Figured grazing incidence mirrors from reheated float glass

Simon E. Labov

High throughput grazing incidence mirrors have been fabricated from float glass using a new inexpensivetechnique. An array of twelve such mirrors including grazing angles from 3 to 150 has been fabricated andtested. Optical measurement of this array shows a line spread function of 6.3-min of arc FWHM with 90%energy enclosed within 13 min of arc. Three-mirror arrays are used to provide 1-D focusing for a diffuseextreme ultraviolet spectrometer. This paper presents the fabrication techniques and testing proceduresused, as well as the mirror performance results.

1. Introduction

This paper describes the development of a highthroughput, low-cost grazing incidence mirror system.The mirror system consists of an array of curved glassplates which provide 1-D focusing over a large geomet-rical area. Each module behaves much like an achro-matic cylindrical lens which can focus x-ray and ex-treme ultraviolet (EUV) radiation. Two such systemscan be mounted perpendicular to each other to provide2-D focusing as described by Kirkpatrick and Baez.1

The mirrors for this system were fabricated by soft-ening float glass to mold it close to the proper figure.These mirrors were then mounted into modules andtested. This simple process is easily reproducible andtherefore lends itself to mass production. Once themandrels and module design are complete, many mod-ules can be fabricated with minimal effort for eachadditional module. The materials and processing areinexpensive, and the natural smoothness of the floatglass requires no polishing.

The modules described in this paper were designedand built as part of a diffuse spectrometer for theEUV.2 In that instrument a 1-D mirror array providesfocusing in the spectral direction while allowing largeangles to enter the instrument in the perpendiculardirection without reducing the spectral resolution.The lack of 2-D focusing results in no imaging in thespectrometer, but the gain in solid angle is crucial tothe diffuse spectrometer's sensitivity. In particular,

The author is with University of California, Berkeley, Space Sci-ences Laboratory, Berkeley, California 94720.

Received 23 September 1987.0003-6935/88/081465-05$02.00/0.© 1988 Optical Society of America.

the mirror system was designed with a very short focalratio (f/No. = 1) to utilize a large solid angle. Twoperpendicular modules could not provide adequatefocusing into such a fast beam. The second arrayalways defocuses the first and the amount of defocus-ing increases as the focal length is reduced.

II. Mirror Array Design

Linear mirror arrays have been discussed by Kirk-patrick and Baez,1 Van Speybroeck et al.,3 Weisskopf,4Schmidt,5 Kast,6 Angel,7 Schmidt,8 and Fabricant, Co-hen and Gorenstein 9 whose success in bending glassplates is described elsewhere in this issue. A coordi-nate system with the origin at the intersection of theoptical axis (the z axis) and the focal plane ideallydescribes the geometry of a 1-D system. All the focus-ing takes place in the x direction and the focal length fis taken as the distance from the focal plane to themiddle of the array.

The equation for mirror i is given by

X2

p2

2pi(1)

An ideal mirror will focus on-axis light perfectly, butthe image degrades as the incident angle increases. Amirror of height Az will produce an image width Axhwhere

AXh Az tanOn (2)

and Oin is the incident angle. There is also a contribu-tion to the image size due to the width of the array. Ifthe array stretches from x = -D/2 to D/2 and most ofthe area is filled with mirrors, the array width contri-bution to the image size can be estimated by

Ax, - tanOjn tan 0j, + tan-, -2 2f)

(3)

For an array on only one side of the optical axis, the

15 April 1988 / Vol. 27, No. 8 / APPLIED OPTICS 1465

image degrades faster when the mirror graze angles areincreasing (in > 0) than when they are decreasing (in< 0). If both sides of an array are used, the absolutevalue of Sin should be used in Eq. (3).

The total angular resolution can be estimated byadding Eqs. (2) and (3) in quadrature and dividing bythe focal length:

A _ {(z tan~in) 2 + [D tan~in, tan(Oi, + tan- 1 D )]21t t + (4)

which improves as the height of the mirrors (Az) or thearray width (D) decreases relative to the focal length.Decreasing the height of the mirrors, however, meansthat more mirrors will be necessary to fill a givenaperture. Since each mirror has a finite thickness andmay require baffles, the throughput decreases with thenumber of mirrors used. As a result, one must alwayschoose a compromise between better angular resolu-tion and throughput. Similarly, if a fast optical sys-tem is desired, smaller focal ratios will also compro-mise the resolution. Note that the focal ratio is alsodirectly related to the highest photon energy which canbe focused:

D 12/- 2f/No. = tan(2amax), (5)

where max is the critical graze angle at the shortestwavelength desired.

