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III. INSTRUMENTATION AND
APPLICATIONS.OPTICAL AND PHYSICAL
PROPERTIES OF DIFFRACTION GRATINGS FOR
THE V.U.V. AND SOFT X-RAY REGION
T. Namioka, W. Hunter
To cite this version:
//l.
INS TRUMENTA TION AND A PPL ICA TIONS.
OPTICAL AND PHYSICAL PROPERTIES OF DIFFRACTION GRATINGS
FOR THE V.U.V. AND SOFT X-RAY
REGION
T. NAMIOKA
Research Institute for Scientific Measurements, Tohoku University, Sendai 980, Japan
and
W. R. HUNTER
E. 0. Hulburt Center for Space Research, U. S. Naval Research Laboratory
Washington, D.C. 20375, U.S.A.
Rtkumd. - Cette contribution est une synthtse des discussions sur la table ronde consacrke a 1'Ctude et i la realisation des rkseaux de diffraction ainsi qu'a leurs proprietks physiques calculCes et mesurees. Dans une premitre partie les methodes de production sont dkcrites en detail ; la seconde partie met l'accent sur les probltmes de caracterisation de la lumitre parasite, sur les mesures d'effica- cites et les anomalies. Pour terminer les performances des rkseaux, souhaitables pour les experiences
a venir, sont passees en revue.
Abstract. - This paper is a report of a panel discussion on (1) design and production of various types of gratings and (2) predicted and observed physical properties of gratings. In the first part the emphasis is laid on methods and techniques of producing gratings, and in the second part various problems on stray light characteristics, grating efficiencies, and grating anomalies are taken up. Some future requirements for gratings are also discussed.
1. Introduction. - This paper is a summary of the proceedings of Panel 1 oi the optical and physical properties of diffraction gratings for the V.U.V. and soft X-ray region. The panel was organized by T. Namioka (Tohoku University, Sendai) in collabo- ration with the Programme Committee and was presided over by W. R. Hunter (Naval Research Laboratory, Washington, D.C.). The panel members were, in alphabetical order, E. T. Arakawa (Oak Ridge National Laboratory, Oak Ridge), P. Dhez (University of Paris-Sud, Orsay), J. H. Dijkstra (Space Research Laboratory, Utrecht), J. Flamand (Jobin Yvon, Longjumeau), M. C . Hutley (National Physical Laboratory, Teddington), R. Petit (Univer- sity of Marseille 111, Marseille), D. Rudolph (Univer- sity of Gottingen, Gottingen), and R. J. Speer (Impe- rial College, London).
The purpose of the panel was to summarize the present state of knowledge on gratings and to discuss future requirements of gratings. Since the designs of devices using ruled and holographic gratings were taken up by Panel 3 at this conference, the main
emphasis is laid on : (1) methods of producing gratings of various types and (2) the physical properties of gratings. In the first part, various methods are pre-
sented for producing special types of gratings, such as blazed holographic gratings, laminar gratings, toroidal gratings, and transmission gratings. In the second part, problems of stray light, grating effi- ciencies at normal and grazing incidence, cut-off wavelength, polarization, and grating anomalies are discussed and various theoretical consequences are reported. Finally, futuie requirements for gratings in laboratory work and space research are also discussed. The material supplied by the participants has been integrated into groups covering different subjects rather than being presented individually. Thus no panel member is specifically associated with their contribution, although in many cases the association is obvious.
One should remember, in discussing polarization effects and diffraction gratings, that in American terminology the electric vector is perpendicular and parallel to the plane of incidence for S- and p-polari-
zation, respectively. The French school refers these two directions to the grating grooves. Thus for clas- sical grating mountings, radiation polarized parallel to the groove is the parallel component although it is perpendicular to the plane of incidence. In the case of conical diffraction, where the grooves are parallel
C4-170 T. NAMIOKA A N D W. R. HUNTER
to the plane of reference (see Section 4.7), the two conventions are the same.
2. Production of gratings. - 2.1 BLAZED HOLO-
GRAPHIC GRATINGS.
-
2.1.1 Extension of Sheridon's method to concave gratings. - Sheridon [l] described in 1968 a method of producing plane holographic gratings with triangular groove profiles. This method has been extended by Hutley and Johnson [2] to the manufacture of blazed gratings on concave spherical substrates. In this case the photoresist is exposed to spherical standing waves of which the center of curvature lies on the Rowland circle. This may then be considered as a holographic grating constructed from two coincident point sources; one real and one virtual. Since both sources lie on the Rowland circle it follows that the horizontal focal curve will be identical to that of the equivalent ruled grating and the two types are interchangeable. However, the grooves on the holographic grating are curved and therefore the shape of the focused image differs from that of a conventional grating. For example the focus is perfect in autocollimation at the wavelength of manufacture.Judging from spot diagrams it appears that blazed concave holographic gratings are compatible with instruments designed to accept ruled concave gratings and that the quality of the image is not worse, but often much better, than that of a conventionally ruled grating. Their advantage over ruled gratings is that the level of stray light is significantly lower. Their advan- tage over other aberration corrected gratings, except for stigmatic ruled concave gratings produced by Harada et al. [3], is that their efficiency is significantly higher.
2.1 . 2 Use of optical guided waves. - A method has been developed by Cowan and Arakawa [4, 51 of producing asymmetric (blazed) groove profiles on holographic gratings by use of the interference bet- ween an ordinary wave and an optical guided wave.
A layer (400 nm) of photoresist (n = 1.64), acting as both a recording medium and a waveguide, was spin coated onto one face of an equilateral glass prism (n = 1.5) that had been coated with a thin (35 nm) coupling Ag layer. The prism was illuminated with a collimated beam of light (488 nm) from an argon ion laser, as shown in figure l. Part of this beam excited a guided mode (TM,, 463.3 nm) within the photoresist, and part was reflected internally within the prism, passing through the photoresist as an ordinary wave (297.6 nm). Interference of these waves produced, upon development, an asymmetric groove profile (blaze angle = 22.2O, grating constant = 748 nm, see Fig. 2).
The most recent efficiency measurements on a grating produced in this way gave an on- to off-blaze intensity ratio of approximately 6 : 1, with a peak intensity at about 510 nm for p-polarized light.
REFLECTED
PHOTORESIST
TRANSMITTED BEAM
FIG. l . - Method of forming interference between the ordinary wave and the surface wave.
TRANSMITTED BEAM WAVEFRONTS SURFACE MODE
WAVEFRONTS
T
FIG. 2. - Interference fringes resulting from the reflected beam and the surface wave.
The production of a blazed profile could be achieved using a grating coupler in place of the prism coupler described above. Studies of this type are in progress at Oak Ridge National Laboratory.
2.1.3 Use of ion etching. - At the 4th V.U.V. Radiation Physics Conference in 1974 Hutley [6] showed that it was possible to produce blazed holo- graphic gratings whose efficiencies were as high as those of the best ruled gratings and were particularly suitable for wavelengths down to 100 nm.
