Classifications

• • G—PHYSICS
  • G03—PHOTOGRAPHY; CINEMATOGRAPHY; ANALOGOUS TECHNIQUES USING WAVES OTHER THAN OPTICAL WAVES; ELECTROGRAPHY; HOLOGRAPHY
  • G03F—PHOTOMECHANICAL PRODUCTION OF TEXTURED OR PATTERNED SURFACES, e.g. FOR PRINTING, FOR PROCESSING OF SEMICONDUCTOR DEVICES; MATERIALS THEREFOR; ORIGINALS THEREFOR; APPARATUS SPECIALLY ADAPTED THEREFOR
  • G03F7/00—Photomechanical, e.g. photolithographic, production of textured or patterned surfaces, e.g. printing surfaces; Materials therefor, e.g. comprising photoresists; Apparatus specially adapted therefor
  • G03F7/70—Microphotolithographic exposure; Apparatus therefor
  • G03F7/70216—Mask projection systems
  • G03F7/70283—Mask effects on the imaging process
• • G—PHYSICS
  • G03—PHOTOGRAPHY; CINEMATOGRAPHY; ANALOGOUS TECHNIQUES USING WAVES OTHER THAN OPTICAL WAVES; ELECTROGRAPHY; HOLOGRAPHY
  • G03F—PHOTOMECHANICAL PRODUCTION OF TEXTURED OR PATTERNED SURFACES, e.g. FOR PRINTING, FOR PROCESSING OF SEMICONDUCTOR DEVICES; MATERIALS THEREFOR; ORIGINALS THEREFOR; APPARATUS SPECIALLY ADAPTED THEREFOR
  • G03F7/00—Photomechanical, e.g. photolithographic, production of textured or patterned surfaces, e.g. printing surfaces; Materials therefor, e.g. comprising photoresists; Apparatus specially adapted therefor
  • G03F7/70—Microphotolithographic exposure; Apparatus therefor
  • G03F7/70483—Information management; Active and passive control; Testing; Wafer monitoring, e.g. pattern monitoring
  • G03F7/70591—Testing optical components
• • G—PHYSICS
  • G03—PHOTOGRAPHY; CINEMATOGRAPHY; ANALOGOUS TECHNIQUES USING WAVES OTHER THAN OPTICAL WAVES; ELECTROGRAPHY; HOLOGRAPHY
  • G03F—PHOTOMECHANICAL PRODUCTION OF TEXTURED OR PATTERNED SURFACES, e.g. FOR PRINTING, FOR PROCESSING OF SEMICONDUCTOR DEVICES; MATERIALS THEREFOR; ORIGINALS THEREFOR; APPARATUS SPECIALLY ADAPTED THEREFOR
  • G03F1/00—Originals for photomechanical production of textured or patterned surfaces, e.g., masks, photo-masks, reticles; Mask blanks or pellicles therefor; Containers specially adapted therefor; Preparation thereof
  • G03F1/60—Substrates
• • G—PHYSICS
  • G03—PHOTOGRAPHY; CINEMATOGRAPHY; ANALOGOUS TECHNIQUES USING WAVES OTHER THAN OPTICAL WAVES; ELECTROGRAPHY; HOLOGRAPHY
  • G03F—PHOTOMECHANICAL PRODUCTION OF TEXTURED OR PATTERNED SURFACES, e.g. FOR PRINTING, FOR PROCESSING OF SEMICONDUCTOR DEVICES; MATERIALS THEREFOR; ORIGINALS THEREFOR; APPARATUS SPECIALLY ADAPTED THEREFOR
  • G03F1/00—Originals for photomechanical production of textured or patterned surfaces, e.g., masks, photo-masks, reticles; Mask blanks or pellicles therefor; Containers specially adapted therefor; Preparation thereof
  • G03F1/68—Preparation processes not covered by groups G03F1/20 - G03F1/50
  • G03F1/72—Repair or correction of mask defects
• • G—PHYSICS
  • G03—PHOTOGRAPHY; CINEMATOGRAPHY; ANALOGOUS TECHNIQUES USING WAVES OTHER THAN OPTICAL WAVES; ELECTROGRAPHY; HOLOGRAPHY
  • G03F—PHOTOMECHANICAL PRODUCTION OF TEXTURED OR PATTERNED SURFACES, e.g. FOR PRINTING, FOR PROCESSING OF SEMICONDUCTOR DEVICES; MATERIALS THEREFOR; ORIGINALS THEREFOR; APPARATUS SPECIALLY ADAPTED THEREFOR
  • G03F7/00—Photomechanical, e.g. photolithographic, production of textured or patterned surfaces, e.g. printing surfaces; Materials therefor, e.g. comprising photoresists; Apparatus specially adapted therefor
  • G03F7/0002—Lithographic processes using patterning methods other than those involving the exposure to radiation, e.g. by stamping
• • G—PHYSICS
  • G03—PHOTOGRAPHY; CINEMATOGRAPHY; ANALOGOUS TECHNIQUES USING WAVES OTHER THAN OPTICAL WAVES; ELECTROGRAPHY; HOLOGRAPHY
  • G03F—PHOTOMECHANICAL PRODUCTION OF TEXTURED OR PATTERNED SURFACES, e.g. FOR PRINTING, FOR PROCESSING OF SEMICONDUCTOR DEVICES; MATERIALS THEREFOR; ORIGINALS THEREFOR; APPARATUS SPECIALLY ADAPTED THEREFOR
  • G03F7/00—Photomechanical, e.g. photolithographic, production of textured or patterned surfaces, e.g. printing surfaces; Materials therefor, e.g. comprising photoresists; Apparatus specially adapted therefor
  • G03F7/70—Microphotolithographic exposure; Apparatus therefor
  • G03F7/70216—Mask projection systems
  • G03F7/70316—Details of optical elements, e.g. of Bragg reflectors, extreme ultraviolet [EUV] multilayer or bilayer mirrors or diffractive optical elements
• • G—PHYSICS
  • G21—NUCLEAR PHYSICS; NUCLEAR ENGINEERING
  • G21K—TECHNIQUES FOR HANDLING PARTICLES OR IONISING RADIATION NOT OTHERWISE PROVIDED FOR; IRRADIATION DEVICES; GAMMA RAY OR X-RAY MICROSCOPES
  • G21K1/00—Arrangements for handling particles or ionising radiation, e.g. focusing or moderating
  • G21K1/06—Arrangements for handling particles or ionising radiation, e.g. focusing or moderating using diffraction, refraction or reflection, e.g. monochromators
  • G21K1/062—Devices having a multilayer structure
• • H—ELECTRICITY
  • H01—ELECTRIC ELEMENTS
  • H01L—SEMICONDUCTOR DEVICES NOT COVERED BY CLASS H10
  • H01L22/00—Testing or measuring during manufacture or treatment; Reliability measurements, i.e. testing of parts without further processing to modify the parts as such; Structural arrangements therefor
  • H01L22/20—Sequence of activities consisting of a plurality of measurements, corrections, marking or sorting steps
• • B—PERFORMING OPERATIONS; TRANSPORTING
  • B82—NANOTECHNOLOGY
  • B82Y—SPECIFIC USES OR APPLICATIONS OF NANOSTRUCTURES; MEASUREMENT OR ANALYSIS OF NANOSTRUCTURES; MANUFACTURE OR TREATMENT OF NANOSTRUCTURES
  • B82Y40/00—Manufacture or treatment of nanostructures
• • H—ELECTRICITY
  • H01—ELECTRIC ELEMENTS
  • H01L—SEMICONDUCTOR DEVICES NOT COVERED BY CLASS H10
  • H01L22/00—Testing or measuring during manufacture or treatment; Reliability measurements, i.e. testing of parts without further processing to modify the parts as such; Structural arrangements therefor
  • H01L22/10—Measuring as part of the manufacturing process
  • H01L22/12—Measuring as part of the manufacturing process for structural parameters, e.g. thickness, line width, refractive index, temperature, warp, bond strength, defects, optical inspection, electrical measurement of structural dimensions, metallurgic measurement of diffusions

Definitions

• the present invention relates to the field of generating a predetermined three- dimensional contour of an optical component and/or a wafer.
• Optical systems have to fulfil increasing demands with respect to their performance. For example, the size of optical components of telescopes steadily increases in order to collect a limited number of photons originating from far distant objects. Slightest deviations of optical components from their predefined surface forms result in a reduction of optical capabilities of the telescopes or generally of optical systems.
• microscopes have for example to resolve objects having a very low contrast compared to their surroundings. Moreover, it is requested that projection systems of lithography devices resolve smaller and smaller structures.
• Integrated circuits are another kind of devices which has to also fulfil steadily increasing demands. According to Moore's law the minimum dimension of structural elements generated on wafers in order to fabricate ICs continuously shrinks.
• EUV extreme ultraviolet
• EUV optical elements as for example mirrors or photolithographic masks
• the tolerable deviation of the substrates of these optical elements regarding the planarity is only a portion of a wavelength of the exposure wavelength in order to not significantly disturb the phase front of the electromagnetic wave reflected from a multi-layer structure arranged on a surface of the substrate.
• Larger deviations of the planarity of the substrate of EUV mirrors and masks may lead to variations of the optical intensity distribution in the photoresist due to a (partial) constructive or destructive addition of the wave front in the photoresist of the wafer.
• the variations of the optical intensity may result in the fabrication of defective semiconductor devices as for examples ICs.
• EUV substrates as supplied from the manufacturer may not even fulfil the planarity condition for EUV mirrors and masks. Further, the manufacturing process of mirrors and masks which forms a multilayer structure and fine patterns on one surface of the multilayer structure, respectively, may even deteriorate the planarity of the substrate.
• a curvature of the substrate of a photolithographic mask may also lead to imaging errors of an EUV mask.
• the US 2008 / o 322 206 Ai describes a method to improve the planarity of a manufactured photolithographic mask.
• this document proposes forming expansion stress and/or compaction stress generation portions in a predetermined region of the substrate which includes the curved region.
• the expansion stress and compaction stress generation portions are generated by focussing femtosecond laser pulses in this region which locally modify the bonding state of the substrate.
• wafers sometimes bend during the manufacturing process so that it is difficult to clamp the wafers with a vacuum chuck.
• the bending seems to be caused by stress which is introduced into the wafer during the manufacturing process of ICs.
• wafer bending occurring during the processing of the wafer can only be lowered by reducing the stress induced by various processing steps of the manufacturing process.
• the processing steps have to be modified. This is an involved task due the highly complex sequence of processing steps required to fabricate modern ICs. Additionally, it may also be necessary to tolerate performances losses in the electrical function of the fabricated ICs.
• layers are at the moment developed which can be applied to the rear surface of a wafer and which reduce the bending effect of various processing steps. Adding an additional layer to the rear wafer side would introduce further processing step in an already involved manufacturing process of modern ICs.
• a method for generating a predetermined three- dimensional contour of an optical component and/or a wafer comprises: (a) determining a deviation of an existing three-dimensional contour of the optical component from the predetermined three-dimensional contour; (b) calculating at least one three- dimensional arrangement of laser pulses having one or more parameter sets defining the laser pulses for correcting the determined existing deviation of the three- dimensional contour from the predetermined three-dimensional contour; and (c) applying the calculated at least one three-dimensional arrangement of laser pulses on the optical component and/or the wafer for generating the predetermined three- dimensional contour.
• the three-dimensional (3D) approach of the present invention establishes a relation between the laser beam parameters and the parameters of the 3D arrangement of laser pulses on the one hand and the volume deformations induced by a 3D arrangement of laser pulses in an optical component and/or a wafer on the other hand.
• the establishment of such a relation allows the determination of parameters of the laser beam and the 3D arrangement of laser pulses necessary of the generation of a predetermined 3D contour of the optical component and/or the wafer.
• the approach of the inventive method for the calculation of a 3D arrangement of laser pulses takes into account the 3D nature of an optical components and/or wafers. Therefore, it overcomes most of the drawbacks of a two-dimensional approximation as well as of a phenomenological approach typically used up to now.
• the inventive method allows calculating one or more 3D arrangements of laser pulses having a single or several parameter sets defining the laser pulses which correct a determined deviation of an optical component and/or a wafer from its predetermined 3D contour.
• the inventive method enables the simultaneous correction of deviations of a 3D contour parallel and perpendicular to a surface of an optical component and/or a wafer.
• method step (b) further comprises determining elementary volume deformations induced in the optical component and/or the wafer by predefined three-dimensional arrangements of laser pulses having predefined parameter sets and/or having predefined distances of laser pulses in three directions in the arrangements of laser pulses, wherein the three directions are not within a plane.
• step (b) further comprises determining elementary volume deformations induced in the optical component and/or the wafer by sequentially applying predefined three-dimensional arrangements of laser pulses, the predefined three-dimensional arrangements of laser pulses comprising laser pulses having predefined parameter sets and/or having predefined distances of laser pulses in three directions in the three-dimensional arrangements of laser pulses, wherein the three directions are not within a plane.
