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`(19) —d
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`Europaisches Patentamt
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`European Patent Office
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`Office européen des brevets
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`(11)
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`EP 1 376 185 A2
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`(12)
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`EUROPEAN PATENT APPLICATION
`
`(48) Date of publication:
`02.01.2004 Bulletin 2004/01
`
`(51) Int cl.7; GO2B 7/18
`
`(21) Application number: 03013982.8
`
`(22) Date offiling: 20.06.2003
`
`(84) Designated Contracting States:
`AT BE BG CH CY CZ DE DKEE ESFI FR GB GR
`HU IE ITLILU MC NL PT RO SE SISK TR
`
`(71) Applicant: Nikon Corporation
`Tokyo 100-8331 (JP)
`
`Designated Extension States:
`AL LT LV MK
`
`(72) Inventor: Sogard, Michael
`Menlo Park, CA 94025 (US)
`
`(80) Priority: 20.06.2002 US 390293 P
`04.06.2003 US 455750 P
`
`(74) Representative: Viering, Jentschura & Partner
`Patent- und Rechtsanwéalte,
`Steinsdorfstr. 6
`
`80538 Miinchen (DE)
`
`(54) Minimizing thermal distortion effects on EUV mirror
`
`has a high heattransfer coefficient even if its pressure
`(57)=Amirroris provided with throughholes, or chan-
`is not too high.
`In some applications the gap may be
`nels, formed through its main body and a coolant pipe
`of a heat-conductive material is inserted in each of the
`filled with a heat-conductive fluid. It may be preferable,
`depending upon the circumstances, to form these chan-
`nels proximally to the surface on which radiation is made
`incident. Additionally, the surface of the side of the mirror
`opposite the reflective side may be heated by auxiliary
`heat sources.
`
`channels for passing a cooling fluid inside. The outer
`wall of the coolant pipe does not contact the inner wall
`of the channel, and thereis left a gap in between. The
`gap contains aheat-conducting gas such as helium. The
`
`
`gap is of a width of less than 100 um such that the gas
`
`EP1376185A2
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`FIG. 1A
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`Printed by Jouve, 75001 PARIS (FR)
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`EP 1 376 185 A2
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`Description
`
`BACKGROUND OF THE INVENTION
`
`[0001] This invention is in the technical field of mirrors such as extreme ultraviolet (EUV) mirrors and relates in
`particular to the problem of minimizing thermal distortion effects on such a mirror.
`[0002] The EUV mirror absorbssignificant amounts of heat from the EUV radiation primarily because its reflectivity
`is not very high. The absorbed heat causesdistortion of the mirror surface from thermal expansion effects. The distortion
`in turn leads to optical aberrations and hence must be minimized as much as possible by whatever means. The thermal
`distortion has been modeled and is generally considered to consist of (1) a local distortion which is associated with
`the local temperature distortion, and (2) a so-called "global distortion" which is related to thermal stresses which alter
`the shape of the entire mirror. Both of these typesof distortion must be appropriately dealt with. The present invention
`relates to the problem of global distortion.
`[0003] The global distortion depends both on the thermal and mechanical boundary conditions and constraints, and
`on the details of the mirror design and illumination pattern. In some studies, it is assumed that the back of the mirror
`is in contact with a temperature reservoir. This may be expected to minimize distortion because a maximum amount
`of heat can thus be removed, but this may be true only with regard to the local distortions. Since the heated front
`surface coupled with the cooled back surface creates a temperature gradient in the direction approximately perpen-
`dicular to the mirror surface, this may lead to "bowing" of the mirror.
`If the mirror is held kinematically, nothing will
`prevent such a bowing effect. Thus, such a cooling technique should be avoided, unless the back of the mirror can be
`attached to a base plate which is a part of the temperature reservoir and is rigid enough to prevent the mirror from
`bowing. Local distortions will still remain.
`
`SUMMARYOF THE INVENTION
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`[0004] An EUV mirror embodying this invention may be characterized as having throughholes, or channels, formed
`within the mirror member and a coolantpipe of a heat-conductive material inserted in each of the channels for passing
`a cooling fluid inside, its outer wall not contacting the wall of the channel but leaving a gap in between. It may be
`preferable to form these channels proximally to the surface on which light in made incident. The gap contains a gas
`with high thermal conductivity such as helium which is maintained below atmospheric pressure. The gap is of a width
`of less than 100 um such that the gas hasahigh heattransfer coefficient even if its pressure is not too high. The
`coolant flow may behighly turbulent, to further increase heat transfer from the mirror. Additionally, the mirror may be
`heated on its rear surface, the surface opposite the reflective surface, to eliminate residual global thermal distortion.
