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Vacuum ultra violet monochromators
Nuclear Instruments and Methods 172 (1980) 123-131 © North-Holland Publishing Company
VACUUM ULTRA VIOLET MONOCHROMATORS * Malcolm R. HOWELLS The National Synchrotron Light Source, Brookhaven National Laboratory, Upton, N. Y. 11973, U.S.A.
The field of UV monochromators for synchrotron radiation work is surveyed with special emphasis on the 10-300 A region. The three main classes of instrument considered are the constant deviation Rowland circle types, the plane grating types and the simple rotation toroidal grating monochromator (TGM). It is argued that the Rowland designs are unsatisfactory in various ways and that their good resolution properties can be matched by plane grating devices operating with modern dedicated storage ring sources. For low resolution applications the use of toroidal grating instruments is preferred. Since the design of TGMs is a relatively young art, a good deal of explanation and discussion is included on the theoretical basis of the procedure. Some detailed consideration of the most important resolution determining aberrations is given and some particular examples and ray traces are shown.
Table 1 summarizes the position. We see that for the range 3 0 0 - 1 8 0 0 A, where normal incidence optics can be used and optical systems with low aberrations can be designed, that most requirements can be met. It is the grazing incidence region ( 1 0 - 3 0 0 A) where most problems arise. There is a bewildering variety of desirable features that every user would like to have plus more mundane pressures like shortage or money and time and questions about whether to commit valuable scientific manpower to the task of acquiring expertise in UV optics. The waters are further muddied by the influence of external factors like source geometry (especially size) beamline length and so on and by an understandable desire that an expensive
1. Introduction The general field of spectrometers for the ultra violet has been covered at great length in the past and much invaluable material is on record [ 1 - 5 ] . Here we limit our attention to monochromators rather than spectrographs and consider only those which are of use in the vacuum ultra violet region (say 1 0 - 1 8 0 0 A) for Synchrotron Radiation (SR) work. We further restrict our consideration to those which use reflection gratings although it is clear that the traditional domination of the field by these devices is being increasingly challenged by the development of transmission gratings of various kinds [6,71. Even with these limitations the number of different optical arrangements that have been used is still very large so we will make some attempt to see how the general classes of Rowland circle, plane grating and the new toroidal grating monochromators (TGMs) have succeeded in the past. We then make some fairly subjective evaluations of which approaches hold the best promise for the future and discuss in more detail the basis of these approaches and how they might be implemented.
Table 1 Survey of technologies Wavelength range (X)
Technology
Remarks
Over 1200 300-1200
Windows possible Normal incidence Optics O.K. Refl. Au >~few % Grazing incidence Optics needed
Few problems Small aberrations Standard designs O.K. Bad aberrations Standard designs meet some of the requirements, no design satisfies all Extra difficult region
45-300
2. Present technologies Some areas of our wavelength range of interest are covered better than others by existing methods.
1-45
* Research supported by the U.S. Department of Energy. I
123
Crystals and transmission devices compete with reflection gratings
IV. XUV MONOCHROMATOR SYSTEMS
124
M.R. Howells / Vacuum Ultra Violet Monochromators
grazing monochromator should provide some coverage of the normal incidence range as well. In the next section we try to catalogue this complex set of requirements to form a basis for evaluating the available design approaches.
3. User requirements Table 2 attempts to collect together a plausible set of requirements and to give very rough assessments of how well the three classes of grazing monochromator considered can handle the requirements. ~ Before studying the scorecard, we may note that the light gathering power of a monochromator (line 2) is expressed as the brightness, solid angle X area. This quantity is really the traditional two dimensional "Helmholtz invariant" [8] which is invariant through a loss-free optical system. It is a function of resolving power for all cases and is the only reasonable way to compare the light gathered by a Rowland instrument (large solid angle, small aperlure (slit)) and that by a plane grating instrument (small solid angle, large aperture). Turning to table 2 we can best absorb a message from this if we view it as a trial of the Rowland circle approach. We can identify two primary difficulties which the special situation of SR raises for this case. These are No. 5: fixed in and out directions and (normally) No. 6 rejection of higher diffracted orders. These are caused by the immovability and spectral
continuity of the synchrotron source. Various secondary difficulties are caused in turn by these. In particular, the only way to accommodate No. 5 is by extra reflections and special mechanisms. This gives problems of transmission efficiency and complications of manufacture and use. One way No. 6 has been solved in Rowland designs has been by the use of predispersers [10]. This adds further to the difficulties with large numbers of reflections, increased sensitivity to contamination and inconvenient and expensive mechanisms. These disadvantages of the Rowland circle approach make us turn back to its major advantage, i.e., its capability for good resolution. This is based on the fact that certain aberrations vanish on the Rowland circle. We especially need a small wavelength bandpass at low wavelength where a small 2 ~ is required. Consequently we look at the resolution achieved by use of the Rowland Circle in the difficult 1 0 - 1 0 0 A range. If we continue to confine our attention to instruments with constant deviation then around 0.04 A (or a resolving power of 2 5 0 - 2 5 0 0 ) seems to be the lower limit. This is not an aberration limit but seems to arise indirectly from the transmission difficulties mentioned above and the punishing loss of flux that occurs as both slits are closed down to widths of several microns. There is also a rather limited demand for resolution better than 0.04 N. In the light of these considerations we must ask ourselves whether it is necessary to continue to wrestle with the formidable problems of the constant
Table 2 User requirements and available grazing monochromators
1. High transmission-few reflections, good vacuum 2. Large collection solid angle X area (~tendue) 3. Capability for good resolution 4. Wide wavelenght coverage 5. Fixed in and out directions 6. Good rejection of second and higher diffracted orders 7. Highly polarized output 8. Converfience factors simple mechanism easy to operate low cost horizontal output Examples:
Rowland circle
Plane grating
Toroidal grating
poor good good possible hard hard various
medium good source limited easy easy fairly easy various
good very good poor possible easy poor various
poor poor poor various Grasshopper [9] Jaegle [10] Codling/Mitchell [ 11 ]
good good medium various Kunz/Dietrich [ 12] Miyake [13-15] Flipper [ 16]
very good good good various Madden Ederer [ 17] J.Y. LHT 30 [18] Wisconsin 'TGM' [ 19 ]
M.R. Howells / Vacuum Ultra Violet Monochromators
deviation Rowland mount or whether we cannot achieve as good or better resolution using plane grating instruments. These offer the possibility of fewer reflections, simpler mechanisms, cheaper gratings but a resolving power which depends on the source size (see later). Thus the suitability of the plane grating principle is dependent in part on the source size. This has been discussed recently by Pruett [20]. A recent design study of a plane grating instrument [21] for use at the Brookhaven storage ring has indicated a source size limited resolution (with a 1200 lines/mm grating) of less than 0.025 A for the range 2 5 - 1 0 0 N. This is excellent resolution and shows that with a present generation, dedicated storage ring plane grating monochromators (PGMs) look very good.
NkX = sin a + sin/3,
125
(1)
and for fixed in and out directions we also have a -/3 = 20.