• the parameters of the laser pulses and the parameters of the 3D arrangement of laser pulses are calculated for correcting a determined deviation of a 3D contour of an optical component and/or a wafer from their predetermined contours. This approach avoids the determination of deformation volumes from material parameters and complex calculations.
• a first 3D arrangement of laser pulses is applied to the optical component and/or the wafer in which a (linear) density of laser pulses is higher in the x-direction than in the y-direction.
• a second 3D arrangement of laser pulses is applied to the optical component and/or the wafer in which the (linear) density of laser pulses is higher in the y-direction than in the x-direction.
• the density may be kept fixed within the 3D arrangement of laser pulses and a parameter of the laser pulses may be varied.
• calculating the at least one three-dimensional arrangement of laser pulses comprises: (d) setting up a target functional comprising deviations of the existing three-dimensional contour of the optical component and/or the wafer from the predetermined three-dimensional contour and volume deformations induced by the at least one three-dimensional arrangement of laser pulses; and (e) minimizing the target functional by varying the at least one three-dimensional arrangement of laser pulses.
• Varying the at least one three-dimensional arrangement of laser pulses comprises varying at least one parameter of the laser pulses.
• a further aspect comprises using a Lagrange variational principle for minimizing the target functional.
• the parameter set defining the laser pulses comprises: an energy of the laser beam, a pulse length, a repetition rate, a number of pulses directed onto one location of the optical component and/or the wafer, a beam polarization, a numerical aperture, a focus size, a beam shape, and/or an astigmatism.
• parameters of the at least one three-dimensional arrangement of laser pulses comprises a size of the arrangement in three directions and a distance between locations of two or more incident laser pulses in three dimensions.
• different laser pulses generate pixels which do not overlap in an area normal to a beam direction in the three-dimensional arrangement of laser pulses.
• laser pulses generate two or more pixels which partially or essentially fully overlap in the area normal to the beam direction in the three-dimensional arrangement of laser pulses.
• two or more pixels overlapping in the area normal to the beam direction fully or partially overlap in the beam direction.
• the effect of the arrangement of laser pulses on an optical component and/or the wafer can be calculated by summing up the effects of the individual pixels.
• Yet another aspect further comprises modifying a mass density and/or an optical transmission of the optical component and/or the wafer by a stress distribution induced by a strain distribution caused by laser pulses, and wherein the stress distribution and the strain distribution are connected by Hooke's law.
• a further favourable aspect comprises introducing a three-dimensional grid with nodes N. penetrating the optical component and/or the wafer, the three-dimensional grid nodes N. defining elementary volumes a .
• the terms elementary volumes and elementary cells are used as synonyms.
• a displacement ⁇ . of the three-dimensional grid node N i is a function of the parameter set of the laser pulses and/or the parameters of the at least one arrangement of laser pulses.
• Another aspect further comprises determining at least one three-dimensional writing density a a and a plurality of writing mode signatures MSTM , wherein MS. denotes the components of a single writing mode signature and the index m counts the plurality of writing mode signatures. In case of several three-dimensional writing densities a m a the index m also counts the number of writing densities.
• the three-dimensional writing density a a in an elementary volume a and the writing mode signature MS. introduce a displacement ⁇ ⁇ given by
• optical component and/or the wafer and wherein ⁇ ⁇ ] ⁇ ] are projections of equilib- rium deformation displacements for unity writing to the writing mode signature MS. .
• This equation connects parameters of the laser pulses and/or the 3D arrangement of laser pulses with their effect onto the optical component and/or the wafer.
• the inventive method use material parameters of the optical component and/or the wafer and basic physical law in order to determine the effect that a beam of light pulses induces in a substrate material of the optical component and/or the wafer.
• a three-dimensional writing density a a of an arrangement of laser pulses across the optical component and/or the wafer induces a displacement ⁇ ⁇ of the grid nodes N n of the optical component and/or the wafer given by
• both tensors comprise material parameters of the optical component and/or the wafer, and wherein e lt are equilibrium deformation displacements of the grid nodes N i of the elementary volume a for a unity writing density-
• the three-dimensional writing densities a m of R arrangements of laser pulses across the optical component and/or the wafer superimposed in an elementary volume a introduce a displacement ⁇ ⁇ given by
• the at least one writing density of the laser beam is below a damage threshold of the optical component and/or the wafer. In an alternative aspect, the at least one writing density of the laser beam is above the damage threshold of the optical component and/or the wafer.
• Another aspect further comprises determining a displacement ⁇ ⁇ of the three- dimensional grid N n in an elementary volume a of the optical component and/or the wafer without interactions of the elementary volume a with other elementary volumes of the optical component and/or the wafer. Still a further aspect comprises determining a total deformation of the optical component and/or the wafer by combining the elementary volumes of the optical component and/or the wafer by combining the elemen- tary volumes and minimizing an accumulated potential energy of the combined elementary volumes.
• the approach of the last aspect facilitates the complex calculation of the parameters of the laser pulses and the 3D arrangement of laser pulses. Furthermore, as the effects of the laser pulses are small in the material of an optical component and/or the wafer and as the experimental resolution is presently limited this approach may be the only possible one.
• step (a) further comprises measuring the existing three- dimensional contour of the optical component and/or the wafer.
• measuring the existing three-dimensional contour further comprises using a contact profilometer, a pseudo-contact profilometer, a non-contact profilometer, an interferometer, a white light interferometer, a confocal microscope, a photomask metrology tool, a scanning electron microscope and / or a combination of these devices.
• step (a) further comprises determining deviations ⁇ . of the three-dimensional contour of the optical component and/or the wafer at positions i from the difference of the determined existing locations ⁇ p ei and the predetermined locations ⁇ rec,e ' .
• step (b) further comprises minimizing the target functional wherein ⁇ . is the displacement at the
• a beneficial aspect further comprises applying laser pulses to the optical component and/or the wafer which can essentially transmit the optical component and/or the wafer.
• Another aspect further comprises selecting a wavelength of the laser pulses so that a photon energy of the laser pulses is lower than a band gap energy of the optical component and/or the wafer.
• the photon energy of the laser pulses is lower than 0.95, preferred lower than 0.9, more preferred lower than 0.8, and most preferred lower than 0.7 the band gap energy of the optical component and/or the wafer.
• the photon energy of the laser pulses is lower than 0.95, preferred lower than 0.9, more preferred lower than 0.8, and most preferred lower than 0.7 the band gap energy of the material of a processed wafer having the lowest band gap energy, and wherein the processed wafer comprises one or more integrated circuits or at least a part of the one or more integrated circuits.
• the optical component comprise an optical component for extreme ultraviolet wavelength radiation, in particular a mirror or a photolithographic mask for extreme ultraviolet radiation.
• the optical component for extreme ultraviolet wavelength radiation comprises a transparent conductive coating on a rear surface which is opposite to a front surface having a multilayer structure, and wherein the transparent conductive layer is optically transparent in the near infrared, the visible, and/or the near ultraviolet wavelength range.
• the transparent conductive coating comprises tin oxide, indium tin oxide, antimony tin oxide, aluminium zinc oxide, or a combination of these materials.
• step (a) comprises determining a flatness deviation of the rear surface of the optical component for extreme ultraviolet radiation
• step c. comprises applying the calculated at least one three-dimensional arrangement of laser pulses for flattening the rear surface of the optical component for extreme ultraviolet radiation.
• step (a) comprises determining the flatness deviation of the rear surface of a photolithographic mask for extreme ultraviolet radiation after arranging the multilayer structure and an absorber layer on the front surface, but before patterning the absorber layer
• step (c) comprises applying the calculated at least one three-dimensional arrangement of laser pulses for flattening the rear surface of the photolithographic mask for extreme ultraviolet radiation.
• step (a) comprises determining the flatness deviation of the rear surface of the optical component for extreme ultraviolet radiation after removing the multilayer structure from a peripheral area of the front surface, and step (c) comprises applying the calculated at least one three-dimensional arrangement of laser pulses for flattening the rear surface of the optical component for extreme ultraviolet radiation.
• step (a) comprises determining the flatness deviation of the multilayer structure of the optical component for extreme ultraviolet radiation from a predetermined flatness of the multilayer structure, and step (c) comprises applying the calculated at least one three-dimensional arrangement of laser pulses for generating the predetermined flatness of the multilayer structure of the optical component for extreme ultraviolet radiation.
• the inventive method can be applied between various steps in the fabrication process of an EUV mask in order to avoid the generation of defects during the mask fabrication process.
• step (a) comprises determining the deviation of pattern elements of a transmissive photolithographic mask from a predetermined pattern, and step (c) comprises applying the calculated at least one three-dimensional arrangement of laser pulses for generating the predetermined pattern.
• step (a) comprises determining the deviation of an optical transmission across a transmissive photolithographic mask from a predetermined optical transmission, and step c. comprises applying the calculated at least one three-dimensional arrangement of laser pulses for generating the predetermined optical transmission.
• step (a) comprises determining the deviation of pattern elements and the optical transmission across a transmissive photolithographic mask from the predetermined pattern and the predetermined optical transmission
• step (c) comprises applying the calculated at least one three-dimensional arrangement of laser pulses for simultaneously generating the predetermined pattern and the predetermined optical transmission across the transmissive photolithographic mask.
• step (a) comprises determining the deviation of a flat optical component from a predetermined three-dimensional optical contour
• step (c) comprises applying the calculated at least one three-dimensional arrangement of laser pulses for generating the predetermined three-dimensional optical contour of the originally flat optical component.
• the inventive method has the potential to fabricate specific optical components from a given blank.
• step (a) comprises determining a deviation of a spherical contour of an optical component from a predetermined aspherical contour
• step (c) comprises applying the calculated at least one three-dimensional arrangement of laser pulses for generating the predetermined aspherical contour of the optical component.
• inventive method allows fabricating of specific optical components from standard cost-effective optical components. If the 3D contour of the standard optical component is provided by the manufacturer of the standard component, step (a) of the inventive method has not to be performed.
• step (a) comprises determining the deviation of a contact surface of a frustrated total internal reflection shutter from a predetermined contact surface of the frustrated total internal reflection shutter, and step c. comprises applying the calculated at least one three-dimensional arrangement of laser pulses for generating the predetermined contact surface of the frustrated total internal reflection shutter.
• step (a) comprises determining the deviation of the three- dimensional contour of a nanoimprint lithography template from the predetermined three-dimensional contour of the nanoimprint lithography template, and step c. comprises applying the calculated at least one three-dimensional arrangement of laser pulses for generating the predetermined three-dimensional contour of the nanomim- print lithography template.
• the wafer comprises a semiconductor material, or the wafer comprises compound semiconductor material.
• the semiconductor material comprises silicon.
• the wafer comprises at least a portion of one or more integrated circuits.
• wafers are a second kind of components or devices to which the above defined method can be applied.
• wafers may bend during the fabrication steps in a semiconductor factory. The bending gets worse with shrinking structural elements and more complex ICs.
• the method described above can be used within the production process of ICs to correct a bending of processed or partly processed wafers accumulated during the manufacturing process in a semiconductor factory.
• the defined method can at least partly avoid performance losses of future ICs.
• the wafer comprises silicon and the wavelength of the laser pulses are in the range of 1.0 ⁇ - 8.0 ⁇ , preferred 1.3 ⁇ - .o ⁇ , and most preferred from 1.5 ⁇ - 6.0 ⁇ .
• a further aspect comprises selecting a wavelength for the laser pulses so that the energy of the laser pulses is close to the band gap energy of the wafer.
• the wafer comprises silicon and the wavelength of the laser pulses is ⁇ 1300 nm, more preferred ⁇ 1200 nm, and most preferred ⁇ 1100 nm.
• the introduction of the calculated at least one three- dimensional arrangement of laser pulses into the wafer bends the wafer.
• the introduction of the calculated at least one three-dimensional arrangement of laser pulses into the wafer compensates the bending of the wafer occurring during processing the wafer.
• the at least one three-dimensional arrangement of laser pulses is asymmetrically introduced into the wafer with respect to the depth of the wafer.
• the wafer comprises a micro-electromechanical system
• MEMS microelectron et al.
• a further aspect comprises introducing the calculated at least one three-dimensional arrangement of laser pulses into the wafer through a front side of the wafer, wherein the wafer is processed on the front side to generate one or more integrated circuits.
• the energy of the laser pulses is lower than the material of the integrated circuits having the lowest band gap energy.
• Another aspect comprises introducing the calculated at least one three-dimensional arrangement of laser pulses into the wafer through a rear side of the wafer, which is opposite to the front side on which the wafer is processed to generate one or more integrated circuits.
• a further beneficial aspect further comprises: (f) determining a remaining deviation of a generated three-dimensional contour of the optical component and/or the wafer from the predetermined three-dimensional contour; (g) ending the method if the remaining deviation is smaller than a predetermined threshold; and (h) repeating steps (b) and (c) if the remaining deviation is larger or equal than the predetermined threshold.