`In other applications the gap may contain a thermally conductive fluid maintained at a pressure comparable to the
`ambient pressure surrounding the mirror.
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`BRIEF DESCRIPTION OF THE DRAWING
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`[0005] The invention, together with further objects and advantages thereof, may best be understood with reference
`to the following description taken in conjunction with the accompanying drawings in which:
`
`Fig. 1A is a schematic optical diagram of a representative embodimentof an X-ray microlithography system com-
`prising at least one multilayer-film reflective optical element according to any of the embodiments of this invention,
`Fig 1B is a detailed view of the projection-optical system of the microlithography system shownin Fig. 1A, and
`Fig. 1C is aschematic optical diagram of another representative embodiment of an X-ray microlithography system
`comprising at least one multilayer-film reflective optical element (including a reflective reticle) according to any of
`the embodiments of this invention;
`
`Fig. 2 is a process flow diagram illustrating an exemplary process by which semiconductor devices are fabricated
`by using the apparatus shownin Fig. 1 according to the present invention;
`
`Fig. 3 is a flowchart of the wafer processing step shownin Fig. 2 in the case of fabricating semiconductor devices
`according to the present invention;
`
`Fig. 4 is a schematic drawing of an EUV mirror system embodying this invention, including a EUV mirror of which
`a schematic sectional view is included;
`
`Fig. 5A is a schematic sectional view of one of the cooling channels and Fig. 5B is a schematic sectional view of
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`a cooling channel having more than one pipe (two pipes) therethrough;
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`EP 1 376 185 A2
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`Fig. 6 is a graph showing the relationship between the heattransfer coefficient of helium gas in a gap and its
`pressurefor different gap widths;
`
`Fig. 7 showsthe heat transfer coefficient of the coolant within the cooling tube for conditions ofa full turbulent flow
`and a laminarflow;
`
`Fig. 8A is a schematic sectional view of a cooling system embodying the invention;
`
`Fig. 8B is a schematic sectional view of the cooling system of Fig. 8A;
`
`Fig. 8C is a schematic sectional view of another cooling system embodying the invention;
`
`Fig. 8D a schematic sectional view of another embodiment of the cooling channels;
`
`Figs. 9A and 9B are schematic sectional viewsofstill another cooling system embodying the invention, Fig. 9A
`being a sectional view taken along line 9A-9A of Fig. 9B and Fig. 9B is a sectional view taken along line 9B-9B of
`Fig. 9A;
`
`Fig. 10 defines the terms and coordinate system describing the temperature distribution in the mirror;
`
`Fig. 11 is a schematic which defines cooling planes relative to the cooling channels;
`
`Fig. 12 shows an embodimentof a control system which provides appropriate heating to the back surface of the
`mirror;
`
`Fig. 13 shows an embodimentof a radiant heating system for the back surface of the mirror;
`
`Fig. 14 shows an embediment of a heating system with a remote radiation source which is projected onto the back
`of the mirror;
`
`Fig. 15 shows another embodimentof part of a projection heating system for the back surface of the mirror; and
`
`Fig. 16 is a schematic illustration of another embodiment of a projection heating system which can provide real
`time correction to the radiation intensity distribution at the back of the mirror.
`
`DETAILED DESCRIPTION OF THE INVENTION
`
`Fig. 4 shows schematically an EUV mirror system 10 embodying this invention, including an EUV mirror 20
`[0006]
`with a front surface 22 and a back surface 24. The front surface 22 is where EUV illumination is intended to impinge.
`The mirror surface is normally curved and is assumed to possess, at least to a near approximation, axial symmetry.
