Analysis of a new class of grazing incidence spectroscopictelescope
James C. Green and Stuart Bowyer
The throughput and imaging properties of one of a new class of grazing incidence spectroscopic telescope areexamined with a Monte Carlo ray tracing technique. The results are compared with Wolter Schwarzschildtype II telescopes of similar size. The image quality of this telescope is comparable, and the control of the off-axis light is superior to the Wolter Schwarzschild design.
1. Introduction
A new class of grazing incidence spectroscopic tele-scope was introduced by Hettrick and Bowyerl (here-after referred to as Paper 1). This class of telescopeuses primary mirror surfaces that are parabolas ofrevolution. Its secondary mirrors are conic sections ofrevolution that are confocal and concentric with theprimary mirror. There are two types of this telescope,hereafter designated HB-I and HB-II. The HB-I tele-scope has a secondary mirror that is an ellipse of revo-lution and produces a converging beam. It could beused, for example, to feed variable line-spaced grat-ings.2 The HB-II secondary mirror is an hyperbolaand produces a beam diverging from a virtual focus.One possible use would be to feed a Rowland circlespectrometer or similar device. Both types cause theon-axis light, that is, light parallel to the optic axis, tocome to a focus between the primary and secondarymirrors. This point is referred to hereafter as theinitial focus.
This paper focuses on the HB-I telescope. Theimaging and throughput characteristics of this designwill be examined and compared with the WolterSchwarzschild type II, which is the telescope with themost similar application. The striking new feature ofthis design will also be evaluated, i.e., the ability torestrict the field of view without attenuation of the on-axis and near on-axis light.
The authors are with University of California, Space SciencesLaboratory, Berkeley, California 94720.
The surface equations of the HB telescopes can bemade coma-free by strictly obeying the Abbe sine rule.In Paper 1, a general description of such a surface wasgiven. The following surface equations provide analternative but equivalent representation of the HB-Itelescope which has the advantage that the use of thisset of parameters (f, c, e, and 3) makes the physicalstructure of these telescopes more easily understood.For the HB-I telescope
r = f sinfl,
Z =-f +f sin2s + + c sin2 cos(A-Yic
c 4 e !~ 2
Z2 = d cosfl,
r2 = d sink,
-=-sin - + -- (1-+-c sin - +c cos - 1+c d f 2 f( 2 2/
(1)
(2)
(3)
(4)
(5)
where r is the distance from the optical axis to themirror surface, z is the distance along the optical axisfrom the system focus, and # is the angle between theoptical axis and the optical path for an on-axis ray fromthe system focus to the point (r,z). Subscripts 1 and 2refer to the primary and secondary mirrors, respective-ly. In Fig. 1 we show a cutaway view of an HB-Itelescope with the various quantities explicitly la-beled. The effective focal length of the system is f, andthe dimensionless parameters c and e are positive realnumbers. A more physical understanding of the pa-rameters c and e can be obtained if we allow (2 + c)/(1+ c) = c/(1 + c) = 1, and cos2i3/2 = 1. The surfaceequations then reduce to those of a confocal parabolaand ellipse. The location of the initial focus (IF) ofthis reduced system is
Fig. 1. Cutaway view of the HB-I telescope indicating the quanti-ties used in the surface equations. The mirror surfaces closely
approximate a parabola and an ellipse.
where the distance to the initial focus is measured fromthe system focus. The eccentricity of the secondarymirror EC can be expressed as
EC = [1 + = _ 1-. (7)
The designs explored in this paper have values of cthat range from a few hundred to a few thousand. Theangle a is typically only a few degrees. Thus theapproximations above are very good, and the coma-free mirror surfaces are only minimally perturbed fromconic sections. For example, in our calculations theparabolic approximation for initial focus coincidedwith the actual location of the initial focus of a coma-free system as determined by ray tracing to within 1part in 104. Therefore, as stated in Paper 1, the coma-free design offers very little improvement in imagingover the parabola-ellipse design. However, with mod-ern fabrication techniques it is just as easy to producethe coma-free telescopes as the parabola-ellipse tele-scopes.3 Hence all telescopes discussed in this paperuse the coma-free surface equations.