Once the width of the array, the focal length, and themirror length have been defined, it is necessary todefine the positions of the individual mirrors, i.e., tofind the values of pi. To find the array parameterswhich maximize throughput, however, one must firstconsider how the system will be baffled. A mirrorarray requires baffles to prevent unfocused rays fromreaching the imaging portion of the focal plane. Inmany situations, a large range of incident angles willenter the mirror system, and one must be certain thatno ghost images are formed. Ghosts can be caused byrays which miss the mirrors entirely, or rays whichmake multiple reflections including reflections fromthe mirrors' back sides. The baffling system used forthe mirrors discussed here follows a scheme similar tothe one described by Kast.6 Each mirror has twobaffles, one over the top of each mirror, and a second infront of each mirror somewhere between the top andthe bottom of the mirror. For a given length of the topbaffle and position of the middle baffle, the length ofthe middle baffle is chosen so that no unreflected rayscan reach the ghost-free region of the focal plane. Foreach mirror, these baffle parameters and the mirrorposition are then varied to find highest throughputover the range of incident angles to be used.

Although the above procedure gives the maximumthroughput for a given ghost-free focal plane region,these baffles will always decrease the system efficien-cy. This loss due to the baffles must be considered inthe trade-off of angular resolution (determined by themirror length Az) and throughput. Shorter mirrorsobscure more area and also require more baffling. Ifthe range of incident angles is limited, however, the

baffling requirement can be reduced or eliminated. Inany case, the range of incident angles and region ofghost-free images should be determined so that bafflelosses can be considered in design trade-offs.

Ill. Fabrication and Alignment

Once the mirror array parameters have been de-fined, the next step is to fabricate the mirror mandrelsor molds. A numerically controlled milling machinecould be programmed to produce surfaces followingEq. (1), but producing the desired curve over a signifi-cant length involves technical difficulties and substan-tial cost. An alternative method, which was used tofabricate the mirrors described here, is as follows.

A large milling machine was fitted with a large flycutter a distance R from the axis. When the head ofthe machine is tilted at an angle 0, the tool will cut theedge of an ellipse into a block passed under it along thedirection of the head tilt. Note that if 0 = 900, the toolwill cut the edge of a circle with the radius R. Theresulting ellipse is described by

Z 2 = (R2+ X

2) sin 2O. (6)

To test the focusing properties of a given section ofellipse, Eq. (5) was rotated and translated into the (x,z)coordinate system so the end points match that of thedesired off-axis parabola section of Eq. (1). The el-lipse was then ray traced to determine the width of thefocused image. The offset from the center of the el-lipse to the section used and the angle 0 were varied tofind the elliptical section with the best focus. Theellipse with the largest curvature in the system de-scribed here has a theoretical spot size of <15 sec of arc.In the diffuse EUV spectrometer, the minimum spec-tral line size is more than four times larger than this,therefore the elliptical curves were deemed adequate.Furthermore, ray tracing of off-axis rays being focusedby these elliptical curves shows that the image sizeincreases with increasing incident angle (in) in thesame way as the parabolic mirrors [Eq. (2)].

The actual R of the fly cutter used (12.5 cm) wasadjusted to compensate for different thermal expan-sions of the glass mirrors and the stainless steel man-drels. Once a mandrel was cut, no polishing or buffingwas attempted. The residual machine marks do some-times transfer from the mandrel to the back side of theglass, but there has been no evidence that these marksaffect the front surface.

Float glass was used for its naturally smooth surfacewhich requires no polishing to reduce x-ray scatteringand therefore provides a substantial savings of cost.The float glass used here was 1.8 mm thick and wasground to a rough finish on the back side to reduceundesired reflections. Each glass piece was positionedon a mandrel and then heated to 6500C for 2 h andallowed to cool to room temperature overnight. Thistemperature proved to be hot enough to allow the glassto soften, but not so hot as to allow the glass to flow anddisfigure the surface. If the glass does flow, it willoften stick to the stainless steel and then crack andsplinter during cooling. With the procedure used

1466 APPLIED OPTICS / Vol. 27, No. 8 / 15 April 1988

here, the glass did not stick to the stainless steel and noparting agent was necessary.