`Throughholes (serving as cooling channels) 30 are provided through the mirror 20,
`lying in a plane approximately
`perpendicular to the axis of symmetry. The cooling channels 30 may serve to cause a cooling fluid (not shown) to flow
`therethrough. Since the coolant may be approximately of an atmospheric pressure or more while the mirror 20 typically
`operates in a vacuum environment, this pressure difference tends to cause some distortion of the mirror 20. One
`method of eliminating the effects of such distortion is to preliminarily polish the mirror 20 with the channels 30 over-
`pressurized by a specified amount such as one atmosphere, such thatthis distortion effect will be preliminarily polished
`out. However, to completely eliminate the distortion effect, this method must include the pressure drop suffered by the
`fluid along the channel length, which is caused bythe fluid's viscosity. Also shownis an auxiliary source of heat 90
`which heats the surface of the mirror opposite the reflective side. This heat source creates thermal stresses in the
`mirror opposite to those created bythe illumination irradiating the reflective side, thereby canceling their effects.
`[0007] Another approach is, as shown in Fig. 5A, to provide a pipe (or a conduit) 32 inside each cooling channel 30.
`The pipe 32 is made of a material with high thermal conductivity and a cooling fluid 34 flows therethrough. As shawn
`in Fig. 5A, the channel 30 is sufficiently oversized that the pipe 32 therein does not contact the inner wall of the channel
`30, leaving an annular gap 36 therebetween. The gap 36 between the inner wall of the channel 30 and the pipe 32 is
`filled with a gas of high thermal conductivity, such as helium gas, for conducting heat between the material of the mirror
`20 and the coolantinside the pipe 32. It now goes without saying that the cross-sectional shape of the channels need
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`EP 1 376 185 A2
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`not be circular and that one channel may contain two or more pipes inside, as shown in Fig. 5B.
`[0008]
`Fig. 6 showsthe relationship between the heattransfer coefficient of helium gas in a gap and its pressure for
`different gap widths. The heat transfer coefficient increases with pressure in a lower pressure range but the rate of
`variation decreases at higher pressures. Fig. 6 also showsthat the heat transfer coefficient remains relatively high
`even at lower pressures if the gap width is small. For a gap of the order of a few tens of microns or less, the helium
`gas may be maintained at a fraction of one atmosphere with little reduction in thermal conductivity, thereby reducing
`pressure distortion of the mirror 20. For example, a pressure of 10 kPais just one tenth of atmospheric pressure, yet
`relatively high thermal conductivities remain attainable at that pressure. For the purpose of the present invention, the
`width of the gap is set less than 100um and preferably less than 50um.
`[0009]
`If the mirror material above the channel 30 is approximated by a plate of thickness t,, the deformation from
`the pressureP in the channel maybe estimated from the theory of bending of elastic beams underload. If the width
`of the channel (the channel dimension approximately parallel to the mirror surface) is c, the maximum deformation d
`of the plate normal to the plane of the mirror is given by
`
`d= Pc"/326t,°,
`
`(1)
`
`whereE is Young's modulus for the mirror material. The deformation of the mirror is linearly proportional to the pressure
`inthe channel, so reducing the pressure by a factor of 10 also reduces the deformation by a factor of 10. Alternatively,
`Eq. 1 showsthat for a given amount of deformation, reducing P by a factor of 10 means the distance t, may be reduced
`by the factor 1013 = 2,15, so the channels may be located closer to the mirror surface.
`[0010]
`Fig. 7 showsthe heattransfer coefficient of a typical cooling fluid (Fluorinert FC-3283, manufactured by 3M
`Corp. of Minneapolis, MN) as a function of Reynolds numberfor the cases of laminar flow and full turbulent flow. The
`Reynolds number R is defined by R = pDv/n, where p is the coolant density, D is the coolant pipe inner diameter, v is
`the coolant flow velocity along the pipe axis, and 7 is the coolant viscosity. Creating full turbulent flow conditions in-
`creasesthe heat transfer by several orders of magnitude compared to the case of laminar flow. The expressions used
`to calculate these results are described in the article "Some fundamentals of cooled mirrors for synchrotron radiation
`beam lines" by M. Howells in Optical Engineering 35, 1187(1995), the contents of which are incorporated herein by
`reference. However, associated with turbulent flow are pressurefluctuations in the fluid which can create vibrations.
`Such behavior is absentin laminar flow, so laminar flow conditions have typically been used in the prior art when high
`mechanical stability of the mirror surface is required. Because the coolantpipe is mechanically isolated from the mirror
`by the gap, turbulent flow may be employed with the present invention, thereby permitting higher heat transfers to be
`achieved. Although the cooling fluid is typically a liquid, it might also be a gas.