(2)
Hence if we know N and 0 we have -1 F Arkx ]
L2~osO]+ 0 ,
¢J=sin
a = sin -1 L2 cos 0J + 0 ,
(2a)
(2b)
and the configuration is well defined. (2) Another result of working with constant deviation is that we see the so called "horizon" effect, i.e. there is a limit XH to the wavelength scan which occurs when a or ~ = 90 °. In fact 2 cos 20
XH-
4. TGMs and PGMs Now that we have argued that TGMs and PGMs are a promising way to meet a large number of the requirements of table 2 let us look at some of the features they have in common and discuss some design concepts. The general requirement of fixed deviation plus our choice of the simple rotation scanning principle for both TGMs and PGMs implies that we must consider the inward and outward direction as fixed. This is shown in fig. 1. Using the notation and sign convention shown in that figure we may write down a description of our system: (1) We have the grating equation
~
DIFFRACTIONGRATING WITH N GROOVES/MM
N~
(3)
(3) If we use the light source as an entrance slit then its finite size (s) and distance (r) provide a limit on the wavelength resolution AXs given by S COS a
A X s - Nkr
(4)
(4) From eqs. (1) and (2) we see that NkX = 2 cos 0 sin(0 + 13). Now since (0 +/3) = ~bthe angle the grating has turned from zero order, we have X proportional to sin qS. Hence we can have an exactly linear wavelength scale if we/use a sine bar drive with q5= 0 at zero order. This fits well to the simple rotation scanning principle we are talking about. (5) The magnification of a TGM is easily seen to be given by: Magn. -
/"'COS O~ ?' COS ,~
'~ ~
•
/
INCIDENT RAY (WAVELENGTHS)
CONSIDEREDPOSITIVEIN THE COUNTERCLOCKWISE DIRECTIONMEASURED FROM THE NORMAL, 0 IS COUNTED4"
ORDER:klS +lFIPl
Ang. Magn. = cos a/cos ~. These general statements apply equally well to both PGMs and TGMs but they give no indication of how, from a design point of view, one might choose N and 0. It turns out that we have very little freedom to manipulate 0 which is determined entirely by the short wavelength performance that is required. The margin of possible error is rather wide in practice since reflectance [22,23] varies considerably with deposition method and contamination state even for nominally identical surfaces. Consequently there is IV. XUV MONOCHROMATOR SYSTEMS
M.R. Howells / Vacuum Ultra Violet Monoehromators
126
I
I
I
I
I
I
[
100 90 :~
I
.7"7 [] / ' J . ~
/"
s0
x
-
./
~-
70-
uJ ._1
60-
"' >
50 _
0
I
.o/+
~ -
7. 7"
u.U" 40
/.x
0 p.
30-
/
O
20
+/
10-
[]
0 90
/
88
86
84
B 82
I 80
I 76
78
I 74
I 72
70
ANGLE OF iNCIDENCE (e °) Fig. 2. Measured data from various sources giving approximate cut-off wavelengths as a function of grazing angle. X J.B. West et al. actual monitor threshold. ~ 0 (5%) measured from Pt (V. Rehn private comm.). ® E = 4.8 × 103/0 (eV and deg) Pt: X-ray range. - . . . . . Calculated threshold from n and k values (ref. 31). Calculated threshold from n and k values (ref. 32).
z~0 (5%) measured for Pt [Romer (ref. 4, p. 73]. n Henke 0 (5%) measured for Au. • 0 (5%) measured for Au (ref. 1, p. 36). @0 (5%) measured for Au (ref. 34). + 0 (5%) measured for Au (ref. 33). Since the scatter is so great there seems little point in separating the Au and Pt values.
considerable spread in the available data. However, fig. 2 gives enough o f an overview o f actual measurements to enable a reasonable choice to be made for any particular spectrometer. The choice of N is best made on the basis of the horizon wavelength [eq. (3)]. The best choice is the maximum groove density (best resolution) that gives a XH which is larger than the longest wavelength of interest by a reasonable margin. We can choose to cover a desired range with more than one grating. For example if we had a design with N - - 1200 lines/mm 0 = 87.5 ° then XH ~ 32 3, So we might hope to cover 1 0 - 2 5 A. If we had a design that worked from an aberration and focus point o f view over this range then we could also have the full scheme: N=12001ines/mm;
3,=10-25A,
XH = 3 2 A ;
N = 6001ines/mm,
X= 20-50A,
XH = 63 A;
N = 3001ines/mm,
X=40-100A,
XH = 1 2 7 A .
This is because NX is the same and hence also c~ and [eq. (1)]. The outstanding parameter left to choose is the order k. For PGMs the issues are limited but for TGMs there are a number o f questions. These concern the collection aperture, r and r' values, the source size resolution limit and magnification behaviour.
5. Toroidal grating analysis Following normal practice we can define a suitable coordinate system (figs. 1 and 2) and write down an expression for the optical path F of the ray APB. (For simplicity we confine ourselves to the case of A in the x,y plane.) This can be expressed as a power series in the aperture co-ordinates y,z. ,
~-~
y2
F = .ZJFi] = r + r' + (sin c~ + sin/3)y + Q o II
+C3o ~y a Z4
X
z2 y4 +Co2 +c12y z 2 +C4o y2z2
+ Co4 R-g + - ~ -
C22 + higher order t e r m s .
(5)
Each Fq (Fi]=Ciiyiz]/R i+]-l) corresponds to some particular type of aberration. Following Haber we can derive expressions for the Cii (see appendix) but here we have factored the R dependence out and expressed the Ci]'s as functions only o f r*, r*', p*, a and /3 where the * indicates that the length is expressed in units of R. This means the Cq's are functions only of the shape, not the size of the configuration. Fermats principle tells us that the conditions for a kth order diffracted image at B are
3F' 0y
- NkX
and
3F' 0z
- 0.
M.R. Howells / Vacuum Ultra Violet Monochromators If we define F ' : Fo + F, where Fo : r + r' + (sin ~ + sin/3)y then this becomes OF/3y = OF/Oz = 0. Combining these with eq. (5) we obtain the grating eq. (1) and the condition that all the Cii's are zero, i.e., no aberrations. When OF/Oy or OF/Oz 4= O, then we have aberrations and the rays arriving in the image plane are displaced by some amount from the ideal Gaussian image point B. The significance of the nonzero differentials [17,25] is that of a change in the appropriate direction cosine (7) of the emergent ray from the "correct" value i.e.,
OF/Oy = 6(cos 3'y) = d(sin 13) = cos/5 d/3 = cos ~3 AY/r' ; the significance of &Y is shown in fig. 2. It is the displacement of the ray AP when it arrives in the image plane. Thus
r' OF/Oy
Ay =-
COS
(6)
Similarly
&Z = r' OF/Oz.
(7)
We also have from eq. (1) an expression for the reciprocal linear dispersion AX
A Y = r'Nk '
(8)
thus we find 1 OF AXArk 0y ' and if we imagine an overall AX comprising a lot of &Xo's arising from the various aberration terms then we have
AX6
5.1. Astigmatism The primary purpose of using a toroidal grating is to reduce astigmatism. This is achieved at a wavelength corresponding to angles c~ and/5 if we set Co2 [see eq. (14)] equal to zero, i.e. COSO~ + COS~
P*
'
1/r* + 1/r*'
(9)
We can readily show that if a - t 3 = 2 0 > 1 5 0 ° then (cos a + cos/3) is constant within about 1-2%. Thus if astigmatism is corrected at one wavelength then the correction is good to sufficient accuracy over the whole range.