OPTICAL AND PHYSICAL PROPERTIES OF DIFFRACTION GRATINGS C4-171
gratings may also be used at grazing incidence for much shorter wavelengths and have shown promising results with X-rays down to 0.23 nm.
Since these gratings are etched directly into the glass substrate they are more robust than both replicas and ruled masters. They can withstand high radiation densities, may be cleaned and recoated and can be baked at higher temperatures than other gratings.
2.2 LAMINAR GRATINGS. - Holographically made laminar gratings for the vacuum ultraviolet and soft X-ray region have been developed at the Optical Laboratory of the Gottingen Observatory [7]. The work has been done, in part, in cooperation with the Imperial College, London, and the Deutsches Elek- tronen-Synchrotron (DESY), Hamburg.
A grating mask for the production of phase gratings
can be formed on a glass substrate by recording the interference fringe pattern in photoresist whose thick- ness is less than the grating constant. Coating of a metal layer, e.g. chromium, on this mask and subse- quent stripping produces a grating with a square wave (laminar) profile in metal on glass. Other methods, e.g. combination with etching processes, can also be used in producing a laminar grating. Since the gratings are made in metal on glass without any remaining organic material, the gratings can be heated to meet the requirements for ultrahigh-vacuum operations and are well suited for use with radiation of high intensity, such as from synchrotrons and storage rings.
The manufacturing technique of Rudolph and Schmahl provides an accurate means of controlling the groove profile of laminar gratings, especially of the groove step height, so as to produce phase can- cellation in the zero order. Thus it is possible to set the zero order minimum at the designed position.
Until recently the efficiency of gratings produced in this way is greater than that of most ruled gratings at grazing incidence (see Section 4.3 .3). Experimental
results [8] prove that holographic X-ray gratings have lower stray light levels than ruled ones and show a marked increase in signal-to-noise ratio.
New design possibilities for X-ray gratings arise from the fact that in principle a holographically- formed grating mask can be formed with a variable spacing on any surface capable of intersecting the interference fringe pattern without shadowing. An example of such a grating is given in Section 2.3.
2.3 TOROIDAL GRATINGS. - In contrast to the rather restricted range of possibilities for conven- tionally ruled gratings, holographic gratings may be recorded on substrates of any curvature.
At grazing incidence there may be typically 104 waves of primary astigmatism in Rowland circle mountings of conventional concave gratings [9].
An illustrative calculation is given in table I. This is a very large aberration by optical lens design standards, and, for one element auto-imaging gratings at X-ray wavelengths, the initial correction comes from a geometrical optics consideration of the right choice of substrate curvature.
In practice the A type toric surface, shown in figure 3, is close to the ideal ellipsoid with the same conjugates, for pupil values normally encountered in Rowland circle grazing incidence mountings. Speer and his colleagues at Imperial College (I.C.) have therefore favored this choice of substrate, partly because of the relative ease of manufacture (compared to ellipsoids), and in part because of a particularly simple focal law [9] that applies when the stigmatic wavelength is to be changed (in contrast to the (( B ))
type toric surface in figure 3, where the stigamatic wavelength is fixed and not variable).
In a joint I.C./Gottingen collaboration, gratings have now been produced at 600/mm and 1 200/mm on toric substrates with primary radii of 2 m [l01 and 5 m [l11 and the secondary radii of 5.64 mm and
Example calculation for f/125, 5 Meter radius, Rowland circle grazing incidence mounting
/z=5OA n =
+
l orderlld = 1 200 lineslmm
a = 20
p
= 6.590Distance of entrance slit from grating pole
r = R sin a = 17.45 cm Focal distance r' = R sin
p
= 57.39 cm R = 500 cm xmax = 112 = 0.5 cm ymax = W/2 = 2.0 cm W = 4.0 cm (ruled width) l = 1.0 cm (ruling height)Zero Defocus at r' = R cos
p
z 2 X 104 waves Primary astigmatism
z 2 X 103 waves Astigmatic curvature
Zero Coma
z 2 X 102 waves -
z 3 waves
Zero -
T. NAMIOKA AND W. R. HUNTER
FIG. 3. - Schematic for the two possible toric substrates. cc A D-
H E I G H T S
0 .o 0 . 1 - 0 . 1 0 . 0 0 . 0
type upper. cc B ))-type lower.
I
25 mm, respectively, designed for correction within the wavelength range 1 nm ,( 1 5 10 nm.
Some of the fabrication techniques and X-ray properties of these I.C./Gottingen gratings are des- cribed below.
(a) Shaping of the toroidal substrate. The most recent 5-m toroidal polishing laps were made by turning to the minor radius, a metal bar elastically deformed to the desired concave major radius. After machining this (cylindrical) surface, the bars were released, each springing back to form the convex toroidal lap required. These laps were subsequently optically polished and proved for major radius by optically contacting with the GML5M 5-m radius [l21
concave spherical test sphere along the meridional section (principal plane).
(b) Theoretical performance. The 2-m toroidal
grating performance has been described in detail elsewhere [9, 101. Figure 4 shows the potential level of first order astigmatism correction achievable by the most recent 5-m series, using plane wave construc- tion. This relatiqely simple grating system thus use- fully combines 40 pm of spatial resolution at the object (in the plane orthogonal to the dispersion axis) with 0.002 nm of spectral resolution, with a significant gain in speed over crossed slit systems.
(c) Results to date. Despite the rather poor field imaging characteristics expected on theoretical grounds, the I.C. group has been able to obtain the result shown in figure 5 in initial X-ray imaging tests
0.250
FIG. 4. - Exact computer ray trace for 5 000 mm X 25 mm radii
cc A >)-type toric at 1 200 grooves/mm. The image plane is the Rowland cylinder and the image field is 2 mm long in the dispersion
axis X 0.5 mm. Two 03-axis Object points were displaced
+
1UO pm from a central on-axis object point. 1, = 2.160 nm (1s-2 'P), O,,,,and I, = 2.180 nm (IS-2 3P), OV1,. (All dimensions in mm.)
FIG. 5. - X-ray test image from toroidal reIay mirror. Conjungates as shown in ref. [Ill. Wavelength, 4.4 nm. Q2 plate. Aperture (transverse to chief ray); 2-mm diameter. Diffraction width at
4.4 nm ; 2 pm.
at 44
A
using the substrate alone. This corresponds to the nA = 0 (zero order) condition of grating use, and under these conditions a (angle of grazing inci- dence) = 40 with equal conjugatesThe coarse and fine object grids used were 4/mm and 33/mm, respectively. Surprisingly they were able to resolve at 44
A
the 10-pm thick fine wires in the 33/mm grid easily over a field 200 X 200 pm2, with the prin-cipal limitat~on bang Q2 plate grain.
