In-focus monochromator: theory and experiment of a new grazing incidence mounting Michael C. Hettrick Applied Optics Vol. 29, Issue 31, pp. 4531-4535 (1990) © 1990 Optical Society of America. One print or electronic copy may be made for personal use only. Systematic reproduction and distribution, duplication of any material in this paper for a fee or for commercial purposes, or modifications of the content of this paper are prohibited. In-focus monochromator: theory and experiment of a new grazing incidence mounting designs requiring a prohibitively large exponential variation in spacing.10 The combined rotation and translation of a (varied­space) grating is a new mounting; hence the grating Michael C. Hettrick zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA itself must be derived from a new focusing condition rather than as an improvement in any classical grating. Hettrick Scientific, Inc., P.O. Box 8046, Kensington, Cali­ The extent to which the optical aberrations are controlled fornia 94707. is best analyzed if the local groove density of the grating is Received 6 June 1990. expanded as a power series in the grating aperture: Sponsored by William R. Hunter, Springfield, Virginia. 0003­6935/90/314531­05$02.00/0. © 1990 Optical Society of America. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA where σ 0 is the nominal groove spacing at the grating center A varied­space grating mounted to both rotate and trans­ (w = 0); N2, Ns, N4, etc. are varied­space constants; and w is late constitutes a practical single element fixed slit mono­ the ruled width coordinate as projected on the plane tangent chromator which is in focus at all wavelengths. Keywords: to the grating at its center. Monochromators, diffraction gratings, grazing incidence, x­ The wavelength diffracted through an infinitesimally nar­ ray optics. row exit slit by the grooves in the vicinity of coordinate w will differ from that diffracted by the grating center by an No self­focusing reflection grating has hitherto delivered amount spectral images which remain in focus to first power in the aperture over a continuum of wavelengths diffracted be­ tween fixed entrance and exit slits. Rowland circle de­ where m is the spectral order, and F2, Fs, F4, etc. are the signs1,2 require movable slits, whereas plane grating mono­ Fermat aberration coefficients. The image is considered to 3­7 chromators employ auxiliary mirrors which lower the be in­focus at the exit slit if the first power aberration F2 = 0. throughput and introduce figuring and alignment errors. Finite F3 values result in a comatic image whose profile is Fixed slit designs where a concave grating simply rotates to asymmetrical, while the third power F4 term is spherical select the wavelength are practically in focus at near normal aberration, present even along the classical Rowland circle. 8 incidence but suffer from severe defocusing at grazing inci­ To minimize the total amount of translation required over a dence.3,9 finite spectral region, we first consider rotation only of the These limitations are overcome with a new optical design, grating to select two wavelengths, λ 1 and λ 2, and adjust the whose essential features are shown in Fig. 1. A reflection design parameters to minimize the aberrations there. Even grating consists of grooves whose spacings vary continuously with the constraint of fixed slits, F2 and F 3 may be made to across its ruled width. To select the desired transmitted vanish at both wavelengths if the grating is concave (0 < R < wavelength, the grating is rotated about an axis fixed in ∞ ), resulting in the following focusing condition: space, while simultaneously being translated along its sur­ face in the direction of its ruled width. Due to the varied spacing, the translation provides a new set of effective grat­ ing parameters where the principal ray strikes the grating surface. The freedom to choose the amount of translation permits each wavelength to be brought into an exact focus (to first power) at the fixed exit slit. The novelty of this scheme may be appreciated from the fact that such a translation where T = (cos2α )/r ­ (cosα )/R,zyxwvutsrqponmlkjihgfedcbaZYXWVUTS TzyxwvutsrqponmlkjihgfedcbaZYXWVUTS ´ =zyxwvutsrqponmlkjihgfedcbaZYXWV (cos2β )/r' ­zyxwvutsrqponmlkjihgfedcb (cosβ )/R, zyxwvutsrqpo would have no effect on the properties of a conventional (equally spaced) grating. This degree of freedom inherent in varied­space gratings has