• the optical component is optically transparent in the near infrared, the visible and/or the near ultraviolet wavelength range.
• step (c) of the inventive method can actually be performed on the optical component and/or the wafer.
• an apparatus for generating a predetermined three- dimensional contour of an optical component and/or a wafer comprises: (a) a metrology tool operable to determine a deviation of an existing three-dimensional contour of the optical component and/or the wafer from the predetermined three-dimensional contour; (b) a computing unit operable to calculate at least one three-dimensional arrangement of laser pulses having one or more parameter sets defining the laser pulses for correcting the determined existing deviation of the three-dimensional contour from the predetermined three-dimensional contour; and (c) a light source operable to apply the calculated at least one three-dimensional arrangement of laser pulses for generating the predetermined three-dimensional contour of the optical component and/or the wafer.
• the metrology tool comprises a contact profilometer, a pseudo- contact profilometer, a non-contact profilometer, an interferometer, a white light interferometer, a confocal microscope, a photomask metrology tool, and/or a scanning electron microscope.
• the computing unit comprises a microprocessor, a general purpose processor, a special purpose processor, a central processing unit and/or a combination thereof.
• the light source comprises a laser source operable to generate a beam of ultra-short laser pulses and a scanning means operable to generate the at least one arrangement of laser pulses.
• the laser source comprises a Ti: sapphire laser system and/or a YAG laser system doped with at least one element of the group which comprises: neodymium (Nd), thulium (Tm), holmium (Ho), and erbium (Er).
• the apparatus is adapted to be integrated in a semiconductor manufacturing equipment in a semiconductor factory.
• Fig. 1 shows in cross-section a schematic view of a transmissive photolithographic mask
• Fig. 2 shows in cross-section a schematic view of a reflective photolithographic mask; schematically depicts a cross-sectional view of an extreme ultraviolet (EUV) mask blank having an uneven rear surface; schematically depicts a cross-sectional view of a confocal microscope for a determination of an existing three-dimensional contour (3D) of an optical component; schematically represents a block diagram of an apparatus for calculating a three-dimensional (3D) arrangement of laser pulses and for applying the calculated 3D arrangement of laser pulses to an optical component for generating a predetermined contour; schematically depicts a configuration for applying a calculated 3D arrangement of laser pulses to an EUV mask; schematically depicts an expansion of an initially flat mask blank perpendicular to a mask blank surface by applying two two-dimensional (2D) arrangements of laser pulses generating pixels which locally expand the mask blank predominantly in vertical direction; schematically presents the deformation induced in the elementary area by the application of a of laser pulse leading to an essentially isotropic expansion in a plane perpendicular to the beam
• FIG. 8 represents the deformation of a mask blank in beam direction induced by the 2D arrangement of laser pulses using laser pulses resulting in the deformation of an elementary area as indicated in Fig. 9; shows a cross-section of a rotational symmetric 3D target contour to be generated; presents the 3D target contour of Fig. 13 as a 2D representation on a mask blank; illustrates a calculated 2D arrangement of laser pulses or writing map used to perform an expansion of the mask blank of Fig. 14 in x-direction; represents a calculated 2D arrangement of laser pulses used to perform an expansion of the mask blank of Fig.
• FIG. 19 schematically illustrates the fabrication of a lens with a large focal length from an initially flat mask blank by the application of a 3D arrangement of laser pulses; shows a 2D representation of a 3D target contour having a profile of a saddle; presents a calculated 2D arrangement of laser pulses used to expand an initially flat mask blank in horizontal direction (x-direction) close to the front surface of the mask blank; shows a calculated 2D arrangement of laser pulses used to expand the initially flat mask blank in vertical direction (y-direction) close to the front surface of the mask blank; depicts a calculated 2D writing mode used to expand the initially flat mask blank in horizontal direction close to the rear mask blank surface; represents a calculated 2D writing mode used to expand the initially flat mask blank in vertical direction close to the rear mask blank surface; shows the measured 3D contour of the mask blank induced by the application of the 3D arrangement comprising the writing maps of Figs.
• FIG. 23 to 26 presents a measured 3D contour of a mask blank; shows the measured 3D contour of Fig. 27 after applying a TPS fit; represents a calculated 2D arrangement of laser pulses or writing map for expanding the 3D contour of Fig. 27 in horizontal direction in order to remove the 3D contour, i.e. to flatten the mask blank surface; depicts a calculated 2D arrangement of laser pulses for expanding the 3D contour of Fig. 27 in vertical direction in order to remove the 3D contour, i.e. to flatten the mask blank surface; presents the 3D contour of Fig. 27 after applying the 3D arrangement of laser pulses comprising the writing maps of Figs. 29 and 30; shows the 3D contour of Fig.
• FIG. 31 after reducing the height scale from 8 ⁇ (- 4000 nm to +4000nm) to 1 ⁇ (-500 nm to +500 nm); represents the 3D contour of Fig. 32 after applying a smoothing TPS fit; presents an overview of the transmission spectrum of silicon (Si) in the wavelength range from 0.1 ⁇ to 100 ⁇ ; depicts the transmission spectrum of Si in the wavelength range from 1 ⁇ to 9 ⁇ and additionally shows the temperature behaviour of the transmission in this wavelength range; shows the variation of the absorption coefficient across the wavelength range from 250 nm to 1400 nm; illustrates the focus range of a Gaussian beam and represents a cylinder in this range;
• Fig. 38 depicts the energy deposited by a laser pulse in the cylinder of Fig. 37 in the
• Fig. 39 represents a regular pulse train, wherein the individual pulses have a pulse width t Pulse and period ⁇ ;
• Fig. 40 shows the transmission of Si in the band edge range, i.e. in the wavelength range from 950 nm to 1200 nm;
• Fig. 41 present the pulse energy estimated to heat up the Gaussian beam focus of
• Fig. 37 to a temperature exceeding the melting point of Si
• Fig. 42 shows a schematic cross-section through a wafer prior to the beginning of a process to fabricate integrated circuits on the front surface of the wafer
• Fig. 43 schematically presents the wafer of Fig. 34 after some or all of the processing steps have been performed and indicates the locations two three- dimensional (3D) arrangements of laser pulses to be introduced into the processed wafer;
• Fig. 44 schematically depicts the processed wafer of Fig. 35 after the two 3D arrangements of laser pulses have been introduced into the processed wafer;
• Fig- 45 presents a flow chart of the inventive method
• Figs. 46a and 46b represent a flow chart of using the inventive method during the manufacturing process of EUV mask in order to generate a predetermined 3D contour of the EUV mask
• Fig. 47 schematically illustrates an elementary volume or an elementary cell of an optical component.
• the present invention is described by taking photolithographic masks and wafers as examples.
• the present invention can also be applied to optical elements used in telescopes, microscopes and/or laser system to just mention a few examples.
• the operation wavelength range of optical components is not restricted to the wavelengths used in lithography. Rather, the optical components processed with the inventive method across the entire optical range.
• the present invention is not restricted to silicon wafers as described below. Rather, it can be applied to all kind of wafers during the fabrication processes of any kind of integrated circuits.
• the method defined in the third section of this description can be applied to all components whose material(s) is (are) transparent for a specific wavelength range of the electromagnetic spectrum. If a material of a component fulfils this requirement pixels can be introduced into the material of the component.
• the transparent wavelength range needs only to be small ( ⁇ 100 nm) since laser systems are typically utilized to generate the laser pulses for writing pixels into the material of the component. It is presently preferred that the transparent wavelength range of a material of a component is in the ultra-violet, the visible and/or the infrared part of the electromagnetic spectrum as numerous laser sources are available for these spectral ranges.
• Fig. l represents a schematic cross-section view of a transmissive photolithographic mask loo.
• the mask ⁇ comprises a substrate no having a first or front surface 130 and a second or rear surface 140.
• the substrate 110 has to be transparent for the wavelength used for the illumination of a photoresist on a wafer.
• the exposure wavelength may be in the deep ultraviolet (DUV) spectral range of the electromagnetic spectrum, in particular around 193 nm.
• the substrate material comprises typically quartz.
• the substrate has typical lateral dimensions of 152 mm x 152 mm and a thickness or height of essentially 6.35 mm.
• the substrate 110 of the photolithographic mask 100 has on its front surface 130 pattern elements 120 normally fabricated from chromium which form on the photoresist the predetermined structure elements from which semiconductor devices are produced.
• the portion of the substrate 110 of the photolithographic mask 100 carrying pattern elements 120 is called active area 150 of the mask, whereas the boundary portion or the peripheral part which does not have pattern elements 120 is called non-active area 160.
• a laser beam 170 at the exposure wavelength illuminates the substrate 110 of the mask 100 through the second or rear surface 140 of the substrate 110.
• Fig. 2 shows a schematic cross-sectional view of a reflective photolithographic mask 200 for future use in the extreme ultraviolet (EUV) spectral range of the electromagnetic spectrum, in particular for an exposure wavelength in the range of 10 nm to 15 nm.
• the mask 200 is a reflective optical component based on a multi-layer mirror structure 255.
• the multi-layer mirror system 255 of the photolithographic mask 200 is deposited on a front substrate surface 230 of a suitable substrate 210, such as a fused silica substrate.
• a suitable substrate 210 such as a fused silica substrate.
• Other transparent dielectrics, glass materials or semiconducting materials may also be applied as substrates for EUV photolithographic masks as for example ZERODUR®, ULE® or CLEAR- CERAM®.
• the multi-layer mirror system 255 comprises for example approximately 40 pairs of alternating molybdenum (Mo) 250 and silicon (Si) layers 260.
• the thickness of each Mo layer 250 is 4.15 nm and that of the Si layer 260 amounts 2.80 nm.
• a capping layer 270 of silicon with a native oxide of 7 nm depth is arranged on top of the structure 255.
• the Mo layers 250 act as scattering layers, whereas the silicon layers 260 function as separation layers.
• For the scattering layers instead of Mo other elements with a high Z number may utilized, such as cobalt (Co), nickel (Ni), tungsten (W), rhenium (Re) and iridium (Ir).
• the multi-layer structure 255 on the substrate 210 acts as a mirror for EUV electromagnetic radiation.
• a buffer structure 280 and an absorbing structure 290 are additionally deposited on the capping layer 270.
• the buffer layer 280 may be deposited to protect the multi-layer mirror structure 255 during processing, for example during etching of the absorbing structure 290.
• Possible buffer structure materials are for example fused silica (Si0 2 ), silicon-oxygen- nitride (SiON), ruthenium (Ru), chromium (Cr), and/or chromium nitride (CrN).
• the absorbing structure 290 comprises a material having a large absorption constant for photons in the EUV wavelength range.
• Cr chromium
• CrN tantalum nitride
• a thickness of about 50 nm is sufficient to absorb essentially all EUV photons 295 incident on the absorbing structure 290. In contrast, the majority of the photons 295 incident on the capping layer 270 is reflected as photons 297.
• the substrate 210 of the EUV mask 200 also has lateral dimensions of 152 mm x 152 mm and a thickness or height of essentially 6.35 mm.
• the rear surface 225 of the substrate 210 or the rear substrate surface 225 has a thin metallic coating 220.
• this coating 220 comprises chromium.
• the metallic coating 220 is used to fix the photolithographic mask 200 at the EUV scanner by the application of electrostatic forces.
• energy from light pulses of a light beam is locally deposited in the substrate 210 of the EUV mask 200.
• the light pulses of the light beam can neither penetrate the front substrate surface 230, as they are absorbed by the multi-layer mirror structure 255, nor the rear substrate surface 225, as they are also absorbed by the metallic coating 220 on the rear substrate surface 225.
• the metallic coating 220 on the rear substrate surface 225 of the EUV mask 200 is replaced by a transparent conductive coating 222, as it is shown in Fig. 2.
• a transparent conductive coating 222 may for example comprise indium tin oxide (ITO).
• ITO indium tin oxide
• transparent conductive coating means that the coating 222 is optically transparent for light in the near-infrared, and/or the visible, and/or the near-ultraviolet wavelength range.
• transparent conductive coatings are for example fluorine tin oxide (FTO) and/or aluminium zinc oxide (AZO) and/or antimony tin oxide (ATO). These materials can easily be applied to the rear substrate surface 225 of a fused silica substrate 210 and have a conductivity which is high enough to fix the mask 200 to the EUV scanner.
• the transparent conductive coating 222 enables to irradiate the completely manufactured EUV mask 200 with light pulses of a laser beam through the rear substrate surface 225.
• the transparent conductive coating 222 also allows applying ultra-short laser pulses to the EUV mask 200 during its manufacturing process if the coating 222 is arranged on the mask blank or on the rear substrate surface 225 in a first step of the mask manufacturing process.