`[0011]
`To illustrate the advantage of the present invention a calculation was done, comparing the thermal resistance
`between the mirror material and the coolant flowing through a channel under laminar flow conditions (prior art), and
`the thermal resistance associated with heat flow from the mirror material through a 50 tum He gas gap, a copper tube,
`and into the coolant flowing within the tube under full turbulent flow conditions. For a given temperature difference
`between two points, heat flow is inversely proportional to the intervening thermal resistance, so a lower thermalresist-
`ance permits higher heat transfer. The copper tube was assumed to have 4 mm inner diameter and 0.5 mm wall
`thickness. Based on the information presented in Figs. 6 and 7, the heat transfer coefficients for the He gas and the
`coolant flowing under turbulent conditions were assumedto be 1000 W/m2-K, and the heat transfer coefficient for the
`coolantflowing underlaminar conditions was assumed to be 58 W/m2-K. Underthese conditions the thermal resistance
`associated with the prior art case was found to be approximately 17 times larger than that associated with the present
`invention. Therefore heat transfers out of the mirror material can be greater than an order of magnitude higher with
`the present invention.
`[0012] A preferred position for providing the channels 30is difficult to determine because there are many factors to
`be taken into consideration. A model calculation was carried out for a mirror in the shapeof a circular disk of radius r
`being heated uniformly from one side (on the top surface). The channels 30 are assumedto be in a plane (the "cooling
`plane") at distance t, from the top surface which is uniformly heated and at distance tg from the opposite surface, the
`total thickness of the mirror being t = ty + tp. Thus, a uniform temperature gradient is assumed over a distance(in the
`direction of the thickness) of t; and the temperature is assumed to be constant over the adjacent distance of tp. The
`temperature gradient causes the entire disk to deform and the question is how the deformation changesast, is varied.
`[0013] The deformation was calculated by stress and strain formulas in Table 24 of Roark and Young ("Roark's
`Formulas for Stress and Strain," McGraw-Hill, 6th ed.), the contents of which are incorporated herein by reference,
`which were applied by using the assumption that the disk is separated at the cooling plane into a thermally stressed
`top plate and a bottom plate which is deformed by the application of moments along its periphery into a spherical cap.
`Thermal stress alone will deform the upper plate into a spherical cap. An application of additional moments along its
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`EP 1 376 185 A2
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`periphery will change its radius of curvature to match that of the lower plate such that they join together. A final defor-
`mation is calculated on the further requirement that the stress of the bottom surface of the top plate be equal to that
`of the top surface of the bottom plate. The final displacement y of the plate center thus calculated is in the form of y =
`(r2yAT/2\/(ty4t5) where ¥ is the coefficient of thermal expansion and ATindicates the temperature difference through
`the thicknessof the top plate. This equation suggests that it does not matter where the cooling plane is. However, for
`a plate heated on one side by a constant heat flux F and cooled by a temperature reservoir on the other side, the
`temperature difference AT betweenthe two sides scales with the quantity AT=Ft,/K whereK is the thermal conductivity
`of the material according to Carslaw and Jaeger in Section III.8 of "Conduction of Heat in Solids," Oxford, the contents
`of which are incorporated herein by reference. It is assumed here and belowthat the heatflux F represents that part
`of the incident flux which is absorbed by the surface. Therefore, the deformation y is in fact proportional to ty, and the
`conclusion hereby obtained is that the cooling plane should be as close to the heated surface as possible.
`[0014] There are factors that limit how small t; can be made.
`If the cooling network involves channels machined
`within the mirror, they must be far enough from the mirror surface such that they will not influence the mirror figure. As
`explained above, the helium gas in the gap 36 is maintained at a low pressure to reducedistortion of the mirror surface,
`while the gap is small enough to ensure a high heat transfer coefficient. However,if the channels are very close to the
`mirror surface, the material directly above the channels may experiencea slightly larger temperature gradient than
`material lying in a plane midway between adjacent channels. This could contribute to local temperature distortion of
`the mirror. In the above cited paper, Howells teachesthat t, should be at least several times the channel width to avoid
`local distortion effects. He also teaches that forces from polishing and coolant pressure dictate a practical range for t,
`of approximately 0.5 mm to 5 mm. Since coolant pressureis significantly reduced in this invention, smaller values of
`t, may be allowed.
