Con-X Grating Designs:
"sub-aperture" and "separate sector images"

Simulated zeroth-order images for different designs, left to right:

• full-aperture scatter-dominated mirror ("sxt");
  
• the same mirror sub-apertured to use only
  the top and bottom quadrants ("sub");
  and
• the same sub-apertured mirror where the quadrants have been
  divided into 5 sectors each and the sector images are separated ("ssi")
  and abutted in the focal plane.

These mirror configurations are combined with dispersive gratings with
the dispersion axis going from left to right, specifically the HEG-angle
green dashed line was used in simulations below.
In the right case ("ssi") the red line segments define a "filament path"
that is used by the analysis software to shear the data and obtain a
high-resolution line response function (LRF.)

Summary

This note summarizes the "sub-aperturing" (SubAp) and "separate sector images"(SSI) techniques that can be used with mirror-grating systems to get higher resolving power; these techniques are explicitly included in Web Cash's Con-X Off-plane grating design. "Sup-aperturing" here refers to using less than the full 360 degrees of mirror/grating, in particular not using the azimuths which have their scatter largely in the dispersion direction. "Separate sector images" means offsetting the images in the focal plane from different azimuthal mirror-grating ranges or sectors. This can be achieved most simply (conceptually) by physically displacing azimuthal mirror/grating sectors to separate the images. Although these offsets could be arbitrary, it is useful and cute to offset the sectors so that the sector images form a "C" pattern in the focal plane.

These resolution improving techniques can be applied to reflection (in-plane) and transmission mirror-grating instruments as well; the main assumptions for the schemes to be useful are that:

• i) the PSF/mirror blur is primarilly in the plane of reflection (e.g., dominated by surface wiggles, "scatter")
  and
• ii) the grating adds little or no additional blur/scatter in the dispersion direction.

Assumption (i) is being taken as a given but does need to be confirmed for specific mirror fabrications and system parameters (e.g., a large aspect reconstruction error could dominate the blur.) The term "scatter" is used here to refer to the effect of surface normal variations which produces greater in-plane than out-of-plane blur in the focal plane by roughly the ratio Focal_length / R_mirror as shown in this cartoon:

Assumption (ii) is considered trivially satisfied by an Off-plane design as the dispersion axis is largely normal to the in-plane scatter of the Off-plane reflection grating. However, assumption (ii) can also be met by transmission gratings (e.g. the HEG and MEG have very low scatter) and could be met by a reflection grating whose surface/figure properties are much better than the mirror's.

It is important to note that these resolution improvements apply only to a point source: the resolution will degrade as the source is larger, say a few arc seconds. Note that the standard RGS design has the finest effective period of these configurations and so is least effected by source size and mirror blur size (other things, like diffraction order, being equal.) Also, for the SSI scheme there are additional complications as the source size gets larger than the individual sector images (spatial overlap of the sector images occurs for objects larger than 30 arc seconds diameter - roughly E0102 size.) These ideas and results are demonstrated below with ray-traces and anayses.

Some properties of the three grating spectrometer designs are summarized in this table:

Grating Type	Flight Heritage	Physical Period	Effective Period	Can use Sub-aperture or Separate Sector Images schemes?	Weight	Importance of Alignment and Figure	Efficiency 60A - 8A, 0.2 - 1.5 keV
"trans- mission"	HETG, LETG (Chandra)	fine, 5000-11000 l/mm	same as physical, 2000A-900A	OK, via mirror sector spatial offsets OR by grating roll	~10s kg	loose tolerances	20% first orders total; could be 30% using phasing material e.g., Si bars
"reflection" e.g., in-plane	RGS 1&2 (XMM)	coarse, ~500 l/mm, ~20,000A	fine, p_eff = sin(a)*p , ~550A	OK if grating has low scatter	~100s(?) kg	as important as mirror	groove shape and reflectivity determined
"Off-plane" (in reflection)	none?	fine, 5500 l/mm	~same as physical, 1800A, divided by order.	OK, via mirror sector spatial offsets, and/or grating tilts(?)	~100s(?) kg	not as bad as mirror	uses higher orders: e.g., blazed for 2nd order at 0.7keV(~18A). Groove shape and reflectivity determined.