111. Analysis
A representative selection of telescopes was chosenfor study. For comparative purposes, all the tele-scopes examined have a 1-m o.d. opening aperture.The inner radius of the opening annulus is 30 cm,yielding a geometric collecting area of 5027 cm2. Thischoice of annulus resulted from a compromise betweenthe total effective area and the actual size of the sec-ondary mirror. A larger annulus will yield a largercollecting area, but it forces the secondary mirror to bephysically smaller. Since the secondary mirror musthave its interior surface cut and polished, it is essentialthat it be a reasonable size. While no figure of meritwas drawn, the choice of opening aperture used hererepresents a reasonable compromise. The length ofeach telescope, from front aperture to system focus, is4.5 m. They were constrained to leave 2 m of space forinstrumentation between the back of the secondary-mirror and the system focus. The maximum allowablegraze angle on any surface was limited to 200.
In our calculations, a range of effective focal lengthswas examined. For a given focal length, there aremany possible designs reflecting varying choices of cand e. The various designs involve trade-offs betweeneffective area and imaging capabilities. Telescopeswith a less effective area have shorter mirror surfacesand consequently a smaller variation in the graze an-
Table 1. Effective Areas of Telescopes DIscussed In this Paper
On-axis effective area (cm2 )Telescope type Focal length (m) at iooA 1216 A
gles over those surfaces. This allows such telescopesto image better. However, since the effective area iscommonly the primary design criterion, the parame-ters c and e for each focal length were chosen to maxi-mize the effective area at 100 A within given physicalconstraints.
In Table I, we list the telescopes that were ray tracedand the on-axis effective area of each telescope. Theeffective area presented is calculated by multiplyingthe mirror's geometric collecting area times the tele-scope's reflective throughput at the given wavelength.Two Wolter Schwarzchild type II telescopes (WS-IIaand WS-IIb) have been analyzed in'the study as com-parisons. Telescope WS-IIa has been designed tomaximize its effective area, and telescope WS-IIb hasbeen designed to have the same throughput as tele-scope HB-Ib. Both conform to the same physicalconstraints as the HB telescopes and have been de-signed with the same focal length as telescope HB-Ib.
Each telescope was analyzed using a Monte Carloray tracing technique. Incident rays of a given off-axisangle were placed randomly on the opening aperture.Using the principles of geometrical optics, the ray waspropagated through the system, and the angles of re-flection off each mirror surface were recorded for re-flectivity calculations. Each ray was checked to see ifit would vignet, strike the primary mirror twice, or notpass through a field stop at the initial focus. All rayswhich successfully traversed the system were thenbrought to a focus in a plane parallel to the focal plane.The reflectivity of each surface was then calculatedusing the Fresnel equations and the complex indices ofrefraction,4 assuming that the mirror surfaces are gold.To reduce the overall computational effort, thethroughput of the system was calculated using area-weighted graze angles. More rigorous calculations ofthe throughput were made in several cases by calculat-ing the throughput of a larger number of rays and thenobtaining an overall average. The results from thiscalculation differed from those using the area-weight-ed gaze angle approach by <1%.
Analysis of each of the telescopes was carried outwith three circular field stops placed at the initial focusof the telescope with radii of 1, 2, and 3 mm. Thesefield stop sizes were chosen because they yielded fieldsof view appropriate to typical spectroscopic applica-tions and allow for the control of off-axis light withoutattenuation of the on-axis light.
In Fig. 2, we display the throughput of telescope HB-Ib as a function of off-axis angle for the three differentfield stops. The throughput has been normalized bythe mirror reflectivity, so that the on-axis through-put is 1. The sharp decline in throughput with off-axisangle is almost exclusively due to the field stop. Thechange in mirror graze angles is slight over the range ofoff-axis angle displayed, and vignetting is minimal.(The maximum vignetting observed in this analysiswas <3%.) The vignetting is small because most of therays that would miss the secondary do not passthrough the field stop. It should be noted that thecutoff is sharper for the smaller field stops. Thisreflects improved imaging of the primary mirror whenthe light is close to on-axis.
HB-I telescopes with different focal lengths havesimilar throughput curves. The location of the off-axis cutoff (defined herein as the point of 1% through-put) is a function of the primary mirror's plate scaleand, therefore, of the focal length. In general, tele-scopes with longer focal lengths require larger fieldstops to create the same field of view. This increase inaperture size results in a slightly less sharp cutoff.