5.2. Focus Assuming we have a value for p* we must now proceed to design a configuration which will optimise the resolution determining aberrations over some chosen wavelength range. The most important ones in practice turn out to be focus (C2oy2/R) and astigmatic coma (C1 zYz2 /R 2) or 1 2y
1 z2
A~k20 N~ R C2o
cos
_ 1 OFii_ 1 Cifiy.i_lz.i] N k Oy Ark
giving the important conclusion that the aberration lhnited resolution is independent of R! Since the resolution of practical TGMs is normally aberration limited we may ask why we should not have a very small R and hence a small monochromator. The answer is that although increasing R does not change the resolution, it does (via eq. (8)) increase the corresponding slit widths. If we assume that the slit length is about 1 - 2 mm for all cases then we conclude that the 6tendue is proportional to R. Consequently other things being equal, the light collected is proportional to R. We can imagine that it might sometimes be attractive to reduce the size and hence the cost of a monochromator substantially if the intended light source were bright enough.
127
and
m)kl2
Ark R 2 C12.
From the form of the terms it is obvious that, as always, one can get a better image by reducing the aperture. In particular reducing the ruled width (2y) improves focus and reducing the groove length (2z) improves astigmatic coma. We have two degrees of freedom to work with, namely choice of r* and r*'. In fig. 3 we show schematically how the term C2o varies with wavelength. Following Lepere [26] (who treated the aberration corrected case) we can see that we can fix on two wavelengths Xl and X2 and arrange to have a good focus at these wavelengths. Suppose that N and 0 are
z
Y ~
~
~
x
AO-= r B
OB= r/ Y
A Fig. 3. Co-ordinate systems used to discuss the "perfect ray" APB to the Gaussian focus B and actual ray APB'. IV. XUV MONOCHROMATOR SYSTEMS
128
M.R. Howells / Vacuum Ultra Violet Monoehromators
)~1
given and cq and ~1 correspond to and a2 and t32 to X2. Then putting C2o = 0 at Xt and 7,2 gives
-
1 I (CO~;a 1-
2
cos
2
+ ~ -v
-
5.3. Astigmatic coma
If imperfect focus were the only important aberration then the o p t i m u m solution would be close at hand. However, we have already used our only two degrees o f freedom and we have not yet addressed the contribution from AX12. To get a feeling for the form of this we proceed as follows: set 3C2o/3X = 0 with the assumption (see earlier) that cos a + cos/3 = const. This gives us a condition that is satisfied at the wavelength (~-rnin) for which 6"2o has a minimum, i.e. -
l/
as well. In both cases the lines joining the dots of the ray trace corresponds to lines o f constant y on the grating and the very heavy lines to the top and bottom edge o f the grating. By using eqs. (6) and (7) we can compute the amount of Y and Z deflection and understand the origin of the parabolic curves. We can also see the effect of increasing Ymax or Zmax on the size and shape of the image in the Y, Z system. Returning to the design question let us try to move the point P to reduce C12. We know that at P, eq. (10) holds. If in addition we insist that Ct2 = 0, i.e. P lies on the X axis of fig. 4 then we find from eq. 05)
sin/3 cos [3
(10)
r*'
7,1:-7--1 I-sin a /l i T _ c o : a ) + sin/3( 1
I
I
I
I
1--
cos_/3) ]
[
I
z= ZMA X Z=0
.5-Z
If we consider the terms containing p* we can see that C12 is independent of p* if eq. (10) is satisfied. In fact we see that the C~2 curves corresponding to various p* values pass through a point P at ~-min as shown in fig. 3. Clearly we would like P to lie on the X axis and we might hope that a p* value corresponding to a fairly horizontal C12 curve might also correspond to an acceptable amount of astigmatism correction. Before we address this question let us examine the nature o f astigmatic coma [27] in a little more detail. Its effect is similar to coma for the case o f near zero astigmatism (see fig. 5) and more like spectrum line curvature if significant astigmatism is present. Fig. 6 shows a realistic case where some defocus is present
I
1.5
Now consider the form of C12 C12
.
C20
Fig. 4. Schematic variation of C2 o and C12 with wavelength.
2
r*
I I i
cos/32)1 = 0 .
Clearly one can solve these for r* and r*'. By this process one can easily arrange to have an approximate focus over the range between 7.L and XR. For spectroscopy where a given AX is required one would choose ~k1 and Xz symetrically between XL and ?'RFor experiments where a given 2xE is required Xl and X2 would be pushed toward the important short wavelength end o f the range.
sin a cos a
~R
I I I
\,
al ) + (~ cOs2/31 ___ OS,l) =0 77
co .)
-
~L
) ~ ~ _ _
Y=YMAX -y = SMALL
0-GAUSSIAN IMAGE
%5
•',Z=2C12 r* yz L~Y= C12 (CO@~) z 2 -1.5 --2 -2
Z~Z2 hi" y2
I -1.5
I -1
I -.5
I 0 Y
I .5
T 1
Ay
I 1.5
Fig. 5. Toroidal grating image with C 12 the dominant aberration. The r' value is chosen for exact focus and the astigmatism is small. Eq. (6), (7) and (15) are used to compute ~Z and Ay and show how a parabolic curve arises for each y value. Y and Z are in 100's of um and the parameters are R =100, p=6.1, r=38.5, r'=17.14 (all cm), 0=150 ° , n = 1200 lines/mm, X = 150 A, k = -1.
M.R. Howells / Vacuum Ultra Violet Monoehromators
sin ~ r .2
sin
129
-- ORDER
(r*')2
'-I- ORDER
• ~L
~R
I I
I I
/C20
Solving this with eq. (10) leads to the conditions o~ = - / 3 ,
r* = r * ' ,
and
~,min= 0 1
So that P in fact lies at the origin of fig. 3. Here we have adopted a second extreme approach. Previously we saw how to optimise C2o whilst ignoring C12. Now we have found a scheme that turns out to optimise C12 rather well but largely ignores C2o. We can still use one degree o f freedom to determine one wavelength of exact focus, XF say (see fig. 6). This involves choosing a value for r* = r*'. To do this we make use of the parameter A=X/XH. From eq. (3) we can rewrite eqs. (2a) and (2b) as /3 = s i n - l ( A cos 0) - 0 , a = s i n - l ( A cos 0) + 0 . Substituting these into eq. (12) and setting Cz0 = 0 for the wavelength AF we find
I
Pl
"2 Fig. 7. Schematic of the variation with wavelength of C20 and C a2 for tlre case r = r'. Note the point P has moved to the origin and Xmin = 0. The instrument can now be operated in either positive or negative order.
from astigmatic coma is much reduced and defocus dominates. Eq. (11) is an interesting equation containing a number o f fruitful cases which are discussed elsewhere [28,29].