`[0015] The coolant 34 flowing within the cooling pipe 36 may be in a turbulent state for higher heat transfer, but the
`consequent vibrations should not perturb the mirror surface. Successful application of this invention requires that the
`cooling pipes be mechanically isolated from the mirror while remaining surrounded by a controlled atmosphere of He
`gas. Some embodiments are described which satisfy these requirements. Figs. 8A and 8B show schematic views of
`a mirror 800 with a series of cooling pipes 810 running through parallel channels 815 in the mirror. Each pipe is centered
`in its channel, and the channels are sealed at the mirror periphery by compliant seals 820 made of rubber or some
`other visco-elastic material. The compliant seals will help to isolate and damp any vibrations the pipe experiences from
`turbulent fluid motion. He gas or some other high thermal conductivity gas is admitted to each channel through an inlet
`825 which is connected to a gas manifold 830. The gas pressurein the manifold 830 and the mirror channels 815 is
`controlled by means of a He gas supply 835, a vacuum pump 840, a valve 845 to switch between them, a pressure
`gauge 850, and a gas pressure controller (not shown). The coolant pipes run to a coolant manifold (not shown) which
`directs coolant to them from a reservoir which may be part of a closed loop system involving a heat exchanger and
`temperature controller (not shown). Such coolant systems are well knownin the art.
`[0016]
`Fig. 8C shows acoolantpipe 810 within a channel 815. The channel has visco-elastic gas tight seals 820 at
`the mirror periphery. In addition, the pipe is centered within the channel by means of a helical coil 822 of visco-elastic
`material. This coil will also help to damp any vibrations the pipe experiences from turbulentfluid motion in the coolant.
`Other designs, employing spacers madeof visco-elastic material may also be used.
`[0017]
`Fig. 8D shows a cross section of several channels 815a and coolant pipes 810a with a rectangular rather
`than circular cross section. The paper by Howells teachesthat a tall, narrow rectangular geometry provides superior
`heattransfer capabilities over a circular cross section, when overall heat transfer is limited by convection in the coolant.
`When overall heat transfer is limited by conduction in the mirror material, the shape of the channels is less important.
`[0018]
`Figs. 9A and 9B show an embodiment where coolant pipes 910 are further mechanically isolated from the
`mirror 800. The pipes make no mechanical contact with the mirror. They are positioned relative to the channels 915
`by a rigid framework (not shown) which is referenced to the mirror or its mounts but isolated from it by visco-elastic
`damping units. The coolant pipes are enclosed in a He gas manifold 930. A bellows 925 or other flexible connection
`servesto isolate the pipes from any stresses associated with the visco-elastic gas seals 920 at the ports of the manifold
`930. The pipes connect to coolant manifolds 960 and the rest of the coolant system (not shown). The gas manifolds
`are sealed to the mirror through compliant visco-elastic seals 932. The gas pressure in the manifold 930 and the mirror
`channels 915 is controlled by means of a He gas supply 935, a vacuum pump 940, a valve 945 to switch between
`them, and a pressure gauge 950. The only mechanical connection to the mirror in this case is at the mirror periphery.
`[0019] As mentioned previously, the He gas pressure is maintained at a small fraction of an atmosphere, so pressure
`distortion of the mirror surface is small. Also, vibrations of the coolant pipe caused by turbulent flow are isolated from
`the mirror by visco-elastic seals and spacers. Nevertheless, some vibrations could be transmitted to the mirror through
`the gas as sound waves. This contribution is expected to be very small, as a simple argument shows. Sound waves
`generated in the coolant propagate through the coolant tube and then through the gas. Since the gap between the
`coolant pipe and the mirror is small, we treat the sound propagation as being one-dimensional. At the outer surfaceof
`the coolant pipe, part of the sound wavesare transmitted into the gas and the rest is reflected back into the pipe. For
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`EP 1 376 185 A2
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`simplicity we assume thatall of the transmitted energy is subsequently absorbed by the mirror. The fraction of vibrational
`energy transmitted into the gas dependson the relative density of the two media as well as their speeds of sound and
`is given approximately by
`
`transmitted fraction = 4pgasCgas!/Ppipepipe’
`
`(2)
`
`where p is the density of the medium and c is the speed of sound. This expression is derived in Section 33.6 of B.
