have been extensively selected for application tox-ray astronomy in space.4–7 The optical configura-tion of these telescopes consists of nested double-reflection mirrors operating at grazing-incidenceangles low enough to ensure high reflectivity fromthe coating material, normally gold. Recently, moresophisticated coatings, based on multilayer struc-tures, have been proposed and investigated for fu-ture space missions.8–10 In type I Wolter mirrors,the x-ray radiation from distant sources is first re-flected by a parabolic surface and then by a hyper-boloid, both with cylindrical symmetry around theoptical axis. The two surfaces are arranged in acoaxial configuration and share a common focus.The second focus of the hyperboloid is the imagefocus. The type I Wolter design is known to allowAbbe’s condition11,12 to be approximately satisfied,thus ensuring low coma aberration. An evolution ofthe Wolter design, the Wolter–Schwarzshild config-uration,13,14 exactly satisfies Abbe’s condition.
More recently, a variation of the type I Wolterdesign already proposed for other applications,15,16
in which the parabolic surface is replaced by anellipsoid, has found application for collecting the
radiation at 13.5 nm emitted from a small hotplasma used as a source in extreme-ultraviolet(EUV) microlithography,17–20 currently considereda promising technology in the semiconductor indus-try for the next generation of lithographic tools. Thehot plasma21–23 in an EUV lithography source is gen-erated by an electric discharge [discharge-producedplasma (DPP source)] or by a laser beam [laser-produced plasma (LPP source)] on a target consistingof lithium, xenon, or tin, the last one currently beingthe most promising. The emission from the source isroughly isotropic and, in DPP sources, is limited by thedischarge electrodes to an angle of approximately 60°or more from the optical axis.
The purpose of the collector in EUV sources is totransfer the largest possible amount of in-band poweremitted from the plasma to the next optical stage, theilluminator, of the lithographic tool. With reference toFig. 1, the radiation from the source is first reflectedby the hyperbolic surface, then reflected by the ellip-tical surface and finally focused to an image or inter-mediate focus. As in the type I Wolter telescope, theelliptical and the hyperbolic surfaces share a commonfocus. The output optical specification of the collector,in terms of numerical aperture and etendue, mustmatch the input optical requirements for the illumi-nator. The collector is then designed to maximize theoverall efficiency, defined as the ratio between thein-band power at the intermediate focus and the totalin-band power radiated by the source in 2� sr, whilematching the optical specification of the illuminatoron one side and withstanding the thermal load andthe debris from the plasma source on the other side.Indeed, the power requirement for in-band radia-
F. E. Zocchi ([email protected]) is with Media LarioTechnologies, Località Pascolo, 23842 Bosisio Parini (LC), Italy.
Received 18 May 2006; revised 31 July 2006; accepted 31 July2006; posted 7 August 2006 (Doc. ID 71131).
tion at the intermediate focus is currently increasingfrom the original 115 W (Ref. 24) toward 180 W andmore25 because of the expected increase in exposuredose required to achieve the desired resolution andlinewidth roughness of the pattern transferred ontothe wafer. Since the maximum conversion efficiencyof both the DPP and LPP sources is limited to a fewpercent,21–23 and since the reflectivity of normal-incidence mirrors in the illuminator and the projec-tion optics box cannot exceed approximately 70%(Refs. 26–29) for each of the six to eight mirrors30,31 ormore32 along the optical path to the wafer plane, thecollector must withstand thermal loads in the rangeof several kilowatts. Deformations induced by suchhigh thermal loads on the thin metal shell that themirrors are made of33,34 may jeopardize the stabilityand the quality of the output beam of the collectoreven in the presence of integrated cooling systems onthe back surface of the mirrors.35
It is clear from the above discussion that any im-provement in the collector efficiency has benefits forrelaxing the effort in developing extremely powerfulsources, improving the optical quality and stability ofthe collector output and increasing the collector life-time. For a given maximum collection angle on thesource side, the collector efficiency is determinedmainly by the reflectivity of the coating on the opticalsurface of the mirrors. Figure 2 shows some examplesof the dependence of the reflectivity on the grazing-incidence angle for some selected materials at awavelength of 13.5 nm.36 Since each ray experiencestwo reflections, the overall reflectivity is given by theproduct of the reflectivity of each of the two reflec-tions. It is easy to show that the overall reflectivity ismaximized when the two grazing-incidence angles,and thus the reflectivity of the two reflections, areequal,35 at least for the kind of dependence on thegrazing-incidence angle shown in Fig. 2. This condi-tion cannot be satisfied for all rays in a type I Wolterdesign. Indeed, for each mirror, the two grazing-incidence angles can be made equal for one ray atmost.