(11)
This gives us our monochromator configuration. The ray trace (fig. 7) shows as expected that the curvature [
~
5.4. Overall optimisation
, cos0(1 - c o s 2 0 A } ) r = -,,/(1 - A} cos20)
2
t
I
I
I
I
Whilst the two extreme approaches we have discussed here are physically revealing, neither of them leads to the best TGM designs. To get the best design
I
2
1.5--
I
[
I
I
I
I
I
1.5 -
LkZ = 2 r " C12 yz 1 -.5 A Y -
~jji ~I~y
C12 Z + C20
max ,z max_ 1 -Ymin Zmax .5 -
Z
0 -- So that
_.~L . . . . . . . ~,~ I=~"-GAUSSIAN '~\ IMAGE
-.5-
r 1 /kz \2 ~ zxy=K|C121~i ÷C202y ~ / t ~ 2C12r'y / =v • ",,.1~,/" -1 or " ~
-1.5--
-2
Z~Z2 =,
Z=O
BOTTOM EDGE OF GRATING
-
Z
0
*.5 _
K [ - C?0 2y'K] 4C12r*2 y 2 t ~Y
-1
--
-1.5
I
I
I
I
I
I
I
-1.5
-1
-.5
0
.5
1
1.5
Y
Fig. 6. More realistic image for the same grating as fig. 5. This time r ' = 17.5 cm the other parameters are the same. Now both defocussing and astigmatism are significant. &Z and ~ Y are computed as before with the assumption that 2~Z is determined by C02 and A y by C2o and C12. Again the observed parabolic shape is explained and the role of the two aberrations becomes clear.
-2 -2
I
[
I
I
I
I
I
-1.5
-1
-.5
0
.5
1
1.5
TOROIDAL DIFFRACTION GRATING INT. DIST.
Fig. 8. Image produced by the same toroidal grating as shown in figs. 5 and 6. This time configured with conjugates r = r' = 26.41 cm. These correspond to choosing XF = 0.15X H giving ;~F = 167 A. The image is ray traced for X = 250 A which is considerably out of focus. The dominant influence of C2o (defocus) and the small effect of C12 are clearly seen. IV. XUV MONOCHROMATOR SYSTEMS
M.R. Howells / Vacuum Ultra Violet Monochromators
130
we must search parameter space with a computer program to find the best performance at a sequence of chosen wavelengths within the range of interest. This will represent the best compromise between control of C2o and ClZ which are the main resolution determining aberrations. Once the configuration is established the actual resolution/flux trade-off can be established by choice of the grating width and height. The method used at Brookhaven for doing this is discussed in another contribution to this meeting
[30]. We might legitimately ask how the adequacy of these theoretical and computational design procedures is checked. It is essential to have a good ray trace program for this purpose and fortunately modern microprocessor technology makes this reasonably easy to achieve on a self contained basis. Our approach has actually been to have more than one program using completely different methods of calculation. Since we are dealing with low resolving power, poor imaging, systems it is safe to assume that the geometrical optics approximation is valid.
Appendix Here we list a few of the terms of the expansion
(5). Focussing term C2oy2 /R,
[(cos os4 +
C20
\ r*'
--
cos )l
Coma terin Caoya/R 2, _ 1 Fsin a / c o s 2 a _ cos a ]
C
]
r* sin {cos2
+~
r
~ r*~
t3)l
(13)
-cos./l
Astigmatism term Co2z2/R,
l_[t
Co2 = 2 L\r
cosq ~w ] + ( ) ,
c~,/3); •
(14)
Astigmatic coma term C 1 j z 2 /R 2, 1 Fsin a t 1
c,2
cos a_)
0"
sm (rl'-~ cos + -~p* ]
(is)
The remaining terms are of little importance for approximate focussing situations. However, they can be found in ref. 24.
References [ 1] J.A.R. Samson, Techniques of vacuum ultra violet spectroscopy (Wiley, New York, 1967). [2] A.N. Zaldel and E.Ya. Schreider, Vacuum ultra violet spectroscopy, (Halstead Press, New York, 1970) (translated from Russian). [3] F. Wuilleumier and Y. Farge, eds., Synchrotron radiation instrumentation and new developments (NorthHolland, Amsterdam, 1978). [4] Ed. G.V. Marr and I.H. Munro, in Int. Symp. for Synchrotron radiation users, Daresbury, Jan. 1973 (DNPL/ R26). [5] M. Pouey, J. de Phys. Coll. C4 Supp. Vol. 7 (1978) 188. [6] E. Kallne, H.W. Schnopper, J.P. Delvaille, L.P. van Speybroeck and R.Z. Bachrach, Nucl. Instr. and Meth. 152 (1978) 73. [7] Proc. New York Acad. Sci. Conf. on Soft X-ray microscopy, New York, 1979. [8] J.F. James and R.S. Sternberg, Design of optical spectrometers (Chapman Hall, London, 1969). [9] F.C. Brown, R.Z. Bachrach and N. Lieu, Nucl. Instr. and Meth. 152 (1978) 73. [10] P. Dhez, P. Jaegle, F.J. WuRIeumier, E. Kallne, V. Schmidt, M. Bertand and A. Carillou, Nncl. Instr. and Meth. 152 (1978) 85. [11] K. Codling and P. Michel, J. Phys. E 3 (1970) 685. [12] H. Dietrich and C. Kunz, Rev. Sci. instr. 43 (1972) 434. [13] K.P. Miyake, R. Kato and H. Yamashita, Sci. Light 18 (1969) 39. [14] J.B. West, K. Codling and G.V. Marr, J. Phys. E 7 (1974) 137. [15] M.R. Howells, D. Norman, G.P. Williams and J.B. West, J. Phys. E 11 (1978) 199. [16] W. Eberhardt, G. Kaltsolien and C. Kunz, Nucl. Instr. and Meth. 152 (1978) 81. [17] R.P. Madden and D.L. Ederer, J. Opt. Soc. Am. 62 (1972) 722A and private communication. We gratefully acknowledge the receipt of an unpublished manuscript detailing the design of the original toroidal grating monochromator at NBS. [18] Jobin Yvon publication, see also ref. 26. [19] B. Tonner, these Proceedings, p. 133. [20] C. Pruett, private communication. [21] M.R. I-Iowells and W.R. McKinney, BNL Rept. No. 26027 (1979). [22] G.P. Williams and M.R. Howells, BNL Rept. No. 26121 (1979). [23] V. Rehn and V.O. Jones, Opt. Eng. 17 (1978) 504. [24] H. Haber, J. Opt. Soc. Am. 40 (1950) 153. [25] T. Namoika, J. Opt. Soc. Am. 49 (1959) 446. [26] D. Lepere, Nouv. Rev. Opt. 6 (1975) 173. [27] W.T. Welford, in Progress in optics, Vol. 4 (NorthHolland, Amsterdam, 1965).
M.R. Howells / Vacuum Ultra Violet Monochromators
[28] M.R. Howells and T.J. Aggus, BNL Rept. No. 26341 (1979). [29] M.R. Howells, in preparation. [30] W.R. McKinney and M.R. Howells, these Proceedings, p. 149.
131
[31] M.J. Hagemann, W. Gudat and C. Kunz, DESY SR-74/7 (1974) table 5 and 6. [32] A. Lukirski and E. Savinov, Opt. Spect. 14 (1963) 152. [33] Malina and Cash, Appl. Opt. 17 (1978) 3309.