`Yavor and A. Detlaf, Handbook of Physics, Mir Publishers, 1975, the contents of which are incorporated herein by
`reference, for the case where pgasCgas << PpipeCpipe- For He gas, pgas~ 0.0166 kg/m? and Cgas ~ 1000 m/sec, at 20 °C
`and a pressure of 10 kPa.If the coolantpipe is made of copper, the corresponding quantities are p,ipg ~ 8930 kg/m
`And Cping ~ 3750 m/sec. As a result the fraction of vibrational energy transmitted into the gas, and assumecto ultimately
`be absorbed by the mirror, is only approximately 2x10-6. Therefore the gas layer is very effective in decoupling vibrations
`of the coolant pipe from the mirror.
`[0020] Although the embodiments described above show straight coolant pipes lying within straight channels in the
`mirror, other geometries are possible as well. If there is an advantage for example in having curved pipes within similarly
`curved channels, the mirror could be fabricated in two parts, split at a plane wherethe pipeswill lie. Channels in the
`two mating surfaces could be suitably machined, the pipes inserted, and the two halves bondedtogether. For some
`geometries, separation of the mirror into more than two pieces may be required.
`[0021]
`In some cases a thermally conductive fluid may be substituted for the gas in the gap between the coolant
`pipe and the mirror channel. The fluid is chosen to maintain its liquid and thermal properties at the operating temperature
`and ambient pressure of the mirror. For example the mirror might operate at atmospheric pressure, so the fluid would
`be maintained at atmospheric pressure. The coolant within the pipe could then have pressures different from an at-
`mosphere without perturbing the mirror surface. The pressure required to provide a given flow of fluid through a pipe
`of a given diameter, and therefore a specified heat transfer, is a strong function of both the flow rate and the pipe
`diameter, so the pressure within the pipe will usually differ from the ambient pressure. In addition, fluid flowing in a
`pipe suffers a significant pressure drop along the length of the pipe because ofthe fluid's viscosity. The effects of this
`pressure gradient on mirror distortion would thus be avoided. Vibrations associated with coolant turbulence within the
`pipe would not be asisolated from the mirror as in the case of a gasfilling the gap, but some lesser vibration isolation
`remains. Assuming the fluid in the gap to be water of density 1000 kg/m® and speed of sound approximately 1500 m/
`sec, and acopper pipe, Eq. 2 estimatesthe fraction of vibrational energy transmitted to the mirror to be approximately
`0.18. Forexample, the thermal conductivity of water exceeds that of He gas, so enhanced cooling is possible. However,
`the fluid heat capacity will be much greater than that of a the gas, and this may complicate temperature control of the
`mirror unless the fluid is circulated. By keeping the gap small, the heat capacity of the fluid can be minimized.
`[0022]
`Providing a plane of cooling channels near the heated surface of the mirror reduces the global thermal de-
`formation of the mirror as shown above. Any residual undesired global thermal deformation may be eliminated by
`applying a distribution of heat sources to the bottom of the mirror, so that thermal stresses in the mirror below and
`abovethe cooling plane cancel out. A fairly general statement can be made about these heat sources, using an argu-
`ment similar to the one above concerning the optinum placementof the cooling plane. Again, treating the mirror as a
`flat plate, divide the plate at the cooling plane into an upper plate and a lower plate. Assume the temperature distribution
`at the top surface is given by T,(x, y), and the temperature distribution in the cooling plane is given by T,(x, y), as
`shown in Fig. 10. In a steady state situation the temperature distribution in an intermediate plane at height z can be
`written as
`
`T(x, y) = T.0% y) + (71% y) - To y))z/ty =T(% y) + ATY(% y)ZAy.
`
`(3)
`
`[0023] Theterm T,(x, y) will cause thermal distortion in the x-y plane but not in z. Only the thermal gradient term AT,
`(x, y)z/t, will cause out-of-plane deformation of the plate. According to a theorem by J. Goodier in ASME J. Applied
`Mechanics 24, 467(1957), the contents of which are incorporated herein by reference, the out-of-plane deformation
`can be canceled by applying a distributed load q,(x, y) in the z direction over the surface of the plate:
`
`a,(x,Go|at Spe(x.y).
`
`Bt; (Oo
`
`(4)
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`EP 1 376 185 A2
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`where E is Young's modulus and v is Poisson's ratio for the plate material.