Experiments to determine the corresponding limit- ing performance in the diffracted X-ray image are now in progress at Tmperial College- with 70 pm X 70 pm
OPTICAL A N D PHYSICAL PROPERTIES OF DIFFRACTION GRATINGS C4- 173
2.4 SOFT X-RAY TRANSMISSION GRATINGS. - The production of transmission gratings at the Space Research Laboratory (SRL), Utrecht [13-151 is essen- tially an application of interferographic techniques. Their experience with X-ray heliography by means of zone plates and the results of spectral investigation with transmission gratings in Skylab (ATM) stimu- lated this production. General use in laboratory V.U.V.-spectrometers was considered to be practi- cable as well [16].
The main advantages of transmission gratings are reasonable production cost, manufacturable by labo- ratory means, no heavy constructional requirements for resolving powers of about 100, wavelength inde- pendent transmission and efficiency (see below howe- ver), free choice of line spacing, wirelslit ratio about 1 (low even orders).
The SRL process of manufacturing transmission gratings comprises seven steps in some 125 mani- pulations : interferographic printing of grating pattern into photoresist on nickel substrate on glass flat (5 X 5 cm2) followed by gold electroplating, contact
printing of two grids for supporting of grating struc- ture and mounting edge, each followed by gold plating, selective etching of the glass flat, cementing of grating elements onto subframes and aligning of elements onto the main mounting frame. The main problems in this process are as follows. (a) Standing waves in the resist layer [ l 7 result in exposed layers parallel to the substrate which peel off during development and cause thin grating wires. The solution of this problem is not yet clear. (b) Reproducibility : laser, photo- resist, electroplating bath and mercury light source all degrade gradually with time. Regular checking, filtering, and light intensity integration are necessary. Over exposure is necessary anyhow. (c) Utmost cleanliness of surfaces is a must : besides clean benches a plasma-etching apparatus is a requirement.
Large area gratings required in combination with X-ray grazing incidence telescopes have to be composed of a substantial number of elements. Consequently the total intensity and resulting reso- lution are determined by the individual elements.
Acoustic damage during rocket launching neces- sitates protection measures. If no damping shield or evacuated canister can be accomodated, a thin plastic cover must be used, reducing the advantage' of an open structure.
Optical properties have been investigated by Dijk- stra et al. [l51 in some detail. They have found methods of aberration correction, especially of coma cor- rection, for space applications by introducing a variable spacing and rotation of the pattern at an appropriate angle.
More details are to be found in references [13-151. 3. Stray light. - 3.1 SOURCES OF STRAY LIGHT IN
GRATINGS 118, 191. - A perfect, infinite plane grating
would diffract light only into those directions defined
by the grating equation. All light detected in any other position will confuse the spectrum (or the spectroscopist) and may be regarded as stray light. Even a perfect grating of finite size will diffract some light away from the main orders through Fraunhoffer diffraction of the aperture but the level of such light may be reduced by suitable apodization (e.g. by underfilling the grating aperture) at some cost to the resolution.
Ghosts and grass arise respectively from periodic and random variations on the grating surface and particularly, in ruled gratings, from mechanical errors in the positioning of the grooves. Both can be completely eliminated in holographic gratings although ghosts can be generated if proper care is not taken during manufacture.
Diffuse scattering arises from roughness of the grating surface and is common not only to both types of gratings but also to other optical elements such as mirrors. Light can also be scattered through interaction with surface plasmons on the grating surface. This may only be of academic interest but it can clearly be seen when the grating is illuminated with a laser.
Each type of stray light has a different spatial distribution (and probably a different wavelength dependence). A ghost forms a spurious spectral image, grass is restricted to the plane of dispersion and is seen as a thin line joining the orders. Diffuse and surface plasmon scattering, on the other hand, send light out of the plane of dispersion. Because of this variation of angular dependence, the proportion of stray light relative to the diffracted order that is detected will depend upon the geometry and parti- cularly the slit dimensions, of the instrument. The relative importance of the different forms of stray light also depend upon whether it is the total flux that is measured (photoelectric instruments) or its intensity (photographic measurements). In view of this it is virtually impossible to perform unambiguous measu- rements of stray light and difficult even to compare different gratings except in the instrument in which they are to be used.
3.2 EVALUATION OF STRAY LIGHT. - Quantitative measurements of stray light levels are not easily made, especially as a function of wavelength. However, a number of qualitative and semi-quantitative methods are available for evaluating the stray light characte- ristics of gratings.
C4- 174 T. NAMIOKA AND W. R. HUNTER
common method is to record, photoelectrically, the profile of a laser line diffracted by the test grating or to compare such a profile with that recorded with a high-quality mirror, both measured under exactly the same conditions. A high dynamic range of 107 is required in this case. Flamand and his colleagues at Jobin Yvon were able to find a certain correlation between stray light of mirrors, by comparing their line profile measurements with soft X-ray scattering measurements on the same mirrors done by de Korte and Laine [20] of Huygens Laboratory, Leiden University. A practical stray light test is to measure the spectrum of an emission line source by mounting the test grating in a monochromator or a spectrograph
iff which it is to be used [21]. In this case the useful to stray light ratio is the ratio of line intensity to the intensity measured between the lines. An example is shown in figure 6. The figure shows the He spectrum
FIG. 6. - Helium spectrum recorded with a toroidal holographic grating.
recorded with a toroidal holographic grating in a Lepere V.U.V. monochromator [22]. If this type of measurement is extended to the vicinity of the zero order, the amount and the rate of increase in signal gives a very good idea of the quality of the grating as far as stray light is concerned. An example of such a measurement can be seen in figure 7 of reference [23]. At Imperial College, stray light is measured routi- nely but is difficult to quantify at X-ray wavelengths [24]. In practice comparisons between ruled and holographic gratings, for example, are only useful provided identical grating radii and recording geome- try are used. Generally the holographic gratings they have examined have much lower scattered light at soft X-ray wavelengths than their mechanically ruled counterparts. This is most probably due to the tech- niques used for producing holographic gratings, which result in a generally smoother profile.
4. Grating efficiencies. - 4.1 PROBLEMS ASSOCIATED
WITH EFFICIENCY MEASUREMENTS. - Efficiency measu- rements have shown that a grating usually does not have a uniform efficiency over its surface [23], and if it is used in a monochromator to disperse the radiation, the beam emerging from the monochromator will not
have a uniform intensity over its cross section. If this beam is used to measure the efficiency of a test grating, an unfolding process must be used to separate the non-uniformities of the two gratings. A simpler
procedure is to reduce the cross section of the beam so that only a small portion of the test grating is illuminated. Under these conditions, beam non- uniformities are not important and an efficiency map of a grating surface can be obtained by moving the grating through the beam while recording the inten- sity of a given order. A device for this purpose has been described [25] and is presently in use at the U.S. Naval Research Laboratory (NRL) to measure grating efficiencies in the spectral region from 2 000
A
to about 160A.