• An EUV mirror may have the structure of the photolithographic mask 200 unless it does not have the buffer layer 280 and the absorbing structure 290. Therefore, also for an EUV mirror the metallic coating 220 may be replaced by a transparent conductive coating 222, so that the light pulses are able to enter into the substrate from the rear substrate surface (not shown in Fig. 2).
• the diagram 300 of Fig. 3 schematically shows a mask blank or a substrate 310 of an EUV mask 300 where the rear substrate surface 340 is not flat but has two deviations 360 and 370 from a predetermined surface contour 350.
• the chuck may flatten the rear surface 340 of the blank 300 which results in a distortion of the front surface 330.
• the multilayer structure 255 is deposited on an uneven front surface 330 of the substrate 310. This will finally lead to an uneven surface of the multilayer structure 255.
• the absorbing structure 290 of an EUV mask 200 is typically generated by patterning a layer of absorbing material with an electron beam writer.
• the EUV mask 200 is not fixed by an electrostatic chuck.
• the not flat rear surface 340 of the mask blank 310 is essentially not changed during the patterning process. Therefore, an EUV mask having a rear substrate surface contour 340 as indicated in Fig. 3 will have a different shape when it is fixed on an electrostatic chuck during an illumination process. This results in distortions of the image placement of a wafer.
• Fig. 4 schematically illustrates an example of a metrology tool 400 which can be used to determine a two-dimensional (2D) or a 3D image of an optical component or a wafer.
• the confocal microscope 400 of Fig. 4 is only an example of a class of metrology tools which can be applied to measure the 3D contour of an optical component or of a wafer.
• a profilometer can be used to analyse a surface profile or a 2D or 3D microscopic or sub-microscopic contour of an optical component as well as of a wafer.
• Profilometers are available which utilize contact or pseudo-contact methods.
• Stylus profilometers or mechanical profilometers, atomic force microscopes, scanning tunnelling microscopes are examples of this type of profilometer.
• a number of non- contacting profilometers which apply for example optical methods are also available. Examples of this type of profilometers are vertical scanning interferometers, white-light interferometers, phase shift interferometers and/or differential interference contrast microscopes.
• a confocal microscope 400 uses a point illumination of a pinhole in an optically conjugate plane in front of a detector to eliminate out-of-focus signal components.
• the point illumination is generated with a light source 420, an optical fiber 425 which couples the light of the light source 420 through the pinhole 430 via the objective 440 into a focal plane 450 of an photomask 410.
• Light reflected from the focal plane 450 is directed by a dichroic mirror 435 to the pinhole 465 which is arranged in front of the detector 475.
• the portion of light which is reflected from outside of the focal plane 450 is significantly smaller than light reflected from the focal plane 450. Furthermore, light reflected from outside the focal plane is not fo- cussed onto the pinhole 465.
• the light source 420 may be a laser source. If an optical fiber 425 is used for guiding the electromagnetic radiation from the light source to the pinhole 430, the pinhole 430 may be omitted as the fiber 425 has a similar effect as the pinhole 430.
• the detector 475 may for example be a photodiode, an avalanche photodiode, or a photo- multiplier.
• a photomask or generally an optical component 410 may be arranged on a sample stage 405.
• the sample stage 405 may be movable and is moved by a scan unit 480 in a plane which is perpendicular to the incident light beam (x- and/or y-direction).
• a scan unit 480 for moving the sample stage 405 (not shown in Fig. 4).
• micromanipulators or servomotors can be used for moving the sample stage 405 (not shown in Fig. 4).
• the focal plane can be scanned through the optical component 410.
• the sample stage is fixed and the light beam 455 is scanned across the photomask 410. This implementation is called confocal laser scanning microscope (CLSM) (not depicted in Fig. 4).
• CLSM confocal laser scanning microscope
• a control unit 470 controls the light source 420, the detector 675 as well as the scan unit 480. It is also possible that the control unit 470 controls the movement of the objective 440 (not shown in Fig. 4). Further, the control unit 470 may be embedded in a computer system having at least a processor, a storage unit, I/O (input/output) unit(s), and a display (not represented in Fig. 4). The control unit or the computer system stores data received from the detector 475. Moreover, a processing unit of the control unit or the processor of the computer system (also not shown in Fig. 4) are able to process the measurement data of the detector 475. Finally, the control unit 470 or the computer system may have a connection to the apparatus 500 of Fig. 5 in order to transmit the measured and/or the processed data to the apparatus 500.
• the resolution of the confocal microscope 400 in lateral direction is limited by diffraction. This means that the lateral resolution depends on the diameter or opening of the pinholes 430 and 465, the numerical aperture (NA) of the objective 440 as well as on the wavelength of the light source 420. In lateral direction the resolution is in the range of the wavelength of the light source 420.
• the resolution in the beam direction is not diffraction limited, but limited by the determination of the position of the maximum intensity. The resolution in beam direction reaches the one digit nanometer range.
• Fig. 5 depicts a schematic block diagram of an apparatus 500 which can be used to calculate and to apply an arrangement of laser pulses to an optical component and/or a wafer.
• the principle is illustrated for the example the masks of Figs. 1 and 2.
• Other examples of optical components may be templates used in the nanoimprint technique and/or lenses of an optical system. Further examples are wafers on which integrated circuits (ICs) are generated.
• the apparatus 500 comprises a chuck 520 which may be movable in three dimensions.
• the optical component 510 for example a mask 510, may be fixed to the chuck 520 by using various techniques as for example clamping.
• the photolithographic mask 510 may be the transmissive mask 100 of Fig. 1 mounted upside down, so that its rear substrate surface 140 is directed towards the objective 540.
• the apparatus 500 includes a pulse laser source 530 which produces a beam or a light beam 535 of pulses or light pulses.
• the laser source 530 generates light pulses of variable duration.
• the pulse duration may be as low as 10 fs but may also be continuously increased up to 100 ps.
• the pulse energy of the light pulses generated by the pulsed laser source 530 can also be adjusted across a huge range reaching from 0.001 ⁇ J per pulse up to 10 mJ per pulse.
• the repetition rate of the light pulses comprises the range from 1 Hz to 100 MHz.
• the light pulses may be generated by a Ti:sapphire laser operating at a wavelength of about 800 nm.
• the methods described in the following are not limited to this laser type, principally all laser types may be used having a photon energy which is smaller than the band gap to the optical component 510 and which are able to generate ultra-short pulses with durations in the femtosecond range. This criterion also holds for wafers.
• the emission wavelength of the apparatus 500 has to be adapted the band gap of the optical component 510 or of a wafer.
• YAG laser systems doped with at least one of the elements neo- dymium (Nd), thulium (Tm), holmium (Ho), and erbium (Er) can also be used.
• dye laser systems may also be applied.
• the apparatus 500 may also comprise more than one pulse laser source 530 of the same type or of different types (not shown in Fig. 5).
• the requirement that the band gap of the optical component 510 is larger than the emission wavelength of the laser source used to apply ultra-short laser pulses is typically equivalent to the requirement that the optical component 510 is optically transparent in the near-infrared, the visible, and/or the near-ultraviolet wavelength range. In these wavelength ranges laser sources are available which can generate ultra-short pulses having a large energy density.
• the following table represents an overview of laser beam parameters of a frequency- doubled Nd-YAG laser system which is used in an embodiment of the inventive method.
• the pulse density refers to a 2D density in a plane perpendicular to the beam direction (lateral plane).
• a laser pulse does not generate a pixel in form of a sphere but creates a pixel having a form similar to an ellipsoid or a spheroid.
• the length of the spheroid is in a range of 1 ⁇ to 50 ⁇ and its width or thickness extends for examples from 0.5 ⁇ to 10 ⁇ . Characteristic lengths to thickness ratios are in the range of 2 to 10.
• pixel densities in the lateral plane are 10 3 to 10 7 pixels per square mm.
• One- dimensional pixel densities in beam direction cover a range of 1 to 100 pixel(s) per mm.
• the following tables indicate parameters of differently influencing the density and/or the optical transmission distribution of the substrate no of the photomask 100.
• Table 2 presents parameters of an embodiment using a frequency-doubled Nd-YAG laser system for a mode of introducing or writing pixels called standard process window (std PW).
• Table 3 summarizes parameters of a mode called low registration process window (LowReg PW) again of an embodiment using a frequency-doubled Nd-YAG laser system.
• This operation mode of the laser system 530 uses light pulses having a lower energy than the std PW, but introduces a higher pixel density.
• NoReg PW no registration process window
• Table 5 presents parameters of a mode called pixelless process window (pixelless PW).
• pixelless PW pixelless process window
• a Ti: Sapphire laser system is used for the modification of the density and/or the optical transmission distribution of an optical componnt. This laser system generates ultra-short laser pulses at an emission wavelength of about 800 nm. The pulse density of the pixelless PW is very high.
• the steering mirror 590 directs the pulsed laser beam 535 into the focusing objective 540.
• the objective 540 focuses the pulsed laser beam 535 through the rear substrate surface into the substrate of the photolithographic mask 510.
• the NA (numerical aperture) of the applied objective depends on the predetermined spot size of the focal point and the position of the focal point within the substrate of the transmissive mask 510 relative to the rear substrate surface. As indicated in table 1, the NA of the objective 540 may be up to 0.9 which results in a focal point spot diameter of essentially 1 ⁇ and a maximum intensity of essentially 10 20 W/cm 2 .
• the apparatus 500 also includes a controller 580 and a computer 560 which manage the translations of the two-axis positioning stage of the sample stage 520 in the plane perpendicular to the laser beam (x- and y-direction).
• the controller 580 and the computer 560 also control the translation of the objective 540 perpendicular to the plane of the chuck 520 (z-direction) via the one-axis positioning stage 550 to which the objective 540 is fixed.
• the chuck 520 may be equipped with a three-axis positioning system in order to move the optical component 510 or the mask 510 to the target location and the objective 540 may be fixed, or the chuck 520 may be fixed and the objective 540 may be moveable in three dimensions.
• both the objective 540 and the chuck 520 with three-axis positioning systems.
• manual positioning stages can also be used for the movement of the optical component 510 as for example the transmissive mask to the target location of the pulsed laser beam 535 in x-, y- and z-directions and/or the objective 540 may have manual positioning stages for a movement in three dimensions.
• the apparatus 500 may also provide a viewing system including a detector 565 as for example a CCD (charge-coupled device) camera which receives light from an illumination source arranged to the chuck 520 (not shown in Fig. 5) via the dichroic mirror 545.
• the viewing system facilitates navigation of the optical component 510 to a target position. Further, the viewing system may also be used to observe the formation of a modified area in the optical component 510 by writing the arrangement of laser pulses with the laser beam 535 of the light source 530 into the optical component 510.
• the computer 560 may be a microprocessor, a general purpose processor, a special purpose processor, a CPU (central processing unit), a GPU (graphic processing unit), or the like.
• the computer 560 may be arranged in the controller 580, or may be a separate unit such as a PC (personal computer), a workstation, a mainframe, etc.
• the computer 560 may further comprise I/O (input/output) units like a keyboard, a touchpad, a mouse, a video/graphic display, a printer, etc.
• the computer 560 may comprise one or several connection ports through which it may send a request for experimental data to the control unit 470 of the confocal microscope 400 of Fig. 4.
• the computer 560 may also comprise a volatile and/or a non-volatile memory.
• the computer 560 may be realized in hardware, software, firmware, or any combination thereof.
• the computer 560 may control the laser source 530 (not indicated in Fig. 5).
• a non-volatile storage of the computer 560 may store a three-dimensional contour of an optical component.
• the computer 560 may receive experimental raw data from the control unit 470 of the confocal microscope 400. Alternatively, it may receive processed data from the control unit 470.
• the computing unit of the computer 560 determines from the predetermined 3D contour of the optical component 510 and the data received from the control unit 470 one or several deviations of the measured or existing 3D contour of an optical component from its predetermined 3D contour. It is also possible that the supplier of an optical component provides the deviation of the existing 3D contour from a predetermined contour.
• the computer 560 calculates from these data and from algorithms explained in section 5.3 Ctheoretical background") at least one arrangement of laser pulses which is appropriate to bring the existing 3D contour of the optical component 510 in accordance with its predetermined 3D contour.
• the diagram 600 of Fig. 6 schematically shows the application of one or more arrangements of laser pulses to an EUV mask 610.
• the EUV mask 610 has on its front surface an EUV pattern 615 and on its rear surface an optically transparent conductive coating 665.
• the mask 610 is arranged in a specific housing 620 and mounted with the mask pattern 615 down on brackets 625.
• the housing 620 is sealed which a protection glass window 630.
• the specific housing 620 allows controlling the environment of the mask 610. In particular, it may enable to evacuate the housing 620 (not indicated in Fig. 6).
• the laser beam 650 is focused by an objective 640 into the substrate 660 of the EUV mask 610.
• An exchangeable compensating glass plate 670 adjusts the focus of the laser beam 650 to the depth in the substrate 660 of the mask 610.