To compare the performance of the HB-I and WS-IIdesigns, we plotted the throughput of telescopes HB-Ib, WS-IIa, and WS-IIb as a function of wavelength inFig. 3. The sharp wavelength cutoff of the HB-I is aclear advantage over WS-II designs. The problem ofdata confusion with Wolter-Schwarzschild telescopesdue to their extended short-wavelength response hasbeen addressed elsewhere and a solution proposed,5but the resultant short-wavelength cutoff is not nearlyas clear as for the HB-I design. The sharp cutoff seenwith the HB-I design is a direct consequence of thetelescope's geometry. Because the HB-I brings thebeam to a focus twice, the HB-I requires larger reflec-tion graze angles than the WS-II. These larger grazeangles cause the sharper cutoff and strongly affect themirror's reflectivity.
It is important to understand how graze angles in thetwo different types of telescope are effected by thephysical constraints imposed on the design. The grazeangles on the primary and secondary mirrors (G1 andG2) for any single ray can be expressed by the followingequations:
For the HB-I telescope
tan(2G) = L '
2 62 4 2 ( )
For the WS-II telescope
tan(2G,) R 6
G2 , G- tan~l-)
In these equations, L is the axial distance betweenthe points where the ray strikes the primary and the
.5-
.4-
.2
0 2 4 6 8 10 12 14Off Ai Agle (A-eminutes)
Fig. 2. Normalized throughput of telescope HB-Ib-as a function ofoff-axis angle. Three different field stops have been inserted at theinitial focus of the telescope: A represents the mirror throughputwith a 1-mm field stop; B has a 2-mm field stop; and C has a 3-mm
field stop.
!.,
.01100 10
Wavelength (A) 1000
Fig. 3. Overall throughput as a function of wavelength for tele-scopes HB-Ib (solid line), WS-IIa (dashed line), and WS-IIb (dottedline). It is assumed that the telescopes are employed in a modewhich requires strict control of off-axis light. Hence 60% transmis-sion collimators have been included in the WS-II telescope through-puts. Note the much sharper short wavelength cutoff concurrentwith high throughput for the HB-I design. The variations inthroughput at a wavelength longer than the cutoff are primarily the
result of the gold coating.
secondary, L2 is the axial distance between where theray strikes the secondary and the focus, is the radialdisplacement from the optic axis of the point where theray strikes the secondary, and R is the radial location ofthe point where the ray strikes the primary. Whilethese equations are only exact for a given ray, thevalues of L1, L2, R, and 6 do not vary greatly from ray toray for most designs. Therefore, we can develop anunderstanding for the average graze angles by lettingL1, L2, R, and 5 be fixed for all rays. Allowing L = L1 +L2 total length of device, we make the followingobservations:
Fig. 4. Imaging capabilities of the HB-I telescopes. Blur sizes arecalculated by ray tracing. The HB-I telescope can be seen to becoma-free by the slope of the lines in this plot. The slight bending ofthe lines is due to the presence of a 2-mm aperture at the initial focus.
(1) In the limit where L >> R and L >> 6, the equa-tions approach each other asymptotically. There-fore, as LIR increases, the graze angles on the twotelescopes become similar.
(2) An HB-I telescope with large Li (long primarymirror) can have a smaller G, than a WS-II with ashorter L, but the same L. Thus an HB-I and WS-IIcan be designed with the same length, opening diame-ter, and throughput by reversing the values of the grazeangles (i.e., the primary graze angle on the HB-I equalsthe secondary graze angle on the WS-II, and the sec-ondary graze angle of the HB-I equals the primarygraze angle on the WS-II). Telescopes HB-Ib andWS-IIb are examples of such a design. However, thisdoes not represent an optimal design for the WS-II interms of minimizing graze angles. For a given L and R,a WS-II design can always be found with a slightlyhigher reflectivity than an HB-I design. If, however,LIR >> 1, the difference in telescope throughput isminimal.