Recently there has been a mounting interest in gratings used at grazing incidence brought about in part by an increased emphasis on solar spectroscopy at wavelengths well below 100 nm. Very few quanti- tative measurements of grating efficiencies have been made at grazing incidence from 160 i%to longer
wavelengths, consequently one knows only vaguely what to expect from conventional gratings, and can only guess at the properties of holographic gratings at high angles of incidence. A series of measurements are underway at NRL to determine the properties of gratings at grazing incidence [26]. A number of interesting results are arising from these measure- ments, not the least of which are the problems encoun- tered in measuring concave grating efficiencies at grazing incidence due to aberrations [27].
The geometry of the NRL device does not conform to any conventional grating mounting, and resembles a reflectometer with a detector capable of rotating around the grating at a fixed distance to measure the
FIG. 7. - Results of ray tracings showing the image height (top) and width (bottom) in cm for a 1 m radius grating at 800 angle of incidence. The numbers to the right of each curve give the j-number
OPTICAL A N D PHYSICAL PROPERTIES O F D I F F R A C T I O N G R A T I N G S C4- 175
different orders. The main effect of aberrations in this geometry is a widening of the diffracted beam in the direction of dispersion, an effect investigated by ray tracing.
Figure 7 shows the results of the ray tracing for a l-m radius of curvature grating at 800 angle of inci- dence and for a detector distance of 26.6 cm. The abscissa is the dimensionless produce ngA, where n is the order number, g is the groove density in grooves/ cm, and A is the wavelength in cm. The upper part of the figure shows that the height of the beam changes very little with ngL, but the bottom part shows that the width can change drastically. Each part contains five curves corresponding to the f-numbers 300, 600,
1 000, 104, and cc (parallel radiation), of the illumi- nating beam. The curves for f-104 and cc coincide in the upper part of the figure.
The width of the beam is large at small and large ngA and reaches a minimum at that angle of diffraction where the horizontal focus occurs. As the angle of incidence increases, the width of the beam at zeroth order usually increases and, at the horizontal focus, reaches an even deeper minimum. With the geometry of the NRL device, the effect is greatest for small radius gratings and usually becomes less as the radius increases.
Figure 8 shows the diffracted beam,widths in terms of ngA for grating radii of 0.5 m (solid lines), 1 m
n'g'
X
FIG. 8. - Diffracted image widths in terms of ng3, for grating radii of 0.5 m
-.
1.0 m ---.
2.0 m and 3.0 m . . . The shaded area represents the detector width. The image width is read from the vertical scale : i.e.. for ngi. = 0.01. the image width for a 0.5 m radius grating is 0.002 9 5 n'q'i' 0.016 2. The horl- zontal arrows show the limit of order overlappins. The left arrowindicates the largest ngi. value for which overlap O C C L I ~ S .
(dashed lines), 2 m (solid lines with circles), and 3 m (dotted lines) for an f-300 illuminating beam. The shaded strip represents the equivalent width of the detector used with the NRL device. This figure also shows that overlapping of diffracted beams occurs if the horizontal distance between the lines defining the beam widths spans less than an octave in ngA.
4.2 HOLOGRAPHIC GRATINGS. - The grating team
at Jobin Yvon has found experimentally that maxi- mum relative efficiencies of about 30
%
in the first order could be obtained at wavelengths below 100 nm for holographic gratings having an optical modu- lation depth of about one quarter of the blaze wave- length, 1, and that the experimental result agreed quite well with the value of 33%
calculated from a simple scalar theory (Bessel function) for a grating with sinusoidal grooves whose depth was less than 0.3&.
Rigorous electromagnetic theories developed by Petit and his colleagues [28] confirmed these findings for near normal incidence. Therefore, for- mation of shallow grooves, having the depth of several nm to several tens nm, is required for a holo- graphic grating to have an optimum efficiency in the V.U.V. region.Efficiency measuring devices at Jobin Yvon and MATRA, complete with test chambers, V.U.V. light sources, and monochromators [29], permit absolute efficiency measurements down to 25.6 nm. The signal-to-noise ratio obtainable with the Jobin Yvon device will be improved by a factor of 10 to 20 in the V.U.V. region, especially at wavelengths below 58.4 nm, by the use of a new toroidal holographic grating monochromator 1221. The light source pre- sently in use is of the Damany type, low-pressure discharge in gas with magnetic confinement, but use of a sliding spark source will extend the capability of the device down to 15 nm. A joint J-Y/LURE (Uni- versity of Paris-Sud) collaboration has made it possible to evaluate efficiencies of ruled and holo- graphic concave gratings down to 4 nm [30] (see also Section 4.3.2).
4.3 COMPARISON BETWEEN RULED AND HOLOGRA-
PHIC GRATINGS. - 4.3.1 Normal incidence. - Effi-
C4- 176 T. N A M I O K A A N D W. R. HUNTER
conventional grating, if properly blazed, has higher efficiency than the holographic grating. Some of the results obtained have been published for gratings used in normal incidence [23, 311.
4.3.2 Grazing incidence (XUV gratings). - Infor- mation on the efficiency and diffuse stray light are still relatively scarce for gratings used at grazing incidence. In order to determine optimum conditions of a spectromonochromator for XUV users at LURE,
Dhez and 'his colleagues have tested several gratings, all coated with platinum, using their test system [32] and a channeltron detector. The gratings tested were four Bausch and Lomb replica gratings (two identical ones with 576 grooves/mm groove density and 10 blaze angle ; one with 1 200 grooveslmm and 10 blaze ; and one with 2 400 grooves/mm and 20 blaze) and five Jobin Yvon holographic gratings having pseudo- sinusoidal profiles ; three with 1 200 grooves/mm and two with 2 400 grooves/mm ; each one with a different depth modulation rate. With the test system the angle of grazing incidence could be varied between 4" and 200 and a single or double grating mode could be chosen. Their aim was to evaluate the gratings and the incidence adapted to each experiment at LURE. The use of synchrotron radiation permitted acquisi- tion of data continuously over an extended wave- length range. Curves given below are corrected for the ACO spectral intensity and normalized.
Similar curves to that in figure 9 have been obtained
for all the ruled gratings and a cut-off has been observed at increasingly high wavelengths as the grazing incidence angle was increased. Wavelengths corresponding to Symmetrical Reff ection on the Facet, i.e. so-called blaze wavelength, are indicated in the figure by black points. For the 2 400 grooves/mm gratings the theoretical blaze wavelength lies before the rise in the curve. For other gratings it lies on the rise or at the top but keeps the same relative position regardless of changes in
v.
Curves for the two suppo- sedly identical 576 grooves/mm gratings showed them to be not identical ; one grating showed an extra bump at the position corresponding to the blaze wavelength before the main rise. Information thus obtained confirms the necessity of testing each grating before attempting any intensity measurement. Usual for- mulas for blaze and cut-off wavelengths give only a rough estimate.Figure 10 compares the intensities in the orders
+
1 and - 1 obtained with the 1 200 grooves/rnrn grating that was placed so as to give blaze on the positive order. All of the ruled gratings gave about the same intensity ratio. That the ratio was inverted as the blaze direction was reversed demonstrates the blaze effect of the gratings. This fact implies that these ruled gratings can be used in either order with a good efficiency merely by reversing the blaze direction.FIG. 9. - Variation in cut-off wavelength vs. grazing incidence angle v .