Simulations

Note: The values of HPD used here are for the complete PSF and as such include blur effects like aspect reconstruction errors, detector pixel size effects, etc. Likewise any 1-D FWHM or E/dE values are also based on a complete instrument with all these effects.

Although primarilly for demonstration, the quantitative results here (e.g., E/dE) can be interpreted via scaling as equivalent to some Con-X relevant situations:

Configuration Description	Simulation Scale factor?	Scaled PSF HPD	Scaled Alignment, aspect, etc.	Scaled Eff. Period	Scaled Pixel size
"hrma" literally: Chandra HEG w/HRMA	n/a	0.83"	~0.8"	2000A	24 um

Configuration Description	Simulation Scale factor?	Scaled PSF HPD	Scaled Alignment, aspect, etc.	Scaled Eff. Period	Scaled Pixel size
"sxt" literally: Chandra HEG w/ x18 HRMA scatter	n/a	5.6"	~0.8"	2000A	24 um
interpret as: Con-X RGS in first order with 14" HPD PSF	x2.5	14"	~2"	800A	60 um

Configuration Description	Simulation Scale factor?	Scaled PSF HPD	Scaled Alignment, aspect, etc.	Scaled Eff. Period	Scaled Pixel size
"sxt15as" literally: Chandra HEG w/ x48 HRMA scatter	n/a	14.9"	~0.8"	2000A	24 um
interpret as: Con-X OPG or TG in first order with 15" HPD PSF	x1	15"	~0.8"	2000A	24 um

For the same mirror, LRF's when different schemes are used are shown on linear scale (left) and
on a log scale (right); click for full image.
The red curve is the fully illuminated mirror (sxt), the blue curve is the SubAp (sub)
case, and the black is the separate sector images (SSI) case (ssi, five 18 degree sectors.)
(The light purple is SSI case with only the central 15 degree sector
contributing, ssi15.)

Without further ado, the resolving powers, E/dE, obtained for different amounts of additional scatter (different HPD values) and different instrument configurations (e.g., full aperture, SubAp, SSI, etc.) are given in the table below. There are links to the observed 15.014 A line LRF (dark green) with the core Gaussian fit to it shown also (in purple, this fit is used to get dE ~ 2.35*sigma.) The LRFs for the 5.6" HPD configurations are plotted together (images above, both log and linear y-axis versions) and clearly show the improvement in FWHM and wing-reduction that the SubAp and SSI designs offer over the full-aperture version.

Simulation / Configuration	E/dE m=+1	Link to LRF plot w/Gaussian fit to core (purple) at 15.014 A, 0.825796 keV and FWHM/HPD	Comments/Comparison
0.8" HPD:	-	-	-
hrma	1475	LRF plot 1.1	Essentially Chandra performance: very close to Gaussian LRF. sub-aperturing this makes only 10-20% reduction in FWHM since mirror "scatter" is not the dominant blur term.
-	-	-	-
5.6" HPD:	-	-	-
sxt	570	LRF plot 0.42	Should be similar to Con-X baseline RGS: E/dE = 260 at 15A - why is my simulated E/dE larger? (See "Comparison ..." below.)
sub	670	LRF plot 0.35	E/dE and wings improved a bit [core counts ~ 42,000.]
sub_cdnarrow	900	LRF plot ...	sub with narrow cross-dispersion extraction: core counts reduced to 60% from "sub" but gives improved E/dE and wings. [core counts ~ 24,000.]
ssi	~900	LRF plot 0.26	E/dE improved and wings greatly reduced [core counts 56,600.]
ssi15	1250	LRF plot 0.19	This is just the central 15 degree sector. Note that the "scattering" is small compared to other terms which widen the "waist" of the PSF and give a LRF close to a Gaussian.
-	-	-	-
15" HPD:	-	-	-
sxt15as	455	LRF plot 0.19	PSF dominated by the "scatter" and there are large wings on the 1-D LRF but a narrow core is evident due to underlying arc second quality of other terms.
sub15as	540	LRF plot 0.16	Wings are reduced by subaperturing. A narrow cross-dispersion extraction as in "sub_cdnarrow" above would improve wings and E/dE at a similar cost in counts in the core of the LRF.
ssi15as	~680	LRF plot 0.13	Should approximate Cash's "O-P n=1" value: 1000. Correcting Cash's 1000 value for period (x0.9) and Roland Spacing (x0.86) effects gives 775 which isn't so far from the simulated value here. (See "Comparison ..." below.)
ssi15as15	850	LRF plot 0.10	As in "ssi15" above this is just the central 15 degree sector image. Beacuse the "scatter" is so large the LRF is non-Gaussian showing characteristic wings, though greatly reduced from the full aperture case, sxt15as.