`[0024]
`Similarly, if the temperature distribution on the bottom surfaceis given by To(x, y), we can describe the tem-
`perature at an intermediate plane by
`
`T(x, y) = Tg (x, ¥) + (Tg, y) - Talx, y))Z/Aty = TE (X, y) + ATalX,y)Z/to.
`
`(5)
`
`[0025] Again the lower plate can be flattened by the application of a distributed load qa(x, y), where
`
`Et,
`(@
`@
`aloy)= sq") ox?By? WAT,(x,y).
`
`(6)
`
`By adjusting the temperature distribution T(x, y) so that q4(x, y) + qo(x, y) = 0, we can cancel out the thermal
`[0026]
`stresses in the upper and lower parts of the plate, so that the entire plate (mirror) remainsflat. This condition can be
`shown to be equivalent to
`
`2
`AT2(x, y) = 3 AT, (XY).
`ty
`
`(7)
`
`So the temperaturedifference ATo(x, y) required between the cooling plane and the bottom plate is less than
`[0027]
`that in the upper part of the plate bythe ratio (t4/t)2, and since we saw abovethat the best cooling geometry minimizes
`t, comparedto to, this ratio will typically be quite small. Furthermore, if the variation in the temperature differences with
`position in x and y is relatively small, we can assume the heat absorbedat the surfaces which creates the temperature
`differences scales approximately like AT, = F,t,/K and AT» = Foto/K, as described in the Carslaw and Jaeger reference
`above. Then the heat absorbed at the bottom surface of the plate which is required to flatten the plate will scale with
`the heat absorbedatthe top of the platelike
`
`F(x, y)= (ty/ta)°F4(x, y).
`
`(8)
`
`is less than ty.
`Thus, relatively little additional heat should be neededto flatten the plate if t,
`[0028]
`The result in Eq. 7 is fairly general, since it does not require the cooling plane to be isothermal. However,if
`[0029]
`cooling is accomplished by circulating coolant through channels, the approximation of a single plane, in which cooling
`takes place, may not be justified. A more realistic situation is shownin Fig. 11. Unless a very high cooling rate is needed,
`the temperature in the coolant channels is notlikely to be significantly different at the top and bottom of the channels.
`Therefore, the temperaturedistributions at the cooling planes 1 and 2 should be almostidentical. Then, the arguments
`made above should apply to this case as well. The distances t, and tp are more accurately defined as in Fig. 11 now.
`[0030]
`Eq. 7 is an approximation, because it neglects any variation in t, associated with the mirror surface curvature.
`Therefore, a correction must be determined either empirically or by modeling. Similarly, some correction to Eq. 8 must
`be determined either empirically or by modeling to compensatefor variations in both t, and the variation in temperature
`differences with position on the mirror.
`If the mirror curvature is gentle and the temperature differences don't change
`very much, however, Eqs. 7 and 8 should give reasonable approximations to the final relationships.
`[0031]
`Fig. 12 shows an embodimentof a control system which provides appropriate heating to the back of the mirror
`1200, to eliminate global thermal distortion. The radiation 1210 incident on the mirror surface is monitored by a detector
`1220 or detectors, and the absorbedradiation flux F4(x, y) determined from the Known absorption properties of the
`mirror. Alternatively, a separate monitor (not shown) could measure the surface temperature distribution of the mirror.
`Acontroller 1230 uses the absorbedsurface flux, or surface temperature, along with information describing the coolant
`channel 1240 conditions to determine from models the required absorbed heatflux Fo(x, y) on the back of the mirror.
`The controller then controls a heat source ("heater") 1250 to provide the necessary heat flux. The models in the con-
`troller will be based on the relations in Eqs. 7 and 8, along with corrections to them.
`[0032] The heat source could provide radiant heating of the mirror back from close proximity. Or the heat source
`could be in contact with the mirror back and heatit directly. Or the heat source could be a remote source of radiation
`which is modulated appropriately and projected onto the mirror back by means of an optical system. Fig. 13 shows an
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`EP 1 376 185 A2
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`embodiment of a radiant heating system. It consists of a number of bodieslying in a flat plane, temperatures of which
`may be adjusted independently. For example, each body may be heated by an array of resistive heaters of which
`electrical current is controlled. Or they may be heated by a flow of fluid heated remotely by heaters (not shown