I I l I
0 100 200
x
( A )
FIG. 10. - Comparison between the intensities in the positive and negative first orders. The grating was placed so as to give blaze
OPTICAL AND PHYSICAL PROPERTIES OF DIFFRACTION GRATINGS
The same type of measurement has been performed with the holographic gratings. A variation in the cut- off wavelength with grazing incidence angle was also observed but changes took place more slowly as compared to the case of the ruled gratings. This may be due to the difference in the groove profiles of the two types of gratings. The ratio between the peak intensity and the noise intensity before the rise was about four times larger with the ruled gratings than with the holographic gratings. One of the 1 200 groo- ves/mm holographic gratings showed a high intensity ratio between the positive and negative orders that was equivalent to the value obtained with the ruled gratings, demonstrating the possibility of having a blaze effect with a pseudo-sinusoidal profile, like that of a phase grating suitable to measurements in She X-ray region [33].
The results thus obtained are summarized below. Characteristics of gratings at grazing incidence are peculiar to each one, requiring thorough tests before making any intensity measurement. The observed behavior of cut-off wavelength vs. angle of incidence indicates that an angle of grazing incidence as high as 200 cannot be used to avoid order overlapping for wavelengths up to 20 nm. A good signal-to-noise
ratio observed with some of the holographic gratings and the possibility of having a blaze effect with a pseudo-sinusoidal groove profile seem to indicate that holographic gratings can be used also in XUV experiments where low noise is the main factor.
Measurements made at NRL [26] show that at grazing incidence holographic gratings have a quite uniform efficiency vs. wavelength characteristic and show a large number of orders whose efficiency decreases with increasing order number, as at normal incidence. Conventional gratings have an efficiency maximum at the blaze wavelength and show very few orders. Efficiency values are lower than expected. At 800 angle of incidence and at 30 nm, the maximum efficiency measured at NRL as of this writing is 30
%,
from a conventional grating of 3-m radius of curva- ture and 400 grooves/mm. Most gratings measured had efficiencies on the order of 10%
or lower.4 . 3 . 3 Grazing incidence (X-ray gratings). -' Since the 4th V.U.V. Radiation Physics Conference in 1974, the newer holographic gratings have continued to improve their diffraction performance to even shorter wavelengths relative to conventionally ruled types. This point is illustrated in figure 11, which is based on the data obtained at Imperial College (I.C.) in collaboration with University of Gottingen. In this diagram are plotted the average and highest values recorded for the peak 4.4 nm first order diffraction efficiency, against year of manufacture. The data are restricted to the two most common groove densi- ties, namely 600 grooves/mm and 1 200 grooves/mm. The total number of samples is about 25.
Figure 12 shows the same parameter plotted for all
FIG 1 1. - Progress In the l.C /Gott~ngen holographic grazing ~ncidence grating development program, as measured by average (square histogram) and highest (star) values recorded for first order
peak 4.4 nm efficiency.
.
All gmtihjs tested up b July ,977Ftrst order efficiency vs line frequency
FIG. 12. - First order efficiency vs. groove density for all gratings (ruled and holographic) examined up to July 1977.
useable gratings examined at I.C. up to July 1977, but now arranged according to groove density. The unweighted averages of these data, now separated according to whether ruled or holographic yields the interesting result shown in figure 13. Thus, in broad generalization, holographic and ruled are seen to be approximately comparable in first order X-ray dif- fraction performance at this stage in their develop- ment. This is a remarkable achievement for the holographic technique considering that the develop- ment of this type of X-ray reflection grating only seriously got under way in 1973.
T. NAMIOKA AND W. R. HUNTER
FIG. 13. - Averages of data shown in figure 12, but separated according to c( ruled D or (( holographic n.
E % l0 9 8 7 6 S 4 3 2 I
factor of 10. Thus, on the average the luminosity- resolution product is tending to stay about constant. The loss in speed with increasing groove density must have its origins in the progressive loss of control over the ideal groove profile as the groove density increases, and is a difficulty that is common to all types of gratings.
Current research activity of the joint I.C./Gottingen collaboration in this areas is directed towards improv- ing efficiency at higher groove densities (e.g. 1 200 grooves/rnm and 2 400 grooves/mm), down to the practical short wavelength limits of grazing incidence mountings, say 0.5 nm.
Figure 14 summarizes the data of I.C. up to Sep- tember 1977, where the selection is for the best gratings in each groove density group. The grating origin is given in table 11, together with catalogue number where this is available. Very recently Speer and his colleagues have been able to examine a series of remarkable concave shallow blazed original rulings of I-m radius and 1 200 grooves/mm ruled by B. Bach of Hyperfine Inc., Fairport, New York. At the time of this writing these gratings are among the fastest
+
ho~ogrqphlc they have seen at this particular groove density from* r e ruled any source.
Number o f results svcroged
given for -<h p,nt Because replica series-stability is of considerable
.l0
interest to users as well as manufacturers, in figure 15b are reproduced the only X-ray data available of this
*
16type known to date. Six widely spaced replicas were compared in this experiment, undertaken by I.C. collaboratively with Bausch Lomb. Despite the inhe- rent difficulties they were able to establish that
* I first order performance was maintained to within a
factor of two along the series. Zero order reflectivity
.l 4; remained constant to within much closer limits as
ANGLE OF GRAZING INCIDENCE
300 600 /ines/mm IZOO 1800 2400 shown in figure 15a.
FIG. 15. - 0 and
+
1 order efficiencies vs. ' grazing angle for 6 widely spaced replicas off the same master ruling [34] (Bausch& Lomb 2517, 1 200 grooves/mm, 2 m to fit Hilger E580 spectro-
graph). Coating indicated Pt or Au. Measurement wavelength 4.4 nm. The original ruling was convex and 2-m radlus, Bausch
& Lomb catalogue 35-52-40-700. Imper~al College data reproduced
by permission from Bausch & Lomb.
4.4 TRANSMISSION GRATINGS. - A preliminary efficiency test was performed with a 1 000 grooves/mm grating, and is shown in figure 16. The collimation set-up is shown schematically also. The results looked quite promising.
