. R/R sun S = 1.5 = 12 Å S = 1.5 = 20 Å S = 1.8 = 12 Å S = 1.8 = 20 Å TN-0013, Rev. D Page 12 of 18

16 5. DUST CONTAMINATION 5.1 THE MIE MODEL FOR SCATTER BY DUST PARTICLES Dust contamination of reflecting surfaces is known to have a significant impact on stray-light performance of telescopes and other optical instrumentation. This is particularly true for the measurement of low signals in the sun s corona close to the solar limb. Many careful studies have been performed, notably a series of four fairly modern (1992) articles by Paul Spyak and William Wolfe 8. Part I of the series compares measurements of scattered light due to spherical particles on mirror surfaces with predictions by Mie theory. Mie theory is a rigorous derivation of bulk (volume) scatter by suspended spherical particles over a broad range of particle sizes ranging from sub-wavelength dimensions to particles much larger than the wavelength of light. Spyak and Wolfe used a variation of this theory that has been adjusted for the combined effects of forward and backward scatter from a thin layer of spherical particles on a highly reflective surface. The modified Mie model fit the spherical particle data quite well for both visible light, and infrared radiation out to10 m. Part II of the series showed that the same modified Mie model also compared well against observed scatter by dust in spite of the spherical-particle assumption. ASAP includes a built-in version of this same modified Mie model. The parameters that must be supplied by the user include the wavelength of light, the fraction of the mirror covered by particles, and a function that describes the distribution of particle sizes. The particle size distribution is the only variable that presents any sort of problem in this analysis. This function is by no means a constant in either time or location. We have not selected a site for ATST, and have no direct measure of particle size distributions at any of the sites under consideration. The usual approach under these circumstances is to use the MIL- STD-1246A distribution derived for clean room specification. It gives a formula for the number of particles per square foot (n) with diameters greater than s micrometers as a function of c, the cleanliness or contamination level : log n 2 log c log s (Equation 9) The derivative of this function is just what is needed for the ASAP model, but is there any reason to expect that the distribution of particles that accumulate on clean-room surfaces is anything like what we can expect inside a telescope enclosure? Varsik et al have looked at particle distributions at the Apache Point Observatory on Sacramento Peak in New Mexico 9. They found that their distributions were similar to MIL-STD-1246A for fairly clean mirrors, but a simple power law often fit dirty mirrors better. They also noted large variations in particle size distributions in their study of airborne dust. The character of the distribution changed as a function of season, and was also different during periods of high dust levels. We will proceed with this analysis using MIL-STD-1246A for lack of anything better since we plan to keep the ATST mirror as clean as possible, particularly when doing observations of the sun s corona. 5.2 RESULTS FOR A CLEAN MIRROR The initial analysis was performed assuming a fractional coverage on the mirror surface of (0.01%) as suggested in Buffington and Jackson s conclusions 1. This can be related to the contamination level via the relationship shown graphically in Figure 11. This corresponds to a contamination level around 240, which might be a reasonable expectation right after cleaning. TN-0013, Rev. D Page 13 of 18

17 Figure 11. This figure shows the relationship between contamination level and fractional coverage of an optical surface. The dotted red line shows the contamination level corresponding to our 0.01% assumption. (Figure provided by Gary Peterson 10.) The full Mie calculation is complex and time consuming. Evaluating the function in ASAP at approximately 700 angles took about seven hours on a 1 GHz-class computer. For this reason, it is more efficient to calculate the model once for a given set of parameters, and fit the result to a simpler BSDF model before performing the Monte Carlo simulation. Note that Spyak and Wolfe showed in Part IV of their series 8 that, like polished glass, scatter from dust-contaminated mirrors is shift invariant (see section 4.2 above). Thus we are justified in using the normal incidence calculation for our small scatter angles, and the shift-invariant Harvey function to model this scatter mechanism. Since Mie scatter scales directly with percent coverage, it is only necessary to do the full calculation and fit once for a given wavelength and particle-size distribution. Other levels of contamination can be scaled from this one result. Figure 12 shows the resulting BSDF at normal incidence, 1 m wavelength, and for a coverage fraction of The figure also shows a fit to the sum of two Harvey functions. Buffington and Jackson noted that the apparent power-law relationship (or the sum of two power laws in this fit) cannot continue indefinitely into the small-angle regime if the TIS is to remain finite. When the Mie model is pushed to very small angles as it was in Figure 12, the theory does predict the necessary roll-off, and it can be approximated very well using the extended Harvey function with the l parameter (Equation 3 in section 4.2). In the visible and near IR, however, this roll-off is not observable in the scatter observations of TN-0013, Rev. D Page 14 of 18