previously been left unexploited, except for theoretical 1 November 1990 / Vol. 29, No. 31 / APPLIED OPTICS 4531 Fig. 1. Basic optical configuration of the monochromator. The upper portion shows a section taken across the meridional plane of the grating. A reflection grating rotates about a fixed axis (open circle) while translating along its surface in the direction of its varied groove spac­ ing. The fixed principal ray is indicated by dark lines. The grating position is drawn solid for a typical wavelength and dashed for two extreme wavelengths at opposite ends of the spectral range of a concave grating embodiment. The bottom portion is a top view of the grating surface, schematically showing the varied spacing. is extremely high in the immediate vicinity of the two wave­ lengths, λ 1 and λ 2. However, it degrades rapidly elsewhere, dominated by a large amount of defocusing (curve 200). The maximum defocus of 0.5 A is nearly as poor as the 0.7 A resulting from an equally spaced grating design9 of the same aperture, system length, angular deviation, and groove den­ sity (curves 100,102, and 104). The key to removal of defocusing aberrations, and hence the usefulness of the present grating design, is a translation of the grating at all wavelengths other than λ 1 and λ 2. Con­ sidering simple linear translation in the direction of the tangent plane at the grating center results in the following substitutions: wherezyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA R is the grating radius of curvature, r is the object distance,zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA r´ is the image distance, and α and β are the angles of incidence and diffraction, respectively, relative to the grating surface normal. All values are measured from the fixed axis of rotation, which for simplicity is assumed to intersect the grating at its center. For example, a design tailored to extreme UV wavelengths may have as input parameters σ 0 = 1/1500 mm, R = 10 m, constant deviation α + β = 164°, m = +1,λ 1= 100Å,and λ 2 = 200 A. Equations (3)­(12) then provide the design parame­ ters r = 1011.488 mm, r' = 964.542 mm, N2 = ­1.63766 mm"2, and N3 = +0.00267255 mm ­3 . These differ substantially from those of a conventional concave grating monochroma­ where tor. Using the above parameters and a 50­mm illuminated aperture, curves 200, 202, and 204 of Fig. 2 are the individual Aw being the amount of translation in the direction of the optical aberrations of Eq. (2) with only grating rotation to select the wavelength. As constrained above, the resolution decreasing ruled width. The fixed principal ray now strikes zyxwvutsrqpo 4532 APPLIED OPTICS / Vol. 29, No. 31 / 1 November 1990 Fig. 2. Results of Fermat calculations using parameters for a graz­ ing incidence monochromator: (a) first­order aberration of defo­ cusing; (b) second­order aberration of coma; (c) third­order spheri­ cal aberration; and (d) grating surface translation. Curves 100­106 are for a classicial equally spaced spherical grating, which simply rotates about its pole to select the wavelength. Curves 400­406 are optimized for the new focusing condition, where a varied­space concave grating rotates about a fixed pole and translates along its ruled width. Given the same grating width, the new device exhibits a factor of 200 higher spectral resolution, limited only by spherical aberration. All aberrations are extrema (calculated from the edge of the grating aperture). Fig. 3. Measured line profiles of a prototype in­focus monochroma­ tor, employing a single spherical grating reflection. Typical slit widths were 5­10zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGF urn. These traces are not plotted on the same scale; however, the measured FWHM is indicated for each profile. The full grating aperture of 45 mm was used for all traces, except for the top and bottom traces where the aperture was stopped to ~36 mm to provide a centered illumination. zyxwvutsrqponmlkjihgfed 1 November 1990 / Vol. 29, No. 31 / APPLIED OPTICS 4533 remove a residual amount of defocusing, the required accura­ the ruled width coordinatezyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA w = ∆w, and w* is measured cy of translation is modest. relative to this new pole. Experimental verification of the above new theory has • Using the same numerical parameters previously given been accomplished by construction of a breadboard develop­ and choosing N4 = 0, curves 300, 