FIG. 14. - Summary of the best results from the I.C. grazing incidence grating research and development program [9], arranged according to groove density. Ordinate : efficiency in the
+
1st order at 4.4 nm. Abscissa : effective angle of grazing incidence. See table I1 for typesOPTICAL A N D PHYSICAL PROPERTIES OF DIFFRACTION GRATINGS
Reference, origin and design for the 22 gratings yieldingpeak values oj'the parameter E , in figure 4 Code (Fig. 14) - 17 Radius 2 M Plane 2 M 2 M 2 M 2 M 2 M 5 M Plane 2 M 2 M 2 M 1 M 5 M 5 M 5 M Plane Plane 5 M 2 M 2 M Plane Origin B & L J.Y. B & L B & L B & L B & L B & L N.P.L. GottingenIIC GottingenIIC GottingenIIC GottingenIIC B & L GottingenIIC GottingenIIC GottingenIIC N.P.L. N.P.L. N.P.L. B & L B & L N.P.L. Catalogue No. 35-52-37-800 35-52-40-700 35-52-40-700 35-52-40-400 35-62-41 -800 35-52-41 -800 -
Recently efficiency tests have been performed at the Stanford Synchrotron Radiation Project. Radia- tion emerging from a monochromator is diffracted through the grating and detected in a channel multi- plier array. The results are plotted in figure 17 in comparison with the calculated efficiency of the ATM- combination (Skylab). This test showed anomalies at 8 nm as a consequence of transmission through the gold wires.
Serial No. - 2553-6- 1-1 Shallow Blazed. Ruled. Holographic. Shallow Blazed. Ruled. Pt. Shallow Blazed. Ruled. Pt. Shallow Blazed. Ruled. Pt. Shallow Blazed. Ruled. Pt. Shallow Blazed. Ruled. Au. Laminar. Ruled. Au. Laminar. Holographic. Au. Laminar. Holographic. Au. Laminar. Holographic. Au. Laminar. ~ o l o ~ r a ~ h i c . Au. Shallow Blazed. Ruled. Au. Laminar. Holographic. Au. Laminar. Holographic. Au. Laminar. Holographic. Au. Shallow Blaze. Holographic. Au. Shallow Blaze. Holographic. Au. Laminar. Ruled. Original. Au. Shallow Blazed. Ruled. Pt. Shallow Blazed. Ruled. Pt. Laminar. Ruled. Original. Au. Further X-ray tests will be executed'by the Space Research Laboratory, Utrecht, on behalf of the HEAO-B and EXOSAT projects during 1977. Long beam equipment will be used (X-ray source, vacuum tube 300 m and 70 m respectively, vacuum tank with telescope, grating and detector).
C4- 180 T. NAMIOKA AND W. R. HUNTER 1000 I P / r n r n G R A T l N C
[qr;;--
C ;
- +
1
, - , -P R O -P ILS. >L,. ~ L . = . :oi.*>ta-
.-
-1. X , i 8 l j 5%!
.;
8 , I . 3;
1 1
;
(1,
"" m b e c , G;
.: C." L... ..C...'
. . .
...
:a......
...FIG. 16. - Efficiency of a 1 000 grooves/mm transmission grating. The inset gives a scheme of the coarse mechanical collimator.
FIG. 17. - Calculated efficiency of ATM-Skylab-telescope-grating and measured efficiency of single grating element for HEAO-B
(1 440 grooves/mm and l 000 grooves/mm, respectively). Note the constructive interference by phase-shift through gold wires.
the grating efficiencies for the p- and S-components of incident light, the grating in a monochromator introduces some polarization into the diffracted light beam, the amount depending on the geometry of the system and on the type of grating (here, p and S stand
for light polarized with the electric vector parallel and perpendicular to the plane of incidence at the grating, respectively). A knowledge of this polari-
zation is important, and since it cannot, at this time, be calculated theoretically it is necessary to measure directly the polarization introduced by the grating. The polarization produced by a 2.2-m grazing- incidence (820 from the grating normal) scanning monochromator has been measured by Arakawa and Williams [35] using a calibrated triple reflection polarizer as the analyzer. A condensed spark dis-
charge source was located at the movable slit while the analyzer was attached to the stationary slit.
Results have been obtained over the spectral range from 20 to 160 nm for three separate gratings over- coated with gold and two aluminized gratings in the monochromator. In all cases the polarizaf on (P = R,/ R,, where R, and R, are the reflectances from the grating for p- and S-light) was greater than 0.8 for wavelengths less than 40 nm and approached unity below 20 nm. The polarization for the gold-coated gratings did not change appreciably during several years of use whereas that for the aluminized gratings
was altered drastically, presumably due to conta- mination by polymerized pump oil [35].
Hunter and his colleagues made measurements at 121.6 nm and at incidence angles up to 870 on the polarization produced by Jobin Yvon 2 400 grooves/ mm plane holographic gratings that were coated, at Goddard Space Flight Center, with A1
+
MgF,. They are not sure the 87O measurements mean too much although the data fit in with those at smaller angles. They found that the polarization, R,/R,, increased to five at 87O, although the polarization to be expected from a mirror coated with the same coating was the opposite, R,/R, = 2.5 (or R,/R, = 0.4). 4.6 THEORETICAL CONSEQUENCES. - Electroma- gnetic theories of gratings [36] have been developed by Petit and his colleagues to determine the efficiencies of gratings when the grating constant d is of the same order as the wavelength 2. For perfectly conducting gratings it appears that the use of such theories is generally not necessary if I / d is less than 0.25. This conclusion is wrong when gratings are used in the i vacuum ultraviolet because the assumption of infinite conductivity is of course absurd ; even for a plane mirror we have to take into account the polarization as soon as we depart from the normal incidence. In other words, in the vacuum ultraviolet, polarization effects may be substantial even though the ratio p = /Z/d is quite small. Unfortunately, using electro- magnetic theory, the computation time is roughly inversely proportional to p 3 , therefore numerical studies in the V.U.V. are much more expensive than in the visible region and a grant of the C.N.E.S. (Centre National d'Etudes Spatiales) has been neces- sary for Petit et al. to perform a systematic exploita- tion of their computer programs in the V.U.V. The results have been presented in their final report [37]. From this report, and some subsequent work done by Petit et al., may be inferred the following general rules for plane gratings.(a) In the V.U.V. region, for high efficiencies the echelette grating is a better performer than the holo- graphic grating [37-411.
(b) To obtain the best efficiency the echelette grating must be used close to a certain mounting that Petit and his colleagues RWF (Reflected Wave on the Facet) or, in French, ORF (Onde Rtfltchie sur la Facette). For this mounting [37, 381, the incident and diffracted beams are symmetrical with respect to the plane of the larger facet of the grooves (see Fig. 18). This condition is equivalent to using the grating at the blaze wavelength. The efficiency can be predicted by a very simple formula within a relative accuracy of the order of 10
%.
OPTICAL A N D PHYSICAL PROPERTIES OF DIFFRACTION GRATINGS C4-181
X FIG. 18. -The R.W.F. mounting. The incident unit wave vector U
and fhe diffracted unit wave vector U-, are perpendicular to the
grooves and symmetrical with respect to the normal on the large facet. This is equivalent to the blaze condition. 0 and 0' are, respec- tively, the angle of incidence and the angle of diffraction for the
- l order.
( d ) In many practical mountings, and especially
near grazing incidence, we have to take into account the polarization of the incident light 1371.
(e) Anomalies of all kinds (Rayleigh anomalies, resonance or surface plasmon anomalies) do not seem very important especially when using an RWF mounting even if the angle of incidence is as high as 750.