Once cool, the mirror pieces were cleaned, testedoptically, and coated with a metallic surface for highEUV reflectivity. The mirrors and the midmirror baf-fles were mounted in holders which had long slots cutinto each side. Small plastic shims provide a bufferbetween the glass and the aluminum housing, and thinmetal shims were used to position each mirror. Thealignment was achieved by moving each mirror to fo-cus a collimated beam of light onto a target on the focalplane. Once the proper shims were all in place, theywere held captive by four end plates. A single coverplate which included all the top baffles for the arraywas adjusted to eliminate any ghosts in the imagingportion of the focal plane.

IV. Testing and Results

The mirrors have been tested optically with a setupsimilar to the one used for alignment. A collimatedbeam of light was focused by the mirrors, and a narrowslit was moved linearly along the focal plane. A photo-sensitive diode was mounted behind the slit to measurethe light intensity. The results for a single mirror areshown in Figs. 1 and 2. The mirror used in this testwas one near the outside edge of the array with anaverage graze angle of 140. The mirror is 5 cm high,has a focal length of 22.5 cm, and its profile deviatesfrom a straight line by 0.16 mm (radius of curvature 2 m). The relative intensity is shown in Fig./1 as afunction of angle. The total intensity enclosed withina given angular distance from the optical axis is shownin Fig. 2. This mirror has a full width. at half-maxi-mum (FWHM) of 4.2 min of arc, a half-energy width(HEW) of 3.3 min of arc, and 90% of its energy within13.0 min of arc. The angles shown in Fig. 2 are theangles from the center of the image, i.e., these anglesare the halfwidth of an image which encloses a givenenergy level. The results of the same test performedon an array of twelve mirrors on one side of the opticalaxis are shown in Figs. 3 and 4. For this array, theFWHM is 6.3 min of arc, the HEW is 3.4 min of arc, and90% of the energy is contained within 12.4 min of arc.

Individual mirrors were also tested to determinehow the bending process changed the scattering prop-erties of the glass. Both a flat and a bent glass samplewere placed at a 70 incident graze angle to a monochro-matic beam of 237-A radiation. The number of countsobserved as a function of angular distance from thespecular reflection is shown in Fig. 5. The reflectionfrom the bent sample is indicated by the solid line, andthe flat sample by the dotted line. Even though thebent glass was not positioned for optimal focusing, alarge reduction in the image width can be seen in Fig. 5.The number of counts in each image is nearly the same.The average scattered light intensity within 20 of thespecular reflection from the flat mirror is <1.8 X 10-3/deg and <4.8 X 103/deg from the bent mirror. Onlyupper limits to the scattering can be obtained becausethe amount of scattered light inherent to the measure-ment apparatus is difficult to determine. One can

7

6

5

4

3

2

0

-40 -30 -20 -10 0 10Arc minutes

20 30 40

Fig. 1. Image profile for a single mirror element. The boxes are thedata points and the lines simply connect the points. Measurement

errors are smaller than the plotting symbols.

.6

.4

00 2 4 6 8 10 12 14 16 18 20

Fig. 2. Energy enclosed as a function of distance from the center ofthe image for a single mirror element. The circles are the datapoints and the lines simply connect the points. Measurement errors

are smaller than the plotting symbols.

conclude, however, that the nonspecular reflectionfrom the bent glass is no more than three times greaterthan the untreated float glass, and both glass sampleshave quite low levels of nonspecular reflectivity.