IV. XUV MONOCHROMATOR SYSTEMS
VACUUM ULTRA VIOLET MONOCHROMATORS * Malcolm R. HOWELLS The National Synchrotron Light Source, Brookhaven National Laboratory, Upton, N. Y. 11973, U.S.A.
The field of UV monochromators for synchrotron radiation work is surveyed with special emphasis on the 10-300 A region. The three main classes of instrument considered are the constant deviation Rowland circle types, the plane grating types and the simple rotation toroidal grating monochromator (TGM). It is argued that the Rowland designs are unsatisfactory in various ways and that their good resolution properties can be matched by plane grating devices operating with modern dedicated storage ring sources. For low resolution applications the use of toroidal grating instruments is preferred. Since the design of TGMs is a relatively young art, a good deal of explanation and discussion is included on the theoretical basis of the procedure. Some detailed consideration of the most important resolution determining aberrations is given and some particular examples and ray traces are shown.
Table 1 summarizes the position. We see that for the range 3 0 0 - 1 8 0 0 A, where normal incidence optics can be used and optical systems with low aberrations can be designed, that most requirements can be met. It is the grazing incidence region ( 1 0 - 3 0 0 A) where most problems arise. There is a bewildering variety of desirable features that every user would like to have plus more mundane pressures like shortage or money and time and questions about whether to commit valuable scientific manpower to the task of acquiring expertise in UV optics. The waters are further muddied by the influence of external factors like source geometry (especially size) beamline length and so on and by an understandable desire that an expensive
1. Introduction The general field of spectrometers for the ultra violet has been covered at great length in the past and much invaluable material is on record [ 1 - 5 ] . Here we limit our attention to monochromators rather than spectrographs and consider only those which are of use in the vacuum ultra violet region (say 1 0 - 1 8 0 0 A) for Synchrotron Radiation (SR) work. We further restrict our consideration to those which use reflection gratings although it is clear that the traditional domination of the field by these devices is being increasingly challenged by the development of transmission gratings of various kinds [6,71. Even with these limitations the number of different optical arrangements that have been used is still very large so we will make some attempt to see how the general classes of Rowland circle, plane grating and the new toroidal grating monochromators (TGMs) have succeeded in the past. We then make some fairly subjective evaluations of which approaches hold the best promise for the future and discuss in more detail the basis of these approaches and how they might be implemented.
Table 1 Survey of technologies Wavelength range (X)
Technology
Remarks
Over 1200 300-1200
Windows possible Normal incidence Optics O.K. Refl. Au >~few % Grazing incidence Optics needed
Few problems Small aberrations Standard designs O.K. Bad aberrations Standard designs meet some of the requirements, no design satisfies all Extra difficult region
45-300
2. Present technologies Some areas of our wavelength range of interest are covered better than others by existing methods.
1-45
* Research supported by the U.S. Department of Energy. I
123
Crystals and transmission devices compete with reflection gratings
IV. XUV MONOCHROMATOR SYSTEMS
124
M.R. Howells / Vacuum Ultra Violet Monochromators
grazing monochromator should provide some coverage of the normal incidence range as well. In the next section we try to catalogue this complex set of requirements to form a basis for evaluating the available design approaches.
3. User requirements Table 2 attempts to collect together a plausible set of requirements and to give very rough assessments of how well the three classes of grazing monochromator considered can handle the requirements. ~ Before studying the scorecard, we may note that the light gathering power of a monochromator (line 2) is expressed as the brightness, solid angle X area. This quantity is really the traditional two dimensional "Helmholtz invariant" [8] which is invariant through a loss-free optical system. It is a function of resolving power for all cases and is the only reasonable way to compare the light gathered by a Rowland instrument (large solid angle, small aperlure (slit)) and that by a plane grating instrument (small solid angle, large aperture). Turning to table 2 we can best absorb a message from this if we view it as a trial of the Rowland circle approach. We can identify two primary difficulties which the special situation of SR raises for this case. These are No. 5: fixed in and out directions and (normally) No. 6 rejection of higher diffracted orders. These are caused by the immovability and spectral
continuity of the synchrotron source. Various secondary difficulties are caused in turn by these. In particular, the only way to accommodate No. 5 is by extra reflections and special mechanisms. This gives problems of transmission efficiency and complications of manufacture and use. One way No. 6 has been solved in Rowland designs has been by the use of predispersers [10]. This adds further to the difficulties with large numbers of reflections, increased sensitivity to contamination and inconvenient and expensive mechanisms. These disadvantages of the Rowland circle approach make us turn back to its major advantage, i.e., its capability for good resolution. This is based on the fact that certain aberrations vanish on the Rowland circle. We especially need a small wavelength bandpass at low wavelength where a small 2 ~ is required. Consequently we look at the resolution achieved by use of the Rowland Circle in the difficult 1 0 - 1 0 0 A range. If we continue to confine our attention to instruments with constant deviation then around 0.04 A (or a resolving power of 2 5 0 - 2 5 0 0 ) seems to be the lower limit. This is not an aberration limit but seems to arise indirectly from the transmission difficulties mentioned above and the punishing loss of flux that occurs as both slits are closed down to widths of several microns. There is also a rather limited demand for resolution better than 0.04 N. In the light of these considerations we must ask ourselves whether it is necessary to continue to wrestle with the formidable problems of the constant
Table 2 User requirements and available grazing monochromators
1. High transmission-few reflections, good vacuum 2. Large collection solid angle X area (~tendue) 3. Capability for good resolution 4. Wide wavelenght coverage 5. Fixed in and out directions 6. Good rejection of second and higher diffracted orders 7. Highly polarized output 8. Converfience factors simple mechanism easy to operate low cost horizontal output Examples:
Rowland circle
Plane grating
Toroidal grating
poor good good possible hard hard various
medium good source limited easy easy fairly easy various
good very good poor possible easy poor various
poor poor poor various Grasshopper [9] Jaegle [10] Codling/Mitchell [ 11 ]
good good medium various Kunz/Dietrich [ 12] Miyake [13-15] Flipper [ 16]
very good good good various Madden Ederer [ 17] J.Y. LHT 30 [18] Wisconsin 'TGM' [ 19 ]
M.R. Howells / Vacuum Ultra Violet Monochromators
deviation Rowland mount or whether we cannot achieve as good or better resolution using plane grating instruments. These offer the possibility of fewer reflections, simpler mechanisms, cheaper gratings but a resolving power which depends on the source size (see later). Thus the suitability of the plane grating principle is dependent in part on the source size. This has been discussed recently by Pruett [20]. A recent design study of a plane grating instrument [21] for use at the Brookhaven storage ring has indicated a source size limited resolution (with a 1200 lines/mm grating) of less than 0.025 A for the range 2 5 - 1 0 0 N. This is excellent resolution and shows that with a present generation, dedicated storage ring plane grating monochromators (PGMs) look very good.
NkX = sin a + sin/3,
125
(1)
and for fixed in and out directions we also have a -/3 = 20.