In this paper a new design of a double-reflectioncollector mirror is proposed, in which the above con-dition is satisfied for all rays collected by each mirror.The theoretical treatment and the description of thedesign is given in Section 2 where the expected effi-ciency of a nested collector based on the new mirror
design is also compared to the efficiency of a type IWolter collector. Although Abbe’s condition is not sat-isfied in the proposed design, coma aberration is ofconcern only to the extent that it affects the collectorefficiency. Because of the finite size of the plasmasource and possibly the shape errors of the collectormirrors, the relative contribution of coma aberrationis expected to be negligible.
For completeness the design of a double-reflectionmirror with equal reflection angles is also briefly out-lined in Section 3 for the case of an object at infinity.
2. Collector Design
In the geometry shown in Fig. 3, a ray emitted fromthe object or source focus S is reflected at point P onthe first surface, reflected at point Q on the secondsurface, and finally focused to the image or interme-diate focus (IF). Cylindrical symmetry around theoptical axis is assumed. The positions of the sourceand the image focus define the vector, 2c � IF � S oflength 2c. The ray path is described by the threeadjacent vectors �1u1 � P � S, �2u2 � IF � Q, and�3u3 � Q � P of length �1, �2, and �3, respectively. Thedirection of each vector is defined by the unit vectorsu1, u2, and u3 forming angles �1, �2, and �3 measuredcounterclockwise with respect to the optical axis. Ifwe assign the three vectors �1u1, �2u2, and �3u3 asfunctions of a parameter t, the geometry of the cross
Fig. 1. Conceptual optical layout of a type I Wolter collector forEUV plasma sources. Cylindrical symmetry around the opticalaxis is assumed.
Fig. 2. Theoretical reflectivity of selected materials at 13.5 nm(Ref. 36).
Fig. 3. Geometry and conventions of the two-reflection mirror.Cylindrical symmetry around the optical axis is assumed.
sections of the two surfaces is defined with respect toS by the tips of the vectors �1u1 and �1u1 � �3u3. Theprecise identification of t with one of the geometricalquantity in Fig. 3 does not need to be specified at thisstage. It can be chosen to be the arc length along thecross section of one of the two surfaces or to be one ofthe angles, for example, the emission angle �1, as willbe done later. For the present we require only thatthe functional dependences of �1, �2, �3, u1, u2, and u3on t is differentiable.
However, the three vectors �1u1, �2u2, and �3u3 can-not be arbitrarily defined as functions of t. Obviously,the three vectors must satisfy the following relation:
�1u1 � �2u2 � �3u3 � 2c. (1)
In addition, for a spherical wave emitted from thesource S and reflected by the two surfaces to be fo-cused to the image focus IF, the optical path must bethe same for all the rays. If 2a is the constant lengthof the optical path, the following condition must besatisfied:
�1 � �2 � �3 � 2a. (2)
Finally, reflection conditions at points P and Q arealso required. By inspection of Fig. 3, the reflectioncondition at point Q can be written as
d��2u2�dt · �u2 � u3� � 0, (3)
where the dot denotes scalar product. Equation (3)says that the unit vectors u2 and u3 have equal pro-jections along the tangent to the cross section of thesecond surface at point Q.
A similar relation can also be written at point P.However, it is easy to show that the reflection condi-tion at P is not independent from Eqs. (1), (2), and (3).Indeed, by taking the derivative of both Eqs. (1) and(2), and by noting that d�1�dt � u1 · d��1u1��dt, wehave
d��1u1�dt �
d��2u2�dt �
d��3u3�dt � 0, (4)
d��1u1�dt · u1 �
d��2u2�dt · u2 �
d��3u3�dt · u3 � 0. (5)
By solving for the third term in Eq. (4), substituting itin Eq. (5), and taking into account Eq. (3), we get
d��1u1�dt · �u1 � u3� � 0, (6)
which is the reflection condition at P. Conversely, Eq.(2), that is, all rays from the source to the image havea constant optical path length, is a consequence of thereflection conditions (3) and (6) on both mirrors.