• the housing 620 does not have any movable parts.
• the illustrations 700 of Fig. 7 schematically represent a local variation of a height of a mask blank 710.
• the mask blank 710 is flat in its initial state.
• a 2D arrangement of laser pulses 720 the height of the surface is locally modified.
• a height variation or a vertical deformation or a deformation in z- direction is a combination of an expansion or contraction in z-direction and a bending.
• a local deformation generates a localized effect, whereas a bending results in a global effect.
• a local deformation in z-direction 730 can be the result of the application of a 2D or D3 arrangement of laser pulses 720 and can also result from an interaction of a local area having a local deformation with an area in which a 2D or D3 arrangement of laser pulses has been written which is expressed by Poisson's ratio (see section 5.3: "theoretical background").
• the theoretical considerations in section 5.3 also reveal that in most cases a bending in z-direction and a (local) deformation in z-direction can be decoupled as they have a significantly different magnitude.
• Fig. 7 clearly indicates that the flatness control of an optical component requires controlling of both, the bending and the surface contour induced by arrangements of laser pulses.
• the generation of a predefined bending or the correction of a determined bending of an optical component is a good example to demonstrate the capabilities of a 3D deformation model which leads to the calculation of 3D arrangements of laser pulses.
• the exemplary optical component used in the experiments described in the following is a 6 inch (152 mm) mask blank. Most of the results are also applicable to many other optical components having other shapes which allow the application of 3D arrangements of laser pulses into the volume of the optical components, i.e. optical components which are optically transparent in the near-infrared, the visible, and/or the near- ultraviolet wavelength range.
• the application of different 2D arrangements of laser pulses is performed in planes parallel to the mask blank front and rear surface in different depths from the respective surface and using different writing modes, i.e. using 2D arrangements of laser pulses which have different parameters for the laser pulses and/or different pulse densities.
• the sizes of the 2D arrangements comprise the entire active area of the mask blank.
• the deformations resulting from the arrangements were measured in an area of 146 mm x 146 mm, wherein the measurement grid was about 30 x 30 nodes.
• the confocal microscope 400 used in the application examples described in this section is a Micro Epsilon IFS 3401-10 confocal sensor with a working distance of 10 mm and a resolution in beam direction or in z-direction of 400 nm.
• the noise of the measurement sensor is around 1000 nm.
• 100 distance samplings have been added in order to reach a distance sampling repeatability (in z-direction) of about 100 nm.
• the positioning stage (of Schneeberger) of the objective 440 of Fig. 4 has a repeatability in z-direction of approximately 200 nm. In order to reduce the contribution of the positioning stage to the variance in z-direction five sequential scans are acquired. Thus, the repeatability of the determination of an existing 3D contour of an optical component could be reduced to about 200 nm.
• a tilt of a mask blank is always mathematically corrected.
• the shape of a laser pulse is an efficient parameter to introduce an anisotropic expansion of a mask blank, or generally an optical component.
• 2D arrangements of laser pulses are written in a mask blank close to its upper surface.
• a 2D arrangement of laser pulses leading to an isotropic expansion in the lateral plane is written in a mask blank.
• An isotropic expansion in a plane perpendicular to the beam direction can for example be generated by laser pulses having a circular shape.
• the ellipse 820 of Fig. 8 describes the resulting expansion of the mask blank.
• the two lines 860 and 870 show the magnitude of the expansion versus the direction in space.
• the two axes 860 and 870 illustrate the main axes of the expansion.
• the eccentricity of the ellipse 820 is very low which indicates that the expansion of the mask blank in the lateral plane is almost uniform.
• the square indicates an elementary area 810 of the mask blank prior to writing an arrangement of laser pulses in the elementary area 810.
• the elementary area 810 is a 2D projection of an elementary volume a introduced in section 5.3.
• an isotropic expansion is introduced in the elementary area 810 by writing a homogeneous 2D arrangement of laser pulses in the mask blank.
• the deformation of the elementary area 810 is described by eight parameters which are the length and the directions of the arrows 830 at the four corner points of the elementary area 810.
• the parallelepiped 840 indicates the deformations introduced in the elementary area 810 by a homogeneous 2D arrangement of laser pulses having laser pulses with an essentially a circular beam shape.
• the square 850 represents a normalized deformation of the elementary area 810 by an ideal isotropic expansion. Fig. 8 reveals that an isotropic 2D arrangement of laser pulses having an essentially circular shape results essentially in an isotropic expansion of the elementary area 810.
• Fig. 9 depicts the situation when a homogeneous 2D arrangement of laser pulses is written in a mask blank, wherein the laser pulses result in an essentially lD expansion of the elementary area 910.
• a strongly anisotropic expansion as shown in Fig. 9 can for example be generated by a laser beam having a slit-like beam shape as for example generated by a cylinder lens.
• an anisotropic expansion of an elementary area 910 can also be generated by writing overlapping pulses in the mask blank which lead to pixels overlapping in one direction or create an anisotropic pixel.
• the anisotropic expansion of the mask blank is symbolized by the ellipse 920.
• the ellipse 920 has a large eccentricity and is almost degenerated to a line. .
• the arrows 930 again symbolize the deformations induced at the corner points in the elementary area 910.
• the anisotropic beam shape of laser pulses results in a high anisotropic or a nearly lD deformation of the elementary area 910.
• the square 950 represents a normalized deformation of the elementary area 910.
• Fig. 10 shows a mask blank having dimensions of 152 mm x 152 mm.
• a stripe with a width of approximately 39 mm is covered by a 2D arrangement of laser pulses.
• a single layer of the 2D arrangement of laser pulses is written close to the upper surface of the mask blank which facilitates the interpretation of the obtained results.
• the optical transmission of the mask blank is changed (reduced) in the vertical stripe by essentially 4.5 %.
• the change of the optical transmission or the increase of the optical attenuation introduced in an optical component by writing pixels into the optical component is a historic measure to quantify the amount of the writing of pixels into an optical component.
• An attenuation of 4.5 % means an aggressive writing of pixels in the stripe covered by the 2D arrangement of laser pulses.
• Fig. 11 presents the height variation induced by the homogeneous 2D arrangement of Fig. 8, wherein the laser pulses have parameters which essentially lead to an isotropic expansion.
• Fig. 12 depicts the variation in beam direction or in z-direction induced by the homogeneous 2D arrangement of laser pulses of Fig. 9, wherein the laser pulses have parameters which result in a strongly anisotropic expansion.
• the height variation scales from -3000 nm to +3000 nm resulting in a total height change of the mask blank of about 6 ⁇ across a length of 152 mm.
• the presented method allows to control the surface contour at a nanometer level which results in an inaccuracy in the per mille range. A preciseness in the nanometer range is necessary at the treatment of EUV masks as already deviations of their 3D contour from a predetermined contour lead to aberrations which significantly reduce the capability of this optical component.
• a comparison of Figs. 11 and 12 confirms the assumption that it is not possible to use an isotropic expansion 820 in an elementary area 810 in order to generate a bending which has different curvatures in different directions.
• the pixels are introduced far from the mask blank surface and the contribution from the pixel writing to a change of the surface profile or contour is low.
• the surface profile is modified by less than 1 nm.
• the ratio of surface change to bending is less than 1 % in this example. Consequently, an anisotropic or a strongly isotropic expansion of an elementary area is well suited in order to generate an anisotropic or an almost iD bending of an optical component with minimal effect of the surface contour of the optical element.
• the defined method can generate a predetermined 3D contour or a predetermined bending profile which has a rotational symmetry even if the initial optical component does not have a rotational symmetry.
• a mask blank is again used as an initial optical component.
• the mask blank has again dimensions of 152 mm x 152 mm.
• Fig. 13 shows the cross-section of the target profile.
• the units in the x-direction are ⁇ and are nm in the y-direction.
• the center portion of the mask blank is almost flat. The total bending from the edge of the central portion of the mask blank to the outer edge symmetrically raises by approximately 500 nm.
• Fig. 10 the bending is adapted to the symmetry of the mask blank. This restriction is removed in the following example. This means a rotationally symmetric bending is applied to a square mask blank.
• This target is selected as the flat top 3D contour of Fig. 13 has in different parts of the bending profile different bending curvatures in the x- and the y-direction. This means that the 3D contour of Fig. 13 cannot be obtained with an arrangement of laser pulses having one expansion property.
• Fig. 14 presents a 2D map representing the 3D target contour of Fig. 13. As indicated in the scale on the right side of the mask blank, the overall height variation amounts to 900 nm. A 10 x 10 equidistant step grid is used for computations. The profile of the mask blank plane is sampled at 29 x 29 locations.
• the 2D arrangements of laser pulses for two selected writing modes are calculated using information on known 2D mode signatures.
• this is a reasonable assumption because the contribution of the 2D arrangements of laser pulses to an expansion in the beam direction or in z-direction is about 0.01 nm per layer of 2D arrangement of laser pulses or per writing layer. This contribution is negligible.
• Fig. 15 shows the calculated writing map or the 2D arrangement of laser pulses used for the expansion of the mask blank in x-direction
• Fig. 16 presents the corresponding writing map for the expansion of the mask blank in the y-direction
• Fig. 15 illustrates that the 2D arrangement of laser pulses or the 2D writing map responsible for the expansion in x-direction has a higher density in the area of the mask blank where the resulting 3D contour has to have a maximum bending curvature in x- direction. This symmetry is also reflected in Fig. 16 presenting the 2D writing map for the expansion in y-direction.
• the 2D arrangements of laser pulses or the 2D writing maps show a 180 0 rotational symmetry, because the mask blank and mode signatures of the two 2D writing modes have this symmetry.
• the 2D writing modes of Figs. 15 and 16 show a kind of 90 0 rotational symmetry.
• the symmetry is not perfect due to the absence of a 90 0 symmetry between the writing mode signatures of Figs. 15 and 16.
• the two 2D writing modes of Figs. 15 and 16 forming a 3D arrangement of laser pulses are separated in vertical direction by approximately 15 ⁇ .
• the resulting change of the 3D contour of the bottom surface of the mask blank is depicted in Fig. 17.
• the variation range of the 3D contour of the mask blank after writing the 3D arrangement of laser pulses comprising the writing layers of the two writing modes of Figs. 15 and 16 extends from -800 nm to +1500 nm or 2.3 ⁇ as it is indicated by the scale on the right side of Fig. 17. This differs by approximately a factor of 2.5 with respect to the 3D target contour presented in Fig. 13. This results from a mistake in the normalization for the conversion of the mode signature (MS) from a 2D to a 3D representation.
• MS mode signature
• the measurement data contain a noise magnitude of about 200 nm. Measurements are performed on the initial mask blank prior the application of the 3D arrangement of laser pulses comprising the writing maps indicated in Figs. 15 and 16. The resulting change of the 3D contour inherently contains a noise component with a magnitude of 300 nm.
• a smoothing thin plate spline (TPS) fit is applied which removes data variations having a magnitude smaller than 300 nm.
• Fig. 18 shows a TPS fit of the measured bending of the mask blank induced by the 3D arrangement of laser pulses.
• Fig. 13 two cross section plots across the modified mask blank are prepared at the positions indicated by the lines 1810 and 1820.
• Fig. 19 presents a comparison between the 3D target contour 1910 with the two cross- section plots at positions 1810 and 1820 of Fig. 18.
• the 3D target contour 1910 uses the scale indicated at the y-direction.
• the two cross-section plots 1810 and 1820 use the scale indicated on the right side of Fig. 19 which has a different scale than that of the y- axis. This is due to the above mentioned normalization problem.
• the target profile 1910 contains a small elevation in the center of rotational symmetry.
• the bending resulting from the pixel writing cannot have this elevation, as only expanding writing modes are used which are arranged in one side of the mask body. From the presented example it is estimated that curvatures leading to a height variation in the range from about 1 nm to 50 ⁇ can be induced in an optical component by writing one or more 3D arrangements of laser pulses in the volume of the optical component.
• the variation of the surface contour is again below 1 nm.
• the process of bending an optical component allows for example to fabricate lenses 2010 which have an extremely large focal length from an initially flat optical component 2000.
• a 3D arrangement of laser pulses 2020 is written close to the rear surface 2070 of the optical component 2000 which induce a bending with a rotational symmetry, and thus transforming the flat component 2000 into a lens 2010 having a large focal length.
• Parallel light 2040 incident on the front surface 2060 of the lens 2010 leaves the rear surface 2070 of the lens 2010 as slightly divergent light 2050.
• parallel light incident on the rear surface 2070 leaves the front surface slightly convergent (not indicated in Fig. 20).
• the next application example presents the generation of an optical component having a predetermined arbitrary shape.
• a target is selected with a shape which has curvatures with two different signs in different directions.
• This specific 3D contour cannot be performed by using a single plane 2D arrangement of laser pulses which predominantly induces an expansion in the plane of the 2D arrangement of laser pulses.
• a single 3D contour having curvatures with different signs can be computed from a second order polynom.