302, and 304 of Fig. 2 result mental version of this monochromator. To demonstrate the from numerically iterating Eqs. (14), (15), and (18) to elimi­ grazing incidence performance and hence the applicability of nate defocusing at all wavelengths. All wavelengths are now this design to short wavelengths, the included angle (a +zyxwvutsrqponmlkji β )of sharply in focus, the new limit to the optical resolution being the principal ray was chosen to be 140° (20° graze at zero spherical aberration. As this term is proportional to the order). The numerical parameters were chosen to enable third power of the grating aperture, ~63% of the total dif­ operation in the ultraviolet and visible regions of the spec­ fracted energy is enclosed within an image width which is trum so as to allow alignment and testing in atmosphere. only one­fourth of the extreme aberration plotted in Fig. The resulting groove density was 200 g/mm at the grating 2(c). The resulting spectral resolution is thus ~0.003 Å. center, the radius of curvature of the spherical grating was A further correction is available by use of nonzero values 1001 mm, the object distance was 301.5 mm, image distance for N4. From Eq. (16) it is clear that this term will signifi­ was 316.6 mm, and the full grating aperture was 45 mm. For cantly change the substituted value ofN*3as the grating is this demonstration, the grating translation was provided by translated (∆wzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA ≠ 0). Curve 402 reveals the elimination of a manual micrometer and ball slide, and the rotation driven the coma at a third wavelength near the spectrum center by ­7 ­4 by a precision lead screw and wavelength bar. At each the choice ofzyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA N4 = ­6.99 Χ 10 mm . In practice the coma wavelength tested, the translation was experimentally ad­ becomes negligible at all wavelengths, resulting in a highly justed to maximize the detected power. Then a high resolu­ symmetrical image whose remaining spherical aberration tion scan across the spectral line was traced on a chart (curve 404) may be deconvolved from a spectrum by the use recorder. Two light sources were used: a low pressure mer­ of accurate modeling techniques. cury lamp and a He­Ne laser. The mercury discharge was Because the grating radius and rotation provide for the ~5 mm in diameter and placed behind the entrance slit, broad selection of wavelength, the amount of space variation whereas the collimated laser pencil beam ~0.6 mm in diame­ required is small and easily accomplished with present tech­ ter was simply diffracted by the entrance slit to illuminate nology. As plotted in Fig. 2(d), the maximum amount of the full grating aperture. translation is only 25 mm. A 75­mm ruled grating width, Figure 3 shows the wavelength profiles obtained for three with an aperture stop (Fig. 1) to constrain an exactly fixed strong emission lines of the Hg lamp and the red line of the beam direction, will provide the assumed 50­mm illuminated He­Ne laser. These traces reveal a symmetrical centrally aperture. Alternatively, the full 75 mm may be utilized at all peaked in­focus image at each wavelength. As listed in wavelengths if the incident beam overilluminates the grating Table I, which includes an additional three (weaker) Hg and a ±15% deviation is allowed in the direction of the ray lines, all measured resolutions are attributed entirely to ei­ diffracted from the center of the aperture. ther the finite slit widths or the physical diffraction­limited The small amount of translation also enables the use of a resolution (9000 grooves full aperture). In agreement with simple linear translation stage, the resulting vertical move­ both geometrical (Fermat) calculations and numerical ray ment of the intersection point of the concave grating surface tracing of the line profiles, the obtained resolution of <1 A is with the principal ray being unimportant. This maintains approximately a factor of 4 less than the extremum spherical fixed directions for the incident and diffracted rays. Fur­ aberration calculated to be 3­5 A for wavelengths of 2534 to zyxwvutsrqp thermore, because the grating translation functions only to zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Table I. Predicted and Measured Monochromator Performancea a zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM m, spectral order; X, wavelength; Aw (theory), predicted grating translation; Aw (actual), measured grating translation relative to 0­μ m reading for optimized imaging at X = 6328 Å; ∆λ d, diffraction­ limited resolution, assuming a full grating aperture; s0, entrance slit width; si, exit slit width; ∆λ 0, entrance slit­limited resolution; ∆λ i, exit slit­limited resolution; AX (actual), measured FWHM of traced line profile. h Grating width stopped to 36 mm, centered at the rotation axis. 4534 APPLIED OPTICS / Vol. 29, No. 31 / 1 November 1990 6328 A. A few of the profiles in Fig. 3 can be seen to have a and translation of a varied­space grating, one embodiment of slightly higher wing to one side of the peak. This is due to the which has been described in this Communication. grating translation (9 mm, as given in Table I), which weights The author thanks George Hirst for fabrication of the the aperture and thus the spherical aberration to one side of grating and loan of a detector used in the measurements the grating pole. This effect is removed by stopping this reported here. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONM small fraction of the aperture, resulting in the symmetrical profiles shown in the top and bottom traces of Fig. 3. The References diffraction­limited profile at 6328 A is evidenced by the 1. H. A. Rowland, "On Concave Gratings for Optical Purposes," presence of subsidiary maxima. Philos. Mag. 16, 197­210 (1883). The measured FWHM of these profiles, using slit widths 2. F. C. Brown, R. C. Bachrach, and N. Lien, "The SSRL Ultrahigh of 5­10zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA μ m, represents resolving powers λ /zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA ∆λ = 5000­12,000 Vacuum Grazing Incidence Monochromator: Design Charac­ for this grazing incidence monochromator of only 0.6 m in teristics and Operating Experience," Nucl. Instrum. Methods length and a meridional aperture of 0.05 rad. A convention­ 152, 73­79 (1978). al spherical grating would exhibit a defocusing of 35 A at the 3. M. C. Hettrick, "High Resolution Gratings for the Soft X­Ray," center of this spectral region (λ /∆λ = 125). Nucl. Instrum. Methods A266, 404­413 (1988). The ability to maintain the spectral focus for large grating 4. M. Itou, T. Harada, and T. Kita, "Soft X­Ray Monochromator apertures provides higher throughput than previous de­ with a Varied­Space Plane Grating for Synchrotron Radiation: signs,3­9 making the in­focus monochromator (IFM) particu­ Design and Evaluation," Appl. Opt. 28, 146­152 (1989). larly advantageous when operated with soft x­ray radiation. 5. H. Dietrich and C. Kunz, "A Grazing Incidence Vacuum Ultravi­ When used with conventional laboratory sources (electron olet Monochromator with Fixed Exit Slit," Rev. Sci. Instrum. bombardment, spark gap, Penning, and hollow cathodes), a 43, 434­442 (1972). single element IFM as discussed above is appropriate. How­ 6. W. R. Hunter, R. T. Williams, J. C. Rife, J. P. Kirkland, and M. ever, to maintain bright images when operating with low N. Kabler, "A Grating/Crystal Monochromator for the Spectral emittance sources, such as laser­produced plasmas and syn­ Range 5 eV to 5 KeV," Nucl. Instrum. Methods 195, 141­153 chrotron radiation, stigmatic versions of the monochromator (1982). are required. This may be accomplished without degrada­ 7. H. Petersen, "The Plane Grating and Elliptical Mirror: A New tion of spectral resolution by the insertion of a mirror which Optical Configuration for Monochromators," Opt. Commun. 40, provides focusing in the direction normal to the grating 402­406(1982). dispersion. For example, a 1­m long stigmatic IFM having a 8. T. Namioka, "Theory of the Concave Grating III. Seya­Na­ geometrical collection aperature of ~ 1 Χ 1 0 ­ 3 sr would pro­ mioka Monochromator," J. Opt. Soc. Am. 49, 951­961 (1959). vide an optical resolving power of 103 in the 35­500­A wave­ 9. M. C. Hettrick and J. H. Underwood, "Stigmatic High Through­ length region. put Monochromator for Soft X Rays," Appl. Opt. 25, 4228­4231 Commercial models of the IFM are being developed by (1986). Hettrick Scientific, Inc., which holds a license for manufac­ 10. D. E. Aspnes, "High­Efficiency Concave­Grating Monochroma­ ture, sale, and use of these devices. U.S. and foreign patents tor with Wavelength­Independent Focusing Characteristics," J. are pending on the general technique of combined rotation Opt. Soc. Am. 72, 1056­1061 (1982). 1 November 1990 / Vol. 29, No. 31 / APPLIED OPTICS 4535 zyxwvutsrqp