Equations (1), (2), and (3) or (6) are a system offour scalar equations in six unknowns, that is, thethree lengths �1, �2, and �3 and the three angles �1,�2, and �3. Since we need one free parameter todefine the position of a point along the mirror profile,we are left with the possibility of adding anotherconstraint. As discussed in Section 1, we chose thetwo grazing-incidence angles �13 � ��1 � �3��2 and�23 � ��3 � �2��2 to be equal:
�1 � �3 � �3 � �2. (7)
The differential Eq. (3) cannot be directly integrated.However, a linear combination of Eqs. (3) and (6) can,provided relation (7) is satisfied. By calculating thederivative in Eq. (3),
��2
du2
dt � u2
d�2
dt � · �u2 � u3� � 0, (8)
and noticing that u2 · u2 � 1 and u2 · du2�dt � 0,we get
d�2
dt �1 � u2 · u3� � �2
du2
dt · u3. (9)
Since u2 and u3 have components �cos �2, sin �2� and�cos �3, sin �3�, respectively, Eq. (9) implies
1�2
d�2
dt � cot��3 � �2
2 �d�2
dt . (10)
A similar relation can be obtained starting from (6),
1�1
d�1
dt � cot��3 � �2
2 �d�1
dt . (11)
From Eq. (7) we can eliminate �3 � ��1 � �2��2 in bothEqs. (10) and (11), which become
In addition to Eq. (15), the other relations thatdefine the mirror geometry can easily be derived fromEqs. (1), (2), and (7). By substituting �3 from Eq. (2)into Eq. (1), we obtain
�1�u1 � u3� � �2�u2 � u3� � 2au3 � 2c. (16)
By multiplying Eq. (16) by �u2 � u3�, and noting thatit is orthogonal to �u2 � u3�, it gives
where we have used relation (7) expressed in the formu1 · u2 � u2 · u3. Instead, if we multiply Eq. (16) by�u1 � u3�, which is normal to �u1 � u3�, we obtain
Finally, by projecting Eq. (16) along the direction ofthe optical axis, we find the third and last equation ofthe following system:
�2�1 � k sin�4��2 � �1
4 �,�1 � �2 � 2c
cos �2 � cos �1
cos��1 � �2� � 1,
�1 cos �1 � �2 cos �2
��2a � �1 � �2�cos��1 � �2
2 �� 2c. (20)
If �1, a, c, and k are given, these are three equationswith three unknowns, �1, �2, and �2, that can be solvednumerically. The resulting profile is then rotatedaround the optical axis to obtain the axial symmetrictwo-surface mirror. The surfaces defined by Eqs. (20)cannot be described by second-order algebraic equa-tions. In particular, these surfaces are not generatedby conic sections and do not have a common focus ashappens in two-reflection systems consisting of on-axis or off-axis confocal ellipsoids or hyperboloids.37,38
One convenient manufacturing process for nestedgrazing-incidence mirrors is based on electroform-ing,33,34 by which the mirror is obtained by galvanicreplication from a negative master. In this case, it isappropriate to extend the two reflecting surfaces un-til they join at a given point. In this way the twosurfaces can be manufactured in a monolithic struc-ture, thus avoiding the need for further relativealignment. The values �1,R and |�2,R| of the angles �1and |�2| at the intersection point R are the minimumangles at both the source and the image focus. Since�3 � 0 at R, assuming that c is assigned, the length
�1,R and �2,R are known and the constants a and kare determined by relation (2) and the first of rela-tions (20):
a ��1,R � �2,R
2 , (21)
k � �1,R�2,R sin4��2,R � �1,R
4 �. (22)
When �1 is allowed to increase from its minimumvalue �1,R, relations (20) give the shape of both sur-faces of the mirror. The maximum value of �1 is ar-bitrary to a certain extent. A convenient choice issuch that the minimum distance of the mirror fromthe source focus is some prescribed value �̄1 so that aspherical region of radius �̄1 around the source is leftfree for the hardware required to mitigate the debrisfrom the plasma source. Alternatively, to ease themounting of the mirror on a common supportingstructure, the maximum value for �1 is chosen suchthat all the mirrors end at the same horizontal coor-dinate on the side of the image focus.
The outer mirrors can then be calculated itera-tively as follows. The vertex R� of the second mirror isdefined by the intersection of the rays through pointsA and B. These rays also define the minimum values�1,R� and |�2,R�| of the angles �1 and |�2| and thecorresponding length of �1,R� and �2,R�. The above pro-cedure can then be applied to calculate the new con-stant values a� and k� from Eqs. (21) and (22) and themirror shape from relations (20). The process canthen be iterated to cover the desired numerical aper-ture with a proper number of nested mirrors.