• the following polynom is selected:
• equation 61 describes a 3D contour having a saddle shape with different magnitudes of curvature in x- and in y-direction.
• the term x ⁇ y is added to equation 61 that slightly rotates the saddle around the vertical axis.
• a quantitative representation of a single pixel is a writing mode signature, abbreviated as MS for mode signature, which describes the amount of a resulting expansion in case of using pixels of a single type (cf. also section 5.3).
• MS for mode signature describes the amount of a resulting expansion in case of using pixels of a single type (cf. also section 5.3).
• One writing mode uniformly processes an area of 1 mm x 1 mm.
• Figs. 22 and 24 show calculated writing maps or 2D arrangements of laser pulses for writing modes which essentially lead to an expansion in x-direction.
• Figs. 23 and 25 represent computed 2D writing maps for writing modes which predominantly result in an expansion in y-direction.
• the 2D arrangements of laser pulses of Figs. 22 and 23 are written close to the rear or bottom surface of a mask blank.
• the 2D writing modes of Figs. 24 and 25 are introduced close to the front or upper surface of the mask blank.
• the term "close to” describes a range of 100 ⁇ to 800 ⁇ from the respective surface.
• the two 2D writing modes close to the respective surface of the mask blank are separated in beam direction by a range of 15 ⁇ to 50 ⁇ .
• the four writing maps of Figs. 22 to 25 form the 3D arrangement of laser pulses used to generate the 3D target contour.
• equation 61 details of the 2D writing maps and the 3D contour or the target bending profile. This is a consequence of the minor rotation introduced by the 3D contour of equation 61 that destroys a covariation between the 3D contour, the shape of a mask blank (or generally an optical component in its initial state) and the 2D writing modes forming the 3D arrangement of laser pulses.
• the specific 3D contour i.e. the saddle having different curvatures in the horizontal and the vertical direction can only be obtained by a 3D arrangement of laser pulses which combines the different writing modes, wherein in each case two of different writing modes are applied close to each surface of the mask blank.
• Fig. 26 presents the measured 3D contour introduced in the mask blank by the 3D arrangement of laser pulses containing the arrangement of writing modes of Figs. 22 to 25.
• the coefficients of this function are derived from the measured data. The computed numerical values for the coefficients are:
• the generated 3D contour or the induced bending of the mask blank is a result of the combination of the deformations induced by all 2D arrangements of pixels of different modes which lead to a possible bending error of about 20 % to 30%.
• the inventive method is applied to demonstrate its ability to improve the flatness of a surface of an optical component by applying a 3D arrangement of laser pulses in the optical component.
• a degraded quartz plate or mask blank having a pronounced bending is used in this application example in order to have a reasonable signal to noise ratio.
• the specification of ASAHI for the flatness of a mask blank defines that a flatness deviation is ⁇ 2 ⁇ and ⁇ 0.5 ⁇ for advanced mask blanks.
• Fig. 27 shows a measured 3D contour or a 3D profile of a mask blank.
• the total variation of the height covers a range of 8 ⁇ (from -4000 nm to +4000 nm) so that the measured mask blank does not fulfill any of the ASAHI specifications.
• the measured 3D contour or the surface curvature does not change its sign across the mask blank. Therefore, the surface of the mask blank can be corrected by applying a 3D arrangement of laser pulses close to the front or upper surface.
• a flatness deviation of about 300 nm fulfills the ASAHI specification of advanced mask blanks, and (b) as discussed above, the noise of the metrology tool used to determine the 3D contour of the mask blank is in the range of 300 nm.
• a smoothing thin plate spline (TPS) fit is applied to the 3D contour of Fig. 27 which removes variations of the 3D contour having a magnitude of less than 300 nm.
• Fig. 28 presents mask blank of Fig. 27 after performing a TPS fit of the measured 3D contour.
• TPS thin plate spline
• Figs. 29 and 30 depict the two writing modes contained in the 3D arrangement of laser pulses and used for flattening the 3D contour of Fig. 28.
• a grid of 9 x 9 x 3 is used across the mask blank.
• the two writing modes may induce a variation of the optical transmission in the range of o % to 5 %.
• Both 2D arrangements of laser pulses predominantly locally expand the mask blank in a plane perpendicular to the laser beam.
• a simulation shows that it is possible to flatten up to 94 % of the 3D contour of the smoothed TPS fit.
• Fig. 31 shows the 3D contour of Figs. 27 and 28 after the application of the 3D arrangement which contains two writing modes.
• the flatness variation is reduced from a range of 6500 nm with a standard deviation (io) of 1000 nm of Fig. 27 to a flatness deviation of about 1300 nm with a standard deviation of about 250 nm. This means that the 3D contour of Fig. 27 could be flattened by a factor of 4.
• Fig. 32 represents again the measured 3D contour of Fig. 31 wherein the scale of the height is reduced from 8 ⁇ (-4000 nm to +4000 nm) to 1 ⁇ (-500 nm to +500 nm).
• Fig. 32 reveals that the writing mode responsible for the expansion of the mask blank in y-direction shows some undershoot and that there is some room for improvement.
• the expansion of the mask blank in x-direction is near to the target expansion in this direction.
• the measured 3D contour contains a significant contribution of the measurement noise of Fig. 27. It is again assumed that the initial 3D contour of the mask blank has a smooth curvature. Further, it is not possible to induce a front surface modulation with a high frequency by applying a 3D arrangement of laser pulses as far as 6 mm (or 10 6 nm) from the front surface (or 350 nm from the rear surface of the mask blank). Based on these assumptions a smoothing TPS fit is applied to the measured 3D contour of Fig. 32 in order to eliminate the measurement noise of the metrology tool from Fig. 32.
• Fig- 33 presents the residual 3D contour of Fig. 32 after the smoothing TPS fit.
• the 3D contour shows a height variation of about 700 nm with a standard deviation of 180 nm.
• the correct way to compare the result of the application of the 3D arrangement of laser pulses is to compare TPS data of the initial 3D contour and the TPS data of the remaining 3D contour after fmalization of the application of the 3D arrangement of laser pulses. This approach allows to some extent to eliminate the noise of the metrology tool 400 and to distill the characteristics of the initial and the remaining 3D contour.
• the standard deviation of the 3D contour could be reduced from 1000 nm to 180 nm which results in an improvement of 82 %. This means that the initially rejected mask blank (as it does not fulfill the ASAHI specification for regular mask blanks) is brought in the range of the ASAHI specification for advanced mask blanks.
• the described method can improve the 3D contour by more than one order of magnitude, and thus bringing an existing 3D contour in close accordance with a predetermined 3D contour.
• the defined method is applied to write one or more three-dimensional (3D) arrangement of laser pulses into a wafer.
• parameters of the laser pulses are estimated which can be used to write pixel in a silicon wafer.
• Silicon is used in the following example as it is a popular semiconductor material.
• the use of the method defined in this application is not restricted to silicon (Si) wafers. Rather, the method can also be used for wafers which comprise different semiconductor materials, as for example germanium (Ge). Further, it can also be applied to wafers comprising compound semiconductors having two or more semiconducting elements. Examples of binary compound semiconductors are gallium arsenide (GaAs), indium phosphide (InP), and gallium nitride (GaN), just to name a few.
• GaAs gallium arsenide
• InP indium phosphide
• GaN gallium nitride
• wafer comprises the input to a semiconductor manufacturing process, i.e. typically a semiconductor disc and the output of the semiconductor process, i.e. a fabricated component just before it is separated from the wafer. Moreover, the term “wafer” comprises all intermediate stages of the wafer in the semiconductor manufacturing process.
• Fig- 34 presents an overview of the transmission spectrum of silicon for the wavelength range of 0.1 ⁇ to 100 ⁇ in a semi-logarithmic representation, i.e. covering four orders or magnitude for the wavelength.
• This spectrum has been taken from the Internet: http://micro.com/technical-notes/bk7-quartz-ge-si.
• the band gap energy of the indirect transition of pure Si is approximately 1.14 eV at 300 K which corresponds to a wavelength of 1.09 ⁇ . This band gap energy is indicated by the shape increase of the transmission in the wavelength region around 1 ⁇ .
• the absorption coefficient of silicon is high in the wavelength range from o.i ⁇ to 1.0 ⁇ .
• Silicon is used as a detector material for ultraviolet and visible radiation.
• Silicon is transparent for electromagnetic radiation in the wavelength ranges from approximately l.i ⁇ to 6.0 ⁇ and silicon and again from approximately 40 ⁇ to 100 ⁇ .
• the arrow in Fig. 34 indicates the interesting wavelength range from 1.2 ⁇ to 7.0 ⁇ which can be used to for writing pixels into pure silicon.
• Fig- 35 depicts the wavelength range of 1 ⁇ to 9 ⁇ with a higher spectral resolution. This curve has been taken from the Internet: http: / / micro.com/technical-notes/bk7- quartz-ge-si. Fig. 35 additionally indicates the variation of the transmission as a function of temperature. It clearly shows that the optical transmission significantly reduces when heating silicon. This effect is the more pronounced for the larger wavelength. Using a wavelength of around 5 ⁇ may be beneficial.
• Fig. 36 shows the absorption coefficient of Si as a function of the wavelength in the range of 200 nm to 1400 nm in a semi-logarithmic representation.
• This diagram again indicates that silicon can be used as a detector material for electromagnetic radiation in the wavelength range from 200 nm to approximately 1000 nm.
• the absorption drops by more than 8 orders of magnitudes.
• Fig. 35 suggests to that wavelengths in the range of 5 ⁇ are preferred for the writing of pixels into a Si wafer with respect to the requirement that Si needs to be transparent for the electromagnetic radiation or for the laser pulses used to introduce pixel in Si.
• looking at the absorption coefficient of Si as shown in Fig. 36 it seems that the wavelength of the laser pulses needs to stay below 1300 ⁇ .
• the generation of local volumes in which the melting point is surpassed requires very or ultra-short laser pulse. Only by applying one or more ultra-short laser pulses a heating of a small Si volume can be generated which is fast enough so that the generated heat cannot significantly dissipate in a volume surrounding the focus spot of the laser pulse.
• NA denotes the numerical aperture of the optical system used to focus the laser pulses into the cylinder of Fig.37. Further, the approximation is used that for small angles sin a « a .
• the angle a is half of the total angle spread ⁇ or the Gaussian beam.
• the depth of focus is two times the Raleigh range z R , wherein the Raleigh range or length is given by:
• equation (64) is used for the last transformation
• the specific heat capacity is given by:
• a denotes the absorption coefficient of the Si crystal which is represented in Fig. 36.
• the intensity loss is equivalent to the energy loss of the Gaussian beam within its focus:
• W 0 is the energy arriving at the Gaussian focus and W ⁇ b) is the energy leaving the focus or the cylinder of length b. It is now assumed that the product a - b ⁇ ⁇ , i.e. only a portion of energy of a laser pulse is absorbed in the cylinder in the Gaussian beam focus so that
• Fig. 39 illustrates a pulse train of a laser system which has pulses of a pulse length or a pulse width At Pulse .
• the mean power emitted by laser system in the pulse train of Fig. 39 is given by:
• P Pube is the pulse power.
• the pulse power can be expressed as a function of the mean power of the laser system, its frequency and the pulse width of the individual pulses:
• the pulse energy W 0 of an incident laser pulse can be expressed pulse power and the pulse width:
• Gaussian focus is absorbed by the Si crystal within in the focus:
• Equation (69) describes the power required to increase the temperature within the Gaussian focus at least by the temperature difference ⁇ .
• Equation (76) expresses the energy provided by a laser pulse. Equating equation (69) with equation (76) leads to: o A T P 2 ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ 0
• the required mean power of a laser required to locally melt a Si crystal in the Gaussian focus can be estimated from the required temperature difference ⁇ : ⁇ ( ⁇ 0 )- ⁇ - ( ⁇ ) 2
• equations (78) and (79) it is indicated that the absorption coefficient of Si strongly depends on the wavelength of the laser system applied to introduce pixels into the Si crystal, as it can be seen from Fig. 36.
• NA 0.8.
• a focusing optic can be utilized using a solid immersion lens (SIL) which would allow the NA to exceed 2.
• SIL solid immersion lens
• Fig. 40 shows the variation of the internal optical transmission of a pure Si wafer having a thickness of 775 ⁇ in the range of its band edge.
• a wavelength of the laser pulses used to write pixels in the Si wafer has to be selected which on the one hand low enough so that the photons of the laser pulses are absorbed in the Si wafer and in particular in the Gaussian beam focus.
• the wavelength of the laser pulses has to be high enough in order to have a reasonable transmission through the whole wafer thickness so that pixels can be placed at predetermined locations.
• Fig. 41 represents the pulse energy as a function of the wavelength around the band edge of a pure Si crystal. The pulse energy amounts 0.4 ⁇ J as can be taken from Fig. 41 for a wavelength of 1050 nm.