Comments on E/dE values

Comparison with the baseline RGS design

As noted by the scaling above, the "sxt" E/dE value above, 570 at 15A, should be roughly equivalent to the baseline Con-X RGS (800A effective period) with a 14" HPD PSF. However, the Con-X predicted E/dE is shown in plots as 260 at 15A - why the difference ? Perhaps the narrowness of the mirror PSF in 1-D is not appreciated: the 14" equivalent HPD PSF here is giving a core LRF with FWHM of order 6" equivalent (that is FWHM/HPD ~ 0.42 as given in the table.) Note that the 1-D FWHM is quite a bit smaller than the HPD in this case.

Perhaps however, the PSF blur is not dominated by the simple mirror "scattering" terms but includes substantial alignment (and even aspect reconstruction) terms as well - these might be relatively large and still not effect the HPD very much yet be important for the 1-D FWHM especially if sub-aperturing or the SSI scheme are used. In this case the PSF and 1-D projection shape are closer to Gaussians and the 1-D FWHM is more similar to the PSF's HPD value - note that this is the case for the Chandra response ("hrma" above) with FWHM/HPD ~ 1.0. The in-plane grating scatter/figure will also provide additional blur that the SubAp or SSI schemes cannot remove so that keeping FWHM/HPD ~ 1 may be the most appropriate/realistic choice and leads to the E/dE ~ 260 estimate at 15A.

Based on this simple comparison one might expect better baseline RGS resolving powers than the current predictions by up to a factor of two or more if mirror alignment and aspect etc. can be at the few arc second level.

Comparison with the Off-plane proposed design

The "ssi15as" value above, ~680, is to be compared with Cash's Off-plane design of an 1818A grating with a 15 arc sec HPD PSF - which shows E/dE of 1000 at 15A in first order. To compare these two numbers, scale my 680 value by 1.1 to 750 as if the period were 1818A. My simulation also has the Rowland Spacing being 0.86 times the focal length since the grating is mounted behind the mirror. Perhaps Cash's value does not include such a factor, taking out this factor as well then gives my simulated E/dE as 870 which is not too far from his 1000.

Note however, that the "15as" simulations are for a PSF of Chandra quality with additional "scatter" to increase the HPD to 15 arc seconds. Thus the underlying alignment, aspect, etc. must be at the < 1 arc second level - whereas Cash's plot is labeled with gratings and alignment of "2 arc seconds."

Note that 15A (0.83 keV) is on the short wavelength side of the nominal Off-plane first-order range (0.2 to 0.6 keV) and I would expect the resolving power at 30A (0.41 keV) to be about twice the value at 15A. Reading off his Off-plane plot gives, however, E/dE of order 3000 at 0.4 keV which surprises me. It's possible that the E/dE falls as one goes away from the center of the arc of dispersion which would explain the factor of 3 in E/dE between 0.7 keV first-order and 0.7 keV second order - but then the value 3000 seems too high, I'd expect 1000 to at most 2000 instead...

Based on this (limited comparison) it would seem the Off-plane predictions for E/dE are too high by 20-50% (Rowland spacing factor, 1st vs 2nd order values mentioned above) or more (aligment etc. is really around 1" vs the stated 2")... Knowing more specifics of the design assumptions would help understand the comparison.