18 BSDF M1 Microroughness and Dust Contamination Harvey Fit to Mie Data 1.E+00 1.E-01 1.E-02 1.E-03 1.E-04 1.E-05 1.E Sine Theta Harvey 1 Harvey 2 b b0 l s %TIS 1 b b0 l s % TIS 2 % Sum 7.000E E Figure 12. The figure shows the modified Mie theory predictions for about 700 angles, plotted in the usual loglog form. The red and blue curves correspond to two different Harvey models, the sum of which fits the Mie data fairly well (green curve). This figure does show the roll-off expected, and the Harvey l parameter has been used to fit it. Spyak and Wolfe because of the difficulty making measurements at very small angles relative to the specular direction ( ~ 0.4 degrees). Can we believe the theory, even though this roll-off is not observed? As the gray area in Figure 12 shows (and as Buffington and Jackson also pointed out) the behavior of the BSDF function in this region is critical to accurate estimates of scatter due to dust in ATST because this is the very part of the curve that will dominate near-limb coronal observations. The best observational evidence in support of the rolloff comes from Spyak and Wolfe s scatter observations in Part III of their series where scatter is studied at 10 m. At this wavelength the roll-off is easily observed, and fits Mie theory quite well. The effects of the largest particles dominate this low-angle end of the BSDF. Here, refraction is not an issue because of the size of the particles relative to a wavelength of light. The scatter is essentially caused by diffraction around these large particles that look like small circular holes in the aperture 11. The roll-off occurs because there is a large-particle cut-off in the size distribution (240 micrometers in this distribution). Hence, there is every reason to expect a similar roll-off as we scale down to shorter wavelengths, given that Mie theory is grounded in a rigorous treatment of diffractive scatter as a function of wavelength. Figure 13 shows the results at the usual limb positions for a coverage fraction of at 1 m. The figure also shows the best case scatter curve due to microroughness (S = 1.5, and a 12 Ångstrom finish) for comparison. The two are roughly comparable at this contamination level, given the uncertainty of many of the assumptions in both curves. If a fractional coverage of (0.01%) is reasonable to expect right after cleaning, how long can we expect the mirror to last in this state in the open air? Data are available from an emissivity study done for the Gemini project at the United Kingdom Infra-Red Telescope (UKIRT). Dust was allowed to accumulate on samples located near the primary mirror, with emissivity measurements made periodically TN-0013, Rev. D Page 15 of 18

19 Relative Emissivity (%) Ratio to On-Disk Irradiance M1 Microroughness and Dust Contamination at infrared wavelengths. The rate of change in emissivity can be inferred from a plot of emissivity versus time. For the UKIRT data, this represents a change of percent per hour (Figure 14). Mie at 0.01% Compared to Microroughness 1.0E E E E E E % coverage 12 A Microroughness 4.0E E E E E Distance from Sun Center (solar radii) Figure 13. This plot shows a comparison of the most likely case microroughness result (blue) and a mirror with 0.01% of its surface covered by dust with the MIL-STD-1246A particle-size distribution (red). UKIRT Emissivity versus Time y = x Hours Figure 14. The blue diamonds are measurements of relative emissivity made at various times as dust accumulated on sample surfaces. A linear fit through these points suggests a rate of change of percent per hour. TN-0013, Rev. D Page 16 of 18