The HB telescopes were developed explicitly forapplications in which off-axis light might be con-trolled. Hence the overall effectiveness of the HB-Itelescope should be compared with that of a telescopewith an alternative means of controlling off-axis light.This could be accomplished with a Wolter Swarzschilddesign by introducing a collimator whose maximumthroughput is by necessity less than one because of theblockage due to the collimator. Alternatively, controlof off-axis light could be obtained by placing an aper-ture at the system focus of a WS-II telescope, but thisapproach will require at least two more mirror reflec-tions to provide comma-free imaging. These extrareflections would substantially reduce the effectivearea of the system.6 To compare the performance ofthe HB-I and WS-II designs we have plotted in Fig. 3the throughputs of telescopes HB-Ib, WS-IIa, andWS-IIb as a function of wavelength. In the compari-son we included the effects of a 60% transmission colli-
-10
1 10Of-Axis Angle (arcmin)
Fig.5. Comparison of the imaging capabilities of the HB-I and WS-II telescopes. Throughput maximized HB-I and WS-II telescopes
exhibit very similar imaging performances.
mator when calculating the throughput of telescopesWS-IIa and WS-IIb.
In Fig. 4, we show the imaging capabilities of theHB-I as rms blur size vs off-axis angle for the variousfocal lengths plotted in the Gaussian focal plane. Thetelescopes all have a 2-mm radius field stop at theirinitial foci. It should be noted that the plot is not astraight line as would be the case for WS-II telescopes;the slight downturn in the curve at the large off-axisangle is due to the field stop. As it cuts out the light, itacts as a selector which only permits rays that arespatially close to each other to pass, and this causes areduction in the size of the final image. Plots of theray-traced images show that this effect does not distortthe shape of the image, although it can result in a shiftin the location of the image centroid transverse to thesource displacement. As expected, ray traces of HB-Itelescopes without field stops at their initial focus yieldstraight lines on the log-log graph.
In Fig. 5, we plot the quantities as in Fig. 4 fortelescopes HB-Ib, WS-IIa, and WS-IIb from Table I.We can see that the WS-II design provides somewhatbetter imaging than the HB-I design, but both providebetter than 1-sec of arc imaging over a 2-min of arc field.In general, imaging of the HB-I is poorer due to greatervariation in graze angles; by bringing the beam to aninitial focus, the HB-I design causes a faster beam tostrike the secondary mirror, and consequently a largerrange of graze angles exists for the telescope through-put.
As is the case for Wolter Schwarzschild designs,optimally curved focal surfaces for the HB-I telescopeswere found to have unrealistically small radii of curva-ture (r < 1 cm) for use with realistic detectors. Allimaging analyses have, therefore, been done with flat-focal surfaces. The optimum position for a flat detec-tor was examined by moving the detector plane of eachtelescope until the on-axis image blur was 1 sec of arc.The field of focus was then defined as twice the maxi-mum off-axis angle at which the image size is less than
I . . . I . . . I . 0. . I I 1 . . I . . . I 0 . . I
Field f Focus (arminutes)
Fig. 6. Field of focus for the HB-I and WS-II telescopes. The fieldof focus is the field of view within which the image blur is less than orequal to the on-axis blur. Note that all telescopes can provide better
than 1-sec of arc imaging over a 3-min of arc field.
or equal to 1 sec of arc. In general, the field of focus istwice the off-axis angle at which the image size is equalto the size of the on-axis image. This field of focus fora 1-sec of arc blur has beeen tabulated for telescopes HB-Ia through WS-IIb in Table II. Again we see thattelescopes HB-Ib and WS-IIa have very similar perfor-mance.
When designing telescopes, one often chooses a fieldof focus and then computes the on-axis image sizewhich results. For purposes of comparing the twodesigns, in Fig. 6 we plot the image size vs the field offocus for telescopes HB-Ib, WS-IIa, and WS-IIb. Op-timally designed HB-I and WS-II designs show onlyslight differences in their imaging capabilities.
The plate scale of a telescope is defined as the spatialdisplacement of an image centroid as a function of off-axis angle. Ideally, this displacement should be asimple multiple of the off-axis angle. For the tele-scopes investigated in this paper, the displacement ofangle ratio was constant, <1%.
In Fig. 7, we plot the resolution of the HB-I tele-scopes in the Gaussian focal plane (in which the on-axis image if vanishingly small) as a function of off-axisangle. The resolution is expressed as the minimumangular separation between two objects so that theimages are resolvable as separate using the Rayleighcriterion. Therefore, the resolution is approximately
HB-Id
10
40
AO 3 HB-1.