( f ) It is not certain that the classical mountings used in spectroscopy are the most efficient ones, i.e., diffract the maximum amount of radiant energy into the desired spectral orders. A numerical study has
shown that the utilization of conical diffraction results in better efficiencies. This prediction has been recently confirmed by experiments performed in collaboration with the U.S. Naval Research Laboratory [42] (see also Section 4.7).
( g ) Finally, people who want to work in the Littrow
or near Littrow mounting to angles as large as those encountered in the Seya-Namioka mounting will find useful information in references [40, 41, 431.
4.7 CONICAL DIFFRACTION MOUNTING. - In the classical RWF diffraction mounting, illustrated in figure 18, the incident wave vector, U, and the normal
to the grating, y, define a reference plane which also contains A, the normal to the large facet of the grating. The grating rulings are perpendicular to this plane where they intersect the plane. This reference plane coincides with the plane of the Rowland circle for a concave grating used in the classical RWF diffraction mounting. If the grating rulings of a plane grating are not perpendicular to the reference plane, the different orders are diffracted onto a cone, hence the designa- tion, conical diffraction [28]. As was mentioned in section 4.6, the use of plane gratings in conical
Marseille (Faculte des Sciences de St-JCr6me). Start- ing from the classical RWF diffraction mounting, if one rotates the blazed grating around A , then the grating normal no longer lies in the reference plane, however, the minus first order, the incident beam, and A remain in the reference plane. Otherwise it would not have been possible to make any efficiency measurements with the NRL instrument because its detector is restricted to motion in a plane around the grating.
Special attention was devoted to the particular case in which the plane of the small facet is parallel to the reference plane, and therefore parallel to the incident wave vector, as shown in figure 19. This is equivalent
FIG. 19. - Grating configuration with the incident wave vector k parallel t o the small facet. The wavelength L and angle cp are varied in such a way that I = 2 d sin a cos cp, where dis the groove spacing.
A is the normal to the large facet.
to measuring the grating with the large facet perpen- dicular to the reference plane. For short wavelengths, less than 30 nm, the experiment shows that the effi- ciencies are larger, sometimes by a factor of three, as compared to those measured using the classical diffraction mounting, where the grooves are perpen- dicular to the reference plane.
It should be noted that when the plane of the large facet is perpendicular to the reference plane, the angles of incidence and diffraction of the minus first order, as projected onto the reference plane are equal. This property permits a very simple mechanical design for a scanning monochromator.
Such efficiency enhancements have also been observ- ed in other configurations where the grooves are neither parallel nor perpendicular to the reference plane. Unfortunately, the aberrations arising when concave gratings are used in the conical mounting have so far defied attempts to design a useful focussing monochromator. Possibly plane gratings could be used in connection with highly collimated sources, such as synchrotrons or storage rings, to make useful, highly efficient predispersers.
diffraction has been investigated experimentally at
C4- 1 82 T. NAMIOKA AND W. R. HUNTER over a comparatively narrow wavelength range.
Grating anomalies were first observed experimentally by Wood [44] in 1902 and became known as Wood's anomalies because no explanation was available using the grating theory of that time. The strength of an anomaly and where it occurs in the spectrum depend upon the groove profile of the grating, the nature of the surface of the grating and the design of the instru- ment in which it is used. Anomalies have the effect of introducing spurious features into the spectrum and it is therefore most important that they should be measured before an instrument is used. Just as ghosts from ruled gratings can be mistaken for emission lines, so can anomalies be mistaken for absorption bands, narrow emission continua, or absorption edges. It is therefore highly desirable for many applications to have accurate knowledge of anomalies.
5.1 POSITION OF A N ANOMALY. - In this section Hutley's account of anomalies and his way of calcu- lating the position of an anomaly are given.
Anomalies occur when the incident radiation causes some mode of electromagnetic vibration to be set up in the surface of the grating. The simplest example of this is a diffracted order propagating along the grat- ing surface. In this case the anomaly can be explained in terms of the redistribution of light as an order
passes offover the horizon and its position is determined by the angle of incidence and the grating constant. Other forms of surface mode are surface plasmons which are longitudinal electromagnetic vibrations in the free electron gas of a metal surface, and guided waves which may propagate in a dielectric layer on a metal surface. The positions of these depend upon the nature of the grating surface and the groove profile in addition to the geometry of diffraction.
It is possible to calculate very simply a first approxi- mation to the position of an anomaly. In doing so it is convenient to bear in mind that the function of a grating is to change the momentum of a photon, and it does this by adding a quantum of momentum of hld to the component of the photon momentum in the plane of the grating (d is the grating constant). An anomaly can occur whenever a linear combination of the photon momentum (resolved in the plane of the grating) and the momentum of the grating can be made to match the momentum of the surface mode. In the case of a Rayleigh type of anomaly, the surface mode is simply an order diffracted along the grating surface, and the condition for matching of momenta is nothing more than the application of the grating equation with an angle of diffraction of 900. If the mode is a surface plasmon then its momentum is a function of the optical properties of the surface of the grating and is equal to
where E ~ ( w ) = n2 - k2 and n and k are the real and
imaginary parts of the refractive index. The magnitude of this momentum corresponds, for most metals, to that of a vacuum ultraviolet photon. However, because the photon momentum may be augmented by that of the grating, coupling may be induced between even visible photons and surface plasmons.
In the case of guided waves in a dielectric layer the momentum of the mode is
where t is the thickness of the layer and 8 is given by
Here, m is the mode number, n is the refractive index,
and c p , , and cp,, are phase changes on reflection at
the surfaces of the layer. In fact, cp, and cp, cannot be calculated directly-as they depend on 6.' For such modes an iterative calculation is necessary.
In practice the grating may well perturb the propa- gation of the mode so the calculation of the precise shape and position of anomalies requires a detailed knowledge of the groove profile and the optical properties of the surface. However, by matching the momenta of the incident photon, the grating and the surface mode, it is possible to estimate the position of an anomaly with sufficient accuracy to indicate whether a given feature of a spectrum is real or due to an anomaly.
5.2 GRATING ANOMALIES AND SURFACE WAVES. - Grating anomalies have been studied by many workers over the years but the explanation of these anomalies in terms of surface plasmon and optical guided wave resonances is relatively recent. Over the past ten years Arakawa and his colleagues at Oak Ridge National Laboratory have studied grating anomalies using both conventionally ruled reflection gratings [45-471 and holographic reflection gratings [48]. Agreement between theory and experiment has been excellent, both with and without dielectric overlayers on the gratings. In addition, the results of these studies have enabled them to explain changes in the polarization spectra produced by gratings which have been observed during actual usage in a monochromator over an extended period of time [49]. The following is the account of grating anomalies given by Arakawa and his colleagues.