The efficiency of three-mirror arrays from the dif-fuse EUV spectrometer is shown in Figure 6. Thesemeasurements were made with a 10-min of arc beam,and the incident angle at each wavelength was set tocoincide with the angle incident on the mirrors in thediffuse EUV spectrometer configuration. The trian-gles in Fig. 6 show the efficiency of the rhodium-coatedmirror array which was measured with in = 3.10 at 68A, dropping to 00 at 150 A, and back up to 2.60 at 237 A.The squares indicate platinum-coated mirrors mea-sured with °i, = 1.70 at 237 A, dropping to 0° at 304 A,and up to 3.20 at 445 A. Similarly, the circles indicateplatinum-coated mirrors measured with Sin = 2.7° at

15 April 1988 / Vol. 27, No. 8 / APPLIED OPTICS 1467

Ie

bj

aI

IIIII

} I T I . I I l l I I . . . I I l

\i i

;r'\

jA,

I . . . . I . . .

-30 -20 -10 0Arc minutes

2 . . ,-

10 20 30

Fig. 3. Image profile for an array of twelve mirrors. The boxes arethe data points and the lines simply connect the points. Measure-

ment errors are smaller than the plotting symbols.

I ''I|'l ||l ' '' IjIL 4-6-|' 4>- -;~ ~s1(

0 2 4 6 6 10 12 14 16 18 20Arc minutes

Fig. 4. Energy enclosed as a function of distance from the center ofthe image for an array of twelve mirrors. The circles are the datapoints and the lines simply connect the points. Measurement errors

are smaller than the plotting symbols.

445 A, down to 0 at 584 A, and up to 1.40 at 660 A.The points shown are average efficiency over the entirelength of the array, and therefore include the lossesdue to the baffles and those due to the finite thicknessof the individual mirrors.

Finally, the focusing properties of these mirror ar-rays was tested with EUV radiation. This was doneindirectly by measuring the spectral linewidths of theassembled diffuse EUV spectrometer. Each wave-length of light enters the mirror array at a differentangle. The collimator limits the range of angles withinone wavelength to 40 min of arc. Therefore, even on-axis light will be blurred by a minimum angle which is afunction of wavelength.

The measured image widths are shown in Fig. 7 as afunction of Sin The triangular points were measuredat wavelengths between 80 and 240 A (short), the

10 -

-120 -10 - 80 -0 -40 -20 0 20 40 0A-cco

00 100 120 140

Fig. 5. Histogram of the detector image produced by a beam ofmonochromatic 237-A radiation reflected off two different samplesat 70 grazing angle. The dotted histogram indicates the image froman undisturbed sample of float glass, and the solid lines show theimage from a bent glass sample. The total number of counts in each

image is nearly the same.

.7

.6

.5

.4

.3

.2

.1

50 100 150 200 250 300 350 400 450Waccl-egth ()

500 050 600 650 700

Fig. 6. Efficiency of the diffuse EUV spectrometer mirror arrays asa function of wavelength and incident angle. Each point is anaverage over the entire length of the mirror array and thereforeincludes losses due to baffles. The triangles show the efficiencies formirrors coated with rhodium, and the squares and circles indicateplatinum-coated arrays. The incident angle (i) for each point

changes with wavelength as described in the text.

squares from 220 to 440 A (medium), and the circlesfrom 420 to 650 A (long). This instrument has beenflown on a Nike Black Brant sounding rocket, and theopen symbols indicate measurements obtained beforethe launch, while the solid symbols show the postflightdata. The error bars only indicate the statistical er-rors in the line profile fitting; the total error in themeasurements is indicated by the scatter in Fig. 7.Within this scatter there is no indication of changes inoptical resolution as a result of the sounding rocketflight and recovery. The image size predicted by Eq.(4) has been calculated, added in quadrature to the

1468 APPLIED OPTICS / Vol. 27, No. 8 / 15 April 1988

350

300

250

200

2 10.S

7- IS

100 I_

50

.11I lj i iii~~~~~~~~~~~~~1

- 1 . I I I l I .. - _ . . . . .. . . I } .i

I....'' ................... ......

.8,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, . . .... . . . . . . . . ... ....... ... ..... . ., I, . . . . . . . . . I l . i

.11-1 ...... 1-1.11-

i

i

14

- ----- --- -r

Is

I

I

._

0 .5 1 1.0 26. (dcg-m)

2.5 3 3.0 4

Fig. 7. Resolution as a function of incident angle (i) of the diffuse EUV spectrometer mirror arrays. The triangles indicate measurementsobtained with the 80-240-A system, and the squares and circles with the 220-440-A and the 420-650-A systems, respectively. Open symbolsindicate measurements taken before a sounding rocket launch, and solid symbols show postflight results. The solid, dotted, and dashed lines

indicate the resolution predicted for the short, medium, and long wavelength systems, respectively.

collimator-induced blur, and the results are shown inFig. 7. The solid, dotted, and dashed curves indicatethe resolution predicted for the short, medium, andlong wavelength systems, respectively.