(2)
Hence if we know N and 0 we have -1 F Arkx ]
L2~osO]+ 0 ,
¢J=sin
a = sin -1 L2 cos 0J + 0 ,
(2a)
(2b)
and the configuration is well defined. (2) Another result of working with constant deviation is that we see the so called "horizon" effect, i.e. there is a limit XH to the wavelength scan which occurs when a or ~ = 90 °. In fact 2 cos 20
XH-
4. TGMs and PGMs Now that we have argued that TGMs and PGMs are a promising way to meet a large number of the requirements of table 2 let us look at some of the features they have in common and discuss some design concepts. The general requirement of fixed deviation plus our choice of the simple rotation scanning principle for both TGMs and PGMs implies that we must consider the inward and outward direction as fixed. This is shown in fig. 1. Using the notation and sign convention shown in that figure we may write down a description of our system: (1) We have the grating equation
~
DIFFRACTIONGRATING WITH N GROOVES/MM
N~
(3)
(3) If we use the light source as an entrance slit then its finite size (s) and distance (r) provide a limit on the wavelength resolution AXs given by S COS a
A X s - Nkr
(4)
(4) From eqs. (1) and (2) we see that NkX = 2 cos 0 sin(0 + 13). Now since (0 +/3) = ~bthe angle the grating has turned from zero order, we have X proportional to sin qS. Hence we can have an exactly linear wavelength scale if we/use a sine bar drive with q5= 0 at zero order. This fits well to the simple rotation scanning principle we are talking about. (5) The magnification of a TGM is easily seen to be given by: Magn. -
/"'COS O~ ?' COS ,~
'~ ~
•
/
INCIDENT RAY (WAVELENGTHS)
CONSIDEREDPOSITIVEIN THE COUNTERCLOCKWISE DIRECTIONMEASURED FROM THE NORMAL, 0 IS COUNTED4"
ORDER:klS +lFIPl
Ang. Magn. = cos a/cos ~. These general statements apply equally well to both PGMs and TGMs but they give no indication of how, from a design point of view, one might choose N and 0. It turns out that we have very little freedom to manipulate 0 which is determined entirely by the short wavelength performance that is required. The margin of possible error is rather wide in practice since reflectance [22,23] varies considerably with deposition method and contamination state even for nominally identical surfaces. Consequently there is IV. XUV MONOCHROMATOR SYSTEMS
M.R. Howells / Vacuum Ultra Violet Monoehromators
126
I
I
I
I
I
I
[
100 90 :~
I
.7"7 [] / ' J . ~
/"
s0
x
-
./
~-
70-
uJ ._1
60-
"' >
50 _
0
I
.o/+
~ -
7. 7"
u.U" 40
/.x
0 p.
30-
/
O
20
+/
10-
[]
0 90
/
88
86
84
B 82
I 80
I 76
78
I 74
I 72
70
ANGLE OF iNCIDENCE (e °) Fig. 2. Measured data from various sources giving approximate cut-off wavelengths as a function of grazing angle. X J.B. West et al. actual monitor threshold. ~ 0 (5%) measured from Pt (V. Rehn private comm.). ® E = 4.8 × 103/0 (eV and deg) Pt: X-ray range. - . . . . . Calculated threshold from n and k values (ref. 31). Calculated threshold from n and k values (ref. 32).
z~0 (5%) measured for Pt [Romer (ref. 4, p. 73]. n Henke 0 (5%) measured for Au. • 0 (5%) measured for Au (ref. 1, p. 36). @0 (5%) measured for Au (ref. 34). + 0 (5%) measured for Au (ref. 33). Since the scatter is so great there seems little point in separating the Au and Pt values.
considerable spread in the available data. However, fig. 2 gives enough o f an overview o f actual measurements to enable a reasonable choice to be made for any particular spectrometer. The choice of N is best made on the basis of the horizon wavelength [eq. (3)]. The best choice is the maximum groove density (best resolution) that gives a XH which is larger than the longest wavelength of interest by a reasonable margin. We can choose to cover a desired range with more than one grating. For example if we had a design with N - - 1200 lines/mm 0 = 87.5 ° then XH ~ 32 3, So we might hope to cover 1 0 - 2 5 A. If we had a design that worked from an aberration and focus point o f view over this range then we could also have the full scheme: N=12001ines/mm;
3,=10-25A,
XH = 3 2 A ;
N = 6001ines/mm,
X= 20-50A,
XH = 63 A;
N = 3001ines/mm,
X=40-100A,
XH = 1 2 7 A .
This is because NX is the same and hence also c~ and [eq. (1)]. The outstanding parameter left to choose is the order k. For PGMs the issues are limited but for TGMs there are a number o f questions. These concern the collection aperture, r and r' values, the source size resolution limit and magnification behaviour.
5. Toroidal grating analysis Following normal practice we can define a suitable coordinate system (figs. 1 and 2) and write down an expression for the optical path F of the ray APB. (For simplicity we confine ourselves to the case of A in the x,y plane.) This can be expressed as a power series in the aperture co-ordinates y,z. ,
~-~
y2
F = .ZJFi] = r + r' + (sin c~ + sin/3)y + Q o II
+C3o ~y a Z4
X
z2 y4 +Co2 +c12y z 2 +C4o y2z2
+ Co4 R-g + - ~ -
C22 + higher order t e r m s .
(5)
Each Fq (Fi]=Ciiyiz]/R i+]-l) corresponds to some particular type of aberration. Following Haber we can derive expressions for the Cii (see appendix) but here we have factored the R dependence out and expressed the Ci]'s as functions only o f r*, r*', p*, a and /3 where the * indicates that the length is expressed in units of R. This means the Cq's are functions only of the shape, not the size of the configuration. Fermats principle tells us that the conditions for a kth order diffracted image at B are
3F' 0y
- NkX
and
3F' 0z
- 0.
M.R. Howells / Vacuum Ultra Violet Monochromators If we define F ' : Fo + F, where Fo : r + r' + (sin ~ + sin/3)y then this becomes OF/3y = OF/Oz = 0. Combining these with eq. (5) we obtain the grating eq. (1) and the condition that all the Cii's are zero, i.e., no aberrations. When OF/Oy or OF/Oz 4= O, then we have aberrations and the rays arriving in the image plane are displaced by some amount from the ideal Gaussian image point B. The significance of the nonzero differentials [17,25] is that of a change in the appropriate direction cosine (7) of the emergent ray from the "correct" value i.e.,
OF/Oy = 6(cos 3'y) = d(sin 13) = cos/5 d/3 = cos ~3 AY/r' ; the significance of &Y is shown in fig. 2. It is the displacement of the ray AP when it arrives in the image plane. Thus
r' OF/Oy
Ay =-
COS
(6)
Similarly
&Z = r' OF/Oz.
(7)
We also have from eq. (1) an expression for the reciprocal linear dispersion AX
A Y = r'Nk '
(8)
thus we find 1 OF AXArk 0y ' and if we imagine an overall AX comprising a lot of &Xo's arising from the various aberration terms then we have
AX6
5.1. Astigmatism The primary purpose of using a toroidal grating is to reduce astigmatism. This is achieved at a wavelength corresponding to angles c~ and/5 if we set Co2 [see eq. (14)] equal to zero, i.e. COSO~ + COS~
P*
'
1/r* + 1/r*'
(9)
We can readily show that if a - t 3 = 2 0 > 1 5 0 ° then (cos a + cos/3) is constant within about 1-2%. Thus if astigmatism is corrected at one wavelength then the correction is good to sufficient accuracy over the whole range.