30
HB-Ia
10
1 2 3 4 5 8 7 0 00ff-Axis Angle (arcmin)
Fig. 7. Resolution of the HB-I telescope in the Gaussian focalplane. A 2-mm aperture is located at the initial focus of eachtelescope. Resolution is not given for small off-axis angles wherefabrication limitations dominate the performance characteristics.
40
HO-1a
35-
30-
.0 25
20-
2
15
5 0
0 1 2 3 4 5 .5 7 0 0 100ff-Axis Angle (remin)8 9
Fig. 8. Comparison of the resolutions of the HB-I and WI-II tele-scopes. As before, throughput maximized HB-I and WS-II tele-
scope provide almost identical imaging performance.
equal to the image blur divided by the plate scale. Theresolution has not been plotted for very small off-axisangles since the ray-trace algorithm assumes that themirror surfaces are perfect, while actual resolutionvery close to the axis would be dominated by mirrorimperfections and detector resolution. Since theplate scale does not vary with off-axis angle, one canassume that very close to axis real telescopes withlarger focal lengths will exhibit better resolution thanthose with shorter focal lengths. In Fig. 8, we comparethe resolution of the HB-I and WS-II designs. Againthe WS-II and HB-I designs are very similar when theeffective area of each is maximized.
V. Conclusions
We compared the HB-I telescope to the WS-II tele-scope. The HB-I and WS-II designs exhibit nearequal performance in resolution, plate scale, and
throughput. In principle, the imaging characteristicsof the HB-I design are marginally poorer than those ofthe WS-II design. However, in actual use, the twodesigns would produce very similar results. The ma-jor design advantage of the HB-I design becomes ap-parent when it is incorporated into a system that re-quires strict control of off-axis light. The HB-I designallows control of off-axis light without the addition ofother elements to the system, and compared to WS-IIdesigns, a system designed with a HB-I telescope offersmore total effective area for the same imaging capabili-ties, or conversely, better imaging for the same totaleffective area.
We would like to thank Chris Martin and DaveFinley for useful discussions. The research was per-formed under NASA grant 5-420.
Both authors hold appointments in other depart-ments at Berkeley: James Green in Physics and Stu-art Bowyer in Astronomy.
References
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3. S. Bowyer, J. Green, D. Finley, and R. F. Malina, "DiamondTurned Grazing Incidence Mirrors for the Extreme Ultraviolet:Ten Years of Fabrication and Peformance," in Proceedings,Grazing Incidence Workshop, Annapolis, MD (1985).
4. J. H. Weaver, C. Krafka, D. W. Lynch, and E. E. Koch, OpticalProperties of Metals, Vols. l and 2, DESY F41 Hasylab 81/01 and81/05 (1981).
5. D. S. Finley, P. Jelinsky, S. Bowyer, and R. F. Malina, "AnExtreme Ultraviolet Telescope with no Soft X-Ray Response,"Proc. Soc. Photo-Opt. Instrum. Eng. 628, 23 (1986) forthcoming.
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OLD PHENOMENON TAKES ON NEW MEANING AS CRYSTAL SURFACE PROBEResearch in the NBS Center for Chemical Physics has cleared upan old misunderstanding in electron spectroscopy, and therebydeveloped a useful new tool for surface science. Spectralpeaks in x-ray photoelectron spectroscopy (XPS) and Augerelectron spectroscopy (AES) of crystals are enhanced indirections along the principal axes of the crystal, aphenomenon long thought to be related to the "channeling" ofelectrons along the rows of atoms in the crystal. NBSresearcher Bill Egelhoff has demonstrated that, in fact, thisXPS "searchlight effect" is not due to the deep structure ofthe crystal, but rather to interference effects caused by atomson or near the surface of the crystal. As a result, thesearchlight effect is now seen as a surface probe which givesdirect, easily analyzed information on the structure and, inparticular, the depth of features in the first few atomiclayers of a crystal. The searchlight effect, says Egelhoff,can be an important complement to other surface science toolsfor such problems as developing more efficient catalysts, orunderstanding the electrical nature of metal/semiconductorcontacts.