Polarization anomalies may be studied in a variety of ways. Figure 20 shows the p-polarized (electric vector parallel to the plane of incidence) spectra obtained when light from a tungsten lamp was dif- fracted by a Bausch and Lomb replica concave grating for various angles between the entrance and exit slits of the monochromator [45]. Figure 21, also obtained
OPTICAL AND PHYSICAL PROPERTIES O F DIFFRACTION GRATINGS C4- 1 83
I
WAVELENGTH (A')
FIG. 20. - p-polarized spectra of tungsten lamp diffracted by concave grating for .varying angles between entrance and exist
slits.
l I l I I
8000 7000 6000 5000 4000
x (H,
FIG. 21. - First order (on-blaze) diffracted spectra of AI substrate coated with varying thicknesses of A1,0,.
Al,O, overcoating the aluminized grating used in the where Seya-Namioka geometry. The most concise way to
describe these polarization anomalies is through the D = Q(K,) = Q() ( o l c ) sin a
+
( 2 m / 6 )1)
dispersion relations of the non-radiative surface The angle of incidence a is given as a function of angle plasmon resonance' which are for the 8 the half-angle between the source and detector, anomalies. It can be shown l451 that the probabilityiy
the grating equation for the photon-surface plasmon-photon interaction
has the typical Lorentzian form, a = 0
+
sin- l (nn' c / 6 o cos 8 ) , ( 2 )In this expression
K , = (o/c) sin a
+
(2 7 4 6 )is the propagation vector of the surface plasmon and expresses momentum conservation at the grating surface. a is the angle of incidence measured from the normal to the grating surface and 6 is the grating constant. o = c
I
kI
is the incident photon frequency, Q(K,) and y ( ~ ; ) are the frequency and damping rate, respectively, of the surface plasmon, and n is a positive or negative integer. The condition for an intermediate state resonance is that the energy of the virtual sur- face plasmon should be equal to the energy of the incident photon, i.e.,o = SZ (1)
where n' is the ordel: of the diffracted light. Using this in the equation for K,, one obtains the surface plas-
mon dispersion relation n
K , =
-
( 2 n+
n')+
6In this equation 6, 8, and n' are known, and o can be calculated from the measured wavelength of an anomalous peak. From these parameters the second term in eq. (3) is obtained. The first term in eq. (3) is found by substituting the proper value of the integer n. Initially this is done by assuming a positive or
C4-184 T. NAMIOKA A N D W. R. HUNTER
to be very close to the light line, o = c
I
kI.
Each n value corresponds to a branch of the experimental data, and once an n for a given branch is found, theothers follow in sequence. Figure 22 illustrates the principles involved. The experimental points were obtained [45] from data such as that in figure 20, while the dashed curves give the theoretical dispersion curves for surface plasmons on a smooth surface with o = SZ in
FIG. 22. - (a) Feynman diagram of photon-surface-plasmon inter- action. (b) Dispersion curves of surface plasmons in AI and Au.
where eI(o) is the real part of the dielectric constant of the metal of the grating surface. If
I
&,(m)I
+ CO,eq. (4) reduces to o = c I k
l,
the expression for the light line, and also the dispersion relation for Rayleigh anomalies. Rayleigh's equations for the anomalies did not take into account the finite value of &,(m) for the metal surface of the grating. Examination of figure 22 shows that this approximation applies in the visible for AI but breaks down in the vacuum ultra- violet.It is seen in figure 22 that up to the limit of the experimental data, theory and experiment agree well for Au. For energies larger than
-
2 eV in Al, the experimental points lie below the theoretical curve defined by eq. (4). This deviation can be shown to be due to a thin layer of A1203 on the Al. For relatively thin dielectric layers on the grating surface the classical (retarded) surface plasmon dispersion relation is obtained as a solution of Maxwell's equations applied to a semi-infinite metal slab bounded by a dielectric layer of finite thickness. The result [46] isK, ,v]K, tanh K, z
+
K,& =
- v -
K, ylc, -t K, tanh K, z (5)
where
E = &(m) = dielectric constant of the metal = E ,
+
i ~ , ,V] = v](o) = dielectric constant of the dielectric layer = V],
+
iv,, and z = dielectric layer thickness. Figure 23 illustrates typical dispersion curve resultsFIG. 23. - Dispersion curves of surface plasmons in (1) AI grating coated with the natural A1,0, layer and (2) AI coated with
35 nm MgF,. Dashed lines, theoretical curves.
involving thin dielectric overlayers. Similar agree- ment between the experimental and theoretical curves has been obtained for the dispersion of surface plas- mons in multiple metal and dielectric layers on concave diffraction gratings [47]. A treatment of A1,0, and pump oil layers on aluminized gratings showed that both the Al,03 and the oil contributed to the pola- rization anomalies observed, but that changes of polarization with time are due to progressive conta- mination by pump oil with use of the grating in the monochromator [46, 47, 491. Drastic changes have been observed in the polarization and efficiencies as the gratings become contaminated with thin layers of polymerized silicone diffusion pump oil, and the Wood's anomalies can be used to estimate the thick- ness of this contaminant layer.
OPTICAL AND PHYSICAL PROPERTIES OF DIFFRACTION GRATINGS C4-185
orders) only a single low order mode was excited. layer of photoresist 410 nm thick, the results shown As expected the anomaly existed only for p-polari- in figure 25 were obtained. In addition to a surface zation. Furthermore, while a ruled grating often plasmon mode, resonances now occur corresponding exhibits peaks in the on-blaze spectra, and dips off- to the generation of optical guided modes in the blaze, only a dip was observed with the non-blazed photoresist layer. The allowed modes in this asym- grating studied here. After coating the grating with a metric wave guide are described by
q 2 - &(l) 112 2
-
&(2) 112(Dwlc) ( E ~ - q2)112 - tan-
[&,
- q2]
- tanp1 F:[ v q2]
= ma (S modes)(Dolc) (eF - q2)112 - tan-'
FIG. 24. - First order p- and S-polarized diffraction spectrum for an aluminum coated grating made holographically. The dip in the p-polarized spectrum is due to the TM, mode (surface plasmon).
FIG. 25. - Same spectrum as figure 24 but with the grating coated with a 410 nm layer of photoresist. The mode numbers corres- ponding to the dips are indicated in the figure. Subsidary structure in p-polarization is due to weak TM ,(n = 1) and TM
,
(n = 2) modes.where D is the thickness of the photoresist layer and E, is its dielectric constant, E(') and E'') are the dielectric
constants of the adjacent materials, in this case A1 and air, respectively, and m is an integer = 0, 1, 2,.
. ..
The coupling into the waveguide is through the periodicity of the grating such that
q = sin 8
+
(1246)for a resonance. The theoretical dispersion curves for the optical guided modes in the photoresist and the experimental points obtained [48] are shown in figure 26. It is seen that, again, good agreement is obtained between experiment and theory.
K : , ~ ~ C ~ - ' I
FIG. 26. - Dispersion relations for optical guided modes on an AI surface coated with (a) a 5-nm natural oxide layer and (b) a 410-nm layer of dielectric (photoresist) : Circled data points obtained from first order diffraction spectra, other points from zero order reflection
spectra. E is the photon energy in eV.