5.2. Focus Assuming we have a value for p* we must now proceed to design a configuration which will optimise the resolution determining aberrations over some chosen wavelength range. The most important ones in practice turn out to be focus (C2oy2/R) and astigmatic coma (C1 zYz2 /R 2) or 1 2y
1 z2
A~k20 N~ R C2o
cos
_ 1 OFii_ 1 Cifiy.i_lz.i] N k Oy Ark
giving the important conclusion that the aberration lhnited resolution is independent of R! Since the resolution of practical TGMs is normally aberration limited we may ask why we should not have a very small R and hence a small monochromator. The answer is that although increasing R does not change the resolution, it does (via eq. (8)) increase the corresponding slit widths. If we assume that the slit length is about 1 - 2 mm for all cases then we conclude that the 6tendue is proportional to R. Consequently other things being equal, the light collected is proportional to R. We can imagine that it might sometimes be attractive to reduce the size and hence the cost of a monochromator substantially if the intended light source were bright enough.
127
and
m)kl2
Ark R 2 C12.
From the form of the terms it is obvious that, as always, one can get a better image by reducing the aperture. In particular reducing the ruled width (2y) improves focus and reducing the groove length (2z) improves astigmatic coma. We have two degrees of freedom to work with, namely choice of r* and r*'. In fig. 3 we show schematically how the term C2o varies with wavelength. Following Lepere [26] (who treated the aberration corrected case) we can see that we can fix on two wavelengths Xl and X2 and arrange to have a good focus at these wavelengths. Suppose that N and 0 are
z
Y ~
~
~
x
AO-= r B
OB= r/ Y
A Fig. 3. Co-ordinate systems used to discuss the "perfect ray" APB to the Gaussian focus B and actual ray APB'. IV. XUV MONOCHROMATOR SYSTEMS
128
M.R. Howells / Vacuum Ultra Violet Monoehromators
)~1
given and cq and ~1 correspond to and a2 and t32 to X2. Then putting C2o = 0 at Xt and 7,2 gives
-
1 I (CO~;a 1-
2
cos
2
+ ~ -v
-
5.3. Astigmatic coma
If imperfect focus were the only important aberration then the o p t i m u m solution would be close at hand. However, we have already used our only two degrees o f freedom and we have not yet addressed the contribution from AX12. To get a feeling for the form of this we proceed as follows: set 3C2o/3X = 0 with the assumption (see earlier) that cos a + cos/3 = const. This gives us a condition that is satisfied at the wavelength (~-rnin) for which 6"2o has a minimum, i.e. -
l/
as well. In both cases the lines joining the dots of the ray trace corresponds to lines o f constant y on the grating and the very heavy lines to the top and bottom edge o f the grating. By using eqs. (6) and (7) we can compute the amount of Y and Z deflection and understand the origin of the parabolic curves. We can also see the effect of increasing Ymax or Zmax on the size and shape of the image in the Y, Z system. Returning to the design question let us try to move the point P to reduce C12. We know that at P, eq. (10) holds. If in addition we insist that Ct2 = 0, i.e. P lies on the X axis of fig. 4 then we find from eq. 05)
sin/3 cos [3
(10)
r*'
7,1:-7--1 I-sin a /l i T _ c o : a ) + sin/3( 1
I
I
I
I
1--
cos_/3) ]
[
I
z= ZMA X Z=0
.5-Z
If we consider the terms containing p* we can see that C12 is independent of p* if eq. (10) is satisfied. In fact we see that the C~2 curves corresponding to various p* values pass through a point P at ~-min as shown in fig. 3. Clearly we would like P to lie on the X axis and we might hope that a p* value corresponding to a fairly horizontal C12 curve might also correspond to an acceptable amount of astigmatism correction. Before we address this question let us examine the nature o f astigmatic coma [27] in a little more detail. Its effect is similar to coma for the case o f near zero astigmatism (see fig. 5) and more like spectrum line curvature if significant astigmatism is present. Fig. 6 shows a realistic case where some defocus is present
I
1.5
Now consider the form of C12 C12
.
C20
Fig. 4. Schematic variation of C2 o and C12 with wavelength.
2
r*
I I i
cos/32)1 = 0 .
Clearly one can solve these for r* and r*'. By this process one can easily arrange to have an approximate focus over the range between 7.L and XR. For spectroscopy where a given AX is required one would choose ~k1 and Xz symetrically between XL and ?'RFor experiments where a given 2xE is required Xl and X2 would be pushed toward the important short wavelength end o f the range.
sin a cos a
~R
I I I
\,
al ) + (~ cOs2/31 ___ OS,l) =0 77
co .)
-
~L
) ~ ~ _ _
Y=YMAX -y = SMALL
0-GAUSSIAN IMAGE
%5
•',Z=2C12 r* yz L~Y= C12 (CO@~) z 2 -1.5 --2 -2
Z~Z2 hi" y2
I -1.5
I -1
I -.5
I 0 Y
I .5
T 1
Ay
I 1.5
Fig. 5. Toroidal grating image with C 12 the dominant aberration. The r' value is chosen for exact focus and the astigmatism is small. Eq. (6), (7) and (15) are used to compute ~Z and Ay and show how a parabolic curve arises for each y value. Y and Z are in 100's of um and the parameters are R =100, p=6.1, r=38.5, r'=17.14 (all cm), 0=150 ° , n = 1200 lines/mm, X = 150 A, k = -1.
M.R. Howells / Vacuum Ultra Violet Monoehromators
sin ~ r .2
sin
129
-- ORDER
(r*')2
'-I- ORDER
• ~L
~R
I I
I I
/C20
Solving this with eq. (10) leads to the conditions o~ = - / 3 ,
r* = r * ' ,
and
~,min= 0 1
So that P in fact lies at the origin of fig. 3. Here we have adopted a second extreme approach. Previously we saw how to optimise C2o whilst ignoring C12. Now we have found a scheme that turns out to optimise C12 rather well but largely ignores C2o. We can still use one degree o f freedom to determine one wavelength of exact focus, XF say (see fig. 6). This involves choosing a value for r* = r*'. To do this we make use of the parameter A=X/XH. From eq. (3) we can rewrite eqs. (2a) and (2b) as /3 = s i n - l ( A cos 0) - 0 , a = s i n - l ( A cos 0) + 0 . Substituting these into eq. (12) and setting Cz0 = 0 for the wavelength AF we find
I
Pl
"2 Fig. 7. Schematic of the variation with wavelength of C20 and C a2 for tlre case r = r'. Note the point P has moved to the origin and Xmin = 0. The instrument can now be operated in either positive or negative order.
from astigmatic coma is much reduced and defocus dominates. Eq. (11) is an interesting equation containing a number o f fruitful cases which are discussed elsewhere [28,29].