• the pulse energy rises to approximately 1 ⁇ J for a laser pulse having a wavelength of about 1100 nm.
• the estimated energy range for writing pixels in a Si crystal is similar to the one for introducing pixel in a substrate of a photolithographic mask, as is indicated in Table 1.
• the two dots in Fig. 41 indicate that laser systems are available which are able to supply enough pulse energy to locally melt a silicon crystal.
• the HE1060 of Fig. 41 is an Nd-YAG laser system which emits laser pulses at 1064 ⁇ . It can supply sufficient energy to locally melt a Si crystal within a wafer within a Gaussian beam focus. The melted Si volume within a wafer will solidify as amorphous silicon which has a mass density which is 1.8 % smaller than that of crystalline silicon (see Table 6). Hence, the pixels will induce stress in the surrounding Si crystal which can be used to alter the surface contour of a Si wafer.
• the assessment for the laser parameters given above is only a rough estimation.
• a more accurate estimation of the laser parameters to write pixels into a Si wafer can be obtained by performing a finite element calculation of how incoming laser pulses propagate through the Si wafer, and in particular how the temperature within the Gaussian focus dynamically changes the optical properties of the Si crystal. For example, as can be seen in Fig. 35, the absorption in silicon increases as a function of the temperature of the Si crystal. Further, a more accurate estimation also considers how the energy deposited in the Gaussian beam focus dissipates into the surrounding crystal material.
• the estimation presented above is a very first order approximation that shows that arrangements of pixels can also be written in a defined manner into a Si crystal or more generally in a semiconductor crystal.
• the diagram 4200 of Fig. 42 schematically depicts a cross-section through a wafer 4210 at the beginning of a process to fabricate integrated circuits on the front surface 4420 of the wafer.
• the dotted line 4250 indicates the half of the depth of the wafer 4210 and is also called neutral plane.
• the wafer 4210 is fixed with a chuck, for example a vacuum chuck, on its rear surface 4230.
• the wafer 4210 is a silicon wafer.
• Fig. 42 shows that the wafer 4210 is essentially plan prior the start of the processing steps to fabricate one of more integrated circuits (ICs) on its front surface 4220.
• Fig. 43 schematically illustrates the situation after some processing steps have been performed in order to generate the ICs 4360, 4370 and 4380.
• the processing of the front surface 4320 has induced stress in the wafer 4310 which results in a bending of the wafer 4310.
• ICs are fabricated on the front surface 4320 of the wafer 4310.
• the person skilled in the art will appreciate that any kind of ICs can be fabricated on the wafer 4310.
• the method defined in this application can be applied to any manufacturing process which causes stress in the wafer 4310.
• the magnitude of the bending depends on the wafer, its dimensions and the IC(s) to be fabricated on the wafer.
• the bending may be in the range of up to 100 ⁇ and may be even larger for unfavorable conditions.
• the bending effect tends to increase with advanced storage technologies, as for example 3DNand.
• the apparatus 400 of Fig. 4 can be used to measure the amount of bending of the wafer 4310. It is preferred to use the front surface 4320 of the wafer 4310 to determine the amount of wafer bending. This allows measuring the wafer 4310 as it is fixed to the chuck. In an alternative approach, the rear surface 4330 of the wafer 4310 which has no profile or topography due various processing steps can also be used to determine the bending effect of the wafer 4310.
• the calculated 3D arrangements of laser pulses are written into the upper part of the wafer 4310 which generates 3D arrangements of pixels in the wafer 4310 which are indicated by the dots 4390.
• the calculated 3D arrange- ments of laser pulses take into account the symmetry of the problem, i.e. the bending the wafer 4310.
• Fig. 44 schematically represents the wafer 4410 after finalization of the pixel writing process.
• the bending of the wafer 4310 of Fig. 43 is essentially removed for the wafer 4410.
• a laser system can be used for generating the ultra-short laser pulses which emits at a wavelength of around 2 ⁇ .
• Laser which emit in this wavelength range are for example Ho-YAG und Tm-YAG laser systems.
• Nd-YAG laser systems emitting at a wavelength of 1064 nm which are presently used to cut dies from a wafer. This is beneficial as an already available laser system can be applied for the introduction of the 3D arrangements of laser pulses in the wafer 4310.
• the parameters of the laser pulses and the 3D arrangements of laser pulses are selected so that the pixels locally induce stress into the Si crystal which reduces the mass density of the wafer 4410 in order to counteract the stress introduced by various processing steps and leading to the bending effect.
• the laser pulses through the front surface 4320 of the wafer 4310 as this can be performed with a minimum handling effort of the wafer 4310. In doing so, it has to be taken into account that the processing steps may change, i.e. reduce the band gap of the materials of the ICs 4360, 4370 and 4380 with respect to the pure silicon wafer 4310. This means that it is necessary to select a wavelength for the laser pulses so that the radiation can essentially transmit the ICs 4360, 4370 and 4380 without any absorption. In an alternative approach, it is also possible to introduce the 3D arrangements of laser pulses through the rear substrate surface 4330.
• the 3D arrangements of pixels are arranged in the substrate of the ICs 4360, 4370 and 4380 but not in their functional layers. Therefore, the pixel writing process does not induce any damages to the ICs 4360, 4370 and 4380.
• Fig. 45 depicts a flow chart 4500 of an embodiment of the inventive method.
• the method begins at block 4510.
• a first step 4520 an existing 3D contour of an optical component or a wafer is measured.
• the metrology tool 400 depicted in Fig. 4 can perform this step.
• a deviation of the existing 3D contour from a predetermined 3D contour is determined.
• This method step can for example be executed by the computer 560 or the computer system 560 of Fig. 5.
• step 4540 a 3D arrangement of laser pulses is calculated from the deviation determined in step 4530.
• the computer 560 can again perform this calculation.
• the calculated 3D arrangement of laser pulses is applied on the optical component.
• the laser system 500 of Fig. 5 can execute this method step.
• step 4560 the 3D contour generated by the application of the 3D arrangement of laser pulses is measured. Then in step 4570, a remaining deviation of the generated 3D contour from the predetermined 3D contour is determined. At decision block 4580 it is decided if the remaining deviation between the generated and the predetermined 3D contour is smaller than or equal to a predefined threshold. If this is the case, the method ends at block 4590. If the remaining deviation is still larger than the predefined threshold, the method proceeds to block 3740 and a new 3D arrangement of laser pulses is calculated which brings the generated 3D contour in closer accordance with the predetermined 3D contour.
• the flow chart 4600 of Figs. 46a and 46b present, as an example, the application of the defined method during the manufacturing process of EUV masks.
• the defined method can be applied to other optical components as well as to wafers during the manufacturing process of ICs on the wafers.
• An EUV mask blank and an EUV mask have to be ideally flat. If the rear surface of the EUV mask blank is not flat within a range of 1 ⁇ it is differently gripped by a mechanical chuck during the manufacturing process of the EUV mask and/or by an electrostatic chuck of a scanner during the application of the EUV mask in an illumination process.
• the different surface profiles due to the different gripping of the EUV mask results in topographic errors which cannot be fully compensated if their shape is different for the two mounting principles. Furthermore, if an EUV mask is not perfectly flat, it is bended each time if it is mounted to a chuck which additionally might result in particle generation. It is therefore mandatory that an EUV mask blank is perfectly flat at the beginning of an EUV mask manufacturing process and is kept flat during the various steps of the manufacturing process.
• the flow chart 4600 of Figs. 46a and 46b assume that a mask blank entering the manufacturing process is ideally flat.
• the manufacturing process begins at block 4605 of Fig. 46a.
• the multilayer structure 255 of Fig. 2 is arranged on a mask blank or on a substrate 210.
• step 4615 a deviation of the flatness of the rear substrate surface 225 is determined caused by the manufacturing process steps of block 4610.
• decision block 4620 it is decided if the deviation of the flatness of the rear substrate surface 225 is smaller than or equal to predefined threshold. If this is the case, the method proceeds to step 4640. If this condition is not fulfilled, the method proceeds to block 4625, where a 3D arrangement of laser pulses is calculated which flattens the rear substrate surface 225.
• step 4630 the calculated 3D arrangement of laser pulses is applied to the substrate 210 of the EUV mask.
• decision block 4635 it is checked whether the rear substrate surface 225 now meets the flatness requirement. If this is still not the case, steps 4625 and 4830 are repeated. Method steps 4625 and 4630 are repeated until the determined deviation of the EUV mask flatness is below a predetermined threshold. Then the method proceeds to block 4640.
• step 4640 the absorber layer is patterned and the peripheral part of the multilayer structure 255 is removed.
• step 4645 the deviation of the flatness caused by the processes of block 4640 are determined. It is or course also possible to determine the flatness variation individually caused by each of the processes of block 4640.
• decision block 4645 it is decided if the flatness variation of a deviation of the flatness is smaller than or equal to the predefined threshold. If this condition is fulfilled, the method ends at block 4655. If this is not the case, at step 4660 of Fig. 46b, a second 3D arrangement of laser pulses is calculated to eliminate or to at least reduce the flatness variation caused by manufacturing steps of block 4640.
• step 4665 the calculated 3D arrangement of laser pulses are applied on the substrate 210 of the EUV mask 200. Moreover, at decision block 4670 it is checked if the deviation of the flatness of the rear substrate surface 225 is below the predetermined threshold. If this is true, the method ends at block 4675. If the condition is not fulfilled, the method proceeds to block 4660 and repeats steps 4660 and 4665. Steps 4660 and 4665 are repeated until the determined flatness of the rear substrate surface of an EUV mask is below a predetermined threshold. 6. Theoretical background
• 3D three-dimensional
• optical component and/or the wafer it is possible to introduce pixels of different kinds in the optical component and/or the wafer in order to predominantly cause a lateral variation or to predominantly vary the height of the optical element and/or the wafer. Both, the lateral and the variation in z-direction can be performed as local and/or global variations.
• optical component and optical element are used as synonyms in this application.
• pixels introduce stress into an optical element or a wafer by locally changing the density of the optical element material or wafer material.
• Expanding pixels and contracting pixels can be introduced into the optical element or can be written into the optical element by varying several parameters of the laser beam of the ultra-short laser source, such as the pulse duration and/or the pulse energy, and/or the beam shape, and/or repetition rate.
• the density of the optical element material or the wafer material is locally reduced, whereas by writing contracting pixels the density of the optical element material or the wafer material is locally increased.
• the writing of pixels into an optical element or a wafer is characterized by two set of parameters, (a) A first set defines the 3D dimensions of an individual pixel, (b) A second set defines an assembly of individual pixels in a 3D arrangement.
• Laser beam parameters or writing mode or writing mode signature The writing of individual pixels into the optical element or the wafer with different physical conditions results in different kinds of pixels having different properties and/or different shapes. Parameters which characterize the 3D dimensions and the effects of an individual pixel are: (I) The pulse power of the light beam of the laser source which changes the size of a pixel. Reducing the laser pulse energy results in smaller pixels. Actually, for the pixel generation, the minimum pulse energy is limited by the breakdown threshold of the material of the optical element or the wafer. For pix- elless writing there is no well-defined threshold. It is beneficial to use pulse energies just below the breakdown threshold in order to obtain a high efficient pixelless writing process.
• the pulse duration of the laser beam pulses also influences the size of an individual pixel.
• the number of laser pulses applied to a single location of the optical element also affects the pixel dimensions and thus the effects of an individual pixel.
• the polarization of the beam of ultra-short light pulses has an influence on the lateral effects of an individual pixel.
• the NA (numerical aperture) of the light beam and/or the beam divergence affects the breakdown threshold of the material of the optical element or the wafer. Thus, the NA has to be controlled in combination with the pulse energy. Further, the NA has an essential influence on the size of an individual pixel in the beam direction (z-direction).
• the wavelength of the laser light pulses also affects the effects of an individual pixel with respect to a specific material of the optical element of the wafer.
• the lateral shape of the light pulse induces the lateral form of the generated pixel, and thus influences the deformation induced by the pixel in the optical element or the wafer.
• (b) 3D arrangement of pixels Besides the individual pixels (determined by the laser beam parameters) the assembly of the pixels in the optical element or the wafer defines the change of the 3D contour generated by a 3D arrangement of pixels.
• a 3D arrangement of pixels is characterized by six parameters. Three parameters determine the dimensions of the pixel arrangement in the x-, y-, and z- direction. Further, three parameters specify the linear assembly and thus the linear pixel density in the x-, y-, and z-direction. The writing of a different linear pixel density in different directions parallel to the surface of the optical element or the wafer leads in particular in case of overlapping or partly overlapping pixels to a nonsymmetrical expansion of the optical element or the wafer.
• the size of a pixel is smaller than half of the distance to a neighboring pixel.
• pixels can also be written with a laser beam wherein a distance between neighboring pixels is much less than the individual pixel size.
• a quasi-continuous structure is written into the optical element.