20 Ratio to On-Disk Irradiance M1 Microroughness and Dust Contamination In work performed in support of the Hubble Space Telescope design effort, Facey and Nonnenmacher 12 showed that percent coverage of a surface by dust could be inferred from emissivity data by assuming that the effective emissivity of dust over the range of typical dust-particle sizes was 0.5. This result allows us to assume a change in percent coverage for Mauna Kea of about percent per hour. With this information, we can return to the ASAP model and show how accumulating dust will affect scattered light performance as a function of time. The results after one day and one week at this rate are shown in Figure 15, along with the clean mirror and RMS microroughness results. Scatter w ith Accumulation 1.8E E E E E E E E-05 Microroughness 0.01% Coverage 1 Day (0.063%) 1 Week (0.38%) 2.0E E Distance from Sun Center (solar radii) Figure 15. The figure shows the effects of dust accumulation on the primary mirror after one day and one week, assuming the UKIRT accumulation rate of percent per hour. 6. CONCLUSIONS AND RECOMMENDATIONS The foregoing model analysis suggests for microroughness we can be reasonably optimistic about obtaining a mirror that exceeds the likely scientific requirement, and is close to even the most aggressive scattered light goal. The following two points should be kept in mind, however, when we get closer letting a contract for the task of polishing M1: Since meeting our science requirement for scattered light is critical to success, the project might want to consider incentive payments or other means of encouraging the selected vendor to produce the best polish possible within their cost-benefits envelope. Care should be taken when writing the primary mirror polishing specification. As this study notes, the microroughness alone is not sufficient to completely characterize the scatter behavior of the mirror. It is likely that the vendor will measure the surface profile in such a way that both height and slope statistics can be derived over the known bandwidth of the observations. This would give enough information to obtain the full power spectral density function, and a complete scattered-light characterization of the mirror. We should investigate this possibility during preliminary discussions with mirror vendors. Scattering due to dust contamination of the primary mirror would appear to be the most serious stray-light concern for coronagraphic observations. The accumulation of dust on the primary quickly overwhelms TN-0013, Rev. D Page 17 of 18

21 the effects of surface microroughness from the polishing process. The project can take the following actions to mitigate this: Frequent cleaning of the primary mirror perhaps even daily cleanings may be necessary to control dust accumulation. The project should provide facilities that make it operationally safe and easy to orient the primary mirror for access by personnel performing CO 2 -snow cleaning or some equally effective method. Easy in-situ soap-and-water cleaning of the primary should also be investigated during the design phase since the UKIRT/Gemini study showed that mirrors subjected to CO 2 cleaning did not return to their original just cleaned condition when too much time elapsed between cleanings. A soap-and-water cleaning should be schedule a few times per year. From a systems perspective, all efforts to minimize the effects of telescope and dome seeing need to be viewed in the context of dust mitigation. The combined requirements for good seeing and minimum dust contamination have the potential to be in conflict. The same is true for the airflow schemes that may be used to control the temperature of the primary mirror. 7. REFERENCES 1. Buffington, A., and Jackson, B. V., Some Stray-Light-Reduction Design Considerations for ATST (18 April 2002, revised 17 July 2002). 2. LaBonte, B., Sky Brightness Measurements at Haleakala, (1999, University of Hawaii, Institute for Astronomy preprint). 3. Beckers, J., et al, Interim Report of the Feasibility Study for a Large Optical/Infrared Solar Telescope. (July 1997). 4. Barducci, A., et al, The solar aureola: theory and observations, (Astronomy and Astrophysics 240, , 1990). 5. Stover, John C., Optical Scattering Measurement and Analysis, SPIE Press, 1995, Bellingham, WA, 2 nd ed Church, Eugene L., Fractal Surface Finish, (Applied Optics 27, No. 8, 15 April 1998.) 7. Peterson, Gary L., The Effect of Mirror Surface Figure Errors on the Point Spread Function of the Gemini Telescope, (30April 1993, Gemini Report CON-BRO-G0003). 8. Spyak, P., and Wolfe, W., Scatter from Particulate Contaminated Mirrors, (Optical Engineering 31, No 8, , August 1992). 9. Varsik, J., Siegmun, W., and Berger, D., Telescope Mirror Contamination and Airborne Dust (Unpublished, but available from John Varsik at BBSO). 10. Peterson, Gary. In-Field and Near-Field Stray Light Calculations, UCSD Presentation for ATST (July 16, 2002). 11. van de Hulst, H. C., Light Scattering by Small Particles, John Wiley and Sons, Inc., New York, NY, Facey, T. A., and Nonnemacher, A. L., Measurement of Total Hemispherical Emissivity of Contaminated Mirror Surfaces, Stray Light and Contamination in Optical Systems, SPIE Vol. 967, pp , TN-0013, Rev. D Page 18 of 18