Figure 4 shows the optical layout of a nested col-lector consisting of 15 double-reflection mirrors, eachwith a thickness of 2 mm. The design assumes a focallength 2c of 1500 mm, a minimum distance �̄1 be-tween the optics and the source focus of 110 mm, andminimum and maximum angles of the radiation atthe intermediate focus of 1.5° and 8°, respectively.The corresponding minimum and maximum collectedangles are 9.2° and 86.8°, equivalent to 5.3 sr (takinginto account the obscurations from the shell thick-ness). As mentioned in Section 1, the collection effi-ciency of the collector is defined as the ratio betweenthe power at the image or intermediate focus and thepower emitted from the source in 2� sr. For an iso-tropic point source, the collection efficiency of eachmirror is given by
� ���1,R
�1,A
R��13�R��23�sin �1d�1 ���1,R
�1,A
R2��1 � �2��1�4 �
sin �1d�1, (23)
where R��� is the mirror reflectivity at the grazing-incidence angle �. Assuming a reflective coating ofruthenium with theoretical reflectivity, the total col-lection efficiency for the collector in Fig. 4 is 50.9%.
This value should be compared with the calculatedefficiency of 40.1% for a collector design based on atype I Wolter configuration matching the same bound-ary conditions in terms of focal length, angles at theintermediate focus, and maximum collected angle.
The total reflectivity experienced by each ray isplotted in Fig. 5 as a function of the emission angle.The design with an equal reflection angle is moreeffective than the type I Wolter design at a largeemission angle only. Indeed, the inner mirrors collecta small angular range so that the gain in reflectivitywhen the two grazing-incidence angles are equal islimited.
A few comments are worthwhile about the designof both a type I Wolter collector and the proposedconfiguration. First, it seems impossible to design atype I Wolter collector that, in addition to the givenfocal length, angles at the intermediate focus, andmaximum collected angle, also satisfies the conditionof a minimum distance of 110 mm from the sourcefocus. Indeed, the example discussed above has aminimum distance of 68.6 mm and it is not compliantwith the specification. Second, the designer of a typeI Wolter collector has the freedom to adjust the shapeof each mirror by specifying one free parameter foreach nested mirror. On the other hand, this freedomis not available in the proposed design. Third, in bothdesigns the mirrors do not end at the same horizontalcoordinate and this increases the complexity of themechanical design of the support structure.
3. Telescope Design
For the sake of completeness, in this section a two-reflection design for an object at infinity is brieflysummarized. In this case u1 is parallel to the opticalaxis and �1 � 0, as shown in Fig. 6. Only the projec-tion of Eq. (1) on the optical axis is applicable,
�1 � �2u1 · u2 � �3u1 · u3 � 2c. (24)
Instead, Eqs. (2) and (3) are still valid. Also in thiscase, the reflection condition in P is a consequenceof Eqs. (2), (3), and (24). As before, by taking thederivative of Eqs. (2) and (24) and noting thatdu1�dt � 0, we get
Fig. 4. Optical layout of a nested collector based on the mirror design proposed in the paper.
Fig. 5. Total reflectivity experienced by each ray as a function ofthe emission angle for the design of Fig. 4 and for a type I Wolterdesign.
which is the reflection condition at P.We chose again the two grazing-incidence angles
�13 � ��3�2 and �23 � ��3 � �2��2 to be equal, giving�3 � �2�2. Thus, Eq. (10) gives
1�2
d�2
dt � �cot��2
4 �d�2
dt , (26)
which, upon integration, implies
�2 � k sin�4��2
4 �. (27)
If we choose �2 as the independent variable, Eq. (27)gives �2 and we are left to solve the linear system ofEqs. (2) and (24) in the unknown �1 and �3, or
�1 � �3 cos��2�2� � 2c � �2 cos��2�,
�1 � �3 � 2a � �2. (28)
As before, if c is assigned, the constants a and k aredetermined by Eqs. (2) and (27) once the minimumvalue |�2,R| of the angle |�2| at point R is given,
a ��1,R � �2,R
2 , (29)
k � �2,R sin4��2,R
4 �. (30)
The process for the determination of the first andsubsequent mirrors is then identical to that describedfor the collector in Section 2.
It is worth noting that double-reflection conicalmirrors for x-ray telescopes39,40 also satisfy the con-dition of equal reflection angles on the two surfaces.In this case, however, axial rays do not come to apoint geometric focus and the optics is not correctedfor on-axis spherical aberration.
4. Conclusions
The design of double-reflection mirrors with equalgrazing-incidence angles is effective in increasing theefficiency of collectors for EUV microlithography atlarge emission angles. The increasing demand for ahigh power level needed for high-volume manufac-turing tools requires pushing the performance of thesubsystems to the physical limits. For collectors thisimplies, among other things, increasing the collectedsolid angle and improving the overall reflectivity. Inthis direction, the proposed design has a collectionefficiency about 27% greater than a particular type IWolter collector design used as a reference for theselected example specifications.
The author thanks Enrico Benedetti for performingthe numerical calculations and preparing Fig. 4.
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