• a very low laser pulse power is used for this kind of pixels. The advantage of using such a kind of pixels is that they do not cause light scattering, but only changes the optical transmission of the optical element. This structure having no visible pixels in the visible spectral range resembles more a layer in the bulk of the material of the optical element with different physical properties.
• this layer is homogeneous enough, it less affects UV or DUV beam properties, no scattering occurs, and the layer does not introduce any artificial periodicity, and hence does not generate any diffraction effects. Directing the laser beam onto the optical element having such laser beam parameters is called pixel-less writing.
• the two parameters sets of a writing mode and a writing mode density or simply writing density are not independent from each other. For example, the effect of smaller pixels can be compensated by increasing the pixel density.
• a pixel arrangement comprises pixels written with a single set of laser beam parameters or with a single writing mode.
• a writing mode can also contain two or more pixels generated with two or more laser beam parameters.
• two or more pixel arrangements can be introduced in the optical element. The various pixel arrangements can have different positions within the optical element, can partially overlap or can completely overlap.
• the effects of ultra-short light pulses having different parameters onto the optical element has to be known.
• ultra-short light pulses having different parameters and/or writing modes have to be determined for the whole optical element.
• the volume of the optical element is partitioned into elementary volumes elements or elementary cells.
• the elementary volumes may be selected arbitrarily, it is preferred to use elementary volumes which have straight lines as edges and planes as faces or lateral surfaces.
• the corner points of the elementary volumes form 3D grid points across the volume of the optical element.
• Preferred elementary volumes are parallelepipeds, cuboids or tetrahedrons.
• Within one elementary volume pixels are written having a single writing mode and writing mode density.
• the solution of the direct problem would compute the deformation of the elementary volumes from the action of the laser beam, i.e. it would compute the volume defor- mations of the elementary volumes as a function of the writing mode or the laser beam parameters and the writing mode density or the writing density.
• the inverse problem has to be solved. This means, it is the problem to compute a 3D map of writing modes or laser beam parameter and writing mode densities or writing densities that transform a determined 3D contour of an optical element to its predetermined 3D contour.
• Optical elements are normally made of very rigid materials.
• lenses of optical systems and photomasks for photolithographic illumination systems are preferably made of quartz.
• the deformations necessary for the correction of deviations from a predetermined 3D contour are very small in amplitude and do typically not exceed the elastic limit of the material of the optical element.
• the complete optical element can be described as a system of cuboids, where every cuboid has a constant writing mode and writing mode density.
• Cuboids can be characterized by a new or modified equilibrium shape having new or modified physical parameters of the optical element such as Young's modulus, Poisson's ratio, etc.
• Young's modulus Young's modulus
• Poisson's ratio Young's ratio
• Both approaches give a linear dependence of the displacements on the writing mode densities at a given distribution of writing modes. Both approaches allow the computation of the modifications of the elementary volumes induced by the laser beam based on the knowledge of the writing modes and the writing mode densities.
• a writing mode is a specific pixel generated by one or several overlapping or partly overlapping laser pulses, wherein the pixels are generated with a discrete parameter set of the laser beam and the problem has to be formulated as an optimization problem.
• a target functional has to be constructed that has a minimum (or a supremum) at the desired displacement field, which transforms the determined 3D contour of the optical element or the wafer to its predetermined 3D contour.
• Optimized writing modes or optimized laser beam parameters and writing mode densities produce the displacement field that minimizes the target functional and thus generates the predetermined 3D contour of the optical element or the wafer.
• An optimization of writing modes can be performed using a MLS approach (Moving Least Squares); but a complete enumeration of the discrete writing mode map can be very difficult in case of small elementary cuboids.
• the discrete parameters of a 3D mode map can be converted to a continuous analog with the assumption that one elementary cuboid can have a superposition of the different writing modes with different writing mode densities.
• the target functional is a square of the residual deviation of the determined 3D contour from its predetermined 3D contour. Then, the variational formalism will result in a linear problem.
• the 3D infinitesimal strain tensor ⁇ ( ⁇ , ⁇ , ⁇ ) has components ⁇ ⁇ ( ⁇ , ⁇ , ⁇ ) and the stress tensor a(x,y,z) having components ⁇ ⁇ (x, y, z) .
• the elasticity tensor H(x,y,z) is a 4 th -order tensor field.
• the elasticity tensor has only the two independent components E(x, y, z) and ⁇ ( ⁇ , y, z) , and is of the form: 1 - ⁇ ⁇ ⁇ 0 0 0 0
• E(x,y,z) denotes Young's modulus and ⁇ , ⁇ , ⁇ ) is Poisson's ratio.
• ⁇ ( ⁇ , y, z) denotes a change of the x-coordinate of a point within the optical element or the wafer with respect to its position prior the application of laser pulse(s).
• the infinitesimal strain tensor field ⁇ ⁇ and can be derived from the displacement vector field u(x, y, z) by means of Cauchy formulas. The infinitesimal strain can then be written:
• the infinitesimal strain is a second order tensor and only looks like a vector due to the used engineering notation.
• the Lagrange variational principle is used for the determination of the writing modes and the writing mode densities from the potential energy of the optical element or the wafer.
• the overall potential energy P of a deformation resulting from the introduction of pixels into the optical element or the wafer can be presented in form of the volume integral of the potential energy density P(x, y, z) :
• the potential energ density resulting from the stress tensor field a t (x,y,z) caused by a strain tensor field is proportional to the integral of the scalar product of the vector components both quantities.
• the potential energy density P(x, y, z) caused by the stress vector field a t (x,y,z) resulting from the strain vectors field £ t (x,y,z) is thus given by:
• K x - L y - M z elementary volumes are parallelepipeds, cuboids or cubes or elementary tetrahedra.
• K x is the number of elementary volumes (as for examples cuboids) in the x-direction
• L y and M z are the numbers of elementary volumes (or for examples cuboids) in y- und in z-direction, respectively.
• the optical element is split in K x - L y - M z elementary cuboids of identical size.
• the total potential energy P is the sum of the potential energies P a of the individual cuboids a.
• P ⁇ P a do
• the potential energy P a of an elementary volume or an elementary cuboid a is obtained by integrating the potential energy density across the volume V a of the elementary cuboid a:
• the index a counts all the elementary cuboids a into which the optical element is separated. It is assumed that each cuboid or each elementary volume a is so small that the second order tensor field H(x, y, z) of the elasticity tensor can be substituted by a constant tensor H" within an individual elementary volume or cuboid a.
• the diagram 4700 of Fig. 47 illustrates an elementary cuboid a .
• the cuboid has 8 corner points numbered from o to 7.
• the side lengths of the cuboid in Fig. 47 are defined as 2 - m x , 2 - m y and 2 ⁇ m z , respectively.
• the vector field u(x, y, z) can be substituted by its linear approximation.
• the vector field u(x, y, z) is represented by a linear interpolation of the displacements of the corner points of the elementary volume.
• index a denotes the elementary volume into which the optical element is separated.
• index g denotes non-perturbed cube corner coordinates N" or 3D grid nodes, i.e. locations prior to the application of light pulses or laser pulses into of the optical element: ga ga
• the displacements of the coordinates of the corner points caused by writing of pixels into the optical element are defined by:
• the potential energy P a of the elementary volume a will be represented as a function of the corner displacements of the cuboid a.
• the components of the potential energy matrix p k a l have the following matrix form
• the elements of the potential matrix ⁇ '" do not depend on the amount and the type of writing pixels in the elementary volume a
• Equations 16 and 27 are derived on the supposition that the deformation of the elementary volume a starts from an undisturbed state.
• the equilibrium coordinates of the corner points of the elementary volume a after the introduction of pixels in the optical element or wafer are called w ki ⁇ From symmetry reasons it is obvious that a change of the potential energy resulting from a deformation from an original or not deformed state of an elementary volume a to the new equilibrium state w ki expressed by the volume deformation u k a i is opposite to a modification due to a deformation from the new equilibrium state w k a i back to the original or not deformed state.
• equation 27 of the potential energy density is converted to:
• the potential energy of the overall optical element is the sum over all elementary volumes a :
• each internal node coordinate ⁇ ⁇ is presented eight times in the components for all the adjacent nodes.
• the nodes are enumerated from left to right (x-direction), then in rows from bottom to top (y-direction) and finally in planes from bottom to top (z-direction).
• K x , L y and M z are the maximum number of elementary volumes a in the optical element in the x-direction (rows), the y-direction (columns) and the z-direction (planes), respectively.
• An enumeration matrix Q" s t is introduced which connects the coordinates of the internal nodes ⁇ . with the displacement of the corner points of the elementary volume a :
• Equation 41 is the starting point for the description of all the different aspects of the inventive principle discussed in the preceding section. It is important to note that the determinant of the matrix P t f is zero due to the invariance of the potential energy of the overall optical element or the wafer versus a rotation and a translational movement. This condition is always automatically fulfilled by adding a condition with respect to the resulting translational movement and the rotation of the optical element or the wafer. This means that it is always possible to calculate the inverse matrix (second order tensor) (p ⁇ ) 1 of the normalized matrix P? .
• equation 41 allows the computation of the deformations resulting from locally directing the laser beam into a portion of the optical element or a wafer for known equilibrium deformations of the elementary volume a. The relation is expressed by:
• the resulting volume deformations are expressed in an accurate terminology for the writing mode signature (or the laser beam parameter) and the writing mode density.
• a uniform writing means that the writing density is constant in each of the x-, y-, and z- direction in the elementary volume a .
• the writing density comprises three linear densities which can individually be modified.
• the writing density in the elementary volume a is also called writing mode density a a .
• a change of the coordinates of the corner points of the elementary volume a for a unity writing density is denoted as e" t . It is now assumed that one single writing mode is used for every elementary volume a of the optical element and only varying the writing density a a . This means that the same laser beam parameters are used for every elementary volume a and that only the writing density or pixel density is modified. Equation 44 can be rewritten in a form:
• tion 45 is of the form:
• Equation 46 supposes that the writing of the laser beam is performed with a fixed writing mode or a fixed set of laser beam parameters and with a fixed writing mode density a a .
• the deformation property of the writing mode is coded in equilibrium volume deformations e lt for unity writing, i.e. for a single writing density a a .
• the 24 components of the volume deformations at the eight corner points of an elementary volume a are functions of 18 independent ones.
• Three virtual translations of an elementary volume a of the optical element as well as three rotations of the elementary volume a of the optical element around three independent coordinates do not contribute to the potential energy.
• Using the 18 independent components it is possible to construct an orthonormal set of 18 unity vectors n' from the 24 equilibrium volume deformations e lt .
• N l (n' ,e J ) is a matrix which converts the basis e j to the basis n' .
• the orthonormal set of unity vectors n' fulfills the following equations:
• the writing mode signature represents a property of the writing tool or of the laser system for a selected type of the writing and for a set of laser beam parameters having a predefined set of physical and geometrical properties.
• Equation 52 provides a clear instruction how to determine the mode signature MS i from experimental data: Pixels are written in the optical element having a selected writing mode or a parameter set of the laser beam. Then the resulting displacements ⁇ ⁇ are determined. Finally, the inverse matrix of equation 53 is multiplied with the determined displacements ⁇ ⁇ . The following equation expresses the last step:
• equation 55 allows the determination of the displacements ⁇ ⁇ which results for any set of writing maps or writing densities a m a if the mode signature MS * , is known for all writing modes.
• the 3D contour can be measured at the set of locations.
• Deviations ⁇ ] of the set of locations from predetermined ⁇ p " "ecfe ' locations of the optical element can be determined according to:
• the deviations ⁇ ] have to be corrected by locally directing a laser beam onto the optical element.
• the volume deformations introduced in the elementary volume a in the optical element are additive to all deviations A(p j .
• the displacements are only known at the nodes ⁇ ⁇ of the elementary volumes a of the optical element.
• a matrix can be generated which transforms the magnitudes of the displacements at the nodes ⁇ . to the magnitude at the desired location ⁇ . of the 3D contour of the optical element. This transformation can be executed by using the equation 40:
• ⁇ ⁇ is the result of a linear interpolation or of a linear combination of computed displacements ⁇ ; at locations X ; , Y ; , Z ; .
• the matrix My has a dimension of
• the resulting 3D contour at a selected location i in the optical element or the wafer after the application of ultra-short laser pulses is:
• the parameters of the writing mode(s) are selected and the writing mode density/densities in the elementary volume a are identified which minimize the target functional ⁇ , i.e. it has to be solved: ⁇ ⁇ ( ⁇ + ⁇ ) ⁇ (59)
• the displacements ⁇ . are varied in order to bring the determining 3D contour ⁇ p/ ei in accordance with the predetermined 3D contour (p redet of the optical element.
• a Tikhonov regularization has been added to the target functional (last term in equation 60) to make sure that the result will define a physically reasonable solution.
• the regularization coefficients ⁇ have to be selected to be small enough, so that they do not to introduce a significant change to the solution.

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