The sections below provide some commentary and additional information on the different simulation and analysis cases.

Mirror w/scatter

Encircled-energy curves for the simulated cases of HPD 5.6" and 15" were made and are shown above. Note that the only difference between these is increasing the MARX P1Blur, H1Blur, etc. terms - leaving all other blur contributions to the PSF at the one arcsecond level.

Sub-aperture effect

A sub-apertured simulation was accomplished by using the XRCF shutters in the MARX s/w to keep only the top and bottom quadrants of the HRMA for all 4 mirror shells. The images here clearly show the expected result of scattering in two quadrants. Note that the "waist" of the 5.6" HPD PSF (left) is relatively broader than the waist of the 15" HPD PSF (right) - this is because the underlying other terms are the same arc second size in either case and so proportionally smaller in the 15" HPD case leading to a crisper waist.

* * * The sub-apertured version of a mirror has reduced wings and somewhat improved FWHM. If spectral extraction is done using less than the full cross-dispersion width (indicated by the extent of the red lines in the figure at right), however, then the E/dE and size of the wings are both improved. For example, the "sub_cdnarrow" case in the table above reduces the number of core counts (area of Gaussian approximation to core) to 60% (42,000 counts to 24,000) but greatly improves E/dE and the wings - approaching the SSI result. So, quadrant sub-aperturing gives the option of extraction within a limited cross-dispersion width and might be a good compromise between a full mirror and the more complex SSI designs especially for brighter sources.

Separate sector images scheme

In order to simulate the separate sector image scheme an IDL routine was used to modify the MARX event values: using "cheat" information in the marx2fits output file it is possible to determine the azimuth each photon came from. All photons from a given azimuth range, or sector, were then offset in dispersion and cross-dispersion by a specified amount, e.g., each sector could be offset just in cross-dispersion to "stack" the images. The nominal Cash-like offset pattern which has the advantage of continuity of the LRF response is autogenerated in the code by just specifying the number of sectors, total azimuth range, and spacing between sector images. The red "path" is also created automatically. The SSI effect is applied by the IDL routine: fil_ssi_apply.pro

The extraction of the LRF from the SSI data is done by "shearing" the data in the cross-dispersion direction so that the "path" becomes a straight line, as shown at right for the zero-order of the "ssi" simulation. Note that in analyzing ssi simulated data it is important to get the path well aligned with the image and the roll angle needs to be well calibrated also - it's possible that these effects are adding a little extra blur in my simulations but they may represent real-world effects.

Two examples of using the SSI scheme with extended sources are shown below. For slightly extended sources the sector images are still separate and the resolving power degrades with source size. As the source gets larger than the separation of the sectors there is some messy spatial mixing as the "Cas A" example below shows!

At left, an E0102-sized ring seen with Con-X-performance SubAp mirror (two opposite 90 degree quadrants/sectors used.) Dispersion direction is roughly along the NE in this Sky-coordinate color-intensity image. At right is shown the zeroth-order image for the same ring in the SSI design (dispersion running along the green dashed line, roughly left-to-right). The ring diameter (30" equiv.) is just less than the spacing between the sector images and so there is little sector-sector overlap. This is an event plot - if a color intensity plot were made each of the sector images would look similar to the color image at left.

At right a Cas A sized object (ring with equiv. inner diameter 75", outer diameter 112" - of order the size of the bright ejecta emission in Cas A) is observed with the SSI designed instrument. Note that the 5 sector images are quite overlapped giving additional complexity to spatial-spectral analysis...

Details of Simulations

Simulations of the effects of SubAp and SSI are made using MARX, custom IDL s/w, and ISIS. Specifically, for the the SSI "LRF" the IDL s/w defines a path that recovers the high spectral resolution in analysis (using "filament analysis" a la dd's N132D analysis s/w.) The simulation process and explicit commands are detailed in the file:

• marx_ssi_sims.txt .

The filament analysis output tables are in:

All Sim Tables

(locally available at MIT only)