(11)
This gives us our monochromator configuration. The ray trace (fig. 7) shows as expected that the curvature [
~
5.4. Overall optimisation
, cos0(1 - c o s 2 0 A } ) r = -,,/(1 - A} cos20)
2
t
I
I
I
I
Whilst the two extreme approaches we have discussed here are physically revealing, neither of them leads to the best TGM designs. To get the best design
I
2
1.5--
I
[
I
I
I
I
I
1.5 -
LkZ = 2 r " C12 yz 1 -.5 A Y -
~jji ~I~y
C12 Z + C20
max ,z max_ 1 -Ymin Zmax .5 -
Z
0 -- So that
_.~L . . . . . . . ~,~ I=~"-GAUSSIAN '~\ IMAGE
-.5-
r 1 /kz \2 ~ zxy=K|C121~i ÷C202y ~ / t ~ 2C12r'y / =v • ",,.1~,/" -1 or " ~
-1.5--
-2
Z~Z2 =,
Z=O
BOTTOM EDGE OF GRATING
-
Z
0
*.5 _
K [ - C?0 2y'K] 4C12r*2 y 2 t ~Y
-1
--
-1.5
I
I
I
I
I
I
I
-1.5
-1
-.5
0
.5
1
1.5
Y
Fig. 6. More realistic image for the same grating as fig. 5. This time r ' = 17.5 cm the other parameters are the same. Now both defocussing and astigmatism are significant. &Z and ~ Y are computed as before with the assumption that 2~Z is determined by C02 and A y by C2o and C12. Again the observed parabolic shape is explained and the role of the two aberrations becomes clear.
-2 -2
I
[
I
I
I
I
I
-1.5
-1
-.5
0
.5
1
1.5
TOROIDAL DIFFRACTION GRATING INT. DIST.
Fig. 8. Image produced by the same toroidal grating as shown in figs. 5 and 6. This time configured with conjugates r = r' = 26.41 cm. These correspond to choosing XF = 0.15X H giving ;~F = 167 A. The image is ray traced for X = 250 A which is considerably out of focus. The dominant influence of C2o (defocus) and the small effect of C12 are clearly seen. IV. XUV MONOCHROMATOR SYSTEMS
M.R. Howells / Vacuum Ultra Violet Monochromators
130
we must search parameter space with a computer program to find the best performance at a sequence of chosen wavelengths within the range of interest. This will represent the best compromise between control of C2o and ClZ which are the main resolution determining aberrations. Once the configuration is established the actual resolution/flux trade-off can be established by choice of the grating width and height. The method used at Brookhaven for doing this is discussed in another contribution to this meeting
[30]. We might legitimately ask how the adequacy of these theoretical and computational design procedures is checked. It is essential to have a good ray trace program for this purpose and fortunately modern microprocessor technology makes this reasonably easy to achieve on a self contained basis. Our approach has actually been to have more than one program using completely different methods of calculation. Since we are dealing with low resolving power, poor imaging, systems it is safe to assume that the geometrical optics approximation is valid.
Appendix Here we list a few of the terms of the expansion
(5). Focussing term C2oy2 /R,
[(cos os4 +
C20
\ r*'
--
cos )l
Coma terin Caoya/R 2, _ 1 Fsin a / c o s 2 a _ cos a ]
C
]
r* sin {cos2
+~
r
~ r*~
t3)l
(13)
-cos./l
Astigmatism term Co2z2/R,
l_[t
Co2 = 2 L\r
cosq ~w ] + ( ) ,
c~,/3); •
(14)
Astigmatic coma term C 1 j z 2 /R 2, 1 Fsin a t 1
c,2
cos a_)
0"
sm (rl'-~ cos + -~p* ]
(is)
The remaining terms are of little importance for approximate focussing situations. However, they can be found in ref. 24.
References [ 1] J.A.R. Samson, Techniques of vacuum ultra violet spectroscopy (Wiley, New York, 1967). [2] A.N. Zaldel and E.Ya. Schreider, Vacuum ultra violet spectroscopy, (Halstead Press, New York, 1970) (translated from Russian). [3] F. Wuilleumier and Y. Farge, eds., Synchrotron radiation instrumentation and new developments (NorthHolland, Amsterdam, 1978). [4] Ed. G.V. Marr and I.H. Munro, in Int. Symp. for Synchrotron radiation users, Daresbury, Jan. 1973 (DNPL/ R26). [5] M. Pouey, J. de Phys. Coll. C4 Supp. Vol. 7 (1978) 188. [6] E. Kallne, H.W. Schnopper, J.P. Delvaille, L.P. van Speybroeck and R.Z. Bachrach, Nucl. Instr. and Meth. 152 (1978) 73. [7] Proc. New York Acad. Sci. Conf. on Soft X-ray microscopy, New York, 1979. [8] J.F. James and R.S. Sternberg, Design of optical spectrometers (Chapman Hall, London, 1969). [9] F.C. Brown, R.Z. Bachrach and N. Lieu, Nucl. Instr. and Meth. 152 (1978) 73. [10] P. Dhez, P. Jaegle, F.J. WuRIeumier, E. Kallne, V. Schmidt, M. Bertand and A. Carillou, Nncl. Instr. and Meth. 152 (1978) 85. [11] K. Codling and P. Michel, J. Phys. E 3 (1970) 685. [12] H. Dietrich and C. Kunz, Rev. Sci. instr. 43 (1972) 434. [13] K.P. Miyake, R. Kato and H. Yamashita, Sci. Light 18 (1969) 39. [14] J.B. West, K. Codling and G.V. Marr, J. Phys. E 7 (1974) 137. [15] M.R. Howells, D. Norman, G.P. Williams and J.B. West, J. Phys. E 11 (1978) 199. [16] W. Eberhardt, G. Kaltsolien and C. Kunz, Nucl. Instr. and Meth. 152 (1978) 81. [17] R.P. Madden and D.L. Ederer, J. Opt. Soc. Am. 62 (1972) 722A and private communication. We gratefully acknowledge the receipt of an unpublished manuscript detailing the design of the original toroidal grating monochromator at NBS. [18] Jobin Yvon publication, see also ref. 26. [19] B. Tonner, these Proceedings, p. 133. [20] C. Pruett, private communication. [21] M.R. I-Iowells and W.R. McKinney, BNL Rept. No. 26027 (1979). [22] G.P. Williams and M.R. Howells, BNL Rept. No. 26121 (1979). [23] V. Rehn and V.O. Jones, Opt. Eng. 17 (1978) 504. [24] H. Haber, J. Opt. Soc. Am. 40 (1950) 153. [25] T. Namoika, J. Opt. Soc. Am. 49 (1959) 446. [26] D. Lepere, Nouv. Rev. Opt. 6 (1975) 173. [27] W.T. Welford, in Progress in optics, Vol. 4 (NorthHolland, Amsterdam, 1965).
M.R. Howells / Vacuum Ultra Violet Monochromators
[28] M.R. Howells and T.J. Aggus, BNL Rept. No. 26341 (1979). [29] M.R. Howells, in preparation. [30] W.R. McKinney and M.R. Howells, these Proceedings, p. 149.
131
[31] M.J. Hagemann, W. Gudat and C. Kunz, DESY SR-74/7 (1974) table 5 and 6. [32] A. Lukirski and E. Savinov, Opt. Spect. 14 (1963) 152. [33] Malina and Cash, Appl. Opt. 17 (1978) 3309.
IV. XUV MONOCHROMATOR SYSTEMS