V. Conclusions

From inspection of the bent glass and its images, itappears that the glass curvature is uneven. It seemsthat the glass is not forming the optical figure whichwas cut into the mandrel. However, the deviationswere not large enough to affect the performance of thediffuse EUV spectrometer, and their cause remainsunknown. Likely possibilities include deformation ofthe stainless steel mandrel during heating and incom-plete glass softening. Errors from both of these effectscan probably be reduced by further refinements of thefabrication procedure.

Since the amount of mirror curvature necessary isreduced as the mirror position moves closer to theoptical axis, the inside mirrors produce better images.This effect was observed visually during alignmentand is suggested by the similar focusing properties ofone outside mirror and an entire mirror array. If theinside mirrors were not better than the outside ones,the errors in alignment would broaden the array focusto a much larger degree than observed. Consequentlyif better imaging is desired, the technique here may becapable of providing it if larger focal ratios are used (fiNo. > 1). In the x-ray region, smaller graze angles andconsequently smaller curvatures are required. If theglass heating process does not greatly increase the x-ray scattering, this process may be capable of provid-ing x-ray mirrors with much higher resolution.

In summary, a method of bending float glass mirrorsby heating has been developed and a moderate resolu-tion system has been produced at low cost. This sys-tem was built, aligned, and tested as part of the diffuse

EUV spectrometer and successfully survived a sound-ing rocket launch and recovery.

I wish to thank Glenn Stark for his work in develop-ing this technique, and Anne Miller, Avrim Blum, Ai-leen Corelli, and Stuart Bowyer for their assistanceand support. This work was supported by NASAGraduate Student Researchers grant NGT-05-003-805and NASA grant NGR 05-003-450.

This material was presented as paper 830-22 at theConference on Grazing Incidence Optics for Astro-nomical and Laboratory Applications, sponsored bySPIE, the International Society for Optical Engineer-ing, 17-19 Aug. 1987, San Diego, CA.

References1. P. Kirkpatrick and A. V. Baez, "Formation of Optical Images by

X-Rays," J. Opt. Soc. Am. 38, 766 (1948).2. S. Labov, S. Bowyer, and C. Martin, "A Spectrometer to Measure

the Diffuse, Astronomical Extreme Ultraviolet Background,"Proc. Soc. Photo-Opt. Instrum. Eng. 627, 379 (1986).

3. L. P. Van Speybroeck, R. C. Chase, and T. F. Zehnpfennig,"Orthogonal Mirror Telescopes for X-Ray Astronomy," Appl.Opt. 10, 945 (1971).

4. M. C. Weisskopf, "Design of Grazing-Incidence X-Ray Tele-scopes. Part 1," Appl. Opt. 12, 1436 (1973).

5. W. K. H. Schmidt, "A Proposed X-Ray Focusing Device withWide Field of View for Use in X-Ray Astronomy," Nucl. Instumn.Methods 127, 285 (1975).

6. J. W. Kast, "Scanning Kirkpatrick-Baez X-Ray Telescope toMaximize Effective Area and Eliminate Spurious Images; De-sign," Appl. Opt. 14, 537 (1975).

7. J. R. P. Angel, "Lobster Eyes as X-Ray Telescopes," Astrophys. J.233, 364 (1979).

8. W. K. H. Schmidt, "Wide Angle X-Ray Optics for Use in Astrono-my," Space Sci. Rev. 30, 615 (1981).

9. D. G. Fabricant, L. M. Cohen, and P. Gorenstein, "X-Ray Perfor-mance of the LAMAR Protoflight Mirror," Appl. Opt. 27, 1456(1988).

15 April 1988 / Vol. 27, No. 8 / APPLIED OPTICS 1469

71a9I

.2

II

120

100

60

60

40 ~ ~ ~ ~ ~ ~ ~ ~~

20